...(1983). the recent past, several studies have investigated the seasonal behavior monthly stock market returns of a number of countries. If the seasonal effects are prominent and systematic in the stock markets, then speculators and portfolio managers can engage in playing games in derivative markets such as futures, options, and mutual funds portfolio rebalancing as these stock markets are deemed to be inefficient. To date, many applied researchers were primarily concerned about the seasonality, but not the nature of the seasonality. Identifying the exact nature of seasonality would undoubtedly improve prediction of stock returns and model specification of financial markets in empirical studies.

Since the work of Hyllerberg et al. (1990) and Hyllerberg (1995), it has been recognized that the nature of seasonality can either be deterministic or stochastic. A stochastic process can, in turn, be either stationary or non-stationary. As has been argued before, it is important to detect the exact nature of seasonality for empirical modeling and prediction purposes. It is well-known that in empirical economics and finance researchers prefer to use seasonally adjusted data series. The choice of adjustment process, however, depends on the nature of seasonality. Moreover, we believe that investigating the various nature of seasonal pattern in the stock returns of international securities markets will enhance the understanding of these markets. Although a number of procedures have been developed over the past ten years to analyze seasonal behavior of time series data by testing for the presence of seasonal unit roots and for seasonal stability, little attention, to our knowledge, has been paid to their application. This unexplored issue is addressed in this paper.

The primary objective of this paper is to take a significant step forward from the existing literature to examine the nature seasonal behavior of the monthly stock return series of several OECD countries and emerging economies. The Beaulieu-Miron's (1993) and the Franses' (1991) procedures are used for testing multiple unit roots at the monthly seasonal frequencies, followed by Canova-Hansen's (1995) procedure for testing for stability of seasonal patterns. It is important to investigate whether application of these tests would reveal various nature of seasonality in stock returns. This paper does not attempt to examine the source of seasonality detected by these tests. This is an important issue which warrants future research.

In the past, many studies have used the parametric and non-parametric tests to examine the stock market seasonality; see, for example, Hollander and Wolfe (1973). Rozeff and Kinney (1976) have investigated the existence of seasonality in the monthly returns on the equally-weighted index of the New York Stock Exchange (NYSE) from 1904-1974 and have found that the seasonal effects were significant in the NYSE rates of return, in particular the January effect. A summary of earlier empirical work in this area can be found in Kendall (1953).

We briefly discuss some of the previous work on testing and the source of seasonal effects, particularly the January effect, although the latter is not addressed in this paper. An equally-weighted index is a simple average of the prices of all firms listed on the NYSE, giving small firms greater weight than their share of the market value. Thus, finding a January effect only in the equally-weighted index suggests that it is primarily a small firm phenomenon. Keim (1983) also discovered that this phenomenon is related to abnormally high returns on small firm stocks by examining the relationship between monthly returns and market values of the NYSE common stocks. On the other hand, Roll (1983) pointed out that the January effect was due to tax-loss selling at the end of the tax year. The tax-loss selling hypothesis states that there is a downward pressure on the prices of these stocks, which decline during the year as investors attempt to realize their losses against their taxable income. Roll provided evidence that small firm stocks are affected more by tax-loss selling than are large firm stocks. (See also Reinganum, 1983.) Brown et al. (1985) examined the Australian stock market seasonality and reported the evidence of December-January and July-August seasonal effects, with the latter due to a June-July tax year.

Gultekin and Gultekin (1983) examined the presence of stock market seasonality in sixteen industrial countries. Their evidence shows strong seasonalities in the stock market due to January returns, which is exceptionally large in fifteen of sixteen countries. The January and April effects in the NZ stock market were tested and were not found to be significant (Raj and Thurston, 1994). Portfolio rebalancing was found to cause high January returns in the Canadian stock market (Anthanassakos, 1992). Chan (1986) found that the source of January seasonal in stock returns is long-term loss. See Ligon (1997) and French and Trapani (1994) for similar empirical studies.

Beaulieu-Miron and Franses proposed procedures separately to test the null of unit roots at the zero and monthly seasonal frequencies against the alternative of stationarity. On the other hand, Canova-Hansen proposed the LM procedures to test for the null of stability of seasonal intercepts against the alternative of seasonal unit roots and/or non-constant seasonal intercepts. To find strong evidence for or against the presence of multiple roots in monthly series, it is important to be able to test the null of unit roots at different frequencies against the stationary alternatives and vice versa. If the tests results lead to the same conclusion, one can deduce whether the seasonal root is stationary with a high degree of reliability. In case of contradictory findings, one may have to do further analysis or collect more information to reach a reliable conclusion. In this paper, these new tests are applied to thirteen seasonally unadjusted monthly stock returns series.

Canova and Hansen (1995) used their procedure for testing the null of stationarity against the alternative of unit roots at the seasonal frequencies in the monthly stock return series. Their results show that Japan and the UK stock returns have a unit root at the annual frequency, which is due to June returns. To understand clearly the nature of the stochastic seasonality in different securities, the Beaulieu-Miron and the Franses tests are applied to test for the null of presence of multiple unit roots in the stock returns, followed by the Canova-Hansen (1995) stability tests to confirm the presence of multiple roots at monthly seasonal frequencies.

Tests for Stationary Stochastic Seasonality and Stability of Deterministic Seasonality

In this section, we discuss briefly: (i) the procedures for testing the null hypothesis of unit roots at zero and seasonal frequencies against stationary alternatives and (ii) the procedures for testing the null of stationary stochastic seasonality against non-stationary seasonality, and the null of constant deterministic seasonal intercepts in monthly time series. Applying these tests we believe would detect the nature of seasonality and whether it is deterministic, stationary, or non-stationary.

Testing for Null Hypotheses of Monthly Unit Roots

The following procedures are applied for testing unit roots at zero and seasonal frequencies in monthly time series:

The Beaulieu-Miron Test

Beaulieu and Miron (1993) extended the procedure Hylleberg, Engle, Granger-Yoo (1990) (HEGY) for testing seasonal unit roots in quarterly time series data to those for monthly time series data. The testing procedure is described as follows.

Consider the following model

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where:

[x.sub.t] = The set of fixed regressors, including an intercept and/or a linear trend;

[d.sub.t] = A set of deterministic seasonal components, [[DELTA].sub.12][y.sub.t] = [y.sub.t] - [y.sub.t-12],

[Y.sub.1t] = (1...

NOTE: All illustrations and photos have been removed from this article.



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