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Conditional nonlinear asset pricing kernels and the size and book-to-market effects Burke, Stephen Dean 2002

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Conditional Nonlinear Asset Pr ic ing Kernels and the Size and Book-to-Market Effects by Stephen Dean Burke B.Comm., The University of British Columbia, 1991 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in The Faculty of Graduate Studies (Finance Department, Faculty of Commerce and Business Administration) We accept this thesis as conforming, to the required standard The University of Br i t i sh Columbia March 2002 © Stephen Dean Burke, 2002 In presenting this thesis in partial fulfilment of the requirements for an ad-vanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Finance, Faculty of Commerce and Business Administration The University of British Columbia Vancouver, Canada Date Abstract We develop and test asset pricing model formulations that are simultaneously conditional and nonlinear. Formulations based upon five popular asset pricing models are tested against the widely studied Fama and French (1993) twenty-five size and book-to-market sorted port-folios. Test results indicate that the conditional nonlinear specification of the Fama and French (1993) three state variable model (FF3) is the only specification not rejected by the data and thus capable of pricing the "size" and "book-to-market" effects simultaneously. The pricing performance of the FF3 conditional nonlinear pricing kernel is corifirmed by robustness tests on out-of-sample data as well as tests with alternative instrumental and conditioning variables. While Bansal and Viswanathan (1993) and Chapman (1997) find unconditional nonlinear pricing kernels sufficient to capture the size effect alone, our results indicate that similar unconditional nonlinear pricing kernels considered here do not price the size and book-to-market effects simultaneously. However, nested model tests indicate that, in isolation, both conditioning information and nonlinearity significantly improve the pricing kernel performance for all five asset pricing models. The success of the conditional nonlinear FF3 model also suggests that the combination of conditioning and nonlinearity is critical to pricing kernel design. Implications for both academic researchers and practitioners are considered. ii Contents Abstract ii Contents iii List of Tables vi List of Figures viii Acknowledgements x 1 Introduction 1 2 Literature Review 7 3 Methodology 13 3.1 Conditional Nonlinear Asset Pricing Models 13 3.2 Estimating and Testing the Conditional Nonlinear Asset Pricing Models . . . 17 4 Data 25 4.1 The Portfolio Returns 25 4.2 Instrumental and Conditioning Variables 27 iii 4.3 The Asset Pricing Models 29 5 Empirical Results 32 5.1 Linear Model Results 32 5.2 Unconditional Nonlinear Model Results 34 5.3 Conditional Linear Model Results 36 5.4 Conditional Nonlinear Model Results 38 6 The Conditional Second Order F F 3 Model: Robustness Tests 39 6.1 Specification Tests on Out-of-Sample Portfolio Returns 40 6.2 Re-estimation with An Alternative Instrumental Variables Set 41 7 Conditioning with CAY Rather Than TERM AA 8 How Close a Substitute is CAY for TERM as a Conditioning Variable? 47 9 The Conditional Second Order F F 3 Model: Further Discussion 50 9.1 Qualitative Review 50 9.2 The Role of Term Spread Conditioning Information 52 9.2.1 Theoretical Motivation for Conditioning Information in General . . . 52 9.2.2 Support for Term Spread as the Conditioning Variable 54 9.3 The Role of Nonlinearity 57 iv 9.3.1 Theoretical Motivation for Nonlinearity in General 57 9.3.2 Support for Nonlinearity in the FF3 State Variables 61 10 Concluding Remarks 64 10.1 Summary 64 10.2 Implications for Academic Researchers 65 10.3 Implications for Practitioners 67 Bibliography 70 A Tables 80 B Figures 107 v List of Tables A . l Summary Statistics 81 A.2 First Order (Linear) Models 82 A.3 Price Errors from the First Order (Linear) Models 83 A.4 Second Order Polynomial Models 84 A.5 Price Errors from the Second Order Polynomial Models 85 A.6 Third Order Polynomial Models 86 A.7 Price Errors from the Third Order Polynomial Models 87 A.8 Term Spread Conditional First Order Models 88 A.9 Price Errors from the Term Spread Conditional First Order Models 89 A. 10 Term Spread Conditional Second Order Models 90 A. 11 Price Errors from the Term Spread Conditional Second Order Models . . . . 91 A.12 Fama French 3 Factor Model Out of Sample Tests 92 A. 13 Out of Sample Price Errors from the Fama French Model Specifications . . . 93 vi A. 14 Term Spread Conditional Second Order Models with Alternative Instrumental Variables 94 A. 15 Price Errors from the Term Spread Conditional Second Order Models with Alternative Instrumental Variables 95 A. 16 CAY Conditional First Order Models 96 A. 17 Price Errors from the CAY Conditional First Order Models 97 A. 18 CAY Conditional Second Order Models 98 A. 19 Price Errors from the CAY Conditional Second Order Models 99 A.20 Fama French 3 Factor Model Out of Sample Tests (CAY) 100 A.21 Out of Sample Price Errors from the Fama French Model Specifications (CAY)lOl A.22 Term Spread Conditional First Order Models with Substitute Instrumental Variable CAY 102 A.23 Term Spread Conditional Second Order Models with Substitute Instrumental Variable CAY 103 A.24 CAY Conditional First Order Models with Substitute Instrumental Variable TERM 104 A.25 CAY Conditional Second Order Models with Substitute Instrumental Vari-able TERM 105 A.26 Testing the Statistical Significance of Variable Means Across TERM Envi-ronments 106 vii List of Figures B . l Correlation Coefficients for the Fama French 25 Portfolios 108 B.2 The Choice of In and Out of Sample Portfolio Subsets 109 B.3 First Order (Linear) Models 110 B.4 First Order (Linear) Models I l l B.5 Second Order Polynomial Models 112 B.6 Second Order Polynomial Models 113 B.7 Third Order Polynomial Models 114 B.8 Third Order Polynomial Models 115 B.9 Term Spread Conditional First Order Models 116 B.10 Term Spread Conditional First Order Models 117 B . l l Term Spread Conditional Second Order Models 118 B.l2 Term Spread Conditional Second Order Models 119 B.13 Term Spread Conditional Second Order FF3 Model, Returns-Weighted . . . 120 B.14 Term Spread Conditional Second Order FF3 Model, Optimal-Weighted . . . 121 viii B.15 Variable Means Across TERM Environments 122 B.16 Principal Components Analysis of Portfolio Returns 123 B.17 Comparing Term Spread and Log Consumption-Wealth Variables 124 B.18 CAY Conditional First Order Models 125 B.19 CAY Conditional First Order Models 126 B.20 CAY Conditional Second Order Models 127 B.21 CAY Conditional Second Order Models 128 B.22 CAY Conditional Second Order FF3 Model, Returns-Weighted 129 B.23 CAY Conditional Second Order FF3 Model, Optimal-Weighted 130 ix Acknowledgements I gratefully acknowledge the financial support of a University Graduate Fellowship from the University of British Columbia. This thesis has benefited from the many helpful comments of David Chapman, Glen Donaldson, Adlai Fisher, Robert Heinkel, Alan Kraus, Brendan McCabe, James Nason and the seminar participants at the University of British Columbia. Any errors or omissions that remain are my responsibility alone. This thesis represents the end product of what has been a long and sometimes arduous journey. I wish to acknowledge the unconditional love and support of my wife, Bronwyn, that has made this journey possible. I dedicate this thesis to her. Chapter 1 Introduction Several fundamental works in asset pricing theory such as Merton (1973) and Ross (1976) posit that expected financial asset returns are explained by a few relevant state variables. Only risks related to these state variables are relevant in determining prices; all other risks are not priced because they are diversifiable. However, the asset pricing models are frequently rejected by the data when estimated and analyzed in their unconditional, linear form. This is especially true for the Sharpe (1964), Lintner (1965), and Mossin (1966) Capital Asset Pricing Model (CAPM). To remedy this poor performance, several researchers propose con-ditional linear formulations of the models1. Alternatively, Bansal and Viswanathan (1993) propose unconditional nonlinear model formulations2. In this thesis, we develop and test asset pricing model formulations that are simultaneously conditional and nonlinear3. Our conditional nonlinear formulations nest unconditional linear, conditional linear, and uncon-ditional nonlinear formulations as special cases. This model nesting helps illuminate the marginal contribution of, and interaction between, both conditioning information and non-linearity in the formulations. To provide a broader view of the role of conditioning and nonlinearity in asset pricing, we 1See, for example, Ferson et al. (1987), Bollerslev et al . (1988), Harvey (1989), Shanken (1990), He et al. (1996) , and Ferson and Harvey (1998, 1999). 2 Nonl inear approximations to asset pricing kernels are also investigated by Bansal et al. (1993), Chapman (1997) and Ghysels (1998). 3 I n recent work, Di t tmar (2001) prices industry sorted portfolios using a conditional nonlinear C A P M pricing kernel. 1 consider formulations based upon five different models: the Sharpe (1964), Lintner (1965), and Mossin (1966) C A P M ; the consumption-based capital asset pricing model, C C A P M , motivated by Lucas (1978); a nonseparable consumption pricing model, NS-CCAPM, gen-erally based upon the habit formation models of Constantinides (1990) and Ferson and Constantinides (1991); the Cochrane (1996) investment-based asset pricing model, labeled COCHRANE; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Collectively, the five models represent asset pricing based upon wealth, consumption, investment, and empiri-cally determined state variables. Al l the models utilize only a few state variables and thus remain relatively parsimonious even in their conditional nonlinear formulations. The widely studied Fama and French (1993) twenty-five portfolios sorted by market value of equity (ME) and book value to market value of equity (B/M) serve as a formidable test of the various conditional and nonlinear formulations of the five asset pricing models. We use these characteristic sorted portfolios to test the asset pricing models against a specific alternative hypothesis that expected returns are affected by the non-risk asset specific char-acteristics. The empirical literature refers to the apparent pricing influence of the M E and B / M characteristics as the "size" and "book-to-market" effects respectively4. We test the pricing restrictions implied by the models using the pricing kernel method derived from the work of Hansen (1982), Hansen and Richard (1987), Hansen and Jagan-nathan (1991) and Cochrane (1996)5. Jagannathan and Wang (2001) argue that the pricing kernel method is more general than, and equally asymptotically efficient as, the classical beta methods such as the Fama and MacBeth (1973) two-step method. The pricing kernel method also more readily accommodates conditional and nonlinear model formulations. We consider five formulations (or specifications) applied to each of the five asset pricing models. Nonlinearity is introduced into the pricing kernel using sets of second and third order orthonormal polynomials in the state variables following Chapman (1997). Conditioning information is modeled by scaling the state variables with a lagged conditioning variable as 4 The size and book-to-market effects are discussed in the literature review of Chapter 2. 5 Elsewhere in the literature, the pricing kernel is also referred to as the stochastic discount factor or SDF. 2 discussed in Shanken (1990) and Cochrane (1996). Thus, the five specifications considered for each model are: unconditional linear, unconditional second order nonlinear, unconditional third order nonlinear, conditional linear, and conditional second order nonlinear. In total, we estimate twenty-five model/specification combinations, many of which are nested. For the main body of our empirical work, the term spread (TERM) is used as the conditioning variable6. However, in Chapter 7 of the thesis we consider the Lettau and Ludvigson (2001a) log consumption-wealth variable (CAY) as an alternative to TERM7. The results are qualitatively similar using this alternate conditioning variable. As a form of robustness test, each of the twenty-five model/specification combinations is estimated and tested using both the returns-weighted generalized method of moments (GMM) of Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) and the optimal-weighted G M M of Hansen (1982). Following Hansen and Singleton (1982), instru-mental variables are used in the G M M estimation to add pricing restrictions related to the predictability in asset returns. For each model/specification combination, a battery of pric-ing kernel specification tests are examined including: x 2 specification tests, Andrews (1993) supremum Lagrange multiplier (supLM) tests for instability and structural change, informal Hansen and Jagannathan (1991) lower standard deviation bound tests, and Wald tests for pricing errors attributable to individual assets or groups of assets. In the results we report several interesting findings. As a base comparison point, the unconditional linear specifications for all five models are rejected on the size and book-to-market sorted portfolios. Furthermore, all of the models except FF3 fail the Hansen and Jagannathan (1991) lower standard deviation bound tests. The rejection of the FF3 model in particular is curious given that this model utilizes (in addition to market premia) the SMB (small minus large) and HML (high minus low) factor mimicking portfolio returns as state variables. However, this result is consistent with the findings for the FF3 unconditional linear pricing kernel tested in Hodrick and Zhang (2000) as well as the Fama and French (1993) rejection using the Gibbons et al. (1989) F-statistic for two-step beta method regressions. 6 Usually defined to be the long term bond yield minus the Treasury b i l l yield, the term spread variable proxies for information contained in the shape of the term structure of interest rates. 7 W e thank Mar t i n Lettau and Sydney Ludvigson for providing the log consumption-wealth variable v ia download from their web pages. 3 More surprising are the results for the unconditional second order nonlinear and uncon-ditional third order nonlinear model specifications. Nested model tests indicate that the unconditional nonlinear specifications offer a statistically significant improvement over the linear formulations. Furthermore, all of the models except C A P M pass the Hansen and Jagannathan (1991) lower standard deviation bound tests for the unconditional third order specifications. However, both unconditional nonlinear specifications for all five asset pricing models are rejected by the sample data. These results may appear at odds with the findings of Bansal and Viswanathan (1993) and Chapman (1997) who use unconditional nonlinear kernels to price the size effect alone. In results not reported here, we find that both (second and third order) unconditional nonlinear specifications for all five asset pricing models are not rejected by the set of size decile and fixed income portfolios considered in Chapman (1997). Evidently, the combination of size and book to market effects presents a significantly more difficult asset-pricing challenge than the size effect alone. Consistent with related findings by He et al. (1996) and Hodrick and Zhang (2000), the TERM conditional linear specifications for all five models are rejected by the data. Tests for the nested unconditional linear model specifications indicate that conditioning the pricing kernels with the TERM variable provides a statistically significant improvement in pricing performance. This finding is echoed in the improved Hansen and Jagannathan (1991) lower bound tests for most models. Similar to our findings for nonlinearity, the conditioning information contained within the lagged conditioning variable appears to be an important element in the pricing kernel, but not sufficient for good pricing performance. We also find that substituting CAY for TERM produces qualitatively similar results. In particular, consistent with the Hodrick and Zhang (2000) but contrary to Lettau and Ludvigson (2001b), the CAY conditional linear C A P M and C C A P M are both rejected by our sample data. In the final set of results, we report that the TERM conditional second order nonlinear specifications are rejected for C A P M , C C A P M , NS-CCAPM, and COCHRANE. However, neither the returns-weighted nor the optimal-weighted estimation for the TERM conditional nonlinear FF3 is rejected by the data. Interestingly, all five models pass the Hansen and 4 Jagannathan (1991) lower bounds tests. Furthermore, nested model tests indicate that the conditional nonlinear specifications offer a statistically significant improvement over both the conditional linear formulations and the unconditional nonlinear formulations. The combi-nation of conditioning information and nonlinearity significantly improves the asset pricing performance of all the models. In the case of the conditional nonlinear FF3 model, this improvement is sufficient to price the size and book-to-market effects simultaneously. Here again, substitution of CAY for TERM as a conditioning variable produces qualitatively similar results. The returns-weighted and optimal-weighted TERM conditional nonlinear FF3 model estimations are subjected to out-of-sample robustness tests. Using return data not used in the estimation stages, these tests fail to reject either of the two estimations. These results mitigate concerns that the performance of the conditional nonlinear FF3 pricing kernels is due to over-fitting or factor dredging as discussed in Lo and MacKinlay (1990) and Fama (1991). Furthermore, we show that the conditional nonlinear FF3 model is not rejected by the data when we re-estimate with an alternative instrumental variables set. Qualitative inspection of the conditional nonlinear FF3 pricing kernels reveals a high degree of nonlinearity resulting from both the interaction between the SMB and HML state variables as well as the interaction between the conditioning variable and SMB and HML. For instance, the pricing kernel is increasing (decreasing) in the premia to SMB for small (large) values of TERM. A l l of these pricing kernel features would be absent in an unconditional linear formulation. Implications of our results for academic researchers are manifold. Clearly, caution is war-ranted for theoreticians and empiricists reaching (unnecessarily) for modeling assumptions intended to produce elegant, but less effective, unconditional linear pricing rules. Further, our failure to reject the conditional nonlinear FF3 model on characteristic (size and book-to-price) sorted portfolios supports the rational asset pricing hypothesis in the popular "rational factor pricing vs. irrational characteristic pricing" debate in the literature. Three fundamental messages for practitioners also emerge from our work. First, improved cost of capital calculations for capital budgeting decisions may be achieved through the in-5 corporation of conditioning information and nonlinearity in the model of expected firm asset returns. Second, higher returns to certain investment styles (e.g. small capitalization value stocks) previously deemed "anomalous" are likely artifacts of a misspecified unconditional linear asset pricing model used to discount strategy returns. Furthermore, high Sharpe ra-tios associated with tactical style rotation strategies do not imply higher expected utility for these strategies given the likelihood that the strategies entail maximal allocation to certain risks at times when the average investor derives the least utility from bearing a given unit of that risk. Naturally, performance measurement should be adapted to reflect these issues as well. In our final discussion of the conditional nonlinear FF3 pricing kernel, we propose several a priori reasons for expecting such a pricing kernel to succeed on our sample data. First, we review theoretical motivations for both conditioning information and nonlinearity in asset pricing kernels. Then to be more specific, we analyze the sample return data to help identify the features of the data which favor the use of conditioning information and nonlinearity. The rest of this thesis is organized as follows. A brief review of the literature is given in Chapter 2. In Chapter 3, we develop the canonical form for a conditional nonlinear asset pricing model and propose estimation and testing methods. The data is described in Chapter 4. Chapter 5 presents the estimation and testing results for the twenty-five model/specification combinations. Robustness tests using out-of-sample portfolios and an alternative instruments set for the conditional nonlinear FF3 model estimations are presented in Chapter 6. Chapter 7 considers CAY as a substitute for the conditioning variable TERM. The similarity between CAY and TERM as conditioning variables is explored in Chapter 8. In Chapter 9, we provide further discussion regarding the theory and intuition behind the success of the conditional nonlinear FF3 model. Finally, Chapter 10 concludes with a summary of results and implications for academic researchers and practitioners. 6 Chapter 2 Literature Review This thesis is closely related to two bodies of the empirical research literature. The first body of literature consists of work examining the role of either conditioning information or nonlinearity in the formulation of asset pricing models. The second area of relevant empirical research includes work contributing to the debate regarding the consistency of various asset pricing models with the apparent size and book-to-market effects in the cross-section of expected stock returns. As in this thesis, these two areas of research are not completely separate from one another. Discussion relating this thesis to these two bodies of literature follows. The early foundations of asset pricing theory such as the Sharpe (1964), Lintner (1965), and Mossin (1966) C A P M or Ross (1976) arbitrage pricing theory (APT) yield static or unconditional models for use in describing the cross-section of expected stock returns. To remedy the poor performance of these early unconditional linear models, several researchers propose conditional linear formulations of the models. Varying degrees of implementation complexity for the use of conditioning information are evidenced in the works of Ferson et al. (1987), BoUerslev et al. (1988), Harvey (1989), Shanken (1990), He et al. (1996), Cochrane (1996), Jagannathan and Wang (1996), Ferson and Harvey (1998, 1999), and Lettau and Ludvigson (2001b) among others. In this thesis, we use a conditioning variable, term spread, to simply scale various polynomial orders of the given state variables. In this regard, our 7 work with conditioning information is most similar to that of Shanken (1990) and Cochrane (1996). Scaling the state variables in this fashion is equivalent to permitting either the coefficients on these state variables or the associated risk premia to be time-varying. Following a related line of research, Bansal and Viswanathan (1993) suggest that un-conditional nonlinear model formulations are more general, but still valid, interpretations of the asset pricing theories of Merton (1973) and Ross (1976). Nonlinear approximations to asset pricing kernels are also investigated by Bansal et al. (1993), Chapman (1997), Ghysels (1998), and Dittmar (2001). While Bansal and Viswanathan (1993) use neural networks to approximate their nonlinear A P T pricing kernel, Bansal et al. (1993) and Ghysels (1998) rather use low-order polynomial series expansions with some orders removed to reduce multi-collinearity problems. Similarly, Dittmar (2001) employs low-order polynomial series expan-sions but does not remove any terms. However, Chapman (1997) suggests that orthonormal polynomials are more efficient than the types of constructions used by Bansal et al. (1993), Ghysels (1998), and Dittmar (2001). In this thesis, we choose to follow Chapman (1997) and construct pricing kernels from sets of orthonormal polynomials. As will become evident in Chapter 3 below, working with polynomials (orthonormal or otherwise) has the added advantage that it permits us to model a pricing kernel that is nonlinear in the state variables but linear in its estimated parameters. This feature significantly simplifies the estimation procedure and is not shared by the neural network approximated kernels of Bansal and Viswanathan (1993). One final note on this vast body of literature involves the mixing of conditioning in-formation and nonlinearity within the pricing kernel. To our knowledge, Dittmar (2001) is the only work other than ours that blends conditioning with nonlinearity. The pricing kernels considered by Dittmar (2001) are particularly interesting because their risk factors are endogenously determined and because preferences are used to restrict the pricing kernel. However, in this thesis we consider the mixture of conditioning and nonlinearity in a broader context of five different asset pricing models, rather than the C A P M alone1. Further, while Dittmar (2001) chooses to price industry sorted portfolios, our work focuses on the widely 1 Di t tmar (2001) also considers a conditional linear F F 3 and a conditional power ut i l i ty model synthesized from Brown and Gibbons (1985) and Campbell (1996). 8 popularized size and book-to-market effects. Another important difference is that none of the pricing kernels estimated by Dittmar (2001) satisfy the Hansen and Jagannathan (1991) lower volatility bounds, while in this thesis we report satisfaction of the bounds for all our conditional nonlinear kernels. Traditional asset pricing model formulations are usually augmented with conditioning information and/or nonlinearity in order to improve asset pricing performance. Perhaps one of the most perplexing empirical asset pricing problems is posed by the size and book-to-market effects. The empirical research examining these two effects represents the second body of literature that is closely related to our work. Early evidence regarding the size effect is provided by Banz (1981) who finds average stock returns are decreasing in firm market value. More recently, Chapman (1997) inves-tigates several consumption-based models for their ability to properly price the size effect. Chapman (1997) finds that his second and third order polynomial approximated pricing ker-nels based upon the C C A P M and NS-CCAPM models are well specified for simultaneously pricing Treasury bills, corporate bonds, and several deciles of size sorted portfolios. Bansal and Viswanathan (1993) also report success capturing the size effect, but using artificial neural network approximated pricing kernels based upon three state variables: the nominal market return, the nominal Treasury bill yield to maturity, and the nominal yield spread between nine-month and three month Treasury bills. Among the many model/specification combinations examined in this thesis, we include unconditional nonlinear specifications very similar to the C C A P M and NS-CCAPM models considered by Chapman (1997), only we report less pricing success. By the book-to-market effect we refer to the findings of Stattman (1980), Rosenberg et al. (1985), DeBondt and Thaler (1987), Keim (1988) and Fama and French (1992) which document a positive relationship between average returns and the ratio of book value to the market value of equity. Daniel and Titman (1997) and Davis et al. (2000) also focus on the book-to-market effect alone. In both studies, the authors form portfolios by triple sorting stocks on market value of equity (ME), book-to-market value ratio (B/M), and risk loadings. The aim is to find variation in risk loadings unrelated to the B / M characteristic and thus 9 the tests focus on the book-to-market effect in isolation. While Daniel and Titman (1997) find the Fama and French (1993) three state variable model is inconsistent with the data, Davis et al. (2000) reverse this result using a data set with a longer history. Lakonishok et al. (1994) report superior returns to portfolios generated by following what they call value or contrarian strategies, selecting portfolios based on measures such as B / M . These authors find little support for the view that these strategies are fundamentally riskier, and thus find the superior returns difficult to reconcile with traditional asset pricing theories. The size and book-to-market effects are originally studied together by Fama and French (1992) who find that in explaining the cross-section of asset returns, betas are overwhelmed by the two characteristic variables ME and B / M . While synthesis of the empirical literature is complicated by the variation in samples dates, portfolio sets and econometric testing meth-ods, formal asset pricing tests involving the size and book-to-market effects together have generally failed. Fama and French (1993) use the time-series regression approach of Black et al. (1972) and report that the F-statistic of Gibbons et al. (1989) rejects both the FF3 model and the FF3 model augmented with two bond-market state variables. This F-statistic based result for the size and book-to-market portfolios is confirmed with similar tests, but longer sample periods, in Fama and French (1996) and Davis et al. (2000). Brennan et al. (1998) use Fama and MacBeth (1973) regressions on individual securities, rather than sorted portfolios, and find that size and book-to-market characteristics have marginal explanatory power relative to the Fama and French (1993) FF3 model and an A P T model constructed from Connor and Korajczyk (1988) principal component factors. In contrast, Li et al. (1999) report much more promising results that the size and book-to-market sorted portfolios do not reject a multi-sector investment-growth asset pricing model. Recently, Lettau and Ludvigson (2001b) also report success pricing the size and book-to-market effects using beta methods and G M M to estimate versions of a conditional linear C A P M and C C A P M . The authors condition using a log consumption-wealth variable, CAY, that they develop in earlier work (Lettau and Ludvigson, 2001a). However, Hodrick and Zhang (2000) find contradicting results using pricing kernel methods to estimate a conditional linear C A P M and C C A P M with the CAY as a conditioning variable. One key difference between the works is the fact that Hodrick and Zhang (2000) also price Treasury bill returns 10 with the size and book-to-market portfolios. This forces the pricing kernel to price risky and riskless assets simultaneously, a very difficult test of any asset pricing model. Note that including a riskless asset in the portfolio set places a mean restriction on the pricing kernel. As Dittmar (2001) notes, Dahlquist and Soderland (1999) find that imposing this restriction is important when evaluating pricing kernel performance. Interestingly, we report in Chapter 7 below that substituting CAY for TERM in our conditional linear C A P M and C C A P M formulations produces pricing kernels that are still rejected by the sample data. Our sample includes T-bills, corporate bonds, and numerous "managed portfolios" created by the use of instrumental variables2. This forces the pricing kernel to price not only the size and book-to-market effects, but also fixed income returns and predictable variation in returns. Note also that Lettau and Ludvigson (2001b) caution that small sample bias in iterated G M M is more acute as the number of cross-section observations grows in relation to the time-series sample size (Ferson and Foerster, 1994; Hansen et al., 1996). Using the pricing kernel approach to testing asset pricing models, He et al. (1996) and Hodrick and Zhang (2000) also reject unconditional and conditional linear versions of the Fama and French (1993) three state variable model on the size and book-to-market sorted portfolios. Note that He et al. (1996) use instrumental variables G M M and thus include moment conditions for managed portfolio returns in the asset set. This forces the model to not only price the size and book-to-market effects, but also to price predictable variation captured by the instrumental variables. In this thesis, we attempt to maximize the difficulty of the asset pricing tests by both adding Treasury bill and bond returns to our asset set and by using instrumental variables to generate and include managed portfolio returns. In summary, other than the results reported in this thesis, we know of no unconditional nonlinear or conditional nonlinear model formulations that have been applied to price the size and book-to-market sorted portfolios successfully3. We use the pricing kernel method, 2 T h e role of instrumental variables and managed portfolio returns in testing asset pricing models is discussed in Chapter 3. 3 B a n s a l and Viswanathan (1993), Chapman (1997), and Ghysels (1998) price size sorted portfolios and Ghysels (1998) and Di t tmar (2001) price industry sorted portfolios. We recently became aware of the size and book-to-market pricing success of the L i et al. (1999) unconditional linear investment-growth model. 11 rather than the beta method, since this more general method easily accomodates estimation of our conditional nonlinear model specifications. Furthermore, Jagannathan and Wang (2001) show that the pricing kernel method has the same asymptotic precision as the beta method for the purpose of estimating risk premia. Finally, while pricing the size and book-to-market effects simultaneously we attempt to maximize the power of the tests by including fixed income and managed portfolio returns in the sample asset set. 12 Chapter 3 Methodology 3.1 Conditional Nonlinear Asset Pricing Models We develop a conditional nonlinear asset pricing model closely following the work of Bansal and Viswanathan (1993). Consider a discrete time representative agent economy where N assets are traded at time t with payoff to the assets received at time t + 1. Let Qt be the information set available to the agent at time t, where fis C 0,t for all s <t. Assuming no short sale constraints, the first-order conditions for the agent's investment decision are1: E [ m t i t + l R i u + i \ n t } = 1 i = l , . . . , iV (3.1) where mttt+i is the representative agent's intertemporal marginal rate of substitution between time t and t + 1 consumption and where Ri,t,t+\ is the gross return on asset i for the same period. Within these first-order conditions, the marginal rate of intertemporal substitution, m t > t + i , can be replaced with its projection on the space of all one-period payoffs. Let represent this projection. Equation (3.1) may then be re-written to provide the following ^ee Lucas (1978),Breeden (1979), and Stulz (1981) among others. 13 conditions: E[p*t+1Riu+l\nt} = 1 i = l,...,N. (3.2) Hansen and Jagannathan (1991) show that this projection, p*+i, has the minimum variance in the class of all pricing kernels and can be expressed as a general linear combination of the iV asset one-period payoffs represented by the vector .R^+i = [Ri,t,t+i]iLi-N = ^ ait+iRijt+i, (3-3) where the conditional weighting vector at+i = [a^+i]^ satisfies: a t + 1 = [£?[J2{IT+1J2tit+1|n(]]-1. (3.4) So far, we have a pricing kernel with as many factors as there are assets being priced. From this point, traditional linear factor-pricing is derived by imposing a restriction that is a linear combination of only a designated set of factor payoffs. For their nonlinear A P T model, Bansal and Viswanathan (1993) rather choose to impose a sufficient statistic restriction on the one-period intertemporal marginal rate of substitution at time t + l 2 . A similar approach is followed here in our development of a conditional nonlinear asset pricing model. Before imposing the sufficient statistic restriction, we use the law of iterated expectations to re-write the agent's first-order conditions in equation (3.1) as: E[E[mt,t+1\nt+1}Riu+l\Qt} = 1 i = 1 , N . (3.5) We now impose a sufficient statistic restriction that the conditional expectation of the in-tertemporal marginal rate of substitution between t and t + 1 is a function of £ t + 1 = [£I,H-I, £M,H-I] an M-dimensional vector of basis variables and ct = [c^t,cr, i t] an L-dimensional vector of conditioning variables. We require both that £ m G and ct e Qt. 2 B a n s a l and Viswanathan (1993) also consider the separate case of adding a non-negativity restriction to the sufficient statistic restriction (footnote 9 on page 1238). Imposing non-negativity complicates the model estimation procedure significantly. 14 The conditional expectation of r n t t t + i is then written: E[mttt+1\nt+1} = E[mtit+1\Zt+1,ct] = H(£t+1,ct) (3.6) where K and L are low numbers and H(-) is a well behaved function among a class of flexible functional forms. In practice, the vector of basis variables, £ m , will constitute a set of factors associated with a given asset pricing model. The vector of conditioning variables, ct, will consist of macroeconomic variables thought to capture predictable time variation in state variable betas3. The exact specification of the conditional nonlinear pricing kernel, H(-), is unknown and must be approximated. In this thesis, we follow the work of Chapman (1997) and use low-order orthonormal polynomials to approximate the pricing kernel. In our approximation of a q-th order conditional nonlinear asset pricing kernel, we begin by creating a new vector for each state variable, denoted £ ' m , consisting of all orders of that state variable: £ i , t + i = - i = 1, M. (3.7) The conditioning information is then incorporated by scaling each of these orders of the state variable by each conditioning variable, creating an expanded basis for each state variable denoted as follows: ^ + 1 = ^ + 1 ® ^ ] i = l , . . . , A f (3.8) where "<g>" represents the Kronecker product operator. Shanken (1990), Cochrane (1996) and Chapman (1997) note that scaling the state variables in this fashion is equivalent to permitting the coefficients on these state variables to be time-varying. To eliminate collinearity between the terms in the expanded basis for each state variable, * i , t + i > w e find a n orthonormal basis for the T x qL matrix \1>' formed by our time series sample for the vector where t — 1, ...,T. Using Theorem 2.5.2 in Golub and Van Loan (2000), it can be shown that there exists the following "thin" singular value decomposition 3 F o r the use of conditioning variables that we describe below, Cochrane (1996) notes that their role may also be interpreted as capturing predictable time variation in the state variable risk premia. 15 for * t 9 : V^PlSiVi i=l,...,M (3.9) where Pf is an T x qL orthonormal basis for the range of with PfJPf = I; Si is a qL x <TL diagonal matrix with nonnegative diagonal elements (singular values) in decreasing order; and V; is a unitary matrix whose columns are the singular vectors4. This is called a "thin" decomposition because we have computed only the first qL columns of Pf and conform the other matrices. We then approximate the conditional nonlinear pricing kernel using orthonormal poly-nomial expansions from the columns of each Pf matrix, where i = 1,...,M. Omitting cross-product terms for reasons of parsimony, the specification of the conditional nonlinear pricing kernel for time t + 1 is given by: M H(£t+1,ct) = G ( P ? i t + 1 , P q M t t + v 6o, 0 t T , ® T M ) = Go + ®JPl+i (3-10) t=i where G(-) is a well behaved function among a class of flexible functional forms, 9o is a scalar intercept term, 0, is a q(L+l) x 1 vector of coefficients associated with the i-ih state variable, and Pft+i is the z-th state variable's q(L +1) x 1 vector of orthonormal polynomial terms of order 1 to q, including conditional terms. To simplify notation, let 0 T = {0 O , ®J,©M} and Pt+i = {1, P\tt+\i •••! P9M,t+i} s u c n that the condensed expression for the pricing kernel is: G(Pt+l;G) = e r P t + 1 . (3.11) Finally, substitution of this conditional nonlinear pricing kernel into the representative agent's first order conditions yields: E[G(Pt+1; e)Rtit+1 | a] = E[(@JPt+1)RtM1 \ Slt] = 1 (3.12) where 1 is an JV x 1 vector of ones. Equation (3.12) represents the canonical form of the conditional nonlinear asset pricing models investigated in this thesis. 4The matrix Vi is called unitary because VjV\ = ViVj, = I. 16 For example, consider a conditional nonlinear version of the Sharpe (1964), Lintner (1965) and Mossin (1966) C A P M using second order polynomials (q = 2) and lagged term spread as the conditioning variable. In this example, we have a single ( M = 1) state variable, equal to the market premia from t to t + 1, and a single (L = 1) conditioning variable, ct, equal to the term spread at t. The time t + 1 basis for the state variable is £ 2 + 1 = [£ t+ i , £ 2 + 1 ] which, with conditioning, yields the expanded basis: = £ ? + i ® [ l , < * ] = [&+i>£t 2 +i> £S2+IQ]T-A singular value decomposition of SI>2 produces the orthonormal polynomial matrix P2 with T rows and 4 columns. Finally, the time t + 1 approximate pricing kernel for this conditional C A P M model specification is: G ( P 2 + 1 ; 0 ) = O o + e [ P 2 + 1 where 9o is a scalar intercept term and 0 i is a 4 x 1 vector of coefficients associated with the expanded basis for the market premia. The parameters 0 = { 0 O , &J} can be estimated from sample data using the methods described in the subsequent section. 3.2 Estimating and Testing the Conditional Nonlinear Asset Pricing Models Following Bansal and Viswanathan (1993), Bansal et al. (1993) and Chapman (1997) we estimate the pricing kernel using the generalized method of moments (GMM). The first order conditions described by equation (3.12) map into the following set of moment conditions: E[G(Pt+1; e)(Rtit+l ® Zt)] = E[l ® Zt] (3.13) 17 where Zt is a 1 x K vector of instrumental variables known to the representative agent at time t (i.e., Zt G 0*). Cochrane (1996) notes that equation (3.13) is derived from equation (3.12) by multiplying both sides of equation (3.12) by Zt and then taking unconditional expectations. Conversely, if equation (3.13) holds for all choices of Zt G fit, then equation (3.12) holds. In equation (3.13), the Kronecker product of returns with instrumental variables, _RM+1<g> Zt, creates a vector of "managed portfolio" returns that the model must price5. For a par-ticular managed portfolio, say R i i t i t + i Z j i t , the return may be viewed as though it is generated by a strategy of investing more or less in Ritt,t+i according to the signal in Z j i t . As Cochrane (1996) explains, this use of instrumental variables effectively expands the pay-off space to represent some of the predictability in the product of portfolio returns and the pricing ker-nel. An econometric benefit of using the instrumental variables is the multiplicative effect on the number of moment conditions, increasing the number of degrees of freedom in the estimation. As in Hansen (1982) and Cochrane (1996), let ET = T~L Y^[=i denote the sample mean and denote the sample moments gT: gT(P, R, Z ; 0 ) = ET[G(Pt+l] 0 ) ( J R t , t + i <g> Zt)\ - ET[1 ® Zt\. (3.14) Effectively, we can view gT is an NK x 1 vector of portfolio pricing errors. The objective of the G M M estimation is to choose 0 to minimize a weighted sum of squared pricing errors. Let W denote an NK x NK weighting matrix. The G M M sample objective function is then written: M®) = 9T(P, R, Z; &)TWgT(P, R, Z; 0 ) . (3.15) Notice that equations (3.11), (3.14) and (3.15) together imply that the parameters, 0 , enter the objective function linearly, facilitating an analytic solution to the minimization 5 Typica] ly , the first element of Zt is simply the constant 1 reflecting the moment conditions for the original set of portfolios. 18 problem. Let © denote the estimate of 0 . The first order conditions to the minimization of equation (3.15) are: dgAP^e) P f R z £ = Q a® Following Hansen (1982) and Cochrane (1996), we simplify the notation by letting DT denote the gradient of the sample moment of the pricing errors with respect to the parameters, found equal to: DT(P, R,Z,S) = % r ( P g Q Z ; e ) = ET[(Rt+1 ® Zt)Pj+1}. (3.17) Substituting equations (3.14) and (3.17) into equation (3.16) yields the following analytical solution for the parameter estimates: 0 = {DlWDT)-lDJTWET[l ® Zt\. (3.18) Hansen (1982) shows that 0 is asymptotically normal with variance-covariance matrix Var(&) = T-l{DlWDTylDJTWSTWDT{DlWDT)-1 (3.19) where ST is a consistent estimate of the long-run covariance matrix of the model pricing errors6. Following Chapman (1997), we use a heteroskedasticity and autocorrelation con-sistent (HAC) estimator of S proposed by Andrews (1991)7. Following Hansen (1982), we assume that returns, Rt+\ <8> Zt, are stationary and the long-run covariance matrix, ST, is positive definite. Before proceeding further, we must choose a specification for W, the weighting matrix. Hansen (1982) shows that if W is chosen to equal STX then the coefficient estimates of 6 M o r e formally, the long-run covariance matrix of the model pricing errors is: oo S= E[ut"I-j] j = — oo where ut = [G(Pt,®)(Rt-i,t ® Zt-i) — 1 ® Zt-i] denotes the vector of model pricing errors at time t. Typical ly, a heteroskedasticity and autocorrelation consistent ( H A C ) estimate of this long-run covariance matrix is substituted in place of the true S. 7 T h e Andrews (1991) H A C estimator is the spectral density of the price errors evaluated at the zero frequency using a quadratic spectral kernel. More detailed discussion of this and other H A C estimators used in the G M M context is provided in Matyas (1999), Chapter 3. 19 equation (3.18) are "optimal" and their variance-covariance matrix reduces to: Var(S) = T~\DlST-lDT)-\ (3.20) The coefficient estimates obtained using W = S T - 1 are "optimal" in the sense that they have the smallest asymptotic covariance. Since ST is a function of the parameters, an iterative procedure must be used to determine ST and the optimal 0 simultaneously. As in Chapman (1997), we iterate on equation (3.18) and the corresponding pricing error covariance matrix estimate, ST, until the parameters, 0 , converge8. Kocherlakota (1990) reports Monte Carlo results indicating that the iterative G M M estimator performs better than the conventional two stage estimator in small samples. Alternatively, Hansen and Jagannathan (1997) propose setting the weighting matrix equal to the inverse of the second-moment matrix of returns. For the instrumental variables G M M used here, this involves setting W = ET[(Rt,t+i®Zt)(Rt,t+i®Zt)T]~1 • We must assume that this matrix is non-singular. Cochrane (1996) expresses concerns that the second-moment ma-trix of returns may be nearly singular, leading to inversion problems. We did not encounter this problem with the asset sets chosen for our estimations. Since this Hansen and Jagannathan (1997) specification for W is independent of the parameters, iteration is not required to find the solution. Additionally, this weight matrix does not share the tendency of the Hansen (1982) optimal weight matrix to assign large weights to assets with small variances in their pricing errors and assign small weights to assets with large variances in their pricing errors. This tendency of the optimal weight matrix is particularly undesirable if we suspect that the assets we are most interested in pricing have the larger pricing error variances. After estimating the pricing kernel associated with a given model specification, we wish to test whether the associated pricing errors are equal to zero. Using the results in Hansen 8 Optimizations are deemed to have converged when either a l l parameter deltas are less than one percent or 20 iterations have been completed. 20 (1982) , Cochrane (1996) shows that the variance-covariance matrix for the pricing errors is: Var(gT) = T ^ J - DT(DlW DT)~L DlW\ST[I - U> r(L>JWDT^D^W] (3 .21) where J is an NK x NK identity matrix. For a returns-weighted estimation where W = ET[(Rt,t+i <8> Zt){Rttt+i ® ^<)T] _ 1) w e test whether all pricing errors are zero using the following statistic: JT(®RW) = 9A®Rw)T[Var{gT)}+gT(®RW) ~ XNK-Q(L+i)M+i (3-22) where RW stands for returns-weighted, NK is the number of pricing errors, q(L + l)M +1 is the number of estimated parameters, and [-]+ represents the pseudo-inverse operator9. The test statistic in equation (3 .22) represents a squared version of the minimum mean-square distance, from the candidate pricing kernel to the family of admissible discount factors, as discussed in Hansen and Jagannathan (1997) . For pricing kernels estimated using the optimal-weighting matrix, W = S^1, the spec-ification test in equation (3 .22) reduces to the Hansen ( 1982) JT test of overidentifying restrictions: TJT(@ow) = 9T(®ow)TS^lgT(@ow) ~ X M - , ( L + I ) M + I (3-23) where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. There are several motivations for considering both Hansen's ( 1982) optimal-weighted and Hansen and Jagannathan's (1997) returns-weighted estimations for each candidate asset pricing model specification. The optimal-weighted estimations are efficient while the returns-weighted estimations are not. Further, the optimal-weighted estimations are consistent with the construction of the nested model and Wald price error tests described below. The asymptotic distributions for these tests under the returns-weighted estimation are unknown. However, Ferson and Foerster (1994) report that optimal-weighted G M M may have poor 9 The rank of Var(gT) is equal to NK — q(L + 1)M + 1 while the dimensions are NK x NK. This means Var(gT) is singular. Cochrane (1996) proposes the use of the psuedo-inverse to compute the specification test statistic of equation (3.22). We compute the pseudo-inverse of Var(gT) using the M A T L A B pinv(-) command. 21 finite sample properties. Additionally, since the optimal-weighted G M M weighting matrix is the inverse of the estimated pricing errors covariance matrix, Cochrane (1996) and Chapman (1997) note that this approach may favor pricing kernel estimates with more volatile pricing errors. In contrast, the returns-weighted approach uses a weight matrix that is invariant to parameter choice; this approach does not favor models with highly volatile pricing errors and should produce more stable results (Cochrane, 2000). Finally, Ahn and Gadarowski (1999) report Monte Carlo simulation results indicating that the size of the Hansen and Jagannathan (1997) HJ-distance specification test, a test closely related to our returns-weighted x2 test, is poor in finite samples. Clearly, it is difficult to motivate using only one approach at the expense of the other. Cochrane (1996) notes that Hansen's (1982) test of overidentifying restrictions may also be used to test a model against specific alternatives. In this thesis, we test for nested model specifications by comparing the minimized TJT(@OW) statistics for the full and nested models. The nested model is simply a restricted version of the full model. Cochrane (1996) notes the importance of using the same optimal-weighting matrix for both TJT(®OW) test statistics in order for them to be comparable10. The test statistic for nested models is: TJ T (0ow) r e s t r i c t ed ~ TM©o^)unres t r ic ted ~ X n u m b e r o f restrictions. ( 3 ' 2 4 ) A commonly used, but less formal, test of the consistency of a candidate pricing kernel involves visually comparing the mean and standard deviation of the estimated pricing kernel to the Hansen and Jagannathan (1991) lower standard deviation bound. In the context of the instrumental variables estimation approach used here, for a given pricing kernel mean, G = ET[G(Pt+u ©)], the standard deviation of the pricing kernel, OG = ET[(G(Pt+i; 0 ) — G ) ( G ( P m ; 0 ) - G) ] 1 / 2 , must satisfy the following bound: O-G > [(Er[l ® Zt] - GET[Rt:t+1 ® Zt])TW(ET[l ® Zt\ - GET[Rt,t+\ ® Zt}}^2 (3.25) where W = ET[(Rtit+1 <g> Zt)(Rt,t+i <8> Zt)T], the inverse of the second moment matrix of 1 0 O n e model may achieve a lower TJT(€>0w) statistic simply by inflating ST rather than by producing smaller pricing errors. 22 returns11. This particular version of the lower bound is a simple modification of equation (12) on page 234 of Hansen and Jagannathan (1991). Typically, a candidate pricing kernel that fails this consistency test is rejected as invalid. The above x2 specification tests measure the overall pricing ability of the candidate kernel specification. Ghysels and Hall (1990) find that such tests have a tendency not to reject the asset pricing model in cases where it is inappropriate to fix the parameter vector, 0, over the sample period. Ghysels (1998) advocates using the Andrews (1993) supremum Lagrange Multiplier (supLM) test to examine for structural shifts in the parameters of models estimated by G M M . Using the supLM test, Ghysels (1998) finds that conditional asset pricing models are more prone to this form of misspecification and often produce larger pricing errors than unconditional models. While the supLM test is designed to test the null of parameter stability against the alternative hypothesis of a single structural break at an unknown time, Andrews (1993) shows that this test has power against more general forms of structural instability. Let 7T € (0,1) denote the change point associated with the time TTT structural change alternative. The associated Lagrange Multiplier statistic, LMT(IT), is calculated as follows: LMt(TT) = -7^^gT(®0w)STlDT(DlST1DT)-1D^ST1gT(@ow) (3.26) 7T(1 — 7T) where 9T(®OW) = 7f J } G ( P t + i ; eX^M+i ® Zt)] - - J > ® Zt}. (3.27) and S T and D T are as before. Following Ghysels (1998) and Hodrick and Zhang (2000) LMT(TT) statistics are evaluated at 5% increments between 20% and 80% of the sample. The largest of these LMT(ir) statistics is the supLM statistic. Inference is performed using the supLM statistic distribution presented in Table 1 of Andrews (1993). Having thus far considered several specification tests, we also wish to test for the signifi-cance of individual pricing errors or groups of pricing errors using Wald type tests. Chapman 1 1 A review of several more formal tests based on the Hansen-Jagannathan lower standard deviation bound is provided by Burnside (1994). 23 (1997) notes that this is admissible only for the optimal-weighted estimations. In this case, the pricing errors' variance-covariance matrix of equation (3.21) reduces to: Var(gT(G0w)) = T ' ~ 1 [5 r - DT^DIS^DT^DI]. (3.28) For each asset return set consisting of the basic asset and all managed portfolios of that asset, we are interested in testing the hypothesis that the set of associated pricing errors are zero. For the NK x 1 pricing error vector, gT(@ow), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K — 1)} of the vector12. The hypothesis is then tested using the set of restrictions. V(i)gT(@0w) — 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(i) - [V(i)gT-0]J[V(i)Var(gT)V(i)T][V(i)gT-0] (3.29) = [V{i)gTV[V^Var{gT)Vi,Y}\V^gT] ~ q { r e s t r i c t i o n s . For every estimated specification, this Wald test is performed for the ./V asset sets consisting of a basic asset and its associated managed portfolios. Finally, we can verify the robustness of a candidate pricing kernel specification by measur-ing its performance on assets not included in the estimation. Using a new set of Q assets, the specification test statistics in equations (3.22) and (3.23) are recomputed following Hansen (1982). However, these out of sample versions of the test statistics are both distributed x 2 with Q degrees of freedom since no new parameters are estimated. Additionally, we use the Wald test as described in equation (3.29) above to test for the significance of sets of pricing errors associated with the out of sample assets. Recall that there are N basic assets and K — 1 managed versions of each asset. 24 Chapter 4 Data In this section, we report specific details regarding the source and preparatory calculations for the sample data including: portfolio returns, instrumental variables, conditioning variables, and the state variables associated with candidate asset pricing models. A l l data is quarterly and covers the period from Q2, 1959 to Q4, 1999. 4.1 The Portfolio Returns We investigate the ability of conditional nonlinear asset pricing kernels to capture the size and book-to-market effects using the Fama and French (1993) twenty-five portfolios. These portfolios are sorted by five quintiles in market value of equity (ME) and five quintiles in the ratio of book value to market value of equity (B/M). The portfolios are labeled SxBy where x represents the ME quintile and y represents the B / M quintile. Nominal monthly returns for the 25 M E and B / M sorted portfolios are calculated following the methods detailed in Fama and French (1993)1. Real quarterly portfolio returns are computed by compounding the monthly nominal returns and subtracting the logarithmic first difference in the quarterly Consumer Price Index series (all urban, all items, seasonally adjusted) available on Thomson Financial Datastream using code USCP....E. 1 We thank Kenneth French for providing this data via download from his web page. 25 To ensure the pricing kernel is robust to pricing a broader class of assets, we expand our basic set of portfolio returns to include the three month holding period return on three month Treasury bills, a high grade corporate bond portfolio, and a government bond portfolio. The three-month holding period return on three month Treasury bills is computed from the three month Treasury bill discount yield series available from the U.S. Federal Reserve web site under the code tbsmSm. Nominal monthly returns for the high grade corporate bond portfolio and the government bond portfolio in Ibbotson Associates (2001) are compounded into nominal quarterly returns. As with the Fama-French portfolio returns, real quarterly returns for the corporate bond (CORP) and Treasury bill (TBILL) portfolios are computed by subtracting the logarithmic first difference in the quarterly Consumer Price Index series from each nominal series. As evidenced in the correlation coefficient surface depicted in Figure B . l , adjacent Fama-French 25 portfolios exhibit high contemporaneous return correlation. Consistent with Chap-man's (1997) findings for deciles of size sorted portfolios, this leads to numerical instability of the inversion of the second moment matrix of returns. This problem is only exacerbated when the size of the second moment matrix of returns expands multiplicatively with the addition of managed portfolios, created by the product of basic portfolio returns and instrumental vari-ables. Furthermore, Cochrane (1996) reports that optimal G M M estimates behave badly as the covariance matrix of model pricing errors expands in dimension. We choose to estimate and test each candidate model specification using a subset of the portfolios which captures the cross-sectional diversity in the full set of portfolios. More specifically, we work with a basic set of portfolios consisting of S1B1 (small capitalization, growth), 5155 (small cap-italization, value), 5551 (large capitalization, growth), 5555 (large capitalization, value), 5353 (middle capitalization, average growth/value), three month Treasury bills (TBILL), and corporate bond (CORP) returns. This portfolio subset is depicted graphically in Figure B.2 under the "In Sample" label. Panel B in Table A . l provides basic descriptive statistics for quarterly inflation and the real quarterly returns to these portfolios. A second non-overlapping subset of portfolios is chosen to serve as an out-of-sample set of basic portfolios. This data is used to test the robustness of valid pricing kernels following the methods described above in Chapter 3. This portfolio subset consists of the 5153, 5252, 26 5254, 5351, 5355, 5452, 5454, 5553, and government bond (GOVT) portfolio returns. This portfolio subset is depicted graphically in Figure B.2 under the "Out of Sample" label. Panel E in Table A . l provides basic descriptive statistics for the real quarterly returns to these portfolios. 4.2 Instrumental and Conditioning Variables Following Chapman (1997), we consider three instrumental variables similar to a subset of the ones used by Ferson and Constantinides (1991): the credit spread, denoted DEF, is measured by the difference between the yields on a portfolio of BAA-rated bonds and a portfolio of AAA-rated bonds constructed by Moody's Investor Services2, the Standard and Poor's (S&P) 500 composite stock index dividend yield, denoted DIV, available from Thomson Financial Datastream using the code SkPCOMP(DY), and the annual growth rate in the U.S. Federal Reserve Board's monthly index of total industrial production, denoted AIP, available from Thomson Financial Datastream using the code USINPRODG. A l l three instrument series are standardized to have zero unconditional means and unit variances. This standardization is necessary since the relative scale of the instrumental variable effects the relative weightings ascribed to the managed portfolios in the G M M objective function when using the inverse of the second moment matrix of returns3. Earlier empirical work has demonstrated the predictive power of the credit spread (Fer-son and Harvey, 1991; Chen, 1991), dividend yield (Fama and French, 1988; Campbell and Shiller, 1988), and industrial production growth (Balvers et al., 1990; Chen, 1991; Pesaran and Timmermann, 1995) in forecasting equity and fixed income returns. As discussed above, the instrumental variables expand the set of moment conditions to include pricing errors for portfolios managed according to predictive information in the instruments. This aug-mentation of the original asset set provides for a more powerful test of the model under 2 T h e B A A - r a t e d and A A A - r a t e d bond portfolio yield series are available for download from the U .S. Federal Reserve's web site www.federalreserve.gov or from Thomson Financial Datastream using the codes FRCBBAA and FECB AAA. 3 Cochrane (1996) (footnote 10, page 594) discusses the choice of instrument scale when using the Hansen and Jagannathan (1991) weight matrix for G M M estimation and testing. 27 consideration. Furthermore, Chapman (1997) notes that this particular set of three instru-mental variables "reflect variations in the stock market, bond market, and the real economy." Panel C in Table A . l provides basic descriptive statistics for the standardized versions of the quarterly time series for these instruments. In addition to the instrumental variables, we also require the use of conditioning variables to scale the state variable(s) in the conditional versions of the asset pricing models. For reasons of parsimony, we consider only the term spread as measured by the difference between the yield on a portfolio of all Treasury bonds over ten years to maturity and the yield on a one year constant maturity Treasury note. Both yield series are provided by the U.S. Federal Reserve4. We label the term spread variable as TERM. The predictive value of the shape of the term structure is supported by the the work of Keim and Stambaugh (1986), Campbell (1987), and Fama and French (1989) among many others. Panel D in Table A . l provides basic descriptive statistics for the term spread variable expressed in decimal, rather than percentage, form. While TERM is established in the empirical literature as an effective macroeconomic forecasting variable, many well motivated alternative conditioning variables could be used in its place. For example, Lettau and Ludvigson (2001a) propose a log consumption-wealth variable and find it is both a strong predictor of aggregate stock returns (Lettau and Lud-vigson, 2001a) as well as a useful conditioning variable for explaining the cross-section of average stock returns (Lettau and Ludvigson, 2001b). Lettau and Ludvigson (2001a) develop an economic framework which implies a cointegrated relationship between consumption, as-set holdings, and labor income. The authors define CAY to be the deviations from this shared trend (see Equation (12) on page 823 of Lettau and Ludvigson (2001a))5. To check whether our various conditional model results are likely to be an artifact of our choice of TERM as our conditioning variable, in Chapter 7 we replicate all the conditional model tests using CAY in place of TERM. 4 T h e yield on a portfolio of al l Treasury bonds over ten years to maturi ty and the yield on a one year constant maturity Treasury note are both available for download from the U.S . Federal Reserve's web site www.federalreserve.gov wi th reference codes tcmlOp and tcmly, respectively. 5 W e thank Mar t i n Lettau and Sydney Ludvigson for providing the log consumption-wealth variable via download from their web page, http:/ /www.newyorkfed.org/rmaghome/economist/ lettau/lettau.html. 28 4.3 The Asset Pricing Models To facilitate a broad evaluation of the conditional nonlinear asset pricing kernel approach, we consider five different specifications for each of five asset pricing models. The five specifica-tions that we investigate are summarized as follows: i) sets of first order (linear) orthonormal polynomials in the model state variables, ii) sets of second order orthonormal polynomials in the model state variables, iii) sets of third order polynomials in the state variables, iv) sets of first order polynomials in the model state variables and conditional state variables, and v) sets of second order polynomials in the model state variables and conditional state variables6'7. Naturally, the set of state variables varies across the five asset pricing models. The first model we consider is the Sharpe (1964), Lintner (1965) and Mossin (1966) C A P M . The one original state variable for the C A P M model is the market premia, MKT. We begin with the monthly excess returns on the value weighted market portfolio of NYSE, A M E X , and (after 1972) Nasdaq stocks as a proxy for the market premia8. Excess monthly returns are calculated using the one month Treasury bill yield from Ibbotson Associates (2001). Quarterly excess returns are computed by compounding the monthly returns. Uncon-ditional nonlinear specifications of the C A P M pricing kernel are constructed using columns drawn from the orthonormal transformation of the state variable matrix [MKT, MKT2, MKT3] where MKTq represents the g-th power operator applied to the column vector on an element by element basis. Conditional nonlinear specifications of the C A P M are constructed using columns drawn from the state variable matrix [MKT, MKT2, MKT * TERM, MKT2 * TERM] where * represents an element by element vector multiplication operator9. For the second model, we consider a representation of the consumption capital asset pric-ing model (CCAPM) of Rubinstein (1976), Breeden and Litzenberger (1978), and Breeden 6 Deta i l s concerning the construction of the sets of orthonormal polynomials from the original state vari-ables is provided above in Chapter 3. 7 Sets of th i rd order polynomials in the model state variables and conditional state variables were not tried due to a problematic loss in degrees of freedom for these specifications. 8 W e thank Kenneth French for providing this data v ia download from his web page. 9 A s noted above, for reasons of parsimony the conditional nonlinear specifications do not employ as high an order polynomial terms as the unconditional nonlinear specifications. 29 (1979). Following the empirical work of Breeden et al. (1989), Chapman (1997), and Hodrick and Zhang (2000) we employ versions of the model which use a single state variable for real consumption growth. We use A C to denote real quarterly consumption growth constructed using the U.S. personal consumption expenditures on nondurable items reported by the U.S. Department of Commerce and available from Thomson Financial Datastream using the code USCONNDRB. The calculation of AC is made on a per capita basis by dividing real con-sumption by the resident population of the U.S. reported by the Organization for Economic Co-operation and Development (OECD) and available from Thomson Financial Datastream using the code USOCFTPP10. Unconditional nonlinear and conditional nonlinear speci-fications of the C C A P M are constructed using columns drawn from orthonormalized state variable matrices constructed analogously to those described above for the C A P M . A natural extension of the C C A P M is the case where utility of consumption is not time-separable. The third asset pricing model we consider is a non-separable consumption capital asset pricing model (NS-CCAPM) generally based upon the concept of habit formation ex-amined by Constantinides (1990) and Ferson and Constantinides (1991). Following Chapman (1997), we model NS-CCAPM using A C from above in addition to AC shifted one quarter ahead, denoted AC+i. Unconditional nonlinear specifications of the NS-CCAPM pricing kernel are constructed using columns drawn from the orthonormal transformation of the two state variable matrices [AC, A C 2 , A C 3 ] and [ A C + i , A C | j , AC+J. Conditional nonlinear specifications of the NS-CCAPM are constructed using columns drawn from orthonormal transformations of the two state variable matrices [AC, A C 2 , A C *TERM, AC2 *TERM] and [AC+i, A C 2 X , AC+i * TERM, AC2+l*TERM]. To diversify the model set, we choose the investment-based asset pricing model, denoted COCHRANE, developed by Cochrane (1996) to serve as our fourth model. The formal Cochrane (1996) model utilizes factors that are returns to phsysical investment. These fac-tor returns must be inferred from investment data using an assumed production function. However, investment return is approximately proportional to growth in investment in the model and Cochrane (1996) reports that a investment growth model performs equally well. 1 0 T h e population figure from this source is available only on an annual basis. Quarterly estimates of the population are produced using a simple linear interpolation between annual figures that are attributed to the second quarter of the year for which they are reported. 30 Thus, following the empirical work of Cochrane (1996) and Hodrick and Zhang (2000), we use two state variables to represent investment growth: i) the quarterly growth rate in real non-residential investment, denoted NRINV and ii) the quarterly growth rate in real residential investment, denoted RINV. Nominal quarterly investment growth rates are computed from the U.S. Department of Commerce index of nonresidential private fixed investment and index of residential private fixed investment available from Thomson Financial Datastream using the codes USIVFN..E and USIVFR..E. Real quarterly investment growth rates are com-puted by subtracting the logarithmic first difference in the quarterly Consumer Price Index series from each nominal series. Unconditional nonlinear and conditional nonlinear specifica-tions of the COCHRANE model are constructed using columns drawn from orthonormalized state variable matrices constructed analogously to those described for the NS-CCAPM. While the C A P M , C C A P M , NSCAPM, and COCHRANE models are all based upon economic theories, for our fifth and final model we consider the empirical asset pricing model of Fama and French (1993). This model is commonly referred to as an "empirical" one because it utilizes the returns to (zero-cost mimicking) portfolios as state variables. While the FF3 model is widely used for risk adjustment in the empirical research literature, many financial economists caution that it does not represent an asset pricing theory because the model employs empirically determined state variables. However, using a dynamic general equilibrium production economy where stock returns are characterized by an intertemporal C A P M , Gomes et al. (2001) explicitly link expected stock returns to firm characteristics such as size and the book-to-market ratio. Further, Berk et al. (1999) establish a similar link using a partial equilibrium model of the firm making optimal project investment decisions. We consider the Fama and French (1993) three state variable model (hereafter, FF3) consisting of the market premium, MKT, the SMB (small minus big) factor, and HML (high minus low) factor. Quarterly returns for SMB and HML are calculated following the methods in Fama and French (1993)11. Unconditional nonlinear and conditional nonlinear specifications of the FF3 model are constructed using columns drawn from three orthonor-malized state variable matrices constructed analogously to those described above for the NS-CCAPM. 1 1 We thank Kenneth French for providing this data via download from his web page. 31 Chapter 5 Empirical Results We examine the roles of nonlinearity, conditioning, and conditional nonlinearity in the context of the five asset pricing models described above: C A P M , C C A P M , NS-CCAPM, COCHRANE, and FF3. For this model set, we review five sets of progressively more com-plex model specifications in order to better identify the relative contributions of nonlinearity and conditioning information in pricing the size and book-to-market effects1. For each spec-ification of every model, the estimation and testing is performed using two approaches: the returns-weighted G M M of Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) and the optimal-weighted G M M of Hansen (1982). 5.1 Linear Model Results As a base point, we first assess linear (first order polynomial) kernel specifications for the five asset pricing models. The model specification tests are reported in the five columns of Table A.2. The specification tests of equation (3.22) are labeled "Returns-weighted \ 2 test" in Panel A of the table. Hansen's (1982) test of overidentifying restrictions, equation (3.23) above, is labeled "Optimal-weighted x2 test" in Panel B of the table. Both x2 tests are designed to test the null hypothesis that all pricing errors are equal to zero. The small p-X A summary of the five specifications is provided above in Chapter 4. 32 values reported in both Panel A and Panel B of Table A.2 lead us to reject the null hypothesis at the five percent level for all five linear models. Finally, the "supLM test statistic" in Panel B represents the Andrews (1993) test for structural breaks in the parameter estimates. The x2 specification test results for the linear models are corroborated by the informal Hansen and Jagannathan (1991) standard deviation bound tests. The means and standard deviations of the linear pricing kernels are reported in Table A.2 and plotted in Figure B.3 versus the Hansen and Jagannathan (1991) lower bound of equation (3.25). FF3 is the only model for which the linear specification passes the standard deviation bound test using both the returns-weighted and the optimal-weighted estimation procedures. However, even this kernel fails both x2 specification tests. The poor performance of the linear FF3 model pricing kernel, when applied to the Fama and French (1993) ME and B / M sorted portfolios, is particularly troubling given that the FF3 utilizes (in addition to MKT) the SMB and HML mimicking portfolio returns as state variables. This result is consistent with related findings for the FF3 model linear pricing kernel tested in Hodrick and Zhang (2000). An additional insight into the failure of the linear models is provided by inspecting groups of individual pricing errors. Figure B.4 provides a graphical representation of the linear models' average pricing errors for each basic asset and managed portfolio. While the average pricing errors for most of the models do not appear to be large, the specification test rejections indicate that the variance for the pricing errors must be large, i.e., the small average pricing error is achieved via the time averaging of large positive and large negative price errors. Further investigation of the price errors is provided in Table A.3 where we report the Wald statistics of equation (3.29) for asset groups consisting of one basic asset and all managed portfolios of that asset. The null hypothesis is that the pricing errors for a given basic asset and associated set of managed portfolios are all zero. The null hypothesis is rejected for the S1B5 portfolio and its associated set of managed portfolios in all models except the C A P M . S1B5 represents the smallest capitalization quintile and highest book-to-market quintile stocks. These "small cap. value stocks" have produced the highest real returns (3.8% per quarter) over the sample period and appear most problematic for the 33 unconditional linear pricing kernels. 5.2 Unconditional Nonlinear Model Results Given the poor performance of the linear specifications, we next investigate the effectiveness of adding nonlinearity to the pricing kernels. Table A.4 summarizes the specification test results for the pricing kernels constructed from sets of second order orthonormal polyno-mials. Overall, the small p-values for both the returns-weighted and optimal-weighted x2 tests indicate misspecification for all five asset pricing models. For the NS-CCAPM and COCHRANE models, the average pricing errors depicted in Figure B.6 appear very large. The Wald tests for asset groups consisting of one basic asset and all managed portfolios of that asset are reported in Table A.5 and generally indicate that the SlBh (small cap. value) and S5B5 (large cap. value) portfolios are most problematic. While the overall performance of the second order specifications is poor, we are also in-terested in asking whether these specifications are an improvement over the linear versions. Tests for the nested linear models using the x2 statistic of equation (3.24) produce mixed results2. For the C A P M , C C A P M , and FF3 we do not reject the nested linear model in-dicating that the second order terms offer statistically significant improvement in only the NS-CCAPM and COCHRANE models. One area of improvement for the second order specifications is the Hansen and Jagan-nathan (1991) lower standard deviation bound test. From Figure B.5 it is evident that, in comparison to the linear specifications, a larger number of pricing kernels he above the lower standard deviation bound. However, failure of the specification tests and mixed results for nesting tests indicate that nonlinearity in this form is inadequate for pricing the Fama and French (1993) ME and B / M sorted portfolios. Following Chapman (1997), we next consider constructing pricing kernels using sets of 2 R e c a l l from the discussion in Chapter (3.2) that the unrestricted weight matrix must be used for both the restricted and unrestricted specifications. A s a result, the difference in x2's between the two specifications is not consistent wi th using x2 statistics reported in Table A .2 for the restricted model. 34 third order orthonormal polynomials. Table A.6 summarizes the specification test results for these pricing kernels. In general, the sizes of both the returns-weighted and optimal-weighted x2 test statistics are much smaller than those observed for the second order models3. However, the returns-weighted x2 test p-values reported in Panel A indicate misspecification for all five asset pricing models. Interestingly, the optimal-weighted x2 p-value for the FF3 model reported in Panel B indicates that this pricing kernel is not rejected at the five percent level. However, the average pricing errors depicted in Figure B.8 and the Wald pricing error tests provided in Table A.7 still indicate serious pricing problems for the third order specification of the FF3 model. As before, we also wish to assess the incremental value of increasing the pricing kernel complexity. Nesting tests using the x2 statistic of equation (3.24) reject the nested first order and second order specifications for all five models. These results appear as Panels C and D in Table A.6. This improvement is echoed in the Hansen and Jagannathan (1991) lower standard deviation bound tests depicted in Figure B.7. A l l of the pricing kernels, except for the optimal-weighted C A P M , lie above the lower standard deviation bound. Furthermore, all five models also pass the Andrews (1993) test for structural breaks in the parameters. Clearly, the nonlinearity introduced by the third order specifications offers a significant improvement to the pricing for all models considered here. The results for our unconditional nonlinear pricing kernel at first appear at odds with results from Bansal and Viswanathan (1993) and Chapman (1997). For instance, Chapman (1997) reports success pricing the well documented size effect using second, third, and fourth order orthonormal polynomial approximated pricing kernels for the models that we have labelled C C A P M and N S - C C A P M 4 . In related work, Bansal and Viswanathan (1993) find that an artificial neural network approximated pricing kernel also adequately prices the size effect using three state variables: the nominal market return, the nominal Treasury bill yield to maturity, and the nominal yield spread between nine-month and three-month Treasury 3 N o t e that only the returns-weighted \ 2 values are comparable across models and specifications. 4 T h e success of consumption-based models reported by Chapman (1997) contrasts wi th most of the other empirical work involving consumption. The consumption-based capital asset pricing models ( C C A P M ) derived from Lucas (1978) are also rejected by the data as reported by Mehra and Prescott (1985), Hansen and Singleton (1982), and Breeden et al. (1989). However, most researchers attribute the poor performance of the C C A P M to the use of poor proxies for consumption. Breeden et al . (1989) discuss some of the limitations of the consumption data. 35 bills. Note however, that both Bansal and Viswanathan (1993) and Chapman (1997) test the ability of unconditional nonlinear kernels to price only the size effect. In results not reported here, we find that both the unconditional second and the third order specifications for all five of our models pass both the returns-weighted and optimal-weighted x2 tests when applied to the set of size decile and fixed income portfolios considered in Chapman (1997)5. These findings for the size effect in isolation contrast sharply with the near uniform rejection of both the second and the third order specifications for all five models that we report above for the size and book-to-market sorted portfolios. Evidently, the combination of size and book-to-market effects presents a significantly more difficult asset pricing challenge than the size effect alone. 5.3 Conditional Linear Model Results As a prelude to considering conditional nonlinear models, we evaluate the performance of conditional linear specifications for the five models. We condition using the lagged term spread variable, denoted TERM, defined above in Chapter 4. The returns-weighted and optimal-weighted \ 2 test statistics and corresponding p-values reported in Panels A and B of Table A.8 indicate a rejection of the pricing kernels for all five models. The average pricing errors depicted in Figure B.10 reveal very large average errors for the COCHRANE model in particular. However, the Wald pricing error tests provided in Table A.9 indicate serious individual pricing error problems across all models except FF3. Similar to the findings for all the unconditional specifications, the SlBb (small cap. value) and SbBb (large cap. value) appear the most difficult to price. Overall, the failure of our conditional linear model specifications corroborate with the related literature. Hodrick and Zhang (2000) use the cyclical component of Gross National Product (GNP) as measured by the Hodrick and Prescott (1997) filter and the consumption-f o l l o w i n g Chapman (1997), we use a basic set of portfolios consisting of the T B I L L and C O R P series described above i n addition to the 1st, 5th and 10th deciles of the size sorted portfolios available from the Center for Research in Security Prices. We use the same sample period, instrumental variables, state variables and model compositions as in our work above. Tabulated results for our specification tests based on these size sorted portfolios are available upon request from the author. 36 wealth series of Lettau and Ludvigson (2001a) as conditioning variables in several asset pric-ing models applied to pricing the Fama and French (1993) twenty-five ME and B / M sorted portfolios and a Treasury bill portfolio. While Hodrick and Zhang's conditional specifica-tions appear to price better than unconditional versions, the authors find, in out of sample tests, that the estimated models do not price managed portfolios constructed using term spread as an information variable. In related work, He et al. (1996) also attempt to price the Fama and French (1993) twenty-five ME and B / M sorted portfolios using a generalization of the Harvey (1989) specification of conditional asset pricing models. He et al. (1996) test and reject conditional versions of the Fama and French (1993) three and five state variable models when applied to the size and book-to-market pricing problem. In comparison to unconditional linear models, has conditioning information improved pricing performance? Nesting tests using the x2 statistic of equation (3.24) reject the nested unconditional linear specifications for all five models. The lagged TERM spread variable appears to contain statistically significant pricing information. This result is not surprising given the previous work of Keim and Stambaugh (1986), Campbell (1987), Fama and French (1989) and others who find term spread to be useful in forecasting the risk premia for equity and fixed income markets. The Hansen and Jagannathan (1991) lower standard deviation bound tests depicted in Figure B.9 reveal an improvement for the C A P M and C C A P M models under return-weighted estimation. However, for the optimal-weighted estimations, adding conditioning information to the linear specifications improves only the COCHRANE model. In summary, adding conditioning information to linear specifications produces results parallel to those found for adding nonlinear terms; while the incremental complexity fails to solve the overall pricing problem, a significant improvement in pricing performance is observed. Both conditioning information and nonlinearity in the pricing kernel specifications appear important, even though neither enhancement considered individually is sufficient to salvage the asset pricing models. 37 5.4 Conditional Nonlinear Model Results The final specification we consider represents a blending of both conditioning information and nonlinearity in the pricing kernels of the five models. More specifically, we construct pricing kernels from sets of second order orthonormal polynomials that include unconditional terms as well as terms scaled by the term spread variable TERM6,7. While the x2 tests for the C A P M , C C A P M , NS-CCAPM, and COCHRANE models reject the conditional second order models, both the returns-weighted and optimal-weighted tests fail to reject the FF3 kernel. The FF3 kernel also passes the Andrews (1993) structural break test for the parameters. The average price error graphs of Figure B.12 depict relatively small errors for the FF3 model in comparison to the four other models. More formally, the Wald tests for individual asset sets reported in Table A. 11 fail to reveal any statistically significant individual pricing errors for the FF3 model. While the FF3 model is the only one to pass both x2 tests, the parameter stability tests, and all the Wald pricing error tests, we are still interested in the incremental improvement provided by the conditional second order specifications for the broader model set. The first area of improvement is depicted in Figure B . l l ; both the returns-weighted and the optimal-weighted estimations of all five models produce pricing kernels that satisfy the Hansen and Jagannathan (1991) lower standard-deviation bound8. Additionally, the nesting tests of Panel C in Table A. 10 reject the nested unconditional second order specifications for all but the NS-CCAPM models. For all five models, the nesting tests of Panel D in Table A.10 reject the nested conditional linear specifications. Taken together, the two nesting tests indicate that both conditioning and nonlinearity are important elements in the pricing kernel specifications. 6 M o r e exact details of the kernel construction methods are provided above i n Chapters 3 and 4. 7 A s noted before, we do not consider conditional third order polynomials for reasons of parsimony. 8 A s mentioned previously, this result contrasts sharply wi th Di t tmar (2001) who reports conditional nonlinear pricing kernels that do not satisfy the Hansen and Jagannathan (1991) lower bounds established using industry group portfolios. 38 Chapter 6 The Conditional Second Order F F 3 Model: Robustness Tests The conditional second order specification of the FF3 pricing kernel warrants further inves-tigation based upon the promising test results reported in the previous section including: • a failure to reject the pricing kernel based upon the returns-weighted x2 test, • a failure to reject the pricing kernel based upon the optimal-weighted x2 test of overi-dentifying restrictions, • for both the returns-weighted and optimal-weighted estimations, satisfaction of the Hansen and Jagannathan (1991) lower standard deviation bound, • a failure to reject the pricing kernel based upon the supLM parameter stability tests, • a failure to reject the null of no pricing error for each of the seven Wald tests for individual asset subsets consisting of a basic asset (TBILL, CORP, SlBl, 5155, 5353, 5551, or 5555) and associated managed portfolios. These results indicate that, for the sample period Q2-1959 to Q4-1999, the conditional second order specification of the FF3 pricing kernel cannot be rejected when applied to the 39 basic asset set (TBILL, CORP, 5151, 5155, 5353, 5551, 5555) and the accompanying managed portfolio set generated by the instrumental variables (DEF, DIV, AIP). However, before lending further interpretation to these results, we investigate their robustness. We can not rule out the possibility that our results are an artifact of the sample data we have chosen to work with. While no "silver bullet" exists for dealing with the risk of Type II statistical error, several ancillary tests may be used to gauge how robust the results are to alternative data choices. In particular, the following two subsections address in turn: i) specification tests using the conditional nonlinear FF3 parameterizations estimated in the previous chapter applied to out-of-sample portfolio returns, and ii) specification tests based upon new estimations of the conditional nonlinear FF3 model using an alternative instrumental variables set. Overall, the results provided below indicate that our failure to reject the conditional nonlinear FF3 model in Chapter 5 above is not an artifact of our particular choice of portfolio returns or instrumental variables. In the subsequent chapter, Chapter 7, we investigate the possibility that our results are an artifact of our particular choice of conditioning variable, TERM. 6.1 Specification Tests on Out-of-Sample Portfolio Re-turns Recall from Chapter 4 that, in order to improve the stability of our results, only five of the twenty-five Fama and French (1993) M E and B / M E sorted portfolios are used in estimation and testing1. While we have attempted to choose five portfolios representing the broadest cross-section in portfolio characteristics (see Figure B.2), the possibility remains that the conditional second order specification of the FF3 pricing kernel may not adequately price the unused Fama and French (1993) portfolios. With this concern in mind, we utilize a new subset of basic assets (5153, 5252, 5254, 5351, 5355, 5452, 5454, 5553, and GOVT) to perform several tests on both the returns-weighted and the optimal-weighted pricing kernel estimations while holding the respective optimal parameterizations fixed. 1 I n Chapter 4, we provide a more detailed motivation for the use of a smaller basic asset subset. 40 Using the new basic asset set and the same instrumental variables, the x2 specification test statistics in equations (3.22) and (3.23) are recomputed and reported in Table A.12. The out-of-sample specification test results for the conditional second order specification of the FF3 pricing kernel appear in the column labeled "TERM 2nd." The large p-values, 0.9251 and 0.3586 for the returns-weighted and optimal-weighted estimations respectively, indicate that we cannot reject the conditional second order specification of the FF3 pricing kernel based upon the pricing of the out-of-sample portfolios. Furthermore, Wald pricing error tests provided in Table A.13 fail to reject the null of no pricing errors for individual assets and associated managed portfolios in the alternative portfolio set. Both the returns-weighted and optimal-weighted pricing kernels perform very well when applied to pricing restrictions not included in the original estimations. 6.2 Re-estimation with A n Alternative Instrumental Variables Set The second question we pose is whether the failure to reject the conditional nonlinear FF3 model is simply the result of a fortuitous choice of instrumental variables. In the theoretical development of Chapter 3 above, recall that the representative agent's first order conditions are mapped into the following set of moment conditions: E[G(Pt+i; Q)(Rt,t+l ® Zt)\ = E[l ® Zt] where Zt is a 1 x K vector of instrumental variables known to the representative agent at time t (i.e., Zt € H (). Note that, in theory, this mapping holds for any Zt G flt observed by the agent. In practice, the econometrician must choose a relatively small set of well motivated instrumental variables and hope to capture as much of the agent's relevant information set as possible. In our empirical work above, we follow Chapman (1997) using a set of three instrumental variables: term spread, DEF; the Standard and Poor's composite stock index dividend yield, DIV, and the annual growth rate in the U.S. Federal Reserve Board's monthly index of total industrial production, AIP. Chapman (1997) comments that these 41 three instruments represent information from the fixed income market, the equity market, and the real economy respectively. To assess the sensitivity of our conditional nonlinear FF3 test results to choice of instru-mental variables, we propose an alternative set of three instruments: the discount yield for the one month Treasury bill from Ibbotson Associates (2001) observed at quarterly intervals, denoted TBY1; the quarterly return on the Standard and Poor's 500 composite stock index from Thomson Financial Datastream, denoted SPRET; and the cyclical component of the natural logarithm of the U.S. Industrial Production Index from Thomson Financial Datas-tream. The Hodrick and Prescott (1997) filter is used to estimate this cyclical component from monthly industrial production data using a smoothing parameter of 6400. Monthly rather than quarterly frequency industrial production data is used to improve filtering by providing more data points for any given window of the cycle. A quarterly series for the cycli-cal component, IPCYC, is created by extracting observations from the monthly series at quarterly intervals. As before, all three instruments are lagged one quarter and standardized to have zero unconditional means and unit variances. Earlier empirical work has demonstrated the predictive power of the Treasury bill yield (Fama and Schwert, 1977; Campbell, 1987; Chen, 1991; Ferson and Harvey, 1993), the lagged market return or momentum (Jegadeesh and Titman, 1993; Conrad and Kaul, 1998), and the cyclical component of industrial production (Hodrick and Zhang, 2000) in forecasting equity and fixed income returns. Note that, as with the original instrument set, these three variables also represent information from the fixed income market, the equity market, and the real economy respectively. Using the original basic asset set and this new instrumental variables set, the x2 specifica-tion test statistics in equations (3.22) and (3.23) are recomputed and reported in Table A.14. The p-values, 0.1489 and 0.2990 for the returns-weighted and optimal-weighted estimations respectively, indicate that we cannot reject the conditional second order specification of the FF3 pricing kernel estimated using the new instrumental variables set. Wald pricing error tests provided in Table A. 15 fail to reject the null of no pricing errors for individual assets and associated managed portfolios in the alternative portfolio set. Furthermore, the both 42 returns-weighted and optimal-weighted estimations pass the Andrews (1993) supLM tests for structural shifts in the parameters. In summary, the conditional nonlinear FF3 pricing kernels perform very well when estimated with the alternative instrumental variables set (TBY1, SPRET, IPCYC). 43 Chapter 7 Conditioning with C A Y Rather Than T E R M The evidence presented thus far indicates that the failure to reject the conditional nonlinear FF3 model is not simply an artifact of our choice of basic portfolios or instrumental variables set. Is it possible that our results hinge critically on a fortuitous choice of conditioning variable, TERM? Recently, Lettau and Ludvigson (2001a) propose a log consumption-wealth variable, CAY, and demonstrate that a wide class of optimal models of consumer behavior imply that CAY will be a predictor of expected asset returns. Furthermore, Lettau and Ludvigson (2001b) report success pricing the size and book-to-market effects using CAY conditional linear versions of the C A P M and the C C A P M . Interestingly, Hodrick and Zhang (2000) find contradicting results using pricing kernel methods to estimate a conditional linear C A P M and C C A P M with the CAY as a conditioning variable. In this chapter, we examine CAY as an alternative to TERM as a conditioning variable. Choosing to test CAY as an alternative to TERM as a conditioning variable serves two distinct purposes. First, we may assess whether the failure to reject the conditional nonlinear FF3 model is robust to a change in conditioning variable. Second, CAY conditional linear versions of the C A P M and C C A P M may be tested to determine whether our rejection of the TERM conditional versions of these models owes to limitations of TERM as a conditioning 44 variable. We hope this second set of results helps shed light on the contradictory set of reports by Lettau and Ludvigson (2001b) and Hodrick and Zhang (2000) regarding the effectiveness of CAY conditional linear C A P M and C C A P M in pricing the size and book-to-market effects. In summary, results presented below indicate that while a CAY conditional nonlinear FF3 is not reject by the data, CAY conditional linear and conditional nonlinear versions of the C A P M and C C A P M are indeed rejected by our sample data. We first report the performance of CAY conditional linear specifications for all five asset pricing models. The returns-weighted and optimal-weighted x2 test statistics and corresponding p-values reported in Panels A and B of Table A. 16 indicate a rejection of the pricing kernels for both the C A P M and C C A P M and most other models. The Wald pricing error tests provided in Table A. 17 indicate serious individual pricing error problems for the CAY conditional C A P M in particular. Overall, substituting CAY for TERM in our conditional linear C A P M and C C A P M formulations produces pricing kernels that are still rejected by the sample data. In this regard, our results are more supportive of those found in Hodrick and Zhang (2000) than they are of the findings of Lettau and Ludvigson (2001b). However, note that unlike Lettau and Ludvigson (2001b) our sample includes T-bills, corporate bonds, and numerous managed portfolios created by the use of instrumental variables. This forces the pricing kernel to price not only the size and book-to-market effects, but also fixed income returns and predictable variation in returns. Recall also that Lettau and Ludvigson (2001b) caution that small sample bias in our iterated G M M is more acute as the number of cross-section observations grows in relation to the time-series sample size (Ferson and Foerster, 1994; Hansen et al., 1996). Next, to examine whether the failure to reject the conditional nonlinear FF3 model is robust to a change in conditioning variable, we report the performance of CAY conditional nonlinear specifications for the FF3 and the four other asset pricing models. The returns-weighted and optimal-weighted x2 test statistics and corresponding p-values reported in Panels A and B of Table A. 18 indicate a rejection of the pricing kernels for all except the FF3 model estimations. Additionally, the Andrews (1993) supLM tests indicate that parameter stability is not a problem for the CAY conditional nonlinear FF3 model. Finally, the Wald pricing error tests provided in Table A. 19 reveal no significant pricing error problems with 45 the CAY conditional nonlinear FF3. To further assess the robustness of the CAY conditional nonlinear FF3, we apply the estimated pricing kernels to out of sample data. Following the work presented above for the TERM conditional FF3, we utilize a the subset of basic assets (5153, S2B2, 5254, S3B1, 5355, 5452, 5454, 5553, and GOVT) to test the returns-weighted and the optimal-weighted pricing kernel estimations while holding the respective optimal parameterizations fixed. Using the new basic asset set and the same instrumental variables, the x2 specification test statistics in equations (3.22) and (3.23) are recomputed and reported in Table A.20. The out-of-sample specification test results for the CAY conditional second order specification of the FF3 pricing kernel appear in the column labeled "CAY 2nd." The large p-value, 0.3559 for the returns-weighted estimation indicates that we cannot reject the conditional second order specification of the FF3 pricing kernel based upon the pricing of the out-of-sample portfolios. However, the optimal-weighted estimation is rejected by the out-of-sample data with a p-value of 0.0029. For both the returns-weighted and optimal-weighted estimations, the Wald pricing error tests provided in Table A.21 fail to reject the null of no pricing errors for individual assets and associated managed portfolios in the alternative portfolio set. While the CAY conditional nonlinear FF3 model prices the size and book-to-market effects reasonably well, it does appear somewhat less robust than the TERM conditional version. Overall, substituting CAY for TERM in our conditional nonlinear FF3 formulation produces pricing kernels that perform nearly as well as the originals. These results indicate that the success of the conditional nonlinear FF3 does not likely hinge critically on a fortuitous or fluke choice of conditioning variable. In summary, results presented here and in the preceding section support the view that the original TERM conditional nonlinear FF3 success in pricing the size and book-to-market effects is not an artifact of our particular choice of portfolio returns, instrumental variables, or conditioning variable. 46 Chapter 8 How Close a Substitute is C A Y for T E R M as a Conditioning Variable? When used by econometricians as conditioning variables, both CAY and TERM are in-tended to summarize changes over time in asset pricing relevant information such as the stage of the business cycle or level of aggregate risk aversion. In the preceding chapters, we report that both TERM conditional and CAY conditional nonlinear FF3 pricing kernels are capable of pricing the size and book-to-market effects. To what extent might CAY and TERM be viewed as informational substitutes for each other? Figure B.l7 depicts time series plots of TERM and CAY together for the sample period Q2, 1959 to Q4, 1999. To ease graphical comparison, both variables are standardized to have zero unconditional means and unit variances. Visual inspection appears to reveal a high degree of correlation between the two variables. In fact, the contemporaneous correlation between the two over the sample period is 30.88%. One method of testing whether TERM captures the conditional asset pricing information in CAY is to include CAY as an instrumental variable in tests of the TERM conditional models. This effectively changes the set of moment conditions to include pricing errors for portfolios managed according to predictive information in CAY. Naturally, the reverse comparisons may also be made by including TERM as an instrumental variable in tests 47 of the CAY conditional models. Below, we report results for these reciprocal tests. To preserve the original number of moment conditions we drop the credit spread, DEF, as an instrumental variable to make room for either TERM or CAY as an instrumental variable in the CAY and TERM conditional model estimations respectively. We first report the performance of TERM conditional linear specifications for all five as-set pricing models using the following three instrumental variables: log consumption-wealth variable, CAY; the S&P 500 composite stock index dividend yield, DIV; and the annual growth rate in industrial production, AIP. The returns-weighted and optimal-weighted \ 2 test statistics and corresponding p-values reported in Panels A and B of Table A.22 indicate a rejection of the pricing kernels for all models except the optimal-weighted FF3. This one exception is close to rejection and may be the result of an expanded S? matrix as discussed in Chapter 3 above rather than the result of better pricing performance. Next, the performance of TERM conditional nonlinear specifications for all five asset pricing models using the same three instrumental variables (CAY, DIV, AIP) are con-sidered. The returns-weighted and optimal-weighted x2 test statistics and corresponding p-values reported in Panels A and B of Table A.23 indicate a failure to reject either the returns-weighted or the optimal-weighted TERM conditional nonlinear FF3 formulations. It appears as though in the conditional nonlinear FF3 model context, the TERM condition-ing variable is capable of capturing the asset pricing relevant information in CAY. Finally, we report the performance of CAY conditional linear and nonlinear specifications for all five asset pricing models using the following three instrumental variables: term spread, TERM; the S&P 500 composite stock index dividend yield, DIV; and the annual growth rate in industrial production, AIP. The returns-weighted and optimal-weighted x2 test statistics and corresponding p-values reported in Panels A and B of Tables A.24 and A.25 and indicate a rejection of the pricing kernels for all models, both conditional linear and conditional nonlinear. In particular, for the conditional linear and nonlinear FF3 model contexts, the CAY conditioning variable is not capable of capturing the asset pricing relevant information in TERM. We consider a small subset of the tests or metrics that exist for comparing the asset 48 pricing information content of TERM to that of CAY. In sample, both TERM and CAY appear adequate as conditioning variables in the context of the conditional nonlinear FF3 model applied to price the size and book-to-market effects. Interestingly, TERM conditional models appear to perform marginally better in tests on the out of sample data. Furthermore, the TERM conditional nonlinear FF3 model is capable of pricing, among other assets, managed portfolios created from using CAY as an instrumental variable. The reverse is not true of the CAY conditional nonlinear FF3 model applied to, among other assets, managed portfolios created from using TERM as an instrumental variable. Ideally, the econometrician would likely wish to include both CAY and TERM as conditioning variables. However, if the econometrician is forced to choose between the two for reasons of parsimony, the results present here motivate a preference for TERM in this particular context. 49 Chapter 9 The Conditional Second Order FF3 Model: Further Discussion We consider collectively all of the evidence presented above and distill the results into three fundamental findings. First, incorporating conditioning information into the pricing kernel on its own, and in the presence of nonlinearity, contributes significantly to pricing perfor-mance for a broad cross-section of asset pricing models. Second, incorporating nonlinearity into the pricing kernel on its own, and in the presence of conditioning information, also con-tributes significantly to pricing performance for a broad cross-section of asset pricing models. And finally, the TERM conditional nonlinear FF3 model in particular is not rejected by the enigmatic Fama and French (1993) twenty-five size and book-to-market sorted portfolios. While great attention thus far has been paid to the statistical substantiation of these results, little intuitive explanation or theoretical motivation has been provided for these results. In this section, we attempt to address these issues. 9.1 Qualitative Review What is it about the TERM conditional nonlinear FF3 model that makes it so effective? A closer look "inside" the kernel is revealing. We use 3-dimensional graphical analysis of the 50 pricing kernel response to varying levels of the state variables and conditioning variable to highlight the greatest sources of nonlinearity as well as areas of interaction between nonlinearity and conditioning information. Several insights into the conditional second order FF3 pricing kernel are found in the charts sets provided in Figures B.13 (returns-weighted) and B. 14 (optimal-weighted). The value of the FF3 kernel is a function of three state variables (MKT, SMB, HML) and one conditioning variable (TERM). Each 3-dimensional chart depicted in Figure B.13 or B.14 depicts the simulated kernel value derived from holding two variables (state or conditioning) constant at their mean level while permitting the other two variables to vary over the ranges [-10%, 10%] for MKT, SMB, and HML or [-2%, 2%] for TERM1. First, note that the returns-weighted pricing kernel of Figure B.13 appears qualitatively very similar to the optimal-weighted pricing kernel of Figure B.14. We are reassured that the two different estimation techniques have identified qualitatively similar pricing kernels. Since the two different estimations of the pricing kernel are so similar, we simplify the discussion by making observations on the returns-weighted pricing kernel alone. The three left column charts in Figure B.13 depict the greatest pricing kernel nonlinearity in response to the SMB and HML state variables. In comparison, the pricing kernel value appears to be almost linear in the market premia, MKT. Holding MKT return constant at its sample mean value, there appears to be significant nonlinear interaction between HML and SMB. In particular, the pricing kernel is relatively lower when both HML and SMB are simultaneously at extremes, but it matters not whether these are positive or negative extremes or both of the same sign. In economic terms, the intertemporal marginal rate of substitution is relatively lower when both the relative return premia between small and large capitalization stocks and the relative return premia between value and growth stocks are large is absolute terms. These conditions are likely to exist during times of substantial style rotation or "change of leadership" in the market. While the net interaction effects between MKT and SMB or MKT and HML are modest, observe that the TERM variable produces very interesting interaction effects with x T h e sample ranges for these variables are [-26.3%,22.9%], [-12.9%, 14.9%], [-19.1%, 15.7%], and [-4.0%, 3.9%] for MKT, SMB, HML and TERM respectively. 51 both SMB and HML. For example, the pricing kernel is decreasing in SMB for small values of TERM and vice versa for large values of TERM. In economic terms, this implies that in positive (negative) term structure shape environments the intertemporal marginal rate of substitution is decreasing (increasing) in the relative return premia between small and large capitalization stocks. Clearly, an unconditional version of this model would be unable to capture this form of a reversal in the pricing relationship. Given that the term spread tends to be highest just prior to or during economic expansion (Kessel, 1965; Fama, 1986), one interpretation of this relationship is that the average risk premia on risky assets is lower (higher) during "good times" (recession) if riskier assets like small capitalization stocks happen to be outperforming less risky large capitalization issues. Conversely, the average risk premia on risky assets is higher (lower) during "good times" (recession) if riskier assets like small capitalization stocks happen to be underperforming less risky large capitalization issues. 9.2 The Role of Term Spread Conditioning Informa-tion In reference to traditional asset pricing theory, would one a priori expect conditioning infor-mation to play a fundamental role in explaining the cross-section of expected stock returns? Furthermore, would one have any a priori reason to expect term spread to prove effective as a lone conditioning variable? We address these two questions in turn below. 9.2.1 Theoretical Motivation for Conditioning Information in Gen-eral We summarize five commonly cited theoretical based motivations for the use of conditional asset pricing models. One common thread to the five explanations is that they motivate a link between expected asset returns and the business cycle. In much of what follows, we borrow extensively from the intuitive explanations provided by Chen (1991). 52 In early work, Fama (1970) argues that in a multi-period economy, investors will have an incentive to hedge against stochastic shifts in consumption and the investment opportunity set. As Chen (1991) notes, this implies that state variables that are correlated with changes in consumption and the investment opportunity set will represent priced risks in the economy. Consistent with Merton (1973) and Cox et al. (1985), an asset's expected returns will be affected by the its covariance with these state variables. Many empiricists (Campbell, 1987; Harvey, 1988; Fama and French, 1989; Chen, 1991; Lettau and Ludvigson, 2001a) adopt a business cycle interpretation for these changes in the consumption and the investment opportunity set. Secondly, the intertemporal general equilibrium models of Merton (1973), Lucas (1978), and Breeden (1979) predict that consumption depends on wealth and not income. This implies consumption smoothing by the representative agent (consumers). In particular, consumers smooth consumption by saving more (less) in times when consumption is high (low) relative to wealth. As Fama and French (1989) note, if the supply of capital-investment opportunities does not fluctuate in sync with consumption, then higher desired savings will inevitably lead to lower expected asset returns. Note that consumption smoothing is not really separate from the preceding paragraph's point regarding hedging stochastic shifts in the opportunity set. Indeed, consumption smoothing may be interpreted as one example of hedging stochastic shifts in consumption while assuming that the investment opportunity set is relatively stable. A third theoretical motivation for conditional asset pricing models relates to aggregate risk aversion. In the intertemporal general equilibrium models of Merton (1973), Rubinstein (1976), Breeden (1979) and Cox et al. (1985) the market risk premium is a positive function of the aggregate risk aversion parameter. Cyclical variation in aggregate risk aversion is one implication of the habit formation models of Sundaresan (1989), Constantinides (1990), and Campbell and Cochrane (1999) which are driven by an endogenously determined subsistence level. Essentially, these models imply that in states where consumption is low (high) relative to the subsistence level, risk aversion and thus asset risk premia will be high (low). Chen (1991) provides an explicit example of how this relationship arises in the context of a HARA class utility function that incorporates a subsistence level of consumption. 53 Business cycle induced changes in the expected productivity of capital represent a fourth theoretical motivation for conditional asset pricing models. For instance, the stochastic constant-returns-to-scale economy of Cox et al. (1985) implies that a higher productivity of capital will lead to higher nominal expected risky asset returns. Chen (1991) develops a special case of the Abel (1988) exchange economy where both the nominal and excess expected market return are increasing in the expected future production level. In such an economy, a business cycle measure proxying for the expected growth rate of aggregate production should be positively correlated with the expected market premium. Finally, business cycle conditional changes in the uncertainty of the productivity of capital is another motivation for conditional asset pricing models. For example, for the same special case of the Abel (1988) exchange economy mentioned above, Chen (1991) notes that the expected market premium will be positively related to any measure proxying for conditional uncertainty of the production technology. To the extent that this uncertainty is influenced by the broad movements in the business cycle, one more explanation for business cycle related time variation in expected risky asset returns is obtained. In summary, we have reviewed five theoretical based motivations for the use of condi-tional, rather than static, asset pricing models. As mentioned above, one common thread to the five explanations is that they may be used to motivate a link between expected asset returns and the business cycle. In the context of the research presented in this thesis, the fundamental question that remains is whether our conditioning variable, term spread, is a reasonable proxy for information regarding the business cycle. This is the question to which we now turn. 9.2.2 Support for Term Spread as the Conditioning Variable In Chapter 5 above, every unconditional linear model specification and all but one uncon-ditional nonlinear model specification (NS CCAPM) is rejected by its TERM conditional counterpart. We argue that the improved asset pricing performance achieved by conditioning with term spread is not at all surprising, but rather consistent with both asset pricing theory 54 and the extant empirical research involving the term structure. In the previous subsection, we offer five theoretical motivations for a link between business cycle proxies and expected risky asset returns. To the extent that term spread is an effective business cycle proxy, it is also then a theoretically motivated forecaster of expected risky asset returns. Indeed, review of the empirical literature reveals evidence that term spread is very closely related to the business cycle. Furthermore, an extensive body of empirical research indicates that term spread is a highly effective forecaster of risky asset returns, exactly what theory would suggest of a business cycle proxy. Empirical research linking the term spread to the business cycle dates back at least as far as Kessel (1965) who finds that the term spread is small immediately before a recession and large immediately before and during economic recovery. In the often cited work of Fama (1986), the author reports a positive sloped term structure during expansion and a humped or negative sloped term structure during recessions. Further, both Fama (1990) and Jensen et al. (1996) report that the term spread is counter-cyclical, i.e., decreases near peaks in the economic activity and increases near economic troughs. In related work, both Estrella and Hardouvelis (1991) and Chen (1991) find that a positive term spread is associated with a future increase in real economic activity. In summary, many empirical studies conducted over the past thirty-five years demonstrate a strong link between term spread and the business cycle. According to the conditional asset pricing theory reviewed in the previous subsection, the strong relationship between term spread and the business cycle should in turn lead to a strong relationship between term spread and expected risky asset returns. In fact, the predictive value of the term spread is strongly supported by the the work of Keim and Stambaugh (1986), Campbell (1987), Harvey (1988), Fama and French (1989), Chen (1991), and Patelis (1997) among many others. Clearly, both asset pricing theory and empirical evidence strongly support the use of term spread as a conditioning variable in asset pricing models. How consistent is our particular sample data set with the theory and empirical evidence mentioned above? As mentioned above, business cycle effects like consumption smooth-55 ing and changes in aggregate risk aversion generally imply that expected returns should be counter-cyclical. Furthermore, asset pricing theory suggests a positive relationship between expected returns and the expected future productivity of capital. Since empirical evidence indicates that term spread is both counter-cyclical (Fama, 1990; Jensen et al., 1996) and positively associated with a future increase in real economic activity (Estrella and Hardou-velis, 1991; Chen, 1991), one would a priori expect a positive relationship between our term spread variable, TERM, and the sample portfolio returns2. We next provide a simple test of this hypothesis for our sample data set. The lagged term spread variable, TERM, is used to separate all sample period observa-tions into one of two states: 1) periods for which TERM equals or exceeds its sample mean, and 2) periods for which TERM is less than its sample mean. Columns two through six in Table A.26 report the full sample mean, high TERM state mean, low TERM state mean, t-statistic for difference between these two means, and the associated one-tailed p-value for this t-statistic. The t-statistic is used to test the null hypothesis that mean basic asset returns are equal across high and low T E R M periods. The t-statistic is computed as follows: t-StatistiC = —. „ 1 1 r = ~ t(nH-l) + (nL-l) ^Var(r?)/(nH - 1) + Var(r[)/(nL - 1) ( ) + ( ] where ff is the mean return to asset i for all high TERM state periods, nH is the number of high periods, and Var(rf) is the variance of asset i's return in the high periods. The sample moments for the nL low state returns, rf, are defined similarly. Consistent with asset pricing theory and term spread's demonstrated role as a business cycle proxy, sample average real quarterly returns to our basic set of portfolios (TBILL, CORP, SlBl, 5155, 5353, 5551, 5555) are higher (lower) immediately following quarters where TERM is larger (smaller) than average. For the equity portfolios, the difference in subsample means is large in economic terms, ranging from to 3.09% to 3.90% per quarter for the 5353 and 5151 portfolios respectively. The one-tailed p-value reported for the t-statistics in Table A.26 indicate that all the mean differences are statistically significant. A visual representation of the TERM state (high or low) conditional returns for all twenty-decent empirical tests reported in Duffee (2001) reject consumption smoothing as an explanation for the ability of term spread to predict risky asset returns. 56 five of the Fama and French (1993) size and book-to-market sorted portfolios is provided in Figure B.15. The pattern of higher average sample returns following high TERM state periods is consistent across all twenty-five portfolios. In summary, we propose that both asset pricing theory and empirical findings help explain why incorporating conditioning information into the pricing kernel contributes significantly to pricing performance for a broad cross-section of asset pricing models. Furthermore, we motivate TERM as a predictor of risky asset returns and then find it is indeed highly effective as such for our particular sample of portfolio returns. Finally, these perspectives on the role of conditioning information, and TERM as a conditioning variable in particular, suggest that our failure to reject the TERM conditional nonlinear FF3 model is not simply a chance artifact of the sample data. 9.3 The Role of Nonlinearity In reference to traditional asset pricing theory, would one a priori expect pricing kernel nonlinearity in the state variables to play a fundamental role in explaining the cross-section of expected stock returns? Furthermore, would one have any a priori reason to expect nonlinear terms in the FF3 model state variables to prove effective in pricing the twenty-five Fama and French (1993) size and book-to-market sorted portfolios? We address these two questions in turn below. 9.3.1 Theoretical Motivation for Nonlinearity in General We summarize several theoretical based motivations for the use of pricing kernels that are nonlinear in their given state variables. In general, we argue that nonlinearity is the rule rather than the exception for many of the traditional asset pricing theories. For many of these theories, strict and often-times unrealistic assumptions must be layered upon the most general form of the theory to obtain linear pricing rules. While these modified representations often lend tractability to empirical application and testing, they should not be mistaken for 57 the only possible representations of the traditional asset pricing theories. To begin, consider the Sharpe (1964), Lintner (1965), and Mossin (1966) C A P M . This equilibrium model implies an expected return equation that is linear in the market risk premium, but requires the assumption of either quadratic utility or multivariate normal asset returns (Huang and Litzenberger, 1988). By design, either of these two assumptions lead investors to care only about the mean and variance of market returns and the covariance of security returns. The former assumption is undesirable because it implies that financial assets are inferior goods (Arrow, 1970; Pratt, 1964). Further, the latter assumption is soundly rejected by the empirical evidence (Campbell et al., 1997). The dubious nature of the traditional C A P M assumptions and the poor empirical per-formance of the model have lead many researchers to consider equilibrium models where investors care about higher return moments (i.e., skewness, kurtosis, etc.) and co-moments (i.e., co-skewness, co-kurtosis, etc.). Rubenstein (1973) and Kraus and Litzenberger (1976) were the first to propose extensions to the traditional C A P M to account for investor pref-erence over higher moments in the asset return distribution. Harvey and Siddique (2000) use Taylor series expansion to obtain a skewness pricing kernel that is quadratic in the mar-ket return. As Dittmar (2001) notes, since "the coskewness of a random variable x with another random variable y can be represented as a function of Cov(x,y) and Cov(x,y2), the quadratic pricing kernel is consistent with a the three-moment C A P M . " Indeed, Kraus and Litzenberger (1976) and Jurczenko and Maillet (1996) show how it is possible to use the quadratic market model as a consistent data generating process in the three-moment C A P M . In these theoretical variations on the C A P M , expected returns are a nonlinear function of a single state variable, either market return or market excess return. Naturally, multi-moment versions of the C A P M are not the only asset pricing models for which the pricing kernels are nonlinear in the state variables. Consider the Rubinstein (1976), Breeden and Litzenberger (1978), and Breeden (1979) consumption capital asset pricing model (CCAPM). Breeden et al. (1989) propose a Taylor series approximation, or alternatively a set of distributional assumptions, to obtain a C C A P M pricing kernel that is linear in the C C A P M state variable, growth in per capital aggregate consumption. Alter-58 natively, Hansen and Singleton (1982) assume constant relative risk aversion to arrive at an (unconditional) linear pricing kernel for the C C A P M . Finally, Brown and Gibbons (1985) develop a more general version of the C C A P M assuming power utility for the representative investor. This leads to a pricing kernel function with growth in per capital aggregate con-sumption that is raised to a power equal to the coefficient of relative risk aversion3. For this model, the pricing kernel will be linear in the state variable only for the case of log utility where the coefficient of relative risk aversion is unity. Many reasonable alternative utility function choices for the C C A P M will lead to pricing kernels that are nonlinear functions of the consumption growth state variable. One popular modification to the C C A P M is to assume that utility is non-separable over time. The habit formation models of Constantinides (1990), Ferson and Constantinides (1991) and Campbell and Cochrane (1999) are well-known examples of this tact. The Euler equations from these models imply pricing kernels which are much more complex functions of consumption growth than what is obtained for time-separable utility versions of the C C A P M . Simply put, the pricing kernels derived Constantinides (1990), Ferson and Constantinides (1991) and Campbell and Cochrane (1999) are not linear functions of consumption growth, but rather nonlinear functions of current and sometimes past consumption growth. Similarly, nonlinearity is also found in investment-based asset pricing model such as that proposed by Cochrane (1996). This particular model utilizes factors that are returns to physical investment. However, these factor returns must be inferred from the model's true state variables, capital investment data, using an assumed production function. As a result, the investment returns are a nonlinear function of the capital investment state variables. A second layer of nonlinearity arises in the pricing kernel which itself may be a nonlinear function of the inferred investment returns. Cochrane (1996) demonstrates that the pricing kernel will be linear in the inferred investment returns (but not in the investment growth state variables) for the case of log utility and Cobb-Douglas production functions. However, the author's empirical work suggested that the pricing performance of the investment-based asset pricing model does not depend critically on the exact functional form of the pricing kernel. 3Brown and Gibbons (1985) also propose replacing aggregate consumption with some proxy for the market portfolio in order to avoid empirical measurement problems commonly associated with consumption. 59 Nevertheless, under many alternative assumptions for the utility function and production functions, this investment-based asset pricing model implies a pricing kernel that is nonlinear in the underlying state variables. Consider also the popular empirical asset pricing model of Fama and French (1993), the FF3 model. The proposed model implies a pricing kernel that is a linear function of the premia associated with the market portfolio, the SMB (small minus big) market capitaliza-tion factor, and the HML (high minus low) book-to-market factor. Fama and French (1992) assert that the linear factor structure for the FF3 is consistent with the multifactor asset pricing models of Merton (1973) and Ross (1976). However, note that linearity in the Merton (1973) context is obtained only for less general model variations utilizing either a restricted class of utility functions (Merton, 1971) or the assumption of log-normally distributed re-turns (Merton, 1972, 1973). In the case of the Ross (1976) arbitrage pricing theory, APT , the linear factor structure is a direct result of the starting assumption of a linear structure for the returns generating equation. Indeed, Bansal and Viswanathan (1993) provide the theoretical foundation for a nonlinear version of the A P T which rather rests upon the sufficient statistic restriction that for each time period the conditional expectation of the pricing kernel is a function of low-dimensional vector of state variables. In theory, a nonlinear version of the FF3 rests on a similar sufficient statistic assumption for the market, SMB, and HML state variables. Finally, Bansal and Viswanathan (1993) note that even if one assumes that stocks are linear pay-offs in the state variables, then derivative securities on those stocks will necessarily be nonlinear in the state variables. In related work, Dybvig and Ingersoll (1982) show that applying a linear C A P M model to price derivative securities leads to a violation of arbitrage. Since a pricing kernel should in theory be capable of pricing all risky assets in an economy, one might expect a nonlinear functional form for the pricing kernel to be the rule rather than the exception for many asset pricing theories. 60 9.3.2 Support for Nonlinearity in the FF3 State Variables In Table A.6 of Chapter 5, we report that the 3rd order polynomial specifications for all five asset pricing models (CAPM, C C A P M , NS C C A P M , COCHRANE, and FF3) reject both the linear and second order polynomial specifications in nested model tests. We also report specification test results indicating that the TERM conditional second order FF3 pricing kernel in particular is capable of simultaneously pricing fixed income, managed, and size and book-to-market sorted portfolios. We have already provided theoretical motivation for both conditional asset pricing models and for the specific conditioning variable TERM in particular. Our empirical examination of TERM's effectiveness in forecasting time-variation in our sample portfolio returns is supportive of its use as a conditioning variable. Finally, we have provided numerous theoretical motivations for the general use of nonlinearity in asset pricing kernels. What remains to be addressed is a theoretical motivation for the FF3 model in particular and some examination of why a (conditional) nonlinear version of this model might prove so effective. At the most general level, there exists theoretical support for the FF3 model. In par-ticular, Gomes et al. (2001) use a dynamic general equilibrium production economy where stock returns are characterized by an intertemporal C A P M to explicitly link expected stock returns to firm characteristics such as size and the book-to-market ratio. In their model, size and book-to-market appear to predict stock returns because they are correlated with the true conditional market beta of returns. In similar work, Berk et al. (1999) employ a partial equilibrium model of the firm's optimal investment choices which drive changes in the firm's assets and growth options. In the Berk et al. (1999) model, book-to-market has a role in explaining the cross-section of expected stock returns because it proxies for changes in a firm's systematic risk levels. Additionally, market value of equity (size) has a role in explaining the cross-section of expected stock returns because it proxies for the state variable in their model that describes the relative importance of existing assets and growth options. In related work, Brennan et al. (2001) posit a simple model of time varying investment opportunities in which the SMB and HML factors will covary with changes in the invest-ment opportunity set. While the Berk et al. (1999) and Gomes et al. (2001) models generally 61 cast SMB and HML as proxies for unobservable changes in betas, the Brennan et al. (2001) model motivates SMB and HML as proxies for time variation in the investment opportu-nity set. Finally, Liew and Vassalou (2000) note that their findings that SMB and HML are predictive of Gross Domestic Product supports a risk-based explanation for the asset pricing performance of SMB and HML. Overall, the role of the FF3 model in explaining the cross-sectional of expected stock returns is, at the very least, consistent with plausible dynamic asset pricing theories. The effectiveness of the FF3 model in our sample data set is not entirely surprising in light of the strong relationship between the Fama and French (1993) size and book-to-market sorted portfolios being priced and the SMB and HML state variables being used to price these portfolios. Indeed, some readers may be more surprised by the inability of the linear and conditional linear FF3 specifications to adequately price the size and book-to-price portfolios. Thus while it may be natural to assume a priori that SMB and HML will be effective state variables for pricing these particular portfolios, the curiosity lies in the necessity for pricing kernel nonlinearity in these state variables. In order to illuminate the features of the size and book-to-market sorted portfolio returns which necessitate nonlinearity in the FF3 state variables, we consider a principal components analysis (PCA) of the Fama and French (1993) twenty-five portfolio returns for our sample period. PCA is a dimension-reducing method of multivariate statistics that we use to identify common comovements in returns across the twenty-five size and book-to-market portfolios. Since many of the portfolio return series will tend to "move together" over time, PCA may be used to replace the twenty-five return series with a small number of new time-series. These new series are called the principal components and are linear combinations of the original portfolios return series. This set of principal components forms an orthogonal basis for the space of the twenty-five portfolio returns. For our data set, the first three principal components account for over 95% of the total variance of the original data. To be precise, the first, second and third principal components account for 87.3%, 4.0% and 3.8% of the total variance respectively. Each principal com-ponent is represented by a vector twenty-five elements long. In Figure B.16, we graph the 62 first and second principal components after reforming the vectors into the familiar five-by-five grid for size and book-to-market. These first two principal components represent the common co-movements that explain the greatest amount of total variance in the twenty-five portfolio returns. Notice in the top graph of Figure B.16 that common return co-movements are non-monotonic across B / M (book-to-market) quintiles. The interpretation is that in a typical bull (bear) market quarter, the "largest" principal component of returns is best de-scribed as decreasing (increasing) in B / M over the first three B / M quintiles and increasing (decreasing) thereafter4. This pattern is true moving across all five ME (size) quintiles. From an asset pricing perspective, it is difficult to imagine how a linear function of the HML state variable will price this nonlinear element of the twenty-five portfolio returns. Similarly, in the bottom graph of Figure B.16 the second "largest" principal component of returns reveals a systematic unevenness in return levels moving across the B / M quintiles for each given ME quintile. In particular, for each of the five B / M quintiles larger gaps in return occur when moving from the first to the second ME quintiles and to some extent when moving from the fourth to the fifth" ME quintiles. This qualitative inspection of the Fama and French (1993) twenty-five portfolio returns hints that returns for a given B / M quintile are not simply a linear increasing (or decreasing) function in ME; this favors design of the FF3 model pricing kernel where the SMB state variable enters nonlinearally. 4 B y the term "largest", we intend to imply that the variance of this principal component is the maximum for all possible choices of the axis that this principal component represents. 63 Chapter 10 Concluding Remarks 10.1 Summary In this thesis, we develop and test asset pricing model formulations that are simultaneously conditional and nonlinear. Formulations based upon the C A P M , C C A P M , NS-CCAPM, COCHRANE and FF3 asset pricing models are tested against the widely studied Fama and French (1993) twenty-five size and book-to-market sorted portfolios. In total, twenty-five model/specification combinations are estimated using instrumental variables G M M with both returns-weighted and optimal-weighted procedures. The battery of specification test results indicate that the conditional nonlinear specification of the FF3 model is the only one not rejected by the data and thus capable of pricing the size and book-to-market effects simultaneously. The pricing performance of the FF3 conditional nonlinear pricing kernel is confirmed by robustness tests on out-of-sample data as well as tests with an alternative instrumental variables set and an alternative conditioning variable. While Bansal and Viswanathan (1993) and Chapman (1997) find unconditional nonlinear pricing kernels sufficient to capture the size effect alone, our results indicate that pricing the size and book-to-market effects simultaneously presents a far more difficult challenge. For the broad cross-section of unconditional nonlinear pricing kernels tested here, nonlinearity on its own does not adequately price the size and book-to-market sorted portfolios. Further-64 more, the conditional nonlinear specifications for the C A P M , C C A P M , NS-CCAPM, and COCHRANE models also fail to price these effects. Apparently, these four theoretical-based asset pricing models are not salvageable using the conditioning information and nonlinearity explored in this thesis. However, nested model tests indicate that, in isolation, both term spread conditioning information and nonlinearity improve the pricing kernel performance for all five asset pricing models. The success of the conditional nonlinear FF3 model suggests that the combination of conditioning and nonlinearity is critical to pricing kernel design. Perhaps in future research, alternative specifications for the conditioning information and/or the form of nonlinearity will lead to different results for the four theoretical-based asset pricing models. As a first step in this research direction, we test the Lettau and Ludvigson (2001a) log consumption-wealth variable, CAY, as an alternative to TERM as a conditioning variable. Lettau and Ludvigson (2001b) report success pricing the size and book-to-market effects using CAY conditional linear versions of the C A P M and the C C A P M . However, Hodrick and Zhang (2000) find contradicting results using pricing kernel methods to estimate a conditional linear C A P M and C C A P M with the CAY as a conditioning variable. Our results indicate that while a CAY conditional nonlinear FF3 is not reject by the data, CAY conditional linear and conditional nonlinear versions of the other four models, including the C A P M and C C A P M , are rejected by our sample data. This latter finding is supportive of those reported in Hodrick and Zhang (2000) rather than the findings of Lettau and Ludvigson (2001b). The former finding suggests that the success of the conditional nonlinear FF3 model does not rest crucially on our choice of conditioning variable. 10.2 Implications for Academic Researchers The theoretical and empirical implications of our work for. financial economic researchers are manifold. To begin with the most trivial note, our results add to the weight of empirical evidence supporting conditional rather than unconditional models of asset pricing theory. Of course, for many theorists and empiricists alike, this may already be a foregone conclusion. 65 More importantly, in the context of a diverse set of asset pricing models we argue that nonlinearity appears to be a significant and unavoidable element in risky asset pricing. This bodes poorly for new theories which might potentially employ utility or return distribution assumptions to obtain a linear pricing rule. Additionally, the message for empiricists is to beware assumptions motivated soley for the purpose of yielding linearity from a given theory. While this tact often lends tractibility to a model, our work indicates that the cost is often a significant loss in the descriptive power of the model. The results of our thesis also contribute to the factor versus characteristic debate in the literature (Daniel and Titman, 1997; Brennan et al., 1998; Davis et al., 2000). Many empiri-cists claim that the failure of modern asset pricing models to properly price characteristic sorted portfolios such as the Fama and French (1993) twenty-five is a sign of behavioral bias or market inefficiency. We argue that many previous tests of popular asset pricing theories unfairly disadvantage the theoretical models as a result of the imposition of linear and or unconditional constraints on the empirical representations of the models. In this thesis, we discuss the substantial theoretical motivation that supports lifting these types of constraints. Our empirical findings and the work of others (Bansal and Viswanathan, 1993; Bansal et al., 1993; Chapman, 1997; Dittmar, 2001) are also supportive of this direction. To the extent that theoretically consistent innovations to pricing kernel construction manage to salvage previously rejected rational asset pricing theories, the need for recourse to behavioral theories or claims of market inefficiency will be diminished. With respect to the success of the conditional nonlinear FF3 pricing kernel in particular, more theoretical and empirical work is warranted. Why do the state variables in the FF3 model price assets effectively. What do these variables really proxy for? Why and how do the premia on these state variables vary with the business cycle? Certainly, the theoretical frameworks proposed by Gomes et al. (2001), Berk et al. (1999) and Brennan et al. (2001) are promising steps in this direction. Following Liew and Vassalou (2000), more theoretical and empirical work examining the link between SMB and HML and the business cycle may lend greater interpretation to what many financial economists currently view as dubious, empirically derived factors. 66 10.3 Implications for Practitioners Practitioners will most often employ the use of asset pricing models in one of three contexts: i) cost of capital calculations, ii) selecting investment portfolios, iii) evaluating ex-post port-folio performance. Our work has implications for each of these applications of asset pricing theory. With each new asset pricing theory comes a new expected return equation. While most corporate managers are now well aware that the C A P M is not a suitable guide to estimat-ing the cost of equity capital, no clear alternative has yet emerged in the M B A class-room. The conditional second order FF3 model that we fail to reject in this thesis may be some-what complex for common practice. However, several practical implications are clear from graphical inspection of the pricing kernel discussed in Chapter 9 above. For example, where a linear FF3 model would normally underestimate the required expected return for small capitalization value stocks, a nonlinear FF3 model will assign a higher estimate to the fair cost of equity capital for such firms. Furthermore, the corporate managers are well advised to condition upon business cycle proxies when making their cost of capital estimates. Our empirical work generally indicates that the cost of capital is counter-cyclical, i.e., higher during recession and lower during economic expansions. These considerations for nonlinearity and conditioning information extend naturally into the practical problems of portfolio construction and performance measurement. To begin, the pervasiveness of static mean-variance based portfolio construction may wane as practitioners adjust their views of asset pricing to reflect empirical findings like those reported here. For instance, the large difference in expected returns conditional upon the stage of the business cycle may lead many investors to tactically re-allocate funds between risky and riskless assets conditional on indicators such as the term spread. However, according to conditional asset pricing models, investors following such a strategy would not be generating abnormal returns. Rather, proper performance measurement would attribute the increased nominal returns with a tendency to increase risk bearing at points in the business cycle when the intertemporal margin rate of substitution is highest. Interestingly though, if an investor believes her personal intertemporal marginal rate of substitution does not vary to 67 the extremes implied by the market overall, the proposed tactical asset allocation strategy will increase her expected utility. The implications for portfolio construction and performance measurement are also felt in the field of style investing. Our findings regarding the Fama and French (1993) size and book-to-market sorted portfolios are particularly pertinent given the pervasive industry use of the Fama and French (1992) size and book-to-market characteristics as style classification criteria. If expected returns are truly linear in size and book-to-market exposure for example, then two portfolios with the same average size and average book-to-market exposures will have the same expected rates of return. However, if expected returns are nonlinear in these risk exposures, then it is possible to have the same exposures on average but different the-oretically fair expected returns. The implication for professional portfolio managers seeking to maximize Sharpe ratios is to identify portfolio combinations with the highest expected return for a given average exposure set. However, in response to this clients and fiducia-ries must recognize that fair performance measurement may only be achieved by using the nonlinear asset pricing model to risk adjust returns and by avoiding the reliance on average portfolio exposures and identifying exposures on an asset-by-asset basis. Finally, the features of our conditional nonlinear FF3 pricing kernel imply that tactical style rotation will also be a rewarding portfolio construction strategy when evaluated using nominal returns or Sharpe ratios as metrics. We draw this conclusion from the two qualitative features of the pricing kernel revealed in Figures B.13 and B.14. First, the influence of SMB and HML on the pricing kernel varies greatly as the level of TERM varies. And second, these TERM driven variations are different for SMB and HML. In practical economic terms, this implies that expected returns on different capitalization quintiles and different book-to-market quintiles vary over time in a predictable way (Jensen et al., 1997; Kao and Shumaker, 1999; Copeland and Copeland, 1999). Furthermore, these variations are not completely in sync. However, higher prospective nominal returns or Sharpe ratios associated with style rotation do not imply higher expected utility in the context of our conditional nonlinear FF3 asset pricing model. 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Review of Financial Studies, 2:73-89. 79 Appendix Tables Table A . l : Summary Statistics All data is quarterly and covers the period from Q2, 1959 to Q4, 1999. Columns two through eight report the sample mean, standard deviation, and autocorrelations at quarterly lags one, two, three, four, eight, and twelve. The portfolio return series in Panels B and E are in excess of the quarterly inflation rate. All three instrument series in Panel C are standardized to have zero unconditional means and unit variances. Variable definitions are provided in Section 4. Autocorrelation Series Mean Std. Dev. 1 2 3 4 8 12 Panel A: State Variables MKT 0.0175 0.0829 0.0417 -0.1465 -0.0592 -0.0074 -0.0498 -0.0042 A C 0.0023 0.0091 0.1814 0.1194 0.1526 0.0344 -0.0448 -0.0861 SMB 0.0096 0.0216 0.4527 0.3810 0.2458 0.1245 -0.2118 -0.0198 HML 0.0057 0.0503 0.5480 0.2716 0.1243 -0.0354 -0.1650 -0.3014 NRINV 0.0043 0.0564 -0.0063 0.1933 -0.0183 0.2163 0.0935 0.1549 RINV 0.0101 0.0533 0.0520 0.0108 0.0296 0.1502 0.0260 0.0123 Panel B: Inflation and Real Returns Inflation 0.0108 0.0080 0.7526 0.7067 0.7208 0.6150 0.3317 0.2563 TBILL 0.0041 0.0066 0.6363 0.6104 0.6379 0.5104 0.2297 0.1755 CORP 0.0076 0.0514 0.0606 0.0846 0.1340 0.0609 -0.0367 -0.0354 SlBl 0.0159 0.1564 -0.0030 -0.0104 -0.0589 0.0325 -0.0651 0.0596 S1B5 0.0380 0.1275 -0.0253 -0.1237 -0.0848 0.1926 0.0789 0.1189 S3B3 0.0244 0.0988 -0.0057 -0.0927 -0.0270 -0.0070 -0.0243 0.0127 S5B1 0.0215 0.0936 0.0865 -0.1266 0.0084 0.0564 -0.0506 -0.0086 S5B5 0.0237 0.0833 0.0761 -0.1318 -0.0150 0.0408 -0.0109 0.0761 Panel C: Instrumental Variables DEF 0.0099 0.0045 0.9067 0.8337 0.7692 0.6985 0.4835 0.3282 DIV 0.0352 0.0105 0.9548 0.8979 0.8517 0.8066 0.7107 0.6432 AIP 0.0352 0.0487 0.8627 0.6231 0.3613 0.1027 -0.2587 -0.0690 Panel D: Conditioning Variables TERM 0.0054 0.0132 0.8453 0.7489 0.7217 0.6440 0.3372 0.1237 CAY 0.6134 0.0117 0.8310 0.6779 0.5620 0.4792 0.1529 -0.1053 Panel E: Real Returns for Out of Sample Tests GOVT 0.0071 0.0554 0.0047 0.0881 0.1155 0.0729 -0.0269 -0.0289 S1B3 0.0281 0.1252 -0.0213 -0.0311 -0.0607 0.0763 -0.0006 0.1097 S2B2 0.0247 0.1216 -0.0701 -0.0763 -0.0670 0.0318 -0.0115 0.0589 S2B4 0.0323 0.1062 -0.0010 -0.0517 -0.0480 0.0591 -0.0093 0.0297 S3B1 0.0206 0.1279 -0.0420 -0.1266 -0.0155 -0.0358 -0.1182 0.0165 S3B5 0.0323 0.1066 -0.0459 -0.1002 -0.0392 0.0764 -0.0417 0.0742 S4B2 0.0198 0.1013 0.0058 -0.0773 -0.0901 -0.0258 -0.0847 0.0218 S4B4 0.0292 0.0920 0.0351 -0.1021 -0.0332 -0.0150 -0.0587 -0.0313 S5B3 0.0214 0.0743 0.1014 -0.0756 -0.1079 -0.0022 -0.0069 -0.0452 81 Table A.2: First Order (Linear) Models Columns two through six list results for linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonsepara-ble (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Using W = ET[(Rt,t+i ® Zt)(Rt,t+i ® ^ t ) T ] ~ \ in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = 9T(®Rw)J[Var(gT)\+gT{&RW) ~ x\,K-Q(L+\)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^ 1 . For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(&ow) = 9T(&OW)tS^g-ri&ow) ~ X2NK-Q{L+I)M+I where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. Note that bo ld p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted Returns-weighted x2 test 97.0336 118.7098 113.2256 113.9105 42.3121 Degrees of freedom 26 26 25 25 24 p-value 0.0000 0.0000 0.0000 0.0000 0.0119 Pricing kernel mean 0.9963 0.9963 0.9964 0.9963 0.9962 Pricing kernel standard deviation 0.2000 0.1247 0.2377 0.0624 0.4170 Panel B: Optimal-Weighted Optimal G M M x2 test 121.1785 134.5756 103.8130 146.9256 43.2073 Degrees of freedom 26 26 25 25 24 p-value 0.0000 0.0000 0.0000 0.0000 0.0094 Pricing kernel mean 0.9912 1.0232 0.9951 1.0009 0.9956 Pricing kernel standard deviation 0.1011 0.1466 0.0224 0.0324 0.3666 supLM test statistic 2.4178 38.4393 2.9042 64.9940 4.2066 Number of parameters 2 2 3 3 4 supLM test result pass fail pass fail pass 82 Table A.3: Price Errors from the First Order (Linear) Models Columns two through six list results for linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonsepara-ble (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 , consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product wi th the K instrumental variables. The N basic assets labeled TBILL, CORP, 5 1 5 1 , S1B5, 5 3 5 3 , 5 5 5 1 and S5B5 are described in Section 4. For the NK x 1 pricing error vector, gT(&ow), basic asset i 's set of raw and managed pricing errors are associated wi th elements {i, i + N, ..., i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(&0w) = 0 where V(i) is a diagonal NK x NK matr ix with the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The W a l d test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(0 = [Vir)gTV[V{i)Var{gT)V^][V^gT] ~ rf r e s t r i c t i o n s where the pricing errors' variance-covariance matrix given by: Var(gT{&ow)) = T'^ST - I M - D ^ T ^ T ) - 1 - ^ ] . Note that b o l d p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M N S C C A P M C O C H R A N E F F 3 TBILL group 3.0355 16.0950 15.6595 40.9763 3.6996 0.5519 0.0029 0.0035 0.0000 0.4482 CORP group 1.5086 22.4997 4.7609 13.9332 4.3193 0.8251 0.0002 0.3127 0.0075 0.3645 S l B l group 2.0042 16.0859 3.3908 7.3660 . 2.7551 0.7350 0.0029 0.4947 0.1178 0.5996 S1B5 group 14.7662 22.7060 16.2571 23.4630 3.1217 0.0052 0.0001 0.0027 0.0001 0.5377 5 3 5 3 group 10.2099 23.2600 10.8484 23.1548 3.0437 0.0370 0.0001 0.0283 0.0001 0.5505 S5B1 group 8.5813 21.9748 7.9297 13.3868 4.1555 0.0725 0.0002 0.0942 0.0095 0.3854 5 5 5 5 group 9.4453 20.0868 12.2535 18.2288 3.3364 0.0509 0.0005 0.0156 0.0011 0.5032 83 Table A.4: Second Order Polynomial Models Columns two through six list results for unconditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 . Using W = ET[(Rt,t+i ® Zt)(Rt,t+i ® Z t ) T ] _ 1 , i n Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = 9T(®Rw)T[Var(gT)]+gT(&RW) ~ XNK-Q(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [ ] + represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^.1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(&ow) = 9T(&ow)TS^,1gT(&ow) ~ xl/K-q(L+i)M+i where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested linear models reported i n Panel C is: T J T ( 0 o w ) r e s t r i c t e d - T J T ( 0 o w ) u n r e s t r i c t e d ~ ^number of restrictions-Note that b o l d p-values highlight significance at the 5% level. In Panel B , s u p L M is the Andrews (1993) supremum Lagrange Mult ip l ier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M N S C C A P M C O C H R A N E F F 3 Panel A : Returns-Weighted Returns-weighted x2 test 62.6273 155.0004 70.7400 59.6316 40.5664 Degrees of freedom 25 25 23 23 21 p-value 0 .0000 0.0000 0 .0000 0.0000 0.0063 Pric ing kernel mean 0.9963 0.9963 0.9965 0.9963 0.9959 Pr ic ing kernel standard deviation 0.2137 0.1877 0.4495 0.3896 0.5971 s u p L M test statistic 28.1968 111.3897 9.0889 62585.3850 8.6174 Number of parameters 3 3 5 5 7 s u p L M test result fail fail pass fail pass Panel B : Optimal-Weighted Opt imal G M M x2 test 124.7046 103.3013 46.9185 273.7471 34.2186 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0 .0000 0.0023 0.0000 0.0343 Pric ing kernel mean 0.9937 1.0251 1.0539 0.4081 1.0089 Pric ing kernel standard deviation 0.1086 0.2554 0.9407 2.0640 0.4822 Panel C: Nested First Order Model Test Difference in Opt imal G M M x2 2.6105 2.0739 15.3122 54224.0121 4.5656 Degrees of freedom 1 1 2 2 3 p-value 0.1062 0.1498 0.0005 0 .0000 0.2065 84 Table A.5: Price Errors from the Second Order Polynomial Models Columns two through six list results for unconditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled TBILL, CORP, SlBl, S1B5, S3B3, S5B1 and S5B5 are described in Section 4. For the NK x 1 pricing error vector, gT(&ow), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(&ow) = 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(<) = m)gTV[V{i)Var{gT)Vm\V{i)gT] ~ o f r e s t r i c t i o n s where the pricing errors' variance-covariance matrix given by: Var{gT(&OW)) = T~1[ST - DT{DTS^DT)-lDT]. Note that bold p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M NS C C A P M C O C H R A N E FF3 TBILL group 4.1374 12.9704 4.9971 18.8287 6.4486 0.3877 0.0114 0.2876 0.0008 0.1681 CORP group 2.4420 14.5057 5.8659 18.4295 6.9762 0.6551 0.0058 0.2094 0.0010 0.1372 SlBl group 1.6249 16.1476 3.0039 21.1554 5.5304 0.8043 0.0028 0.5572 0.0003 0.2371 S1B5 group 16.5676 17.5780 5.3668 21.0288 6.0368 0.0023 0.0015 0.2517 0.0003 0.1964 S3B3 group 13.3806 18.0876 5.4184 20.1846 6.2970 0.0096 0.0012 0.2470 0.0005 0.1780 S5B1 group 7.3154 18.3080 5.6618 19.6362 8.8985 0.1201 0.0011 0.2259 0.0006 0.0637 S5B5 group 13.2256 16.5754 6.1390 19.9845 6.3778 0.0102 0.0023 0.1890 0.0005 0.1727 85 Table A.6: Th i rd Order Polynomial Models Columns two through six list results for unconditional third order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, NS-CCAPM; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. Using W = ET[(Rt,t+i ® Zt)(Rtit+1 ® Z t ) T ] _ 1 , in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(@RW) = 9T(®Rw)T[Var(gT)]+gT(&RW) ~ X2NK-G(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(®0w) = gT{®ow)TS^}gT(&0w) ~ xl/K-q(L+i)M+i where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested linear models (Panel C) and nested second order models (Panel D) is: T J T ( 0 O H ' ) r e s t r i c t e d - r J T ( 0 o w ) u n r e s t r i c t e d ~ -^number of restrictions' Note that bold p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted Returns-weighted x2 test 55.2188 62.6688 56.5816 53.6273 36.9546 Degrees of freedom 24 24 21 21 18 p-value 0.0003 0.0000 0.0000 0.0001 0.0053 Pricing kernel mean 0.9962 0.9962 0.9966 0.9962 0.9959 Pricing kernel standard deviation 0.3575 0.3031 0.6359 0.3820 0.5935 Panel B: Optimal-Weighted Optimal G M M x2 test 55.0559 63.1221 40.3339 54.8580 25.3653 Degrees of freedom 24 24 21 21 18 p-value 0.0003 0.0000 0.0068 0.0001 0.1152 Pricing kernel mean 0.9827 0.9645 0.9979 0.9566 0.9701 Pricing kernel standard deviation 0.1615 0.2930 0.9918 0.7990 1.2098 supLM test statistic 3.5063 4.5742 6.7383 6.5385 18.3915 Number of parameters 4 4 7 7 10 supLM test result pass pass pass pass pass Panel C: Nested First Order Model Test Difference in Optimal G M M x2 26.2269 26.3643 37.5458 33.3507 26.4747 Degrees of freedom 2 2 4 4 6 p-value 0.0000 0.0000 0.0000 0.0000 0.0002 Panel D: Nested Second Order Model Test Difference in Optimal G M M x2 31.5087 29.3996 19.5874 1268.5008 25.3714 Degrees of freedom 1 1 2 2 3 p-value 0.0000 0.0000 0.0001 0.0000 0.0000 86 Table A.7: Price Errors from the Thi rd Order Polynomial Models Columns two through six list results for unconditional third order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 , consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product wi th the K instrumental variables. The N basic assets labeled TBILL, CORP, S1B1, S1B5, S3B3, S5B1 and 5 5 S 5 are described in Section 4. For the NK x 1 pricing error vector, gxi&ow), basic asset i's set of raw and managed pricing errors are associated wi th elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(®ow) = 0 where V(i) is a diagonal NK x NK matr ix wi th the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald( i ) = \V(i)gTf\y<i)Var(gT)Vtf]\yM9T] ~ X n u m b e r o f r e S t r i c t i o n s where the pricing errors' variance-covariance matrix given by: Var(gT(&ow)) = T'^ST - DT(D^S^DT)~X£>?]. Note that b o l d p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M N S C C A P M C O C H R A N E F F 3 TBILL group 8.1156 7.7487 2.1687 2.4733 2.9475 0.0874 0.1012 0.7048 0.6494 0.5667 CORP group 4.0905 7.0623 2.4070 2.5408 2.7498 0.3939 0.1326 0.6614 0.6373 0.6005 S l S l group 7.1866 4.2680 2.1143 2.2435 5.5576 0.1264 0.3709 0.7147 0.6911 0.2347 SIB5 group 4.9081 2.1858 1.9461 3.0039 4.6770 0.2969 0.7016 0.7457 0.5572 0.3221 S3£?3 group 2.2216 3.5085 2.2053 2.5886 4.0521 0.6951 0.4766 0.6981 0.6288 0.3990 S5B1 group 10.6138 4.7325 2.6729 3.3406 4.0517 0.0313 0.3159 0.6140 0.5025 0.3990 S5B5 group 0.8754 2.8049 2.1878 2.9728 4.0112 0.9281 0.5910 0.7013 0.5624 0.4045 87 Table A.8: Term Spread Conditional First Order Models Columns two through six list results for TERM conditional linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M , generally based upon the habit formation models of Constantinides (1990) and Ferson and Constantinides (1991); an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 . Using W = ET[{Rt,t+\ ® Zt)(Rt,t+i <S> Z T ) T ] _ 1 , in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = gT{®Rw)T[Var(gT)]+gT(®RW) ~ X / W - , ( L + I ) M + I where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^.1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT{&ow) = gri^ow)1S^,1gT(&ow) ~ X2NK-q(L+i)M+\ where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested unconditional linear models reported in Panel C is: r j T ( 0 O w ) r e s t r i c t e d ~ T - M ® c w ) u n r e s t r i c t e d ~ ^number of restrictions-Note that b o l d p-values highlight significance at the 5% level. In Panel B , s u p L M is the Andrews (1993) supremum Lagrange Mult ip l ier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E F F 3 Panel A : Returns-Weighted Returns-weighted x2 t e s t 83.2190 69.5987 106.1424 96.3996 44.1899 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0.0000 0.0000 0.0000 0.0022 Pr ic ing kernel mean 0.9962 0.9963 0.9963 0.9962 0.9962 Pr ic ing kernel standard deviation 0.3274 0.3186 0.3105 0.1627 0.7554 Panel B : Optimal-Weighted Opt imal G M M x2 test 110.0650 109.6571 65.7350 56.6588 49.3106 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0.0000 0.0000 0.0001 0.0005 Pr ic ing kernel mean 0.9993 0.9987 0.9939 0.9767 0.9327 Pr ic ing kernel standard deviation 0.0371 0.0290 0.0725 0.6786 0.9899 s u p L M test statistic 11.1887 207.3129 4.1086 68.6514 13.5549 Number of parameters 3 3 5 5 7 s u p L M test result pass fail pass fail pass Panel C: Nested Unconditional First Order Mode l Test Difference in Opt imal G M M x2 144.9016 2334.3917 5.5140 11.7376 10.7890 Degrees of freedom 1 1 2 2 3 p-value 0 .0000 0.0000 0.0635 0.0028 0.0129 88 Table A . 9 : P r i c e E r r o r s f r o m the T e r m S p r e a d C o n d i t i o n a l F i r s t O r d e r M o d e l s Columns two through six list results for TERM conditional linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, NS-CCAPM; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled TBILL, CORP, SlBl, 51B5, S3B3, 55B1 and 55B5 are described in Section 4. For the NK x 1 pricing error vector, 9x(®ow), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K— 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(®ow) = 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, ..., i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(i) = [V^gTV[V(i)Var(gT)V^][V^9T] ~ o f r e s t r i c t i o n s where the pricing errors' variance-covariance matrix given by: Var(gT(®0w)) = T _ 1 [ S T - ^ ( - D ? ^ 1 ^ ) " 1 ^ ] . Note that bold p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M NS C C A P M C O C H R A N E FF3 TBILL group 5.3638 12.0818 9.8254 5.3796 4.0135 0.2520 0.0168 0.0435 0.2505 0.4042 CORP group 3.1004 5.9036 5.0257 4.3965 4.5018 0.5412 0.2065 0.2847 0.3550 0.3423 5151 group 3.0488 5.6662 2.5797 3.0441 4.8393 0.5497 0.2255 0.6304 0.5505 0.3042 SIB5 group 18.8419 20.2958 11.1162 4.1664 3.4120 0.0008 0.0004 0.0253 0.3840 0.4914 53B3 group 12.0511 16.8212 6.4673 3.9732 4.0861 0.0170 0.0021 0.1669 0.4096 0.3945 S5B1 group 9.1129 8.4232 6.7634 4.8351 4.7122 0.0583 0.0772 0.1489 0.3046 0.3181 55S5 group 15.6460 14.1474 7.8824 5.3281 4.4845 0.0035 0.0068 0.0960 0.2553 0.3444 89 Table A . 10: Term Spread Conditional Second Order Models Columns two through six list results for TERM conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 . Using W — ET[(Rt,t+i ® Zt)(Rttt+1 ® Zt)T]~l, in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(&RW) = gT(®Rw)T[Var(gT)]+gT(&RW) ~ xliK-G(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1 • For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(&0w) = gT(&ow)TS^gxi&ow) ~ x\iK-q(L+i)M+\. where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested second order models (Panel C) and nested condi-tional linear models (Panel D) is: T - M 0 o w Restricted - T J r ( 0 o w O u r i r e s t r i c t e d ~ ^number of restrictions-Note that b o l d p-values highlight significance at the 5% level. In Panel B , s u p L M is the Andrews (1993) supremum Lagrange Mult ip l ier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M N S C C A P M C O C H R A N E F F 3 Panel A : Returns-Weighted Returns-weighted x2 test 52.5027 77.5451 35.0015 64.2135 17.5631 Degrees of freedom 23 23 19 19 15 p-value 0.0004 0.0000 0.0140 0.0000 0.2863 Pr ic ing kernel mean 0.9962 0.9964 0.9964 0.9963 0.9961 Pr ic ing kernel standard deviation 0.3578 0.6349 1.0466 0.4557 0.9599 Panel B : Optimal-Weighted Opt imal G M M x2 test 95.9950 87.5308 36.1004 33.0413 14.6614 Degrees of freedom 23 23 19 19 15 p-value 0.0000 0.0000 0.0103 0.0238 0.4761 Pr ic ing kernel mean 0.9821 0.9370 1.0055 0.9694 0.9369 Pr ic ing kernel standard deviation 0.2497 0.4335 0.8437 1.2538 1.3492 s u p L M test statistic 68.5889 46.4530 14.8194 13.3061 17.8264 Number of parameters 5 5 9 9 13 s u p L M test result fail fail pass pass pass Panel C: Nested Unconditional Second Order Mode l Test Difference in Opt imal G M M x2 18.9465 61.9788 6.5282 636.1696 26.1822 Degrees of freedom 2 2 4 4 6 p-value 0.0001 0.0000 0.1630 0.0000 0.0002 Panel D: Nested Condit ional First Order Mode l Test Difference in Opt imal G M M x2 57.1861 46.9394 15.5232 55.0895 14.6076 Degrees of freedom 2 2 4 4 6 p-value 0.0000 0.0000 0.0037 0.0000 0.0235 9 0 Table A . 1 1 : Price Errors from the Term Spread Conditional Second Order M o d -els Columns two through six list results for TERM conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled TBILL, CORP, S1B1, 51S5, S3B3, S5B1 and 5555 are described in Section 4. For the NK x 1 pricing error vector, gT(&ow), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V{i)gT(®0w) = 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(0 = [V(i)gTVlV(i)Var(gT)V^][V(i)gT] ~ x n u m b e r o f M i c t i o n s where the pricing errors' variance-covariance matrix given by: Var{gT{®ow)) = T^ST - DT(DT'ST-1DT)-1DT']. Note that bold p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M NS C C A P M C O C H R A N E FF3 TBILL group 4.0597 6.1109 2.5642 2.9662 1.6265 0.3980 0.1910 0.6332 0.5635 0.8040 CORP group 3.1468 6.3427 2.8668 2.9087 1.5547 0.5336 0.1750 0.5804 0.5732 0.8169 SlBl group 18.1385 5.6258 2.4084 3.4939 1.5176 0.0012 0.2289 0.6611 0.4788 0.8235 S1B5 group 10.7617 6.2311 3.6956 3.0114 1.1788 0.0294 0.1825 0.4488 0.5559 0.8816 S3B3 group 7.3281 5.5644 2.9772 3.2008 1.4356 0.1195 0.2341 0.5616 0.5248 0.8380 S5B1 group 9.7424 5.7068 3.4914 3.3120 1.6094 0.0450 0.2221 0.4792 0.5070 0.8071 55B5 group 3.5674 5.5775 3.1636 3.1256 2.1369 0.4677 0.2330 0.5308 0.5370 0.7106 91 Table A . 12: Fama French 3 Factor Model Out of Sample Tests Columns two through eight list out-of-sample results for the unconditional linear, unconditional second order, unconditional third order, conditional linear and conditional second order specifications of the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. This model consists of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Using W = ET[(Rt,t+i ® Zt){Rt,t+i ® ^ t ) T ] _ 1 ! m Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = gT(®Rw)T[Var(gT)]+gT(&RW) ~ XNK where RW stands for returns-weighted, NK is both the number of pricing errors and the number of degrees of freedom and [ ] + represents the pseudo-inverse operator. Note that GRW represents the parameter vector estimated with the original in-sample data. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^.1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(®OW) = 9 r ( ® o w ) T S ^ g ^ & o w ) ~ X / V K where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. Again, note that &OG represents the parameter vector estimated with the original in-sample data. Note that bold p-values indicate significance at the 5% level. 1st 2nd 3rd TERM 1st TERM 2nd Panel A: Returns-Weighted Returns-weighted x2 69.4903 54.2717 50.4472 42.1879 24.5878 Degrees of freedom 36 36 36 36 36 p-value 0.0007 0.0259 0.0556 0.2210 0.9251 Panel B: Optimal-Weighted Optimal G M M x 2 70.7764 66.0744 57.5553 68.0565 38.4626 Degrees of freedom 36 36 36 36 36 p-value 0.0005 0.0016 0.0127 0.0010 0.3586 92 Table A . 13: Out of Sample Price Errors from the Fama French Mode l Specifica-tions Columns two through eight list out-of-sample test results for the unconditional linear, unconditional second order, unconditional third order, conditional linear and conditional second order specifications of the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. This model consists of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each out-of-sample asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled GOVT, S1B3, S2B2, S2B4, S3B1, S3B5, S4B2, S4B4 and S5B3 are described in Section 4. For the NK x 1 pricing error vector, gT(&ow), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(®0w) = 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. Note that ®OG represents the parameter vector estimated with the original in-sample data. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: W a l d W = [V(i)gTV[V{i)Var(gT)Vm[V(i)gT} ~ x 2 u m b e r o f r e s t r i c t k ) n s where the pricing errors' variance-covariance matrix given by: Var(gT(®ow)) = T _ 1 [ S r - DT(DlS^DT)-lDl}. Note that bold p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. 1st 2nd 3rd TERM 1st TERM 2nd G O V T group 4.6113 7.6219 3.1095 3.6427 1.2692 0.3296 0.1065 0.5397 0.4565 0.8666 S1B3 group 2.8514 5.9090 4.9029 3.0230 0.8697 0.5830 0.2061 0.2974 0.5540 0.9289 S2B2 group 3.0766 6.1046 4.9752 3.0316 0.8564 0.5451 0.1915 0.2899 0.5525 0.9307 S2B4 group 3.7501 6.9295 4.7818 3.0499 0.9181 0.4409 0.1397 0.3104 0.5495 0.9220 S3B1 group 3.3205 6.7844 4.9860 3.4333 0.8377 0.5057 0.1477 0.2887 0.4881 0.9333 53B5 group 3.8659 6.2889 5.4092 3.1606 1.4880 0.4245 0.1786 0.2478 0.5313 0.8288 S4B2 group 3.0590 6.4158 4.7953 3.6604 1.1808 0.5480 0.1702 0.3090 0.4539 0.8812 S4B4 group 4.7000 8.7656 4.5761 3.5624 1.3118 0.3195 0.0672 0.3336 0.4685 0.8594 S5B3 group 4.0255 7.9906 4.8779 3.7304 1.2144 0.4026 0.0919 0.3001 0.4437 0.8757 93 Table A . 14: Term Spread Conditional Second Order Models with Alternative Instrumental Variables We propose an alternative set of three instruments: the discount yield for the one month Treasury bill, TBYl, the quarterly return on the Standard and Poor's 500 composite stock index, SPRET, and the Hodrick and Prescott (1997) filter derived cyclical component of the natural logarithm of the U.S. Industrial Production Index, IPCYC. Columns two through six list results for TERM conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. Using W = ET\{Rt,t+i ® Zt)(Rt,t+1 <g> Zt)T}-\ in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(&RW) = 9T(®RW)T[Var(gT)]+gT(®RW) ~ XNK-q(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q{L + l)M + 1 is the number of estimated parameters, and [ ] + represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1, following Hansen (1982). For the optimal-weighted estimations, the test statistic for nested second order models (Panel C) and nested conditional linear models (Panel D) is: rj T(0 O W ') re Stricted ~ r j H 0 ow)unres t r i c t ed ~ X n u m b e r o f restrictions-Note that b o l d p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted Returns-weighted x2 test 60.7707 55.4868 59.9590 67.6415 20.6357 Degrees of freedom 23 23 19 19 15 p-value 0.0000 0.0002 0.0000 0.0000 0.1489 Pricing kernel mean 0.9964 0.9965 0.9963 0.9965 0.9964 Pricing kernel standard deviation 0.3004 0.3601 0.8019 0.3869 0.7257 Panel B: Optimal-Weighted Optimal G M M x2 test 62.2423 75.9870 39.7459 52.8360 17.3397 Degrees of freedom 23 23 19 19 15 p-value 0.0000 0.0000 0.0035 0.0000 0.2990 Pricing kernel mean 1.0008 1.0055 1.1386 0.8319 1.0504 Pricing kernel standard deviation 0.2247 0.1068 1.9878 2.1098 1.0130 supLM test statistic 2.6289 7.1421 9.0747 21.1605 15.0331 Number of parameters 5 5 9 9 13 supLM test result pass pass pass pass pass Panel C: Nested Unconditional Second Order Model Test Difference in Optimal G M M \ 2 11.7430 2.6651 25.1767 40.4541 27.8818 Degrees of freedom 2 2 4 4 6 p-value 0.0028 0.2638 0.0000 0.0000 0.0001 Panel D: Nested Conditional First Order Model Test Difference in Optimal G M M x2 33.7748 2.7539 24.2614 60.5560 25.3383 Degrees of freedom 2 2 4 4 6 p-value 0.0000 0.2524 0.0001 0.0000 0.0003 94 Table A.15: Price Errors from the Term Spread Conditional Second Order Mod-els with Alternative Instrumental Variables We propose an alternative set of three instruments: the discount yield for the one month Treasury b i l l , TBYl; the quarterly return on the Standard and Poor's 500 composite stock index , SPRET, and the Hodrick and Prescott (1997) filter derived cyclical component of the natural logarithm of the U.S . Industrial Product ion Index, IPCYC, Columns two through six list results for TERM conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 , consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled TBILL, CORP, SlBl, S 1 5 5 , S3B3, 5 5 5 1 and 5 5 5 5 are described in Section 4. For the NK x 1 pricing error vector, g-j<{®ow), basic asset i 's set of raw and managed pricing errors are associated wi th elements {i, i + N, i + N{K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(®ow) = 0 where V(i) is a diagonal NK x NK matr ix wi th the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(i) = [V(,)gTY[V{i)Var{gT)V^][V{i)gT\ ~ x n u m b e r o f r e s t r i c t i o n s where the pricing errors' variance-covariance matrix given by: Var{gT{®ow)) = T-\ST - X ^ - D ^ S ^ - D r ) - 1 ^ ] . Note that b o l d p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M N S C C A P M C O C H R A N E F F 3 TBILL group 4.5215 6.4876 4.3016 3.3878 1.6205 0.3400 0.1656 0.3667 0.4951 0.8051 CORP group 3.9044 4.7799 3.7649 3.8806 1.6276 0.4191 0.3106 0.4388 0.4224 0.8038 5 1 5 1 group 4.1104 12.9773 4.4020 2.7873 1.5751 0.3913 0.0114 0.3543 0.5940 0.8133 5 1 5 5 group 6.3429 21.1743 3.8851 8.1828 1.8203 0.1750 0.0003 0.4218 0.0851 0.7688 5 3 5 3 group 3.5236 14.4220 3.7369 5.5803 1.6332 0.4743 0.0061 0.4428 0.2328 0.8028 5 5 5 1 group 3.1565 9.9511 3.7916 5.0296 1.7683 0.5320 0.0413 0.4349 0.2843 0.7783 5 5 5 5 group 3.2781 11.8775 3.8667 6.5529 1.1059 0.5124 0.0183 0.4243 0.1615 0.8933 95 Table A . 16: GAY Conditional First Order Models Columns two through six list results for GAY conditional linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M , generally based upon the habit formation models of Constantinides (1990) and Ferson and Constantinides (1991); an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 . Using W = ET[{Rt,t+i ® Zt){Rt,t+i ® Z T ) T ] _ 1 , in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT{®RW) = 9T(®Rw)T[Var(gT)]+gT(&RW) ~ XNK-9(L+I)M+I where RW stands for returns-weighted, NK is the number of pricing errors, q(L + l)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1 • For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(&ow) = 9T(&OW)TS^.1gT(&ow) ~ X W K - ? ( L + I ) M + I where O W stands for optimal-weighted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested unconditional linear models reported in Panel C is: T J T ( 0 o w ) r e s t r i c t e d ~ T J T ( © o w ) l m r e S t r i c t e d ~ d u m b e r of restrictions-Note that bold p-values highlight significance at the 5% level. In Panel B , s u p L M is the Andrews (1993) supremum Lagrange Mult ip l ier test statistic used to examine for structural shifts i n the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E F F 3 Panel A : Returns-Weighted Returns-weighted x2 test Degrees of freedom p-value 104.7931 25 0.0000 60.9552 25 0.0001 68.0479 23 0 .0000 121.4543 23 0.0000 43.6857 21 0.0026 Pricing kernel mean Pr ic ing kernel standard deviation 0.9962 0.2950 0.9964 0.2235 0.9964 0.2994 0.9961 0.3890 0.9963 0.4838 Panel B : Optimal-Weighted Opt imal G M M x2 test Degrees of freedom p-value 117.1792 25 0.0000 106.3571 25 0.0000 96.9906 23 0.0000 39.5657 23 0 .0172 28.0343 21 0.1392 Pricing kernel mean Pricing kernel standard deviation 1.0005 0.0219 0.9343 0.6242 1.0011 0.2941 0.7169 1.2616 1.0826 0.8885 s u p L M test statistic Number of parameters s u p L M test result 5.5776 3 pass 7.4394 3 pass 8.1589 5 pass 27.4522 5 fail 12.6372 7 pass Panel C: Nested Unconditional Firs t Order Mode l Test Difference in Opt imal G M M x2 Degrees of freedom p-value 217.6265 1 0.0000 22.5802 1 0 .0000 9.4738 2 0.0088 28.9905 2 0.0000 20.5757 3 0.0001 96 Table A . 17: Price Errors from the CAY Conditional First Order Models Columns two through six list results for CAY conditional linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, NS-CCAPM; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled TBILL, CORP, SlBl, S1B5, S3B3, 5551 and 55S5 are described in Section 4. For the NK x 1 pricing error vector, gT(®ow), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(®ow) — 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald«) = [ V ^ Y W ^ V a r ^ V ^ W ^ ] ~ x n u m b e r o f r e s t r i c t i 0 n s where the pricing errors' variance-covariance matrix given by: Var(gT(&ow)) = T~1[ST - DT(DTST'DT)~l£>J]. Note that bold p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M NS C C A P M C O C H R A N E FF3 TBILL group 15.5702 3.9739 1.0621 4.1935 6.4967 0.0037 0.4095 0.9002 0.3804 0.1650 CORP group 3.7750 3.6389 1.8759 4.1523 5.6101 0.4373 0.4571 0.7586 0.3858 0.2302 S l B l group 3.7228 4.0859 2.7553 4.3153 5.3036 0.4448 0.3945 0.5996 0.3650 0.2575 S1B5 group 19.1919 2.6048 11.5348 3.7169 6.7766 0.0007 0.6260 0.0212 0.4457 0.1482 53B3 group 13.2180 3.0724 5.8118 3.8515 6.7915 0.0103 0.5458 0.2136 0.4265 0.1473 55 B l group 9.3686 3.5920 4.4106 4.0060 5.9690 0.0525 0.4640 0.3533 0.4052 0.2015 S5B5 group 18.6392 2.9365 8.3562 3.7836 6.3903 0.0009 0.5685 0.0794 0.4361 0.1718 97 Table A.18: CAY Conditional Second Order Models Columns two through six list results for CAY conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, NS-CCAPM; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. Using W = Er[{Rt,t+i ® Zt){Rt,t+i ® Zt)T]~l, in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = gT(®Rw)T[Var(gT)]+gT(®RW) ~ X2NK-g(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [ ] + represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(®ow) = gA&Ow)1S^lgT(®ow) ~ XNK-q(L+l)M+l where OW stands for optimal-weighted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested second order models (Panel C) and nested condi-tional linear models (Panel D) is: ^ ( © D e r e s t r i c t e d ~ T J T ^ 0 o l v ^ u n r e s t r i c t e d ~ ^number of restrictions-Note that bold p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted Returns-weighted x2 test 50.9386 62.6228 39.5555 38.2348 18.3597 Degrees of freedom 23 23 19 19 15 p-value 0.0007 0.0000 0.0037 0.0055 0.2442 Pricing kernel mean 0.9961 0.9963 0.9962 0.9959 0.9960 Pricing kernel standard deviation 0.4353 0.3246 0.7699 0.6885 0.7610 Panel B: Optimal-Weighted Optimal G M M x2 test 51.5234 79.5818 31.3967 65.0650 13.8636 Degrees of freedom 23 23 19 19 15 p-value 0.0006 0.0000 0.0365 0.0000 0.5359 Pricing kernel mean 1.0339 0.9637 1.1374 0.7164 1.0179 Pricing kernel standard deviation 0.8467 0.3857 1.3401 1.3790 1.1539 supLM test statistic 5.3117 19.9575 26.7674 378.9933 17.6264 Number of parameters 5 5 9 9 13 supLM test result pass fail fail fail pass Panel C: Nested Unconditional Second Order Model Test Difference in Optimal G M M x2 17.0786 55.7084 20.2801 235.5398 38.1944 Degrees of freedom 2 2 4 4 6 p-value 0.0002 0.0000 0.0004 0.0000 0.0000 Panel D: Nested Conditional First Order Model Test Difference in Optimal G M M x2 20.2780 33.1173 23.5967 487.3158 28.3670 Degrees of freedom 2 2 4 4 6 p-value 0.0000 0.0000 0.0001 0.0000 0.0001 98 Table A. 19: Price Errors from the CAY Conditional Second Order Models Columns two through six list results for CAY conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3, consisting of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each asset group consists of the basic asset and all managed portfolios of that asset arising from the product with the K instrumental variables. The N basic assets labeled TBILL, CORP, S1BI, S1B5, S3B3, S5B1 and 55S5 are described in Section 4. For the NK x 1 pricing error vector, 9T(®OW), basic asset i's set of raw and managed pricing errors are associated with elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(®0w) = 0 where V(i) is a diagonal NK x NK matrix with the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(i) = [V^)gTV[V{i)Var{gT)V^\[V{,)9T] ~ X n l l mber of restrictions where the pricing errors' variance-covariance matrix given by: Var{gT{&ow)) = T~1[ST - DT(D^S^DT)~l£>?]. Note that bold p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. C A P M C C A P M NS C C A P M C O C H R A N E FF3 TBILL group 5.6809 9.4246 4.3848 12.1781 2.9282 0.2243 0.0513 0.3564 0.0161 0.5699 CORP group 4.6965 8.2178 4.6289 12.4941 3.1547 0.3199 0.0839 0.3275 0.0140 0.5323 SlBl group 5.8284 7.0085 3.6701 15.4023 1.6774 0.2123 0.1354 0.4525 0.0039 0.7948 S1B5 group 7.0775 4.5534 4.8799 13.1554 2.1231 0.1318 0.3363 0.2998 0.0105 0.7131 S3B3 group 6.0364 6.9982 4.4220 13.1756 2.1748 0.1964 0.1360 0.3519 0.0104 0.7037 S5B1 group 5.8322 5.0372 4.7691 11.7194 2.0390 0.2120 0.2835 0.3118 0.0196 0.7286 S5B5 group 6.3200 5.3394 4.9811 11.9946 2.7288 0.1765 0.2542 0.2892 0.0174 0.6042 99 Table A.20: Fama French 3 Factor Model Out of Sample Tests (CAY) Columns two through eight list out-of-sample results for the unconditional linear, unconditional second order, unconditional third order, conditional linear and conditional second order specifications of the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. This model consists of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Using W = Er[(Rt,t+i ® Zt)(Rttt+y (g> Z T ) T ] _ 1 , in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = gT(®Rw)T[Var(gT)]+gT(GRW) ~ XNK where RW stands for returns-weighted, NK is both the number of pricing errors and the number of degrees of freedom and [•]+ represents the pseudo-inverse operator. Note that &RW represents the parameter vector estimated with the original in-sample data. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^.1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(&0w) = gr(®ow)TS^gj-^ow) ~ XNK where OW stands for optimal-weighted and the x 2 degrees of freedom are as described above. Again, note that &oa represents the parameter vector estimated with the original in-sample data. Note that bo ld p-values indicate significance at the 5% level. CAY 1st CAY 2nd Panel A: Returns-Weighted Returns-weighted x 2 41.1368 38.5291 Degrees of freedom 36 36 p-value 0.2919 0.3559 Panel B: Optimal-Weighted Optimal GMM x 2 52.1388 63.79 Degrees of freedom 36 36 p-value 0.0040 0.0029 100 Table A .21: Out of Sample Price Errors from the Fama French Model Specifica-tions (CAY) Columns two through eight list out-of-sample test results for the unconditional linear, unconditional second order, unconditional th i rd order, conditional linear and conditional second order specifications of the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 . This model consists of the market premium, the SMB (small minus big) size factor, and the HML (high minus low) book-to-market factor. Each out-of-sample asset group consists of the basic asset and all managed portfolios of that asset arising from the product wi th the K instrumental variables. The N basic assets labeled GOVT, S 1 5 3 , S2B2, S2B4, S3B1, S 3 5 5 , S4B2, S4B4 and S5B3 are described in Section 4. For the NK x 1 pricing error vector, gx(®ow), basic asset i's set of raw and managed pricing errors are associated wi th elements {i, i + N, i + N(K — 1)} of the vector. For each asset group, we test the hypothesis that the set of associated pricing errors are zero. The hypothesis is tested using the set of restrictions V(i)gT(&0w) = 0 where V(i) is a diagonal NK x NK matr ix wi th the set {i, i + N, i + N(K — 1)} of diagonal elements equal to 1, and all other elements equal to 0. Note that &OG represents the parameter vector estimated wi th the original in-sample data. The Wald test statistic (Greene, 2000) for this hypothesis is then calculated as follows: Wald(i) = [ V ^ l V ^ V a r ^ V i i Y W ^ ] ~ rf r e s t r i c t i o n s where the pricing errors' variance-covariance matrix given by: Var(gT{®ow)) = T~1[ST - DT(D^ST1 DT)~XD^-\. Note that b o l d p-values indicate asset groups for which the pricing errors are statistically significant at the 5% percent level. CAY 1st CAY 2nd G O V T group 4.3918 2.8223 0.3556 0.5880 S\B3 group 4.8646 1.9080 0.3015 0.7527 5 2 5 2 group 5.3264 2.0377 0.2554 0.7288 5 2 5 4 group 5.4976 2.1298 0.2399 0.7119 5 3 5 1 group 5.0669 1.7592 0.7799 0.7799 5 3 5 5 group 5.3346 1.9130 0.2547 0.7517 5 4 5 2 group 5.1066 1.8883 0.2765 0.7563 5 4 5 4 group 5.0893 2.0923 0.2783 0.7188 5 5 5 3 group 5.1993 1.8933 0.2674 0.7554 101 Table A.22: Term Spread Conditional First Order Models with Substitute In-strumental Variable CAY The log consumption-wealth variable, CAY, replaces credit spread, DEF, as an instrumental variable in the TERM conditional estimations. Columns two through six list results for TERM conditional linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M , generally based upon the habit formation models of Constantinides (1990) and Ferson and Constantinides (1991); an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. Using W = Ex[(Rt,t+i ® Zt){Rt,t+i ® ^ t ) T ] _ 1 i in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(®RW) = 9T(®Rw)T[Var(gT)]+gT(®RW) ~ xliK-G(L+i)M+i where RW stands for re turns-weighted, NK is the number of pricing errors, q(L + \)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1. For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT(&ow) = 9T(&OW)tS^1gT(&ow) ~ XWK- -<J (L+ I )M+ I where OW stands for optimal-weighted and the x 2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested unconditional linear models reported in Panel C is: T J T ( © o w ) r e S t r i c t e d ~ TJT(&ow)unrestricted ~ ^number of restrictions-Note that bold p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted HJ x 2 93.8130 76.7439 139.4569 109.8416 48.6203 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0.0000 0.0000 0.0000 0.0006 Pricing kernel mean 0.9958 0.9961 0.9961 0.9959 0.9957 Pricing kernel standard deviation 0.4799 0.2699 0.3979 0.3947 0.7762 Panel B: Optimal-Weighted G M M x 2 97.8167 115.6417 103.2642 98.8137 29.8489 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0.0000 0.0000 0.0000 0.0951 Pricing kernel mean 0.9959 1.0000 1.0188 1.0284 0.8782 Pricing kernel standard deviation 0.0106 0.1758 0.2843 0.2965 1.3378 supLM test statistic 4.3562 31.1943 28.2889 92.9118 14.8002 Number of parameters 3 3 5 5 7 supLM test result pass fail fail fail pass Panel C: Nested Unconditional First Order Model Test Difference in Optimal G M M x 2 321.8288 7.8629 17.9649 81.9174 32.6420 Degrees of freedom 1 1 2 2 3 p-value 0.0000 0.0050 0.0001 0.0000 0.0000 102 Table A.23: Term Spread Conditional Second Order Models wi th Substitute Instrumental Variable CAY The log consumption-wealth variable, CAY, is added as an instrumental variable in the TERM conditional estimations. Columns two through six list results for TERM conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. Using W = Er[{Rt,t+i ® Zt){Rt,t+i ® ^ t ) T ] _ 1 , m Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT{®RW) = 9T(®Rw)T[Var{gT)]+gT(®RW) ~ xliK-q(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1 ) M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1, following Hansen (1982) . For the optimal-weighted estimations, the test statistic for nested second order models (Panel C) and nested conditional linear models (Panel D) is: •^number of restrictions' Note that bold p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted Returns-weighted x2 test 93.7641 57.1959 63.8320 65.7080 22.4602 Degrees of freedom 23 23 19 19 15 p-value 0.0000 0.0001 0.0000 0.0000 0.0963 Pricing kernel mean 0.9958 0.9960 0.9961 0.9958 0.9958 Pricing kernel standard deviation 0.4751 0.5240 1.1230 0.4440 0.8472 Panel B: Optimal-Weighted G M M x2 123.3420 71.7257 60.5730 41.9224 15.4754 Degrees of freedom 23 23 19 19 15 p-value 0.0000 0.0000 0.0000 0.0018 0.4177 Pricing kernel mean 0.9870 1.0353 0.9871 1.0628 1.0048 Pricing kernel standard deviation 0.0437 0.8305 1.1006 0.5837 1.3174 supLM test statistic 14.2160 54.2357 11.9627 15.5316 10.3341 Number of parameters 5 5 9 9 13 supLM test result pass fail pass pass pass Panel C: Nested Unconditional Second Order Model Test Nested Spec, x2 75.7445 6.6517 47.0844 9.5829 35.3016 Degrees of freedom 2 2 4 4 6 p-value 0.0000 0.0359 0.0000 0.0481 0.0000 Panel D: Nested Conditional First Order Model Test Difference in Optimal G M M x2 30876.7732 17.1208 32.0030 21.0470 10.4333 Degrees of freedom 2 2 4 4 6 p-value 0.0000 0.0002 0.0000 0.0003 0.1076 103 Table A.24: CAY Conditional First Order Models with Substitute Instrumental Variable TERM The term spread, TERM, replaces credit spread, DEF, as an instrumental variable in the CAY conditional estimations. Columns two through six list results for CAY conditional linear specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M , generally based upon the habit formation models of Constantinides (1990) and Ferson and Constantinides (1991); an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, FF3. Using W = Er[(Rt,t+i ® Zt)(Rt,t+i ® ^ t ) T ] _ \ m Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT{®RW) = 9T(®Rw)T[Var(gT)]+gT(&RW) ~ XNK-G(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + l)M + 1 is the number of estimated parameters, and [•] + represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^.1 • For these estimations, the Hansen (1982) JT test of overidentifying restrictions is: TJT{&Ow) = gT{®Ow)TS^gT(&ow) ~ XNK-g(L+l)M+l where OW stands for optimal-weigh ted and the x2 degrees of freedom are as described above. For the optimal-weighted estimations, the test statistic for nested unconditional linear models reported in Panel C is: rJ T (0ow) r e S tr ic ted _ T j r r ( © o w ) u r i r e s t r i c t e d ~ ^number of restrictions' Note that bold p-values highlight significance at the 5% level. In Panel B, supLM is the Andrews (1993) supremum Lagrange Multiplier test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M NS C C A P M C O C H R A N E FF3 Panel A: Returns-Weighted Returns-weighted x2 test 93.2847 111.9254 87.0248 68.4291 57.3840 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0.0000 0.0000 0.0000 0.0000 Pricing kernel mean 0.9962 0.9963 0.9962 0.9959 0.9962 Pricing kernel standard deviation 0.3091 0.2074 0.3712 0.6083 0.4455 Panel B: Optimal-Weighted G M M x2 129.2376 79.9459 97.4379 57.2985 50.6860 Degrees of freedom 25 25 23 23 21 p-value 0.0000 0.0000 0.0000 0.0001 0.0003 Pricing kernel mean 1.0417 0.9995 0.9680 1.0417 1.1019 Pricing kernel standard deviation 0.1665 0.0379 0.6534 0.6838 0.9134 supLM test statistic 5.5387 43.1365 57.3043 6.8618 6.2467 Number of parameters 3 3 5 5 7 supLM test result pass fail fail pass pass Panel C: Nested Unconditional First Order Model Test Difference in Optimal G M M x2 631.5977 24.0389 57.7615 52.1159 47.7349 Degrees of freedom 1 1 2 2 3 p-value 0.0000 0.0000 0.0000 0.0000 0.0000 104 Table A.25: CAY Conditional Second Order Models with Substitute Instrumen-tal Variable TERM The term spread, TERM, replaces credit spread, DEF, as an instrumental variable i n the CAY conditional estimations. Columns two through six list results for CAY conditional second order specifications based upon five different models: the capital asset pricing model, C A P M ; the consumption-based capital asset pricing model, C C A P M ; a nonseparable (habit formation) consumption pricing model, N S - C C A P M ; an investment-based asset pricing model, C O C H R A N E ; and the widely used Fama and French (1993) three state variable empirical asset pricing model, F F 3 . Using W = ET[(Rt,t+i ® Zt)(Rt,t+i ® ^ t ) T ] ~ \ in Panel A we test whether all pricing errors are zero using the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) statistic: JT(&RW) = 9T(®Rw)T[Var(gT)\+gT{QRW) ~ XNK-g(L+i)M+i where RW stands for returns-weighted, NK is the number of pricing errors, q(L + 1)M + 1 is the number of estimated parameters, and [•]+ represents the pseudo-inverse operator. Panel B lists results for pricing kernels estimated using the optimal-weighting matrix, W = S^1, following Hansen (1982). For the optimal-weighted estimations, the test statistic for nested second order models (Panel C) and nested conditional linear models (Panel D) is: T J T ( 0 c w ) r e s t r i c t e d _ TJT(&ow)unrestricted ~ -^number of restrictions' Note that b o l d p-values highlight significance at the 5% level. In Panel B , s u p L M is the Andrews (1993) supremum Lagrange Mul t ip l ie r test statistic used to examine for structural shifts in the model parameters. C A P M C C A P M N S C C A P M C O C H R A N E F F 3 Panel A : Returns-Weighted Returns-weighted x2 test Degrees of freedom p-value 58.1496 23 0.0001 101.8841 23 0.0000 62.8926 19 0.0000 38.2868 19 0.0055 27.3405 15 0.0261 P K mean Pricing kernel standard deviation 0.9960 0.5353 0.9962 0.3787 0.9961 0.7226 0.9963 1.1012 0.9959 0.7902 Panel B : Optimal-Weighted G M M x2 Degrees of freedom p-value 51.0923 23 0.0007 68.3530 23 0.0000 34.5795 19 0.0157 22.2831 19 0.2704 25.6611 15 0.0417 Pricing kernel mean Pricing kernel standard deviation 1.0351 1.3639 1.0076 1.3239 0.8864 1.6951 1.0296 1.3682 0.8827 1.0495 s u p L M test statistic Number of parameters s u p L M test result 15.0149 5 pass 24.3080 5 fail 18.2897 9 pass 39.3241 9 fail 14.4102 13 pass Panel C: Nested Unconditional Second Order Mode l Test Difference in Opt imal G M M x2 Degrees of freedom p-value 15.5077 2 0.0004 62.8645 2 0.0000 35.2218 4 0.0000 32.2214 4 0.0000 58.1270 6 0.0000 Panel D : Nested Condit ional First Order Mode l Test Difference in Opt imal G M M x2 Degrees of freedom p-value 30.0280 2 0.0000 55.1452 2 0.0000 17.0070 4 0.0019 55.8752 4 0.0000 23.0929 6 0.0008 105 Table A.26: Testing the Statistical Significance of Variable Means Across TERM Environments A l l data is quarterly and covers the period from Q2, 1959 to Q4, 1999. The N basic assets labeled TBILL, CORP, SlBl, S 1 B 5 , S3B3, SbBl and S 5 B 5 are described in Section 4. The simple return series axe in excess of the quarterly inflation rate. The lagged term spread variable, TERM, is used to separate all sample period observations into one of two states: 1) periods for which TERM equals or exceeds its sample mean, and 2) periods for which TERM is less than its sample mean. Columns two through six report the full sample mean, high TERM state mean, low TERM state mean, t-statistic for difference between these two means, and the associated one-tailed p-value for this t-statistic. The t-statistic is used to test the null hypothesis that mean basic asset returns are equal across high and low T E R M periods. The t-statistic is computed as follows: where f" is the mean return to asset i for all high TERM state periods, nH is the number of high periods, and Var(rf) is the variance of asset i's return in the high periods. The sample moments for the nL low state returns, r\, are defined similarly. Note that b o l d p-values indicate basic asset for which the are statistically significant at the 5% percent level. t-statistic = nL - 1) t ( n " - l ) + ( n i - l ) Fu l l High TERM periods (n= 86) Low TERM periods (n=77) High Series sample (n = 163) vs. Low t-statistic p-value 3 M t h . T - b i l l 0.0041 0.0076 0.0159 0.0380 0.0244 0.0215 0.0237 0.0052 0.0140 0.0343 0.0560 0.0390 0.0373 0.0405 0.0028 0.0004 -0.0047 0.0179 0.0081 0.0038 0.0050 2.8477 2.0177 1.6904 2.0688 2.2905 2.3304 2.9003 0.0025 0.0226 0.0464 0.0201 0.0116 0.0105 0.0021 Corporate Bonds S l B l Portfolio S 1 B 5 Portfolio S3B3 Portfolio S5B1 Portfolio S5B5 Portfolio 106 Appendix Figures FF 25 Portfolios 5 FF 25 Portfolios Figure B.l: Correlation Coefficients for the Fama French 25 Portfolios The Fama and French (1993) twenty-five portfolios are sorted by five quintiles in market value of equity (ME) and five quintiles in the book-to-market value of equity ratio (B/M). The portfolios are ordered lexigraphically, sorted first by ME quintile, then by B / M quintile. For example, the first five portfolios consist of the five B / M quintiles (increasing in order) of the first size (smallest) quintile. The correlation coefficients are calculating using real quarterly returns for the period from Q2, 1959 to Q4, 1999. 108 In Sample Out of Sample « B / M B / M • 1 5 1 5 + 3 Mth. T-bills + Government Bonds + Corporate Bonds Figure B.2: The Choice of In and Out of Sample Portfolio Subsets The Fama and French (1993) twenty-five portfolios are sorted by five quintiles in market value of equity (ME) and five quintiles in the book-to-market value of equity ratio (B/M). The "In Sample" diagram depicts graphically the M E and B / M characteristics of the Fama and French (1993) portfolios used, together with three month Treasury bills and corporate bonds, in the original estimation and testing of the various specification/model combinations. This subset is designed to capture the cross-sectional diversity in the full set of portfolios. More specifically, the basic set of portfolios consists of 5151 (small capitalization, growth), 5155 (small capitalization, value), 5555 (large capitalization, growth), 5555 (large capitalization, value), S3B3 (middle capitalization, average growth/value), three month Treasury bills (TBILL), and corporate bond (CORP) returns. A second non-overlapping subset of portfolios, depicted in the "Out-of-Sample" diagram, is chosen to serve as an out-of-sample set of basic portfolios. This data is used to test the robustness of valid pricing kernels following the methods described in Subsection 3.2. This portfolio subset consists of the 5153, 5252, 5254, 5351, 5355, 5452, 5454, 5553, and government bond (GOVT) portfolio returns. Note that both the "In-Sample" and "Out-of-Sample" basic assets sets are augmented with "managed portfolios" arising from the use of instrumental variables in the generation of moment conditions. 109 2.5r Returns-Weighted Estimations 1.5 E 1 CD 0.5 01— 0.6 HJ Lower Bound + CAPM o CCAPM NS CCAPM • COCH 0 FF3 0.7 0.8 0.9 1 1.1 Pricing Kernel Mean 1.2 1.3 1.4 Optimal-Weighted Estimations 0.9 1 1.1 Pricing Kernel Mean 1.2 1.3 1.4 Figure B.3: First Order (Linear) Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the unconditional linear pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns-weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the " H J Lower Bound." 110 CAPM o &— LU CD c n co L_ CD 3 O i _ i _ LU CD O) co L_ CD . > < O i— i LU CD CO CO l _ CD . 3 o i_ L_ LU CD CD E CD O i _ i _ LU CD O) CO i CD . 3 10 0 -10 1 1 1 1 -1 1 1 -*- Returns-Weighted •©• Optimal-Weighted " i 10 0 -10 10 FF3 20 25 30 Figure B .4: First Order (Linear) Models The charts depict the sample average pricing errors for the unconditional linear pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets ( S l B l , S1B5, S5B5, S5B5, S3B3, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. I l l Figure B .5: Second Order Polynomial Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the unconditional second order pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the " H J Lower Bound." 112 CAPM 10 0 -10 I I I i i i i -#- Returns-Weighted G Optimal-Weighted " 10 15, 20 CCAPM 25 30 10 I I I i i 0 \ 10 i i i 10 Kie>J5,m. 2 0 2 5 NS CCAPM 30 10 0 -10 * * o r * .0-O-Figure B.6: Second Order Polynomial Models The charts depict the sample average pricing errors for the unconditional second order pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets ( S l £ ? l , S1B5, S5B5, S5B5, S3B3, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. 113 2.5r 1 1-51 T 3 CO 0.5 0 1— 0.6 Returns-Weighted Estimations HJ Lower Bound + CAPM o CCAPM * NS CCAPM • COCH 0 FF3 0.7 0.8 0.9 1 1.1 Pricing Kernel Mean 1.2 1.3 1.4 Optimal-Weighted Estimations 2.5r 0 L- ' 1 1 1 i i i i 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Pricing Kernel Mean Figure B .7 : Third Order Polynomial Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the unconditional third order pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the "HJ Lower Bound." 114 CAPM 0 -10 i 10 0 -10 10 0 -10 I 10 0 -10 1 1 1 1 -*- Returns-Weighted 1 1 1 •©• Optimal-Weighted ~ 10 CC1APM 20 25 30 10 15 20 NS CCAPM 25 30 O - Q - O - O e - e o 10 :6CH 20 25 30 ©• 9 © - O - O ^ Q -0 10 FF3 20 25 30 10 h 0 -10 0 <p E © Q - O - O - O 10 15 20 o-o-o -©. e •© o 25 30 Figure B . 8 : Third Order Polynomial Models The charts depict the sample average pricing errors for the unconditional third order pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets ( S l B l , S1B5, S5B5, S5B5, S3B3, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. 115 Optimal-Weighted Estimations 2.5r Pricing Kernel Mean Figure B .9: Term Spread Conditional First Order Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the TERM conditional linear pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the "HJ Lower Bound." 116 C A P M LU a> D) CD i a) 3 10 0 -10 -*- Returns-Weighted e Optimal-Weighted LU CD D) CO t CD 3 LU CD D) CD CD 3 LU CD cn ro CD 3 10 0 -10 LU CD D) CO L _ CD 3 Figure B .10: Term Spread Conditional First Order Models The charts depict the sample average pricing errors for the TERM conditional linear pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets (SlBl, S1B5, S5Bb, S5B5, S3B3, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. 117 Optimal-Weighted Estimations 2.5r 0 1 ' 1 1 1 i i i i 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Pricing Kernel Mean Figure B . l l : Term Spread Conditional Second Order Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the TERM conditional second order pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the "HJ Lower Bound." 118 CAPM 10 0 -10 * * —*— Rfiturns-Weinhti et r s- eig ted © Optimal-Weighted 3* 10 O i _ UJ 0 d> O) 2 > -10 < ft ft ft>*^* * / 0 25 30 N 7* -*-*\0 • © .Q . o -o -o -© <3- C r©-O-© e.0 X 10 NS CCVM 20 25 30 10 0 -10 o - o ^ e © q 10 ;6CH 20 25 30 10 0 -10 o-o-e-e-G-o-c/ ' " ~ * - * - * * * - * 10 15 20 25 30 Figure B.12: Term Spread Conditional Second Order Models The charts depict the sample average pricing errors for the TERM conditional second order pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets ( S l B l , 51B5, S5B5, S5B5, S3B3, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. 119 SMB(%) -10 -10 MKT (%) MKT (%) -10 Term Spread (%) Figure B.13: Term Spread Conditional Second Order F F 3 Model, Returns-Weighted The estimated parameters for the returns-weighted estimation of the Fama and French (1993) three state variable model (FF3) are used to simulate values of the conditional second order pricing kernel. This pricing kernel is a function of three state variables (MKT, SMB, HML) and one conditioning variable (TERM). Each 3-dimensional chart above depicts the simulated kernel value derived from holding two variables (state or conditioning) constant at their mean level while permitting the other two variables to vary over the ranges [mean - 10%, mean + 10%] for MKT, SMB, and HML or [mean - 2%, mean + 2%] for TERM. 120 S M B ( % ) -10 -10 M K T ( o / 0 ) M K T (%) -10 0 Term Spread (%) Figure B.14: Term Spread Conditional Second Order F F 3 Mode l , Optimal-Weighted The estimated parameters for the optimal-weighted estimation of the Fama and French (1993) three state variable model (FF3) are used to simulate values of the conditional second order pricing kernel. Th i s pricing kernel is a function of three state variables (MKT, SMB, HML) and one conditioning variable (TERM). Each 3-dimensional chart above depicts the simulated kernel value derived from holding two variables (state or conditioning) constant at their mean level while permitting the other two variables to vary over the ranges [mean - 10%, mean + 10%] for M K T , S M B , and H M L or [mean - 2%, mean + 2%] for TERM. 121 Mean Returns for Higher TERM Values ME quintiles B/M quintiles Mean Returns for Lower TERM Values 6. ME quintiles B/M quintiles Figure B.15: Variable Means Across TERM Environments The Fama and French (1993) twenty-five portfolios are sorted by five quintiles i n market value of equity ( M E ) and five quintiles in the book-to-market value of equity ratio ( B / M ) . The simple return series are in excess of the quarterly inflation rate. The lagged term spread variable, TERM, is used to separate all sample period observations into one of two states: 1) periods for which TERM equals or exceeds its sample mean (top chart), and 2) periods for which TERM is less than its sample mean (bottom chart). 122 First Principal Component of Returns ME quintiles B/M quintiles Second Principal Component of Returns 5 0 . ME quintiles B/M quintiles Figure B.16: Principal Components Analysis of Portfolio Returns The Fama and French (1993) twenty-five portfolios are sorted by five quintiles in market value of equity (ME) and five quintiles in the book-to-market value of equity ratio (B /M) . Principal components analysis is used decompose the twenty-five portfolios returns into common collective movements. The first (top chart) and second (bottom chart) principal components capture 87.3% and 4.0% respectively of the covariation in returns. The analysis reveals significant nonlinearity in the common collective movements in returns. 123 Figure B.17: Comparing Term Spread and Log Consumption-Wealth Variables The term spread, TERM, is defined to be the difference between the yield on a portfolio of all Treasury bonds over ten years to maturity and the yield on a one year constant maturity Treasury note. Lettau and Ludvigson (2001a) develop an economic framework which implies a cointegrated relationship between consumption, asset holdings, and labor income. The authors define CAY to be the deviations from this shared trend (see Equation (12) on page 823 of Lettau and Ludvigson (2001a)). Both variables, TERM and CAY, are lagged one quarter in order to be used as conditioning variables. To facilitate easier graphical comparison, both series have been standardized by subtracting their sample means and dividing by their sample standard deviations. 124 2.5 g ro '> CD Q 1.5 CD X J c ro 55 "CD i CD CD £ 0.5 o1— 0.6 Returns-Weighted Estimations — HJ Lower Bound + CAPM O CCAPM * NS-CCAPM • COCHRANE 0 FF3 0.7 0.8 0.9 1 1.1 Pricing Kernel Mean 1.2 1.3 1.4 2.5 Optimal-Weighted Estimations o ro > CD Q ro X I c ro w "CD a i a) C 'o 1.5h 0.5 0.6 0.7 0.8 0.9 1 1.1 Pricing Kernel Mean 1.2 1.3 1.4 Figure B.18: CAY Conditional First Order Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the CAY conditional linear pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the "HJ Lower Bound." 125 CAPM Figure B.19: CAY Conditional First Order Models The charts depict the sample average pricing errors for the CAY conditional linear pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets (5151, 5155, 5555, 5555, 5353, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. 126 Optimal-Weighted Estimations 2.5 r 01 1 1 1 i i i i i 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Pricing Kernel Mean Figure B.20: CAY Conditional Second Order Models The top (bottom) chart depicts plots of the sample mean and standard deviations for the CAY conditional second order pricing kernels from the Jagannathan and Wang (1996) and Hansen and Jagannathan (1997) returns weighted (Hansen (1982) optimal-weighted) estimations. Admissible pricing kernels lie above the "HJ Lower Bound." 127 CAPM LU CD c n CD 3 LU CD c n CO I CD > < CD c n co CD 3 LU CD c n co i CD 3 LU CD c n co L-CD 3 10 0 -10 I 10 0 -10 I 10 0 -10 I 10 0 -10 ( 10 0 -10 Returns-Weighted © Optimal-Weighted 10 c d & M 20 25 ^.®-eo-cr 10 15 20 NS-CCAPM 25 b-©. ©•©• o-o-° 1 0 COCHRANE 2 0 25 v . e © . °-©-o© © © 10 FF3 20 25 r> O - O ^ © - © © - ^ Q ^ ^ e Q~o-©-o ©•© © 10 15 20 30 30 30 30 25 30 Figure B.21: CAY Conditional Second Order Models The charts depict the sample average pricing errors for the CAY conditional second order pricing kernels. For each chart, the first seven plotted observations are associated with the seven basic assets ( S l B l , S1B5, S5B5, S5B5, S3B3, TBILL, CORP). The other twenty one observations are managed portfolios arising from the product between the seven basic assets and the three instrumental variables (DEF, DIV, AIP). For example, observations eight through fourteen are associated with pricing errors for the seven portfolios scaled (managed) with the DEF variable. 128 Figure B . 2 2 : CAY Conditional Second Order F F 3 Model, Returns-Weighted The estimated parameters for the returns-weighted estimation of the Fama and French (1993) three state variable model (FF3) are used to simulate values of the conditional second order pricing kernel. This pricing kernel is a function of three state variables (MKT, SMB, HML) and one conditioning variable (CAY). Each 3-dimensional chart above depicts the simulated kernel value derived from holding two variables (state or conditioning) constant at their mean level while permitting the other two variables to vary over the ranges [mean - 10%, mean + 10%] for MKT, SMB, and HML or [mean - 0.02%, mean + 0.03%] for CAY. 129 Figure B.23: CAY Conditional Second Order FF3 Model, Optimal-Weighted The estimated parameters for the optimal-weighted estimation of the Fama and French (1993) three state variable model (FF3) are used to simulate values of the conditional second order pricing kernel. Th i s pricing kernel is a function of three state variables (MKT, SMB, HML) and one conditioning variable (CAY). Each 3-dimensional chart above depicts the simulated kernel value derived from holding two variables (state or conditioning) constant at their mean level while permitting the other two variables to vary over the ranges [mean - 10%, mean + 10%] for MKT, SMB, and HML or [mean - 0.02%, mean + 0.03%] for CAY. 130 

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