The Monthly Effect in Stock Returns and Conditional Heteroscedasticity
Journal article by Menahem Rosenberg; American Economist, Vol. 48, 2004
Journal Article Excerpt
The Monthly Effect in Stock Returns and Conditional Heteroscedasticity. by Menahem Rosenberg I. Introduction Various seasonal anomalies in stock returns have been researched since Rozeff and Kinney (1976) originally presented the January effect. (1) For example, French (1980) documented the weekend (or Monday) effect, (2) and Ariel (1987) presented the end-of-month (or monthly) effect. In addition, documentation of similar phenomena by Lakonishok and Smidt (1988) using a longer time series and by Jaffe and Westerfield (1985) in foreign equity markets has reduced the possibility that these anomalies are due to sampling error. In this paper, we present a possible structural explanation for Ariel's end-of-month effect. Ariel (1987) reports a higher cumulative return for the first-half of the month than the last-half of the month. Lakonishok and Smidt (1988) confirm this finding using an extended daily time series. In contrast, Wang, Li, and Erickson (1997) show that Mondays in the last two weeks of the month are responsible for the previously reported end-of-month effect. After controlling for Mondays in the last two weeks of the month, they find the coefficient for the end of the month to be insignificant. In other words, the end-of-month anomaly is explained as a weekend anomaly. Most of the research cited above uses daily stock returns data. However, French, Schwert, and Stambaugh (1987) suggest that the daily returns on the S&P 500 index exhibit conditional heteroscedasticity, while Engle and Mustafa (1992) show similar characteristics in individual stock returns. Moreover, Connolly (1989) tests the weekend effect in S&P 500 index returns and rejects the constant variance model in favor of a Generalized Autoregressive Heteroscedastic (GARCH) model. Disregarding such changing volatility and applying a constant variance model can lead to an inefficient though unbiased estimator of mean returns. This could in turn lead to acceptance of a false null hypothesis (see Pindyck and Rubinfeld 1998). In this paper, we use a GARCH model to present evidence that end-of-month returns are lower than beginning-of-month returns. Contrary to Wang, Li, and Erickson (1997), we will show that this result holds, even when we control for Mondays and end-of-month Mondays. We will also present evidence that the monthly effect may be associated with the economic business cycle. Ogden (1990) shows that standardization of payments in the U.S. economy leaves investors with additional funds to commit at the turn of the month. When the money supply is tight (which is a result of stringency in monetary policy of the Federal Reserve Bank), funds availability is negatively correlated with turn-of-month stock returns. The implication here is that the difference between the beginning-of-month mean return and the end-of-month mean return should be significant only in an expansion phase of the business cycle, as we will show. The paper is organized as follows: Section II describes the data and methodology used. Empirical results are presented in Section III. Section IV offers the summary. II. Data and Methodology This paper applies a time-varying GARCH model, as initially suggested by Engle (1982) and then generalized by Bollerslev (1986). The GARCH (1,1) model we apply here incorporates one lagged variance term in the conditional variance. The index return regression model is: (1) [r.sub.t] = [mu] + [alpha][D.sub.t] + [[epsilon].sub.t] [r.sub.t] | [I.sub.t=1] ~ N ([mu], [h.sub.t]) where the conditional variance follows: (2) [h.sub.t] = [[beta].sub.0] + [[beta].sub.1][[epsilon].sup.2.sub.t-1] + [[beta].sub.2][h.sub.t-1] In this model, [r.sub.t] is the daily index return, and [mu] is the daily mean return. [D.sub.t] is a vector of dummy variables that will take a value of one to mark various calendar conditions, such as Monday or the last-half of the month, and otherwise will take a value of zero. Variable [alpha] is the corresponding vector measuring the special calendar date's contribution to the mean return. Hypothesis testing is conducted on each of the [alpha] components that are different from zero. An [alpha] component that is significantly different from zero will point to a calendar date that contributes significantly to the overall mean return, [mu]. In the tests to be reported, we employ the Center for Research in Security Prices (CRSP) daily returns for the period from July 1962 through December 1993, which includes a total of 7,927 daily observations. The daily data also contain the daily calendar dates. This information allows us to mark each daily return for both specific calendar dates and the day of the week. We employ three indices in our analyses: the NYSE value-weighted index, the NYSE equally-weighted index, and the S&P 500 index. Monthly data on U.S. business cycle expansion and contraction is obtained from the National Bureau of Economic Research (NBER). The index return data set, from the period July 1962 through December 1993, contains 378 months. Sixty-two of these months are declared as business cycle contractions, while the remaining 316 months are considered part of a business cycle expansion. Monthly business cycle data are converted into daily data by declaring each day of the month as either a contraction period or an expansion period, according to the NBER determination for that month. The result of this conversion is that 1,304 days out of the total 7,927 days in the data set are in contraction business cycles. The remaining 6,623 days are considered economic expansion periods. Following Wang, Li, and Erickson's (1997) method of investigation, month trading days are divided, as defined by Lakonishok and Smidt (1988). Accordingly, the first-half of the month is defined as the month's first fifteen calendar days when the fifteenth day is a trading day. If the fifteenth day of the month is not a trading day, the next trading day is included in the first-half of the month, and the last-half of the month encompasses the remaining trading days in the month. III. Empirical Results We will first present evidence to support the argument that returns are low during the last-half of the month for all three indices, even when we control for Mondays, as suggested by Wang, Li, and Erickson (1997). Rewriting the return regression (1) with the appropriate dummy variables: (1') [r.sub.t] = [mu] + [[alpha].sub.1] (last half of the month) + [[alpha].sub.2] (all Mondays) + [[alpha].sub.3] (Mondays in last half of the month) + [[epsilon].sub.t] The variable last-half of the month will take a value of one if it is the last-half of the month as defined above, and will take a value of zero otherwise. The variable all Mondays will take a value of one if the calendar day is Monday, and will take a value of zero otherwise. The variable Mondays in last-half of the month will take a value of one if the calendar day is a Monday in the last-half of the month as defined above, and will take a value of zero otherwise. The regression residuals are first modeled with a constant variance and second with a time-varying variance, following equation (2) above in the GARCH (1,1) process. By means of the same numerical optimization used in the conditional GARCH (1,1) variance model, we estimate the constant variance model. Results are presented in Panel A of Table 1 and support Wang, Li, and Erickson's (1997) conclusion for the NYSE value-weighted index. In the estimated constant variance model, the last-half of the month variable is statistically insignificant, while the all Mondays and the Mondays in last-half of the month variables are statistically significant. These results are repeated for the S&P 500 index, but for the NYSE equally-weighted index, the variable last-half of the month is significant with a p-value of 0.033. The results of the conditional GARCH (1,1) variance model are presented in Panel B of Table 1. The hypothesis tested is whether [[alpha].sub.i] (i = 1, 2, 3) equals zero. A significant negative value for any of the [alpha] variables indicates a lower return than the overall mean return--[mu], on those specific calendar dates. The negative value found for the all Mondays ([[alpha].sub.2]) variable (see Panel B of Table 1) confirms the Monday (or weekend) effect previously found in studies such as those by French (1980), Jaffe and Westerfield (1985), and Lakonishok and Smidt (1988). Panel B of Table 1 also shows that for all three indices, the Mondays in the last-half of the month ([[alpha].sub.3]) variable has a significant negative value (the S&P 500 index has the least significant p-value, 0.029). This finding verifies Wang, Li, and Erickson's (1997) results that the Monday returns in the last-half of the month are lower than the overall mean return and, as a result, lower than the Monday returns in the first-half of the month. Table 1 also reports the log-likelihoods for the different models. The likelihood ratio test statistics from tests of the constant variance model (Panel A) against the conditional GARCH (1,1) variance model (Panel B) leads us to reject the constant variance assumption in favor of the GARCH (1,1) variance model. This is consistent with Connolly's (1989) assertion. As shown in Panel B of Table 1, the last-half of the month variable ([[alpha].sub.1]) was found to be negative and significant for all three indices. Its level of significance is lowest for the S&P 500 (with a p-value of 0.0387) and highest for the NYSE equally-weighted index (with a p-value of 0.0156). These results indicate that last-half of the month returns are lower than first-half of the month returns, even though we control for all Mondays and for last-half of the month Mondays, contrary to Wang, Li, and Erickson's (1997) suggestion. Following Ogden (1990), we relate the end-of-month effect to some standard flow patterns in the U.S. economy that increase available and investable liquidity toward the end of the month. This relationship is examined from the perspective of the business cycle, as defined and measured by the NBER. Next, we rewrite the return regression function (1'), dividing the variables in two, to allow us to measure the calendar dates' contributions under two economic regimes: One economic regime is an expansion period. The other economic regime is a contraction period. (1'') [r.sub.t] = [mu] + [delta][D.sub.c] + [[alpha].sup.e.sub.1] (last half of the [month.sup.e]) + [[alpha].sup.e.sub.2] (all [Mondays.sup.e]) + [[alpha].sup.e.sub.3] (Mondays in the last half of the [month.sup.e]) + [[alpha].sup.c.sub.1] (last half of the [month.sup.c]) + [[alpha].sup.c.sub.2] (all [Mondays.sup.c]) + [[alpha].sup.c.sub.3] (Mondays in the last half of the [month.sup.c]) + [[epsilon].sub.t] Variable [D.sub.c] will take a value of one when it is a conctraction period, and otherwise will take a value of zero. Parameter [delta] will measure the contraction period return contribution to the overall mean return [mu]. Hence, the expansion period mean return is [mu], and the contraction period mean return is [mu] + [delta]. The variables last-half of the [month.sup.e], all [Mondays.sup.e], and Mondays in the last-half of the [month.sup.e] are as defined above in (1'), with a superseding condition of taking a value of one only if it is an expansion period. Similarly, the variables last-half of the [month.sup.c], all [Mondays.sup.c], and Mondays in the last-half of the [month.sup.c] will take a value of one as defined in (1'), with a contraction period as a superseding condition. For example, if it is the last-half of the month in an expansion period, then the last-half of the [month.sup.e] will take a value of one, while the last-half of the [month.sup.c] will take a value of zero. The test results are presented in Table 2. As Table 2 shows, the all Mondays ([[alpha].sup.e.sub.2] and [[alpha].sup.c.sub.2]) variable is the only variable that is statistically significant and negative both in expansion and in contraction business cycles. In contrast, it is only in a period of business cycle expansion that the variables last-half of the month ([[alpha].sup.e.sub.2] and [[alpha].sup.c.sub.1]) and Mondays in the last-half of the month ([[alpha].sup.e.sub.3] and [[alpha].sup.c.sub.3]) are negative and significant. That these two variables have significant negative values indicates that the returns in these calendar periods are lower than the overall mean return. For example, the last-half of the month ([[alpha].sup.e.sub.1]) variable is negative with a p-value at or below 2 percent (for the three indices) in an expansion cycle. Therefore, we can conclude that in an expansion cycle, the daily returns in the second-half of the month are lower than the overall mean return. Conversely, in a business cycle contraction, the last-half of the month ([[alpha].sup.c.sub.1]) variable is not significantly different from zero. This leads to the conclusion that last-half of the month returns are not different from the overall mean return in a contractionary business cycle. These conclusions concerning the significant negative value in an expansionary business cycle and the insignificant value in a contractionary business cycle also apply to the Mondays in the last-half of the month variable. IV. Summary In this paper, we analyzed the end of the month effect using an efficient estimation procedure that takes into consideration heteroscedasticity in daily stock returns. As a result, we have shown that stock returns in the last-half of the month are consistently lower than in the first-half of the month. This finding cannot be explained by low returns on Mondays in the last-half of the month, as previously suggested by Wang, Li, and Erickson (1997). Additional insight into the monthly effect is gained when the economic business cycle is considered. We identified and analyzed the data in an expansionary business cycle against those in a business cycle contraction. We found that for two of the calendar dates--the last-half of the month and Mondays in the last-half of the month--stock returns are lower than on other calendar dates only in an expansionary business cycle. On the other hand, in a business cycle contraction, stock returns for the last-half of the month and for Mondays in the last-half of the month are no different from stock returns on any other calendar date. The above results might help in timing transactions, but the differences in return are too small to develop trading strategies based on these patterns. For example, the mean return decrease in a contraction cycle Monday is 0.28 percent (using the NYSE equally-weighted index), while Lakonishok and Smidt (1988) suggest that a one-tick price movement on an average-priced share in the NYSE represents a 0.313 percent return, making the magnitude of the predictable pattern above too small to be exploited for trading profits. Nevertheless, the statistical significance of the calendar dates and the existence of conditional seasonal patterns in daily stock returns should be taken into consideration in research based on daily data such as event studies. One should be aware of and take appropriate measures to correct for the above-detected patterns. TABLE 1Notes (1.) Rozeff and Kinney (1976) were the first to document that average stock index returns in January are higher than in any other month. (2.) French (1980) found that the average return on Monday is low. References Ariel, Robert A. 1987. A Monthly Effect in Stock Returns. Journal of Financial Economics 18(1):161-74. Bollerslev, Tim. 1986. Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics 31(3):307-27. Bollerslev, Tim, Ray Y. Chou, and Kenneth F. Kroner. 1992. ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence. Journal of Econometrics 52(1/2):5-59. Connolly, Robert A. 1989. An Examination of the Robustness of the Weekend Effect. Journal of Financial and Quantitative Analysis 24(2):133-69. Engle, Robert F. 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50(4): 987-1006. Engle, Robert F., and Chowdhury Mustafa. 1992. Implied ARCH Models from Options Prices. Journal of Econometrics 52(1): 289-311. French, Kenneth R. 1980. Stock Returns and the Weekend Effect. Journal of Financial Economics 8(1):55-69. French, Kenneth R., William G. Schwert, and Robert F. Stambaugh. 1987. Expected Stoc Returns and Volatility. Journal of Financial Economics 19(1/2):3-29. Jaffe, Jeffrey E, and Randolph Westerfield.1985. The Weekend Effect in Common Stock Returns: The International Evidence. Journal of Finance 40(2):432-54. Jaffe, Jeffrey E, Randolph Westerfield, and Christopher Ma. 1989. A Twist on the Monday Effect in Stock Prices: Evidence from the U.S. and Foreign Markets. Journal of Banking and Finance 13(SI):641-50. Lakonishok, Josef, and Seymour Smidt. 1988 Are Seasonal Anomalies Real? A Ninety-Year Perspective. Review of Financial Studies 1(4):403-25. Ogden, J. 1990. Turn-of-Month Evaluation of Liquid Profits and Stock Returns: A Common Explanation for the Monthly and January Effect. Journal of Finance 45(4): 1259-72. Pindyck, Robert S., and Daniel L. Rubinfeld. 1998. Econometric Models and Economic Forecasts. 4th ed. New York: McGraw-Hill. Rozeff, Michael S., and William R. Kinney, Jr. 1976. Capital Market Seasonality: The Case of Stock Returns. Journal of Financial Economics 3(4):379-402. Wang, Ko, Yuming Li, and John Erickson. 1997. A New Look at Monday Effect. Journal of Finance 52(5):2171-86. Menahem Rosenberg, Assistant Professor of Finance, Touro College. -1- |
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