Tests for Stochastic Seasonality Applied to Daily ...lc436/papers/test_sto_season.pdf · TESTS FOR STOCHASTIC SEASONALITY APPLIED TO DAILY FINANCIAL TIME SERIES* by I. C. ANDRADE

TESTS FOR STOCHASTIC SEASONALITY APPLIED TODAILY FINANCIAL TIME SERIES*

byI. C. ANDRADE

University of SouthamptonA. D. CLARE

ISMA Centre, University of ReadingR. J. O'BRIEN

andS. H. THOMAS{

University of Southampton

We develop tests for seasonal unit roots for daily data by ex-tending the methodology of Hylleberg et al. (`Seasonal Integrationand Cointegration', Journal of Econometrics, Vol. 44 (1990), No.1^2, pp. 215^238) and apply our tests to UK and US daily stockmarket indices. We also investigate a suggestion by Franses andRomijn (`Periodic Integration in Quarterly MacroeconomicVariables', International Journal of Forecasting, Vol. 9 (1993), No. 4,pp. 467^476) and Franses (À Multivariate Approach to ModellingUnivariate Seasonal Time Series', Journal of Econometrics, Vol. 63(1994), No. 1, pp. 133^151) and create a price series for each day ofthe week and test for cointegration amongst these series. Our MonteCarlo experiments indicate that the Hylleberg et al. procedure is robustto autoregressive conditional heteroscedasticity type errors, while theFranses and Romijn procedure is less so. Finally, we employ Harvey's(Time Series Models, Hemel Hempstead, Harvester Wheatsheaf,1993) basic structural model to test for the presence of stationarystochastic seasonality. Our results suggest that we can reject theexistence of seasonal unit roots at the daily frequency in both of thesemarkets; however, we do ¢nd evidence of stationary stochasticseasonality.

" Introduction

Researchers in ¢nance have documented the existence of deterministicseasonal patterns in daily stock and bond market returns. In particular, ithas been found that equity markets tend to experience abnormally highreturns on Fridays and abnormally low returns on Mondays (see Cross,

ß Blackwell Publishers Ltd and The Victoria University of Manchester, 1999.Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK, and 350 Main Street, Malden, MA 02148, USA.

39

The Manchester School Vol 67 No. 1 January 19991463^6786 39^59

*Manuscript received 24.2.97; ¢nal version received 16.7.97.{We would like to thank Philip Franses, two anonymous referees and participants at the

International Symposium on Forecasting in Toronto, the European EconomicAssociation Conference in Prague and Econometrics Workshops in Southampton andLisbon for useful comments on earlier drafts of this paper. Remaining errors are ourown.

1973; French, 1980). This phenomenon is at odds with the weakest formof the e¤cient markets hypothesis which asserts that in an information-ally and operationally e¤cient market asset prices should not bepredictable using information about past returns. A number ofexplanations have been advanced to explain the `weekend e¡ect'. Someresearchers have suggested that it is due to a `closed market e¡ect' (seeAriel, 1990), while others have investigated the role of settlementsprocedures on daily stock prices (see Board and Sutcli¡e, 1988); othershave investigated whether returns are generated by a process which is dueto calendar time, or to trading time. However, we believe that theconcentration on deterministic seasonal patterns in daily asset returnsmay be misleading if stochastic seasonality exists in daily ¢nancial marketdata. According to Beaulieu and Miron (1993), an examination ofseasonal patterns should begin with a study of stochastic seasonality. Tothis end, we extend the work of Hylleberg et al. (1990) (hereafterHEGY), Osborn (1990), Franses (1991a, 1991b) and Beaulieu and Miron(1993) by developing tests for seasonal unit roots for daily ¢nancialmarket data, generating the appropriate critical values by simulation. Wealso explore the logical implication of tests based upon the HEGYmethodology for seasonal unit roots, namely that there exist ¢veindependent random walks within one series (in the case of daily ¢nancialmarket data), and extend the work of Franses (1991b) and Franses andRomijn (1993) who adapt Johansen's cointegration procedure forquarterly data to daily ¢nancial market data.

In anticipation of our results, we test for the presence of long-runand seasonal unit roots in UK and US daily equity indices over the period31 December 1979 to 17 June 1994, using our generated critical values.We ¢nd that we can reject the hypothesis that there are unit roots at allseasonal frequencies in these series, but that we cannot reject thehypothesis that there is a long-run unit root in both markets. Therefore we¢nd that the appropriate ¢lter to induce stationarity in the series is 1ÿ L

in both cases (where L is the lag operator: LiYt � Ytÿi). We con¢rm this

result using the Franses and Romijn procedure, ¢nding that thecointegrating space for the reconstructed UK and US markets hasdimension n � 4, once again rejecting the hypothesis that seasonal unitroots are present and accepting the existence of one long-run unit root ineach equity market. Having rejected the existence of seasonal unit roots inour data, we found that when we employed Harvey's model using aKalman ¢lter the variance of the random seasonal component is largerthan that of the irregular component and highly signi¢cant for both stockindices. The rest of this paper is organized as follows: in Section 2 weoutline our seasonal unit root tests, the cointegration approach to thestudy of seasonality and Harvey's (1993) model; in Section 3 we presentour results; ¢nally, Section 4 concludes.

40 The Manchester School

ß Blackwell Publishers Ltd and The Victoria University of Manchester, 1999.

á Methodology and Testing Procedures

2.1 Extending the HEGY Methodology to Daily Data

A number of authors have investigated the issue of stochasticseasonality. The existence of d0 zero frequency (or long-run) unitroots and dp unit roots at seasonal frequencies in a time series yt withquarterly periodicity can be tested against a stationary alternativeusing tests due to Osborn (1990) and HEGY. Franses (1991a, 1991b)extends HEGY's testing procedure to the cases of bimonthly andmonthly data. The asymptotic theory for the general case of arbitraryseasonal period is given, in addition to further applications, by Smithand Taylor (1998a, 1998b) and Taylor (1998). Since daily data arecommonly used in empirical work in ¢nancial markets we extendHEGY's procedure in order to test for seasonal unit roots in daily¢nancial market data. Financial economists have documented adeterministic seasonal pattern known as the `weekend e¡ect', whereasset returns are signi¢cantly higher on Fridays and signi¢cantlylower on Mondays (see Cross, 1973; French, 1980; Gibbons and Hess,1981). However, according to Beaulieu and Miron (1993), in anyexamination of seasonality we should begin with a search for seasonalunit roots, since an analysis of deterministic seasonality may bemisleading if seasonal unit roots are present. See Franses et al. (1995)for a formal analysis of misleading inference when there are seasonalunit roots.

There is a substantial literature dedicated to testing for seasonalunit roots in quarterly data (see HEGY; Osborn, 1990; Franses andRomijn, 1993; Franses, 1994; Ghysels et al., 1994; amongst manyothers) and, more recently, in monthly data (see Franses, 1991b; Clareet al., 1995). All these tests have non-standard distributions andtherefore in expanding the HEGY procedure to test for seasonal unitroots in daily data we must derive our own, appropriate criticalvalues. HEGY develop a procedure to test for seasonal unit roots inquarterly data, which allows us to test for separate unit roots at thelong-run and seasonal frequencies and for a wide range ofcombinations of deterministic components under the stationaryalternative. The essential principle of this procedure is given byProposition 1 in HEGY (pp. 221^222). We now adapt this procedureto the test for seasonal unit roots in daily ¢nancial market data, wherethe relevant polynomial for our study is 1ÿ L

5. This polynomial hasone real root equal to 1, and two pairs of complex conjugate rootsa1; a2 � 1

4 �p5ÿ 1�p�ÿ2p5ÿ 10�� 0:309� 0:951i, and, replacing

p5

by ÿp5, a3; a4 � ÿ0:809� 0:588i, where i � p�ÿ1�. From (3.3) inHEGY (p. 222), c�L � with p � 5 can be written as

Tests for Stochastic Seasonality 41


c�L � � l1L c1�L � � l2L1a1

1ÿ 1a2

L

� �c2�L �

� l3L1a2

1ÿ 1a1

L

� �c2�L � � l4L

1a3

1ÿ 1a4

L

� �c3�L �

� l5L1a4

1ÿ 1a3

L

� �c3�L � � c��L �c4�L � �1�

where

c1�L � � �1ÿ zL � L 2��1� �1=z�L � L 2�c2�L � � �1ÿ L ��1� �1=z�L � L 2�c3�L � � �1ÿ L ��1ÿ zL � L 2�c4�L � � 1ÿ L 5

and z � �p5ÿ 1�=2 � 0:618, the `golden mean', and ÿz and 1=z are theroots of L 2 ÿ L ÿ 1. Note that a1 and a2 are the roots of 1ÿ zL � L 2, anda3 and a4 are the roots of 1� �1=z�L � L 2. l2 and l3 are complex conjugatesand we now de¢ne m2=2 � R�l2� and m3=2 � I�l2�; similarly we de¢ne m4

and m5 as twice the real and imaginary parts of the conjugate pair l4; l5.This allows us to write (1) as

c�L � � l1L c1�L � ÿ L ��m3g2 � m2g1� � m2L �L c2�L �ÿ L ��m5g4 � m4g3� � m5L �L c3�L � � c��L �c4�L � �2�

where gi for i � 1; 2; 3; 4, is de¢ned from 1=a1 � a2 � g1 � ig2 and 1=a3 �a4 � g3 � ig4. Thus g1 � R�a1�, g2 � ÿI�a1�, g3 � R�a3� and g4 � ÿI�a3�.Finally, we de¢ne p1 � ÿl1, p2 � m3g2 � m2g1, p3 � m2, p4 � m5g4 � m4g3 andp5 � m5, which in turn allows us to rewrite (2) as

c�L � � ÿp1L c1�L � � �p2 � p3L �L c2�L �� p4 � p5L �L c3�L � � c��L �c4�L � �3�

Expression (3) is comparable to expression (3.4) in HEGY (p. 223) forthe quarterly case. We can now derive the testing procedure for the case ofdaily data. We assume that the data-generating process (DGP) is anautoregressive process given by

c�L �yt � et �4�where et is n.i.d.(0,1). Substituting (4) into (3),

c��L �y4t � p1y1;tÿ1 � p2y2;tÿ1 � p3y2;tÿ2 � p4y3;tÿ1 � p5y3;tÿ2 � et �5�where c��L � is some polynomial function of L with all the roots outsidethe unit circle, and

yit � ci�L �yt i � 1; 4� ÿci�L �yt i � 2; 3



with ci given as in (1). Equation (5) can be estimated using ordinary leastsquares. The augmented version of the test can be obtained in the usualway through the inclusion of additional lags of y4t � c4�L �yt � �1ÿ L 5�yt

on the right-hand side of (5).From Proposition 1 (HEGY, pp. 221^222), testing for a root at a

point yk, where yk is any of the ¢ve solutions of 1ÿ L 5 � 0, is equivalent totesting for lk � 0 in (1), or equivalent to testing for pi � 0 in (5). If wecannot reject the null that p1 � 0 using a one-sided t-type test (see below),then we cannot reject that there is a long-run unit root in the data, andthe ¢lter 1ÿ L is appropriate to induce stationarity at the zero frequencyof the data. For the complex roots corresponding to the seasonal unitroots, the corresponding lk, k � 2 or k � 3, will be zero only if bothcorresponding pi for i � 2; 3, or i � 4; 5, are equal to zero, which suggeststhe use of a joint test for each pair of ps. The corresponding appropriate¢lters are given by c3�L � and c2�L � as de¢ned in (1). Finally, if using thesetests we cannot reject that pi � 0, for all i, then the ¢lter 1ÿ L 5 is theappropriate choice.

The alternative for all these tests is stationarity. For c�1� � 0, thealternative is that c�1� > 0 implying that p1 < 0, and therefore we proposethe use of a one-sided t-type test. For the complex roots, the alternativeto jc�yk�j � 0 is jc�yk�j > 0 for k � 2 and 3 or k � 4 and 5, which suggeststhe use of an F-type test for the joint null. Under the alternativestationarity hypothesis in (5), if pi 6� 0 for i � 2; 3; 4; 5, then this meansthat no seasonal unit roots are present in the data and that the use ofseasonal dummies may be appropriate to model potential stationaryseasonal patterns. p1 6� 0 means that no long-run unit root is present in thedata. Therefore, if pi 6� 0 for all i, the data are stationary.1

The critical values reported in Tables 1 and 2 are generated by MonteCarlo simulation using GAUSS with M � 100;000 replications. They are

Table "Critical Values of the t Ratios for the Null Hypothesis

that p1 is Zero

Auxiliaryregressors 0.01 0.025 0.05 0.10

p1 � 0I, SD, T ÿ3.96 ÿ3.66 ÿ3.40 ÿ3.12I, SD ÿ3.42 ÿ3.11 ÿ2.85 ÿ2.55Notes: I is the constant; SD are seasonal dummies; T is the trend;

N � 800 is the sample size.

1On the other hand, if we accept p1 � 0 and test pi � 0, i � 2; . . . ; 5, we are testing forseasonal unit roots in the ¢rst di¡erence of the data as well as the levels, e.g. in thereturns as well as in the indices.



subject to Monte Carlo sampling error and further approximate theasymptotic distribution by using a sample of N � 800. The DGP used isgiven by (4), and under the alternative we included either a constant andseasonals or a constant, trend and seasonals. Table 1 gives the criticalvalues for the t-type test for the null that H0: p1 � 0;2 Table 2 gives thecritical values for the F-type tests for the pairs of conjugate complex unitroots, i.e. for H0: pi � 0 for i � 2; 3, and H0: pi � 0 for i � 4; 5, and for thenull hypothesis of no unit root seasonality, i.e. H0: pi � 0 for i � 2; 3; 4; 5(or H0: \5

i�2H0i). Kendall and Stuart (1961, pp. 371^378) give a methodfor calculating distribution-free con¢dence intervals; the 95 per centintervals here range from (at worst) �0:03 for Table 1 through �0:04 forthe 0.95 column of Table 2 to (8.74, 8.92) around the 8.85 entry in Table 2.With respect to the sample size approximation, previous studies havefound N � 400 or N � 800 su¤cient: in fact, independent simulations withN � 400 produce results almost invariably lying within the 95 per centcon¢dence interval for N � 800, which suggests that the Monte Carlosampling variability dominates.

These simulations are conducted with n.i.d. disturbances; however, itis well documented that high frequency ¢nancial data exhibit conditionalheteroscedasticity. For example, Franses and Paap (1995) ¢nd periodically

Table áCritical Values for the Null Hypothesis that Sets of p

Parameters are Zero


p2 � p3 � 0I, SD, T 5.58 6.57 7.52 8.69I, SD 5.61 6.64 7.64 8.85

p4 � p5 � 0I, SD, T 5.59 6.61 7.62 8.82I, SD 5.62 6.64 7.64 8.85

p2 � p3 � p4 � p5 � 0I, SD, T 4.84 5.51 6.13 6.87I, SD 4.87 5.54 6.16 6.93

Notes: I is the constant; SD are seasonal dummies; T is the trend;N � 800 is the sample size.

2As HEGY discuss, a two-step procedure can be used, where the ¢rst step is given by a t-typedouble-sided test for p3 � 0 �p5 � 0� and, if this hypothesis cannot be rejected, thesecond step is given by a t-type one-sided test for p2 �p4� against the alternative thatp2 < 0 �p4 < 0�. The drawback of this procedure is that `potentially [it] could lack powerif the ¢rst-step assumption is not warranted' (HEGY, p. 224). In Andrade et al. (1996)we give the critical values for these t-type tests in Tables 2^6.



integrated generalized autoregressive conditional heteroscedasticity(PAR-PIGARCH) models adequate for similar data. It is thus of interestto know how misleading these critical values might be if applied to datawith such errors. Repeating the simulations for IGARCH(1,1) errors withIGARCH parameters a � 0:85 and b � 0:15 gives the sizes and alternativecritical values listed in Appendix A for the I, SD and I, SD, T cases. The95 per cent critical values increase in absolute size, by up to 41 per cent,and the 99 per cent critical values increase by up to 116 per cent. The typeI errors corresponding to the standard critical values were similarlya¡ected. At worst, a nominal 5 per cent test can have size 13.7 per cent,and a nominal 1 per cent test a size of 7.1 per cent. How serious adistortion one considers this to be is a matter of judgement.

2.2 The Franses and Romijn Procedure

An alternative to the HEGY approach uses the fact that under the nullthere exist ¢ve independent random walks for daily data. For quarterlydata, Franses (1991b), Franses and Romijn (1993) and Franses (1994)de¢ne a vector of quarters (VQ) and test whether the four series arecointegrated. For ¢nancial daily data, we can de¢ne a days vector (DV) asxT � �x1T x2T x3T x4T x5T �0, where the prime denotes transpose, and x1T

corresponds to the series of observations of stock prices on Mondays, x2T

to Tuesdays, x3T to Wednesdays, x4T to Thursdays and x5T to Fridays. Theunrestricted vector autoregression (VAR) model used in the Johansenprocedure (see Johansen and Juselius, 1990, for example) can be writtenfor DV as

DxT � G1DxTÿ1 � . . .� Gjÿ1DxTÿj�1 �PxTÿj � eT �6�

The rank of P � n gives the dimension of the cointegrating space and canbe tested using Johansen's maximal eigenvalue and trace test statistics. Ifthere are long-run and seasonal unit roots in yt and 1ÿ L 5 is theappropriate ¢lter to transform the data, then the series of days are notcointegrated and n � 0, i.e. the ¢ve random walks are independent.However, if all the series are stationary, then n � 5. If 0 < n < 5, then theseries are cointegrated and P can be factorized as P � ab0. The columns ofb give the coe¤cients of the cointegrating relationships amongst the series,i.e. the linear combinations b0xT which are stationary.

An important case is when n � 4 and we can write that b0i � 0, wherei is a 5� 1 vector of ones and 0 is the null vector. We can therefore re-standardize (change the base of the cointegration space) the cointegratingvectors in terms of one variable, say Monday, x1T . We can partition b0 asb0 � �b j B�, where b0 is 4� 5, b is a column vector and B is a non-singularsquare matrix of order 4; then, re-standardizing Bÿ1b0 � �Bÿ1b j I�. If



b0i � 0, Bÿ1b0i � 0) Bÿ1b � ÿi, which allows us to write b0 asb0 � �ÿi j I�:

b0 �ÿ1 1 0 0 0ÿ1 0 1 0 0ÿ1 0 0 1 0ÿ1 0 0 0 1

26643775:

An alternative would be to formulate

b0 �ÿ1 1 0 0 00 ÿ1 1 0 00 0 ÿ1 1 00 0 0 ÿ1 1

26643775

That is, the stationary relationships are given by x5T ÿ x4T , x4T ÿ x3T ,x3T ÿ x2T and x2T ÿ x1T . The appropriate ¢lter to stationarize yt is 1ÿ L .There is only a long-run unit root in the data and no seasonal unit roots.The same sort of derivation of relevant restrictions on the coe¤cients of bcan be obtained for the cases when n � 1; 2; 3, where one may also have aperiodically varying di¡erence ¢lter. For this testing procedure, theavailable critical values from Johansen are still appropriate. Weinvestigate the sensitivity of these tests to conditional heteroscedasticityand report these results in Appendix B. We ¢nd that the critical valuesappropriate for this particular IGARCH process are between 34 and 61per cent larger than the corresponding normal values. They are, of course,only indicative of sensitivity to other values of the IGARCH parameters.

2.3 Stationary Stochastic Seasonality

We can test for the existence of stationary stochastic seasonality byemploying the `basic structural model' due to Harvey (1993, Section 5.6)which, he suggests, can be ¢tted with stationary seasonality. This modelcan be adapted for daily data and augmented by ¢xed seasonal e¡ects. Wecan de¢ne the augmented basic structural model as

yt � m� dt � gt � et �7�which decomposes the observations (on di¡erenced natural logarithms inour case) yt into an overall mean m, a ¢xed seasonal e¡ect dt, with dt � dtÿ5and

P4i�0 dtÿi � 0, and a random seasonal component gt, withP4

i�0 gtÿi � ut, ut i.i.d.�0; s2u�, independent of the irregular component et,

which is i.i.d.�0; s2e �. This model (7) can be written in state space form as

follows:

yt � Ztat � et t � 1; 2; . . . ; n

at � T atÿ1 � Zt

�8�



where a0t � �m d1 d2 d3 d4 gt gtÿ1 gtÿ2 gtÿ3� and

T � I 5 OO T�

� �with

T� �ÿ1 ÿ1 ÿ1 ÿ11 0 0 00 1 0 00 0 1 0

26643775

Ip is a unit matrix of order p, O is a null matrix of appropriate dimension,Z0t � �0 0 0 0 0 ut 0 0 0 0�, and Z, i.e. �Z1 Z2 . . . Zn�0, is a repetition of its ¢rst¢ve rows, which are

i; ÿi0

I 4

� �; i;O

� �the null matrix being 5 by 3 and the is columns of 1s of appropriatelengths. The prime denotes transpose. This model has a `random walk'seasonal component as we will now show: if we de¢ne G0t � �gt gtÿ1 gtÿ2 gtÿ3�as a sub-vector of at, from (8) we can write

Gt � T�Gtÿ1 � �ut 0 0 0�0 �9�As gt � ÿgtÿ1 ÿ gtÿ2 ÿ gtÿ3 ÿ gtÿ4 � ut, the ¢ve daily deviations gt, viewedas separate series, are cointegrated with the dimension of the cointegratingspace equal to one, and thus we have four seasonal unit roots. If, instead,we want to consider a stationary random seasonal component in themodel, we introduce a damping factor r, with jrj < 1 (Harvey, 1993,Section 6.5), replacing T� by rT� in (9) and therefore in model (8).Furthermore, either the ¢xed �dt� or the random �gt� seasonal componentscan be written in trigonometric form. For the random component, if wede¢ne lj � 2pj=5, cj � cos�lj� and sj � sin�lj�, then

T� �c1 0 s1 00 c2 0 s2ÿs1 0 c1 00 ÿs2 0 c2

26643775

replacing the original T� matrix in model (8). The last four columns ofthe Z matrix are now all columns of 1s. Again, this version is renderedstationary by the inclusion of a damping factor r. The model thus containsa ¢xed seasonal term dt plus a damped random component, whichcontributes a constant proportion of the variation in yt.



â Results

We now employ the methodologies outlined in Section 2 using 3775 dailyobservations on the UK's Financial Times All Shares Stock Index and onthe USA's S&P 500 Index: from Monday, 31 December 1979, to Friday,17 June 1994. We used the data as provided by Datastream. For publicholidays we were obliged to use the previous trading day's value. We beginby examining the ¢xed seasonal e¡ects in the data. One di¤culty is thenumber of outliers. Removing even considerable numbers of outliers stillleaves the data very erratic, and ad hoc removal becomes di¤cult tojustify. In the results presented below (where necessary) we omit the tenobservations between 19 October 1987 and 30 October 1987 to exclude`Black Monday' and its immediate aftermath.3

Table 3 shows the daily e¡ects in the natural logarithm of the UKindex, detrended by taking ¢rst di¡erences. The individual daily e¡ects areall signi¢cant, and small. One can extract the slight overall drift in the dataas a mean of 0.000578 with signi¢cant deviations for Monday (negative)and, marginally, for Wednesday (positive). The White (1980) hetero-scedastic robust standard errors are provided for the deviations, but theadjustment makes very little di¡erence. The èrrors' contain serialcorrelation and heteroscedasticity, and exhibit non-normality. Thesee¡ects can be reduced, but not eliminated, by deleting outliers. The

Table âDaily Means, Deleting Ten Observations

Mean� 103 jtj Deviation� 103 jRobust tja

UKMonday ÿ0.574 1.7 (Mean) 0.578 4.2Tuesday 0.724 2.4 Monday ÿ1.152 3.8Wednesday 1.075 3.6 Tuesday 0.146 0.5Thursday 0.664 2.2 Wednesday 0.496 1.9Friday 1.003 3.3 Friday 0.424 1.6

USAMonday 0.097 0.3 (Mean) 0.426 1.1Tuesday 0.660 1.9 Monday ÿ0.328 1.0Wednesday 0.973 2.9 Tuesday 0.235 0.8Thursday 0.221 0.6 Wednesday 0.547 1.7Friday 0.176 0.5 Friday ÿ0.249 0.7

Notes: For the UK, SD of dependent variable � 0.008220, `regression' standard error 0.008199,R2 � 0:00503, F�4; 3636� for deviations � 4.60 ( p value 0.001). For the USA, SD of dependentvariable � 0.009169, `regression' standard error 0.009162, R2 � 0:00136, F�4; 3601� fordeviations � 1.23 ( p value 0.298).a Robust t calculated using White's (1980) standard errors.

3The choice of ten observations to omit was made as a matter of judgement after inspectingthe data and the results for Tables 3 and 4.



signi¢cance of the deviations is robust to this process. In this static `model'the main problem with such errors is a loss of e¤ciency. Here, given thelarge number of observations, the daily deviations are detectable, andjointly signi¢cant at 1 per cent. We re-estimated the ¢xed seasonal e¡ectswith ARCH(5) errors, but this had almost no e¡ect on the associatedt values (see Table 4). The ARCH coe¤cients, although signi¢cant, aresmall, ranging from 0.0622 to 0.1150. The pattern is similar in the USA(see Table 3) with returns rising from Monday to Wednesday and thenfalling, except for Friday, which continues the downward trend in theUSA but not in the UK. None of the deviations is signi¢cant at the 5 percent level, although Wednesday is at the 10 per cent level. However, if one¢ts an ARCH(5) error process (see Table 4), there are some changes inthe coe¤cientsöWednesday becomes signi¢cant at 5 per cent, Mondaymarginal at 10 per cent and the deviations are jointly signi¢cant at 10 percent using a likelihood ratio test. There is no evidence of serial correlationin the ARCH residuals. Thus in our sample there is reasonable evidenceof daily e¡ects in the UK stock market, and weaker evidence for the USmarket.

Table ãModel with ARCH Errors

UK USA

Coe¤cient jRobust tja Coe¤cient jRobust tja

Mean 0.00049 4.0 0.00045 3.1Monday ÿ0.00108 4.1 ÿ0.00047 1.6Tuesday 0.00027 1.1 0.00026 0.9Wednesday 0.00039 1.6 0.00061 2.2Friday 0.00045 1.8 ÿ0.00038 1.2ytÿ1 0.139 7.7 n/a ^ytÿ10 0.02986 1.8 n/a ^g0 0.00004 14.7 0.00005 11.4g1 0.09734 3.0 0.05312 2.5g2 0.1037 4.2 0.04801 2.0g3 0.07401 3.0 0.07773 3.4g4 0.1214 4.7 0.1130 3.0g5 0.07007 3.0 0.09940 4.2Log-likelihood 12469.93 ^ 11911.54 ^

BPb 4.514 4.339[0.4780] [0.5017]

BGc 0.972 0.9673[0.4335] [0.4365]

ARCH(5)d 0.4446 0.1413[0.8174] [0.9826]

Notes: a Robust t estimated using analytic formula; the robustness is to non-normality in the quasi-maximum likelihood estimates. See Gourieroux et al. (1984).b BP, Box^Pierce test statistic for white noise residuals calculated using ¢ve lags.c BG, Breusch^Godfrey test statistic �F� for white noise residuals using ¢ve lags.d ARCH(5), F version of the Engle (1982) test.



3.1 Testing for Seasonal Unit Roots

In Table 5 we present the results4 for the HEGY-type test procedure(regression (5)) using two combinations of deterministic componentsöaconstant with seasonal dummies, and a constant with seasonal dummiesand a trend. No augmentation using additional lags was required to whitenthe residuals. However, there are problems of non-normality, whichshould not be surprising in ¢nancial data (see above). We included dummyvariables for certain observations, in particular the period following the1987 crash, which were su¤cient to allow us to accept the hypothesis thatthe residuals were normally distributed. Extensive experiments were madeexcluding up to 50 outliers, detected as large residuals (or, for ARCH,

Table äâææä Daily Observations from â" December "ñæñ to "æ June "ññã on

the UK and US Stock Market Indices

UK USA

Coe¤-cient jtj

Coe¤-cient jtj

Coe¤-cient jtj

Coe¤-cient jtj

I 0.016 2.4 0.003 1.9 0.029 3.1 0.002 1.2D2 0.001 3.4 0.002 3.4 0.001 2.1 0.001 2.1D3 0.002 4.4 0.002 4.4 0.001 2.7 0.001 2.7D4 0.001 3.0 0.001 3.0 0.001 1.0 0.001 1.0D5 0.002 4.1 0.002 4.1 0.001 1.0 0.001 1.0T 0.000 2.0 ^ ^ 0.000 2.9 ^ ^p1 ÿ0.001 2.4 ÿ0.000 1.7 ÿ0.001 3.0 ÿ0.000 1.0p2 0.295 26.7 0.296 26.8 0.286 26.1 0.287 26.2p3 ÿ0.243 22.0 ÿ0.242 22.0 ÿ0.247 22.6 ÿ0.245 22.4p4 0.83 51.0 0.830 51.0 0.748 47.4 0.748 47.3p5 ÿ0.81 49.7 ÿ0.809 50.0 ÿ0.718 45.5 ÿ0.718 45.4

F testsp2^p3 1199.8 1198.4 1203.8 1201.4p4^p5 1522.6 1521.0 1285.3 1282.5p2^p5 5305.1 5300.1 3735.0 3727.0

BPa 0.015 0.025 7.185 6.371[0.999] [0.999] [0.207] [0.272]

BGb 0.406 0.356 1.750 1.571[0.845] [0.878] [0.120] [0.165]

Notes: The auxiliary regression includes I, SD and T, or I and SD. No augmentation with additional lagswas required to whiten the residuals.a BP, Box^Pierce test statistic for white noise residuals calculated using ¢ve lags and approximatelydistributed as w2�5�.b BG, Breusch^Godfrey test statistic for white noise residuals using ¢ve lags and distributed asF�5; 3754� or as F�5; 3755�.

4In Andrade et al. (1996) we report the results of estimating (5) with all ¢ve combinationsof deterministic components. The conclusions of the tests for seasonal unit roots areunchanged.



standardized residuals) in the relationship being ¢tted. The conclusionsfrom the test did not change, and for brevity we report the results for thefull data set. We can see from Table 5 and from the appropriate criticalvalues given in Tables 1 and 2 that we cannot reject the hypothesis of along-run unit root in either the UK or the US stock markets. However, wedo reject the hypothesis that unit root seasonality at all other frequenciesis a feature of these markets, at the 1 per cent and 5 per cent levels ofsigni¢cance. The main implication of these results is that the ¢lter 1ÿ L isadequate to achieve stationarity in these stock market indices.

3.2 The Franses and Romijn Procedure

We now investigate the implication from the HEGY procedure that thereexist ¢ve seasonal unit roots in daily ¢nancial time series and test forcointegration amongst day-of-the-week series. To this end we divide the3775 daily observations on the two stock market indices into ¢ve serieseach formed by the observations corresponding to a day of the week, xiT ,i � 1 (Monday), 2 (Tuesday), 3 (Wednesday), 4 (Thursday) and 5 (Friday).We thus de¢ne a DV (days vector) and use Johansen's test procedure toestimate (6) and test for cointegration amongst these series. We paidparticular attention to determining the lag order of the VAR so that therewas no residual serial correlation, and concluded that a VAR(3) for theUK market and a VAR(2) for the US market were appropriate.5 Usingboth of Johansen's test statistics, we ¢nd that the dimension of thecointegrating space in both cases is n � 4 (see Table 6) and therefore wecan use the method derived in Section 2 to re-standardize b0. (Thisinference is una¡ected if one uses the critical values for the particularIGARCH process in Appendix B.) We obtain b0 � �ÿi j I � to four ¢guresaccuracy for the UK data and three ¢gures for the US data, and sox2T ÿ x1T , x3T ÿ x1T , x4T ÿ x1T and x5T ÿ x1T are stationary. Thesestationary relationships can be written as x5T ÿ x4T , x4T ÿ x3T , x3T ÿ x2T

and x2T ÿ x1T which shows immediately that the appropriate ¢lter toinduce stationarity in the original series yt is 1ÿ L and that no seasonalunit roots are present. These results reinforce the ones obtained using ouralternative methodology above. Franses (1994) shows that one can use thew2 test to test for the �1;ÿ1� cointegrating relations. PcFiml providesw2�4� p values of 0.59 (UK) and 0.12 (US) and acceptance of the null.

3.3 Stationary Seasonality

Having rejected the existence of non-stationary seasonal unit roots in our

5Detailed results are reported in Andrade et al. (1996). The residuals are non-normal, butthe inclusion of event-speci¢c dummies did not change our conclusions.



data we now test for the presence of stationary stochastic seasonality usingHarvey's (1993) basic structural model which we adapt for daily data andaugment with ¢xed seasonal e¡ects (see expressions (7) and (8) above) andestimate using the Kalman ¢lter.

Table åFranses and Romijn's Procedure

UK USA

H0�n�

Eigen-valuel̂i

lmax

(1)ltrace(2)

Eigen-valuel̂i

lmax

(1)ltrace(2)

Osterwald^Lenum95% critical value(1) (2)

n � 4 0.003 1.93 1.93 0.001 0.77 0.77 8.18 8.18n � 3 0.175 144.73 146.66 0.268 235.32 236.09 14.90 17.95n � 2 0.246 212.34 359.00 0.313 282.38 518.47 21.07 31.52n � 1 0.256 221.90 580.90 0.385 366.17 884.65 27.14 48.28n � 0 0.289 257.01 837.90 0.441 437.53 1322.18 33.32 70.60

Notes: lmax and ltrace are Johansen's test statistics ÿT ln�1ÿ l̂i� and ÿTP

i ln�1ÿ l̂i�, respectively. 3775daily observations become 755 observations for each of the ¢ve series corresponding to each dayof the week. UK estimation based on a VAR(3) with constant and US estimation based on aVAR(2) with constant.

Table æEstimates of the Stationary Random Seasonal Model

UK USA

Coefficient� 103 jt ratioj Coefficient� 103 jt ratiojm 0.5781 14.8 0.4279 3.6d1 0.1002 2.1 ÿ0.1856 0.9d2 0.4296 5.8 ÿ0.2615 2.9d3 ÿ1.1720 14.8 ÿ0.3321 3.7d4 0.1390 2.8 0.2288 0.5se 3.5930 (0.2038)a 4.0740 (0.2176)a

su 7.2660 (0.1142)a 8.1940 (0.09877)a

r ÿ0.1657 169.7 ÿ0.05279 53.9Log-likelihood 12356.69 ^ 11808.50 ^

BPb 3.477 4.944[0.6269] [0.4228]

BGc 0.6867 0.9687[0.6335] [0.4354]

ARCH(5)d 191.10 116.6[0.000] [0.000]

Notes: a Standard errors (SEs). All SEs are calculated using numerical second derivatives postestimation.b BP, Box^Pierce test statistic for white noise residuals calculated using ¢ve lags.c BG, Breusch^Godfrey test statistic (F� for white noise residuals using ¢ve lags.d ARCH(5), F version of the Engle (1982) test.p values in square brackets.



Preliminary investigation of the stationary random seasonal modelsis encouraging, in that the seasonal deviations are signi¢cant: see Tables 7and 9 for both UK and US data. The variance of the random seasonalcomponent is larger than that of the irregular component and highlysigni¢cant. The damping parameter on the random seasonal component issmall but signi¢cant, ranging from ÿ0:25 to ÿ0:05 depending on the model

Table ðAutocorrelation Function for the Random Seasonal

Component in Table æ

Lag

r 1 2 3 4 5

ÿ0.1 0.097 ÿ0.0 0.0 ÿ0.0 ÿ0.00.1 ÿ0.097 ÿ0.0 ÿ0.0 ÿ0.0 0.00.2 ÿ0.179 ÿ0.001 ÿ0.0 ÿ0.0 0.00.3 ÿ0.239 ÿ0.006 ÿ0.002 ÿ0.001 0.0020.4 ÿ0.278 ÿ0.015 ÿ0.006 ÿ0.003 0.0100.5 ÿ0.300 ÿ0.030 ÿ0.016 ÿ0.009 0.0310.6 ÿ0.310 ÿ0.052 ÿ0.033 ÿ0.024 0.0780.7 ÿ0.309 ÿ0.080 ÿ0.062 ÿ0.052 0.1680.8 ÿ0.298 ÿ0.118 ÿ0.105 ÿ0.098 0.3280.9 ÿ0.278 ÿ0.172 ÿ0.167 ÿ0.164 0.590

Note: The sign asymmetry is given by the ¢rst two rows. The auto-correlation for lag 5, r5, is equal to r5 and rk�5 � r5rk.

Table ñEstimates of the Trigonometric Form of the Stationary Random Seasonal Model

UK USA

Coefficient� 103 jt ratioj Coefficient� 103 jt ratiojm 0.5780 27.5 0.4252 13.3d1 0.08831 8.4 ÿ0.2018 3.8d2 0.4326 5.6 ÿ0.2516 4.8d3 ÿ1.170 17.4 ÿ0.3331 4.8d4 0.1486 8.0 0.2397 4.4se ÿ0.005632 �4:97� 10ÿ5�a 4.070 �0:3095�asu 8.137 (0.008972)a 8.197 (0.1374)a

r ÿ0.2501 203.3 ÿ0.1167 4.9Log-likelihood 12350.96 ^ 11808.28 ^

BPb 15.79 652620[0.0075] [0.23450]

BGc 2.943 1.096[0.0118] [0.3605]

ARCH(5)d 192.1 116.2[0.000] [0.000]

Notes: As for Table 7.



Table "òAutocorrelation Function for the Random Seasonal

Component in Table ñ

Lag

r 1 2 3 4 5

ÿ0.1 0.025 ÿ0.003 0.0 ÿ0.0 ÿ0.00.1 ÿ0.025 ÿ0.003 ÿ0.0 ÿ0.0 0.00.2 ÿ0.050 ÿ0.010 ÿ0.002 ÿ0.0 0.00.3 ÿ0.075 ÿ0.023 ÿ0.007 ÿ0.002 0.0020.4 ÿ0.100 ÿ0.040 ÿ0.016 ÿ0.006 0.0100.5 ÿ0.125 ÿ0.063 ÿ0.031 ÿ0.016 0.0310.6 ÿ0.150 ÿ0.090 ÿ0.0354 ÿ0.032 0.0780.7 ÿ0.175 ÿ0.122 ÿ0.086 ÿ0.060 0.1680.8 ÿ0.200 ÿ0.160 ÿ0.128 ÿ0.102 0.3280.9 ÿ0.225 ÿ0.202 ÿ0.182 ÿ0.164 0.590

Note: As for Table 8.

Table ""Estimates of the Seasonal Autoregression

with ARCH Errors Model

UK

Coe¤cient jt ratiojMean 0.0004881 4.0Monday ÿ0.0010820 4.1Tuesday 0.0002654 1.1Wednesday 0.0003903 1.6Friday 0.0004475 1.8ytÿ1 0.1390 7.7ytÿ10 0.02986 1.8g0 0.00003537 14.7g1 0.09734 3.0g2 0.1037 4.2g3 0.07401 3.0g4 0.1214 4.7g5 0.07007 3.0Log-likelihood 12469.93 ^

BPa 4.514[0.4780]

BGb 0.9720[0.4335]

ARCH(5)c 0.4446[0.8174]

Notes: t ratios calculated from analytic standard errors.a BP, Box^Pierce test statistic for white noise resi-duals calculated using ¢ve lags.b BG, Breusch^Godfrey test statistic �F� for whitenoise residuals using ¢ve lags.c ARCH(5), F version of the Engle (1982) test.p values in square brackets in each case.



and data. The autocorrelation functions for the damped model are givenin Tables 8 and 10 (trigonometric form) for several di¡erent values of r.The use of only two parameters to represent the stationary randomseasonality does impose a particular shape to this function. Given thesmall values of r observed, and the result that the correlation betweendays a week apart is r5 in either formulation, and thus less than 0.002 forour estimates, it may be that the `damped random seasonal component' isaliasing for serial correlation in the residuals. We found signi¢cantstationary seasonal e¡ects, but with a small estimated damping parameterand with innovations that exhibit signi¢cant non-normality and ARCHe¡ects. Furthermore, in terms of the likelihood, the stationary randomseasonal model ¢ts the data less well than a simple ¢xed seasonality modelwith ARCH errors, as described in Section 3.1 above for the US dataand in Table 11 for the UK data.

ã Conclusions

Financial economists frequently use daily data in their research and in sodoing have identi¢ed deterministic seasonal patterns in daily stock andbond market returns. However, since empirical results can be misleadingin the presence of stochastic seasonality which has not been identi¢ed (seeBeaulieu and Miron, 1993), in this paper we developed tests which willallow researchers in ¢nance to check for seasonal unit roots in daily data.We began by extending the HEGY procedure to the case of daily data andcalculated the appropriate critical values. We then outlined the way inwhich the Franses and Romijn procedure can be used to detect seasonalunit roots in daily ¢nancial market data. Our Monte Carlo experiments,reported in Appendices A and B, indicate that the HEGY procedure isrobust to ARCH type errors, while the Franses and Romijn procedure isless so. We suggest that it is important to test for seasonal unit roots bothfor modelling and for forecasting: inappropriate seasonal di¡erencingimposes seasonal unit roots and gives rise to a noticeably di¡erentforecasting model. Using our own tests and the Franses and Romijnprocedure adapted for daily ¢nancial market data, we tested for seasonalunit roots in the daily prices of the UK's Financial Times All Shares Indexand the USA's S&P 500 Index. Using both procedures, we found thatstochastic seasonality was not a feature of these indices over our sample,but that there was one long-run unit root in both markets. We then testedfor the presence of stationary stochastic seasonality using a stationaryrandom seasonal version of Harvey's (1993) basic structural model. Wefound signi¢cant stationary seasonal e¡ects, but with a small estimateddamping parameter and with innovations that exhibit signi¢cant non-normality and ARCH e¡ects. Furthermore, in terms of the likelihood, the



stationary random seasonal model ¢tted the data less well than a simplemodel with ¢xed seasonal e¡ects and ARCH errors. The main statisticalconclusion from our results is that the ¢lter 1ÿ L was su¤cient to inducestationarity in the stock market indices over this period.

Appendix A:

The Effect of Conditionally Heteroscedastic Disturbances

Tables 1 and 2 were obtained using independent and identically distributedstandard normal deviates. Given the nature of the data, it is relevant to enquirehow robust they are to heteroscedasticity, and non-normal kurtosis. Thecalculations were repeated for the I, SD and I, SD, T cases; Tables 1A and 2A givethe critical values for N � 800 for IGARCH(1,1) errors, with a � 0:85 andb � 0:15.

There are considerable di¡erences when comparison is made with Tables 1and 2, as might be expected, given that the GARCH process, although stationary,has in¢nite variance. However, it may be more relevant to enquire how the size ofthe tests is a¡ected if the normal critical values are used with GARCH errors. Thesame simulation yielded the estimates in Tables 1B and 2B.

Thus at worst a nominal 5 per cent test could have size 13.7 per cent, and anominal 1 per cent test a size of 7.1 per cent. We ¢nd this result reasonablyencouraging.

Table "ACritical Values for IGARCH Errors


I, SD, T ÿ5.02 ÿ4.34 ÿ3.88 ÿ3.45I, SD ÿ4.34 ÿ3.68 ÿ3.25 ÿ2.82Notes: 100,000 replications giving an accuracy (distribution-free 95 per

cent con¢dence interval) of at worst �0:06 for Table 1A, �0:09 forthe 0.95 column of Table 2A and (15.80, 16.64) around the 16.18entry in Table 2A. The GARCH process was run for 50 pre-sampletrials.

Table áACritical Values for IGARCH Errors

Auxiliaryregressors H0 0.90 0.95 0.975 0.99

I, SD, T p2 � p3 � 0 6.77 8.68 11.15 16.18I, SD 6.75 8.66 11.01 15.84I, SD, T p4 � p5 � 0 6.77 8.68 11.02 15.64I, SD 6.74 8.62 11.03 15.52I, SD, T p2 � p3 � 6.13 7.79 10.15 14.86I, SD p4 � p5 � 0 6.13 7.76 10.07 14.64

Notes: As for Table 1A.



Appendix B: IGARCH Errors in Cointegration Testing

Table 6 was obtained using independent and identically distributed standard normaldeviates. As in Appendix A, it is relevant to enquire how robust they are toheteroscedasticity, and non-normal kurtosis. Johansen critical values were obtained

Table "BTail Area Probabilities (%)


I, SD, T 4.4 7.1 10.7 16.7I, SD 3.7 6.3 9.6 15.3

Notes: As for Tables 1A and 2A where applicable, with accuracy between�0:25 per cent (for the largest entry, 19.5 per cent) and �0:12 percent (for the smallest, 3.7 per cent).

Table áBTail Area Probabilities (%)

Auxiliaryregressors H0 0.90 0.95 0.975 0.99

I, SD, T p2 � p3 � 0 16.4 10.8 7.5 5.0I, SD 15.8 10.4 7.2 4.6I, SD, T p4 � p5 � 0 16.2 10.7 7.2 4.8I, SD 15.9 10.5 7.1 4.7I, SD, T p2 � p3 � 19.5 13.7 10.0 7.1I, SD p4 � p5 � 0 18.9 13.4 9.9 7.0

Notes: As for Table 1B.

Table åAJohansen's Test: Sizes, Critical Values for IGARCH Errors

Sizes (%) Critical valuesOsterwald^Lenum95% critical value

H0�n�lmax

(1)ltrace(2)

lmax

(1)ltrace(2) (1) (2)

n � 4 9.8 9.8 10.93 10.93 8.18 8.18(0.1) (0.1)

n � 3 14.2 13.8 21.46 24.78 14.90 17.95(0.2) (0.2)

n � 2 19.3 18.1 31.89 43.30 21.07 31.52(0.3) (0.3)

n � 1 23.8 23.5 42.23 66.43 27.14 48.28(0.4) (0.4)

n � 0 28.4 27.3 53.50 95.21 33.32 70.60(0.4) (0.5)

Notes: lmax and ltrace are Johansen's test statistics ÿT ln�1ÿ l̂i� and ÿTP

i ln�1ÿ l̂i�, respectively.100,000 replications give a sampling error (95 per cent con¢dence interval) for the size estimates of�0:18 per cent to �0:28 per cent. The critical values have 95 per cent distribution-free con¢denceintervals plus/minus at most the ¢gure given in parentheses: thus 10:93� 0:1. The interval is notexactly symmetric.



by simulation. The calculations were carried out for N � 755 for IGARCH(1,1)errors, with a � 0:85 and b � 0:15. N � 755 was chosen as the sample size used forTable 6 as simulating for N � 100, 200, 400, 800 showed that the critical values,while not signi¢cantly di¡erent, appear to be increasing slightly from 400 to 800, asif the asymptote has not been reached. This e¡ect is also noticeable for normalerrors.

We see that the critical values appropriate for this particular IGARCHprocess are between 34 and 61 per cent larger than the corresponding normalvalues; alternatively, inappropriate use of the standard critical values can give asize of between 9.8 and 28.4 per cent. Our empirical conclusions would still stand.Of course, the results for other GARCH and IGARCH processes require furtherinvestigation.

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