MSU LIBRARIES m. \9 RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. EVALUATION OF EFFICIENT MARKET TESTS BASED ON DAILY STOCK RETURNS by Michael D. Atchison A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Finance and Insurance 1982 Gilflo/ ABSTRACT EVALUATION OF EFFICIENT MARKET TESTS BASED ON DAILY STOCK RETURNS By Michael D. Atchison Estimates of ccmnon stock betas (systematic risk) traditionally have been calculated using ordinary least squares (01.3) and monthly stock returns. With the availability of daily returns, betas carputed using OLS and daily stock returns have recently appeared in the literature. However, Scholes and Williams [1977] have established that these betas are biased and inconsistent. The purpose of this paper is to investigate the effect of this bias and inconsistency on predicting security returns and the testing of the efficient market hypothesis. Analytical, very large sample, results were derived establishing the effects of the inconsistency on the prediction of security returns. The effect of the inconsistency on the prediction error was very small. The increase of the mean squared prediction error was less than 0. 10%. A computer simulation based on Merton' 3 continuous time model was utilized to generate daily stock returns which allowed for the veri- fication of the analytical result and the testing of small samples. The same tests conducted using simulated data were performed using daily stock returns from 283 firms listed on the New York Stock Exchange. Each test was performed using OLS and Scholes and Williams estimated betas. Mean squared prediction errors were calculated. The summary statistics of the prediction errors were approximately the same regardless of which estimator was used. A t-test and emula— tions average residuals test was performed on several samples to establish whether or not an investigator would be able to find abnormal performance using the Scholes and Williams estimator more often than using the OLS estimator. Abnormal performance was found equally well employing either estimator. The implication of this study is that OLS estimators can be utilized in future research; furthermore, conclusions of past research, based on OLS estimators , are valid. CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER II. III. IV. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES INTRODUCTION AND OVERVIEW Nonsynchronous Time Periods Overview LITERATURE REVIEW Scholes and Williams EMH Tests Using Daily Return Data Efficient Market Test Procedures BETA BIAS AND MARKET MODEL PREDICTIONS Market Model Predictions SIMULATION MODEL STRUCTURE Verification of the Simulation SIMULATION RESULTS Beta Bias Prediction Results Efficient Market Test Results Cumulative Average Residuals 31 38 59 CHAPTER VI. CHAPTER VII. APPENDIX 1 TEST RESULTS AND CONCLUSIONS FROM TESTS USING NYSE FIRMS Conformance to Scholes & Williams Prediction Results Efficient Market Test Results SUMMARY AND CONCLUSIONS SIMULATION COMPUTER PROGRAMS BIBLIOGRAPHY ii 104 122 127 139 5.6 mm 0 LIST OF TABLES First-Order Autocorrelation Coefficients for the Market: 1971-1975 Daily Returns on Low-Volume Portfolios Regressed on Value Weighted Market Returns Daily Returns on Intermediate-Volume Portfolios Regressed on Value Weighted Market Returns Daily Returns on High-Volume Portfolios Regressed on Value Weighted Market Returns Percent of 50 Firms in Which Cross-Correlations are Statistically Significant Estimates of MSPEOLS — MSPESW The Instantaneous Expected Market Return and Market Variance Estimation Average Number of Transactions for 283 Firms Conformance of Simulated Results to Scholes and Williams' Theory Beta Estimates From Simulated Returns and the Corresponding Biases Beta Estimates from Simulated Returns and the Corresponding Biases for 3 Portfolios Formed by Transactions Times A Comparison of Alternative Parameter Estimates Sample: Firms 1-10 Percentage of 12 Replications Where the Null Hypothesis is Rejected A Comparison of Alternative Parameter Estimates Sample: Firms 11—20 Percentage of 12 Replications Where the Null Hypothesis is Rejected A Comparison of Alternative Parameter Estimates Sample: Firms 1-5 and 16-20 Percentage of 12 Replications Where the Null Hypothesis is Rejected A Comparison of Alternative Parameter Estimates Sample: Firms 6-15 Percentage of 12 Replications Where the Null Hypothesis is Rejected A Comparison of Alternative Parameter Estimates Percentage of 48 Replications Where the Null Hypothesis is Rejected Average T-Statistics Sample: Firms 1-10 Average T—Statistics Sample: Firms 11-20 iii 19 20 21 22 24 37 43 51 58 61 62 82 83 84 85 86 87 88 Average T-Statistics Sample: Firms 1—5 and 16-20 Average T-Statistics Sample: Firms 6—15 Average T-Statistics Number of Replications in Which the S&W CAR Was Closer to Zero Betas for NYSE Firms Using the Value Weighted Market Index Percentage of Firms in Which the Beta Was Adjusted in Accordance With S&W Prediction Results for the 283 NYSE Firms Estimation Period 1000 days, Prediction Period 100 days T-Statistics From Efficient Market Tests Performed Using NYSE Firms iv 89 90 91 103 106 112 114 117 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 LIST OF FIGURES The Relationship Between the Actual and Observed Returns Positive Covariance of Non-Trade Periods Negative Covariance of Non-Trade Periods Equal Non-Trade Periods Division of a Sample Used in a Brown and Warner Test Time Periods Price Generation The Four Prices Generated in Days 1 and 2 Autocorrelation of the Market as Firms Are Added Autocorrelation of the Market Adding a Larger Proportion of Less Actively Traded Securities Bias in Beta Versus Estimation: OLS Bias in Beta Versus Estimation: S&W Bias in Beta Versus Estimation: OLS Bias in Beta Versus Estimation: S&W Bias in Beta Versus Estimation: OLS Bias in Beta Versus Estimation: S&W Bias in Beta Versus Estimation: OLS Bias in Beta Versus Estimation: S&W CAR Plots Firms: 1— CAR Plots Firms: 1— Average and 100 Average and 100 Average and 400 Average and 400 Average Transactions Days Transactions Days Transactions Days Transactions Days Transactions and 1000 Days Average Transactions and 1000 Days Average Transactions and 5200 Days Average Transactions and 5200 Days Number of Mean Squared Prediction Errors 12 Replications Which are Smaller—Estimation Periods: 100 Days and 400 Days Number of Mean Squared Prediction Errors Out of 12 Replications Which are Smaller-Estimation Periods: 1000 Days and 5200 Days Mean Squared Prediction Errors Portfolio Results 10 10 CAR Plots Firms: 11—20 CAR Plots Firms: 11—20 CAR Plots Firms: 1-5 and 16-20 CAR Plots Firms: 1—5 and 16—20 CAR Plots Firms: 6- 15 V Per Per Per Per Per Per Per Per Out Day Day Day Day Day Day Day Day of 11 12 12 30 47 47 54 55 63 64 65 66 67 68 69 70 74 75 76 94 95 96 97 98 99 100 5.19 CAR Plots Firms: 6-15 101 6.1 CAR Plots Equally Weighted Index 120 6.2 CAR Plots Value Weighted Index 121 vi I. INTRODUCTION AND OVERVIEW Estimates of common stock betas (systematic risk) have traditionally been calculated using ordinary least squares (OLS) and monthly returns. These in turn have been used in testing the efficient market hypothesis. With the recent availability of daily returns through CRSP, several studies have employed betas computed using OLS and daily returns (Jaggi [1978] and Brown [1978]). However, Scholes and Williams [1977] and Dimson [1979] have shown that daily betas derived using OLS are both biased and inconsistent. An estimator is unbiased if its expected value is equal to its true value. An estimator is consistent if the esti- mator can be made to lie arbitrarily close to the true value with a probability arbitrarily close to one. In other words, as the sample size gets larger, the estimator approaches the true value (Maddala [1977, Chapter 4]). The purpose of this paper is to investigate the effect of this bias and inconsistency on predicting security returns and the testing of the efficient market hypothesis. The effect will be determined by comparing predictions and test results computed using OLS estimators, to those computed using consistent estimators derived by Scholes and Williams (S&W). Scholes and Williams have established that OLS betas are biased and inconsistent. The effects of the bias and inconsistency on efficient market studies need to be determined. Efficient market studies usually divide a sample of stock market returns into two periods, one period is prior to an "event", and is used to estimate parameters (the alpha and beta for a firm). The second period is used to compute residuals or forecast errors. The errors are the difference between forecasts made using the parameters of the first period and the observations of the second period. The forecasts will be affected by beta and therefore, the effects of biased and inconsistent betas on prediction is the focus of this study. Nonsynchronous Time Periods The OLS bias is a result of nonsynchronous time periods occurring in daily returns. Nonsynchronous time periods arise from individual firm daily returns being computed from the last security transaction of the previous day to the last transaction of the present day. Consequently, the intervals over which daily returns are calculated are not of equal length. This causes serial correlation in daily return data which has been noted in several other research papers, Fisher [1966], Fama [1965], and Schwartz and Whitcomb [1977]. Serial correlation in daily return data is an indication of the common econometric problem of errors in variables, which produces biased and inconsistent OLS estimates of beta and alpha.1 1The errors in variable problem is discussed in Johnston [1972], Chapter 9 and Maddala [1977], Chapter 13. Betas are usually computed using the market model as follows: Rnt = o‘nt + BntRMt + éntr (1'1) where Rnt = return on security n ant = intercept for security n Bnt = slope coefficient for security n RMt = market portfolio return ént = error term for that security n. The error, e is assumed normal with mean zero and nt' variance 02(én ). Merton [1973] has shown that equation t (1.1) is consistent with continuous return theory. The main assumption is that all risky securities have prices distributed as infinitely divisible lognormal random variables. As a result of this assumption, continuously compounded returns Rn on risky securities n = 1, 2, 3...N t as calculated over any time period are joint normally distributed with constant mean un, constant variance 0:, and constant covariance.2 Daily returns are those returns calculated over a time period of one day. However, the data observed, returns calculated from prices quoted in the market (i.e. organized exchanges), are returns calculated from the last transaction of the 2See Scholes and Williams [1977, p. 310) for further discussion of this assumption. See Merton [1973] for other assumptions than the main one listed here. previous day to the last transaction of the present day. The returns on the CRSP daily tapes are calculated in this manner. The observed return period may span more than a day or less than a day. In other words, observed daily returns are measured over nonsynchronous time periods. Since daily returns described by equation (1.1) cannot be observed, there are measurement errors in the data used. Two daily returns have now been established, the daily return in equation (1.1) (the actual return), and the daily return that is observed (the observed return). The actual return is the continuously compounded return that is accumu- lated over a one day holding period. This return cannot be observed. The observed return is the return that is measured from last transaction to last transaction. Figure 1 shows the relationship between these two returns. Snt represents the period during the day t where no transaction takes p1ace3. The observed return period is over the period (t-l—S t—Sn ), and the return is nt-l' t designated R: Equation (1.1) calculated on the observed t. series becomes: Rnt = OLnt + Bnt RMt + wnt (1'2) 3For consistency, the notation used in this paper is the same as that used by Scholes and Williams [1977]. As will be shown in chapter II, returns R: and Rit exhibit . . t serial correlation. .snl an sn3 observed :;_’____I:___\,___fi___?_‘,____Jr s s Rhl ha Rh3 actual Rhl RhZ Rn3 t.= 1 l:= 2 t-= 3 Fig. 1.1 The relationship between the actual and observed returns Scholes and Williams [1977] and Dimson [1979] have shown that the problem described above will affect the estimation of beta. The fact that this problem exists is supported by Schwartz and Whitcomb [1977]. Therefore, the effects of nonsynchronous time periods on estimation have been established in the literature. However, the effects of nonsynchronous time periods on the prediction of security returns have not been esta- blished. Prediction is important because prediction is an integral part of many efficient market studies. The market model (equation 1.2) is typically used to forecast or predict an expected return, and the residual, the difference between the expected return and actual return observed, is used as the basis of isolating abnormal performance. Overview The contribution of this study is to establish the effects of nonsynchronous time periods on prediction of security returns and, in turn, efficient market tests. The theoretical results are derived in chapter III. The market model is a model with stochastic regressors and therefore the results in chapter III are for infinitely large samples. Smaller samples may or may not result in similar conclu- sions. In order to establish the small sample results a computer simulation must be performed. A computer simulation is used to obtain a large enough sample to get results congruous with the theoretical large sample results. Once the large sample size is established, smaller samples can be used to see how conclusions change as the sample size is reduced. The use of simulation requires a model, and once the model is obtained, parameter estimates are necessary. The model used is based on the continuous time model put forth by Merton [1971]. This is the model assumed by S&W [1977]. The only difference is that the simulation model adds nonsynchronous time periods. Actual returns (those without errors) as well as observed returns can be simulated. The actual returns are used to determine the sample size at which the consistent beta estimators converge on the true beta. The derivation of the model is described further in chapter IV as well as the method of obtaining parameter estimates. The effects of nonsynchronous time periods on efficient market tests will be investigated using a procedure similar to that used by Brown and Warner [1980]. An efficient market test using both the inconsistent OLS estimates and the consistent S&W estimates will be conducted using the simulated returns without abnormal performance and with abnormal performance added. The desire is to see if one estimator outperforms the other. Performance is measured by the ability to isolate abnormal performance. The results of this test are discussed in chapter V. Results and conclusions based on simulation must be supported by real data results. This is necessary since simulation is based on a model that may or may not describe actual data. Therefore, the same tests performed using the simulated data were performed using the returns from 283 New York Stock Exchange firms. This was done to see if the simulation results were consistent with the results obtained using real firms. The results of these tests are discussed in chapter VI. As mentioned earlier, Scholes and Williams and others have established that the use of daily returns in the market model results in biased and inconsistent estimates of alpha and beta. The basic uses of these estimates are: l) the evaluation of risk, 2) prediction, and 3) the testing of the efficient market hypothesis. This study addresses the last two uses. Does the fact that the estimates are biased and inconsistent affect predic- tion, and does the fact that the estimates are biased and inconsistent affect the conclusions of efficient market tests? The answers to these questions affect both the interpretation of past research and the design of future research. The conclusions of past research may be invalid. Efficient market tests conducted in the future may or may not have to be adjusted for the bias. This study will attempt to answer the above questions. II. LITERATURE REVIEW Scholes and Williams' [1977] (hereafter, S&W) analysis of beta leads directly to the subsequent analysis of this paper. Therefore, the major portion of the literature review chapter is devoted to the S&W analysis. The other articles included are either in support of S&W or an aid to further analysis. Scholes and Williams Scholes and Williams [1977] assume that at least one trade takes place each day and the periods of non—trading are independent and identically distributed across time.4 These assumptions lead to relationships 2.1, 2.2, and 2.3:5 s 2 2 —O Var (R t) + 2 Var (S ) U , (2.1) where Rit = observed return for firm n at time t 0: = variance of the actual daily return = Var (Rnt), Sn = period of non—trading for security n, un = the average return for security n, and Var (Sn) = the variance of the period of non—trading. 4 This assumption appears reasonable since Hawawini [1980] shows the most siggificant cross correlation of returns occur at lags of -1 day. 5Derivation of these properties are shown in Scholes and Williams [1977, p. 325]. 9 10 Equation (2.1) shows that the variance of the observed return series is greater than that of the actual return series. Consider that the time periods for which these returns are calculated are always changing. One return could be calculated for slightly more than one day (Ril) while the next day's return may be of time period slightly 8 n1 could be greater in magnitude less than one day (R32)' R than the corresponding actual return (Rnl) because R21 has a longer holding period. Riz could be smaller in magnitude than the corresponding actual return (an) because R22 has a shorter holding period. The returns R31 and Riz would be more variable than Rn1 and an but have the same mean. The covariance between the observed return of two securities is as follows: 5 s _ _ _ - Cov (R R ) —o E[max(Sn,Sm) mln (Sn’sm)] o + nt’ mt nm nm where onm = the covariance between the actual returns of security n and security m, E[max(Sn,Sm) - min (Sn,Sm)] = the overlap in no-trade periods for which only one security was traded, and llmdln = the average return for securities m and n. Two situations must be considered in discussing equation 11 (2.2). The first situation is when Cov (Sn,Sm) is positive (see Fig. 2.1), and the second situation is when Cov (Sn'sm) is negative (see Fig. 2.2). Positive covariance indicates that when an increase in Sn occurs, an increase in Sm will more than likely occur. The corresponding period over which the returns are calculated for both securities will decrease and as a result, both returns will decrease. A decrease in Sn and Sm will cause an increase in the observed returns for securities n and m. The observed returns will exhibit a positive covariance due to the fluctuation of Sn and 8m since Cov (Sn'sm) ”hum is positive. Negative covariance implies a decrease in SD which will probably be accompanied by an increase in Sm’ The length of the time periods will vary in opposite directions and cause the observed returns to do likewise. Negative covariance (2 Cov (Sn’sm) unum negative) will reduce the observed covariance. nl n2 n3 Rs n 1 Sal SmZ sm3 { it Rs 0 l 2 3 m Fig. 2.1 Positive Covariance of Non-Trade Periods 12 Snl Snz . sn3 . R3 :1 sml 81:12 81113 RS 0 1 2 3 m Fig. 2.2 Negative Covariance of Non-Trade Periods sn1 an Sn3 c-‘t-n r—JS—q Rs :1 Sml sz Sm3 o 1 2 3 m Fig. 2.3 Equal Non-Trade Periods The other component of equation (2.2), E[max(Sn,Sm) - min (Sn,Sm)]onm, is the expected length of time for which the time periods of R: and R: are not concurrent. During t t this period of time the two returns would not covary and the actual covariance is reduced. If E[max(Sn,Sm) - min _ = . s s (Sn,Sm)] — 0, then Sn SIn (see Fig. 2.3), but Cov (Rnt'Rmt) # Onm because the covariance term 2 Cov (Sn'sm) still has the same properties discussed in the previous paragraph. 13 8 RS ) = O is when Sn = S and The occurrence of Cov (R , nt mt nm S = S for all t. n nt The covariance between the observed return of security n and the prior day's observed return of security m is as follows: COV (RS Rs ) = (E[max(S - S 0)]0 — nt' mt-l n m' nm COV (Sn,Sm) Unum (203) This equation is similar to equation (2.2) except E[max(Sn - Sm,0)] is the expected overlap of the time . s 5 periods for Rnt and Rmt-l' than zero then the two returns have a portion that is con— If E[max(Sn - Sm,0)] is greater current and as a result covary. The other component of the equation Cov (Sn’sm) unum results from the covariance of the length of time for returns R: and RS If Cov (Sn, S ) t mt—l' m is positive and if the length of time for Rit increases then Rit-l decreases. Therefore, the sign of the term Cov (Sn' Sm)unum must be negative. The Scholes and Williams relationship is derived from equation (2.1), (2.2), and (2.3) using: E[max(Sn- Sm) - min (Sn,Sm)] = E[max(Sn-Sm,0)] + E[max(Sm_Sn, 0)] (2.4) Substituting (2.4) into (2.2) leads to: 14 s = _ _ _ . . Cov(RS Rm t) o (E[max(Sn Sm,0)]onm Cov(bn,bm)unmm) nt' nm -(E[max(Sm-Sn,0)lonm-Cov(Sn,Sm)unum) (2.5) Substituting (2.3) into (2.5) results in: s s _ _ s s _ s s Cov(Rnt,Rmt) — Ohm Cov(Rnt,Rmt_l) Cov(Rnt_l,Rmt) (2.6) Let XnM represent the weight of security n in the market index M. Multiplying both sides of equation (2.6) by ng and summing over n - 1, ..., N results in: C0V(Rnt,RMt) — Cov(Rnt,RMt) C°V(Rnt'RMt-l) - Cov(R:t_1,RSt ) (2.7a) where the market return is designated by M. Multiplying both sides of equation (2.7a) by X and summing nM over n = 1, ..., N results in: s s s Var(RMt) = Var(RMt) - 2 Cov(RMt,RMt_l) (2.7) Dividing (2.7a) by (2.7) leads to: s s s Cov(Rnt,RMt) Cov(Rnt,RMt) Cov(Rn t'RMt- l) s s Var(RMt) Var(RMt) Var(RMt) Rs Cov(RSnt_1. RMt) (2.8) Var(R;t) 15 Substituting the following into (2.8): Cov(RS , S ) B = nt RMt ,the observed beta (2.9) n Var(RS ) Mt Cov(RS ,RS ) 33+ = “ts Mt'l , one day lead observed beta (2.10) n Var(RMt) s s _ Cov(R ,R ) BS = nt Mt+1 , one day lag observed beta (2.11) n Var(RS ) Mt Cov(R , ) 8n = nt RMt , the actual beta (2.12) Var(RMt) results in the Scholes and Williams relationship: 85 = 8 (1 + 203) — 85+ - as“ (2.13) n n n n s . where 0 equals the autocorrelation of the observed market. Equation 2.13 is estimated using the estimator _ “ “5+ “5- sw ‘ B0L5 + 8M + 8N l + 283 A 16 where BSW = the S&W estimator, and A As+ AS- . BOLS' BM , 8N are OLS estimators. The magnitude of Scholes & Williams relationship (equation 2.13) depends on the size of OS I Bfi+ and 82-. The market return is an average of individual security returns. An actively traded security would appear to lead the market because it is traded more often than the market average and, therefore, would appear to react to an event prior to a market reaction for the same event. As a result, in equation (2.16) B:+ would dominate. An inactive stock would appear to react more slowly than the market to an event, because it is traded less often than the market average and 8:- would dominate. A security which is traded approximately the same as the market should react to an event at approximately the same time as the market and 82+ s- 3+ 3 s and 8n should be small. If 8n . 8n and D are insignificant the adjustment would also be insignificant. 08 has been computed on a daily basis by both Scholes and Williams [1976, p. 34] and Schwartz and Whitcomb [1977, p. 299]. Both Scholes and Williams, and Schwartz and Whitcomb found the autocorrelation of the value weighted market index 1971-1975 to be .291 while Scholes & Williams found the autocorrelation of the equally weighted market to be .458 (see table 2.1). The significance of Bg+and Bg'depends on the significance of the lead and lag covariances of the security with the market or, alternatively, the lead and lag l7 cross correlations. The denominators of equations 2.9, 2.10, and 2.11 are all equal, therefore, differences can be attributed solely to differences in the numerators. The numerators are comprised of the concurrent, lead, and lag covariances respectively. Hawawini [1980] computed the lead and lag cross correlations for 50 firms with the S&P 500 index for 1 to 20 day leads and lags and found 72% of the 1 day lead cross correlations and 100% of the 1 day lag cross correlations to be statistically significant. As a result of these studies, it appears that these adjustments will make a difference. Scholes and Williams empirically tested their relationship by applying the estimator to five portfolios formed from all stocks listed on the New York Stock Exchange and American Stock Exchange between January 1963 and December 1975. The portfolios were formed based on volume traded. Volume traded was used as a surrogate for the average number of transactions. S&W grouped, for example, the low volume securities together hoping to get mostly less actively traded securities. The results of their test are shown in tables 2.2, 2.3, and 2.4. As can be seen in the tables, the results agree with the theory. In table 2.2, all of the S&W betas are greater than the OLS betas, and in table 2.4, where the more actively traded securities are examined, the reverse is true. If equation 2.13 is rearranged, the difference between the OLS beta and the S&W beta can be computed as follows: 18 B - 8n = Bno:- B:+- B: (2.14) Substituting the averages from tables 2.2, 2.3, and 2.4, the average differences between the OLS estimator of beta and the S&W estimator of beta are —.157, -.111, and .063 respectively. Dimson [1979] relaxed the assumption that the security must be traded once in every period, and defined the market differently from S&W. Scholes and Williams' market is composed of securities traded every day while Dimson's market is the Market Portfolio, or in other words, a portfolio of all risky securities. As a result of this change, Dimson suggests the following beta adjustment (aggregated coefficients method): 8— 2 83k! k=-i n where i = the number of lags and leads. Dimson states that for shares which trade in almost every period, the Scholes and Williams approach has results similar to the aggregated coefficients method. Dimson states that for shares which trade in almost every period, the Scholes and Williams approach has results similar to the aggregated coefficients method. The critical assumption between the two methods would appear to be the assumption of the security trading every day. Looking at Hawawini's study, the significance of the 19 TABLE 2.1 FIRST-ORDER AUTOCORRELATION COEFFICIENTS FOR THE MARKET: 1971-1975. Interval Value-Weighted Equally Weighted Index 1 day .291 .458 2 days .121 .335 1 week .039 .283 2 weeks .109 .289 1 month .083 .134 SOURCE: Scholes and Williams [1976] p. 34. 20 TABLE 2.2 ON VALUE-WEIGHTED MARKET RETURNS DAILY RETURNS ON LOW-VOLUME PORTFOLIOS REGRESSED Year A:= gOLS gsw 88+: 83:5 §S_= ngs AS 1963 0.303 0.544 0.049 0.130 -0.058 1964 0.391 0.561 0.090 0.216 0.122 1965 0.524 0.647 0.045 0.352 0.212 1966 0.426 0.581 0.102 0.391 0.291 1967 0.556 0.651 —0.015 0.267 0.120 1968 0.600 0.775 0.125 0.462 0.266 1969 0.749 0.872 0.183 0.620 0.390 1970 0.679 0.809 0.185 0.565 0.383 1971 0,848 0.993 0.232 0.526 0.308 1972 0.596 0.661 0.121 0.364 0.317 1973 0.499 0.657 0.071 0.435 0.265 1974 0.346 0.431 0.024 0.307 0.284 1975 0.415 0.577 0.057 0.402 0.258 Average 0.674 0.098 0.387 0.243 SOURCE: Scholes and Williams [1977] p. 321. 21 TABLE 2.3 DAILY RETURNS ON INTERMEDIATE-VOLUME PORTFOLIOS REGRESSED ON VALUE-WEIGHTED MARKET RETURNS Year éOLS E3sw 88:5 §8£S 83 1963 0.785 0.905 -0.039 0.005 -0.058 1964 0.754 0.851 0.071 0.233 0.122 1965 1.119 1.202 0.174 0.418 0.212 1966 1.005 1.149 0.248 0.565 0.291 1967 1.123 1.112 0.045 0.212 0.120 1968 1.187 1.274 0.264 0.501 0.266 1969 1.257 1.330 0.421 0.690 0.390 1970 1.248 1.305 0.418 0.638 0.383 1971 1.296 1.386 0.428 0.516 0.308 1972 0.989 1.021 0.299 0.381 0.317 1973 0.983 1.142 0.199 0.564 0.265 1974 0.724 0.830 0.116 0.462 0.284 1975 0.857 0.996 0.171 0.481 0.258 Average 1.115 0.217 0.436 0.243 SOURCE: Scholes and Williams [1977] p. 322. 22 TABLE 2.4 DAILY RETURNS ON HIGH-VOLUME PORTFOLIOS REGRESSED ON VALUE-WEIGHTED MARKET RETURNS Year §0Ls gsw 86:3 égfs AS 1963 1.495 1.336 —0.097 —0.217 —0.058 1964 1.355 1.290 0.199 0.049 0.122 1965 1.597 1.501 0.337 0.204 0.212 1966 1.725 1.564 0.452 0.297 0.291 1967 1.602 1.369 0.219 —0.122 0.120 1968 1.520 1.468 0.449 0.281 0.266 1969 1.531 1.501 0.682 0.458 0.390 1970 1.473 1.437 0.647 0.418 0.383 1971 1.445 1.441 0.535 0.349 0.308 1972 1.314 1.267 0.516 0.240 0.317 1973 1.318 1.314 0.375 0.316 0.265 1974 1.134 1.120 0.303 0.320 0.284 1975 1.174 1.172 0.312 0.290 0.258 Average 1.368 0.380 0.222 0.243 SOURCE: Scholes and Williams [1977] p. 323. 23 lead and lag cross correlations with the market index falls off dramatically after one day. Of the fifty securities tested, only 8% and 22% of the cross correlations were statistically significant for the two-day lead and lag respectively, compared to 72% and 100% for the one day lead and lag (see table 2.5). Therefore, it would appear that Scholes and Williams assumption is reasonable, and the Scholes and Williams adjustment will be used throughout the remainder of this study. Scholes and Williams assume that the major cause of serial correlation is nonsynchronous time periods. Schwartz and Whitcomb [1977] investigate not only the serial correla- tion but also the causes of serial correlation. Schwartz and Whitcomb hypothesize that serial correlation is caused by the following possible reasons: 1) Measurement error 2) The 1/8 effect 3) Impact of Market Makers 4) The Fisher effect. Measurement errors are defined as things such as recording and keypunch errors. The 1/8 effect refers to the rounding error resulting from reporting prices in 1/8 increments. This rounding transforms a smooth price series into a lumpy series, and is analogous to measurement error. Market Makers refer to NYSE specialists. The Fisher effect is attributed to the fact that infrequently traded securities have their last recorded price before the end of the day 24 TABLE 2.5 PERCENT OF 50 FIRMS IN WHICH CROSS-CORRELATIONS ARE STATISTICALLY SIGNIFICANT LAG IN DAYS 0 +1 -1 +2 -2 +3 -3 +4 % of Significant 100 72 100 8 22 8 16 14 Correlation Source: Hawawini [1980] 25 (nonsynchronous time periods). Schwartz and Whitcomb conclude that there is little evidence in support of meas- urement errors, the 1/8 effect, and the impact of market makers, but that there is considerable evidence supporting the Fisher effect and that observed positive index autocor- relation could result from the Fisher effect impacting on thin issues. This conclusion supports S&W's assumption of nonsynchronous time periods. Efficient Market Tests Using Daily Return Data Betas have been shown to be biased, as documented earlier, when estimated using the market model and daily stock returns. However, many studies have used betas estimated with the market model and daily stock returns despite the bias. The incentive of conducting the current study is to investigate the potential impact of using a biased and inconsistent beta on such studies. Brown [1978] tested for the abnormal performance using the market model to obtain expected returns. Daily returns were used in the estimation of beta and alpha. The purpose of his study was to see how fast the market reacted to reports of unusual earnings, and therefore, the cumulative average residuals were inspected. Based on the upward trend of the cumulative average residuals, Brown concluded market inefficiencies exist. However, Brown did not adjust for the bias in alpha and beta, and, as pointed out by Johnston [1972, Chapter 8], biased estimates may cause 26 autocorrelation in the residuals of an OLS regression. The autocorrelation, and in turn, the bias, may account for the upward trend. Jaggi [1978] conducted a study designed to test for market reaction to management forecasts. The expected returns were calculated using the market model, and the parameter estimates were computed using daily returns. Jaggi tested the residuals for significance and assumed a symmetric stable distribution. The symmetric stable distribution was used because residuals resulting from daily returns do not appear normal due to non-normal. kurtosis [Fama and Roll, 1968, 1971]. This may also be due to the bias in the estimates, and Jaggi did not adjust for the bias. Gheyara and Boatsman [1980] tested the market reaction to replacement cost disclosures. This study also used the market model to compute the expected returns, and daily returns were used to estimate beta and alpha. Gheyara and Boatsman computed both OLS estimates and S&W estimates, and found their results to be insensitive to which parameter estimation procedure was selected. They con- cluded the sample firms were drawn from the complete spectrum of trading intensity and the biases were diversi- fied away. However, no tests were conducted to confirm this conclusion. It may also be that the bias does not affect prediction regardless of the spectrum of trading intensity. 27 Beaver, Christie and Griffin [1980] conducted a test similar to Gheyara and Boatsman. Beaver et. al. also recognized the bias problem and used monthly returns to estimate the market model parameters. Beaver states, "Monthly returns, rather than daily data, were used to assess beta because of the nonsynchronous nature of daily data and the attendant problems of beta estimation intro- duced." However, it is not certain that the bias affects prediction and daily returns could have been used. The above four studies point out the need for infor- mation on the bias in OLS estimates and the effect of the bias and inconsistency on the prediction of returns. The intent of the current study is to provide this information. Efficient Market Test Procedures A method for evaluating different efficient market test procedures has been provided by Brown and Warner [1980]. Their desire was to see if one test procedure found abnormal performance better than another. In order to do this, a known abnormal performance was added to every firm's returns in the sample and different procedures were performed to see if the abnormal performance was found. The same test was performed on many samples to see if one procedure isolated abnormal returns more often than another. The current study investigates whether OLS estimators out- perform S&W estimators in the same efficient market test. 28 Therefore, the only difference between the current study and Brown and Warner is that the same procedures with different estimators are being considered rather than different procedures. Two of the tests Brown and Warner [1980] used will be used in this paper: a t-test and a cumulative average residual test (CAR). These two procedures are commonly used and were used in the four papers discussed in the previous section. The procedure is to divide a sample of firms' returns into 3 time periods (see figure 2.4). The first period (days —200 through -90) is used to estimate the parameters of the market model (equation 1.2) using OLS. These parameters are then used in the market model to forecast returns which correspond to the observed returns in periods 2 and 3. Residuals, the difference between forecast and observed returns, are calculated for each day of periods 2 and 3. The residuals from each day are averaged over all the firms in the sample so that there exists an average residual for every day. The average residuals in the second period are used to compute a standard deviation. The standard deviation is divided into the average residual on the event day (day 0) in the third period, which results in a t-statistic as follows (Brown and Warner [1980], equation A.11): 29 N 1/N E Aio i—l N -11 -ll 1 2 l/N( 2 [1/77 E (A_ -( X A,/79))2]) / i=1 t=-89 1t t=-89 1 where N = the number of firms in the sample, and Ait Rit O‘i BiRMt ' and 3i, Bi = either the OLS estimates or the S&W estimates of a and B. If the t-statistic is significant, then the conclusion is that abnormal performance exists. The entire set of third period average residuals is used in the CAR test as follows (Brown and Warner [1980], equation 1): A 1 CARt = CARt_1 + 1/N IIMZ . it i where CARt = the cumulative average residual at time t. These procedures are explained further when they are used. Period 1 used to compute parameter estimates 30 Period 2 used to compute average residual Period 3 used for CAR and t—statistics -200 -90 days -1l -1o 0 10 I Event Day Fig. 2.4 Division of a Sample Used in a Brown and Warner Test III. BETA BIAS AND MARKET MODEL PREDICTIONS Scholes and Williams have established that OLS betas are biased and inconsistent. The effects of the bias and inconsistency on efficient market studies need to be determined. Efficient market studies usually divide a sample of stock market returns into two periods, one period is prior to an "event", and is used to estimate parameters (the alpha and beta for a firm). The second period is used to compute residuals or forecast errors. The errors are the difference between forecasts estimated using the parameters of the first period and the observations of the second period. The forecasts will be affected by beta and therefore, the effects of biased and inconsistent betas on prediction is the focus of this study. Malinvaud [1970] provides a framework for studying the effects of inconsistency on the mean square prediction error. If the following model exists: 5 = + . Rmt Rmt umt (3 1) = + ° Rnt BRmt ent (3 2) where B = a parameter to be estimated, _ s umt - the measurement error of Rmt' and ent the error term for security n 31 32 then the OLS estimate of beta is inconsistent (see also Maddala [1977, Chapter 13]). Prediction is defined as choosing a value RE minimize E(R: for R n )2. and the general aim is to t t' t - Rnt The prediction of R given R5 is: nt m t (3.3) where B = the estimator of B. The resulting error is the following: P _ s ”_ - _ (R - R ) — R (B B) + (bumt e nt nt mt )° (3'4) nt . . . . s . The mean square prediction error conditional on Rmt is: P 2 s _ s 2 ~ 2 2 E[(Rnt Rnt) /Rmt]‘(Rmt’ E[ i. For example, the C matrix for the 2 given in the previous paragraph is 1013847 0 o 7 c = .0069174 .0183044 0 t0046116 ' -.0017428 .0234sg and C C' = Z . Each firm has two time periods each day; the time until the last transaction and the time at the end of the day when no trade takes place. S&W used a poisson distribution for the number.of transactions per day, and, as such, the time between transactions followed an exponential distribution. Oldfield and Rogalski [1980] indicate that time between transactions follow a gamma distribution. An exponential distribution is a special case of a gamma and appears to be adequate for the purpose of this paper. The use of the exponential distribution provided data consistent with the problem described by S&W, therefore, this distribution was considered adequate. Using a poisson generating function 46 both the number of transactions and time between them can be generated. Once the times have been generated the observed returns and actual returns can be computed. The vector Zn' in equation (4.2), is a vector of independent standard normal variates and is generated using the method presented in Naylor [1966, pg. 97]. When the matrix C is multiplied by Zn' a vector of multivariate normal variates is obtained. As a result, the returns are multivariate normal and the returns of all firms must be generated at the same time. Each observed return, however, has a different time period. This problem is handled by generating prices for all firms at each firm's ending transaction time. For example, if there are three firms in the market, each firm will have four prices generated for each day corresponding to the last transaction time of each firm and the end of the day (3 + 1). In figure 4.1, the last transaction times for firms 1, 2, and 3 are .65, .85, and .95 respectively. Therefore, prices must be generated for each firm at time .65 of the day as well as at .85 and .95 (see figure 4.2). A price must also be generated at the end of the day. The prices generated at the four time periods are based on the multivariate distribution CZn’ From the prices in figure 4.2, the observed return for firm 1 in day 2 would be: S R1 = - + - P 2 (1n Pll-.95 1“ Pl.65) (1n P1.9o-.7o 1“ 11-.95) 47 The actual return would be (See figure 4.3): l 2 3 .1. .1. .1. r; r. at 2 2-1.5-1.; ........... 2. 3 nu n9 9 o 11:17 I IT .2 0‘ 1) 1 .L r. AU 9/ 2 2 c l llalnlnll ||||||| l_ '''''''' I' 5 9 1'! - 1 5 3 00 p5 (gr- 1... 1 C2 5 w a... a 1 O 1 r. wk 5 ,0 a 1 1 f. 0 ||||||||||| ll ||||||||| IIII 4.1 Time Periods Fig. J 3 1. 9 9 9 o . u . . a l V. I. ..L 2 3 D. 3. D. 'Iul‘. lllllllllllllllllll rlil 0 0 0 9 9 3 . .. a 3 o 3 3 a, 7 9 O 9 I I I .I n . . 1 Ar 1 0 Av 2 o v T. D. 9 P D. L - I'll. m All 1 . D. 2 m 0 a. 0 7 7 L1 ..2 . 1r . I. 2 3 D. D. D. s S 5 9 9 9 . . . 1 I. .L .L 2 3 P D. D. --l'-‘ s u s a o 8 l - O o s . L! s L. 9 A 5 9 . 5 J . 6 .L 2 . J P a. P h. P 6 8 E III a LII a 1" n 5 2 I P I 1 5 s s s p a a I! o I... 11 2 1. I. D. P D. 0 'II'- lllllllllllllllllll v-|| .90-.7O 9) .85'.65 ’3 Generation Price 4.2 ig. F 48 m a H mxmo :H omumumcoo mooflum uzoa n-. u.o~ cauem.- :~.n~_m one m.v ousmam Amaolfi H“ afllmaalfi fl“ :dvflfldz maulfl .— Nd Om.N.—.m u Gh.lco. mm mm.) u.c—=~15:~V+Am:~IO~—_~vfl mm b w u mm.I~ «A . . . . .- . S» u . a .ma -a aa ea-ee -ea ea ea...ma aa e_-ma a aa aa.u~Ma a a n ma aa ca. mm. N h ms. Om. mm. .— mh.l _ em.) I l r a l m m cum mam modem 49 As a result, the actual returns are always generated over a time period of l and the observed returns over changing time periods. The observed time period in this case was 1.25. The observed time periods depend on the average transactions chosen for each firm. The average number of transactions, a parameter for the poisson generator, affects the covariance between firms which in turn affects the variance and autocorrelation of the market. The time between transactions follows an exponential distribution and therefore the expected time between the last transaction and the end of the day is: E(Sn) = 1/1 n where An = the average number of transactions for firm n. If firm n has a.ln = 2 and firm m has a A = 100, from m equation (2.2) the observed covariance would be approxi— mately 49% less than the actual covariance. Since the market variance is equal to the weighted sum of all the elements of the variance—covariance matrix (see equation 4.4), the observed market variance will be less than the actual market variance. S&W [1977] show that the first order autocorrelation will be: Var(RM) o =l/2( M -1). Var(R§) 50 The greater the differences in the average number of trans- actions, the smaller the observed variance will be, and the smaller the observed variance, the greater the autocorrela— tion of the observed market will be. The average number of transactions will be based on a sample of 283 firms chosen from the New York Stock Exchange.7 The number of trans- actions per day for the first three months of 1980 has been obtained (Woodruff [1981]) for each firm. A summary of the average transactions for those firms is listed in table 4.2. The average number of transactions A is used in a poisson generating function presented in Naylor [1966] as follows: x x+1 Z t.< A < Z t. i=0 1 i=0 1 where x = the Poisson variate A = the average number of transactions ti = —log ri, and ri = a random variate with a uniform distribution The time at which a transaction takes place in a day is the sum of the time between all previous transactions in the day. 7The 283 firms were chosen randomly by Woodruff [1981], but also conformed to the following two criteria: 1) Listed on the New York Stock Exchange 2) Listed in Value Line 51 TABLE 4.2 AVERAGE NUMBER OF TRANSACTIONS FOR 283 FIRMS Class Cumulative Midpoint Percentage Percentage 2.5 4.3 4.3 7.5 12.6 16.9 12.5 12.6 29.6 17.5 11.0 40.5 '22.5 7.6 48.2 27.5 6.0 54.2 32.5 6.0 60.1 37.5 5.0 65.1 42.5 4.7 69.8 47.5 3.3 73.1 52.5 2.3 75.4 57.5 3.0 78.4 62.5 3.3 81.7 67.5 2.3 84.1 72.5 2.0 86.0 77.5 1.7 87.7 82.5 0.3 88.0 87.5 1.3 89.4 92.5 0.7 90.0 97.5 1.0 91.0 102.5 0.7 91.7 107.5 1.0 92.7 112.5 0.3 93.0 117.5 0.7 93.7 122.5 0.3 94.0 127.5 0.7 94.7 132.5 0.0 94.7 137.5 1.3 96.0 142.5 1.0 97.0 147.5 0.3 97.3 152.5 0.0 97.3 157.5 0.0 97.3 162.5 0.0 97.3 167.5 0.0 97.3 172.5 2.7 100.0 minimum average transaction = 2.52/day maximum average transaction = 286.08/day mean = 41.4108 standard deviation = 44.0956 52 The goal in choosing the number of firms is to use the fewest that will adequately represent the market. The fewest number is desired because: 1) The larger the number of securities the more difficult it becomes to estimate a variance-covariance matrix. For example, with 20 firms there are 400 covariances and with 40 firms, 1600. 2) As the number of firms increases, the cost of the simulation increases dramatically. The cost corresponds to the number of calls to the random number generator. The number of calls equals the number of firms squared times the number of days to be simulated. Most of the diversification effect is realized in portfolios of only twelve firms (Francis and Archer [1979], Tinic and West [1979]). The variance of portfolios of twenty or more is attributed mainly to the pairwise covariances (Fama [1976]). Therefore, the selected number of firms in this study is twenty since the pairwise covariances contribute more than the variances of the individual firms to the variance of the portfolio, and a portfolio of twenty will show substantial diversification. The number of firms may also affect the autocorrelation of the simulated market. The autocorrelation of the observed market is affected by two factors, the average transaction time of the firms and their pairwise covariances. If a firm is added that has a 53 small average number of transactions and its pairwise covari- ances with other firms in the portfolio are large, the autocorrelation of the portfolio will be increased. Figures 4.4 and 4.5 show the affects of adding more firms on the correlation of the portfolio. Figure 4.4 shows the results of adding approximately the same percentage of average transactions and approximately the same size covariances. After adding the first 5 or 6 firms, the changes in autocorrelation by adding more firms is slight. Figure 4.5 shows the results of adding firms that have a higher percentage of less than the average number of trans- actions than the firms in figure 4.4. Again, the largest increases to the autocorrelation occur with the first few firms and subsequent changes are smaller. Figure 4.5 does have a higher increase in autocorrelation than does figure 4.4, however, it is apparent that it would take a very large number of firms to approach the autocorrelation of the portfolio made up of the NYSE (the market), and also that the greatest change occurs with a relatively small number of firms. Therefore, adding more than twenty firms would not enhance the study unless a substantial number were added. The two figures (4.4 and 4.5) appear to have a tiered effect because the transaction times were always added in the same order. The first twenty were added from lowest number of transactions to highest, and the second twenty went from lowest to highest. 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Mflfifl mama comm cam moo "coaomeaomm xmo nod mcofiuommcmui ommuo>< monum> mama :« moan h.m ousmam w. .- 1M 0 ' mu.) acoquowncoub H.312 a: c..— 9:] 2: on on a. .cN. . O O O O O O o O c. mu. n. v. noun '70 mama comm cam 3 a m ”scammeacmm xmo Mom mcofiuommcmue omnum>< m:mum> mama so moan m.m musmfim on... mu.) occuuooncauh gunman oc~ . cog cod cod co co .oo . co. . ma. mv. mean 71 for the OLS betas in all cases except at 100 days. There- fore, the S&W beta bias should appear closer to zero, and in fact they do. A definite pattern can be discerned in the plots of the OLS bias whereas a pattern can not be found in the S&W bias. Therefore, the bias has been eliminated using the S&W procedure. Portfolios were made by grouping firms together by transaction times. Three portfolios were made. The first portfolio contained those firms with less than the average number of transactions. The second portfolio contained firms with close to the average number of transactions and the last portfolio contained firms with more than the average number of transactions. Betas and the respective biases were computed for these portfolios and the results are reported in Table 5.2. The results are consistent with the S&W conclusions and are similar to the individual results. Prediction Results The prediction test is designed to substantiate the results in Chapter III. In Chapter III it was shown that for infinitely large samples, the mean squared prediction error using the OLS estimate of beta will be larger than the mean squared prediction error (MSPE) using the S&W estimate of beta. Allen and Hagerman [1980] and Dutta [1975] use mean squared prediction errors as an indication of predictability, which is the method used in the 72 subsequent test. OLS estimates and S&W estimates were computed for alpha and beta starting at sample sizes of 100 and ending at sample sizes of 5200 days. A sample size of 5200 days corresponds roughly to 20.6 years. These estimates were then used to predict the next 100 days, using the market model (equation (1.2)), for each of the twenty firms. One hundred days were used for the prediction period because in subsequent tests a 100 day period is used. The subsequent tests are the efficient market tests fashioned after the Brown and Warner [1980] paper. A 100 day period is neces- sary to allow for enough observations to compute a variance for the residuals. Therefore, to be consistent, a 100 day period was used. The MSPE was then computed over the 100 day prediction period using the two estimates. The MSPE OLS was compared to the MSPE and the smaller MSPE was SW considered the better predictor. Twelve samples of twenty firms each were run by changing either the random generator seed for the estimation period or the prediction period. The results of these runs are shown in figures 5.9 and 5.10. The same runs were done forming portfolios out of the twenty firms. The portfolios were based on average trans- action times. Each run consisted of three portfolios corresponding to less than average, average, and greater than average firms. The results of these runs are shown in figure 5.11. 73 The vertical lines above the zero horizontal line, in figures 5.9, 5.10, and 5.11, represent the number of times out of 12 that the MSPESW was less than the MSPEOLS for each of the twenty firms or three portfolios. The vertical lines below the horizontal represent the number of OLS was less than the MSPESW. As was shown earlier, the difference between these two times out of 12, the MSPE measures is quite small and in some cases the two were virtually the same. Therefore, in some instances, the vertical line is shorter. For example figure 5.10, 5200 days, firm 16, the MSPE's for firm 16 were the same for 6 samples out of the 12 samples. The firms are listed from the lowest number of transactions to the highest, and the transaction times correspond to those in table 5.1. It is not apparent from figures 5.9, 5.10, and 5.11 that either the S&W estimators or the OLS estimators result, on average, in better predictions of returns. However, there does appear to be a trend. The S&W estimators appear to result in better predictions for firms with a low number of transactions and possibly for those with a higher number of transactions. This is far more visible in figure 5.11 which shows the portfolio results. The portfolio results are better because at the individual firm level the major portion of the MSPE is due to the error term (see equation 3.11 and 3.12). As portfolios are formed this error term is reduced. Therefore, based on the simulation, when firms are grouped together such that 12 I MSPESW < 0 I MSPEOLS < 12 S 10 15 20 Firm 1 low high trans. trans. 12 IMSPESW < 0 I MSPBOLS < 12 S 10 15 .20 Firm I low high trans. trans. Figure 5.9 Number of Mean Squared Prediction Errors Out of 12 Replications Which are Smaller - Estimation Periods: 100 days and 400 days 75 12 a MSPESW < o | l l :MspeOLS < 12 low high trans. trans. 12 : MSPESW < 0 ' ) : MSPEOLS< 12 5 10 15 20 Firm . low high trans. trans. Figure 5.10 Number of Mean Squared Prediction Errors Out of 12 Replications Which are Smaller - Estimation Periods: 1000 days and 5200 days 76 100 days 400 days 11 ' 11 5 MSPESW< 5 MSPESW< O "39301.5( . “39301.5( 12 12 1 2 3 1 2 3 portfolio 4 portfolio I 1000 days 5200 days 12 12 9 nspssw< o nspssw< I 12 12 1 2 3 l 2 3 portfolio 0 portfolio I Figure 5.11 Mean Squared Prediction Errors Portfolio Results 77 the majority have a low number or high number of trans- actions, compared to the market average, then the S&W estimator should be used. Efficient Market Test Results The test of efficient market studies is similar to the technique used by Brown and Warner [1980]. The difference between this study and the Brown and Warner study is that Brown and Warner tested different procedures to see if abnormal performance was found using one procedure more often than another, where as this study uses the same procedure with different estimators. The desire is to find out if abnormal performances can be found more often using one method of estimation than another (OLS versus S&W). The market model will be used to predict expected returns. The expected returns will then be subtracted from the observed return to obtain a residual. The residuals are tested for abnormal performance. The S&W beta and alpha and the OLS beta and alpha will be estimated over periods of 100 days to 5200 days. Abnormal performance will be tested over a 100 day period. A 100 day period was used, because 100 days allows for enough days, as will be seen later, to compute a variance for the residuals. Brown and Warner also utilized a 100 month period, therefore, the procedure employed can be the same as the Brown and Warner procedure. Ten firms from the 78 twenty will be used in the abnormal performance test, and a total of 48 different samples will be selected. Only ten firms were used because if all twenty firms were used, the result would be to predict the market with the market, therefore less than twenty firms were desired and 10 were chosen. The abnormal performance will be tested using two different tests, the t-test and a cumulative average residual (CAR) test. The t-test, as derived by Brown and Warner, tests for abnormal performance on an event day (t = 0). The 100 day test period is divided into the time prior to the event (t = —89 through t = -1) and 10 days after the event (t = 1 through t = 10) (see figure 2.4). An average market model residual is calculated on the event day (t = 0) as follows (Brown and Warner [1980], page 253, equation A.9): 110 where N the number of firms in the sample 3 A A s A, = R, - o - R and 1t It i 61 Mt ' di, 8i = either the OLS estimates or the S&W estimates of o and B. 79 The standard deviation of the average market model residual is estimated on the basis of the standard deviation of the residual of each sample security over the period t = -89 to t = —11. At t = 0 the test statistic (Brown and Warner [1980], page 253, equation A.11) is: N l/N z A. i=1 10 (5.1) N -11 -ll l/N( 2 [1/77 E (A, —( Z A./79))2])1/2 i=1 t=—89 1t t=_89 1 The above statistic is distributed Student - t with 78 degrees of freedom. Equation (5.1) is calculated for each of the 48 samples with abnormal performances added at t = 0 of .00, .01, .011, .013, .016, and .02. In other words, if an abnormal performance of .01 is to be added, .01 is added to each firm's return on the event day t = 0. These levels of abnormal performance were chosen in an attempt to obtain t-statistics that were close to the rejection boundary at the three test levels (.05, .025, and .01). For example, at the .025 test level, the boundary for rejection of the null hypothesis is a t-statistic of 2.00. The desire was to pick levels of abnormal performance such that many of the t-statistics were close to 2.00. Tables 5.3 through 5.12 summarize the results of this test. The 48 replications of this test can be divided into 4 sets of 12. The first set of 12 replications was drawn using firms 1 through 10 or those firms with low to 80 average number of transactions. The prediction results (figures 5.9 and 5.10) would indicate that for these 10 firms, on average, the S&W estimators would result in better predictions. Therefore, the expected result of the t-test would be that the S&W estimators would isolate abnormal performance sooner than the OLS estimators. Table 5.3 shows the percentage of replications where abnormal performance was found at each level of abnormal performance and at each test level. The percentage of abnormal performance found is greater for the S&W estimators in some instances but for the majority of the cases the percentages are the same. The average t- statistics, for the first set of twelve replications, are listed in table 5.8 and on average the S&W estimators result in a slightly higher t-statistic. The second set of 12 replications was drawn using firms 11 through 20 or those firms with average to high number of transactions. The prediction results (figures 5.9 and 5.10) would indicate that for these 10 firms, on average, the OLS estimators would result in better predictions. The expected result of the t-test is that the OLS estimators will isolate abnormal performance more often than the S&W estimators. Table 5.4 shows the percentage of replications where abnormal performance was found, and in most cases the percentages are the same. Table 5.9 shows the average t—statistics and on average the OLS estimators result in a slightly higher t-statistic. 81 The third set of replications was drawn using firms 1 through 5, and 16 through 20, or those firms with a low or high number of transactions. The prediction results (figures 5.9 and 5.10) for these 10 firms indicate, on average, better predictions are obtained using the S&W estimators. The expected result of the t-test is that the S&W estimators will isolate abnormal performance more often than the OLS estimator. Table 5.5 shows the percentage of replications where abnormal performance was found. For the most part, the percentages are the same, but there are some instances where the S&W percentage is consistently higher. Table 5.10 shows the average t-statistics and on average the S&W estimators result in greater t-statistics. The last set of replications was drawn using firms 6 through 15 or an average number of transactions. The prediction results (figures 5.9 and 5.10) for these 10 firms indicate, on average, that the OLS estimators result in better predictions. The expected result of the t—test is that the use of the OLS estimators will result in isolating abnormal performance more often than the S&W estimators. Table 5.6 shows some of the OLS percentages are greater than the S&W percentages, but most are equal. Table 5.11, however, shows that the average OLS t—statistic is slightly greater. Tables 5.7 and 5.12 show the aggregated result of all 48 replications. 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However, when the results are not aggregated, the pattern follows that of the predic- tion tests, but the t—statistics of either the OLS estima- tors or the S&W estimators are not that different. The standard deviations are large in comparison to the t-statistics. Therefore, it would not matter which estimates were used. Cumulative Average Residual Test A cumulative average residual test is typically used when there exists only an approximate event date. The CAR's are computed for the days around this date and then plotted against time. The plots are examined to see if a pattern can be observed. If no abnormal performance exists the plot is expected to be flat or display virtually no slope. The CAR's are calculated as follows (Brown and Warner [1980], Equation 1): A l IIMZ CAR = CAR 1+ l/N t t- . it 1 where CARt = the cumulative average residual at time t. The time period covered in this test was a 21 day period from t = -10 to t = +10. 93 The cumulative average residuals were calculated for the same 48 replications that were run in the t-test (the previous section). Figures 5.12 through 5.19 are a few of the CAR plots. Figures 5.12 and 5.13 are two of the replications out of the 12 replications which were made using firms 1 through 10. Figures 5.14 and 5.15, 5.16 and 5.17, and 5.18 and 5.19 represent firms 11 through 20, 1 through 5 and 16 through 20, and 6 through 15, respectively. Parameters of the market model were estimated over periods of 100, 400, 1000 and 5200 days. The upper row of plots are the CAR's where no abnormal performance has been added. The bottom row of plots has an abnormal performance of .001 added to each day of the 21 day period. The x denotes the CAR using S&W parameters and the ° denotes the CAR using OLS parameters. A CAR indicates abnormal performance when the drift of the CAR is greater, in either the positive or negative direction, than expected. The expected drift would be zero if no abnormal performance existed. Therefore, an important aspect is that when no abnormal performance is added, the drift should be small. The focus of this test is to discover which estimator (OLS or S&W) results in CAR's which are closer to zero. Table 5.13 shows the results of the CAR plots. The table is broken down into the 4 groups of 12 replications indicated in the preceeding paragraph and the total of all 48 replications. The data shown are the number of replications in which the CAR on c— c— uss: ccom mo. no. cc. cc. cHIH "magma mucom m=c 33¢ «c. ”so: =:. mo. mo. «c. mo. mo. 96 ‘ I o I. .- i mu m . i mu m o. q. .3 oo. axas econ oouoo "msumm mucom moo oo.m muzoom oo.. oo.- oo... oo.. _o.- 5.- 8.- 3.- I I . o .T o. . .. ..o .. .2. S o . o a 3.. o. o f E. I‘l* I <4 I I . a GM an d U ‘0 MW v m a «a a mm m an. _o. I... .o. I 1 _o. m...” m _o. A “a ‘o .. oo. oo. oo. 8. no. mo. mo. mo. «5. oo. oo. L.5. oo.- ...,... .. .. oo.- o o oo.- 8.. I did i I i u I d K“ ' W O I _o.- s _o.- a . _o... J... 2. _o.- m . m a u c T o. 5N ...p or..- c..: o . LE. 2 o o k..- 2, «Illill: oullfmall n a =dLSdLIEAINNII 3OSOTTT3SI‘A0LHSO3SO7THTIITT SO SOVSOLA LHVLHRQLHO VRRVVV -..HasD.OSJ\..\ .KQWNIQVHNNVD VVGJVOV733303003AA3533..IAOCOOOSAA.)SAA3AAORSOEEOON AADAACAA. VVOVVCVVVV: PHDHCC. 2L. .0; FVVTTCCJCCCNFCRFC. .E “OGRAH HPSE p: 0 0 .U 1 3 3 050 o 10 a. 262 5 5“ 9 3 3 33 19 a 9 BIBLIOGRAPHY Bibliography Allen, S., and R. Hagerman, "Factors Influencing the Forecast Accuracy of the Market Model", State University of New York at Buffalo, unpublished paper, (1980). Arnold, L., Stochastic Differential Equations: Theory and Applications, New York: John Wiley and Sons, Inc., 1974. 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