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Modelling daily returns of New York Stock Exchange by time series with noises having stable distributions Wang, Xiaohua 1993

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We accept this thesis as conformingto the required standardMODELLING DAILY RETURNS OF NEW YORK STOCK EXCHANGEBY TIME SERIESWITH NOISES HAVING STABLE DISTRIBUTIONSbyXIAOHUA WANGM.Eng., Beijing University of Astronautics & Aeronautics, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of StatisticsTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Xiaohua Wang, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignaturDepartment of ,..S'izt-ei'S tASThe University of British ColumbiaVancouver, CanadaDate^.47ttp45.t- 30 ., /993DE-6 (2/88)AbstractThis thesis provides some modelling procedures for the New York Stock Exchange(NYSE) daily data, using stable distributions.The use of stable distributions was motivated by the fact that traditional normalityassumptions are not appropriate for the daily stock returns. Four selected series, twostock indices and two individual industrial returns, were examined. Estimates for thecharacteristic exponent parameter a were obtained. Following the estimation of theexponent parameter a, serial—dependence was studied by fitting autoregressive models,using two different minimization criteria.The final conclusions are the following. The four daily stock returns were stablydistributed with characteristic exponent a less than two. The time series behavior ofthese data was suitably described by autoregressive models.IIContentsAbstract^ iiTable of Contents^ iiiList of Tables vList of Figures^ viAcknowledgement viii3. Introduction 12 Preliminary Data Analysis 52.1 Stock Returns ^ 52.2 Data^ 62.3 Violation of Normality ^ 62.4 Serial Dependence of Daily Stock Returns ^ 83 Properties of the Stable Distribution 103.1 Description ^ 103.2 Tail behaviour of Stable Distributions ^ 113.3 The Importance of the Exponent a 123.4 Estimation of the Exponent a^ 134 Modelling Procedure 154.1 Estimation of a ^ 164.1.1^Estimation of a by Tail Property ^ 164.1.2^Estimation of a by BL Method 17111CONTENTS4.2 Autoregressive Modelling ^  204.2.1 Fitting Autoregressive Models by Using the Yule-Walker Algo-rithm ^  204.2.2 Fitting Autoregressive Models by Minimizing Dispersion ^ 215 Empirical Results^ 235.1 Estimation of a  235.1.1 Estimation of a by Tail Probability — Regression ^ 235.1.2 Estimation of a by the BL Method — Recursive Algorithm^235.2 Autoregressive Modelling ^  245.2.1 Fitting AR Models by Using the Yule-Walker Algorithm ^ 245.2.2 Fitting AR Models by Minimizing Error Dispersion ^ 255.3 Evaluation ^  256 Conclusion and Discussion^ 29A Figures^ 31B Computer Implementation^ 54Bibliography^ 58ivList of Tables1 Summary Statistics ^ 82 Frequency Distribution 93 Estimation of a by Tail Probability ^ 264 Estimation of a and o-a2 by BL Method 265 The Critical Values for Hypothesis Test ^ 266 Autoregressive Modelling: AIC Selection and Regression Coefficients 277 Autoregressive Modelling — Minimizing Dispersion ^ 278 Comparison of Predictions — AR I Models ^ 279 Comparison of Predictions — AR II Models 2810 Comparison of Predictions — Random Walk Models ^ 28VList of Figures1 Time Series Plot-Equal-Weighted Index ^ 322 Time Series Plot-Value-Weighted Index 333 Time Series Plot-AT &T^ 344 Time Series Plot-General Motors ^ 355 Normal Probability Plot-Equal-Weighted Index ^ 366 Normal Probability Plot-Value-Weighted Index 367 Normal Probability Plot-AT &T ^ 378 Normal Probability Plot-General Motors ^ 379 Comparison of Density (I) ^ 3810 Comparison of Density (II) 3911 Autocorrelation Plots ^ 4012 Tail behaviour ^ 4113 Stable Probability Plot -Equal-Weighted Index (I) ^ 4214 Stable Probability Plot-Equal-Weighted Index (II) 4215 Stable Probability Plot -Value-Weighted Index (I) ^ 4316 Stable Probability Plot-Value-Weighted Index (II) 4317 Stable Probability Plot -AT & T (I) ^ 4418 Stable Probability Plot-AT & T (II) 4419 Stable Probability Plot-General Motors (I) ^ 4520 Stable Probability Plot-General Motors (II) 4521 Prediction-Equal-Weighted Index ^ 4622 Prediction-Value-Weighted Index 4723 Prediction-AT & T ^ 4824 Prediction-General Motors 4925 Comparison of Prediction Accuracy-Equal-Weighted Index ^ 50viLIST OF FIGURES26 Comparison of Prediction Accuracy—Value-Weighted Index ^ 5127 Comparison of Prediction Accuracy—AT & T^  5228 Comparison of Prediction Accuracy—General Motors ^  53VIIAcknowledgementI would like to thank Dr. Jian Liu, my supervisor, for the guidance and support whichled to the achievement of this thesis.I would also like to express my gratitude to Dr. Ruben Zamar for his careful readingand many helpful suggestions.Finally, I take this opportunity to thank the Faculty of the Department of Statistics aswell as my classmates for their help and encouragement during my studies.viiiChapter 1IntroductionFinancial markets are an important component of today's economies and the stockmarket is a large segment of the financial market. Research methods concerning stockmarket behaviour are primarily classified as quantitative (or empirical) and qualitative(also called experts' assessment) methods. In order to assure the correctness of thedecision-making processes and to maintain the investors' confidence, most large tradinghouses have sizable research teams engaged in quantitative or empirical research andqualitative assessments.Current studies concentrate largely on:(a) Average returns over relatively long horizons, such as weekly, monthly or quar-terly returns, where various linear time series models with Gaussian distributional as-sumptions are employed, and(b) Returns over shorter horizons such as daily returns, but with emphasis on mod-elling variation patterns or distributional patterns at specific times instead of modellingchanges over time.The advantage of examining returns over longer horizons is that averaging providesus with a much smoother series of observations and potentially better approximation tonormality. One pitfall of this approach is the difficulty of interpreting averaged returnsin the day-to-day decision process. Another drawback is the difficulty of selecting ofappropriate averaging time horizons. From a practical point of view, daily returns havemore appeal since they provide much more timely information. However, daily returns1CHAPTER 1. INTRODUCTION^ 2are more variable and, therefore, more difficult to model and forecast.Traditionally, the prevalent belief was that the distribution of stock returns could beadequately characterized by the normal distribution. For example, the so-called Bache-her (1900)—Osborne (1959) model is based on the assumption that transaction priceschange independently and thus the stock returns can be represented by an independent,identically distributed random walk. The model further assumes that the transactionsare spread fairly uniformly across time, and that the distribution of price changes, dur-ing transactions, has finite variance. Under these conditions, the normal or Gaussianassumption for the daily returns is justified by the central limit theorem.However, the Gaussian hypothesis was seriously questioned by Mandelbrot (1963). Inhis studies, Mandelbrot noted that academic research had neglected the implications ofthe leptokurtosis usually observed in empirical studies. Aware of such unconventionalsample behaviour, Fama (1965), Neiderhoffer and Osborne (1966), and other authorsreported that return series typically exhibit more kurtosis than that predicted by thenormal model. Subsequently, serious doubts about the validity of the normality assump-tion were raised, and this led to the investigation of alternative models.The family of stable distributions was introduced into economics and finance in theearly 1960's by Mandelbrot (1962). The stable distribution model soon became popularand still remains so in the research areas of economics and finance. One probable reasonfor the great popularity of the stable distributions in these areas is the fact that stabledistributions are able to explain sharp spikes or occasional bursts of outliers in observedtime series. Note that the normal distribution is a specific case of stable distributions.The time dependence features of daily stock returns are another interesting topic inthe study of stock behaviour. Some well known authors, for example, Fama (1965)and Mandelbrot (1962), have argued that stock returns can be represented by a randomwalk; However, considering the observations for more recent years, Akgiary (1989) foundthat some daily series exhibit much higher degrees of statistical dependence than thatwhich had been reported in previous studies.CHAPTER 1. INTRODUCTION^ 3The purpose of this thesis is to provide a modelling method for the New York StockExchange (NYSE) daily data which will be shown to have a non-Gaussian distribution,specifically, a heavy-tailed stable distribution. The modelling method contains a distri-butional study and a linear time series modelling. The distributional study concentrateson the estimation of the characteristic exponent parameter. The knowledge of this ex-ponent is very important for distinguishing the family of stable distributions from thespecific category in this family, the normal distributions. On the other hand, we applyan autoregressive model to appropriately model the time dependence structure observedon the daily return data.The proposed methods are then applied to the study of four daily return series, twostock indices and two individual securities from the New York Stock Exchange.The time series models are assessed via a cross-validation scheme by predicting the lasttwo years' observations and comparing the predictions with the actual observations.Autoregressive models and random walk models are compared to justify the previouslyargued serial dependence in daily returns. The comparisons are conducted using thepredictions for 1989 and 1990. Comparisons are obtained by applying three differentcriteria.The following conclusions are obtained: (a) The daily stock returns for New York StockExchange are not normally distributed and they follow stable distributions with expo-nent parameter less then two. The numerical results are given in Tables 3-4 and Figures1-20. (b) The autoregressive process describes the sequential behaviour for the dailyreturns better than the random walk model. The numerical results are given in Tables5-10 and Figures 20-28.The rest of the thesis is organized as follows. Chapter 2 describes the data base andmost importantly, it provides some empirical evidence which motivates the use of stablelaws and autoregressive models. Chapter 3 discusses some important properties of thestable laws which relate to the distributional study. Chapter 4 describes the modellingprocedure and estimation scheme for a linear time series with stable error distributions.CHAPTER 1. INTRODUCTION^ 4Chapter 5 lists the main findings from the application of the modelling procedure dis-cussed in Chapter 4. Finally, Chapter 6 states the final conclusions emerging from ourmodelling methods.Chapter 2Preliminary Data Analysis2.1 Stock ReturnsGenerally, a stock return r(I) is defined as the change in the value of an investment inthe nth security, over some time period from time / — 1 to time /, per dollar of initialinvestment.Stock indices are also considered in addition to the individual stock returns. Stockindices are calculated by using individual returns and they can give an overview of thestock market. Typically, there are two important kinds of stock indices, namely, theequal-weighted daily returns and the value-weighted daily returns. They are calculatedas follows.For the nth security on trading day I, the return R(/) is calculated as the weightedaverage of the returns for the individual stocks in the portfolio:R(I) = [EW,i(I)r,i(I)]1[EW,i(I)].By setting W(I) = 1 for all n, we obtain the so called equal—weighted portfolioindex, which has the same amount of dollars invested in each stock (security). For avalue—weighted portfolio index, the weight 147(/) assigned to the nth security is its totalmarket value.These two stock indices, the equal—weighted index and the value—weighted index, are5CHAPTER 2. PRELIMINARY DATA ANALYSIS^ 6considered to be important indicators because they provide an overview of the movementof all stocks in the market to some degree.2.2 DataThe data were obtained from the Center for Research in Security Prices (CRSP), Gradu-ate School of Business, University of Chicago, since "The CRSP stock files are designedfor research and educational use and have proven to be highly accurate" [1].The two stock indices mentioned previously were chosen for the study since they displaythe overall fluctuation of the stock market. Two daily individual stock returns, theAmerican Telephone and Telegraph (AT & T) and General Motors, are also involved inthe study to gain insight into individual stocks.The data were available for the period January, 1962 to December, 1990. The cur-rent study covered only the last 18 years, from January, 1973 to December, 1990. Thedata from 1973 to 1988 were used in the modelling and estimation processes. The datafrom the remaining years (1989 and 1990) were used to evaluate the forecasting power ofour models. The period from 1973 to 1988 was selected for the modelling step because:(a) since 1973, the stock returns have fluctuated more than they had previously, andthus the data are more closely related to the current situation, (b) this period lasts 16years, which is 4 times 4. Note that in United States, there is an election campaignevery 4 years and usually economic changes follow political changes.2.3 Violation of NormalityBasic summary statistics for the four returns series are listed in Table 1. During the18 year period (1973-1990), the stock returns rose and fell, assuming both positive andnegative values.The coefficients of skewness and kurtosis,^and -y2, are defined as:yi = p3/0-3,^and 72 = p4/0-4 — 3where, p3 and p4 are the third and fourth central moments respectively, and 0-2 is theCHAPTER 2. PRELIMINARY DATA ANALYSIS^ 7variance. Concerning the sample skewness -3,1 and sample kurtosis 5'2, we see the valueof =5'i ranges from -0.13 to -1.61, with a constant variance of 0.001 (varianceRe, 6/n, if nis large enough). The value of 112 ranges from 11.4 to 35.07, with a constant variance of0.005 (varianceP-J 24/n, if n is large enough). The examination of skewness and kurtosisshows strong evidence against the normality assumption. Note that under the hypoth-esis of normally distributed data, the kurtosis and skewness should be equal to zero.One can also analyze the distribution of returns by constructing frequency distribu-tions for the individual series, that is, for each data set, the empirical proportions ofreturns within given standard deviations of the mean can be computed and comparedwith what would be expected if the distributions were exactly normal. The results ofthis process are listed in Table 2, which can be taken as further evidence against thenormality assumption.Figures 1-4 are time series plots of the four daily returns mentioned earlier. Clearly,the series exhibit many spikes during the entire period under consideration. The mostsignificant jump occurred on October 19, 1987, which is often referred to as Black Mon-day.The Q-Q plots (Figures 5-8) of daily returns against the standard normal distribu-tion also clearly demonstrate the inconsistency of the normality assumption. Similarly,Figures 9-10 show the comparison of densities. Obviously, the empirical densities fordaily returns have heavier (longer) tails than the corresponding normal distributions.In summary, we observe the following characteristics in these series:(a) Sharp spikes or occasional bursts.As opposed to yearly or monthly returns, daily returns tend to have more frequentbursts of substantially larger magnitude.(b) Long (heavy) tailed distributions.Daily return series seem to have much longer (heavier) tails compared to the bench-mark, the normal distribution. These empirical behaviours speak against the conven-tional normality assumption. There is therefore a need for a model which attracts moreprobability to extreme observations, i.e., to adopt a heavy-tailed distribution.CHAPTER 2. PRELIMINARY DATA ANALYSIS^8Table 1: Summary StatisticsEqual-WeightedIndexValue-WeighedIndexAT&T General MotorsMean 7.7e-04 4.4e-04 5.4e-04 4.2e-04Max 0.098 0.089 0.164 0.146Min -0.142 -0.180 -0.212 -0.210Standard Deviation 8.6e-03 9.6e-03 12.7e-03 15.7e-03Skewness -1.26 -1.61 -0.48 -0.13Kurtosis 30.08 35.07 11.4 28.50Returning to the time series plots Figures 1-4, we notice that these plots exhibit severalsharp spikes or occasional bursts. These spikes show a pattern which resembles thesample path of stable distributions. The long tail phenomenon observed in density plots(Figures 9-10) suggests that stable laws may be used as an alternative approximationto the daily stock returns.2.4 Serial Dependence of Daily Stock ReturnsThe daily series being studied show strong time dependency. This characteristic is shownin Figure 11, the autocorrelation plots of daily returns. The evidence of dependencesuggests the convenience of using a time series model.CHAPTER 2. PRELIMINARY DATA ANALYSIS^ 9Table 2: Frequency DistributionUnit Normal Equal-WeightedIndexValue-WeightedIndexAT&T General Motors0.5 S 0.3830 0.7497 0.7409 0.8856 0.73621.0 S 0.6826 0.7497 0.7409 0.8856 0.87221.5 S 0.8664 0.9612 0.9480 0.9691 0.94412.0 S 0.9545 0.9826 0.9777 0.9946 0.98542.5 S 0.9876 0.9916 0.9910 0.9946 0.98543.0 S 0.9973 0.9946 0.9955 0.9946 0.99464.0 S 0.9999 0.9975 0.9983 0.9988 0.99885.0 S 0.9999 0.9985 0.9997 0.9993 0.999Chapter 3Properties of the StableDistribution3.1 DescriptionTo apply the stable distributions (aslo called stable Paretian distributions) to the studyof daily stock returns, we first introduce the family of stable distributions.As mentioned in previous chapters, the family of stable distributions was introducedin economics and finance by Mandelbrot (1963) during his study of the empirical dis-tribution of stock price changes. A stable distribution, as Mandelbrot labelled it, isany distribution that is stable or invariant under addition. That is, the distribution ofthe sum of independent, identically distributed stable Paretian variables is itself stableParetian and except for origin and scale, has the same form as the distribution of the in-dividual summands. The Normal distribution is an example of this kind of distribution.Strictly speaking, a random variable Z is said to have a stable distribution if for eachn>l, there exist real numbers an > 0 and bn such that for any independent, identicallydistributed (i.i.d.) random variables Z1,....Zn with the same distribution as Z,Zi + Z2 + • • • 4- Zn Ian Z + bn,^ (1)where "s" denotes "equivalence in distribution".Generally, the stable laws cannot be described by density or distribution functions.10CHAPTER 3. PROPERTIES OF THE STABLE DISTRIBUTION^11However, they can be described by the corresponding characteristic functions (see forinstance Brockwell & Davis (1987)). The form of the characteristic function used hereis:o(u) lexpfiufi — dluN1 — iOsgn(u)tan(?)11 if (2)expliud3 — 411[1 + sgn (u)ln I zi Ii}^if a = 1whereif u 0sgn(u)^lul (3)^0^if u = O.Stable distributions have four parameters: a, #, 0 and d. The parameter a iscalled the characteristic exponent of the distribution and ranges over the interval (0, 2).It determines the height of the extreme tails of the distribution. When a = 2, thedistribution represented is the normal distribution. In fact, by letting a = 2, 9= g andd = (72/2, then we haveb(u) = iutt^cr.2u 2which is the characteristic function of the normal distribution with mean y and varianceu2.When a is in the interval (0, 2), the extreme tails of the stable distribution are higherthan those of the normal distribution. The total probability in the extreme tails getslarger as a gets smaller. The parameter /5' is the location parameter. In the case wherea> 1, 9 is the mean of the distribution. The parameter 0 is an index of skewness, and—1<0<1. If 0 = 0, the distribution is symmetric. The sign of 0 indicates the directionof skewness. The parameter d determines the scale of the distribution. In the case whena = 2, as we have seen, d is one half the variance; if a < 2, the variance is infinite andd only indicates the scale.3.2 Tail behaviour of Stable DistributionsIt has been discovered that the family of stable distributions possesses an importantproperty which is usually called "tail behaviour". Suppose Y is a stable-distributedCHAPTER 3. PROPERTIES OF THE STABLE DISTRIBUTION^12random variable with exponent parameter a. Then we have:y(1Y1 > y) —4 C^(4)as y —4 oo, where 0 < a < 2, or equivalently,P(IY1 > y)--, Cy',where C is a finite positive constant.This tail behaviour is a very important characteristic; it provides researchers a wayto study the family of stable distributions.3.3 The Importance of the Exponent aThe exponent parameter a attracts much more attention than the other three parame-ters 0, d and # in the study of stable distributions. The parameter a is treated as thedecisive parameter. The importance of a can be explained both in economics and instatistics as follows.If the stock market is described by Gaussian process (a = 2), then no sudden changesare expected in the path of the returns. If the market is described by a stable distri-bution with exponent parameter a <2, the returns of a security or an index will oftentend to jump up or down by very large amounts during short periods.The fact that there are a large number of sudden changes in a stable- distributedmarket means that such a market is riskier than a Gaussian market. The variability ofan expected yield is higher in a stable Paretian market with a < 2 than it would bein a Gaussian market, and the probability of large losses is greater. As a result, theparticular value of the exponent parameter a is important to economists.From a statistical point of view, the value of a represents two different but relevanttypes of distributions: normal distributions and the general stable distributions. To geta better description and prediction for the stock returns, different statistical theories canbe applied. If the value of a could be accurately estimated then the most appropriatestatistical model could be chosen.CHAPTER 3. PROPERTIES OF THE STABLE DISTRIBUTION^133.4 Estimation of the Exponent aWhat makes the study of stable laws difficult is that in general a closed form for thecorresponding density and distribution functions is not available.The original estimation of a is derived from the tail property described in section3.2 (Mandelbrot (1962)). Researchers have examined the linear relationship between/n(PIYI > y) and ln(y) to gain information on a. Some other methods, such as esti-mating a by range analysis (Fama (1965)), by sequential variance (Fama (1965)) andby the conditional maximum likelihood method (Booth and Glassman (1987)) have alsobeen reported.The accuracy, the complexity and the interpretation in the estimation of a are ma-jor problems in all the methods used thus far.Brockwell and Liu (1991) provide an effective method for the estimation of the dis-tributional parameters of stable laws. Based on the empirical characteristic functions,the method critically depends on the values of the two real auxiliary variables, u1 andu2. The authors construct an estimator of the exponent a, et, which is a function ofu1 and u2, and prove that this estimator of a is strongly consistent and asymptoticallynormal. The asymptotic variance of ar is also provided.Brockwell and Liu's method can be described briefly as follows.Assume {Xt} is an AR(p) process satisfying:Xt Cb1Xt-1 02Xt-2 — • • • — OpXt—p Zt^(5)where {Zt} is a sequence of independent identically stable distributed random vari-ables with characteristic function i,b(u) described by Equation (2), and the coefficients{OA satisfy:1 — cbiz — c62z2 — • • — OpzP^0,^V1.215.1.Then for any two arbitrary non-zero real values u1 and u2, which satisfy lull 1u21,inl/n10(ui)11— hil/n10(u2)11^6)—  (Injuil — lnludCHAPTER 3. PROPERTIES OF THE STABLE DISTRIBUTION^14Applying the empirical characteristic function of {Xt}, it is found thatini/n11%.,(u1)11— inj/n17Ru211 (7)Iniu21(8)0-2(el — a)—>N(0, cra2).^ (9)(10)where--1-61^21,R.1)12/.10(.01 21i:4.2)1210,4.2)1 ) •^(11)Let Re(c) denote the real part of any complex quantity c, and let /m(c) denote theimaginary part of c. Letwhereandtc, 1)2i- (0 )D2152^0^0^Re(u2 ( Re(7077)(U1)) J'4 11))^4^?2)) .1712(0(12)) )1 n1'1(o) f/ar(Y) = 71' E^— —0(yi —y; (cos(uiXi), sin(uiX;), cos(u2X3), sin(u2X j))'The advantage of the Brockwell and Liu (BL) estimation method is that for anyautoregressive time series with stable distribution, it provides an estimator as well asthe asymptotic variance of the estimator. The calculations for obtaining these quantitiescan be performed easily on today's computers. Note that the values of the two auxiliaryvariables u1 and u2 are of critical importance, and changes in their values could causea large difference in the results of the estimation process. In the application of the BLtheory, my choice of u1 and u2 is such that the asymptotic variance of er is minimized.whereand as n—too,cr2 can be estimated by( = E eiuXjn j=1act,2^ InIU21)2Chapter 4Modelling ProcedureIn chapter 2, we discussed the evidence that daily stock returns are unlikely to be nor-mally distributed. In addition, the evidence of serial correlation as discussed in Chapter2 implies that daily stock returns may be well described by autoregressive models.For an autoregressive stock return series with stably distributed noise, the main mod-elling procedure used in this study consisted of two steps:(a) estimation of exponent parameter a;(b) autoregressive modelling.For comparison purposes, two different methods are used in each step. The linearregression method gives the first estimation of a, which is based on the well known tailproperty (Fama (1965)). A recursive algorithm derived from Brockwell and Liu's theorygives the second estimation.Autoregressive models can be fitted either by applying a modified Yule-Walker algo-rithm or by minimizing the so called "dispersion" defined by Stuck (1978) in his studiesof the error term for the family of stable distributions. A specific version of selectionof Akaikes Information Criterion (AIC) related to stable distributions is used to choosethe order for the autoregressive models.15CHAPTER 4. MODELLING PROCEDURE^ 164.1 Estimation of a4.1.1 Estimation of a by Tail PropertyA straightforward estimate of a may be obtained by studying the tail behaviour for agiven time series.Let {Xt} be a stock returns series. Suppose it is stably distributed with decisive pa-rameter a. According to the tail property, as y—ioo,In(P(IXtl> y), —aln(y)d- C^ (16)here, C is a constant. If we know the log of the empirical tail probability, 1nP(IXt1 > y),and relevant In(y), then we can get the estimate of a by applying Equation (16).Let h(y) be the empirical tail probability. SetYi = Ym + Ay *^ (17)for i = 1, 2, ^Ny, whereAY = (Ym — Ym)/Ary,^ym = max{Xt, t =1, 2, ^Ny},^Ym min{Xt, t =1, 2, ^Ny}.Let Ny be an integer (large enough), where Ny determines the magnitude of pointsbeing searched.The empirical tail probability h(y1) can be calculated byh(y) = ny,/Ny,where ny, is the number of observations of {Xt} which fall outside the interval (—yi, yi).The rough estimation of a is obtained by regressing {In(h(yi))} against {/n(yi)} forthose values of i which satisfy io<i<Ny. We choose the value of io to ensure the limit-ing feature of tail behaviour.n n+^cosiu(xi — X)],Joii=1 (18)CHAPTER 4. MODELLING PROCEDURE^174.1.2 Estimation of a by BL MethodLetU = {(u1, u2), /21, u2 E R1, lull^1u21, u1^0, u2^0}where, u1 and u2 are real numbers. An estimate of a, et and its asymptotic variance?it can be obtained for any point (u1, u2) E U by applying the theory developed byBrockwell and Liu.With the observations {Xj}, from Equation (8), it is then easy to derive:kb(u)I =where, n is the number of observations.Since ex and fra2 are functions of (u1, u2), we can define them as:^et^&(u1, u2)and 6-0,2 =^u2).From Equation (7) and Equations (10)-(13), we have-172114((u2)11 u2) =(19)inluil - //dudand..6•« (u/ u 2 ) = l5t (0)131 (ln1 u1 I — in i u2 D2^(20)whereRels(u2)1^Imli(u2)1 ^)fy =___ ( ^Relikui)1^Imic-b(u1)1 (21)10(u1)12/40,01^1,"40,012/76(ui.)1^it(u2)12/4(u2)1^li(u2)12to(u2)1 ) •As mentioned in Chapter 2, ex and its estimated variance ii,v2 depend on the selectionof u1 and /22. Concerning the stability of a, an iterative algorithm would be as follows.Step 1 Selection of ul and u2 from USuppose the searching of a contains I iteration loops. Define a subset of U, U(i) (i =1,2, ... , /) as:LT(){ {(u12, 41)), k^1,2,...,K, 1 = 1,2,...,L}whereulf1 Auli) * k^ (22)CHAPTER 4. MODELLING PROCEDURE^ 18andu2 + ,64) *^ (23)and where (u12, 4) is the initial point for the ith search. AuP and Atii) are thesearch steps for u1 and u2 respectively.Step 2 Calculation of a and 6!By using the formulas listed in Chapter 3, we can perform the calculations fromEquation (19) and (20), replacing u1 by 42, u2 by uV. This gives us 6(42, uV), andbi(42,uV).Now, letk, 1) =^u21(i)andaa2 (i, k, 1) = a!(uii2 (-u2?).Suppose that a(i, kO, 10) is an estimate of a at the point (t4i20, 4), where k0 and10 satisfy:kO, 10) =^min^{6-!(i, k, 1)}.^(24)i<k<K,1<l<LNote that a(i, kO, 10) is of more interest because its variance is the minimum for the ithsearch in the domain of WO.Step 3 Determination ofSet a positive number Co to be the upper bound for the variability of the estimationwherea(i) =^kO, 10).Iflaw —^ I > co(i)go back to Step 1 and treat (4 „) -2to) as a new initial point, ie., in Equation (23)and Equation (24),(1+1)^(i)U10 - Ulk0CHAPTER 4. MODELLING PROCEDURE^ 19and(i+1)(i)U20 = U210.Then, one can start a new search for the optimal estimation of o-,!, and a.Ifl a(i)^a(i-1)1 < co,stop the iteration process; ei(i) is the desired estimate, in other words,=The estimation of a can be obtained either by regression on tail probability or by therecursive algorithm of BL. The regression is more straightforward and easy to perform,but no clear rule exists for determining the bound for the "tail", ie., for deciding thevalue of io. The only thing known about it) is that it should be large enough to meetthe requirement of limitation.Ihe recursive algorithm can provide the estimation of a, '6, as well as the varianceof the estimation, O.! at the auxiliary values u1 and u2. Considerable calculation isrequired to obtain proper values of u1 and u2. However, with the rule of minimizing O.!and the stopping criterion Co, the optimal estimation of a can be obtained. Further-more, we can test the normality assumption of stock return distribution by studyingthe confidence interval for a. We can see if the distribution of stock returns is normalor not by setting the following hypothesis:Ho : a = 2,: a < 2.From Equation (9), we reject the null hypothesis at the 95% significance level if< 2 — 1.65c,c1V71The null hypothesis of normality can be rejected if the above inequality is true; thisresult would be consistent with previous studies that the stock returns are not normallydistributed.CHAPTER 4. MODELLING PROCEDURE^ 204.2 Autoregressive Modelling4.2.1 Fitting Autoregressive Models by Using the Yule-WalkerAlgorithmNote that the Yule-Walker (YW) algorithm can be used for a stably distributed seriesprovided sample autocorrelations are observed. For the autoregressive process describedin Equation (5), suppose the coefficient vector 4) to be 44) = (6,1, 6,2, , Op)'. Then thematrix form of the YW equations is:(25)where r is the correlation matrix (p x p). r can be written as:1 71 72 7p — 11 7p-2= 72 71 1 7P-37p-1 7p-2 7p-3 •^•^. 1and .7 = (71, 72, 7p)'.Given observations Xi, X2, .. , X, and possible order p (which may or may not equalthe true order p), we can replace the correlation {-yi} by the corresponding correlation{-"•i}, and thus get the so-called Yule-Walker estimator 4)= • • , iskp)i•Because the true order p is unknown, we assume it is bounded by some finite constant.Knight (1989) discussed minimizing Akaike's Information Criterion (AIC) to estimatethe order p for an autoregressive process. His procedure is described below.Let {Xt} be a pth order autoregressive process which is stably distributed with exponenta. Then p may be obtained by applying the Yule-Walker method and minimization AICover the integers 0,1, , (n). AIC is defined as:AIC = nln(er2(k))+ 2k,where o2(k) is the usual estimate of the innovations variance obtained from YWestimate, and k is the number of estimate coefficients, n is the number of observations,CHAPTER 4. MODELLING PROCEDURE^ 21K(n) is chosen as:K(n). (:)(n6),where< 1 — a/2 if c>1and8 < a/2 if a <1.It has been proven that the estimated order f) is weakly consistent.The maximum order of an autoregressive model can be determined if we know thevalue of K(n).^Note that in Knight's study, the expression K(n)^0(726) is not an exact quantity.To compute K(n), setK (n) = [n6 + 1] + K 0, (26)where Ka is a non-negative integer which can be decided by checking the statisticalfeatures of the residuals.The significance of coefficients should be checked by examining the t-values for coef-ficients. General diagnostic checking should also be applied to see if the residuals of theAR model are independent identically distributed.4.2.2 Fitting Autoregressive Models by Minimizing Disper-sionAlternatively, one may fit the autoregressive models by using specific criteria relevantto the stable laws.Let {Xi} be an autoregressive process stably distributed with the exponent parametera. A simple way of fitting the linear model is to find the coefficients fcbil by minimizingerror "dispersion", where the dispersion is defined by:i=p^disp(X) = E ix, - E^.j=p+1^i=1(27)CHAPTER 4. MODELLING PROCEDURE^ 22The optimization is easy to perform if the order p and exponent a are known.It has been shown by Brockwell 8,z Cline (1984) that the minimized predictor=PXn = E:7=1is optimal in the sense that it minimizes the probability of large predictor errors. Fora given order p, one can fit the autoregressive model by minimizing dispersion. Thedetermination of the order, however, is still a problem.Of the two methods of autoregressive modelling provided in this section, it is not clearwhich one can provide more accurate predictions. The two methods are based on dif-ferent optimization principles: minimizing least squares error (Yule-Walker) and mini-mizing error dispersion. Although Brockwell & Cline (1984) showed that for a purelyautoregressive process, the minimum dispersion predictor is exactly the same as theleast squares predictor, in practice, there could be a slight difference in the estimation.From the calculation point of view, the YW algorithm is available in general statisticalpackages; the numerical approximation algorithm of minimizing dispersion is availablein some computer packages.Chapter 5Empirical ResultsIn this chapter, empirical results will be obtained by performing the modelling procedurediscussed in Chapter 4. A brief comparison is also reported.5.1 Estimation of a5.1.1 Estimation of a by Tail Probability — RegressionFigures 12 shows the plots of (1n(y),1n(h(y)) (see section 4.1.1), where h(y) is theempirical tail probability which approximates P(iXti > y) and {Xt} is a stock returnseries. Applying the estimation method related to the tail property, estimations of a forthe four stock series were obtained. Table 3 displays the linear regression result fromthe equationln(h(y)) = —aln(y)+ C.For the four daily returns, the values of el range from 1.90 to 1.59. None of them equalsto two.5.1.2 Estimation of a by the BL Method — Recursive Algo-rithmThe desired estimates of a and cr! were achieved after a number of selections. Thoseselections were made by applying the recursive algorithm described in Chapter 4, andby starting from the origin point (0, 0) for each of the stock series studied. The final23CHAPTER 5. EMPIRICAL RESULTS^ 24results are shown in Table 4. The value of t2 is contained in a very small range. Thevalue of ranges from 1.6 to 1.9 approximately.Furthermore, Table 5 gives critical values for the hypothesis test at the 95% signif-icance level, the hypothesis test is described in section 4.1.2. The null hypothesis ofnormality is rejected since none of the estimated values of a is greater than the corre-sponding critical values.Comparisons of probability plots of normal versus stable Paretian distributions areshown in Figures 13-20 (The stably distributed probability plots have been scaled forconvenience of comparison). The improvement gained by applying stable distributionsis obvious. It seems that the recursive algorithm gives better approximations of thedistribution of stock returns.We conclude the four daily stock returns series studied do not posses normality fea-tures. The series are stably distributed with exponent parameter a less than 2.5.2 Autoregressive Modelling5.2.1 Fitting AR Models by Using the Yule-Walker Algo-rithmWith the estimated a, the employment of the Yule-Walker algorithm requires us to fixthe maximum order K (n) discussed in Chapter 4. We have set K (n) by using Equa-tion (26). The determination of Ko is made by checking the properties of the residualswhich resulted from experimental autoregressive fitting, K(n) is chosen to be theautoregression order which generates uncorrelated fitted residuals.Table 6 shows the values of 8 in the determination of the order p and the results fromautoregression estimation (the coefficients for autoregressive models and related t val-ues (in brackets)). The maximum order of the AR models ranges from 2 to 4. All thecoefficients are significant (the absolute values of t are greater than 2).CHAPTER 5. EMPIRICAL RESULTS^ 25The autocorrelation plots of residuals provide no significant evidence against the as-sumption of independent identically distributed error terms. In other words, the au-toregression results can be reasonably accepted.5.2.2 Fitting AR Models by Minimizing Error DispersionWe obtained alternative linear models by minimizing the dispersion defined by Equa-tion (27). Table 7 lists the numerical results from using this minimization criterion.To facilitate the comparison, the orders for the autoregressive predictors are the sameas those in Table 6. The computer program used for the calculations is listed in Ap-pendix B.Comparing Table 6 and Table 7, we see that for each series, the coefficients estimatedby the two different methods are very close. This may be due to the fact that thevalues of the exponent a are close to two (1.90 and 1.82). This is consistent with Brock-well Sz Cline's result: for purely autoregressive process, the two methods provide thevery similar estimations, and hence AR models are appropriate.The statistical behaviour of the four daily stock returns series studied can be describedby autoregressive models with lags less than 5.5.3 EvaluationIn this study, one of our main interests has been autoregressive modelling. Using thestock returns data from 1989 to 1990, two types of autoregressive models (Tables 6-7)were obtained. These one-step-ahead predictions were compared with those obtainedfrom the random walk model, which is often used in stock studies.The predictions obtained from autoregressive models (by using Yule-Walker algorithm)are shown in Figures 21-24. The numerical results are displayed in Tables 8-10. ByCHAPTER 5. EMPIRICAL RESULTS^ 26Table 3: Estimation of a by Tail ProbabilityYm = 0, ym = 0.1, Ay = 0.001Paramters Stock SeriesEqual-WeightedIndex(io = 30)Value-WeightedIndex(if, = 50)AT&T(i0 = 50)General Motors(i0 = 60)a 1.69 1.92 1.72 1.740 -10.44 -10.53 -9.20 -8.02Table 4: Estimation of a and cr! by BL MethodStock Series u1 u2 & 6-,?,,Equal-Weighted Index -10.50 4.50 1.90 0.49Value-Weighted Index -185.00 -27.25 1.82 1.33AT&T 57.75 -57.50 1.79 0.28General Motors -105.00 1.00 1.90 0.67letting ei denote the error term for the ith prediction, we can compare the accuracy byusing the following three criteria:Effip Ei2,^kik^and maxi<i<ni leij.Obviously, the random walk models are the worst, while the autoregressive modelsobtained by using the Yule-Walker algorithm and by minimizing the error dispersionprovide roughly the same accuracy. Figure 25 also shows the absolute values of theerror terms for each of the three models; the plots in this figures are consistent with thecomparison results listed in Tables 8-10.Table 5: The Critical Values for Hypothesis TestSignificant Equal-Weighted Value-Weighed AT&T General MotorsLevel=95% Index IndexCritical Values^1.98^1.96^1.98^1.97CHAPTER 5. EMPIRICAL RESULTS^ 27Table 6: Autoregressive Modelling: AIC Selection and Regression CoefficientsEqual- WeightedIndexValue- WeightedIndexAT &T General Motors8 0.16 0.04 0.14 0.21K(N) 4 2 4 24 2 4 2'i. 0.34 (42.3) 0.18 (18.6) -0.06 (-4.9) 0.04 (2.5)1k2 -0.05 (-6.2) -0.05 (-5.4) -0.05 (-3.8) -0.04 (-2.7)ti)3 .04 (4.4) -0.05 (-3.8)(-44 0.06 (6.8) -0.09(-6.9)Table 7: Autoregressive Modelling - Minimizing DispersionEqual- WeightedIndexValue- WeightedIndexAT &T General Motors73 4 2 4 2'i 0.35 0.17 -0.02 -0.04(42 -0.04 -0.04 -0.05 -0.04(43 0.04 -0.05 -isk4 0.06 -0.07 -Table 8: Comparison of Predictions - AR I Models(Note: ci denotes error term, m=505Stock Series CriteriaE:r Ei2 Esi:rin icil maxi<i<m HEqual- Weighted Index 0.02 2.12 0.04Value- Weighted Index 0.04 3.06 0.06AT & T 0.13 6.12 0.07General Motors 0.13 5.89 0.08CHAPTER 5. EMPIRICAL RESULTS^ 28Table 9: Comparison of Predictions — AR II ModelsNote: ci denotes error term, m=505)Stock Series Criteria=II e2 Eli:T kil maxi<i<m kilEqual- Weighted Index 0.02 2.05 0.04Value- Weighted Index 0.04 3.06 0.06AT & T 0.12 6.16 0.08General Motors 0.12 5.85 0.08Table 10: Comparison of Predictions — Random Walk Models(Note: ei denotes error term, m=505Stock Series CriteriaE:7 ci2 Etinn Icil maxi<i<mEqual- Weighted Index 0.03 2.59 0.04Value- Weighted Index 0.06 4.12 0.08AT Si T 0.31 9.19 0.15General Motors 0.24 8.33 0.14Chapter 6Conclusion and DiscussionOur study indicates that empirical distributions of daily stock returns series conformbetter to stable Paretian distributions with characteristic exponents a less than twothan to the normal distribution (which is also stable Paretian but with characteristicexponent exactly equal to two). Thus, the stock market is more varying than one wouldexpect under the Gaussian hypothesis.In addition, since the daily returns exhibit significant levels of dependence, the ran-dom walk assumption is not appropriate for the daily returns data.The autoregressive models can be fitted either by using the Yule-Walker algorithmor by minimizing the error dispersion. For the same autoregressive order p, these twoalgorithms provide similar coefficient estimates.In our distributional study, our observation of the tail property led us to estimate thecharacteristic exponent parameter a. The selection of the cut-off point for the "tails"is not clearly determined since the computations are based on the asymptotic feature.Different selection may produce different values of ex.Brockwell and Liu's theory provides an alternative estimate for a. Note however, thatin the calculation of et, the initial value for the two auxiliary variables were randomlychosen to be the origin. Thus the final optimal point (u1, u2) is obtained by local mini-mization rather than global, so the value of ex may not be unique. In addition, further29CHAPTER 6. CONCLUSION AND DISCUSSION^ 30research into the numerical relationships among (u1, u2), ii and era2, is required.The fact that the daily stock returns may be stably distributed with exponent parametera less than two suggests the reconsideration of some methodologies previously used forthe study of daily data. Furthermore, developments in statistical measurement criteriaand modelling methods are needed for the situations when the conventional normalityassumption is violated. All of these topics merit further study.Appendix AFigures31APPENDIX A. FIGURES^ 32—13CDEqual-Weighted Index-0.15^-0;^ 110 -0.05^0.0^0.05^0.10 Figure 1: Time Series Plot—Equal-Weighted IndexValue-Weighted Index-0.15^-0.10^-0.05^0;0; 1 1 0.05APPENDIX A. FIGURES^ 33Figure 2: Time Series Plot—Value-Weighted Index-0.2^ -0.10 —AT&T0;0 0.118 --APPENDIX A. FIGURES^ 34Figure 3: Time Series Plot—AT &TGeneral Motors-0.2^ -0.1^ 0.0^ 0.11 1-13a)APPENDIX A. FIGURES^ 35Figure 4: Time Series Plot—General MotorsAPPENDIX A. FIGURES^ 36Li,1.-9.„----------'-'.-2^0^2Quantiles of Standard NormalFigure 5: Normal Probability Plot-Equal-Weighted IndexLo0x 0a)-c,c O_ ot)a)—Lt,c) o._a) Ocb=lit->....-------.-2^0^2Quantiles of Standard NormalFigure 6: Normal Probability Plot-Value-Weighted IndexAPPENDIX A. FIGURES^ 37T.0^ .i..■-------#cv.9.-2^0^2Quantiles of Standard NormalFigure 7: Normal Probability Plot—AT SiTT0^ . .0.o...,,------,9c49. .-2^0^2Quantiles of Standard NormalFigure 8: Normal Probability Plot—General Motors0CD0CVo I-0.15^-0.10^-0.05^0.0^0.05^0.10^ -0.15^-0.10^-0.05^0.0^0.05^0.10Equal-Weighted Index: empirical Value-Weighted Index: empirical-0.15^-0.10^-0.05^0.0^0.05^0.10^-0.15^-0.10^-0.05^0.0^0.05^0.10Equal-Weighted Index: normal Value-Weighted Index: normalAT & T: empirical General Motors: empirical-0.2^-0.1^0.0^0.1^ -0.2^-0.1^0.0^0.1-0.2^-0.1^0.0^0.1^0.2^-0.2^-0.1^0.0^0.1^0.2AT & T: normal General Motors: normali^„^ ,^0U.co00100•001 •^ '/1'^-/1'0-40721ttit-400-4-D,L4v-14.)1-4bO4-43003000<00LL0< et.000.^1^.^. I^.^.^. 1 • 4  1 1^I^•r^•^. I -110^20LagEqual-Weighted Index10^20LagValue-Weighted Index0^10^20^30 0^10^20^30^Lag LagAT & T General Motors-4 In(y)-8^-6Value-Weighted Index-4-8 In(y)Equal-Weighted Index-6-4 In(y)-8^-6General Motors-4 In(y)-8 -6AT & To .^•^•C? '.,0 •c? •4?.^.:42-4^-2^0exponent paramter=1.69Figure 13: Stable Probability Plot -Equal-Weighted Index (I)LOT92 4--4^-2^0exponent parameter=1.90Figure 14: Stable Probability Plot —Equal-Weighted Index (II)0.ino .9 .r7.4r•....Lo0.ILO0.9.Lo,9odoci60.oAPPENDIX A. FIGURES^ 42ino .0.o z S •..APPENDIX A. FIGURES^ 43Ci;0. .0too .,0e••/OP..-4 -2 0 2 4exponent parameter=1 .92Figure 15: Stable Probability Plot -Value-Weighted Index (I)-4^-2^0^2^4exponent parameter=1 .82Figure 16: Stable Probability Plot—Value -Weighted Index (II)APPENDIX A. FIGURES^ 44,..••••• zpo. .••• • • • 8 ••drT0q09.c‘l9.-4^-2^0^2^4exponent parameter=1.72Figure 17: Stable Probability Plot —AT & T (I)T .00.09•.-4^-2^0^2^4exponent parameter=1 .79Figure 18: Stable Probability Plot—AT & T (II).•-•ci .i0079.(49 -APPENDIX A. FIGURES^ 45-4^-2^0^2^4exponent parameter=1.74Figure 19: Stable Probability Plot-General Motors (I).q0ci/.--.N.9 "-4.-2^0^2^4exponent parameteer=1.90Figure 20: Stable Probability Plot-General Motors (II)APPENDIX A. FIGURES^ 46-OA^tO^0.01 4010 OA VMSV8Figure 21: Prediction—Equal-Weighted IndexaAPPENDIX A. FIGURES^ 474.63^4.01^401^0.03^•.02 441 0.0 461 462^4.04 •On^to^1.92^4.02 4.01^0.0^041^042Figure 22: Prediction-Value-Weighted IndexFigure 23: Prediction—AT T-024 422 OZ2SetasgiaasgiAPPENDIX A. FIGURES^ 48BAPPENDIX A. FIGURES^ 49OAS^.0.04^VA^13.04a19aBBFigure 24: Prediction-General Motorso§0 •0a)3;T-MaCDAPPENDIX A. FIGURES^ 50^Absolute Difference^Absolute Difference0.0^0.01^0.02^0.03^0.04^0.0^0.01^0.02^0.03^0.04Figure 25: Comparison of Prediction Accuracy—Equal-Weighted Index0.3CDaCDAPPENDIX A. FIGURES^ 51Absolute Difference^ Absolute Difference0.0^0.02^0.04^0.06^0.08^0.0 0.01^0.03^0.05Figure 26: Comparison of Prediction Accuracy—Value-Weighted IndexAbsolute Difference0.0^0.05^0.10^0.15Absolute Difference0.0^0.02^0.04^0.06APPENDIX A. FIGURES^ 52Figure 27: Comparison of Prediction Accuracy—AT TAPPENDIX A. FIGURES^ 53Absolute Difference^ Absolute Difference0.0^0.04^0.08^0.12 0.0^0.02^0.04^0.06^0.08Figure 28: Comparison of Prediction Accuracy—General MotorsAppendix BComputer ImplementationMINIMIZATION ERROR DISPERSION by USING UMPOL/DUMPOLALGORITHMc^Variables DeclarationsCINTEGER I, NINTEGER MAXFCN,NOUTREAL FCN, FTOL, FVALUE, S, X(4), XGUESS(4)EXTERNAL FCN,UMACH,UMPOLCc^Inputing the Initial Values of Estimators, XGUESS(N),c^and the Initail Values of FTOL, MAXFCN, and Sc^Inputing the Initial Values of Estimators, XGUESS(N),c^and the Initail Values of FTOL, MAXFCN, and SCPRINT *, 'THE ANALYSIS of GENERAL MOTOR TIME SERIES'N = 2FTOL = 1.0E-5MAXFCN = 500S = 1.0DATA XGUESS 0.02, -0.02OPEN (33, FILE = 'file-coef')54APPENDIX B. COMPUTER IMPLEMENTATION^ 55OPEN (44, FILE = 'file-out')CPRINT *, 'The Number of Estimators, N = ', NPRINT *, ' FTOL = ', FTOLPRINT *, `MAXFCN = ', MAXFCNPRINT *, 'S = ',SPRINT *, 'The Initial Values of Estimators, XGUESS(N) = ', XGUESSCc^Starting the Direct Search Polytope AlgorithmCCALL UMPOL(FCN, N, XGUESS, S, FTOL, MAXFCN, X, FVALUE)Cc^Getting Information of OutputsCCALL UMACH(2,NOUT)Cc^Writting the Final Values of Each Coefficient and Optimal ValueCPRINT *,`FVALUE = ', FVALUEWRITE (33,100) X, FVALUE100^FORMAT (1X, 2(E15.4, 1X), 2X, E15.4)CPRINT *,`The Program Compeleted. Thank You !!!!'cCLOSE (33)CLOSE (44)ENDccc^The SUBROUTINE of the FCNc The subroutine is designed for establishing theobjective function and related calculationsCAPPENDIX B. COMPUTER IMPLEMENTATION^ 56SUBROUTINE FCN (N, X, F)Variables DeclarationsREAL X(N), F, AFREAL AlphaPARAMETER (Alpha = 1.90)INTEGER I, J, K(30), L, M, NDIMENSION Y(5000)PARAMETER (J=4044)Openning the File of Initial ObservationsOPEN (22, FILE = 'file-data')DO 300 L = 1, JREAD (22,*) Y(L)300^CONTINUECc Calculating the Value of the Objective FunctionCAF = 0.0DO 500 M = N+1, JER = Y(M)DO 400 I = 1, NK(I) = M - IER = ER - X(I)*Y(K(I))400^CONTINUEAF = AF+(ABS(ER))**Alpha500 CONTINUEF = AFCc^Saving the Information of Each Iteration by X(N) and FVALUEAPPENDIX B. COMPUTER IMPLEMENTATION^ 57WRITE (44,650) X, F650^FORMAT (1X, 2(E15.4, 1X), 2X, E15.4)PRINT *,`Running the FCN SUBTROUTINE'Closing the Data File for Next IterationCLOSE (22)RETURNENDNote:Here, the file "file-coef" is the file to record the final estimation of coefficient ofminimization, the file "file-data" is the data file which provides the observations ofstock returns. For more information, see IMSL [19].Bibliography[1] Abraham, Bovas and Ledolter, Johannes (1983), Statistical Methods for Forecasting,Wiley Series in Probability and Mathematical Statisticas. John Wiley & Sons.[2] Akgiry, Vedat (1989), "Conditional Heteroscedasticity in Time Series of Stock Re-turns: Evidence and Forecasts", Journal of Business, Vol. 62, No.1,55-80.[3] Anderson, 0.D. (1980), A nalysing Time Series, North-Holland.[4] Bickel and Doksum (1976), Mathematical Statistics: Basic and Selected TopicsOkland: Holden-Day, INC.[5] Blattberg, Robert C. and Gonedes,Nicholas J. (1974), "A Comarison of Stable andStudent Distributions as Statistical Models for Stock Prices.", Journal of Business,Vol. 47 , 24-80[6] Brockwell, Pter J. and Davis, Richard A. (1986), Time Series Theory and Methods,Springer-Verlag.[7] Brockwell, Peter J. and Liu, Jian (1991), "Estimating the Noise Parameters fromObservations of a Linear Process with Stable Innovations", Journal of StatisticalPlanning, Vol.[8] Campbell, J. (1987), "Stock Returns and Term Structure", Journal of FinancialEconomics, Vol. 18, 373-399.[9] Cline, Daren B.H. and Brockwell, Peter J. (1985), "Linear Prediction of ARMAProcesses with Infinite Variance", Stochastic Processes and their Applicatins, Vol.19, 281-296.58BIBLIOGRAPHY^ 59[10] Copeland, Thomas E. and Weston, J. Fred (1988), Financial Theory and CoporatePolicy, Addison-Wesley Publishing Company.[11] Fama, Eugene F. (1965), "The behaviour of Stock-Market Prices", Journal of Busi-ness, Vol. 38, 34-05.[12] Fielitz, B., and Rozell, J. (1983), "Stable Distributions and Mixtures of Distribu-tions Hypothesis for Common Stock Returns", Journal of the American StatisticalAssociation, Vol. 78, 28-36.[13] French, K., Schwert, G. and Stambaugh, R. (1987), "Expected Stock Returns andVolatility", Journal of Financial Economics, Vol. 19, 3 -29.[14] Graupe, Daniel (1984), Time Series Analysis, Identification and Adaptive Filtering,Malabar, Florida: Robert E. Krieger Publishing Company.[15] Grimmett, Geoffrey and Stirzaker, David (1990), Probability and Random Pocess,Oxford; Clarendon Press.[16] Holden, K., Peel, D.A. and Thompson, J.L. (1990), Economic Forecasting: anintroduction, Cambridge: Cambridge Univerity Press.[17] Hsu, Ber-Ann, Miller, Robert B. and Wichern, Dean W. (1974), "On the StableParetian behaviour of Stock-Market Prices", Journal of the American StatisticalAssociation, Vol. 69, NO. 34, 108-113.[18] IMSL Problem-Solving Software Systems, MATH/LIBARY, FORTRAN Subrou-tines for Mathematical Applications, IMSL Inc.[19] Knight, Keith (1989), "Consistency of Akaike's Information Criterion for InfiniteVariance Autoregressive Processes", The Annals of Statistics, Vol. 17, No. 2, 824—840.[20] Shapiro, Alan C. (1990), Modern Corporate Finance, New York: Macmillam Pib-lishing Company.[21] Simkowitz, M., and Beedes, W. (1980), "Asymmetric Stable Distributed securityReturns", Journal of the American Statistical Association, Vol. 75, 306-312.BIBLIOGRAPHY^ 60[22] Tsay, R. (1987), "Conditional Heteroscedastic Time Series Models", Journal of theAmerican Statistical Association, Vol. 82, 590-604.[23] Tucker, Alan L. (1992), "A Reexamination of Finite— and Infinite-Variance Dis-tributions as Models of Daily Stock Returns", Journal of Business & EconomicsStatistics, Vol. 10, No.1 73-81.[24] Whittle, Peter (1983), Prediction and Regulation by Linear Least—Square Mehtods,Oxford: Basil Blackwell Publisher.

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