Conclusiones

Returns are assumed to be identically distributed implying that certain statistical properties of time series data remain invariant over time. This is known as the stationarity hypothesis (Cont, 2001). According to Mandelbrot (1967), stationarity implies that sample moments do not vary substantially from sample to sample. Gibbons and Hess (1981) argue that the assumption of identically distributed returns requires that the mean and variance are constant over time. Giannopoulos (2000) states that changes in these two sample moments are often cited as the reason for excess kurtosis in return distributions. Cont (2001) argues that it is not clear whether this assumption holds as evident from seasonal effects such as the January, weekend and the day of the week effect. Giannopoulos (2000) states that while evidence regarding the non-stationarity of the mean is inconclusive, the non-stationarity of variance is widely recognized.

Two studies indicative of the debate relating to the stationarity of the mean are those of Gibbons and Hess (1981) and Peiró (1994). Gibbons and Hess (1981) state that it is generally assumed that the distribution of stock returns is identical for all days of the week. However, there is increasing evidence that the distribution varies across the days of the week. An often cited example is that of the so called Monday effect whereby Monday returns exhibit a higher mean and variance. The authors investigate the day of the week effect using return data on the S&P 500 Index, value and equally-weighted portfolios constructed from the CRSP database, and individual stocks compromising the DJIA between July 1962 and December 1978. The hypothesis of equality is rejected⁷³ for returns on the S&P 500 Index and both

73 The conventional test for the equality of means is conducted using a dummy regression specification,

0 1 1 2 2 3 3 4 4

t t t t t t

R =α α+ D +αD +αD +αD +ε (Gibbons & Hess, 1981; Kiymaz & Berument, 2003). The coefficients of this specification represent mean returns for each day of the week. By showing that coefficients α₁ through to α₄ are equal, it can be shown that returns are from identical distributions (Mookerjee & Yu, 1999).

portfolios suggesting that the return distribution is not identical across time. Returns for Mondays are lowest although a degree of variation in the mean is also observed for other days of the week. The equality hypothesis is also rejected for all individual stocks compromising the DJIA. Gibbons and Hess’ (1981) findings relating to returns on the aggregates considered and individual stocks suggest that the assumption of identically distributed returns does not hold. The authors conclude that daily seasonality is evident in stock returns and this is manifested by persistently negative mean returns on Mondays.

Peiró (1994) states that one of the most interesting seasonal effects observed is daily seasonality, which manifests itself in a differing distribution across days of the week. The author seeks to establish whether seasonality and day of the week effects are present in six major stock market indices; namely, the DJIA, Nikkei, Financial Times Ordinary Share 30 (FT 30), Commerzbank, Compagnie des Agents de Change (CAC) General and the General index for the period from December 1987 to December 1992.⁷⁴ As in Gibbons and Hess (1981) the standard dummy regression approach is employed to test for seasonality. The null hypothesis of equality is not rejected for the DJIA, Nikkei, Commerzbank and the General indices suggesting that the distribution of returns does not differ in the mean across days of the week for these indices. However, seasonality is observed in returns on the FT 30 and the CAC General indices. For the FT 30 Index, seasonal behaviour is attributed to a strong Monday effect. Peiró (1994) concludes that these findings question the validity of widespread seasonal patterns observed in prior literature. In contrast to Gibbons and Hess’ (1981) findings, these results mostly support the assumption of identically distributed returns.

While Peiró (1994) does not find widespread evidence of seasonality in the mean, the same does not hold for variance. Tests of the equality of variance across days of the week indicate that the null hypothesis of equal variance is rejected for all indices with the exception of the DJIA. This suggests that variance exhibits widespread seasonal effects and is of time-varying nature. Moreover, unlike the ambiguous debate regarding the stationarity of the mean, it is widely accepted that the variance of stock returns is of a time-varying nature (Giannopoulos, 2000). Evidence suggesting the variance is not stationary is found in the literature as early as Bachalier (1914), Mandelbrot (1967) and Praetz (1972). Bachalier (1914) notes that the evidence diverges from his original theoretical formulation in that sample variance differs

74 Indices on exchanges situated in New York, Tokyo, London, Frankfurt, Paris and Madrid.

over time. Mandelbrot (1967) presents a plot of the variance of cotton price changes which indicates that variance differs over time (see Mandelbrot, 1967: Figure 1). It is suggested that seasonal effects, changes in the macroeconomic environment and economic policy are behind changing variance. Praetz (1972) states that it is widely assumed that the variance of returns is constant over time. This assumption however is contradicted by observed extended periods of market activity which are followed by extended periods of inactivity. These transitions in the magnitude of variance are attributed to information clustering around certain dates. Praetz (1972) further notes that evidence suggests that variance varies from year-to-year as market activity varies.

It is however Taylor (2008)⁷⁵ who popularized the notion of time-varying variance in his extensive study of the properties of returns. Taylor (2008) reports that absolute and squared transformations⁷⁶ of US stock return series - both proxies for volatility - exhibit high levels of first order serial correlation and continue to be positively correlated over extended periods of time.⁷⁷ It is suggested that this serial correlation structure is attributable to changes in the variance of returns implying that variance is of a time-varying nature for the January 1966 to December 1976 period. Akgiray (1989) arrives at a similar conclusion. While first order serial correlation in returns on the value-weighted CRSP index is high, it becomes statistically insignificant at longer lags. However, this is not so for absolute and squared returns, which are highly correlated for extended periods of time, as evident from Figure 5.1:

75 This paper references the second edition of Taylor’s seminal work, Modelling Financial Time Series, owing to the unavailability of the original text. The first edition was published in 1986.

76 See Poon (2000), McMillan and Ruiz (2009). The presence of non-linear dependence in returns is interpreted as correlation in volatility and does not in itself imply that returns are serially correlated (Cont, 2001).

77 For example, for the Kodak return series, first order serial correlation for absolute and squared returns is 0.146 and 0.178 respectively. For Alcoa, first order serial correlation is 0.194 and 0.144 respectively. Whereas Taylor (2008) finds that less than 10 percent of correlation coefficients for the linear series are outside the -0.05 and 0.05 interval for 1 to 30 lags, for squared returns, the percentage of series for which correlation coefficients exceed 0.05 is 58 percent.

Source: Akgiray (1989) Figure 5.1: Returns and non-linear transformations of returns

Whereas the serial correlation function of the return series falls below zero after the first lag, the serial correlation functions of absolute and squared returns decay slowly and are still above zero after 60 lags. Akgiray (1989) states that this implies that large price changes are followed by large price changes and small price changes are followed by small price changes of either sign – an example of volatility clustering. Furthermore, the non-linear dependence observed in absolute and squared return series is attributed to changing variance and is cited by Akgiray (1989) as an explanation for leptokurtosis in return distributions. It is suggested that these changes in variance are related to the rate of information arrival, levels of trading activity and corporate financial and operating leverage decisions. Notably, Akgiray (1989) suggests that any model of returns must be compatible with this characteristic (changing variance) and take into account non-linear dependence in returns. Taylor’s (2008) and Akgiray’s (1989) findings confirm the propositions of Mandelbrot (1967) and Praetz (1972) that variance is non-stationary.

The literature suggests that although there is debate surrounding the stationarity of the mean, it is almost a certainty that the variance differs over time. Tang (1997) goes even further.

Using return data on industrial sectors compromising the Hong Kong stock market and returns on the Hong Kong Index (HKI) over the January 1984 and March 1992 period, Tang (1997) finds that seasonality extends into the higher moments of the return distribution implying that the higher moments are stationary. Whether this is related to non-stationarity in the variance warrants further investigation. However, given the evidence

relating to the non-stationarity of the variance, the validity of the assumption of identically distributed returns is questionable; it can be argued that return distributions are stationary in the mean but not in the variance.