Estimating skewness persistence in market returns

Transcription

1 Applied Financial Economics, 1997, 7, Estimating skewness persistence in market returns JATI K. SENGUPTA and YIJUAN ZHENG Department of Economics, ºniversity of California, Santa Barbara, CA 93106, ºSA The conditional returns series for mutual funds and the S&P 500 are analysed to test whether there is persistence in skewness. Three groups of statistical models of market volatility are estimated over the period September 1988 to April 1993 and the empirical evidence provides valuable insights into the skewness persistence. I. INTRODUCTION Recent researches on stock market volatility have found considerable empirical evidence for the persistence of variance in stock market returns. The ARCH and unit root models in different versions analysed by Engle and Bollerslev (1986), Schwert (1989) and Nelson (1991) have usually discussed the conditional variance process as a function of lagged variance, where the skewness of the return distributions has no impact. Yet skewness preference by investors has been empirically documented by a number of researchers. Thus Kraus and Litzenberger (1976) applied the capital asset pricing model in a generalized form to allow for the effect of systematic market skewness on monthly excess returns for 20 portfolios for January 1936 to June 1970 constructed from the New York Stock Exchange (NYSE) stocks and found the regression coefficients of both beta and gamma parameters to be statistically significant. They also found that the higher beta portfolios tend to have more than proportionately higher gammas, where betas reflect the effect of variance and gammas the effect of skewness. This provides a partial explanation for the common empirical finding that the slope of the capital market line is lower than predicted by the traditional mean variance theory. That is, investors have an aversion to increases in market beta as a direct measure of systematic standard deviation and have a preference for increases in beta as a surrogate for proportionally greater increases in systematic positive skewness. More recently Friend and Westerfield (1980), Homaifar and Graddy (1988) and Falk and Levy (1989) found that in the long run, where the market is positively skewed, the marginal rate of substitution of the mean return for skewness is negative, indicating a preference for positive systematic skewness. A second reason for testing whether there is skewness persistence in a return process is the fact that the residuals from the conditional variance equation used in ARCH models fail the normality test very often. Thus Nelson (1991), who has used a more generalized distribution framework than the normal, found for the weekly market indices such as S&P 500 and the value-weighted NYSE index that current returns and future market volatility are negatively correlated. Part of the explanation of the negative mean variance correlation may be due to the effect of the skewness factor. Our object here is twofold: to test the persistence of conditional skewness of market returns and thereby examine the volatility of the skewness trajectory. Just as the variance measure reflects the average fluctuations of the market around its mean, the skewness measure indicates the degree of asymmetry in the market, which has sometimes been measured by good news and bad news effects, or by bullish and bearish expectations. The market here is specified in terms of two indicators: the overall market is analysed in terms of the Standard and Poor (i.e. S&P 500) index and the various components of the market by three groups of mutual funds classified as growth funds, income fund and mixed funds (i.e. growth and income). Since mutual funds are themselves portfolios of several stocks, they represent in a sense different layers of the overall market. II. MODELS OF PERSISTENCE The volatility of market returns is usually estimated in ARCH and generalized ARCH models by decomposing the return series y into its conditional mean and conditional variance y "E y #e ; σ "E e where E y "μ is the conditional mean and E e "σ the conditional variance at time t, both depending on all Routledge 549

2 550 J. K. Sengupta and ½. Zheng past information available up to period t!1. Clearly if one can estimate e through a moving average representation of the conditional variance, σ can be estimated from the observed data on market returns. On the basis of these estimates, Engle and Bollerslev (1986) proposed the generalized ARCH model of market volatility as follows: σ "σ # β σ # α e (1) Two special cases of this formulation are important in applied research. One is the first-order lag case with p"q"1, which yields σ "σ #βσ #αe (2) If the empirical estimates of α and β such that their sum adds up to one or more, in a statistically significant sense, the shocks to volatility are persistent in the sense that the conditional variance tends to be explosive as t goes to infinity. A second variant of Equation 1 drops the regressor variables e and uses p"2 to define a second-order variance process σ "σ #β σ #β σ (3) where σ is a constant denoting unconditional variance. The main reason for using the second-order lagged term σ as the regressor is the fact that the least squares estimates of α in Equation 2 is very often negligible, i.e. close to zero in a statistical sense. For example, the estimates by Sengupta and Park (1990) for two standard market indices, namely the equally weighted return (EWR) and the S&P 500 over the sample period January 1965 to December 1974 based on monthly data are as follows: E¼R y "μ#e ; y "0.0018#e ; (t"0.29) R " σ " #0.986σ!0.0009e (0.88) (46.54) (!0.36) S&P 500 R "0.95 y "0.0018#e ; R " (!0.45) σ " #0.986σ #0.0023e R "0.96 (0.80) (47.87) (1.37) The variance model of market volatility can be generalized in terms of a non-linear process, namely a logistic model σ "σ #σ (α!βσ )#ε (4) where ε is the residual error term. This logistic version is chosen for its several flexible features. First of all, it may be used to test the presence of chaotic dynamics in volatility measured by the conditional variance. In its canonical form the logistic model may be more compactly specified as v /v "h(1!v )#ε (5) where h is the critical parameter which determines the non-linear behaviour of volatility denoted here by v "σ. Note how this form of the logistic equation is the classic chaos equation in the form investigated by Lorenz (1963), Feigenbaum (1983), Gleick (1987) and others. Recent economic applications of chaos theory have been discussed by Barnett and Chen (1988), Brock (1990) and Hsieh (1991). Secondly, the regression coefficient h here acts as a bifurcation parameter in the sense that the qualitative behaviour of variance suddenly changes for values of h exceeding 3. Thus for 1(h(2 we obtain monotonic growth in v converging to the steady state but for h'3 the steady state becomes unstable and a two-period cycle emerges. In fact, the simulation studies by Lorenz (1963) showed that as h increases beyond the value 1# 6, higher and higher even-order cycles emerge, and beyond the value 1# 6 he found the emergence of cycles of very high odd periods. Finally, one could replace the term σ in Equation 4 by the second-order term σ as a proxy and obtain a linear second-order equation as v "α #αv!βv (6) which yields the eigenvalue equation with two roots λ, λ as λ, λ "(1/2)(α$ α!4β) (7) These roots are complex if the condition 4β'α holds. Under these conditions oscillations would occur in the variance series and its convergence (divergence) would depend on whether the modulus of the complex roots were less (greater) than one. By an analogous argument one could specify the models of persistence of conditional skewness defined as "E e. A second-order ARCH type model in skewness may thus be specified as "α #α!β (8) whereas the canonical logistic model in skewness appears as / "h(1! ) (9) The empirical estimates of these models in skewness would provide a direct test of the skewness persistence, if any. Also the skewness persistence may be compared with the variance persistence. Secondly, the impact of skewness on the conditional variance may be estimated by using the skewness term as an explanatory variable in Equation 2. Finally, for the mutual fund returns data one could analyse the impact of the overall market represented by S&P 500 for example on the individual mutual funds with regard to the conditional variance and skewness. Thus if z denotes either

3 Estimating skewness persistence in market returns 551 Table 1. ist of mutual funds: September 1988 to April 1993 Code Ticker symbol Fund name Fund type Price range ($) 01 FDGRX Fidelity Growth Company Fund Growth FMAGX Fidelity Magellan Fund Growth FTRNX Fidelity Trend Fund Growth PRNHX TR-Price New Horizons Growth PRSVX TR-Price Small-Cap Value Growth FIDYX Fin-Prog Dynamics Growth G Artificial sum of six growth funds Growth FGRIX Fidelity Growth & Income Fund G&I FFIDX Fidelity Fund G&I FEQIX Fidelity Equity Income Fund G&I PRGIX TR-Price Growth & Income Fund G&I PRFDX TR-Price Equity Income Fund G&I FIRFX Fin-Prog Industrial G&I GI Artificial sum of six G&I funds G&I FCVSX Fidelity Convertible Securities Income FPURX Fidelity Puritan Income PRHYX TR-Price High Yield Income PRCIX TR-Price New Income Income FIIIX Gin-Prog Industrial Income Income FBDSX Fin-Prog Select Income Income I Artificial sum of six income funds Income SPX S&P 500 daily closing Fund type is according to Stock Guide classification. Equal weighted sum of six same-type funds. the conditional variance (σ "v ) or the conditional skewness (s "E e ) of returns of mutual fund type i and esp denotes the S&P 500 index, the influence of the overall market can be modelled as follows: z "α #α (esp)#α (esp) #ε ε "ρ ε #η (10) Here the residual term η is assumed to be independently and identically distributed with zero mean and finite variance. In this formulation the autocorrelation coefficients ρ may be interpreted in terms of persistance as follows. Let us assume the measured index z representing either volatility or asymmetry is composed of two parts: a long-run or permanent component z reflecting the effects of factors that remain almost constant over time and a short-run or transitory component ε reflecting the influence of short-run conditions, i.e. z "z #ε, z "α #α (esp)#α (esp). We can then write the first line of Equation 10 in an autoregressive form as z "k #ρ z #η (11) where k "(1!ρ )z. This autoregressive form represents a simple description of movements of the variance and skewness of returns in mutual fund i over time that allow incomplete adjustments to deviations from the long-run equilibrium level. The parameter (1!ρ ) measures the speed of adjustment and indicates how quickly the variable z approaches its long-run equilibrium level z. When ρ is large, short-run adjustments erode slowly and the variable z adjusts slowly towards its permanent level. If, on the other hand, ρ is small, short-run effects erode very rapidly. In that case it can be assumed the z series is in long-run equilibrium except for purely random displacements. Thus the higher value of ρ (i.e. close to one) implies the short-run movements are more dominant, whereas the lower values (i.e. close to zero) imply the dominance of the long-run movements. III. DATA The basic data set is the monthly return series for 18 mutual funds divided into three equal groups. The list of these mutual funds is given in Table 1. Monthly return series for the S&P 500 is obtained from the CRSP data set. For the mutual funds the return series for the period September 1988 to April 1993 is obtained from the Dow Jones Retrieval File and the Standard & Poor s Compustat File, and the classification of funds by objectives such as growth, income or mixed was obtained from the Stock Guide published by Standard & Poor. For each fund the monthly return R is computed by the usual method as the monthly percentage capital gain or loss plus the dividend. The conditional mean μ is then estimated by running a linear regression of R on their six lagged values R, R, 2, R in order to eliminate the impact of non-synchronous trading. This is reported in Table 2, which is used to estimate

4 552 J. K. Sengupta and ½. Zheng Table 2. Estimation of return rates: Rt"C #C MA(1)#C MA(2)#C MA(3)#C MA(4)#C MA(5)#C MA(6)#Error Mutual fund No. Group R 01 G !0.2912!0.3175!0.2986! G !0.0835!0.2118!0.1665! G !0.0307!0.1899!0.1915! G !0.0120!0.1620!0.2421!0.3073!0.2897! G !0.1475!0.3432! G !0.3744!0.2888!0.4363! G !0.1030!0.2195!0.2233! G&I !0.1607!0.2152! G&I !0.0376!0.2498!0.3424! G&I !0.0777! G&I !0.1765!0.2544! G&I !0.1890! G&I !0.0016!0.1499!0.3708!0.1966! G&I !0.1693!0.2483! I !0.0773!0.1970! I !0.2133! I !0.0243! I !0.1006!0.3243!0.2492!0.1998! I !0.1614! I ! ! I !0.1988! S&P !0.0732!0.1842!0.2479!0.2210! the residuals e defined in Equation 1. Based on these values of e we calculate the conditional variance series σ "(1/3) el in a recursive moving average form with a fixed period of three months. The three groups of mutual funds, i.e. growth, income and mixed (growth and income) and their group averages are more realistic in decreasing the investor behaviour than the aggregate market indices, such as the value-weighted index or S&P 500, since they are actively traded in the stock market and the fund managers attempt to switch investment between different funds in order to improve their performance. Fig. 1 plots the average of the return variances for the three groups of mutual funds and the S&P 500. IV. ESTIMATES AND RESULTS For skewness persistence we consider three groups of statistical tests. In the first group we compare the skewness persistence with the variance persistence in terms of the linear second-order model (Equation 3). This is the same framework as employed in the standard ARCH model. Tables 3 to 5 persent the comparative estimates. The second group of tests consider the impact on skewness due to the overall market and the lagged variance. Tables 6 to 8 report the statistical estimates in this regard. Finally, we have the estimates of the logistic models (Equations 5 and 9) where h is the critical parameter determining whether there is chaotic instability in the skewness and variance processes. Tables 9 and 10 present the estimates of the canonical logistic models. The main result from the first group of statistical tests is that skewness is more persistent than the conditional variance. This is evident from the fact that the first-order autoregressive coefficient is higher for skewness than for variance. The estimated modulus of the roots is also higher for skewness for each mutual fund. Since the two roots are complex in all cases for both skewness and variance, oscillations are present though these are convergent in most cases. When we consider the interval P "P#3σ where P is the modulus of the roots, the skewness frontier tends to be explosive in all cases except one (no. 20), whereas for the variance process only one fund (no. 19) displays the explosive property, i.e. P '1. The major result for the second group of tests is that the conditional skewness of the mutual fund returns is significantly affected by the overall market. The estimate of the autocorrelation coefficient ρ defined in Equation 10 is uniformly higher for the skewness process than the variance process. This implies that for the skewness process the short-run transitory adjustments erode more slowly than the variance process. This tends to support the earlier

5 Estimating skewness persistence in market returns 553 Fig. 1. Average of the return variances for (a) income fund ( ) v,( )a ;(b)s&p 500 ( ) v,( )a ;(c)growth income fund ( ) a,( )v ;(d)growth fund ( ) a,( )v Table 3. Estimates of the variance and skewness process Mutual fund v "αv!βv "α!β No. Group α β R α β R 01 G G G G G G G G&I G&I G&I G&I G&I G&I G&I I I I I I I I S&P Significant t-value at the 15% level.

6 554 J. K. Sengupta and ½. Zheng Table 4. ¹ests of the second-order variance and skewness processes (modulus) Mutual fund v "αv!βv "α!β No. Group P (std. error) cos Period P (std. error) cos Period 01 G (0.1614) (0.0709) G (0.1380) (0.0834) G (0.1554) (0.0525) G (0.1558) (0.0579) G (0.1462) (0.0532) G (0.1591) (0.0609) G (0.1807) (0.0656) G&I (0.1201) (0.0548) G&I (0.1230) (0.0638) G&I (0.1458) (0.0827) G&I (0.1386) (0.0595) G&I (0.1243) (0.0738) G&I (0.1497) (0.0755) G&I (0.1184) (0.0585) I (0.1431) (0.0523) I (0.1252) (0.0582) I (0.1332) (0.0850) I (0.1795) (0.0780) I (0.0812) (0.0648) I (0.1084) (0.1595) I (0.1384) (0.0609) S&P (0.1189) (0.0749) P "0.25(λ #λ )!0.25(λ!λ ) Table 5. Estimates of the modulus of the roots P "P#kσ Mutual fund v No. Group k"0 k"1 k"2 k"3 k"0 k"1 k"2 k"3 01 G G G G G G G G&I G&I G&I G&I G&I G&I G&I I I I I I I I S&P Note: The estimates are derived from Table 4.

7 Estimating skewness persistence in market returns 555 Table 6. Impact of variance on skewness of returns Mutual fund "α #α #α v No. Group α Standard error α Standard error R 01 G G ! G ! G G G G G&I ! G&I ! G&I ! G&I ! G&I ! G&I G&I ! I ! I ! I ! I ! I ! I ! I ! S&P Table 7. Impacts of market on variance and skewness v "α #α esp#α (esp) #ε μ "α #α esp#α (esp) #ε Mutual fund ε "ρε #η ε "ρε #η No. Group α α α α α α 01 G ! G ! G ! ! G ! G ! ! ! G ! G ! ! G&I ! G&I ! ! ! G&I ! ! ! G&I ! ! ! G&I ! ! ! G&I ! G&I ! ! ! I ! I ! ! I ! ! ! I ! I ! I !0.0018! ! I ! ! Significant t-value at the 10% level.

8 556 J. K. Sengupta and ½. Zheng Table 8. Estimate of persistence in conditional variance and skewness v "f (esp)#ε μ "f (esp)#ε Mutual fund ε "ρε #η ε "ρε #η No. Group ρ Standard error ρ Standard error 01 G G G G G G G G&I G&I G&I G&I G&I G&I G&I I I I I I I I Table 9. Estimates of the standard chaos model (logistic form) Mutual fund Variance Skewness No. Group h Std. error D¼ h Std. error D¼ 01 G G ! G G G G G ! G&I G&I G&I G&I G&I G&I G&I I ! I I ! I I ! I S&P

9 Estimating skewness persistence in market returns 557 Table 10. Estimates of the critical parameter of the logistic form of chaotic dynamics (h "h#kσ ) Mutual fund v No. Group k"0 k"1 k"2 k"3 k"0 k"1 k"2 k"3 01 G G G ! G G G G G&I ! G&I G&I G&I G&I G&I G&I I I ! I I ! I I ! I S&P Note: The estimates of h are obtained from Table 9. conclusion that the skewness is more persistent than the variance. Also it is clear that the conditional variance affects the skewness process, more often negatively. This negative impact is present for all cases of the income fund and most of growth-cum-income funds. The influence of the overall market return represented by S&P 500 on the skewness process of fund returns is most significant in linear terms, whereas for variance the quadratic term represented by the square of S&P 500 is most significant in a statistical sense. The linear impact of S&P 500 on the skewness process is highly significant in a statistical sense for almost all the mutual funds. The major result from the third group of tests is that the possibility of chaotic instability is present in both the variance and the skewness processes. As a matter of fact, the proportion of cases when h "h#σ exceeds the chaotic value 1# 6"3.57 is identical (i.e. 10/22) in both cases, although for the S&P 500 index the chaotic instability sets in at h#2σ for the variance process but not at all for the skewness process. Thus, in terms of the overall market and its volatility measured by the conditional variance, the probability of observing chaotic volatility is not small. This view is consistent with the recent simulation study by Hsieh (1991) who performed a number of statistical tests including the Brock Dechert Scheinkman (BDS) test (Brock et al., 1987) to conclude that the stock market returns are not independently and identically distributed (IID) and this view is consistent with the complexity of chaotic behaviour in stock returns. Sengupta and Zheng (1993) have analysed in some detail elsewhere the complexity of chaotic behaviour in the variance process as revealed through a set of simulated profiles of the variance trajectory. V. CONCLUSIONS Empirical analysis of the mutual fund return data yielded three broad conclusions. First of all, the conditional skewness appears to be more persistent than the conditional variance. Mutual fund skewness is more affected by the overall market index and also by its own variance. The impact of variance on skewness turns out to be negative in most cases, implying the presence of asymmetry. Secondly, the linear second-order models of variance and skewness lead in all cases to the complex roots with oscillations, which tend to be convergent in most cases. The estimated modulus of the roots is found to be higher for skewness than for variance. Finally, for both the skewness and the variance processes there is positive probability for chaotic instability, although for some growth funds the probability is larger for variance than for skewness. REFERENCES Barnett, W. and Chen, P. (1988) The aggregation-theoretic monetary aggregates are chaotic and have strange attractors, in Dynamic Economic Modelling, eds W. Barnett, E. Berndt and H. White, Cambridge University Press, Cambridge.

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