month of the year effect and january effect in pre-wwi stock returns: evidence from a non-linear...

INTERNATIONAL JOURNAL OF FINANCE AND ECONOMICS

Int. J. Fin. Econ. 6: 1–11 (2001)

MONTH OF THE YEAR EFFECT AND JANUARY EFFECT INPRE-WWI STOCK RETURNS: EVIDENCE FROM A NON-LINEAR

GARCH MODELTAUFIQ CHOUDHRY*

School of Management, Uni6ersity of Southampton, Southampton, UK

ABSTRACT

This paper investigates seasonal anomalies in the mean stock returns of Germany, the UK and the US duringpre-World War I (WWI) period. The anomalies studied are month of the year effect and the January effect. Theempirical research is conducted using a non-linear GARCH-t model, and monthly returns. Results obtained provideevidence of the January effect and the month of the year effect on the UK and the US returns. The German returnsshows the month of the year effect but no January effect. Given the lack of tax treatment of capital gains/loss before1914 by these countries, the results fail to provide merit to the tax-loss selling hypothesis of the January effect. Sincewe apply value-weighted returns in all cases, results obtained also fail to provide support for the small firm effect.Copyright © 2001 John Wiley & Sons, Ltd.

KEY WORDS: asymmetric effect; January effect; non-linear GARCH; seasonal anomalies; volatility

JEL CODE: G15

1. INTRODUCTION

In the last few years, empirical investigations of stock market anomalies have been extensive.1 Accordingto Mills and Coutts (1995, p. 79) the persistence of these anomalies in stock markets has put doubts inthe theory of market efficiency and the Capital Asset Pricing Model (CAPM). A popular class ofanomalies in the stock markets focuses on specific time periods or seasonalities, such as the month of theyear effect, the January effect, the day of the week effect, the Monday (weekend) effect, etc. Accordingto Mills and Coutts (1995) one of the most prevalent of these anomalies appears to be the January effect,in which returns are much higher during January than any other month. Since it was first referred to byWachtel (1942), the so-called ‘January effect’ has gone through severe empirical investigations.2 Severaltheories have been put forward regarding seasonal anomalies in the stock market. The month of the yeareffect and the January effect have been mostly explained by the tax-loss selling at the end of the yearhypothesis, size of the firm, insider-trading/information-release hypothesis, January seasonal in therisk-return relationship, and omitted risk factors, etc.3

This paper investigates using a non-linear GARCH model the month of the year effect in the German,UK and US stock returns during the pre-World War I (WWI) period. The main attraction of using thepre-WWI period is the lack of any form of tax treatment of capital gains/loss in all three countries. Alarge part of the empirical work in this field has been conducted using data from the US (and otherdeveloped countries) for most recent periods. The use of stock market data from different time periods inthe study of anomalies has been advocated in several studies, such as Fishe et al. (1993) and Seyhun(1993). According to these studies more proof for or against seasonal anomalies can only be verified byinvestigating markets during different time periods. Pettengill (1986) and Jones et al. (1987) using similarpre-WWI period US stock return and linear ordinary least squares (OLS) method find evidence of a

* Correspondence to: School of Management, University of Southampton, Southampton, SO17 1BJ, UK. Tel.: +44 2380 593887;fax: +44 2380 593844; e-mail: [email protected]

Copyright © 2001 John Wiley & Sons, Ltd.

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significant January effect.4 Their results show that non-tax factors may be responsible for the Januaryeffect.5 This paper differs from the works of Pettengill (1986) and Jones et al. (1987) based on themethodology and the extra countries applied. Application of the pre-WWI data, especially the Germanand the UK returns and the use of a non-linear GARCH model makes this paper unique in the field ofstock market anomalies.

As stated earlier, the empirical investigation in this paper is conducted using a non-linear GeneralizedAutoregressive Conditional Heteroscedasticity (GARCH) model and not the usual linear regressionmethod.6 The GARCH models are capable of capturing the three most empirical features observed instock return data: leptokurtosis, skewness and volatility clustering. Since the studies of Mandlebrot (1963)and Fama (1965), empirical research has found evidence of volatility clustering and kurtosis in changes instock prices. Connolly (1989, p. 134) showed that there is much evidence that stock returns have timevarying volatility and studies of stock market anomalies usually fail to take that into consideration. Whatis not known, according to Connolly, is whether inferences are sensitive to alternative heteroscedasticitycorrections; he advocates the use of the GARCH model in the study of anomalies in the stock markets.7

Further, as indicated by Connolly (1989) and de Jong et al. (1992) most empirical studies dealing withstock market anomalies are usually carried out under the assumption that the error term and hence thereturns follow a normal distribution with constant variance. And according to Connolly (1989) the errorterm from regressions involving stock returns are almost certainly not normally distributed. The problemcreated by fat tailed distribution is that test statistics based on nonrobust standard error estimates cannotbe interpreted in the usual way. The GARCH model takes account of the non-normality of the errordistribution. In this paper a non-linear (asymmetric) version of the GARCH model is applied in order tocapture the non-linear dependencies in the returns volatility along with leptokurtosis, skewness andvolatility clustering.8

2. THEORIES OF SEASONAL ANOMALIES

Several theories have been put forward regarding seasonal anomalies in the stock market. The month ofthe year effect has been mostly explained by the tax-loss selling at the end of the year hypothesis, size ofthe firm, insider-trading/information-release hypothesis, January seasonal in the risk-return relationshipand omitted risk factors, etc.9 According to Seyhun (1993) these theories may be classified into twocategories, one is consistent with the efficient market hypothesis and equilibrium asset pricing models, andthe other indicates a failure of the efficient markets and equilibrium assets pricing model.

The tax-loss selling hypothesis is the most popular in explaining the January effect (Rozeff, 1986;Ritter, 1988; Dahlquist and Sellin, 1994). According to this hypothesis, investors sell their losing stocksbefore year end in order to obtain the tax savings from deducting those losses from capital gains realizedduring the year. The selling pressure in late December is then followed by buying pressure in January asinvestors return to desired portfolio compositions.10 According to Fortune (1991, p. 23) the tax-loss sellinghypothesis is not consistent with the efficient market hypothesis. According to the efficient markethypothesis investors with no capital gains taxes should identify any tendency towards abnormally lowprices in December and should become buyers of stocks oversold in late December. This means thetax-loss selling should affect the ownership of shares but not their price. Other explanations of theJanuary effect that also imply inefficient markets are portfolio rebalancing, which states that windowdressing by institution holders puts pressure on small stocks at turn of the year (Ritter, 1988; Ritter andChopra, 1989). In other words, the high returns in January are caused by systematic shifts in the portfolioholdings of investors at the turn of the year (Haugen and Lakonishok, 1988).

The Chan et al. (1985) hypothesis of omitted risk factors is more consistent with rational investors andefficient capital markets. According to this theory if it is riskier to hold stocks in January than in anyother months of the year, because of some omitted risk factor in that month, then investors should, onaverage, get a higher return in January.11 Along with the omitted risk factor, seasonalities in therisk-return tradeoff is also provided as an explanation for the January effect (Tinic and West, 1984, 1986;

Copyright © 2001 John Wiley & Sons, Ltd. Int. J. Fin. Econ. 6: 1–11 (2001)

MONTH OF THE YEAR EFFECT AND JANUARY EFFECT IN PRE-WWI STOCK RETURNS 3

Hillion and Sirri, 1987). According to this theory, investors require higher returns to take on risk at theend of the year. The suggestion here is that the tradeoff between risk and return is not constantthroughout the year. Specifically, at the turn of the year, investors for some reason give risk a greaterweight. Whatever the reason, higher returns are therefore required by investors to take on risk which,quantitatively, may not have changed. Another reason provided is the information arrival/insider tradinghypothesis, which predicts that informed traders are more likely to trade in January (Williams, 1986;Seyhun, 1988). According to Roll (1983) and Keim (1989) the January effect may be due to econometricand risk mis-measurement problems implying that either the January effect is spurious or that investorscannot trade at these prices. Mills and Coutts (1995) claim that January effect could be due to liquidityconstraints of the economic agents.

Lakonishok and Smidt (1984) and Thaler (1987) claim that the January effect is only a small firmphenomenon. Banz (1981) and Reinganum (1983) found that on average small firms earn higher thanexpected returns. According to the efficient market hypothesis, these high returns should be solely due tohigher risk, but Keim (1983) discovered that these abnormally high returns had a close relation with theJanuary effect. Thaler (1987) claims finding a January effect only in an equal-weighted index suggests thatit is primarily a small firm phenomenon. Since an equal-weighted index is a simple average of the pricesof firms, it gives small firms greater weight than their share of market value. According to Keim (1983)and Reinganum (1983) the January effect in small firms is more prominent in the first 5 days of tradingin January and returns were higher for firms whose prices had declined the previous year. Resultspresented by Seyhun (1993) imply any potential explanation for the January effect ought to include allfirms, not just the small firms.

3. THE DATA AND BASIC STATISTICS

For all three countries monthly stock returns are applied. The data range from January 1870 to December1913 for Germany and the UK and from January 1871 to December 1913 for the US. As indicated earlier,during these periods capital gains/loss in the stock market was not subjected to tax treatment of any kindby the three countries. If a January effect is found during these time periods, it may be concluded thatsome non-tax factor influences the January effect. The stock returns data are basically the first differenceof the log of stock prices. All three stock indices applied do not include dividend yield, thus returnsapplied are basically the capital gains or loss in the stock market. Lakonishok and Smidt (1988), Fishe etal. (1993) and Mills and Coutts (1995) claim that the use of dividend adjusted or unadjusted stock returnsproduces negligible difference in anomalies study results. Furthermore, French et al. (1987, p. 12) indicatesince the ex-dividend days are different for different stocks in a general price index, the fact that returnslack dividends should have little effect on the volatility estimates. The stock price applied is the index ofIndustrial shares for the United Kingdom, and the index of common stock prices for Germany. The USindex is a combination of all industrial and public utilities, and railroad common stocks. The German andthe UK price indices are obtained from the NBER website, while the US index is obtained from HistoricalStatistics of the United States 1789–1945.12 All three indices are value-weighted; firms are given weights

Table 1. Basic statistics

German returns US returnsUK returns

Observations 527527 5150.00120.0007250.0011Mean

Variance 0.00079 0.00022 0.000011648.081aNormality 17.603a176.62a

Skewness −0.450a 0.974a −0.236a

5.689a 5.216a 0.801Kurtosis

a Implies significance at the 1% level. Normality is tested by Jarque–Bera statistics.


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according to their share of market value. Table 1 presents the basic statistics of the three stock returnsseries.13 All three series are found to be non-normal by means of the Jarque–Bera test. This result is notsurprising since all series are found to have excess kurtosis.

4. METHODOLOGY

4.1. Standard GARCH-t Model

According to the generalized ARCH(p, q) model also known as the GARCH(p, q) model theconditional variance of a time series depends upon the squared residuals of the process (Bollerslev, 1986).The GARCH model has the advantage of incorporating heteroscedasticity into the estimationprocedure.14 According to Bollerslev et al. (1992) the GARCH(p, q) model can be viewed as a reducedform of a more complicated dynamic structure for the time varying conditional second order moments.The GARCH model provides a more flexible framework in order to capture various dynamic structuresof conditional variance and it allows simultaneous estimation of several parameters of interest andhypothesis (Chou, 1988). The GARCH model captures the tendency for volatility clustering in thefinancial data.15 Volatility clustering in stock return implies that large (small) price changes follow large(small) price changes of either signs. Stock returns can be represented by the MA(1)-GARCH(p, q)-tmodel as follows:

yt=dxt+et−uet−1 (1)

et/Ct−1� t.d.(0, ht, 6) (2)

ht=v+ %p

j=1

bjht− j+ %q

j=1

aj(et− j)2 (3)

where yt is the stock return considered to be linearly related to a vector of explanatory variables (xt) andan error term (et). The error term depends on past information (Ct−1) and the following inequalityrestrictions v\0, a]0 and bj]0 are imposed to ensure that the conditional variance (ht) is positive.The moving average (MA) term uet−1 is included to capture the effect of non-synchronous trading.According to Susmel and Engle (1994) non-synchronous trading induces negative serial correlation andthe MA term allows for autocorrelation induced by discontinuous trading in the stocks (as suggested byScholes and Williams, 1977). The error terms are assumed to follow a conditional Student-t density (t.d.)with 6 degrees of freedom.16 As Bollerslev (1987) notes, the t-distribution approaches a normaldistribution with variance (ht) as 1/6 approaches zero. Thus, in this model the error distribution may beconditionally heteroscedastic and non-normal. According to Connolly (1989) this is useful because theunconditional leptokurtosis may be traced to non-normality in the conditional error distribution and/orto time varying heteroscedasticity. If the estimate of 6 is greater than 30 but aj, bj are positive, timevarying heteroscedasticity accounts for the non-normal error distribution. If the 6 estimate is less than tenand aj, bj are positive, both non-normality and time-varying heteroscedasticity produce the fat tailed errordistribution. Significance of aj implies the existence of the ARCH process in the error term. Economicinterpretation of the ARCH effects has been provided within both the micro and macro framework.According to Bollerslev et al. (1992, p. 32) and other studies the ARCH effect could be due to clusteringof trade volumes, nominal interest rates, dividend yields, money supply, oil price index, etc.

Along with the leptokurtic distribution of stock returns data, negative correlation between currentreturns and future volatility have been shown by empirical research (Black, 1976; Christie, 1982). Thisnegative effect of current returns on future variance is sometimes called the leverage effect (Bollerslev etal., 1992). The leverage effect is due to the reduction in the equity value which would raise thedebt-to-equity ratio, hence raising the riskiness of the firm as a result of an increase in future volatility.Consequently the future volatility will be negatively related to the current returns on the stock (Black,1976; Christie, 1982). Thus, according to the leverage effect stock returns, volatility tends to be higher



after negative shocks than after positive shocks of a similar size. Glosten et al. (1993) provide analternative explanation for the negative effect; if most of the fluctuations in stock prices are caused byfluctuations in expected future cash flows, and the riskiness of future cash flows does not changeproportionally when investors revise their expectations, the unanticipated changes in stock prices andreturns will be negatively related to unanticipated changes in future volatility. The symmetric (linear)GARCH model presented in Equations (1), (2), and (3) is not able to capture this dynamic patternbecause the conditional variance is only linked to past conditional variances and squared innovations, andhence the sign of returns plays no role in affecting volatilities (Bollerslev et al., 1992). Glosten et al. (1993)provide a modification in the GARCH model that allows positive and negative innovations to returns tohave different impact on conditional variance. This modification involves adding a dummy variable (It)on the innovations in the conditional variance equation, i.e. Equation (3).17 Adding the leverage effectdummy the conditional variance equation of GARCH(1,1)-t becomes:

ht=v+b1ht−1+a1(et−1)2+a2(et−1)2It−1 (3a)

The leverage dummy variable (It) takes the value of one when innovations to returns (et−1) arenegative, and zero otherwise. If the coefficient on the dummy (a2) is positive and significant, this providesevidence of the leverage effect. In other words, a positive and significant dummy indicates that negativeinnovations have a larger effect on returns than positive innovations. Such a result implies non-linear(asymmetric) dependencies in the stock returns and excess returns conditional variances (volatility).

4.2. Month of the Year Effect and the January Effect

The month of the year and the January effect on returns is investigated by including dummy variablesrepresenting all the months of the year in Equation (1)

yt= %12

j=1

diDit+et−uet−1 (1a)

where yt once again is the stock (or excess) returns index, and Dit are dummy variables identifying themonths observations. So D1t is equal to one if t is the month of January, otherwise it is zero, D2t is equalto one when t is February or zero otherwise and so on. The coefficients d1 to d12 in Equation (1a) are themean returns for the 12 months of the year. Equations (2) and (3a) stay the same. Given the statedtheories concerning the January effect and the month of the year effect, the coefficient of the Januarydummy (d1) should be positive and significant in Equation (1a). There are no set expectations regardingthe sign and significance of the other month dummy coefficients. Thus, the asymmetricMA(1)-GARCH(1, 1)-t models applied in the empirical work consist of Equations (1a), (2) and (3a).

5. EMPIRICAL RESULTS

Table 2 present the results from the MA(1)-GARCH-t(1, 1)-GJR model with each month of the yeardummies.18 The ARCH process (volatility clustering) is significant in the UK stock returns only.19 Sincethe size of the ARCH coefficient (a1) is less than unity, shocks to volatility are not explosive. Thecoefficient on the leverage dummy (a2) is significant and positive only in the US returns test. A positiveand significant coefficient indicates that, along with the size, the sign of the error term has an importanteffect on volatility, that is, the conditional variance is higher whenever innovations (et−1) to returns arenegative rather than positive. As stated earlier, this result provides evidence of non-linear dependencies inthe returns volatility as advocated by the theory of leverage effect (Black, 1976; Christie, 1982). In theremaining two cases results fail to show evidence of non-linear dependencies. The MA term coefficient ()is negative and significant in all three returns. Significant MA term may be due to non-synchronoustrading.20 In all tests, the estimate of v is significant and less than ten implying that both fundamentalnon-normality and heteroskedasticity are responsible for the kurtosis in the unconditional errordistribution.


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Table 2. MA(1)-GARCH-t(1,1)-GJR test with month of the year dummies

Coefficients German returns UK returns US returns

d1 0.0029 0.0113a 0.0101a

(0.91) (6.70) (2.13)

d2 0.0047c −0.0009 −0.0002(1.88) (−0.45) (−0.04)

d3 −0.0012 −0.0069a −0.0005(−0.37) (−4.18) (−0.11)

d4 0.0008 0.0008 0.0070c

(0.29) (0.51) (1.66)

d5 0.0009 −0.0018 −0.0060c

(0.35) (−0.95) (−1.66)

d6 −0.0049c −0.0025 −0.0006(−1.68) (−1.33) (−1.22)

d7 −0.0037 −0.0039b 0.0024(−1.10) (−2.06) (0.56)

d8 0.0131a 0.0011 0.0099b

(4.48) (0.68) (2.26)

d9 0.0068b 0.0016 0.0044(2.51) (0.88) (0.99)

d10 −0.0056c −0.0020 −0.0067c

(−1.86) (−1.17) (−1.71)

d11 −0.0018 −0.0006 −0.0002(−0.69) (−0.37) (−0.04)

d12 0.0060b 0.0006 −0.0035(2.05) (0.29) (−0.79)

v 0.000003 0.00003a 0.0002a

(1.14) (2.04) (2.74)

u −0.249a −0.149a −0.386a

(−6.00) (−3.32) (−8.30)

a1 0.0498 0.117a 0.068(1.57) (2.47) (1.36)

a2 0.0280 −0.083 0.300b

(0.77) (−1.30) (2.34)

b1 0.929a 0.760a 0.568a

(41.02) (8.10) (4.79)

6 6.940a 5.788a 9.397b

(3.24) (4.04) (2.31)

L 1547.92 1855.96 1395.44

F-test 2.730*** 5.206*** 1.655*

a, b and c imply significance at the 1%, 5% and 10% levels, respectively. t-Statistics in theparentheses. L, log-likelihood function value.***, ** and * imply rejection of the null (d1=d2= ···=d12) at the 1%, 5% and 10% levels,respectively.

To assess the general descriptive validity of the model, a battery of standard specification tests isemployed. These tests results are presented in Table 3. Specification adequacy of the first two conditionalmoments is verified through serial correlation test of white noise. These tests employ the Ljung–Box Q



Table 3. Diagnostic statistics of the residuals

German returns UK returns US returns

et-raw residualsSkewness −0.44a 0.87a −0.21b

Kurtosis 3.20a 4.98a 1.94a

Ljung–Box (24) 46.04a 47.65a 34.27b

et/ht1/2-standardized residuals

Skewness −0.57a 0.40a −0.090Kurtosis 1.66a 2.60a 0.90a

Ljung–Box (24) 26.72 29.63 25.95

e t2/ht-squared standardized residualsSkewness 5.68a 7.61a 4.01a

Kurtosis 47.88a 89.44a 22.61a

Ljung–Box (24) 15.41 19.55 27.76

a and b imply significance at the 1% and 5% levels respectively.

statistics on the non-normalized residuals (et), standardized (normalized) residuals (et/ht1/2) and

standardized squared residuals (e t2/ht). All series are found to be free of serial correlation (at the 5% level)

except the raw residuals in all cases. Absence of serial correlation in the standardized squared residualsimply the lack of need to encompass a higher order ARCH process. All series are found to be leptokurticand most to be significantly skewed. According to Hsieh (1989) with a correctly specified conditionalvariance, the excess kurtosis in standardized residuals cannot exceed the excess kurtosis in thenon-standardized residuals. In all three cases, the excess kurtosis in standardized residuals is less than inthe non-standardized residuals.

Results presented show significant and positive January coefficient (d1) in the case of the UK and theUS returns (Table 2). In the case of the US significant positive affect is also imposed by April and August.Months of May and October produce significant negative effects. But the largest effect (in absolute terms)is imposed by the month of January. In the case of the UK, besides January the months of March andJuly also significantly affect the stock returns. But, these months impose a negative affect. Once again inabsolute value January produces the largest affect. For Germany, results fail to show a significant Januaryeffect but indicate a significant and positive February effect. Along with February, the German return isalso affected by June, August, September, October and December. Except for June and October theremaining effects are positive. In absolute value, the month of August imposes the largest effect.Seasonality is further tested by checking for equality of all mean returns across the months of the year;a rejection of the null hypothesis of equality will indicate strong seasonality. In all three cases results fromthe F-test (Table 2) rejects the null of mean returns equality across the months of the year. While thisindicates a strong seasonality in returns in all three markets, results show that January is not the onlymonth responsible for it, especially in the case of Germany. Figure 1 presents the mean returns for eachmonth of the year for all three countries. The figure basically summarizes our findings.

Results provide significant evidence of the January effect in the UK and the US stock market21 andsince the index applied is value-weighted, the results may not be attributed as a small firm phenomena.Our results confirm the claim by Seyhun (1993) that any potential explanation for the January effectought to include all firms, not just the small firms. Furthermore, before 1914 capital gains or loss was nottreated as a part of net income in these countries. In other words, before 1914 capital gains in the stockmarket could not be taxed and capital loss could not be deducted from taxation. Based on the taxstructure before 1914, the tax-loss selling hypothesis does not receive much merit in explaining the reasonsbehind the January effect during pre-WWI period. As indicated by Pettengill (1986) non-tax factors maybe responsible for the January effect in the pre-WWI period.

So what do results presented in this paper imply? First, seasonal anomalies in the stock markets are nota phenomenon of recent times but also of earlier periods, as early as last century and it is not just


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Figure 1. Stock returns by month of the year.

restricted to the US market. Second, since empirical tests in this paper employ value-weighted stock indexin both periods, a significant January effect may not be attributed as a small firm phenomenon. Thisparticular result confirms Seyhun (1993) claim that studies seasonal anomalies in stock returns shouldinclude firms of all size. Third, the results fail to find support for the most popular reason provided forthe January effect, that is, the tax-loss selling hypothesis. For the tax-loss selling hypothesis to be a validreason, capital gains in the stock markets need to be taxed as a part of income both on a personal andcorporate level, and capital loss needs to be deducted from income in order to provide tax savings. Lackof tax treatment of capital gains/loss in the stock market of these countries before 1914 fail to providemerit to the tax-loss selling hypothesis. Results presented in this paper clearly indicate that the researchfor reasons behind seasonal anomalies such as the January effect has to go beyond the small firm effectand the tax-loss selling hypothesis. Of course the real reason(s) behind these anomalies in these periodscould be due to any of the other factor(s) cited in the earlier part of the paper. In other words, the mysterybehind the January effect is still a mystery and it calls for more empirical and theoretical research.

6. CONCLUSION

This paper investigates using a non-linear GARCH model seasonal anomalies in the German, UK and USstock markets during pre-WWI period. Application of monthly returns data from pre-WWI period andthe GARCH model makes this paper unique in the field of stock market anomalies. The anomaliesinvestigated are the month of the year effect, and the January effect on the mean of stock returns. Allempirical work is conducted with a non-linear (asymmetric) form of the MA(1)-GARCH(1, 1)-t models,the MA(1)-GARCH(1, 1)-GJR model. The asymmetric GARCH-GJR model allows positive and negativeinnovations to returns to have different impact on conditional variance. In other words, an asymmetricGARCH model takes into consideration the so-called ‘leverage effect’ in stock returns.

Results obtained indicate significant presence of the January effect in the UK and the US returns. Othermonths of the year are also found to be significant affecting the three returns. This result contradicts thenotion that seasonal anomalies in stock markets are features of the markets of today only. Use of thevalue-weighted stock index in all cases implies that results may not be attributed to small firm phenomenaand lack of capital gains/loss tax treatment by these countries before 1914 may not provide much meritto the tax-loss selling hypothesis of the January effect. Our results clearly show the presence of month ofthe year effect and the January effect irrelevant of the tax structure in operation. These results clearlyindicate the need for more research in the field of seasonal anomalies in the stock markets of differentcountries and during different time periods.



ACKNOWLEDGEMENTS

The author thanks the seminar participants at the Department of Accountancy and Finance of theUniversity of Essex, UK, the University of Glasgow, UK, the University of Stirling, UK and LeedsBusiness School, UK. All remaining errors and omissions are the authors responsibility alone.

NOTES

1. See Agrawal and Tandon (1994) and Mills and Coutts (1995) for excellent surveys of the papers investigating stock marketanomalies.

2. See Mills and Coutts (1995) for citations.3. See Seyhun (1993) for an in depth analysis of the reasons behind the stock returns seasonality.4. Schwert (1990) investigates seasonality in the US stock returns between 1802 and 1987. He fails to find evidence of any

significant seasonal anomalies. Similarly, Schultz (1985) using small firm returns from the US during 1900–1929 fails to find asignificant January effect before 1918.

5. Berges et al. (1984) find evidence of the January effect in the Canadian stock returns before the enactment of the Canadiancapital gain tax in 1952.

6. Tests were also conducted by means of the usual linear regressions. Results are not provided in order to save space but areavailable on request.

7. Connolly (1989) and de Jong et al. (1992) apply the GARCH model in their investigations of the day of the week effect in theUS stock market and the Dutch stock market, respectively. Similarly, Clare et al. (1995) apply the GARCH-M model in theirstudy of the UK stock market seasonality.

8. Engle and Ng (1993), Glosten et al. (1993) and Hentschel (1995) provide evidence of an asymmetric effect in stock returnsvolatility.

9. See Banz (1981), Gultekin and Gultekin (1983), Reinganum (1983), Roll (1983), Tinic and West (1984, 1986), Hillion and Sirri(1987), Ritter (1988) and Seyhun (1988, 1993).

10. This analysis is based on the assumption of a January to December tax year as is the case in the US. If the tax year is differentfrom January to December, as in the case of the UK, then the January effect based on tax-loss selling hypothesis may not exist.

11. Seyhun (1993) fails to find evidence for omitted risk factor as a reason for the January effect in the US market between 1926and 1991. Dahlquist and Sellin (1994) also fail to find support for the omitted risk factor as a reason for seasonality in theSwedish stock market. Seyhun (1993) further asserts that the potential explanation for the January effect is more likely to beassociated with the various forms of the price pressure hypothesis, which are inconsistent with market efficiency.

12. NBER obtained the UK stock index from Royal Economic Society, Memo c47, June 1934 and the German index fromViertelyahrshefte Zur Konyunkturforschung, Sonderheft 36. Historical statistics of the US 1789–1945 obtained the US datafrom Cowles and Associates (1939).

13. As indicated by Baillie and Myers (1991, p. 110), with most asset prices it is reasonable to base inference on the change in thelogarithm of price. The augmented Dickey–Fuller test is applied to check for the stochastic structure of the data. All threereturns series are able to reject the null hypothesis of a unit root; in other words, all series are stationary. These results are notprovided in order to space but are available on request. Stationarity after first difference is uninformative about highermoments, which usually results in higher kurtosis.

14. This feature of the GARCH model is desirable because Hodrick and Srivastava (1984) provide evidence that the forecast erroris heteroscedastic.

15. Bera and Higgins (1993) and Bollerslev et al. (1994) provide an excellent analysis of ARCH, GARCH, and related models.Bollerslev et al. (1992) provide a survey of the application of the GARCH and related models in finance.

16. As advocated by Bollerslev (1987) if the series possesses substantial kurtosis, it can be more appropriate to use a conditionalStudent-t density than a conditional normal distribution. The GARCH-t model is able to provide a more robust interpretationof the t-statistics. And, as indicated by Fama (1965), stock returns tend to exhibit non-normal unconditional samplingdistribution in the form of excess kurtosis and skewness. All the return series under study in this paper are found to beleptokurtic and skewed (Table 1).

17. There is more than one GARCH model available that is able to capture the leverage effect. Pagan and Schwert (1990), Engleand Ng (1993), Hentschel (1995) and Fornari and Mele (1996) provide analyses and comparison of symmetric and asymmetricGARCH models. According to Engle and Ng (1993) the Glosten et al. model is the best at parsimoniously capturing thisasymmetric effect.

18. In a GARCH(p, q) model different combinations of p and q may be applied but as indicated by Bollerslev et al. (1992, p. 10)p=q=1 is sufficient for most financial series. Bollerslev (1988) provides a method of selecting the size of p and q in a GARCHmodel. Tests in this paper were also conducted with different combinations of p and q with p=q=2 being the maximum laglength. Results based on log-likelihood function and likelihood ratio test indicates the best combination is p=q=1 for all tests.

19. As stated above the ARCH process could be caused by several different factors. To study the reason(s) behind the ARCHprocess in the UK market is beyond the theme of this paper but is an incentive for further research.

20. The significant MA term may also be due to different news observed by different investors or the same news being interpreteddifferently by different investors. This could create a negative serial correlation, as a result of a process of price adjustmentwhere the price bounces forth and back between centres with different information.

21. Results from the linear regression are not identical but similar. Significant January effect is also found in the UK and the USstock returns. These results are not provided in order to conserve space but are available on request.


T. CHOUDHRY10

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month of the year effect and january effect in pre-wwi stock returns: evidence from a non-linear...

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