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An investigation of the market model when prices are observed with error Gendron, Michel 1984

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AN INVESTIGATION OF Tiffi MARKET MODEL WHEN PRICES ARE OBSERVED WITH ERROR by MICHEL GENDRON B.Sc, Universite Laval, 1977 M.Sc, Univ e r s i t e Laval, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Faculty of Commerce and Business Administration) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1984 © Michel Gendron, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of (Loh\£-fl> CJ£~ The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 o DE-6 (3/81) ABSTRACT The market model, which r e l a t e s s e c u r i t i e s returns to t h e i r systematic r i s k (3), plays a major r o l e i n finance. The estimation of 3 , i n p a r t i c u l a r , i s fundamental to many empirical studies and investment decisions. This d i s s e r t a t i o n develops a model which explains the observed s e r i a l c o r r e l a t i o n s i n returns and the i n t e r v a l i n g e f f e c t s which are inconsistent with the market model assumptions. The model accounts f o r t h i n trading and d i f f e r e n t f r i c t i o n s i n the trading process and has as sp e c i a l cases other models of t h i n trading and f r i c t i o n s presented i n the finance l i t e r a t u r e . The main assumption of the model i s that the prices observed i n the market and used to compute returns d i f f e r by an error from the true prices generated by a Geometric Brownian Motion model, hence i t s name, the error i n prices (EIP) model. Three estimation methods for 3 are examined f o r the EIP model: the Maximum Lik e l i h o o d (ML) method, the Least Squares (LS) method and a method of moments. I t i s suggested to view the EIP model as a missing information model and use the EM algorithm to f i n d the ML estimates of the parameters of the model. The approximate small sample and asymptotic properties of the LS estimate of 3 are derived. I t i s shown that replacing the true covariances by t h e i r sample moments estimates leads to a convenient and f a m i l i a r form for a consistent estimate of 3 • F i n a l l y , some i l l u s t r a t i o n s of six d i f f e r e n t estimation methods for 3 are presented using simulated and r e a l s e c u r i t i e s returns. i i i Table of Contents Page Abstract i i L i s t of Tables v L i s t of Figures v i Acknowledgements v i i INTRODUCTION 1 CHAPTER I: MODELS OF THIN TRADING AND FRICTIONS IN THE TRADING PROCESS 8 1. SW Model 9 2. Dimson Model 16 3. CHMSW Model 18 4. Other Models of F r i c t i o n s 21 CHAPTER I I : EIP MODEL 23 1. Market Model and LS Estimation 23 2. Market Model with Errors i n Prices 25 3. Maximum Lik e l i h o o d Estimation Using the EM Algorithm 28 3.1 Covariance Structure of the Observed Returns 29 3.2 Maximum Lik e l i h o o d Method 3 4 3.3 EM Algorithm 36 3.4 A p p l i c a t i o n of the EM Algorithm to the EIP Model. . . 3 7 4. LS Estimation of Systematic Risk 40 4.1 Generalized EIP Equation 41 4.2 LS Estimator 4 3 4.3 Approximate Small Sample Properties of b 4 5 4.4 Accuracy of Approximate Small Sample Properties . . . 48 4.5 Asymptotic Properties of b 49 4.5.1 Asymptotic Bias 49 4.5.2 Asymptotic Variance 52 4.6 Small Sample and Asymptotic Properties of b Using Centralized Variables 53 4.7 I l l u s t r a t i o n s of the Small Sample Properties 54 5. Methods of Moments 61 i v Page 6. Extended EIP Model 62 6.1 ML Estimation of the Parameters 63 6.2 Estimation of 3 from the Methods of Moments 63 7. EIP Model and Stock Prices Behaviour 64 8. Conclusion 69 CHAPTER I I I : APPLICATIONS OF DIFFERENT 3 ESTIMATORS TO SIMULATED AND REAL DATA 71 1. Applications to Simulated Observations 75 1.1 Description of the Simulation Procedure 76 1.2 Estimation Results f o r the Simulated Observations . . . 82 1.3 Interpretation and Discussion of Simulation Results . . 88 1.4 Computing Remarks f o r the EM Algorithm 91 2. I l l u s t r a t i o n s Using NYSE S e c u r i t i e s 94 2.1 Data Source and Selection C r i t e r i a 94 2.2 Estimation Results from NYSE Data 96 2.3 Interpretation and Discussion of Results from NYSE Data 99 3. Conclusion from Simulated and NYSE Data 101 CONCLUSION 104 Footnotes 107 Bibliography 110 APPENDICES 1. D i s t r i b u t i o n of Observed Returns 113 2. Mean and Variance of Quadratic Forms i n Normal Variables. . . . 115 3. S i m p l i f i e d Forms of the Traces 116 4. Li m i t of the Variance of Some Quadratic Forms 121 5. Consistent Estimate of 3 123 6. Asymptotic Variance of b 124 7. Small Sample Properties Using Centralized Variables 126 8. True 3 and Covariance of Observed Returns 128 9. SW as a Special Case of EIP 130 10. Bid-Ask Spread and the EIP Model. 132 11. Price Adjustment Delays and the EIP Model 134 V L i s t of Tables Table No. Page 1 Parameters Values Used i n the I l l u s t r a t i o n s of the Small Sample Properties 56 2 Small Sample Properties for a Fat Security and a Fat Index 57 3 Small Sample Properties f or a Thin Security and a Fat Index 58 4 Small Sample Properties f o r a Fat Security and a Thin Index 59 5 Small Sample Properties f or a Thin Security and a Thin Index 60 6 $ Estimates f or EIP Model 83 7 3 Estimates f or SW Model 85 8 3 Estimates for CHMSW Model 87 9 Estimates f o r Selected NYSE Se c u r i t i e s 97 v i L i s t of Figures Figure No. Page 1 Relation Between Measured Returns and True Returns i n the SW Model 11 Acknowledgements I want to thank the members of my d i s s e r t a t i o n committee: P i e t de. Jong for h i s most valuable s t a t i s t i c a l suggestions and guidance, Harry Joe for his c a r e f u l reading and d e t a i l e d comments upon the d i s s e r t a t i o n , Rex Thompson for many stimulating discussions, William Welch for h i s p r a c t i c a l suggestions, and my chairman, Eduardo Schwartz, for p a t i e n t l y guiding me through the whole process. F i n a l l y , and most importantly, I thank Helene Crepeau for her p r i c e l e s s professional help and personal support at a l l stages of my program and d i s s e r t a t i o n . 1 INTRODUCTION The market model,which relates s e c u r i t i e s returns to t h e i r systematic r i s k or beta (3), plays a major role i n finance. The estimation of 3 , i n p a r t i c u l a r , i s fundamental to many empirical studies. However, some phenomena inconsistent with the market model assumptions have been observed. There i s , f o r instance, s e r i a l c o r r e l a t i o n i n the observed s e c u r i t i e s and market returns, as w e l l as s e r i a l cross c o r r e l a -t i o n between the s e c u r i t i e s and the market returns. There i s also s e r i a l c o r r e l a t i o n i n the market model res i d u a l s . This has been associated with d i f f e r e n t kinds of measurement errors'*" which cause the least squares (LS) estimate of the market model 3 to be biased and inconsistent. Further-more, these s e r i a l c o r r e l a t i o n s and the bias i n the LS estimate of 3 have been established to vary with the length of the i n t e r v a l over which the returns are measured, hereafter referred to as the measuring i n t e r v a l . In order to better understand the returns behaviour and correct f o r the estimation biases, the market model has been modified i n d i f f e r e n t ways to take account of some market imperfections. The th i n trading e f f e c t and the p r i c e adjustment delays caused by f r i c t i o n s i n the trading process have received s p e c i a l attention. Thin Trading 2 The thinness of a security refers to i t s trading frequency, a th i n security trading infrequently. Because of th i n trading, the c l o s i n g price of a s e c u r i t y f o r a given trading period may correspond to the p r i c e for 2 a transaction that occurred within the period or i n a preceeding period, rather than at the end of i t . This implies that the observed returns for a given set of s e c u r i t i e s w i l l be nonsynchronous. Assuming a continuous p r i c e generating process, the l a s t trading p r i c e of a s e c u r i t y represents the true end of period p r i c e only up to a c e r t a i n error. I t has been shown that t h i n trading can cause s e r i a l c o r r e l a t i o n i n the s e c u r i t i e s returns. Since the market index i s calculated as a weighted average of the s e c u r i -t i e s p r i c e s , i t also s u f f e r s from measurement errors and s e r i a l c o r r e l a -3 t i o n . This brings about the serious s t a t i s t i c a l problem of errors i n variables which causes the LS estimate of $ to be biased and inconsistent. The s o l u t i o n to the t h i n trading problem i s to have the exact true p r i c e at the end of each measuring i n t e r v a l . For t h i s purpose the measur-ing i n t e r v a l could e i t h e r be fixed and quotation p r i c e used instead of the l a s t transaction p r i c e , or be allowed to vary from one transaction to another. Obtaining the quotation p r i c e i s a major obstacle to using f i x e d measuring i n t e r v a l s . Franks et a l . (1977), Schwert (1977) and Marsh (1979) have measured the returns from transaction to transaction. Here again the exact transaction time i s not always av a i l a b l e and the continuous c a l c u l a t i o n of an appropriate index causes major d i f f i c u l t i e s . This type of s o l u t i o n , however, corrects only for t h i n trading. P r i c e Adjustments Delays In addition to t h i n trading, i t has been suggested that some f r i c t i o n s i n the trading process delay the adjustment of the prices and add to the nonsynchronous returns problem. Types of f r i c t i o n most l i k e l y to prevent a l l information from being r e f l e c t e d immediately i n the observed prices i n -3 elude the presence of s p e c i a l i s t s / d e a l e r s which i n t e r f e r e with the p r i c i n g process, transaction costs, and non continuous transmission of information i t s e l f . These delays i n the adjustment of the prices w i l l obviously also cause some discrepancy between the true prices that would be observed i n a perfect market s i t u a t i o n , and the prices a c t u a l l y observed. Cohen et a l . (1983b) have emphasized the importance of the e f f e c t s of these f r i c t i o n s on the s e r i a l c o r r e l a t i o n of returns and the bias i n the LS estimation of 3 . Both th i n trading and pri c e adjustment delays then cause prices to be observed with error, returns to s u f f e r from s e r i a l c o r r e l a t i o n and the LS estimate of 3 to be biased and inconsistent. It has also been observed that s e r i a l c o r r e l a t i o n i n returns and the LS estimate of 3 may change considerably as the estimation i n t e r v a l i s varied."' This i s referred to as the i n t e r v a l i n g e f f e c t . The i n t e r v a l i n g e f f e c t The observed i n t e r v a l i n g e f f e c t on the LS estimate of systematic r i s k has been explained by both the thi n trading and the pri c e adjustment delays. Theobald (1983) studies the a n a l y t i c r e l a t i o n s h i p between non-trading and the i n t e r v a l i n g e f f e c t . He notes that as the measuring i n t e r v a l becomes large r e l a t i v e to the d i f f e r e n c i n g i n t e r v a l i n which successive trades occur, the asymptotic bias introduced by non trading i s reduced. This i s , however, at the expense of the asymptotic variance, implying that there e x i s t s a measuring i n t e r v a l minimizing the mean squared error (MSE). Cohen et a l . (1983b) claim that the i n t e r v a l i n g e f f e c t i s also due to p r i c e adjustment delays. They f i n d that, i n the presence of 4 f i n i t e p r i c e adjustment delays, the asymptotic bias i n the LS estimate of 3 decreases as the measurement i n t e r v a l increases without bound, and that for any measurement i n t e r v a l the bias i n the estimates across s e c u r i -t i e s i s cross s e c t i o n a l l y d i s t r i b u t e d around zero and depends on the r e l a -t i v e magnitude of the p r i c e adjustment delays. I t would then seem d e s i r -able to have a measuring i n t e r v a l as long as possible. However, because only f i n i t e samples are a v a i l a b l e and because the number of observations decreases with increasing measuring i n t e r v a l , we are r e s t r i c t e d i n the choice of the i n t e r v a l . ^ A t y p i c a l approach to the modeling of t h i n trading and p r i c e adjust-ment delays problems has been to make a d d i t i o n a l assumptions to the market model i n order to derive consistent estimators. For instance, Scholes-Williams (1977) (SW) tackle the t h i n trading e f f e c t by assuming that the delays between the time a s e c u r i t y a c t u a l l y traded and the time the prices were recorded are independently and i d e n t i c a l l y d i s t r i b u t e d . They develop a consistent estimator of 3 which uses the returns calculated over any two consecutive i n t e r v a l s reporting a c l o s i n g p r i c e . Dimson (1979) assumes that a s e c u r i t y at each period has a c e r t a i n p r o b a b i l i t y of trad-ing, that i t trades at l e a s t once every 'n' periods and that a c e r t a i n proportion of the market trades each period. He also developed a consistent estimator. Cohen et a l . (1983b) (CHMSW) study the p r i c e adjustment delays. They assume that the observed returns are a weighted average of the true past returns, and derive a general form f o r a consistent estimator i n t h i s framework. B a s i c a l l y , the consistent estimators derived from the CHMSW model for p r i c e adjustment delays, and the SW and Dimson models for t h i n trading use the same p r i n c i p l e ; the security returns are re-5 gressed on synchronous and nonsynchronous market index returns. CHMSW and SW use simple regressions while Dimson uses a multiple regression. This type of procedure was also used by Pogue-Solnik (1974), Ibbotson (1975) and Schwert (1977). Elsewhere Cohen et a l . (1983a) suggest a d i f f e r e n t estimation procedure: i t i s , i n t h e i r own words, "to i n f e r the value of 3 by adjusting LS 3 f o r c r o s s - s e c t i o n a l differences i n the i n t e r v a l i n g e f f e c t as a function of the depth of the market for a secur-i t y . " Fowler,Rorke and Jog (1983) also propose a d i f f e r e n t procedure for estimating 3 i n the presence of t h i n trading. They suggest to multiply i t s LS estimate by a factor which i s the r a t i o of two parameters, each parameter being a function of p r o b a b i l i t i e s that s e c u r i t i e s l a s t trade occurred i n d i f f e r e n t subsamples of the measuring i n t e r v a l . These studies present a common weakness, i n addition to t h e i r r e s t r i c t i v e assumptions. Their analysis i s l i m i t e d to the derivation of a consistent estimator of 3 based on the LS method of estimation; no consideration i s given to small sample properties of the estimators and no other method of estimation i s suggested. This d i s s e r t a t i o n i s concerned with the estimation of a security's systematic r i s k i n the presence of the phenomena described above. It pro-poses a model which d i r e c t l y accounts not only f o r t h i n trading, but also for p r i c e adjustment delays and other f r i c t i o n s i n the trading process. Furthermore, t h i s model explains the observed s e r i a l c o r r e l a t i o n and i n t e r v a l i n g e f f e c t , and the bias i n the LS estimation of 3 • I t i n c o r -porates i n the market model the i n t u i t i v e fact that the observed p r i c e s from which the returns are calculated d i f f e r from the true p r i c e s that would be observed i n a perfect market s i t u a t i o n . This assumption contains 6 i n a sense the assumptions made by the d i f f e r e n t authors. I f there i s a delay between the time the prices are recorded and the time the sec u r i t y traded as assumed i n the SW model or i n Fowler et a l . model, the observed p r i c e s obviously d i f f e r from the true p r i c e s . I f there are some proba-b i l i t i e s of trading f o r a sec u r i t y at each period as i n the Dimson model or i f the observed returns are a weighted average of the past true returns as i n the CHMSW model, the pr i c e s w i l l be observed with an error. In contrast with the previous studies three types of estimators of 3 are studied f o r th i s model: maximum l i k e l i h o o d (ML), least squares (LS) and the method of moments estimators. I f , as i n the papers already men-tioned, consistency i s the most desirable property, ML estimators should be used since they are the best asymptotic estimators. Another advantage of t h i s estimation method i s that not only 3 > but also a l l the other parameters of the model can be e a s i l y estimated. In p r a c t i c e , however, LS estimation i s commonly used f o r 3 • D i f f e r e n t measuring i n t e r v a l s , which may overlap, can be u t i l i z e d i n t h i s estimation. The other type of estimator considered i s then LS estimator. Our concern i s to derive the properties of the LS estimator of 3 i n our framework, for any general way of measuring the returns. Small sample as well as asymptotic proper-t i e s are studied. F i n a l l y , i t i s seen that replacing the true covariances by t h e i r sample moments estimates leads to a convenient form f o r a con-s i s t e n t estimate of 3 • Chapter I reviews some of the more important models of th i n trading and f r i c t i o n s i n the trading process. The SW, Dimson and CHMSW models are described i n d e t a i l . Chapter II presents the market model with errors i n the pr i c e s that w i l l be ref e r r e d to as the EIP or Errors i n Prices 7 model. Three estimation methods for t h i s model are examined. F i r s t , the EM algorithm i s suggested f o r computing the ML estimates of a l l the 9 model parameters. This algorithm i s based on incomplete or missing data. Second, the approximate small sample and the asymptotic properties of the LS estimator of 3 are derived. Third, a method of moments estimator of 3 i s shown to give the same consistent estimates as the SW and CHMSW estimators. F i n a l l y , d i f f e r e n t empirical phenomena which are inconsistent with the market model are explained using the model with errors i n the p r i c e s . Chapter III presents some i l l u s t r a t i o n s of s i x estimation methods described i n the f i r s t two chapters. It i s divided i n two parts. In the f i r s t part the estimation method i s applied to observations generated according to three d i f f e r e n t p r i c e generating processes, while i n the second part i t i s applied to observations taken from the NYSE. 8 CHAPTER I MODELS OF THIN TRADING AND FRICTIONS IN THE TRADING PROCESS In a discussion of microstructure theory and stock prices behaviour, CHMSW (1980) pick out from the l i t e r a t u r e s i x i n t e r r e l a t e d empirical phenomena inconsistent with the t r a d i t i o n a l market model assumptions. These phenomena are: 1) Weak s e r i a l c o r r e l a t i o n i n i n d i v i d u a l s e c u r i t i e s d a i l y returns, with the proportion of s e c u r i t i e s y i e l d i n g s i g n i f i c a n t autocorrelations decreasing as the d i f f e r e n c -ing ( i . e . returns measurement) i n t e r v a l increases, and with predominantly negative sign f o r t h i n s e c u r i t i e s and p o s i t i v e sign f o r " t h i c k " (high v a l u e ) s e c u r i t i e s . 2) P o s i t i v e s e r i a l c r o s s - c o r r e l a t i o n s between se c u r i t y returns and market index returns, with a l e s s e r e f f e c t as the d i f f e r e n c i n g i n t e r v a l i s increased. 3) P o s i t i v e s e r i a l c o r r e l a t i o n i n market index returns, with the e f f e c t smallest for long d i f f e r e n c i n g i n t e r v a l s and those indexes giving the l e a s t weight to thi n s e c u r i t i e s . 4) Autocorrelation of market model residuals which i s weakly p o s i t i v e f o r d a i l y data but which becomes predominantly negative as the d i f f e r e n c i n g i n t e r v a l increases. 5) S e n s i t i v i t y of LS beta estimates to changes i n the d i f -ferencing i n t e r v a l , with t h i n s e c u r i t y betas r i s i n g as the i n t e r v a l i s lengthened and very high value security betas f a l l i n g . 6) Increase i n market model R 2 as the d i f f e r e n c i n g i n t e r v a l i s lengthened, with the largest e f f e c t for t h i n s e c u r i t i e s . On one hand, a t h i n trading element and an i n t e r v a l i n g e f f e c t c l e a r l y emerge from these observations. One the other hand, CHMSW (1980) argue that a l l s i x phenomena "can be at t r i b u t e d to f r i c t i o n i n the trading pro-cess causing a bid-ask spread and price-adjustment delays that d i f f e r 9 systematically across s e c u r i t i e s . " 12 This chapter reviews the two main t h i n trading models, the SW model with i t s extended version i n section 1 and the Dimson model i n section 2, and the CHMSW model of f r i c t i o n s i n the trading process i n section 3. The a b i l i t y of these models to explain some of the phenomena i s also b r i e f l y discussed. F i n a l l y , section 4 mentions other models of f r i c t i o n s i n the trading process. 1. SW Model 13 This model i s based on the continuous time market model which implies y = a + $x + e t = l , . . . , T (1.1) where y t » xfc , e are independently and i d e n t i c a l l y d i s t r i b u t e d ( i i d ) E ( e t | x t) = E(e f c) = 0 a = y - 3 y y x 3 = a /o2 , xy x y : continuously compounded rate of return on the s e c u r i t y from the end of period t to the end of period t +1 calculated as the logarithm of the p r i c e r e l a t i v e s , x^ : continuously compounded rate of return on the market index from the end of period t to the end of period t + 1, e : normal error term, 10 and y and a are the usual mean and covarlance symbols. It i s assumed that i f there are transactions during any i n t e r v a l [ t , t + l ] , the closing price at t + 1 i s determined by the l a s t transac-t i o n . This l a s t transaction occurred at a random point t + 1 - Sy , and the end of the i n t e r v a l . I f there i s no transaction f o r a given i n t e r v a l , no c l o s i n g p r i c e i s reported. When two consecutive i n t e r v a l s have transactions, a. rate of return y i s calculated from the c l o s i n g p r i c e s . The superscript d i f f e r e n t i a t e s the observed rate of return which covers an i n t e r v a l [t - Sy^ , t + 1 - Sy ^] , from the true rate of return y generated by the market model which covers the i n t e r v a l [t , t + 1 ] . The observed market index return x , calculated as a weighted average of the observed s e c u r i t i e s returns, w i l l then l i k e l y d i f f e r from the true market return x usually taken as a weighted average of the true returns. SW claim that the observed returns also obey a - l i n e a r r e l a t i o n , where 0 < Sy t+1 < 1 i s the length of time between the l a s t transaction (1.2) where a * = E[y t] - B*E[x t] 8. = cov(y t , x t)/var(x f c) e has mean 0 and i s uncorrelated with x^_ . Figure 1 reproduced from SW i l l u s t r a t e s the r e l a t i o n between measured returns and true returns. FIGURE 1 Relation Between Measured and True Returns i n the SW Model t-1 t+1 Measured Returns: security y True Returns: se c u r i t y y •k V l * y t ! s y t - i 7 j JT ^ ' > | S y t + 1 I' v * t-Sy f c Y t - 1 t + l - S y t + 1 y t > 12 As mentioned at the beginning of th i s chapter, c e r t a i n empirical observations are inconsistent with the market model assumptions about the returns moments. I t i s i n t e r e s t i n g to look at the observed returns moments predicted by the SW model and see how they compare to the market model moments. SW derive these moments under the a d d i t i o n a l assumption that a l l the non trading periods are i i d over time. The expectation of a s e c u r i t y measured return i s the true return mean E [ y t ] = (1.3) The measured returns variance i s greater than the true variance v a r [ y t ] = U + 2 var (Sy t)/vH a 2 a /u • y y (1.4) - The measured contemporaneous covariances across s e c u r i t i e s d i f f e r from the true ones; f o r instance, l e t z represent another s e c u r i t y cov y > z J t t 1- E max Sy t , -min < Sy fc , Sz f c + 2 cov Sy t , S z t /p v v } a yz y zj yz (1.5) wher e S z t i s analogous to Sy^ _ f o r sec u r i t y p = a /a a yz yz y z - The measured autocovariances of l a g one are negative cov y t ' y t - l - <!var(Sy t)/v2} O2 (1.6) 13 - The measured lag one covariances d i f f e r from zero cov * * h' y t - i J • (E max c + cov . S y t /p v v ) a yz y zj yz "(1.7) F i n a l l y , a l l the covariances for lags greater than one are zero. SW also explain that the observed returns are l i k e l y to appear l e p t o k u r t i c i n contrast to the normal d i s t r i b u t i o n of the true returns. They derive the kurtosis of measured returns where K(y t) = 3 + 0 I yJ 1 + 2 var (Sy f c) terms of order 1/v , y (1.8) and note that from the observed values of V the empirical k u r t o s i s i s y l i k e l y to exceed 3, the kurtosis of a normal v a r i a b l e , so the measured s e c u r i t i e s returns appear l e p t o k u r t i c . A l a s t but important note on the observed returns i s that t h e i r be-haviour as described by the above equations i s d i r e c t l y r e l a t e d to the s e c u r i t i e s ' thinness as determined by the set of Syfc , t=l,...,T . We now turn to the estimation of the market model parameters. SW note that t h e i r LS estimates using measured returns w i l l generally be biased and inconsistent. They show that s e c u r i t i e s trading very frequently or very infrequently have LS estimates biased upward for a and downward for 3 • The biases are reversed f o r s e c u r i t i e s with average trading f r e -14 q u e n c i e s . They d e v e l o p c o m p u t a t i o n a l l y c o n v e n i e n t c o n s i s t e n t e s t i m a t o r s f o r a and 3 T - l T - l a = ( l / T - 2 ) I y - BU/T-2) J x t=2 C t=2 C (1.9) 6 = 4 b - l + b + b + l 1 + 2p l x (1.10) where b , b , b + ^ a r e r e s p e c t i v e l y t h e LS e s t i m a t e s o f t h e r e g r e s s i o n c o e f f i c i e n t o f t h e measured s e c u r i t y r e t u r n s on t h e l a g k , con-temporaneous and l e a d k measured market r e t u r n s , p, i s t h e o r d e r k s e r i a l c o r r e l a t i o n c o e f f i c i e n t o f t h e measured market r e t u r n s , and denotes an e s t i m a t o r . I t i s shown t h a t a and 3 a r e a s y m p t o t i c a l l y e q u i v a l e n t t o t h e i n s t r u m e n t a l v a r i a b l e e s t i m a t o r s w h i c h use as i n s t r u m e n t * ft * * X „ = X , + X + X , ., . 3t t - l t t+1 SW use t h i s f a c t t o d e r i v e t h e a s y m p t o t i c v a r i a n c e o f t h e e s t i m a t o r s a s y v a r ( a ) = ( T - 2 ) _ 1 { p l i m [ / ( T - 2 ) ( 3 - 3 ) ] V + v a r ( e * ) ( l + 2 p * ) } asyvar(3) = ( T - 2 ) _ 1 { v a r (£*) [ l + 2 p * p * x ] / [ b ^ v a r (x* t> ] } where ft * * * p = c o v ( e , 1 ) / v a r ( e ) e t t - l t p 3 x = C O v ( x 3 t - l ' X 3 t ) / [ v a r ( x 3 t - l ) v a r ( x 3 t ) ] 1 / 2 15 bx3x = cov(x* f x* t)/var(x* t) . The SW model has some weaknesses. It i s not convenient for s e c u r i t i e s which s u f f e r severely from t h i n trading since only the returns measured over two consecutive periods where trading occurred can be used. Fowler and Rorke (1980) found that 59 per cent of the TSE s e c u r i t i e s l i s t e d be-tween January 1970 and December 1979 stay one or more months without trading; t h i s i l l u s t r a t e s that t h i s l i m i t a t i o n of the SW model can be serious. The SW model i s also inconsistent with p o s i t i v e s e r i a l c o r r e l a -t i o n i n s e c u r i t i e s returns since the r i g h t hand side of (1.6) w i l l always be negative. It i s f inallyi.inconsistent with s e r i a l c o r r e l a t i o n s extending to more than one period*'"^ since i t predicts a l l covariances greater than one to be zero. R e a l i z i n g the l i m i t a t i o n s of the SW estimator for very t h i n s e c u r i t i e s Fowler and Rorke (1979, 1983) have extended the SW estimation technique for s e c u r i t i e s that trade at l e a s t once every other period or measuring inter-v a l . Assuming that the trading process i s stationary over two periods and that a geometric index i s used, they write the SW extended (SWE) estimator ~ X X X ~ X 3 = ( 2b_, + 2b + 2 b + 1 ) / ( l + 2 ( ^ p l x ) ) where the p r e f i x 2 means that the returns used i n the estimation are measured over two periods. They show that t h i s can be written i n the following more convenient form i n terms of one period returns ^ 5> /\ 3 = (b_ 2 + b_ x + b + b + 1 + b + 2 ) / ( l + 2 p l x + 2 p 2 x ) . 16 16 The SW technique could be extended further but as Dimson (1979) noted i t , a decline i n e f f i c i e n c y due to estimation error i n the regression c o e f f i c i e n t s r e s t r i c t s the number of lead-lag periods which can be used. This e f f i c i e n c y loss however i s yet to be measured. 2. Dimson Model The model developed by Dimson presents a d i f f e r e n t approach to t h i n trading. It assumes that the expected value of an observed p r i c e i s a weighted average of past true p r i c e s . In addition to the continuous time market model, Dimson assumes that - s e c u r i t i e s trade at l e a s t once every n periods, - at time t , the p r o b a b i l i t y of a s e c u r i t y having been most recently traded i n period t-k (k > 0) i s 0^ , for k=l,...,n , - the proportion of the market p o r t f o l i o which was l a s t traded i n period t-k i s (j)^  , - the var i a t e s 6^ and (j)^  are, re s p e c t i v e l y , i i d over the periods; t h i s i s probably why Dimson i n h i s paper does not index these variates with respect to time; h i s notation w i l l be followed for that matter. These assumptions imply that the expected observed p r i c e s , and con-sequently changes (A) i n observed p r i c e s , are weighted averages of the past true p r i c e s , and changes i n prices k^t-k (1.11) t-k (1.12) 17 Since, i n continuous time, the returns are calculated as the logarithm of the p r i c e r e l a t i v e s , the expected observed returns are also a weighted average of the true returns and thejactual security observed returns can be written as y t = X V t - k + V t ' ( 1 ' 1 3 )  Z k=0 R z c and s i m i l a r l y f or the market returns x t = j0 V t - k + u t > ( 1 - 1 4 ) where u and v are mean zero errors uncorrelated with both 0, and t t k Y t - k ' *k 3 n d X t - k ' r e s P e c t i v e l y -Dimson considers a multiple:regression of observed returns on preceed-ing, synchronous and subsequent market returns A . B A A A y t = a +J \ X t + k + £ t ' ( 1 ' 1 5 ) k=-n He substitutes (1.13) and (1.14) f o r the l e f t hand side and r i g h t hand side of (1.15) and then uses the market model r e l a t i o n to write y i n terms of x , . Since both sides of (1.15) are equal, he equates the t K c o e f f i c i e n t s of x^ , for a l l k"",'^  and obtains the following r e l a t i o n t-k ° between the estimate of the true beta and the multiple regression co-e f f i c i e n t s k=-n k In words, an estimate of the true systematic r i s k i s obtained by aggregating the lead, contemporaneous and lag c o e f f i c i e n t s from the 18 multiple regression (1.15). It i s often referred to as the aggregated c o e f f i c i e n t estimator. Note that b which i s a simple regression co-e f f i c i e n t d i f f e r s from $ which comes from a multiple regression. K The Dimson model, which introduces the idea that the observed prices (returns) are a weighted average of the true prices (returns), i s consistent with c o r r e l a t i o n i n returns over many periods. However, a mistake was found i n the de r i v a t i o n of Dimson's estimator of systematic r i s k . Dimson (1983) claims that the bias i n his estimator introduced by th i s mistake i s f o r p r a c t i c a l purposes n e g l i g i b l e . 3. CHMSW Model In t h e i r model the observed returns are a weighted average of the true contemporaneous and past returns which themselves come from the continuous time market model, n V (1.16) k=0 They further assume that - the weights V t ,m fo r security y and V x,n for security z are independent for a l l t , x, m, n, - V i s independent of x and £ for a l l t, T and n, - E(v ) = E(v ) for a l l t, T and m t,m T,m n V t,k ) = 1 for a l l t . k=0 19 From these assumptions, the covariance of observed returns are worked out and i t i s established that a consistent estimator of the true system-a t i c r i s k i s given by 3 = n t k=-n V * , L bx+k k=-n (1.17) where i - s the regression c o e f f i c i e n t of the market returns on the lead k market returns. Note that i n the l i m i t , as the number of observations goes to A A i n f i n i t y , b w i l l be equal to b but for f i n i t e samples these X"T~K. X rC estimates w i l l d i f f e r . CHMSW claim that the SW assumptions translate i n th i s model to y t " V 0 y t + V i a y t - i which gives an estimator of the same form as the SW estimator ^ A A A A 3 = (b_ 1 + b + b + 1 ) / ( l + 2b x_ x) . However, even though the two estimators have the same form, the SW model i s not a s p e c i a l case of the CHMSW model. The following example helps i l l u s t r a t e the difference between the two models. Suppose that for the SW model the l a s t trading day i s always i n the middle of the measuring i n t e r v a l , i . e . Sy i s one ha l f f or a l l t . Then from (1.4) the variance of the observed returns would be the same as the variance of the true returns. Transposing t h i s s p e c i a l case i n the CHMSW model V and t ,o Vt_-^ ^ are both f i x e d and equal to one h a l f . However, the variance of 20 the observed returns i n t h i s case i s one hal f of the variance of the true returns. In the CHMSW model the observed returns are a weighted average of the true returns which are i i d . There i s then a d i v e r s i f i c a t i o n e f f e c t hidden i n t h i s model since the variance of a weighted average of i i d v a r i a b l e s i s smaller than the weighted sum of the variances. There i s no such d i v e r s i f i c a t i o n e f f e c t f or the SW model. The CHMSW and Dimson models are both based on the d i s t i n c t i o n between the a c t u a l l y observed returns and the true unobserved returns. The d i f -ference between the two models i s that for CHMSW the observed returns are a weighted average of the true returns whereas for Dimson the expected observed returns are a weighted average of the true returns. Since Dimson i s only concerned with t h i n trading, every time there i s a transaction and a p r i c e i s observed, i t i s a true p r i c e . Because of infrequent t r a d -ing, i t i s not known when the transaction that gave the end of period p r i c e occurred. The expected observed p r i c e i s a weighted average of the true prices since each transaction that would produce a true p r i c e has a cer-t a i n p r o b a b i l i t y of being the l a s t one to have occurred. On the other hand, CHMSW are concerned with f r i c t i o n s i n the trading process which delay the adjustment of p r i c e s . Consequently, the observed prices are never the true ones and the actual observed returns are an average of the past true returns. This p a r t i c u l a r version of the CHMSW model appears to be inconsistent with negative s e r i a l c o r r e l a t i o n i n the 18 returns. However, i n a more general version, a random variable 9t_^. ^ was included i n the observed return equation to represent the bid-ask spread. 21 t-k,k K t - k ) + e t-k k (1.18) This random va r i a b l e explains the negative s e r i a l c o r r e l a t i o n i n the s e c u r i t i e s returns. The s i x phenomena l i s t e d at the beginning of the chapter are consistent with model (1.18). This was expected since (1.18) was suggested by CHMSW s p e c i f i c a l l y to explain these phenomena. 4. Other Models of F r i c t i o n s Some f r i c t i o n s i n the trading process other than the pri c e adjust-ment delays have received p a r t i c u l a r attention i n other models. Blume and Stambaugh (1983), for instance, take account of the bid-ask spread using both additive and m u l t i p l i c a t i v e p r i c e errors. R o l l (1984) suggests to measure the bid-ask spread by taking twice the squared root of the negative of the s e r i a l covariance of p r i c e changes. Goldman and Beja (1980) study the ro l e of the s p e c i a l i s t s as market makers using a model which accounts f or differences as market makers using a model which accounts f or differences between the equilibrium (true) p r i c e and the observed p r i c e . They assume that the "time path of equilibrium values follow a random walk and that observed market prices follow a continuous stochastic adjustment toward the current equilibrium value." F i n a l l y , i t i s worth noting that the implications of errors i n the measured returns, instead of the p r i c e s , on the estimation of the CAPM parameters have been investigated. Scott and Brown (1980), i n a standard errors i n variables s e t t i n g examine the e f f e c t s of i i d errors i n the r e -turns on LS estimation of systematic r i s k . Assuming a market model r e l a -22 t i o n between the observed returns, where the lagged disturbances are correlated with the current error i n returns (but not the current true market return), and where the disturbances are s e r i a l l y correlated, they show that the LS estimate of 3 can be biased and unstable. 23 CHAPTER II EIP MODEL This chapter generalizes the continuous time market model by assuming that the prices from which the returns are calculated are measured with e r r o r . I t also examines the e f f e c t s of t h i s assumption on the estimation of the model parameters. F i r s t , the continuous time market model i s presented and the adequacy of l e a s t squares (LS) estimation of the parameters i s discussed. The errors i n the pr i c e s are then introduced and three methods of estimation are investigated: the maximum l i k e l i h o o d (ML) method, the LS regression method and the method of moments. Next the EIP model i s extended to i n -clude c o r r e l a t i o n between the errors i n the prices over more than one period, and f i n a l l y i t i s shown how the EIP model explains the s i x empirical phenomena l i s t e d i n Chapter I which are inconsistent with the simple market model,assumptions. 1. Market Model and LS Estimation The model considered i n th i s d i s s e r t a t i o n i s the basic market model 19 i n i t s continuous time form. The pri c e dynamics of the stocks i n the economy are assumed to follow j o i n t d i f f u s i o n processes of the type dP/P = i dt + O dZ (2.1) where P = the current p r i c e of a stock i = the expected rate of return per unit of time, O = the standard deviation of the return per unit of time, 24 dZ = a standard Gauss-Wiener process. This implies that the p r i c e of the stock at the end of T periods. P T , given the current stock p r i c e P , follows a lognormal d i s t r i b u t i o n 20 or equivalently that i t s logarithm (ln) i s normally d i s t r i b u t e d (N) with constant mean and variance and also constant covariance with other stocks' p r i c e s . ln(P /P) ~ N [ ( i - a 2/2)x , a 2T] (2.2) Define the following v a r i a b l e s : p : logarithm of the p r i c e of a given security y at the end of period t, t=l,...,T , pmt : logarithm of the p r i c e of the market index m at the end of period t . Since the continuously compounded rate of return i s given by the logarithm of the p r i c e . r e l a t i v e s , we can write X t = p m t + l " p m t y t = Pt+1 " P t (2.3) (2.4) where from (2.2) (x t , y ) are b i v a r i a t e normally d i s t r i b u t e d (BVN) and are independently and i d e n t i c a l l y d i s t r i b u t e d ( i i d ) over time r \ r U X ~ BVN u I y J 2 a a x xy a a. • xy y for a l l t . (2.5) The marginal d i s t r i b u t i o n of xfc and y are then normal and we can write 25 y t = a + 3xfc + e f c (2.6) where e. ~ N(0,a 2 - a 2 / a2..) t = l,. . . , T E(x„ e ) = 0 for a l l t,s . t s Equation (2.6) follows from the BVN assumption. Consistent with most finance l i t e r a t u r e , i t i s sometimes loosely referred to as the market model. However, i t must be understood that the market model considered here r e f e r s to the generating processes (2.1). Moreover, equation (2.5), implied by (2.1), i s the fundamental s t a t i s t i c a l d e s c r i p t i o n used for estimation purposes. The common estimation method f or the parameters of this model is the LS method. The estimates are e a s i l y computed by regression. In 21 t h i s case, the LS estimates are the same as the ML estimates. 2. Market Model with Errors i n Prices The usual market model was presented i n the preceeding section i n terms of the true returns. However, the true returns cannot be calculated since, as discussed i n Chapter I, the observed prices usually d i f f e r from the true p r i c e s . This i s incorporated i n the market model by w r i t i n g the observed prices as the true prices with an error, •/< pm = pmt + u f c (2.7) P t = P t + V t • ( 2 ' 8 ) For the purpose of ML and LS estimation, we further assume that the vector 2 (u^ , v_) has a BVN d i s t r i b u t i o n with mean (u , u ), variances O and t ' t u v u 2 . a , covariance 0 , and that i t i s l i d over time, v uv 26 u t ~ BVN r 9 \ J I 2 a a u uv a a 2 U V V , for a l l t . (2.9) The observed returns are then, from (2.3) and (2.4) X t = X t + u t + l ' U t y t = y t + v. Hence (2.6) can be written as — v t+1 t (2.10) (2.11) y f c = a + 3xfc + a)t (2.12) where wt • ( v t +i - v - P^t+i-V + et This i s an equation with errors i n the variables (EIV) where the terms 03^ . are not only correlated with the observed independent v a r i a b l e •k x t but are also autocorrelated. Equation (2.12) i s for returns calculated over a unit measuring i n t e r v a l . We now write i t i n a more con-venient way. Define the following vectors and matrices: X : vector of true index returns, Y : vector of true security returns, e : vector of the errors, U : vector of the errors i n the index p r i c e s , V : vector of the errors i n the security p r i c e s , a : vector of parameter a , 27 J : matrix which, f o r the unit measuring i n t e r v a l , gives the corresponding errors i n the v a r i a b l e s . A l l vectors are column vectors; X, Y and a have T components and U and V have T+l components since at least T+l prices are required to c a l c u l a t e T returns. Consequently, J i s T x T + 1 and has the follow-ing form -1 1 0 0 o S \ i S s 1 0 o o s - i s l Equation (2.12) can now be written as ft ft Y = a + BX + JV - BJU + e (2.13) ft where Y = Y + JV X* = X + JU . It w i l l be referred to.as the error i n prices (EIP) equation. The EIP model i s the market model (2.1)-where the prices are observed with erro r . This i s a very general model which includes as s p e c i a l cases many other models. The market model without errors i n the prices and the standard errors i n variables model with independent errors are obviously, as w i l l be shown l a t e r , s p e c i a l cases of the EIP model. It i s shown i n Appendix 9 that the SW model i s i n fact an EIP model where the errors are not normally d i s t r i b u t e d and where a l l SW r e s u l t s can be derived i n an EIP framework. Appendix 10 shows that R o l l ' s measure of bid-ask spread i s also a r e s u l t 28 that can be derived from ah EIP framework. Appendix 11 demonstrates how the EIP model can be used to model pr i c e adjustments delays. The frame-work of the EIP model i s then general and powerful. We now turn to the estimation of the parameters. The f i r s t method that w i l l be discussed here i s the ML method. 3. Maximum Likelihood Estimation Using the EM Algorithm Roughly speaking, the p r i n c i p l e of ML estimation i s to f i n d the values of the parameters which are most l i k e l y to have produced the observed sample. The l i k e l i h o o d function which i s the j o i n t density of the random variables evaluated at the sample values i s maximized with respect to the parameters. The p a r t i c u l a r shape of the covariance matrix of the observed security and market returns prevents from f i n d i n g a simple closed a n a l y t i -c a l s o l u t i o n to the problem of maximizing the l i k e l i h o o d function of the observed returns. We must then resort to numerical methods. The ML e s t i -mates would be e a s i l y obtained i f the true p r i c e s and the errors were observed separately because they have BVN d i s t r i b u t i o n s f or which the ML estimates of the parameters are calculated from simple formulas. Since we only observe the sum of the prices and the errors, we can characterize the EIP model as a missing information model. This suggests the use of the EM algorithm for incomplete data to calculate the ML estimates. In Section 3.1 we describe the covariance structure of the observed returns and the parameters to be estimated and i n Section 3.2 the ML method i f a l l v a r i a b l e s were observed. Section 3.3 presents the EM algorithm which i s applied to the EIP model i n Section 3.4. 3.1 Covariance Structure of the Observed Returns The EIP model implies a - l i n e a r r e l a t i o n between the observed security returns and the observed market returns which are themselves functions of the true returns and the errors i n the p r i c e s . To derive the covariance structure of the observed returns, we f i r s t look at the covariance structure of the true returns and of the err o r s . From the assumption on the p r i c e generating process made i n Section 1, there i s no s e r i a l c o r r e l a t i o n i n the true s e c u r i t i e s and market returns and no s e r i a l c r o s s - c o r r e l a t i o n between the s e c u r i t i e s ' returns and the market returns, i . e . ( y t - u ) ( y s - y ) s = t = 0 S * t (x t - y ) (x t x s = o s = t = 0 (y*. - y ) (x - y ) J t 'y' s ^x = a xy s = t = 0 s + t Similar assumptions have been made for the errors i n the p r i c e s , u and "~ ~ 2 (u. - y )(u - y ) t u s u = a s = t s * t (v. - y ) (v - y ) t v s v = a s = t 30 E (u - y ) (v - y ) = a s = t t u s v uv = 0 s ^ t . Nothing has been said so f a r about the covariance structure between the true returns and the errors i n the p r i c e s . I t i s a somewhat more complicated structure. The true returns i n period t are assumed to be uncorrelated with the erro r i n the prices at t but to be correlated with the error i n the p r i c e at t + l . The ra t i o n a l e f o r t h i s assumption i s as follows. Consider f i r s t a true s e c u r i t y return. I f y i s p o s i t i v e (negative) then the true end of period p r i c e P t + ^ should be higher (lower) than p since from (2.4) Pt+1 = P t + y t '•' However, because of the f r i c t i o n s i n the trading process, t h i n trading or r e l a t e d phenomena, the e f f e c t of the true return for that period w i l l be only p a r t i a l l y r e f l e c t e d i n the observed end of period p r i c e and p t+l which w i l l not be as high (low) as the true p r i c e Pt+-|_ would be. Since the erro r v t + ^ I s given by the difference between P t + ^ a n c ^ P t +2 , i t w i l l be negative. I t i s also l o g i c a l to assume that v t + ^ w i l l be larger i n absolute value, the larger y i s . Suppose, for example, that the only source of measurement error comes from the fact that only a proportion A of the true returns i s r e f l e c t e d i n the observed returns. We would then write y! = Xy,. + ( 1 - A)y t - l 31 or i n terms of the EIP model y t = y t + V * " V-t+1 t where v = (A - l ) y f c v t = (A - 1 ) y t _ 1 • I t i s clear from t h i s example that the true return and the error i n the end of period p r i c e s are correlated. cov(y t , v t + 1 ) = (A - l ) a In t h i s example the covariance i s negative, as expected, since A i s com-prised between 0 and 1. This covariance, however, need not necessarily be negative f o r a l l s e c u r i t i e s . We could construct some examples where the observed p r i c e could be higher than the true p r i c e and a would yv then be p o s i t i v e . Similar reasoning applies to the market returns x and the, errors i n the prices u t+1 F i n a l l y , since y and xfc are rela t e d i n the market model, s i m i l a r covariance structure e x i s t s between y f c and u t + ^ > xfc and v t + ^ • These assumptions can be summarized as follows. (y„ - y ) (v - y ) Jt y s v = a yv s = t + 1 = 0 s / t + 1 (x - y ) (u - y ) t x v s u = a xu s = t + 1 s + t + 1 32 ( y t - y y ) ( u a - y u ) = a yu s = t + 1 = 0 t + 1 (x_ - y ) (v - y ) t x s v = a-XV s = t + 1 = 0 t + 1 From the above discussion, the 4T+2 vector of true prices and observa-t i o n errors R' = (X', Y', U', V ) , where ' denotes the transpose, has the following multivariate normal (MVN) d i s t r i b u t i o n R = X MVN "y T y u • i T ^v ® ^+1 A (2.14) where A = A C C B A = C = a , a x xy a , a xy y a , a xu yu a , a xv yv ® (01)' a , a u uv 2 a , a UV V T+1 33 and ® i s the Kronecker product i ^ , i s a column vector of T ones i s the i d e n t i t y matrix T x T (01) = (0 8 i T , I T ) . Now l e t Z' = (Y*', X*') denote the vector of observed returns, i s shown i n Appendix 1 that the d i s t r i b u t i o n of Z i s the following It Z = f Y* X T MVN y s> i m y T y B i m x T (2.15) where a 2 i + J J ' ( a 2 + a ) y v yv a i+JJ'a +J(0,l)'a +(0,i)J'a xy uv xv yu a I+JJ'a +(0,I)J'a +J(0,l)'a xy uv xv yu a2 l + J J 1 ( a 2 + a ). x u. xu and J J ' has the following form 2 -1 v -1 2: -1-0 -1 '2 0 0 -1 0 0 0 0 0 0 2: : "I -o; •; -1' There are fourteen parameters that could be estimated i n the model, four means, four variances and s i x covariances from which the estimate of 3 i s calc u l a t e d . 34 A major problem usually associated with errors i n variables i s the i d e n t i f i a b i l i t y of the parameters; more than one set of parameters values can give the same d i s t r i b u t i o n of observed returns. This implies that some parameters cannot be disentangled. I t i s the case i n the EIP model 2 2 . . that a and 0" , a and O , are always paired together i n the v yv u xu l i k e l i h o o d function of the observed returns. O f f s e t t i n g changes could 2 be made i n O and O such that d i f f e r e n t values of these parameters u xu would give the same value of the l i k e l i h o o d function. We are then not 2 2 able to separate O and a , a and C ; only t h e i r sum i s l d e n f i -u xu v yv f i e d . The same i s true f o r O and 0 , 0 and a . The other xv uv yu uv 2 2 parameters O , O and a are i d e n t i f i e d . The model's assumptions x y xy can be v e r i f i e d by comparing the sample autocorrelations of observed returns against ft . 22 3.2 Maximum Lik e l i h o o d Method The ML estimates enjoy many nice properties."'' In the class of consistent estimators, f o r instance, the ML estimates are a natural choice since they achieve the Cramer Rao lower bound f o r the variance. On t h i s ground, the ML estimation method i s then at le a s t as good as other consistent estimation methods, such as the SW. The l i k e l i -hood function L(*) of the observed variables i s obtained from (2.15). L(-;X*Y*) = (2TT) 1 / 2 T | f t | 1 / 2 exp <| Y*-y x*-y ' f t "1 Y*-y x*-y x The p a r t i c u l a r form of ft which can be p a r t i t i o n e d i n t r i d i a g o n a l matrices makes the a n a l y t i c a l derivations of ML estimators non t r i v i a l . The EM algorithm f o r incomplete data w i l l be used to compute the ML estimates. As a matter of f a c t , the observed returns can be seen as incomplete data 35 i n comparison to a complete data vector R' = (X',Y',U',V) of true r e -turns and errors i n the p r i c e s . I f t h i s vector R' was a c t u a l l y observed instead of Z then the ML estimates would be e a s i l y obtained under the a d d i t i o n a l assumption that the f i r s t p r i c e of the seri e s are the true p r i c e s . This a d d i t i o n a l assumption implies that u^ and v^ can be dropped and that there are now the same number T of returns and errors i n the p r i c e s . I t w i l l be made f o r a l l matters regarding the ML estima-t i o n method. From (2.14) each p a i r (x t , y f c) , (x t , u 1 ) , ( x t , v ^ , (y f c , u t + ^ ) > (y t 5 v t + l ^ a n c * ^ u t ' v t ^ k-aS a B V N d i s t r i b u t i o n and i s i i d over time. The ML estimates are then computed i n terms of the complete . . . . 23 data s u f f i c i e n t s t a t i s t i c s as follows: y x = I VT ' V ^ y t / T > y u = £ u t / T . i v = r v T ~ 2 £ ( x t - x ) 2 - 2 £ ( y t ~ y ) 2 ~2 ^ u t ~ u ) 2 x 2 £ ( y t _ v ) 2 ' a = , a = , a = , a =—-r^ x T y T U T V T £ (x -x) (y t-y) I (u t-u) (v t-v) „ ^(x f c-x)(u -u) a = , a = , a = xy T uv T xu T I (y t-y) (u + 1 - u ) - I (y t-y) ( v + 1 - v ) • . A £ ( x - x ) ( v t + 1 - v ) a = , a = , a = yu T yv T xv T (2.16) But we do not observe the complete data vector R d i r e c t l y , the observed variables being Y and X . I t i s i n that sense the model can be seen as an incomplete data model and that the use of the EM algorithm to compute the ML estimates from the incomplete data i s j u s t i f i e d . 36 24 3.3 EM Algorithm Each i t e r a t i o n of the algorithm consists of two steps, an expectation step and a maximization step, hence the name EM algorithm. Define R : the complete data vector from the sample space V , Z : the observed data vector from the sample space A , ip .: the vector of parameters, f(R|i|j) : a family of sampling density for the complete data depending on the parameters I J J , g(z|i|;) : the corresponding family of sampling densities for the observed data. The term incomplete data implies the existence of a mapping R to Z(R) from .T to A and that R i s known only to l i e i n T(Z) , the subset of T determined by Z = Z(R) . The complete data s p e c i f i c a t i o n f ( * ) i s rela t e d to the incomplete data s p e c i f i c a t i o n g(*) by • g<z|i|0 = f(x|i|;) dX . r(z) When f(R|i|j) i s of the exponential family form f(R|i|>) = b(R) exp (Tlit (R)')/aOJ>) , where t(R) i s the vector of s u f f i c i e n t s t a t i s t i c s , the EM algorithm goes as follows. E-step At the nth i t e r a t i o n the complete data s u f f i c i e n t s t a t i s t i c s t(R) are estimated by find i n g t h e i r c o n d i t i o n a l expectations given the observed 37 returns, i . e . incomplete data and the previous i t e r a t i o n parameter estimates. t ( n ) = E[t(R) | Z , ^ ( n ) ] . (2.17) M-step The new value of the parameters i f / n + ^ i s taken as the s o l u t i o n to the equations E[t(R) | M = t ( n ) . (2.18) Cn) We note that i f t •represents the s u f f i c i e n t s t a t i s t i c s computed from the complete data then the s o l u t i o n to (2.18) defines the ML estimator of TJ and the parameters are estimated from (2.16). The theory showing the monotone behaviour of the l i k e l i h o o d and the con-vergence of the algorithm i s found i n Dempster et a l . 3.4 A p p l i c a t i o n of the EM algorithm to the EIP model We can write the j o i n t d i s t r i b u t i o n of the complete data and the 25 observed variables using (2.14) and (2.15). X Y U V Y + JV X + JU MVN yy U u 9 I 11 I 12 yy I 21 I 22 (2.19) The c o n d i t i o n a l expectation and variance of the complete data variables 26 given the observed ones are e a s i l y found. 38 X Y U V. Y +JV X +JU u + 1 f 1 12 22 Y + JV - u X + JU - y (2.20) var X Y U V Y +JV X +JU = I -I I 1 I 11 12 22 21 (2.21) We can now proceed with the algorithm. E-step The complete data s u f f i c i e n t s t a t i s t i c s are estimated by find i n g E(I x t | Z ^ ( n ) ) = I E U J Z ^ ^ ) , t t .(n). and E(l x 2 | Z ^ ( n ) ) = I E ( x 2 | Z ^ ( n ) ) = I (Var(x lz,^( n ))+ [ E ( x J z , ^ ( n ) ) ] 2 ) E(X x t y t | z ^ ( n ) ) = I E ( x t y t | z ^ ( n ) ) = I ( C o v ( x t y t | Z 5 ^ ( r l ) ) + E ( x t | z ^ W ) E ( y t | z ^ W ) (n), B, I , ,,,(n). 39 where the RHS of each equation can be obtained from (2.20) and (2.21). We proceed s i m i l a r l y f o r £ y , I y2 I u , £ v £ u2 \ v2 t t c t t t t -I X t Ut+1' I X t V t + l ' I y t U t + l ' J y t V t + l , a n d J U t + l V t + l * t t t t t M-step The estimates of the parameters are calculated from the s u f f i c i e n t s t a t i s t i c s obtained i n the preceeding step. This constitutes one i t e r a -t i o n of the algorithm. S t a r t i n g with i n i t i a l guesses i ] / " ^ f o r the parameters, we obtain from the E step some values t ^ of the s u f f i c i e n t s t a t i s t i c s which (2) are used i n the M step to c a l c u l a t e new estimates ^ of the parameters. The new estimates are then used i n a second i t e r a t i o n and so on. The algorithm i s continued u n t i l the difference between two successive sets of estimates i s small enough, according to some given convergence c r i t e r i o n . The speed of convergence of the algorithm i s a function of the informa-t i o n l o s t due to the missing observations. We then expect i t to be slow for our model. Several methods have been suggested to improve the speed of convergence. In Section 3.2 i t was suggested that the ML estimates should be used preferably to other consistent estimates since they are best asymptotically normal. In t h i s section we have proposed the use of the EM algorithm to f i n d these estimates not only for the systematic r i s k but for a l l the parameters of our model. However, computation constraints can l i m i t the number of observations T that are used with the algorithm. Each i t e r a t i o n requires a number of operations i n v o l v i n g 4T x 4T and 4T x 40 2T matrices and the inversion of a 2Tx2T matrix. Chapter III w i l l present some-computing f i g u r e s . In the next section we look at the LS method which i s commonly used f o r the estimation of systematic r i s k and which does not l i m i t the number of observations to be used. 4. LS Estimation of Systematic Risk The systematic r i s k of a s e c u r i t y i s the market model parameter that f i n a n c i a l e m p i r i c i s t s are most concerned with. Its estimate i s usually calculated from LS regression. The symbol b w i l l always r e f e r to a LS regression estimate of the parameter 3 while the symbol 3 w i l l r e f e r to estimates other than LS. As presented i n Chapter I, the e f f e c t s on b of introducing measurement errors and of varying the measurement i n t e r v a l have been investigated by d i f f e r e n t authors by examining the changes i n i t s asymptotic properties. The small sample properties, however, are more important than the asymptotic properties because i n p r a c t i c e only f i n i t e sample sizes can be used. Furthermore, the i n s t a b i l i t y of 3 over time, as the fundamental c h a r a c t e r i s t i c s of the s e c u r i t i e s evolve, r e s t r i c t s the sample siz e that can be used. In t h i s section, not only the asymptotic properties of b but also i t s approximate small sample properties are derived for our model with errors i n the p r i c e s . The EIP model was presented i n Section 2 for returns measured over a unit i n t e r v a l . Section 4.1 generalizes the EIP equation to accommodate d i f -ferent measuring i n t e r v a l s . In Section 4.2 b i s expressed as a r a t i o of quadratic forms i n mean 0 normal v a r i a b l e s . From t h i s expression, the approximate small sample properties of b are derived i n Section 4.3, 41 and the accuracy of the approximations i s discussed i n Section 4.4. The asymptotic properties of b are also derived i n Section 4.5. Section 4.6 discusses how the Section 4.3 r e s u l t s would be modified i f the returns are not mean zero variables but rather are c e n t r a l i z e d v a r i a b l e s . 4.1 Generalized EIP Equation The EIP equation assumes that the returns are measured over a c e r t a i n unit i n t e r v a l . As mentioned before, i t has been shown for d i f -ferent models that using longer measuring i n t e r v a l s reduces the asymptotic bias of b . Our aim here i s to look at how the small sample properties as w e l l as the asymptotic properties, mean and variance of b , are affected by using d i f f e r e n t measuring i n t e r v a l s , which might even overlap. The f i r s t step i s to write the EIP equation i n a more general way to accommodate the d i f f e r e n t ways to measure the returns. For t h i s purpose, we define the following matrices and v a r i a b l e s . L : matrix which determines how the returns are calculated, K : matrix which, for a given measuring i n t e r v a l , gives the corresponding errors i n the v a r i a b l e s , m : length of the measuring i n t e r v a l or number of periods over which returns are c a l c u l a t e d , k : number of overlapping periods i n two successive returns. A generalized version of (2.13) i s LY* = La + 3LX* + KV - 3KU + Le (2.22) where LY* = LY + KV (2.23) 42 LX* = LX + KU (2.24) K = LJ . (2.25) Each row of L gives a l i n e a r combination of the one period returns ft ft of the vectors X and Y . Since m i s the length of the measuring i n t e r v a l then there are m l ' s per row, and since k i s the number of overlapping periods i n two consecutive returns, there are k l ' s over-lapping for any two consecutive rows. The other elements of L are 0. Each row of K i s determined by the corresponding row of L ; there w i l l be a -1 i n the column of K where the f i r s t 1 appears i n row ' i ' of L ; and there w i l l also be a 1 i n the column immediately following the l a s t column with a 1 i n row i of L . The other elements of K are 0. The following example i l l u s t r a t e s t h i s r e l a t i o n . These matrices correspond to 3 period returns with two periods overlapping. 1 1 l . 0 0 0 -1 0 0 1 0 0 0 0 r. i 1 o 0;' 0 - l 0 0 r o- 0 K = 0 o; l 1 1. 0 q 0 - l 0 0 I 0 .0, 0 0 1 1 1. 0 0 0 -1. 0. 0. 1 This formulation of a "market model" type of r e l a t i o n i s very general. For instance, when L = I then K = J and Y* = a + £X* + JV - 3JU + £ 43 which i s the errors i n the prices equation (2.13). Furthermore, 2 2 i f we set a , a and O equal to 0, then KU and KV are 0 vectors u v uv and we are back to Y = a + BX + e , e ~ N(0 , a 2 I) e which i s the market model equation (2.6) without errors i n the p r i c e s . 4.2 LS Estimator To ease the derivations of the small sample and asymptotic properties of b and sim p l i f y the notation, l e t ' s assume from now on that x , y , u and v are mean 0 v a r i a b l e s . We now rewrite (2.22) i n a form which allows us to express the b conveniently as a r a t i o of quadratic forms i n normal va r i a b l e s . Let A = ( L | K | 0 | 0 ) B = ( 0 | 0 | L | K ) Y * - V X . U - ' Using equations (2.23) and (2.24), we get A<j) = |3B<}> + KV - (3KU + Le . (2.26) The LS estimate i s then calculated as b = (cj)'B'A(j))/((f),B'B(j)) . (2.27) The numerator and denominator of (2.27) are quadratic forms i n normal 44 va r i a b l e s , (j) being a vector of normal variables with mean 0 and covariance matrix W W = (01)0 a yv xy (01)'a 0 1 yv v xy (0i ) a xv x (01)'a a I yu uv (01) a yu (01)'a a I XV uv (01)0-xu (01)'a a I xu u (2.28) It i s convenient f or future purposes to rewrite (j^B'Acj) i n terms of a symmetric matrix. Since <j>'B'A<f) i s a sc a l a r we can write the following cp' B f Ad> = d/A'Bd) = l/2(cj>'A'Bcj)) + l/2(cf>'B'AcJ)) = <J>' (1/2A ,B + 1/2B*A)(J) = <$>'c<$> where C i s symmetric, C = 1/2 0 0 L'L L'K 0 0 K'L K'K L'L L'K 0 0 K'L K'K 0 0 (2.29) and c f i ' B ' B c j ) = cfj'Dd) , where D i s symmetric 45 D = 0 0 0 0 0 0 0 0 0 0 L'L L'K 0 0 K'L K'K (2.30) We can now rewrite the LS estimator of systematic r i s k as b = Q(C)/Q(D) where Q(C) = (j)'C(j) Q(D) = cp' Dcp . 4.3 Approximate Small Sample Properties of b To derive the sample mean and variance of b we would i d e a l l y use i t s d i s t r i b u t i o n . However, since the exact d i s t r i b u t i o n has not been found, we use the following approximations f o r the mean and variance of b i n terms of the mean, variance and covariance of the two quadratic forms Q(C) and Q(D) . The accuracy of these approximations i s d i s -cussed i n the following section. Taking Taylor seri e s expansion of Q(C)/Q(D) around the mean of Q(C) and Q(D) , dropping terms of order higher than two and taking expectations on both sides we f i n d E[Q(C)/Q(D)] ^ E[Q(C)]/E[Q(D)] - Cov[Q(D),Q(D)]/E 2[Q(D)] + Var[Q(D)] E [Q(C)]/E [Q(D)] (2.31) and proceeding s i m i l a r l y f o r the variance we get Var[Q(C)/Q(D)] ~ {E[Q(C)]/E[Q(D)]} {T} 46 where T = Var[Q(C)]/E 2[Q(C)] + Var[Q(D)]/E 2[Q(D)] - 2Cov[Q(C);, Q(D)]/E[Q(C)] E [Q(D) ] } (2.32) Since the cumulant generating function i s known for these quadratic forms, we f i n d t h e i r mean and variance from the f i r s t and second cumulants r e s -p e c t i v e l y (see Appendix 2). We know that and since Hence, E[Q(C)] = tr(WC) E[Q(D)] = tr(WD) Var[Q(C)] = 2tr(WCWC) var[Q(D)] = 2tr(WDWD) . Cov[Q(C),Q(D)] = {Var[Q(C)+Q(D)] Q(C) + Q(D) = Q(C+D) <f)'C<}) + cj>'D(t> = cj>' (C+D)cj). (2.33) (2.34) (2.35) (2.36) - Var[Q(C)] - Var[Q(D)] }/2 (2.37) (2.38) Cov[Q(C),Q(D)] = tr{[W(C+D)] 2} - tr[(WD) 2] - tr[(WC) 2] tr[WCWC+WCWD+WDWC+WDWD] - tr[WCWC] - tr[WDWD] 2tr(WCWD) . (2.39) From these i t i s easy to f i n d expressions for the mean and variance of the LS estimate of B E[b] ~ [tr(WC)/tr(WD)] - 2[tr(WCWD)/tr2(WD)] + 2[tr(WC) t r (WDWD)/tr (WD)] (2.40) 47 Var[b] ~ {tr(WC)/tr(WD)} 2{2[tr(WCWC)/tr 2(WC)] + 2[tr(WDWD)/tr2(wD)] - 4[tr(WCWD)/tr(WC)tr(WD)]} . (2.41) It i s shown i n Appendix 3 that the d i f f e r e n t traces can be written i n the following ways tr(WC) = 0 tr(L'L) + [a + (a +a )/2] t r (K'K) , xy uv yu xv ' tr(WD) = a 2 t r ( L ' L ) + (a 2 + O ) t r (K'K) , X u xu tr(WCWC) = l/2{tr(L'LL'L)(a a + 0 2 0 2 ) + xy xy x y tr(L'KK'L) (a (2a +a +a )+a 2(a 2+a )+a 2(a 2+a ) _ xy uv yu xv x v yv - y u x u + tr(K'KK'K) (a + (a +a )/2) 2+(a 2+a ) ( a 2 + a ) uv yu xv v yv u xu tr(WDWD) = a 2 a 2 t r (L'LL'L) + 2a 2 (a 2 + a ) t r (L'KK'L) + X X X u. x u (a 2 + a ) 2 t r (K'KK'K) , u xu 2 ? tr(WCWD) = a a t r (L'LL'L) + (a (a + a ) + (a + xy x xy u xu uv (a +a )/2)a) t r (L'KK'L) + xv yu x (a +a )(a + (a +a )/2) t r (K'KK'K) u xu uv yu xv For given L , and consequently K , we can evaluate the traces and substitute i n (2.40) and (2.41) to f i n d E(b) and Var(b) i n terms of the d i f f e r e n t covariances of the EIP model. 48 It should be noted that since a and O , a and a always u xu v yv appear as a sum i n these formulas, we do not need to evaluate them separately. This i s somewhat reassuring since, as mentioned i n Section 2, only the sums are i d e n t i f i a b l e , not the parameters separately. 4.4 Accuracy of Approximate Small Sample Properties It would be a formidable task to estimate the information l o s t by dropping the terms higher than 2 i n the Taylor ser i e s expansion. We can nevertheless f i n d some reassurance by looking at two standard cases where the exact small sample properties of b are known. As mentioned e a r l i e r , the usual LS s e t t i n g without errors i n variables can be obtained i f we l e t L = I 2 2 and a = a = o = o = a = a = o = 0 u y uv xu xv yu yv i n our model. The formulas (2.40). and (2.41) f or the expectation and the variance y i e l d E [ b ] - ~ a / a 2 — xy x and Var[b] ~ a 2 / ( T - l ) a 2 ; which are usual expressions for the mean and variance of b . •' A. standard errors i n variables model i s obtained from the generalized EIP equation by se t t i n g a = a = a = a = 0 xu xv yu yv 49 u l " v l = 0 L = I K = I . 29 Exact expressions f o r the mean and variance of b f o r t h i s case have been derived E[b] = (a + a )/(a 2 + a 2) xy uv x u Var[b] = T-2 v 2 , 2 a + a X u a 2 2 a a X u a 4 ( a 2 + a 2 ) 2 X X u It can be v e r i f i e d that the same expressions are obtained from (2.40) and (2.41) r e s p e c t i v e l y , except f o r the denominator of (2.41) where T appears instead of T-2. 4.5 Asymptotic Properties of b For purposes of comparisons with other models, the asymptotic bias of b i s derived. This allows us to investigate the circumstances under which b i s consistent. It i s also shown how the asymptotic variance could be calculated. 4.5.1 Asymptotic bias As already mentioned, the LS estimators are inconsistent when the EIV problem i s present: -1 plimb = plim[((LX*)'(LX*))/T] - [((LX*) 1(LY*))/T] = plim[((LX*)'(LY*))/T]/plim[((LX*)'(LX*))/T] 50 = plim[((LX+KU)'(LY+KV))/T]/plim[((LX+KU)'(LX+KU))/T] = {plim(X ,L ,LY)/T+plim(X ,L ,KV)/T+plim(U'K ,LY)/T+plim(U ,K ,KV)/T} /{plim(X'L 1 LX) /T+plira(X'L'KU) /T+plim(U'K' LX) /T+plira(U 1K' KU) /T} It i s shown i n Appendix 4 that we can go from p r o b a b i l i t y l i m i t s to l i m i t s of expected value and vice-versa and write plim[(X'L'LX)/T] = limE[(X'L'LX)/T] = limE[tr(X'L'LX)/T] T-x>° = limE[tr(XX'L'L)/T] = limtr(a 2L'L/T) m x = l i m a 2 tr(L'L)/T , (2.42) T-x» X plim[(X'L'KU)/T] = limE[tr(UX'L'K)]/T T-*x> = l i m t r [ a (01)'L'K]/T lima . tr [ J ( 0 I ) ' L ' L ] / T T-x» X u lima tr[JJ'LL']/2T m xu lima tr[KK']/2T m xu We proceed s i m i l a r l y f o r the other quadratic forms. 51 plim[(X'L'LY)/T] plim[(Y'L'LY)/T] plim[(U'K'KU)/T] plim[(U'K'KV)/T] plim[(V'K'KV)/T] plim[(X'K'KV)/T] plim[(Y'K'KU)/T] < 4.' plim[(Y'L'KV)/T] We know that T T a) tr(L'L) = £ £ L f , i = l j = l 1 b) there are 'm ones' per row of L , where m i s the length of the measuring i n t e r v a l , and c) the number of rows i n L i s given by the greatest integer i n (T-k)/(m-k) , so that tr(L'L) = [(T-k)/(m-k)] m , (2.43) and consequently l i m t r (L'L)/T = O 2 m/m-k . (2.44) T-H» X = lima t r [ L L ' ] / 2 T = lim a 2 t r [ L L ' ] / 2 T x-w y = lima 2tr[K'K]/2T = lima tr[K'K]/2T T - M X > u v = lima 2tr[K'K]/2T I-HX> V = lima tr[K'K]/2T = lima tr[K'K]/2T x-*» y u = lima tr[K'K]/2T . x^oo y v 52 Since the matrix K has the same number of rows as L and since there i s a -1 and a +1 on each row, we can use the same reasoning to write lim t r (KK')/T = a 2 2/m-k . (2.45) X—oo U The following expression f o r the asymptotic b i s then obtained (2.46) According to (2.46) we would have consistency of the LS estimator i n the following s i t u a t i o n s . 2 i ) When a = a = o = a = a =0: t h i s i s the obvious case uv xv yu u xu without EIV, where b i s unbiased. 2 i i ) When O =-(a + O )/2,a = -a ; this i s the case without" uv xv yu u xu s e r i a l c o r r e l a t i o n i n X" or between X* and Y* . The covariance between the returns and the errors i n the prices counterbalances the e f f e c t of the er r o r s . This can be shown to imply that the error term i n the EIP model w i l l be uncorrelated with X* so the LS estimates are con-s i s t e n t (Appendix 5). i i i ) When the measuring i n t e r v a l 'm' tends to i n f i n i t y ; Cohen et a l . (1983b) also found for t h e i r model that the LS estimators become consistent when the d i f f e r e n c i n g i n t e r v a l i s increased to i n f i n i t y . 4.5.2 Asymptotic variance I t i s shown i n Appendix 6 how the asymptotic variance can be de-rived . Because of i t s complex form, the f i n a l r e s u l t i s not presented here. 53 4.6 Small Sample Properties of b Using Centralized Variables In general, we do not expect the returns to have zero mean. To r e t a i n the convenient properties of the mean zero variables i n th i s chapter, we ce n t r a l i z e the data by subtracting the sample mean from the observations. To do so we multiply the observed returns by the following T by T idempo-tent matrix M M = (I - i i ' / T ) . The equation' (2.19) can be written i n terms of the ce n t r a l i z e d observed variables MY* = gMX* + MJV - 3MJU + Me , (2.47) since Ma = 0 and the general EIP equation becomes LMY = BLMX + LMJV - 3LMJU + LMe (2.48) Proceeding as i n the preceeding sections we define A = ( L | L | 0 | 0 ) B = ( 0 I 0 I L I L ) cj) = • MY MJV MX MJU where § i s a mean zero vector with covariance matrix W = Ma y MJ(OI)'Ma. Ma xy MJ(OI)'Ma M(0I)J'MO yv yv a2 l V M(0l)J'Ma xv Ma xy MJ(OI)'Ma Ma2 M(0I)J'MO yu xv a I uv M(0I)J'Ma xu yu a I uv MJ(OI)'Ma xu a2 i u (2.49) 54 Following Appendix 3 i t i s shown i n Appendix 7 that the same r e s u l t s as those found i n the previous sections hold with the following modifications: mean 0 variables t r LL' t r KK' t r L'LL'L t r K'LL'K t r K'KK'K cen t r a l i z e d variables t r ML'L t r K*'K* t r ML'LML'L t r K*'LML'K* t r K**K*K*'K* where K* = LMJ 4.7 I l l u s t r a t i o n s of the Small Sample Properties In t h i s section we present some i l l u s t r a t i o n s of the approximate small sample bias and MSE of b for s i x d i f f e r e n t sets of parameters and two d i f f e r e n t numbers of observations. These values are calculated from the formulas f o r c e n t r a l i z e d v a r i a b l e s . It can be v e r i f i e d , however, that the difference between the values f o r mean zero variables and for cen t r a l i z e d v a r i a b l e s i s minimal. The sets of parameters correspond to the four possible combinations of a security that i s e i t h e r f a t or t h i n , borrowing from the thi n trading vocabulary, with an index that i s e i t h e r f a t or t h i n . The values of para-meters f o r the d i f f e r e n t s e c u r i t i e s and indexes were chosen to be representa-t i v e of returns c o r r e l a t i o n s for some NYSE s e c u r i t i e s and indexes. Similar parameter values w i l l be used f o r the simulations i n Chapter I I I . The two numbers of observations that are used with each of the six sets of para-meters are s i x t y and one hundred and twenty. Sixty was chosen because 55 i t i s common i n empirical studies to use f i v e years of monthly returns to estimate 3 . It was decided to also use twice that number of observations because one advantage of the LS estimation method over the EM algorithm i s that i t permits the use of more observations. The r e s u l t s are presented i n Tables 2 to 5, one for each of the d i f -ferent sets of parameters. Each table consists of two sections, a and b, for one hundred and twenty and s i x t y observations r e s p e c t i v e l y . Section a presents the MSE and bias i n a v e r t i c a l sequence f or measuring i n t e r v a l s m ranging from 1 to 6 basic units and the number of overlapping periods k varying from 0 to 5, while f o r section b the measuring i n t e r v a l goes from 1 to 4 and the overlapping periods from 0 to 3. The values of the para-meters used i n the i l l u s t r a t i o n s are given i n Table 1. The following observations can be made from the inspection of tables .2 to 5.. - For a given number of overlapping periods . k , as the measuring i n t e r v a l m increases, the bias decreases and the variance increases. The ef f e c t on the MSE depends on the r e l a t i v e s i z e of the change i n the bias and the variance. For the parameters values used here, the increase i n the variance i s larger than the squared decrease i n the bias so the MSE increases, with one exception. For the f a t security and the thi n index, as we go from 1 period returns to 2 period returns with no overlap, the MSE a c t u a l l y decreases. For a fi x e d measuring i n t e r v a l m , as we increase the number of overlapping periods k , the variance decreases and the bias increases. However, the increase i n the bias i s so small compared to the decrease i n the variance that the MSE always decreases. 56 TABLE 1 Parameters Values Used i n the I l l u s t r a t i o n s  of the Small Sample Properties Table 2 Fat Security Fat-Index Table 3 Thin Security Fat Index Table 4 Fat Security Thin Index Table 5 Thin Security Thin Index .0018 .0018 .0018 .0018 .0063 .0063 .0063 .0063 .0018 .0001 .0018 .0001 .0018 .0001 ,0018 .0001 .0007 .0024 .0007 .0024 .00006 .00006 .00006 ,00006 -.000175 -.000175 -.00035 -.00035 -.00002 -.0004 -.00002 -.0004 .00016 -.00016 -.00016 -.00016 -.0004 -.0004 -.0004 -.0004 TABLE 2 Small Sample Properties f o r a Fat Security and a Fat Index 0 m 1 2 3 4 5 6 . 0 2 9 * .048 .069 .091 .113 1.36 + . 0 5 5 * * + .027 + .018 +.013 + .011 + .009 1 .036 + .027 .056 + .018 .077 + .013 .099 + .011 .122 + .009 2 .048 + .018 .068 + .013 .089 + .011 .111 + .009 3 .062 + .014 .081 + .011 .102 + .009 4 .077 + .011 .095 + .009 5 .092 + .009 2 3 4 0 .055 .097 .142 .188 + .056 + .027 + .018 + .014 I .072 + .028 .115 + .018 .160 + .014 2 .099 + .018 .141 + .014 3 .129 + .015 Notes; * MSE ** BIAS TABLE 3 Small Sample Properties for a Thin Security and a Fat Index 0 m l 2 3 4 5 6 . 0 7 0 * - . 1 7 8 * * .074 -.086 .099 -.057 .126 -.043 .155 -.034 .184 -.029 1 .056 -.086 .080 -.057 .107 -.043 .135 -.034 .165 -.029 2 .068 -.058 .093 -.043 .121 -.035 .149 -.029 3 .084 -.044 .109 -.035 .136 -.029 4 .102 -.036 .127 -.030 5 .121 -.030 2 3 4 0 ,111 -.179 .145 -.087 .200 -.058 .260 -.044 1 .107 -.089 .162 -.059 .220 -.045 2 .136 -.061 .191 -.046 3 .172 -.047 es: MSE BIAS TABLE 4 Small Sample Properties f o r a Fat  Security and a Thin Index *1 0 m x 2 3 4 5 6 .155* +.344** .072 + .145 .077 + .093 .092 + .068 .109 + .054 .128 + .045 1 .059 .064 .079 .096 .115 + .147 + .093 + .069 + .055 + .045 2 .057 + .095 .070 + .070 .087 + .055 .105 + .046 3 .065 .080 .096 + .071 + .056 + .046 4 .075 .091 + .057 + .047 .087 5 + .048 2 3 4 0 .196 + .349 .125 + .148 .150 + .095 .186 + .071 1 .100 + .151 .124 + .097 .160 + .072 2 .109 + .100 .141 + .074 3 .130 + .076 Notes: * MSE ** BIAS TABLE 5 60 a) T=120 k 0 b) T = 60 Small Sample Properties for a Thin  Security and a Thin Index m x 2 3 4 5 6 .048* .072 .097 .123 .149 .176 .047** .020 .013 .009 .007 .006 .053 .079 .104 .130 .157 .020 .013 .009 .007 .006 .066 .090 .117 .142 .013 .009 .008 .006 .081 .105 .130 .010 .008 .006 .098 .121 .008 .006 .115 .007 2 3 4 0 .096 .048 .147 .020 .201 .013 .256 .010 1 .107 .021 .163 .013 .218 .010 2 .136 .014 .188 .010 3 .170 .010 Notes: * MSE ** BIAS 61 For a given index, as we go from a f a t sec u r i t y to a t h i n s ecurity the value of b goes down; i f i t s bias i s p o s i t i v e i t w i l l be reduced and might even become negative. Its variance increases. For a given sec u r i t y , as we go from a f a t index to a t h i n index, the value of b i n -creases so i t s bias increases i f i t i s p o s i t i v e and might decrease i f i t i s negative. I t s variance also increases. 5. Method of Moments It i s easy to v e r i f y (see Appendix 8) that f o r the EIP model the true systematic r i s k can be written i n terms of the observed returns parameters as follows: 3 = a / a 2 xy x = [cov(x* , yp + c o v ( x * + 1 + y*) + cov(x*_ 1 , y * ) ] / [Var(x*) + 2cov(x* , x * + 1 ) ] . (2.50) ft R e c a l l i n g that the variance of xfc i s the same for a l l t , we can divide the numerator and the denominator of (5.1) by Var(x*) to obtain the following r e l a t i o n between the true beta and some observed returns betas 3 = [ 3 ^ + 3* + 3 * 1 ] / [ i + 2p] (2.51) ft where 3_^  i s the regression c o e f f i c i e n t of the observed s e c u r i t i e s returns on the l a g one observed market return, ft 3_|_-^  i s the regression c o e f f i c i e n t of the observed s e c u r i t i e s returns on the lead one observed market returns, 62 p = cov(x; :* , x * + 1 ) / [ v a r ( x * ) Var (x* + ]_) ] 1/2 i s the one period s e r i a l c o r r e l a t i o n c o e f f i c i e n t i n the observed market returns• A convenient estimation method for the true systematic r i s k i s to replace the observed returns covariances i n (2.50) by t h e i r sample moments estimates. Since the sample moments estimates are consistent, t h i s gives a consistent estimate of 3 . Doing so (2.51) becomes We recognize here the SW estimate of systematic r i s k . This i s not s u r p r i s -ing since, as shown i n Appendix 1, the SW model i s i n fact a s p e c i a l form of the EIP model. We also recognize here the CHMSW estimate when a l l the weights f o r lags greater than one are zero. Note that the method of moments estimation method does not require the errors i n the prices to be normally d i s t r i b u t e d . 6 . Extended EIP Model In the s p i r i t of Fowler and Rorke (1983) and CHMSW (1983) we can extend the EIP model and assume that the c o r r e l a t i o n between the returns and the errors i n the prices p e r s i s t s f o r more than one period and that there i s s e r i a l c o r r e l a t i o n and cross c o r r e l a t i o n i n the er r o r s . Suppose, for example, that i t p e r s i s t s f o r two periods. The following covariances would then be added to the EIP model. 3 = {b*± + b* + b*^} / {1 + 2p*x> . cov(x t , u t + 2 ) = a xu+2 cov(x t , v t + 2 ) = 0 xv+2 63 cov(y t , u t + 2 cov(y t , v t + 2 cov(u t , u t + 1 cov(v t , v t + 1 cov(u t , v t + 1 cov(v t , u t + 1 = a yu+2 = a yv+2 = a uu+1 a w+1 = a uv+1 = a vu+1 This has serious consequences f o r the d i f f e r e n t estimation procedures. 6.1 ML Estimation of the Parameters These covariances are e a s i l y added to the general covariance structure. If the EM algorithm was used the E step would proceed as i n Section 2 but the M step would not since the exact ML estimate of the new covariances l i k e cov(u , u ,..) and cov(v_ , v ...,) do not have a simple closed form, t t+l t t+l K I t would then be preferable to maximize d i r e c t l y the l i k e l i h o o d func-t i o n of the observed returns, and use the invariance property of the ML estimators to solve for the ML estimates of the true parameters. However, the required computations are a disadvantage for t h i s estimation method. The method of moments i s much more convenient. 6.2 Estimation of 3 from the Method of Moments It can e a s i l y be shown, proceeding as i n Appendix 8, that i f the co-variances between the returns and the errors p e r s i s t over n periods, the true systematic r i s k can be expressed as follows i n terms of moments of observed returns. 64 n n * * P = I c o v ( y ^ x * + k ) / I cov(x* , x* t + k) . (2.52) k=-n k=-n Replacing the d i f f e r e n t covariances by t h e i r sample moments estimates would give a consistent estimate of 3 3 = ( V * 1 / . L +k l k = - n bx+k k=-n This estimate i s i d e n t i c a l to the CHMSW estimate even though the CHMSW model i s not, contrary to the SW model, a s p e c i a l form of the EIP model. Appendix 11 explains how errors i n prices could be used to model the pr i c e adjustments delays and how i t would d i f f e r from the CHMSW model. 7. EIP Model and Stock Prices Behaviour The s u p e r i o r i t y of the EIP model over the market model rests on i t s a b i l i t y to explain the empirical phenomena inconsistent with the market model. These phenomena were r e l a t e d i n Chapter I to the presence of thin trading and f r i c t i o n s i n the trading process. An i n t e r v a l l i n g e f f e c t was also noted. This section v e r i f i e s how we l l the EIP model can explain these empirical phenomena. Each of the s i x phenomena i s f i r s t summarized; an equation then i l l u s t r a t e s why i t i s inconsistent with the market model and f i n a l l y i t i s explained how the EIP model accounts f o r i t . 1) S e r i a l c o r r e l a t i o n i n d a i l y s e c u r i t i e s returns, negative f or t h i n s e c u r i t i e s and p o s i t i v e f o r f a t s e c u r i t i e s , decreasing as the measur-ing i n t e r v a l increases. 65 Market model: EIP model: Cov(y t , y t _ 1 ) / [ v a r ( y t ) var ( y f c _ 1 ) ] = 0 cov(y* , y * ^ ) = - ( q v + V [var(y*) var (y*_ 1> ] h m a 2 + 2(aJ + a ^ ) For t h i n s e c u r i t i e s , i t i s expected that the variance of the errors i n the prices i s more important than the covariance between these errors 2 and the market returns, i . e . 0 i s expected to be l a r g e r i n absolute value than O ; the s e r i a l c o r r e l a t i o n should consequently be negative. 2 For thick s e c u r i t i e s the errors variance i s l i k e l y to be close r to zero so a w i l l be the source of s e r i a l c o r r e l a t i o n which should then yv be negative. Another way to look at t h i s would be to consider a regression of the true s e c u r i t y returns on the errors i n the p r i c e s . For thick stocks, the v a r i a b i l i t y i n the returns i s expected to be greater than i n the errors i n p r i c e s , so the regression c o e f f i c i e n t i s expected to be greater than one i n absolute value and consequently a i s also expected 2 to be greater than O i n absolute value. Similar reasoning could be ° v made for t h i n stocks. As the length of the measuring i n t e r v a l (m) i n -creases, i t i s c l e a r from the EIP r e l a t i o n above that the r e l a t i v e impor-tance of the errors decreases and so does the s e r i a l c o r r e l a t i o n . 2) P o s i t i v e s e r i a l c r o s s - c o r r e l a t i o n s between s e c u r i t i e s and market returns, decreasing as the measuring i n t e r v a l increases. 66 Market model: EIP model: cov(y f c , x t _ 1 ) / [ v a r ( y t ) var (x -j^ ) ] = 0 cov(y , x - ) - ( 0 + o ) Jt t-1 uv xv' [ v a r ( y j var (x* J ] 3 * {[m02 + 2(o2+0 ) ] [m a2+2 (a2+a ) ] }** •'t t-1 y v yv x u xu The covariance between the market p r i c e errors and the s e c u r i t i e s prices errors O w i l l usually be p o s i t i v e since a i s negative, uv xu a i s negative, and O i s usually p o s i t i v e . However, i t i s expected yv xy to be smaller than the absolute value of the covariance between the s e c u r i t i e s p r i c e errors and the market returns a since the s e c u r i t i e s xv errors are more c l o s e l y r e l a t e d to the market movements than to the errors i n the market p r i c e . I t i s also c l e a r from the EIP equation that the cross-correlations decrease as the m increases. 3) P o s i t i v e s e r i a l c o r r e l a t i o n i n market returns, decreasing as the measuring i n t e r v a l increases and as the importance of the t h i n securi-t i e s i n the index decreases. Market model: 1^ cov(x t , x t_ 1)/.[var(x t) var ( x t_ 1) ] 2 = 0 EIP model: ft * 2 cov(x , x ,) -(a + a ) t t-1 u xu ft ft 2 2 [var (x ) var (x .,)] ma + ( a + a ) t t-1 x u xu 6 7 To explain the p o s i t i v e s e r i a l c o r r e l a t i o n , the market can be seen as a very thick security and the Fisher e f f e c t can be invoked. An index giving less weight to t h i n s e c u r i t i e s w i l l have a smaller error component 2 so (a + a ) w i l l be smaller i n absolute value and consequently lower u xu -i y s e r i a l c o r r e l a t i o n w i l l be observed. The presence of an i n t e r v a l l i n g e f f e c t i s explained as i n 1 and 2. 4) Autocorrelation of market model re s i d u a l s , weakly p o s i t i v e f o r d a i l y data but becoming predominantly negative as the measuring i n t e r v a l increases. Market model: EIP model: c o v ( y t - b x t , y t _ 1 - b x t _ 1 ) / [ v a r ( y t - b x t ) v a r ( y -bx = 0 cov(y t -b x f c , y t _ 1 -b x ^ ) ft ft ft ft ft ft [var(y f c - b x f c) var ( y t _ 1 - b x ^ ) ] --(a + a ) -b (a +a ) + b (2a +a +a ) v yv u xu uv xv yu ft ft ft ft ft" ft I [ v a r ( y t ~ b x t> var ( y t _ 1 ~ b x )] ft ft ft where b i s calculated from y and x , 2 i ) (a + a ) > 0 for t h i n s e c u r i t i e s v yv *? ? i i ) b (a + a ) < 0 u xu i i i ) b*(2a + a + a ) < 0 . uv xv yu We would expect the e f f e c t of i ) to be larger than the e f f e c t of i i ) and i i i ) combined and consequently the autocorrelation of the EIP residuals to be negative. The p o s i t i v e autocorrelation f o r d a i l y data residuals seems 68 inconsistent with the EIP model. However, i t i s reasonable to suspect that the extended EIP model might hold for d a i l y data, i n which case some cor r e l a t i o n s would have to be added to the autocorrelation of r e s i d u a l s . These terms are predominantly p o s i t i v e and could explain the p o s i t i v e autocorrelation of residuals f or d a i l y data. 5) Thin s e c u r i t i e s ' LS estimates of beta r i s e and very high value (thick) s e c u r i t i e s ' LS estimates of beta f a l l , as the measuring i n t e r v a l increases. Market model: 3 = a lo 2 xy x EIP model: .u n f * \ m  0 + 2 a + 0 + a o = Lov(,x y y ) _ xy uv xv yu * 2 2 var(x ) m a + 2 ( a + a ) x u xu A good way to understand how t h i s behaviour of the LS estimate of 3 i s consistent with the EIP model i s to present a numerical example. Con-sider a t h i n s e c u r i t y with a true beta of 1, and assume the following values for the parameters: ma = 10, 2a + a + a = -2 xy uv xv yu ma2 = 10, 2(a 2 + a. ) = -1 . X u xu For a u n i t measuring i n t e r v a l (m=l) the LS estimate of 3 i s 8/9, smaller than the true 3 • Consider a s i m i l a r thick security with 2a + a + a = -.5. The uv xv yu LS estimate of 3 i s 9.5/9, which i s greater than the true 3. As m i n -creases, the estimate of 3 w i l l f a l l f o r the thick security and r i s e for 69 the t h i n s e c u r i t y , which i s consistent with what i s observed. 6) Increase i n the market model R 2 as the measuring i n t e r v a l i n -creases, with a larger e f f e c t f o r t h i n s e c u r i t i e s . Market model: „2 D2 2, 2 R = 3 a la x y EIP model: *2 = *2 { a x + 2 < g u + a x u ) / m } K - P ~ -( a z + 2 ( a z + a )/m} y v yv We know that the variance of the observed market returns increases with the measuring i n t e r v a l . F o r . t h i n s e c u r i t i e s the estimate of 3 i n -creases, and the variance of the observed s e c u r i t y returns decreases so * 2 the estimate-of - R increases. For very thick s e c u r i t i e s the estimate of 3 decreases, and the variance of the observed security returns increases. If these combined e f f e c t s are stronger than the increase i n the market returns variance we could observe a decrease i n the estimate of R These s i x empirical phenomena inconsistent with the market model, which some authors have t r i e d to explain by modeling the market micro-structure, can then be explained with the EIP model. 8. Conclusion In t h i s chapter the market model was generalized to take account of measurement errors i n the p r i c e s . The r e s u l t i n g EIP model has been shown to be a very general model. For instance, the SW model of t h i n trading and R o l l ' s model of bid-ask spread were shown to be s p e c i a l forms of the 70 EIP model. I t was also shown how the EIP model can be used to model the f r i c t i o n s i n the trading process. In contrast to the SW, Dimson and CHMSW studies, which only presented a consistent estimator of systematic r i s k , three estimation procedures were examined for the EIP model. Assuming that the errors are normally d i s t r i b u t e d , the EM algorithm was suggested to f i n d the ML estimates f or a l l parameters of the model. The small sample and asymptotic properties of the LS estimate of B were derived for any ways of measuring the returns. A consistent method of moments estimation procedure was investigated for the estimation of systematic r i s k . An extended version of the EIP model was also presented; the choice between the EIP model, i t s extended version or other models can be j u s t i f i e d by inspection of the sample autocorrelations and cross-correlations of lag larger than 1 f o r the sequences of observed s e c u r i t i e s and market returns. F i n a l l y , i t was shown how the EIP model could explain some phenomena which were inconsistent with the market model. 71 CHAPTER III APPLICATIONS OF DIFFERENT 3 ESTIMATORS TO SIMULATED AND REAL DATA The purpose of t h i s chapter i s to compare the estimators of systematic r i s k calculated from s i x d i f f e r e n t estimation methods. These s i x methods are: - Maximum Likelihood, - Scholes William, - Scholes William Extended, - Least Square 1/0, - Least Square 2/1, - Least Square 3/2. The notation m/k ref e r s to the returns measured over m basic periods with k periods overlapping. The chapter consists of two parts; i n the f i r s t part the estimation methods are applied to return observations simulated from three d i f f e r e n t p r i c e generating processes, while i n the second they are applied to return observations taken from the NYSE. The s i x methods of estimating 3> which were presented i n greater d e t a i l i n previous chapters,are summarized here. A l l estimation methods are con-sidered assuming that the EIP model i s the actual p r i c e generating process. ML Estimator The EM algorithm described i n Chapter II i s used to compute the ML estimates of the EIP model parameters from which an estimate of 3 i s calculated. 72 The i n i t i a l values for the algorithm are calculated from the sample moments of observed returns. Let T T T-1 a = I ( x . - x ) Z/T c = I ( x . - x )(y.-y )/T e = £ (y.-y ) ( y i + 1 ~ y - )/T-l T ' T-1 T-1 „ V« - * -2 , r * - * * ft - f t ft - f t b = I (y.-y/> ./T d = £ (x.-x )(x. + 1-x )/T-l f = £ (x. + 1-x .)(y.-y )T-1 T-1 J - v - L ft _ft ft _ft 8 = 2. ( x i - x ) ( y 1 + 1 _ y ) / T - I . Recalling from (2.15) the r e l a t i o n between the covariance of observed returns and the model parameters, the i n i t i a l values are calculated as follows: I f d > 0 I f d < 0 If f > 0 a . = -3/2 d a = 1/2 d a -3/2 f xu X U X V a 2 - 1/2 d a 2 = -3/2 d a = 1/2 f u u uv I f f < 0 I f e > 0 I f e < 0 a = 1/2 f a -3/2 e a = 1/2 e xv yv yv a = -3/2f a 2 = 1/2e a 2 = -3/2e U V V V a = -(g + a ) a = I b + 2 e I yu uv y 1 ' a =|a + 2 d | a = I c + f + g'l . x 1 1 xy 1 b A The choice of the i n i t i a l values f o r the EM algorithm i s of importance because 'inconsistent' i n i t i a l values prevent the algorithm from converging. 73 By 'inconsistent' values we mean values of the covariances which do not produce p o s i t i v e d e f i n i t e covariance matrices. The s t a r t i n g values are v e r i f i e d at the beginning of the algorithm to ensure that the resultant estimated covariance matrix of the vector R of true returns and errors i n the prices i s p o s i t i v e d e f i n i t e . This i s done by inspecting the c o r r e l a -t i o n matrix and s c a l i n g down the covariances as required to make sure that a l l c o r r e l a t i o n c o e f f i c i e n t s are between minus one and one. This procedure i s used f o r i t s s i m p l i c i t y . Since the observed returns are a l i n e a r combination of the true returns and the errors i n the p r i c e s , the estimated covariance matrix of the observed returns w i l l also be p o s i -t i v e d e f i n i t e . The other covariance matrix which must be v e r i f i e d for p o s i t i v e definiteness i s the c o n d i t i o n a l covariance matrix of the true returns given the observed returns and the i n i t i a l values of the parameters. A singular value decomposition of t h i s matrix i s done at the f i r s t i t e r a -—16 _3 t i o n . Eigenvalues of order 10 or smaller, and 10 or larger were found f o r the range of parameters used i n t h i s d i s s e r t a t i o n . Because of the small number of non zero elements i n t h i s matrix, the eigenvalues of —16 order 10 were deemed to be caused by the numerical approximations used i n the decomposition. I f negative eigenvalues were found, new i n i t i a l values were assigned to a l l the covariances by d i v i d i n g them by 10 while the variances values were kept the same, and the tests were repeated. These changes i n the i n i t i a l values have proved to be successful i n a l l cases where they were required, that i s about twenty percent of the time, mostly f o r the EIP generations. These procedures of v e r i f y i n g and co r r e c t i n g f o r 'inconsistent' s t a r t i n g values were used because of t h e i r s i m p l i c i t y . They worked w e l l 74 i n the empirical contexts where 'inconsistent' s t a r t i n g values i n v a r i a b l y led to a decrease of the value of the l i k e l i h o o d function a f t e r i t had increased for some i t e r a t i o n s . This i s i n contradiction with the EM algorithm r e s u l t that the value of the l i k e l i h o o d function increases at each i t e r a t i o n . For 'consistent' s t a r t i n g values the algorithm always converged. The following stopping c r i t e r i a were used with the EM algorithm. When the absolute value of the r e l a t i v e change, from one i t e r a t i o n to the following, was les s than 1% for a l l the i d e n t i f i a b l e parameters, the algorithm was assumed to have converged. Because of computing constraints, i f t h i s condition had not been met by the f o r t i e t h i t e r a t i o n the algorithm was.also-stopped and assumed to have converged. Results about these con-vergence c r i t e r i a w i l l be presented i n the r e s u l t s section. SW Estimator The SW estimates are calculated as follows from (1.10): 3 = {b_ 1 + b + b + 1> /.{l + 2p l x ) . This estimation method uses the method of moments and produces a con-s i s t e n t estimate of 3 under the SW model of t h i n trading where s e c u r i t i e s trade at le a s t once every measuring i n t e r v a l . It i s also a consistent estimator under the CHMSW model when the observed returns depend only on the contemporaneous and lag one true returns, and under the EIP model. SWE Estimator This extended version of the SW estimator proposed by Fowler and 75 Rorke takes care of s e c u r i t i e s that do not trade every period but trade at l e a s t once every other period. Accordingly, lag 2 covariances appear i n the SWE formula ^ ft ft ft ft ft ^ f t /^ f t 3 = ( b _ 2 + b_ 1 + b + b + 1 + b + 2 ) / {1 + 2 p l x + 2 p 2 x> . This estimator i s consistent under the SW model, the CHMSW model where the adjustment delays extend to lag 1 and lag 2 returns, the EIP and extended EIP models.• LS m/k Estimators The LS 1/0 estimator i s biased and inconsistent under the SW, CHMSW and EIP models because of the presence i n these models of the errors i n the independent v a r i a b l e which r e s u l t s i n the w e l l known problem of errors i n v a r i a b l e s . The LS 2/1 estimator i s calculated here since, as seen i n Tables 2 to 5, for c e r t a i n parameter values, i t has lower bias than LS 1/0 even though i t s variance i s higher. For some of these parameter values, t h i s estimate of systematic r i s k could consequently have a lower mean squared error than the LS 1/0 estimates. F i n a l l y , the LS 3/2 estimates are also calculated as another v e r i f i c a t i o n of the bias variance r e l a t i o n s h i p between d i f f e r e n t LS m/k estimators. 1. Applications to Simulated Observations In t h i s section data are generated according to s p e c i f i c models of stock returns so that the d i f f e r e n t estimators of 3 can be compared to the true value. Sets of observations are generated according to the EIP, CHMSW and 76 SW models to avoid bias i n favour of a p a r t i c u l a r estimation method. Each 31 observations set consists of two vectors of monthly returns, a security vector and a market index vector. Computing Aspects of the EM Algorithm The number of observations i n each vector, say T , can be l i m i t e d by computing aspects of the EM algorithm. The algorithm can require large CPU memory since a number of 4Tx4T and 4Tx2T matrices have to be used, and high CPU time since a 2T x 2T matrix i s inverted at each i t e r a t i o n and the number of i t e r a t i o n s can be large due to the slow rate of convergence. Some computing time figures are presented with the r e s u l t s . I t was de-cided to use T equal to 60 observations because i t i s common i n the empirical l i t e r a t u r e to use 5 years of monthly returns to estimate 3 • Twenty was chosen as the number of observation - sets generated for each of the three models because, again, of computing aspects and because the purpose of t h i s chapter i s not to f i n d the empirical d i s t r i b u t i o n of the d i f f e r e n t estimators but to present some i l l u s t r a t i o n s . Following i s a de s c r i p t i o n of how the observations sets from the three models are generated. 1.1 Description of the Simulation Procedure EIP Model F i r s t observation sets are generated according to the EIP model presented i n t h i s d i s s e r t a t i o n i n the following way. i ) For each observation set, s i x t y i i d observations of the normal vector of true returns and errors i n the prices (x,y,u,v) are generated. 77 In addition, one observation of the vector (u,v) independent of the other observations i s generated from a BVN d i s t r i b u t i o n with the same mean and covariance as for u and v . This l a s t vector i s the vector of errors i n the f i r s t p r i ces which, as seen i n Section 3.1, i s independent of a l l the other errors and returns. The following parameter's values are used: y = .016 y = .020 y = .00013 y = .000075 X y u v a = .0018 x .0063 a = .0001 u a = .0007 v a = .0018 xy a = -.00035 xu a = -.00016 yu a = .00006 uv a = -.00002 XV a = -.0004 yv The expected value of the index return y and the variance of x 2 i t s observed returns were chosen from monthly returns on an equally weighted market p o r t f o l i o f o r the NYSE from July 1963 to June 1968. The expected value and the variance of the observed security re-turns were chosen from those parameter values f o r I B M for that same 32 period. The covariance a was chosen to give the security a true 3 2 of 1. a and a were chosen to give a p o s i t i v e s e r i a l c o r r e l a t i o n u xu 33 2 i n the market return. For the secu r i t y , the values of and 0" were selected to give a negative s e r i a l c o r r e l a t i o n i n the security observed returns, which i s frequently observed. F i n a l l y , the security and index observed returns variance were adjusted to keep the observed return's regression R squared c o e f f i c i e n t around 30%. Even though the expected returns and some covariances might seem a l i t t l e high by today's 78 standards, the various simulation conclusions are not r e a l l y affected by these values. For instance, the expected returns do not enter the small sample properties of the LS estimators and the scale of the covariances only have a second order e f f e c t . What r e a l l y matters i s the r e l a t i v e s i z e of the d i f f e r e n t parameter values with respect to one another. F i n a l l y , note that IBM was used as a reference only for 2 2 y and O and not a and a ; consequently, the thinness of IBM y y v yv returns has nothing to do with the thinness of the generated s e c u r i t i e s ' returns. i i ) Sixty correlated observed returns are then constructed from the true returns X ,Y and the errors i n the prices U, V using formula (2.10) and (2.11). X t " X t + u t + l " u t ( 2 ' 1 0 )  Y t = y t + Vt+1 " V t • ( 2 ' 1 1 ) SW Model Observation '" sets were generated according to the SW model of t h i n trading because th i n trading i s a w e l l recognized source of error i n computing returns. Furthermore, the SW model i s widely accepted as a model of t h i n trading. The SW model assumes that the continuous time market model holds but that the actual trading of s e c u r i t i e s i s discontinuous. Because of t h i s , the observed end of period prices might correspond to transactions that occurred within the period rather than at the end of i t . I t i s also 79 assumed that the delays between the time of the l a s t trades and the end of the periods are i i d f o r a l l periods. To simplify the simulations of returns, we further assume that a l l trades during any month are made at the end of a trading day. The following procedure i s used to generate the observed returns. i ) A set of 1220 independent continuously compounded d a i l y returns for a se c u r i t y and a market index are generated from a b i v a r i a t e normal d i s t r i b u t i o n with a c e r t a i n mean and covariance matrix. These returns are taken as the true returns that would be calculated each day i f the s e c u r i t i e s and index were trading continuously. i i ) Assuming that a se c u r i t y has the same p r o b a b i l i t y of trading each day and that there are 20 trading days per month and no weekends, the l a s t trading day i s found f o r each month using a geometric random deviate generator. The p r o b a b i l i t y d i s t r i b u t i o n function f o r the geo-metric d i s t r i b u t i o n i s f(z) = p ( l - p ) Z _ 1 z = l,2,...,T 0 < p < 1 where the deviates z-1 are the number of f a i l u r e s before a success or the number of non trading days before a f i r s t trade. The l a s t trading day f o r month t , say d ( t ) , for 61 months i s determined as follows: d(t) = ( t - l ) 20 + 20 - (z(t) - 1) . The observed returns for month t are then calculated by adding the true d a i l y returns from the l a s t trading day i n month t , d(t) , to the l a s t trading day i n month t+l, d(t+l) , 80 a(t+i) I s=(d(t) i i i ) The index i s a weighted average of the s e c u r i t i e s , of which a proportion i s a c t u a l l y traded each day. However, to si m p l i f y things, the index i s treated here as a s e c u r i t y which does not trade on c e r t a i n days. In other words, the index cannot be " p a r t i a l l y " traded on any given day; i t j u s t trades or does not. This modification of the SW model i s not important since the SW r e s u l t s remain v a l i d even though the index i s treated here as a very f a t se c u r i t y . The same method to compute returns as f o r the s e c u r i t i e s i s repeated f o r the index. The values for the means and covariances were chosen i n the same range as the EIP model values. I t should again.be emphasized that the impor-tant factor here i s the r e l a t i v e size of the parameters with respect to one another and the values of the trading p r o b a b i l i t i e s . The following parameters' values were used to generate the observation sets; note that a l l monthly returns are divided by twenty since d a i l y returns are generated. .061/20 y y = .020/20 .0015/20 a 2 = .0057/20 y .0015/20 . The p r o b a b i l i t y of trading for the index on any given day, p(x) , was set to .9 and the p r o b a b i l i t y of trading for the s e c u r i t y , p(y) to .5. Since the number of days u n t i l f i r s t trade follows a geometric d i s t r i b u t i o n , t h i s implies that the index i s expected to trade on average on the l a s t day xy 81 of the month while the security i s expected to have i t s l a s t trade of the month on the 19th trading day. We then expect the thi n trading problem not to be severe. CHMSW Model F i n a l l y , observed returns are generated according to the CHMSW model. 34 This model was developed to take account of pric e adjustment delays. It assumes that the observed returns are a weighted average of the past and contemporaneous true returns. Although f o r many t h i s i s not a palatable assumption, t h i s model, according to i t s authors, i s consistent with some observed returns' behaviours. The following procedure i s followed i n generating the returns. i ) Vectors of 60 independent true monthly returns f o r a security and an index are generated from a b i v a r i a t e normal d i s t r i b u t i o n with the same monthly mean and covariance matrix as f o r the SW model. i i ) Observed returns are calculated by taking a weighted average of the contemporaneous and past true returns generated i n i ) . Even though i n the CHMSW model the weights can be stoc h a s t i c , which has the e f f e c t of increasing the variance of observed returns, they are kept fixed here. The weights chosen f o r the index are .8 and .2 . This means that 80% of any period observed index return comes from the current period true re-turn and that 20% comes from the previous period true return. The weights for the sec u r i t y are .5, .3, .15 and .05, which means that the true returns for one period a f f e c t the observed returns up to three periods i n the future. 82 Again the length of the measuring i n t e r v a l and consequently the size of the parameters, and the fact that the weights are not stochastic are not r e a l l y relevant here. The most important matter f o r the estimation of 8 i s how protracted are the e f f e c t s of the one period true returns on the future observed returns. 1.2 Estimation Results f o r the Simulated Observations Twenty d i f f e r e n t observation sets were generated for each of the three models and the estimates of 8 were recorded. The r e s u l t s are pre-sented i n Tables 6 to 8, one for each of the three models. Each table shows f or the d i f f e r e n t observation sets generated the estimates of 8 from the s i x estimation methods. To summarize these r e s u l t s f or each estimation method, the average bias and the mean squared deviation (MSE) about the true 8 of the 20 estimates are also presented. An estimate of the variance can be found by subtracting the squared bias from the MSE. The following observations can be made from inspection of these tables. EIP Model A l l three LS estimators produce p o s i t i v e l y biased estimates. The bias decreases but the variance increases as we go from LS 1/0 to LS 2/1 and from LS 2/1 to LS 3/2. The large bias contribution for LS 1/0 gives i t the highest MSE. The MSE f o r LS 2/1 and LS 3/2 i s about h a l f the size of the LS 1/0 MSE, the LS 2/1 MSE being lower. The ML, SW and SWE estimators have small bias estimates compared to the LS estimators; however, t h e i r variance i s larger except for the ML 83 TABLE 6 3 Estimates for EIP Model ML SW SWE LS Vo LS 2 / i LS V , .943 .880 .296 1.175 1.001 .735 1.136 1.195 1.033 1.366 1.291 1.220 1.265 1.325 1.475 1.628 1.456 1.451 1.643 1.570 1.334 1.866 1.693 1.554 .587 .496 -.011 .913 .665 .488 1.396 1.366 1.520 1.652 1.484 1.496 1.157 1.147 .919 1.609 1.328 1.186 .732 .508 .758 1.414 .973 .950 .786 .630 .451 1.257 .873 .744 .968 1.015 .670 1.370 1.168 1.033 1.060 1.131 1.282 1.491 1.275 1.218 1.295 1.281 1.460 1.709 1.438 1.440 .655 .585 .324 1.088 .815 .645 1.567 1.485 3.332 1.631 1.523 1.683 1.118 1.150 .849 1.471 1.287 1.136 .967 .984 1.028 1.200 1.083 1.072 1.157 .975 .503 1.254 1.077 .865 1.047 1.039 .914 1.223 1.124 1.058 .622 .584 .275 1.374 .979 .766 .619 .402 .742 1.522 1.033 .898 BIAS .0360 -.0126 .0409 .4107 .1764 .0799 SD .3088 .3500 .7108 .2353 .2647 .3324 MSE .0919 .1165 .4829 .2112 .0977 .1154 % 100 79 19 44 94 80 estimator which has a smaller variance than LS 3/2. I t must also be pointed out that the variance f o r SWE.-is at least twice as large as for any,other estimator. The ML estimator has the lowest MSE of a l l s i x estimators, s l i g h t l y lower than LS 2/1, because of i t s low b i a s , while SWE has the highest MSE of a l l because of i t s large variance. On a MSE basis, ML i s the best, followed by LS 2/1, LS 3/2 and SW with LS 1/0 and the SWE further behind. SW Model A l l three LS estimators produce negatively biased estimates. This d i f f e r s from the EIP model where the estimates had p o s i t i v e b i a s . As fo the EIP model, LS 1/0 has the largest bias but the smallest variance, LS 2/1 has both the second largest.bias and variance and LS 3/2 has the lowest bias and the largest variance. Since the absolute value of the bias for the LS estimators i s much smaller f o r t h i s model, the variance i s a more important component of the MSE and consequently, t h e i r MSE ranking i s the same as t h e i r variance ranking. The ML, SW and SWE estimators a l l have small bias again, with the bias f o r SWE a l i t t l e b i t larger than for the other two. ML has the smallest variance of the three while SW and SWE have about the same variance with a sli m advantage to SWE. Here also, since the bias i s quite small, the MSE ranking of these three estimators i s the same as the variance ranking. Comparing these estimators with the LS estimators the l a t t e r have lower variance and MSE than the former. Using the MSE c r i t e r i a , LS 1/0, LS 2/1 and LS 3/2 i n that order per form better than the other three estimators for which ML i s the best, followed by SWE and SW. 85 TABLE 7 Estimates for SW Model ML SW SWE LS Vo LS V1 LS y2 1.553 1.577 1.619 1.064 1.338 1.492 1.173 1.270 1.231 .944 1.102 1.178 1.036 1.048 .686 .615 .831 .778 .638 .495 .775 .802 .631 .674 1.229 1.395 1.027 .901 1.138 1.086 .749 .749 1.087 .843 .799 .897 .668 .663 1.346 .823 .767 .883 .845 .858 .947 .953 .892 .910 .794 .687 .858 .590 .627 .682 1.173 1.271 1.471 1.003 1.175 1.285 .737 1.058 1.267 .546 .837 1.060 1.317 1.240 1.385 1.002 1.149 1.222 .992 .710 1.044 1.145 .933 .977 1.409 1.378 .744 1.098 1.217 1.045 .653 .700 .677 .879 .811 .763 1.484 1.563 1.319 1.100 1.357 1.344 .836 .808 1.025 .933 .867 .913 .823 .909 .490 .751 .835 .691 .650 .637 .515 .658 .665 .624 1.211 1.222 1.321 1.178 1.197 1.257 BIAS -.0015 .0119 .0417 -.1086 -.0416 -.0120 SD .2997 .3329 .3257 .1880 .2305 .2502 MSE .0854 .1054 .1025 .0454 .0522 .0596 % 53 43 44 100 87 76 86 CHMSW Model For the LS estimators we observe negative b i a s , as for the SW model, but much la r g e r . The bias and variance rankings are also the same as for the EIP and SW models, the bias decreasing and the variance increasing as we go from LS 1/0 to LS 2/1 and LS 2/1 to LS 3/2. Since the bias i s large, the MSE ranking i s the same as the bias ranking: LS 3/2 has the smallest MSE and LS 1/0 the l a r g e s t . The ML and SW estimators have a negative bias of about the same magnitude as LS 2/1 and LS 3/2 re s p e c t i v e l y , while SWE has only a small bi a s . The variance and MSE are lower f o r ML than SW and for SW than SWE. Comparing the ML, SW and SWE estimators with the LS estimators on a MSE basis, ML, SW, LS 2/1 and LS 3/2 are almost the same, while i t i s larger for LS 1/0 because of i t s bias and even higher f o r SWE because of i t s variance. The MSE ranks LS 3/2 as the best estimator, followed c l o s e l y by ML, SW and LS 2/1, with LS 1/0 and SWE further behind. Summary and General Remarks In summary, the following observations can be made from the simula-t i o n r e s u l t s . For a l l three models, the bias and variance rankings for the LS estimators are the same; the bias decreases and the variance i n -creases as we go from LS 1/0 to LS 2/1 and from LS 2/1 to LS 3/2. The MSE ranking varies a l o t across the models depending on the size of the bias and variance. For the SW model, where the bias i s small and negative, LS 1/0 has the lowest MSE; f o r the EIP model where the bias i s large and 87 TABLE 8 8 Estimates for CHMSW Model ML SW SWE LS V o LS Vi LS V2 .282 .215 -.035 .318 .280 .274 .798 .791 .785 .666 .739 .754 1.039 1.127 1.166 .938 1.045 1.141 1.145 1.040 1.253 .826 .957 1.071 .764 .800 .969 .498 .658 .775 .765 .972 1.369 .703 .865 1.052 .573 .656 .758 .559 .622 .666 .628 .683 .761 .490 .612 .676 .988 1.223 1.558 .883 1.097 1.270 1.141 1.251 1.281 .959 1.128 1.187 .887 1.128 1.828 .900 1.039 1.210 .870 .925 .918 .654 .823 .861 .914 .910 .954 .685 .800 .856 .868 1.125 1.236 .832 1.016 1.093 .183 .131 .244 .199 .157 .168 .578 .587 .584 .547 .569 .588 .863 .967 1.107 .733 .871 .956 .455 .426 .478 .432 .455 .532 .909 .956 1.297 .700 .843 .980 .773 .894 .977 .615 .781 .869 BIAS -.2288 -.1595 -.0256 -.3431 -.2321 -.1524 SD .2574 .3125 .4417 .2055 .2639 .3043 MSE .1154 .1183 .1860 .1579 .1200 .1112 % 96 94 60 70 93 100 88 p o s i t i v e , LS 2/1 has the lowest MSE and for the CHMSW model where the bias i s large and negative, LS 3/2 has the lowest MSE. The ML, SW and SWE estimators have very small bias for the EIP and SW models and so does the SWE f o r the CHMSW model. However, there i s an apparent negative bias i n the ML and SW estimators f o r the CHMSW model. Across the three models the variance and MSE of ML are smaller than the variance and MSE of SW and SWE. For the CHMSW and EIP models, the MSE of SWE i s much la r g e r than that of ML and SW. It should be emphasized that despite the l i m i t e d number of r e p e t i t i o n s , these r e s u l t s are rather i n t e r e s t i n g . They are further supported substant-i a l l y by other simulations using d i f f e r e n t numbers of i t e r a t i o n s and stopping c r i t e r i a , ( n o t reported). These simulations lead to the following a d d i t i o n a l remarks. For the EIP and SW models, the ML, SW and SWE bias i s very close to zero, closer than i s found i n t h i s section f o r SWE, and t h e i r variance i s smaller f or ML than f o r SW and for SW than SWE. For the CHMSW, the bias for the ML and SW are usually c l o s e r to one another than indicated by the r e s u l t s presented here but the other observations are the same. The bias and variance ranking of the LS estimators i s the same for a l l three models. The values of bias and variance are also close to those predicted by the small sample properties f o r the EIP model. 1.3 Interpretation and Discussion of Simulation Results EIP Model For t h i s model, the ML estimator and the SW and SWE method of moments estimators are a l l consistent so i n a large sample the bias should be 89 close to zero. The variance of SWE should be larger than that of SW since more parameters are estimated. This i s a c t u a l l y what i s observed. For the LS estimates the small sample properties predicted from (2.40) and (2.41) may be v e r i f i e d : LS 1/0 LS 2/1 LS 3/2 BIAS .349 .151 .100 SD .2723 .2782 .3134 Note that the bias for the LS estimators i s p o s i t i v e . This i s due to the large p o s i t i v e s e r i a l c o r r e l a t i o n given to the market index. This c o r r e l a -t i o n implies that the variance of the observed returns i s smaller than the variance of the true returns, which increases the 6 estimate. F i n a l l y , these r e s u l t s confirm that the ML estimator performs w e l l under the EIP model. SW Model Given the small sample s i z e , the bias and variance r e s u l t s f o r the d i f f e r e n t estimators are as expected. Since the ML, SW and SWE estimators are consistent f o r t h i s model, we expect the bias to be close to zero. The SWE estimator has a bias of four percent but other preliminary r e s u l t s i n d i c a t e that t h i s i s p a r t i c u l a r to the sample considered here. The variance f o r SW i s also expected to be smaller than f o r SWE since fewer parameters are estimated i n the SW formula. The ML estimator has a lower variance than the SW estimator which i s a method of moments estimator. We know that the SW model i s a s p e c i a l form of the EIP model and that the LS estimates are biased and inconsistent for the EIP model. To deter-90 mine the sign of bias and make a rough estimate of i t s magnitude, we use Appendix 9 and make some si m p l i f y i n g assumptions. Assume that the way the returns are generated can be approximated by s e t t i n g S = .005 x S = .05 y using the fa c t that the p r o b a b i l i t y of trading follows a geometric d i s t r i b u -t i o n . Using Appendix 9 r e s u l t s we f i n d that the LS 1/0 estimate of 3 should be approximately b = .995 . The bias i s then expected to be negative and close to f i v e percent while 10.9 percent i s observed. However, other simulations r e s u l t s are closer to 5 percent than 10 percent. Using the above approximations and assuming that the errors are normal, the following small sample properties are pre-dicted from formulas (2.37) and (2.38). LS 1/0 LS 2/1 LS 3/2 BIAS -.0458 -.0237 -.0164 SD .2231 .2699 .3212 Note that the bias i s larger, except f o r LS 3/2, and the variance lower than predicted. However, given the small sample s i z e , these r e s u l t s are acceptable. As mentioned i n Section 1.1, the SW model returns were generated such that the errors i n the p r i c e represent only a mild problem. It i s then not s u r p r i s i n g that the LS estimators and e s p e c i a l l y LS 1/0 outper-91 forms the other estimators. CHMSW Model By construction there i s p o s i t i v e second order s e r i a l c o r r e l a t i o n i n the returns. Since t h i s i s not accounted for by the ML and SW e s t i -mators, we then expect negative bias i n these estimates. There i s also some t h i r d order p o s i t i v e s e r i a l c o r r e l a t i o n . Because of t h i s t h i r d order s e r i a l c o r r e l a t i o n , we also expect a negative bias i n the SWE e s t i -mator which accounts f o r f i r s t and second order but not t h i r d order s e r i a l c o r r e l a t i o n . But since the magnitude of the t h i r d order c o r r e l a t i o n i s smaller than that of the second order c o r r e l a t i o n , the bias i n SWE i s expected to be smaller than i n ML and SW. The empirical observations con-firm these statements. For the LS estimates, the observed strong negative bias i s also no surprise since these estimates account only for f i r s t order s e r i a l c o r r e l a t i o n . 1.4 Computing Remarks f o r the EM Algorithm F i n a l l y , before moving on to the next section a few points should be mentioned concerning ( i ) the speed of convergence of the EM algorithm and the stopping c r i t e r i a , and ( i i ) the computing time associated with i t . Speed of Convergence and Stopping C r i t e r i a For 31 out of the 60 s e c u r i t i e s generated i n t h i s section, the EM algorithm stopped a f t e r 40 i t e r a t i o n s : 14 for the EIP model, 10 for the SW model and 7 for the CHMSW model. The average number of i t e r a t i o n s required to reach the .01 c r i t e r i a was 23, ranging from 9 to 39. 92 For the EIP model, when the algorithm was stopped, 10 8 estimates were increasing and 10 were decreasing; 10 were increasing or decreasing i n the d i r e c t i o n of the true value of 8 and 10 were moving away from i t . The average of the absolute value of the r e l a t i v e change i n 8 for the l a s t i t e r a t i o n was 0.18%, ranging from 0.00% to 0.59%. Note that the highest r e l a t i v e changes were for small values of 8 , which means that these highest values should not be given too much importance. For the SW model, when the algorithm was stopped, 10 estimates of 8 were increasing and 10 decreasing, 8 were increasing or decreasing i n the d i r e c t i o n of the true value of 8 and 12 were moving away from i t . The average of the absolute value of the r e l a t i v e change i n 8 for the l a s t i t e r a t i o n was 0.074%, ranging from 0.00% to 0.20%. For the CHMSW model, when the algorithm was stopped, 6 estimates of 8 were increasing and 14 were decreasing, 5 were increasing or decreas-ing i n the d i r e c t i o n of the true value of 8 and 15 were moving away from i t . The average of the absolute value of the change i n 8 was 0.12%, ranging from 0.00% to 0.27%. Note that for the CHMSW model we do not t a l k about the r e l a t i v e change i n 8 since some estimates are quite low and the high percentage change that they would produce could be mislead-ing. These figures lead to the following conclusions. The stopping c r i t e r i a do not seem to bias the ML estimate of 8 for the EIP model and SW models since as many estimates were increasing as were decreasing when the algorithm was stopped, with equivalent changes i n value. The calculated variance of the estimate of 8 for these two estimation methods 93 i s probably affected s l i g h t l y by the stopping c r i t e r i a . Roughly the same number of estimates were moving i n the d i r e c t i o n toward and away from the true value, which indicates that the calculated variance would i n -crease i f the algorithm was continued. However, the r e l a t i v e l y small changes i n the l a s t i t e r a t i o n , i n the order of 0.0025 on average for both models, reveal that the bias i n the estimated variance introduced by the stopping c r i t e r i a i s probably quite low. For the CHMSW model, i t seems that the stopping c r i t e r i a introduces an upward bias i n the e s t i -mate of 3 and a downward bias i n i t s variance estimate since more 3 estimates were increasing than decreasing when the algorithm was stopped. Nevertheless, the small si z e of the change i n the 3 estimate i n the l a s t i t e r a t i o n downplays these remarks. Given these r e s u l t s , i t was not deemed necessary to increase the number of i t e r a t i o n s . Computing Time Remarks The f i n a l remarks are devoted to the computing time aspects of the EM algorithm. Most computer runs that were made included 10 s e c u r i t i e s f o r a p a r t i c u l a r model with an average 32 i t e r a t i o n s per se c u r i t y . An average CPU time and CPU VMI memory size used per i t e r a t i o n rather than per s e c u r i t y generated were calculated for each run because of the uneven number of i t e r a t i o n s f or each s e c u r i t y . This was done by d i v i d i n g the t o t a l CPU time and CPU VMI by the t o t a l number of i t e r a t i o n s f or a l l s e c u r i t i e s of the given run. Since each run includes the generation of data and the c a l c u l a t i o n of other estimates of 3 , the average CPU time per i t e r a t i o n overstates the actual time f o r one i t e r a t i o n of the EM 94 algorithm. However, t h i s difference i s very minor since generating the returns and c a l c u l a t i n g the other estimates requires r e l a t i v e l y l i t t l e time and furthermore, t h i s time i s divided among 32 i t e r a t i o n s , more or l e s s . A weighted average of the CPU time and CPU VMI per i t e r a t i o n were then calculated over a l l runs and the r e s u l t s were as follows. For an Amdahl V8 computer the average CPU time and CPU VMI per i t e r a t i o n of the EM algorithm using double p r e c i s i o n were 7.364 seconds and 63.132 page-min. respectively. 2. I l l u s t r a t i o n s Using NYSE S e c u r i t i e s The purpose of t h i s section i s to compare the estimates produced by the s i x methods used i n the previous section f o r r e a l security returns. A sample of three s e c u r i t i e s from the NYSE and two d i f f e r e n t indexes are used. 2.1 Data Source and Selection C r i t e r i a The population of s e c u r i t i e s that was chosen consisted of the s e c u r i t i e s l i s t e d on the Center f o r Research i n Security Prices Monthly Stock Master f i l e , which had p r i c e structures from January 1979 to December 1983. These s e c u r i t i e s were ranked according to t h e i r t o t a l market value as of the 1st of January 19 79. The t o t a l market value of a security's share was taken as a proxy measure for the errors i n the p r i c e s ; i t i s assumed that the s e c u r i t i e s with l a r g e r market value s u f f e r less from errors i n the prices i n the sense that t h e i r f i r s t order auto-c o r r e l a t i o n s and f i r s t order s e r i a l c o r r e l a t i o n s with the index are 95 smaller. S e c u r i t i e s were ranked according to t o t a l market value as of the beginning of the period rather than the end of the period, to avoid the following possible source of bia s . The t o t a l market value at the end of 1983 of a se c u r i t y that did not perform w e l l during the f i v e year period w i l l be smaller than the t o t a l market value of a s i m i l a r s e c u r i t y which did perform w e l l . This means that s e c u r i t i e s with lower t o t a l market values at the end of the period are more l i k e l y to have performed poorly i n the past and that s i z e becomes a proxy not only f o r the errors i n the p r i c e s , but also f o r the performance of a se c u r i t y . Consequently, i t i s preferable to rank s e c u r i t i e s as of the beginning of the period. It was decided to sel e c t f i v e s e c u r i t i e s from the f i r s t , s i x t h and tenth de c i l e s i n order to better capture the di f f e r e n c e s , i f any, across the d e c i l e s . I t was also decided to sel e c t these s e c u r i t i e s at random, rather than choosing s e c u r i t i e s conforming to the expected r e s u l t s . The f i v e s e c u r i t i e s were selected at random using a uniform random deviate generator from each of the f i r s t , s i x t h and tenth de c i l e s of the ranked s e c u r i t i e s . Two s e c u r i t i e s from the f i r s t d e c i l e and one from the f i f t h d e c i l e had numerous consecutive months with missing data and were replaced by other s e c u r i t i e s also drawn at random from the same de c i l e . The monthly returns f o r these s e c u r i t i e s , adjusted for stock s p l i t s and dividends, were taken d i r e c t l y from the monthly master f i l e . F i n a l l y , for each of the three groups, the security with the lowest second order s e r i a l c o r r e l a t i o n was retained. This was done i n an attempt to meet as c l o s e l y as possible the EIP model assumptions. Two indexes were used as proxies to the market index - the value weighted (VW) index and the equally weighted (EW) index. The VW index 96 gives more weight to the larger market value s e c u r i t i e s which, i n general, are expected to s u f f e r less from the errors i n the p r i c e s . We then expect i t to have a smaller s e r i a l c o r r e l a t i o n . Conversely, the EW index, which gives equal weight to a l l s e c u r i t i e s , should have a larger s e r i a l c o r r e l a t i o n . Using a t h i n trading vocabulary, we would expect the VW index to be a f a t index and the EW index to be a t h i n index. In t h i s section, where only three s e c u r i t i e s and two indexes are used, the convergence c r i t e r i a f o r the EM algorithm were changed. The algorithm was not stopped a f t e r 40 i t e r a t i o n s but was continued u n t i l the maximum r e l a t i v e change i n a l l parameters was less than 1% and the r e l a t i v e change i n the (3 estimate was le s s than 0.1%. 2.2 Estimation Results from NYSE Data The 8 estimates under the s i x estimation methods for the three s e c u r i t i e s and the two indexes are presented i n part (a) of Table 9. In part (b), selected s e r i a l c o r r e l a t i o n s and variance calculated from the returns sample moments are presented. F i r s t , consider the EW index. For s e c u r i t y A, Giant Portland CEM Co., the lowest estimate i s SW at 1.18 and the highest i s SWE at 1.34, with the ML roughly halfway between them. For security B, Peabody International Corp., the lowest estimate i s LS 1/0 at 1.24 and the highest i s SWE at 1.62; the three LS estimates are lower than the other three and the ML estimate i s i n between SW and SWE. For s e c u r i t y C, Burroughs Corp., the lowest estimate i s SWE at .71 and the highest i s LS 1/0 at .79, with the three LS estimates larger than the other e s t i -mates; again ML i s i n between SW and SWE. 97 TABLE 9 Estimates f o r Selected NYSE Se c u r i t i e s a) g Estimates EQUALLY WEIGHTED INDEX ML SW SWE LS 1/0 LS 2/1 LS 3/2 A 1.274 1.182 1.343 1.244 1.192 1.239 B 1.570 1.557 1.619 1.236 1.437 1.507 C .719 .729 .707 .787 .780 .760 VALUE WEIGHTED INDEX ML SW SWE LS 1/0 LS 2/1 LS 3/2 A 1.391 1.209 1.294 1.179 1.172 1.212 B 1.972 2.165 2.178 1.211 1.703 1.847 C .884 .805 .862 .900 .870 .867 b) Autocorrelation Estimates from the Method of Moments LAG 1 LAG 2 2/ 2, o (a ) y x A -.231 .025 .0151 B .051 .045 .0105 C .041 -.034 .0055 EW .104 .063 .0256 VW .005 .068 .0019 98 Turning to the 'VW index, f o r security A the lowest estimate i s LS 2/1 at 1.17 and the highest i s the ML at 1.39; the SW i s close to the LS estimates while SWE and e s p e c i a l l y ML are higher. For security B, LS 1/0 i s by f a r the smallest estimate at 1.21, LS 2/1 and LS 3/2 are higher at 1.70 and 1.85, and the ML, SW and SWE are even higher at 1.97, 2.17 and 2.18, res p e c t i v e l y . For sec u r i t y C, LS 1/0 i s the highest at .90 and SW i s the lowest and quite distanced from the others, at .80; LS 2/1, LS 3/2, SWE and ML are close to one another, ranging from .86 to .88 and are not so f a r from LS 1/0. The following i s observed i f one looks at how the estimates behave across indexes f o r a given se c u r i t y . For security A there i s no major changes i n the estimates, except possibly for the ML estimates which go from 1.27 f o r the EW index to 1.39 for the VW index. For sec u r i t y B, the changes are much more dramatic; except f o r LS 1/0 which decreases by .03 from EW to SW, the other estimates show increases from .27 for LS 2/1 to .61 for SW. For sec u r i t y C, a l l estimates increase with SW and ML having the smallest and the largest increases at .08 and .16 r e -spectively. The following observations can be made from part (b) of Table 3.4. Security A has larger variance of observed returns than B, security B has larger variance of observed returns than C, and the EW index has a larger variance of observed returns than VW. Security A has strong negative f i r s t order s e r i a l c o r r e l a t i o n while the other two s e c u r i t i e s have weaker and p o s i t i v e f i r s t order s e r i a l c o r r e l a t i o n . As for the indexes, the f i r s t order s e r i a l c o r r e l a t i o n f o r EW i s p o s i t i v e while i t is v i r t u a l l y 0 for VW. 99 2.3 Interpretation and Discussion of Results from NYSE Data Interesting conclusions emerge from the r e s u l t s of the NYSE data, when analyzed i n view of the expected properties. 3 Estimates and Sample Moments Consider f i r s t the sample moments. Since s e c u r i t i e s A, B and C have smaller, medium and la r g e r t o t a l market value, r e s p e c t i v e l y , the variance of the observed returns i s expected to be larger f o r A than B, and larger f o r B than C. Also, since the t o t a l market value i s taken as a proxy to the errors i n the prices problem, we expect A to have nega-t i v e f i r s t order s e r i a l c o r r e l a t i o n , larger i n absolute value than f o r B and C. We expect B and C to have small p o s i t i v e f i r s t order s e r i a l c o r r e l a t i o n . The EW index was expected to. have a larger variance of observed returns than the VW index, since i t gives more weight to smaller s e c u r i t i e s . The EW index was also expected to have a p o s i t i v e f i r s t order s e r i a l c o r r e l a t i o n because of the Fisher e f f e c t described i n Chapter I I , and t h i s c o r r e l a t i o n to be larger than that of the VW index. The r e s u l t s are i n agreement with expectations. However, i t must be noted that some other s e c u r i t i e s i n the f i r s t sample of 15 did not con-form as well to these expectations as did s e c u r i t i e s A, B and C. The 3 estimate consists b a s i c a l l y of two parts - i t s numerator and i t s denominator. Because the p o s i t i v e f i r s t order s e r i a l c o r r e l a t i o n i n the EW index i s r e l a t i v e l y large, the e f f e c t of correcting f o r i t in the estimation i s to increase the denominator and hence to decrease the 3 estimate. For the VW index, since the f i r s t order s e r i a l c o r r e l a t i o n i s very small, the denominator should not be affected too much by taking 100 account of i t . Similar comments hold for second order s e r i a l c o r r e l a t i o n . For the numerator, the e f f e c t of taking account of f i r s t order c o r r e l a -t i o n depends on the covariance of the index returns with the lag one and lead one security returns, and the e f f e c t of accounting for second order s e r i a l c o r r e l a t i o n depends on the covariance of the index returns with lag two and lead two s e c u r i t y returns. From these comments, the r e l a t i v e s i z e of the LS 1/0, SW and SWE can be explained by inspection of the sample moments of observed returns since these estimators are combinations of sample moments. It must be noted that the change i n the estimate of 8 introduced by accounting for f i r s t order s e r i a l c o r r e l a t i o n cannot be predicted by mere inspection of the index and the s e c u r i t y autocorrelation c o e f f i c i e n t s . The f i r s t order s e r i a l cross c o r r e l a t i o n s must be considered as w e l l . Even though taking account of f i r s t order s e r i a l c o r r e l a t i o n , or using SW instead of LS 1/0 decreases the denominator of the 8 estimate, t h i s resulted i n decreases i n the estimate f o r two out of three of the s e c u r i -t i e s with the EW index. For s e c u r i t y B (C) as we go from LS 1/0 to LS 2/1 and LS 2/1 to LS 3/2, the 8 estimate increases (decreases) i n the d i r e c t i o n of the SW and SWE estimates. For security A, there does not seem to be any pattern i n the LS estimates. For the EW index, where there i s f i r s t order s e r i a l c o r r e l a t i o n , a l l three ML estimates are roughly halfway between the SW and SWE e s t i -mates. For the VW index, where there i s no r e a l f i r s t order s e r i a l cor-r e l a t i o n , the ML estimate for A i s the highest of the s i x ; for B i t i s 101 i n between LS and SW estimates, while for C i t i s closer to the LS 1/0 estimate. Results Summary In summary, i t i s noted that there could be large differences across estimators, e s p e c i a l l y between LS 1/0 and the other ones, and that, i n general, the changes from LS 1/0 to LS 2/1 and LS 2/1 and LS 3/2 are i n the same d i r e c t i o n as the change from LS 3/2 to the SW estimators. I t i s also noted that f o r the estimates of $ using the EW index, the ML i s halfway between the SW and SWE estimates. 3. Conclusion from Simulated and NYSE Data The purpose of the f i r s t section was to i l l u s t r a t e the r e l a t i o n s h i p between s i x d i f f e r e n t estimators under three d i f f e r e n t p r i c e generating processes. For the EIP and SW models, the r e l a t i o n s between the bias and variance of the d i f f e r e n t estimators predicted from Chapter II discussions are confirmed by the empirical r e s u l t s . For the LS estimates, the bias decreases but the variance increases as we go from LS 1/0 to LS 2/1 and from LS 2/1 to LS 3/2. For the ML, SW and SWE estimators, the bias i s low and the variance of the ML i s smaller than f o r SW and SWE. I t i s not s u r p r i s i n g that the SW does not outperform the ML estimator under the SW model since the SW model i s i n fac t a s p e c i a l form of the EIP model. The r e l a t i v e merit of the three LS estimators versus the other three estimators depends on the sev e r i t y of the errors i n the prices problem. For the values of the parameters assumed here f o r the SW model, the errors 102 i n the prices problem are not severe and only a small bias i s introduced i n the LS estimators. Consequently, the LS estimators with t h e i r low variance seem preferable to the other three estimators. For the values of the parameters assumed for the EIP model, the errors in the prices problem are more severe and large bias i n the LS estimators makes the ML estimator preferable to them on a MSE basis. For the CHMSW model, the large bias observed for the d i f f e r e n t e s t i -mators i s consistent with what can be predicted from the p r i c e generating process and the s e r i a l c o r r e l a t i o n that i t implies. It i s also observed that for the values of the parameters considered, LS 1/0 has the smallest variance but large bias makes the other estimators preferable to i t . In the second section, observations from the NYSE are used. Three so-called small, average and large market value s e c u r i t i e s returns to-gether with a f a t (VW) and t h i n (EW) index are used to i l l u s t r a t e the differences between the estimators. The variances and the s e r i a l c o r r e l a t i o n s of the returns are as expected. It i s observed that the LS estimates for s e c u r i t i e s B and C go i n the d i r e c t i o n of the SW and SWE estimates as we go from LS 1/0 to LS 2/1 and from LS 2/1 to LS 3/2. It i s also observed that for the t h i n index a l l three ML estimates are comprised between the SW and the SWE estimates. For the f a t index, one ML estimate i s the highest and the other two are comprised between the LS 1/0 and the SW and SWE estimates. The simulation r e s u l t s suggest the following. The ML estimator performs e f f i c i e n t l y i n the presence of measurement errors. Since p r a c t i c a l contexts indicate the existence of measurement er r o r s , the 103 ML estimator i s recommended. Where there are computing constraints, the method of moments i s a v i a b l e a l t e r n a t i v e , e s p e c i a l l y i f consistency i s viewed as important. I f consistency i s not a major concern, then the LS estimator with an appropriate measuring i n t e r v a l f o r returns i s adequate; LS 2/1 and LS 3/2 performed well for the EIP and CHMSW simulations. When the errors i n prices problem i s minor, as for the SW simula-t i o n s , the LS 1/0 estimator i s suggested. 104 CONCLUSION The market model plays a major r o l e i n finance. The estimation of 8 , i n p a r t i c u l a r , i s fundamental to many empirical studies. However, observed returns have behaviour inconsistent with the market model assumptions. There i s , f o r instance, s e r i a l c o r r e l a t i o n and cross c o r r e l a t i o n i n the s e c u r i t i e s and market returns. These c o r r e l a t i o n s , which vary with the length of the measuring i n t e r v a l , cause the LS estimates of (3 to be biased and inconsistent. In order to better understand the observed returns' behaviour and to correct for the estimation biases, the market model has been modified to take into account market imperfections. The Thin Trading and the Price Adjustment Delays caused by f r i c t i o n s i n the trading pro-cess have received s p e c i a l a t t e n t i o n . While authors have t r i e d to tackle these problems independently, t h i s d i s s e r t a t i o n proposes a model, the EIP model, which d i r e c t l y accounts not only f o r Thin Trading and Price Adjustment Delays, but also for other market imperfections, such as the Bid/Ask spread. Furthermore, t h i s model explains the observed s e r i a l c o r r e l a t i o n s i n returns and the i n t e r v a l l i n g e f f e c t . It adds to the market model the i n t u i t i v e assump-t i o n that the prices from which the returns are calculated are observed with e r r o r . This general assumption allows the EIP model to include as s p e c i a l cases other models such as the SW model of thi n trading and Roll ' s measure of Bid/Ask spread. In contrast with other models, three estimation methods of system-105 a t i c r i s k are studied for the EIP model: the Maximum Likelihood method, the Least Squares method and the method of moments. Assuming that the errors i n the prices are normally d i s t r i b u t e d , i t i s shown how the EM algorithm for missing observations can be used to compute the ML estimate of 3 . LS i s the most common estimation method. The returns used i n computing the LS estimates (of 3) are often measured over d i f f e r e n t i n t e r v a l s which may overlap. The approximate small sample as w e l l as asymptotic properties are then derived for a l l general ways of measuring the returns. The small sample properties not studied by other authors are p a r t i c u l a r l y important since the sample sizes are l i m i t e d by the a v a i l a b i l i t y of the data and the i n s t a b i l i t y of 3 over time. I t i s shown that i n the presence of errors i n the prices i t might be better, i n a mean squared error sense, to use a measuring i n t e r v a l other than one basic period. F i n a l l y , i t i s seen that replacing the covariances by t h e i r sample moments estimates leads to a simple form for a consistent estimate of 3 , which i s b a s i c a l l y the estimation method suggested by other authors. D i f f e r e n t empirical phenomena inconsistent with the market model assumptions are also explained using the EIP model. The l a s t chapter presents some i l l u s t r a t i o n s of the r e l a t i o n s h i p between s i x d i f f e r e n t estimators: the ML estimator; the SW and SWE estimators which are method of moments estimators; LS 1/0; LS 2/1; and LS 3/2 estimators. These i l l u s t r a t i o n s use i n a f i r s t step simulated returns from three models, the errors i n prices or EIP model, the SW model of t h i n trading and the CHMSW model of f r i c t i o n s i n the trading 106 process. In a second step, observed returns from the NYSE are used. These i l l u s t r a t i o n s suggest the following. The ML estimation method performs e f f i c i e n t l y i n the presence of measurement errors and i s recommended. Where there are computing constraints, the method of moments is a v i a b l e a l t e r n a t i v e , e s p e c i a l l y i f consistency i s viewed as important. If consistency i s not a major concern, then the LS estimator with an appropriate measuring i n t e r v a l f or returns i s adequate. When the errors i n prices problem i s minor, the LS 1/0 estimator i s suggested. The EIP model, by i t s simple assumptions, i t s a b i l i t y to explain empirical phenomena and i t s g e n e r a l i t y , represents an improvement to the t r a d i t i o n a l market model and an a l t e r n a t i v e to d i f f e r e n t models of market imperfections. 107 Footnotes 1. See Schwartz and Whitcomb (1977a, 1977b). 2. Fowler, Rorke and Jog (1980) point out that there are two dimensions to the thinness of a sec u r i t y , the trading frequency and the i n -vestor's a b i l i t y to buy or s e l l a large quantity of stock without d i s t o r t i n g the p r i c e as investigated by Scholes (1972). Thin trading i s used i n t h i s d i s s e r t a t i o n to represent the trading frequency, as i t usually does i n other studies including that of Fowler, Rorke and Jog. 3. Scholes-Williams (1977) and Dimson (1979) discuss t h i s r e s u l t . Fisher (1976) pointed out that when s e c u r i t i e s ' prices represent the outcome of transactions that occurred e a r l i e r i n or p r i o r to the period i n question, the index constructed from these s e c u r i t i e s ' p r i c e data w i l l be an average of temporally ordered underlying values of the s e c u r i t i e s ; t h i s induces p o s i t i v e s e r i a l c o r r e l a t i o n i n the calculated index returns. This phenomenon i s known as the Fisher e f f e c t . 4. Johnson (1972) section 9-4 i s a good reference on measurement errors and bias/inconsistency of LS estimates of 8 i n a regression model. 5. Cohen et a l . (1983a) i n footnote 1 provide many references on the in t e r v a l i n g e f f e c t . 6. The changes i n the systematic r i s k of a firm over time also precludes the use of very long i n t e r v a l s . 7. Fowler and Rorke (1983) showed that Dimson i s i n e r r o r . In an un-published reply, Dimson (1982) claims that t h i s error has, for a l l p r a c t i c a l purposes, n e g l i g i b l e e f f e c t s . 8. The MLE estimators are asymptotically normally d i s t r i b u t e d and t h e i r variance achieves the Cramer-Rao lower bound. In that sense, they are the best asymptotically normal estimators. 9. Dempster et a l . (1976) have derived t h i s algorithm. 10. I t has been observed that the frequency of trading of s e c u r i t i e s i s d i r e c t l y r e l a t e d to the t o t a l market value of the firm. See for instance CHMSW (1980). 11. CHMSW (1980), p. 250. 108 12. The Fowler-Rorke-Jog (1983) model i s not presented here since i t has not yet been widely used and would require a rather lengthy d e s c r i p t i o n . The model i s somewhat rel a t e d to the SW model though the estimation method i s quite o r i g i n a l . The interested reader i s referred to t h e i r paper. 13. The continuous time market model w i l l be presented i n greater d e t a i l i n the next chapter. The d e s c r i p t i o n of the models w i l l follow c l o s e l y the d i f f e r e n t papers but the notation i s altered to insure some uniformity. 14. Lindley (1948) examines the conditions under which there exists a l i n e a r r e l a t i o n between va r i a b l e s observed with errors when there i s a l i n e a r r e l a t i o n s h i p between the true v a r i a b l e s . 15. Fama (1976), i n h i s chapter IV and Hawawini i n h i s d i s s e r t a t i o n present some evidence on these c o r r e l a t i o n s . 16. Dimson (1979), p. 205. 17. CHMSW (1983b) i n t h e i r footnote (10) pointed out the way the equation (15) was derived i s not cor r e c t , since a l l terms that depend on x • i J r • ^ S J J m t+k are not equivalenced; f o r instance depends on x m + ^ , and t h i s i s not taken account of. 18. CHMSW (1980) argue that the sign of s e r i a l c o r r e l a t i o n introduced by the weights i s uncertain because they may not be a monotonic func-t i o n due, f o r instance, to the p o s s i b i l i t y of both overshooting and undershooting as new information i s impounded i n the market. While I believe that t h i s could be true i n some instances, t h e i r assumption that E ( v r ) =E(v_ ), or the weights patterns repeat themselves, L , ui i ,m very l i k e l y makes V a monotonic function. 19. See, for instance, Rosenfeld (1980). 20. Rosenfeld (1980) i s also a good source of reference for the d i f f e r e n t stochastic processes of common stocks returns. This proof can be found i n h i s appendix A. 21. For a func t i o n a l r e l a t i o n s h i p which expresses random variables as the sum of two random v a r i a b l e s , l i k e (2.6), the LS estimates are not i n general equal to the ML estimates. However, i t has been shown by Kendall (1961), pp. 14-15 that i f the e i s d i s t r i b u t e d normally, the random o r i g i n of the x's does not a f f e c t the estima-t i o n , so we can regard them as fix e d and the LS estimates are then the same as the ML estimates. 22. See for instance Mood, G r a y b i l l and Boes (1974). 109 23. See Johnson and Kotz (1972), Vol. 4, pp. 103-104. 24. This section follows c l o s e l y section 2 of Dempster et a l . (1976). 25. Recall that, f o r the purpose of the EM algorithm only, the f i r s t observed p r i c e i s assumed to be the true p r i c e . Consequently, the matrices E n and E 2 2 used i n the algorithm w i l l d i f f e r s l i g h t l y from the matrices A and P . The covariance matrix of R w i l l have only 4T elements instead of 4T+2 and the variance of the f i r s t observed return w i l l be 0* + ( a * + a ^ ) instead of 0* + 2 ( a * + c r ^ ) i n the matrix P . 26. See Morrison (1976), p. 92. 27. Dempster et a l . (1976) present one method i n t h e i r paper and Louis (1982) suggests another. 28. See Mood G r a y b i l l Boes (1974), p. 181. 29. See Richardson's equation (2.24). 30. Fowler and Rorke (1983a) have derived a general expression f o r a consistent estimate of 8 f o r s e c u r i t i e s that have "n" periods missed observations. 1 31. The scale of the returns i s not c r u c i a l here. This study could be r e p l i c a t e d with d a i l y data and the r e s u l t s would remain the same. 32. These data can be found i n Fama (1976). 33. Recall from Chapter I discussion of the Fisher e f f e c t that p o s i t i v e s e r i a l c o r r e l a t i o n i s usually observed with index returns. 34. 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"Evidence on the Presence and Causes of S e r i a l C o r r e l a t i o n i n Market Model Residuals," Journal of Fi n a n c i a l and Quantitative Analysis, 12, 291-313. Schwert, G.W. (1977). "Stock Exchange Seats as Ca p i t a l Asset," Journal of F i n a n c i a l Economics, 4, 51-78. Scott, E., S. Brown (1980). "Biased Estimators and Unstable Betas," Journal of Finance, 35, 49-55. Theobald, M. (1983). "The An a l y t i c Relationship Between Inter v a l i n g and Nontrading E f f e c t s i n Continuous Time," Journal of F i n a n c i a l  and Quantitative Analysis, 18, 199-209. 113 APPENDIX 1 D i s t r i b u t i o n of Observed Returns The j o i n t d i s t r i b u t i o n of the vector Z = x* of observed returns i s derived. Since Y and X are l i n e a r combinations of the MVN vector R , t h e i r j o i n t d i s t r i b u t i o n i s also MVN. By d e f i n i t i o n we can write Y* = [ 0 | I | 0 | J ] R X * = [ I | 0 | J | 0 ] R where the dimensions of the matrices I , 0 and J conform to the dimen-sions of the corresponding p a r t i t i o n s of R. Taking expectations we get E[Y*] = U Y ft i t = U Y E[x*] = u x ® i t = y x . Further, E[ (Y* - u ) (Y* - u ) ' ] = [ 0 | I | 0 | J ] A [ 0 | I | 0 | J ] where A, the covariance matrix of R i s given by (2 . 1 1 ) , a 2 I + a J ( 0 I ) ' + a (OI)J' + a 2 J J ' y yv yv v I t i s easy to v e r i f y that J ( 0 I ) ' + (01) J ' = J J ' 114 Hence, S i m i l a r l y , and E[(Y* - y y)(Y* - y y)'] = a* I + (dj + a y v ) JJ' E[(X* - y )(X* - y )'] = a 2 I + (a 2 + a ) JJ' ~ X ~ X X u xu E[(X* - y )(Y* - y )'] = a 1 + a J(0l)'+a (0I)J'+a JJ' ~x ~y xy yu xv uv Combining these r e s u l t s we have X MVN y x ® ± T , ft where ft = a 2 i + (a 2 + a )JJ' y v yv a I + a J(OI)' xy xv + a (Oi)J' + a JJ' yu uv a I + a J(OI)' xy yu + a (OI)J' + a JJ' X V uv a2 i + (a 2 + a )JJ' X u xu 115 APPENDIX 2 Mean and Variance of Quadratic Forms i n Normal Variables From Johnson and Kotz"*"^ the s cumulant feg of quadratic forms Q(C) and Q(D) of mean 0 normal variables i s given by fcs.[Q(C)] = 2 S _ 1 ( S - 1 ) ! tr(WC) S [Q(D)] = 2 S - 1 ( S - 1 ) ! tr(WD) S where t r denotes the trace. The mean and variance, given by f i r s t and second cumulant r e s p e c t i v e l y , are consequently E[Q(C)] = tr(WC) E[Q(C)] = tr(WD) VartQ(C)] = 2tr(WCWC) Var[Q(D)] = 2tr(WDWD) . 116 APPENDIX 3 Sim p l i f i e d Forms f or the Traces Convenient expressions f o r tr(WC), tr(WD), tr(WCWC), tr(WCWD) and tr(WDWD) are derived. The following properties of the trace operator tr(AB) = tr(BA) tr(A) = tr(A') and the following equations w i l l be used frequently i n the derivations: K = LJ t r [ J ( 0 I ) ' M + (OI)J'M] = t r J J ' M tr[J(0I)'M + J(0I)'M'] = trJJ'M* = t r J J ' M (3.1) (3.2) (3.3) and i f M i s symmetric, tr[J(0I)'M] = tr[(0I)J'M] = h t r JJ'M (3.4) We p a r t i t i o n the matrices W, C and D as follows: W = Wl W2 W2' W3 C = 1 2 0 E E 0 D,!= 0 0 0 E where W, (0I)a. yv ( 0 I ) ' V V T + I w = 2 a i„, (Oi)a xy T yu (01)'a a l„,, xv uv T+1 w = 3 a x Z T ( O ' l ) ' a (OI)a xu xu 2 T V T + 1 E = L'L K'L L'K K'K and write the following product of matrices W2E W1E W3E W2*E WD = 0 0 W2E W3E W2EW2E + W1EW3E W3EW2E + W2'EW3E W2EW1E + W1EW2'E W3EW1E + W2'EW2'E WCWD = h 0 W2EW2E + W1EW3E 0 W3EW2E + W2'EW3E WDWD = 0 W2EW3E 0 W3EW3E Let the matrices Gl and G2 be of the following form: al„ (01) c Gl = (01)'d b l T+1 el„ (0I)g G2 = (01)'h f l T+1 118 where a, b, c, d, e, f, g, and h are s c a l a r s . Then the basic product of matrices G1E G2E can be written as aL'L -aL'K bK'L bK'K + c(0I)K'L c(0I)K'K d(0I)L'L dCOD'L'K eL'L eL'K fK'L fK'K + g(0I)K'L g(0I)K'K f hi(0I)'L'L h(0I)'L'K F l + F2 F3 + F4 so that we can write t r ( G l E G2E) = t r F l F 3 + trF2F3 + t r F l F 4 + trF2F4 Evaluating each of these expressions we obtain t r F l F 3 = ae tr[L'LLL] + (af+be) t r [K'LL'K] + bf tr[K'KK'K] trF2F3 = ce tr[(OI)K'LL'K] + cf tr[(OI)K'KK'L] + de tr[(0I)'L'LL'K] + df tr[(01)'L'KK'K] which s i m p l i f i e s to = %(ce + de) t r [K'LL'K] + ^ ( c f + df) t r [K'KK'K] using (3.1) and (3.4). Proceeding s i m i l a r l y as f o r t r F2F3 , we f i n d t r F l F 4 = Js(ag+ah) t r [K'LL'K] + %(bg + bh) t r [K'KK'K] Further, trF2F4 = eg tr[(0I)K*L(0I)K'L] + ch tr[(OI)K'K(OI)'L'L] + dg tr[(0I)'L'L(0I)K'K] + dh tr[(01)'L'K(OI)'L'K] which, when c = d , s i m p l i f i e s to 119 = cg%tr[JJ'L'L(OI)K'L] + c h % t r [ JJ'L'K(OI) 'L'L] = c(g + h) h t r [K'KK'K] using (3.1), (3.2) and (3.3). S i m i l a r l y , when g = h , we have t r F2F4 = g(c + d) ^ trfK'KK'K] . These s i m p l i f i c a t i o n s would be possible f o r the matrices Wl and W2. When c equals d or g equals h , the trace of G1E G2E s i m p l i f i e s to t r G1E G2E = ae tr[L'LL'L] + {af + be + Js[e(c+d) + a(g + h)]} tr[L'KK'L] + {bf + h[f (c + d) +b(g + h)] + k[(c + d) (g + h)]} t r [K'KK'K] (3.5) When neither c equals d or g equals h , the trace of F2F4 cannot be reduced to h[(c+d)(g+h)] t r [K'KK'K] . In F2F4 the c o e f f i c i e n t s of eg and dh are equal and s l i g h t l y lower than ^tr[K'KK'K] and the c o e f f i c i e n t s of dg and ch are equal and s l i g h t l y higher than %tr[K'KK'K]. Consequently, repl a c i n g the d i f f e r e n t traces i n F2F4 by ^tr[K'KK'K] produces small o f f s e t t i n g errors. As a matter of convenience, the trace of F2F4 i s approximated by h[(c+d)(g+h)] t r [K'KK'K] when c i s d i f f e r e n t from d or g i s d i f f e r e n t from h . Replacing the scalars a to g by t h e i r appropriate values i n (3.5), we can now evaluate the d i f f e r e n t traces. trWCWD = V(trW2EW2E + trW2'EW2'E + trWlEW3E + trW3EWlE} = %{trW2EW2E + trWlEW3E} = hid2 + a2 O2} t r [L'LL'L] xy y x + h{o (2a •+a'+:a ) + a 2 ( a 2 + a ) + a 2 ( a 2 + a )} t r [K'LL'K] xy uv. yu xv y u xu x v yv + %{[o + k(o +o ) ] 2 + (a 2 + a ) ( a 2 + a )} t r [K'KK'K] uv yu xv v yv u xu trWCWD = %(trW3EW2E + trW2'EW3E) = trW3EW2E = a 2 a t r [L'LL'L] x xy + {a 2[a +H(o + a ) ] + a (a 2 + a ) } t r [K'LL'K] x L uv ^ v xv yu xy u xu J + [(a 2 + a )(a +h(o +a ))] t r [K'KK'K] u xu uv yu XV trWDWD = t r W3EW3E = a 4 t r [L'LL'L] + 2 a 2 (a 2 + a ) t r [K'LL'K] X X u xu + (a 2 + a ) 2 t r [K'KK'K] . u xu F i n a l l y , the traces of WC and WD are trWC = Js[trW2E + trW2'E] = trW2E a t r [L'L] + [a +%(a +a ) ] t r [ K ' K ] xy uv yu xv trWD = trW3E a 2 t r [L'L] + (a 2 + a ) t r [K'K] X J u x u 1 121 APPENDIX 4 Limit of the Variance of Some Quadratic Forms It i s shown that the l i m i t , as the number of observations T tends to i n f i n i t y , of the variance of some quadratic forms i n normal variables tends to zero. This allows us to go from p r o b a b i l i t y l i m i t s to l i m i t s of expected values f o r these quadratic forms. The r e s u l t s are derived f o r three d i f f e r e n t types of quadratic forms. 1) X'L'LX We know from Appendix 2 that var[Q(L'L)] = 2 t r VL'LVL'L (4.1) where V i s the covariance matrix of the normal v a r i a b l e s . When the vector of normal variables i s X , V = ° l XT and (4.1) becomes var[(X'L'LX)/T] = ~ 2 a 4 t r L'LL'L . (4.2) T X I t i s easy to show that tr(L'LL'L) i s of order T(0(T)) since T T tr(L'LL'L) = I I L'L.. i = l 3=1 1 J and the matrix L'L has non zero elements on a f i n i t e number of diagonals independent of T . Consequently, from (4.2), l i m var[(X'L'LX)/T] = l i m ~ 2 O 2 0(T) = 0. T-x» T-*30 T X 122 For the vector Y , V = a y 1^ , and the same r e s u l t holds, 2) X'L'KU From Appendix 2, 0 L'K 0 L'K 0 L'K var Q = t r V V o 0 J 0 o (4.3) Since (X'U') i s the vector of normal variables considered here V = a I T (0I)a x T xu ( 0 I ) ' C T x u ^ T + l and (4.3) reduces to var(X'LKU) = 2 t r [ ( 0 I ) ' a L'K(0I)*a L'K] xu xu (4.4) Proceeding as i n Appendix 3, the following approximation to (4.4) i s developed 9 T W J - T - V ' W ' V (4.5) var [ (X'L'KU)/T] « 2 O hi t r K'KK'K xu Using (4.5) and the fact that t r K'KK'K i s of order T li m var[ (X'L'KU)/T] = l i m — • y a 2 0(T) = 0 This same r e s u l t holds f o r these quadratic forms X'L'KV, Y'L'KU and Y'L'KV . 3) Q(K'K) Using the same procedure as f o r 1) and 2) lim var[(U'K'KU)/T] = l i m var[(V'K'KV)/T] = 0 123 APPENDIX 5 Consistent Estimate of B 2 When a = (-i)(ef + a ) and a = -a , uv xv yu u xy the error terms i n the EIP model are uncorrelated with the independent variables so that the LS estimate of B i s consistent. From (2.22), (2.34) and (2.37), cov(KV - 8KU + Le , LX*) = cov(LY* - La - 8LX* , LX*) = cov(LY* , LX*) - 8 var (LX*) . Using the r e s u l t s of Appendix 1, cov(LY* , LX*) =L ft 2 L' var(LX*) = L ft22 L* . 2 When a = (-%)(a + a ) and o- = - o , i t follows immediately uv xv yu u xu that cov(KV - 8KU + Le , LX*) = [0] 124 APPENDIX 6 Asymptotic Variance of b We show how the asymptotic variance of the LS estimator 8 can be obtained using the formulas f o r the mean and variance of quadratic forms i n normal v a r i a b l e s . Since i n our p a r t i c u l a r case we can go grom asymptotic expectations to p r o b a b i l i t y l i m i t s , we can write the d e f i n i t i o n of the asymptotic variance as asyvar (3) = T Vim [ /T(3 -3) (3 -3)* Vf ] Furthermore. plim T(B.-3)(3-B)' =plim X*'X* -1 X*'Y*-X*:-'X*8 /f X*'Y*-X*'X*3 /T x*'x* plim[(X*'Y*Y*'X*)/T]+plim[(X*'X*X*'X*)/T]8 ~ 2plim[(X*'X*Y*'X*)/T]8 plim(X*'X*/T) 2 The d i f f e r e n t p r o b a b i l i t y l i m i t s could then be evaluated as follows: E[X*'Y*Y*'X*] fQ(C)l /T Var Q(C) E Q(C) /T /T x*'x*x*'x* • = Var Q(D)1 + • E Q(D)~1 • _ / f _ T _ -X*'Y*X*'X* Var Q(C) +Q(D) / f - Var Q(C) - Var Q(D) _ / T " _ _ / f _ E Q(C) • E Q(D) T T E X*'X* = E Q(D) T T where the r e s u l t s on quadratic forms are given i n Appendix 2. 126 APPENDIX 7 Small Sample Properties Using Centralized Variables This appendix looks at how the r e s u l t s from Appendix 3 are modified by assuming that the returns are ce n t r a l i z e d variables instead of mean zero v a r i a b l e s . Instead of the vector the vector MY MJV MX MJU f \ Y V X U which does not have zero mean, consider which does have zero mean and covariance matrix a 2M a MJ(OI)'M yv a M xy a MJ(0I)'M yu a M(0I)J'M a M yv xy a MJJ'M v a MJJ'M uv a MJ(0I)'M xv a M(0i)J'M a M X V X a MJ(0I)'M xu a M(0I)J'M yu a MJJ'M uv a M(0I)J'M xu 02MJJ'M Proceeding as i n Appendix 3, the basic product of matrices Gl G2 i s written as aML'L aML'L bMJJ'ML'L bMJJ'ML'L f. cM(0I)J'ML'L cM(0I)J'ML'L dMJ(OI)'ML'L dMJ(0I)'ML'L eML'L eML'L fMJJ'ML'L fMJJ'ML'L gM(0I)J'ML'L gM(0I)J*ML'L hMJ(0I)'ML'L hMJ(0I)'ML'L 127 and i t s trace i s found by summing t r F1F3, t r F1F4, t r F2F3 and t r F2F4. Evaluating these traces we f i n d , t r F1F3 = ae t r ML'LML'L + (af + be) t r (ML'K*K*'L) + bf t r K*'K*K*'K* , t r F2F3 = lg(ce + de) t r ML'KK'L + Jg(cf + df) t r (K*'K*K*'K*) , = t r F1F4 , and t r F2F4 = ls(c + d)(g + h) t r K*'K*K*'K* , using the following i d e n t i t i e s where K* = LMJ; ML'LML'L - L'LML'LM L'K*K*'LM - ML'K*K*'L and the same approximations as i n Appendix 3. Comparing these traces with the traces i n Appendix 3 we note that ML'L replaces L'L and K* replaces K. The same r e s u l t s as for mean zero variables then hold with c e n t r a l i z e d variables with ML'L and K* substituted f o r L'L and K. 128 APPENDIX 8 True $ and Covariance of Observed Returns It i s shown how the covariances of observed returns can be combined to obtain the true 8 • From the d e f i n i t i o n s of observed returns, Cov(x* , y*) = Cov(x t + u t + 1 - u f c , y f c + v t + 1 - v f c) = o + 2 a + o + o , xy uv xv yu Cov(x* + 1 , y*) = C o v ( x t + 1 = u t + 2 - u t + 1 , y f c + v t + 1 - v f c) a - a yu uv * ft Cov(x t , y t + ]_) = Cov(x t + u t + 1 - u f c , y f c + 1 + v f c + 2 - v ^ ) - a - a X V uv Var (x t) = Cov(x t + u t + 1 - u t , xfc + u t + 1 - ufc) a 2 + 2(a 2 + a ) , X u xu * ft and Cov(x t , x t + ]_) = Cov(x f c + U f c + 1 - u f c , x f c + ]_ + u t + 2 - u t + ]_) - (a 2 + a ) u xu From these equations, ft ft ft ft Cov(x t , y t ) + C o v ( x t + 1 , y t ) + Cov(x t , y f c + 1 ) Var(x*) + 2 Cov(x* , x* + 1) 129 a + 2 a + a + a - (a + a + a + a ) x y u y x v y u y u u v x v u v a 2 + 2 (a 2 + a ) - 2 (a 2 + a ) X u x u u x u _xy_ 130 APPENDIX 9-SW as an EIP Model This appendix shows that the SW model i s i n f a c t an EIP model. From the SW assumptions, ft V = P - P y t t+1 - S y t + 1 t - Sy t = p - P + T P - P 1 - T P - P I t+1 t L t+l-Sy ^ t + l J 1 t-Sy^ t J J t+1 t = y t + V t + l - V t ( 9 ' 1 } The SW model i s then an EIP model. Note, however, that the V s are not normally d i s t r i b u t e d . A l l the variances and covariances r e s u l t s of SW should be found from the corresponding EIP r e s u l t s . As an i l l u s t r a -ft ft ft t i o n we look at Var(y^) and CovCy^ , z^ _) , where y and z are two d i f f e r e n t s e c u r i t i e s with errors i n the prices v and w r e s p e c t i v e l y . From the EIP model, Var(y*) = a 2 + 2 ( a 2 + a ) (9.2) w t y v yv Cov(y* , z*) = a + 2a + a + a (9.3) t t yz vw zv yw Conditioning on Sy or S as appropriate and using the following r e l a t i o n , Cov(a,b) = E[Cov(a,b | s ) ] + Cov[E(a | s ) , E(b | s ) ] i t i s easy to v e r i f y that under the SW assumptions v a r ( v t + 1 ) = E ( S y t ) a 2 + y2 var (Sy t) , cov(y t , v t + 1 ) = - E ( S y t ) a y c o v ( z t , v t + 1 ) = - E ( S y t ) C y z cov(y t , w t + 1) = - E ( S z t ) a y z cov(v t , wt) = E[min(Sy t , S z t ) ] a y z + yy yz cov(Sy t , Sz t) Using (9.2) and the equations above * 2 2 var(y f c) = CT y + 2 [ y y v a r ( S y t ) ] which i s equivalent to SW equation 6. Using (9.3) and the equations ft ft I cov(y t , z f c) = a y z + 2jE[min(Sy t , S z t ) ] a y z + 2 yyyz cov(Sy t , Sz t) - E ( S y t ) a y z - E ( S z J a t yz a y z + E^[min(Sy t , Sz t) ] - [max(Sy t , Sz f c) ] | + 2 cov(Sy t , s z t ) y y y z which i s equivalent to SW equation 7. The other SW r e s u l t s could be derived s i m i l a r l y using EIP model r e s u l t s . 132 APPENDIX 10 Bid-Ask Spread and the EIP Model In a recent a r t i c l e , Richard R o l l (1984) suggests the following simple i m p l i c i t measure of the e f f e c t i v e bid-ask spread (S) i n an e f f i c i e n t market. S = 2 /-cov(y t , y ^ ) Let R o l l describe the bid-ask problem. "When transactions are c o s t l y to effectuate, a market maker (or dealer) must be compensated; the usual compensation arrangement includes a bid-ask spread, a small region of p r i c e which brackets the underlying value of the asset." This can be i l l u s t r a t e d as follows using R o l l ' s schematic. Spread hs h s ask p r i c e - true p r i c e bid p r i c e In terms of the EIP model, where P t = P t + v t v can take the values + hS or -hS which are equally l i k e l y . The 2 mean of v i s then 0 and i t s variance i s hS . Since the returns y are uncorrelated with v t +-^ w e have from Chapter III r e s u l t s * * 2 Cov(y , y 1) = -(a + a ) J t t - l v yv and solving f o r S S = 2 /-Cov(y* , y*_ 1) Rol l ' s measure of bid-ask spread follows immediately from the EIP framework. 134 APPENDIX 11 Pri c e Adjustment Delays and the EIP Model This appendix shows how the p r i c e adjustment delays (PAD) could be modeled i n an EIP framework. The PAD assumption implies that a portion of the change i n a sec u r i t y p r i c e i n a given period has i t s o r i g i n s i n previous periods. CHMSW have modelled t h i s as follows: where the observed return, or change i n p r i c e , y i s a weighted average of the post and contemporaneous i i d true returns Y t_] c • As mentioned i n Chapter I I , equation (11.1) brings with i t a d i v e r s i f i c a t i o n e f f e c t i n the sense that a weighted average of i i d returns has a lower variance than a one period true return. This i s an undesirable property of the CHMSW model because i f we think of changes i n prices as caused by new information, then any new information, even i f delayed, should have i t s f u l l impact on p r i c e . We suggest to write the observed returns as a function of contempor-aneous and past true changes i n prices as follows: k 't-k (11.1) (11.2) where 6, • t-k k+l are i i d f o r a l l t t-k are i i d f o r a l l t 135 = 0 ,-t-m irrt-1 Here f t k f t k \ + l ' \ represents the f r a c t i o n of the period t-k which gave r i s e to the portion of period t-k return that i s r e f l e c t e d xn y Since 6^ = 0 and ^ m+™ =-'- » a n y true return w i l l be completely r e f l e c t e d i n that same period observed return and the following m-1 observed returns. We can write (11.2) as an EIP model. y t = y t + Vt+1 ~ V t where v t+1 p(t+l) - p ( t + 6j) +p(t) - p ( t - l + 6^ X) + ...+ p(t-m+2) - p ( t - m + l + a 1 1 m + 1 ) m [p(t) - p ( t - l + 6^ X) + ... + p ( t - m + l ) -p(t-m+6 m m ) 

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