Microsoft Word - 13_Olbryś_J.docx DYNAMIC ECONOMETRIC MODELS Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 Joanna Olbryś Bialystok University of Technology ARCH Effect in Classical Market-Timing Models with Lagged Market Variable: the Case of Polish Market† A b s t r a c t. The main goal of this study is to present the regressions of the GARCH versions of classical market-timing models of Polish equity funds. We examine the models with lagged val- ues of the market factor as an additional variable because of the Fisher’s effect1 in the case of the main Warsaw Stock Exchange indexes. The market-timing and selectivity abilities of fund man- agers are evaluated for the period Jan 2003 – June 2011. Results on both the HAC and the GARCH estimates are qualitatively similar, and even better in the case of the simpler HAC meth- od. For this reason, it is not necessary to estimate the GARCH versions of market-timing models in the case of Polish mutual funds, even despite the strong ARCH effects that exist in these models. K e y w o r d s: market-timing, non-trading, ARCH effect, GARCH model. Introduction Market-timing is one strategy by which portfolio managers might attempt to obtain returns in excess of those expected of an unmanaged portfolio. One of the benefits of market-timing is the production of a positively skewed distribu- tion of returns. Treynor and Mazuy (1966) produce a single factor model de- rived from CAPM in which a quadratic term is added to reflect the market- timing. The T-M coefficient measures the co-skewness with the benchmark portfolio. Henriksson and Merton (1981) start from a similar idea, but provide a different interpretation of market-timing ability. Adding a term in the CAPM model that contains a dummy variable based on the difference between market return and the risk-free rate, they permit managers to choose between two levels of market risk: an up-market and a down-market beta (Cogneau, Hübner, 2009). Some other researchers extend market-timing models to multifactor as well as † The author acknowledges the financial support in 2009 – 2011 from the Polish Ministry of Science and Higher Education within the grant no. N N113 173237. 1 Lawrence Fisher’s effect (1966) Joanna Olbryś 186 to conditional versions. In relation to the Polish market, in Olbryś (2009) the usefulness of the conditional multifactor market-timing models for the invest- ment managers’ performance evaluation on the Polish market has been exam- ined. Ferson and Schadt (1996) use a collection of public information variables. In Poland, the suitable variables are: (1) the lagged monthly dividend yield of the WSE stock index, (2) the lagged monthly level of the 1M WIBOR, (3) the lagged monthly measure of the slope of the term structure (Olbryś, 2009, p. 522). The evidence on Polish market shows that the quality and usefulness of these models is rather low. As for the other multifactor models, in light of the empirical results for the Polish market, the influence of the Fama and French’s (1993) size (SMB) and book-to-market (HML) spread variables, and Carhart’s (1997) momentum (WML) factor on the Polish equity funds’ market seems to be rather controversial (Olbryś, 2010a b; 2012). It is worth stressing that the SMB, HML, and WML factors have a diverse explanatory power for the sample of Polish funds. Another important finding is that the investigated funds are not homogeneous regarding the influence of the size, book-to-market and momen- tum factors, despite the fact that all of them are Polish equity open-end mutual funds (Olbryś, 2011a). For this reason, the SMB, HML, and WML factors have not been taken into account as explanatory variables in our models. According to the literature, the methods most widely applied in market- timing models estimation are the two proposed by White (1980) or Newey-West (1987) (see e.g. Ferson, Schadt, 1996; Bollen, Busse, 2001; Romacho, Cortez, 2006; Olbryś, 2010a, b). Some previous publications also describe applications of the GLS procedure with correction for heteroskedasticity (see e.g. Henriks- son, Merton, 1981; Henriksson, 1984) or the Fama-MacBeth cross-sectional regression approach from 1973 (Carhart, 1997). Kao et al. (1998) employ an autoregressive conditional heteroskedastic (ARCH) model, but without testing the ARCH effects. Recent studies in multifactor market-timing models in the case of Polish equity funds by Olbryś (2010a) present possibilities and exam- ples of applying the seemingly unrelated regression method (SUR) which was described by Zellner (1962). The author’s recent research provide evidence of pronounced ARCH effects (Engle, 1982) in the market-timing models of Polish equity open-end mutual funds. For this reason, the main goal of this study is to present the regression results of the new GARCH(p, q) models of these funds. We estimate the GARCH versions of classical market-timing models with lagged values of the market factor as an additional independent variable because of the pronounced Fisher’s effect in the case of the main Warsaw Stock Ex- change indexes. The market-timing and selectivity abilities of fund managers are evaluated for the period January 2003 – June 2011. In comparison to robust Newey-West method results, our findings suggest that the GARCH(p, q) model is suitable but not necessary for such applications. To the best of author’s knowledge, no such investigation has been undertaken for the Polish market. ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 187 The remainder of the paper is organized as follows. Section 1 specifies a methodological background and a brief literature review. First, we stress the validity of the non-trading problem and the Fisher’s effect in the case of market- index returns. Next, we present classical market-timing models with lagged values of the market factor as additional explanatory variable. We also present a brief theoretical framework concerning the ARCH(q) and the GARCH(p,q) models. In the end of Section 1, we describe tests for the ARCH effect in an econometric model. In Section 2, we present the data and methodology in the case of Polish emerging market and discuss the results obtained. Section 3 re- calls the main findings and presents the conclusions. 1. Methodological Background 1.1. Non-trading Problem and the Fisher’s Effect It is worth stressing, that the empirical market microstructure literature is an extensive one recently. High-frequency financial data are important in study- ing a variety of issues related to the trading process and market microstructure (Tsay, 2010, p. 231). For some purposes, such aspects of the market microstruc- ture as non-trading or bid-ask spread effects can be safely ignored. However, for other purposes, market microstructure is central (Campbell et al., 1997). In 1980 Cohen et al. point to various frictions in the trading process that can lead to a distinction between “true” and observed returns. They have focused on the fact that transaction prices differ from what they would otherwise be in a frictionless environment. It has been reported in the literature that some em- pirical phenomena can be attributed to frictions in the trading process (see e.g. Fisher, 1966; Scholes and Williams, 1977; Dimson, 1979), also on the Polish capital market (see e.g. Doman, Doman, 2004; Doman, 2010; Brzeszczyński et al., 2011; Olbryś, 2011b). It is worthwhile to note that two common elements among most of the phenomena are evident, the “interval effect” and the impact of a security’s “thinness”. The evidence that daily market-index returns exhibit a pronounced positive first-order autocorrelation has been called the Fisher’s effect since Lawrence Fisher in 1966 hypothesized its probable cause. Fisher suggested it was caused by a non-trading of the component securities. The ob- served correlation is higher in those indexes that give greater weight to the secu- rities of smaller firms. To detect for the Fisher’s effect in the market-index re- turns, partial autocorrelations functions (PACF) can be calculated. To calculate PACF, first it should be determined (based on the ADF test) that the analyzed index series are stationary. In the next step partial autocorrelations functions for individual stationary processes can be calculated and the significance of the first-order daily serial correlation coefficients 1 can be tested, using the Que- nouille’s test (Kufel, 2009). The evaluation of first-order serial correlation is carried out by testing the null hypothesis: Joanna Olbryś 188 0 1: 0H   (1) If the estimate 1̂ satisfies the inequality 1 1.96 ˆ T   , then we have no reason to reject the null hypothesis (1). The non-trading effect induces potentially serious biases in the moments and co-moments of asset returns such as their means, variances, covariances, betas, and autocorrelation and cross-autocorrelation coefficients (Campbell et al., 1997, p. 84). For this reason, Busse (1999) proposed lagged values of the market factor as an additional independent variable in the regressions of market- timing models using Dimson’s (1979) correction. We emphasize that the Polish market is an emerging market and it can be expected that non-trading problem should be more visible than in other, developed markets. 1.2. Classical Market-Timing Models with Lagged Market Variable The classical parametric Treynor – Mazuy market-timing model with lagged values of the market factor as additional explanatory variable can be expressed as: 2 , 1 , 2 , 1 , ,( ) ,P t P P M t P M t P M t P tr r r r            (2) where tFR , is the one-period return on riskless securities, , , ,P t P t F tr R R  is the excess return on portfolio P in the period t, , , ,M t M t F tr R R  is the excess return on portfolio M in the period t, 1, tMr is the lagged excess return on portfolio M in the period t, P measures the selectivity skills of the manager (Jensen, 1968), P1 is the systematic risk measure of portfolio P to changes in the market fac- tor returns, 2 P is the systematic risk measure of portfolio P to changes in the lagged market factor returns, P measures the market-timing skills of the man- ager of portfolio P (the T-M coefficient), and tP, is a residual term, with the following standard CAPM conditions: ,( ) 0P tE   , , , 1( ) 0P t P tE     . In a way analogous to (2), the classical parametric Henriksson - Merton model with lagged values of the market factor as additional explanatory variable can be expressed as: , 1 , 2 , 1 , , ,P t P P M t P M t P M t P tr r r y            (3) where tPr , , tMr , , 1, tMr , P , 1P , 2 P , ,P t are as in equation (2), P measures the market-timing skills of the manager of portfolio P (the H-M coef- ficient), and , ,max{0, }M t M ty r  . ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 189 If the portfolio manager has the ability to forecast security prices, the inter- cept P in equations (2)–(3) will be positive (Jensen, 1968). Indeed, it repre- sents the average incremental rate of return on the portfolio per unit time which is due solely to the manager’s ability to forecast future security prices. In this way, ˆ P measures the contribution of security selection to portfolio perfor- mance, which corresponds to testing the null hypothesis: 0 : 0PH   (4) i.e., the manager does not have any microforecasting ability. The evaluation of market-timing skills is carried out by testing the null hypothesis: 0 : 0PH   (5) i.e., the manager does not possess any timing ability or does not on his forecast (Henriksson 1984). A negative value for the regression estimate ˆP would im- ply a negative value for market-timing. The size of the estimate ˆP informs us about the manager’s market skills. 1.3. The GARCH(p, q) Model The first model that provides a systematic framework for volatility model- ing is the ARCH model of Engle (1982). Engle proposed the ARCH models to capture the serial correlation of volatility (Campbell et al., 1997, p. 482). Engle suggested the ARCH model as an alternative to the usual time-series process. More recent studies of financial markets suggest that the phenomenon is quite common (Greene, 2002). The basic idea of the ARCH models is that 1) the in- novation t of the regression is serially uncorrelated, but dependent, and 2) the dependence of t can be described by a simple quadratic function of its lagged values (Tsay, 2010). The ARCH(q) regression model is obtained by assuming that the mean of random variable ty , which is drawn from the conditional den- sity function 1( )t tf y y  , is given as tx b , a linear combination of lagged endog- enous and exogenous variables included in the information set 1t  , with b a vector of unknown parameters (Engle, 1982). Formally: 1 2 1 2 0 1 0 ~ ( , ), , ( , , , , ) , 0, 0, 1, , . t t t t t t t q t t t t q i t i i i y N x h y x h h i q                           b b α  (6) Joanna Olbryś 190 where t is the innovation in a linear regression with 2( )V   , q is the order of the ARCH(q) process, α is the vector of unknown parameters, th is the vari- ance function. The null hypothesis of white noise disturbances in (6) is: 0 1: 0qH     (7) The GARCH(p, q) model generalizes the ARCH(q) model of Engle (1982) and is proposed by Bollerslev (1986). The GARCH(p, q) is given by: 1 1 2 1 2 2 0 1 1 0 ~ ( 0 0 1 0 0 1 0 t t t t t t t t t t t q t t t p q p i t i j t j i j i j y N x ,h ), y x , h h( , , , ,h ,h , ,h , , ) h , , ,i , ,q,q , , j , , p, p .                                          b b α β    (8) where t , q , α , th are as in equation (6) and β is a vector of unknown param- eters. In the GARCH(p, q) model, q refers to the number of lags of t and p re- fers to the number of lags of th to include in the model of the regression vari- ance (Adkins, 2010). For 0p  the process reduces to the ARCH(q) process, and for 0p q  , t is simple white noise. The null hypothesis of white noise disturbances in (8) is: 0 1 1: 0; 0q pH           (9) In the ARCH(q) process the conditional variance is specified as a linear function of past sample variances only, whereas the GARCH(p, q) process al- lows the lagged conditional variances to enter as well (Bollerslev 1986). A wide range of GARCH models have now appeared in the econometric literature (e.g. Engle, 2000; Fiszeder, 2009). The parameters of GARCH(p, q) models are almost invariably estimated via Maximum Likelihood (ML) or Quasi-Maximum Likelihood (QML: see Bollerslev, Wooldridge, 1992) methods, which bring up the subject of a suitable choice for the conditional distribution of t . Several likelihood functions are commonly used in ARCH (GARCH) estimation, de- pending on the distributional assumption of t (Tsay, 2010). 1.4. Testing for ARCH Effect in an Econometric Model Before estimating the GARCH(p, q) model it might be useful to test for the ARCH effect. The simplest approach is to examine the squares of the least ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 191 squares residuals. The autocorrelations of the squares of the residuals provide evidence about ARCH effects (Greene, 2002). Two tests are available. The first test is to apply the Ljung-Box statistics ( )Q q (Ljung, Box, 1978). The null hy- pothesis is that the first q lags of ACF of the squares of the least squares resid- uals series are zero. In practice, the choice of q may affect the performance of the ( )Q q statistic. Simulation studies suggest that the choice of ln( )q T , where T is the number of time periods, provides better power performance (Tsay, 2010, p. 33). The second test for conditional heteroskedasticity is the Lagrange multiplier (LM) test of Engle (1982). Lee (1991) found that the LM test of white noise disturbances against GARCH(p, q) disturbances in a linear regression model is equivalent to that against ARCH(q) disturbances. This im- plies that under the null hypothesis of white noise, the GARCH(p, q) effect and the ARCH(q) effect are locally equivalent alternatives. Hence we can proceed by testing the ARCH(q) effect against the GARCH(p, q) effect (Lee, 1991, pp. 269–270). An LM test of ARCH(q) against the hypothesis of no ARCH effects can be carried out by computing 2 2q T R   in the regression of 2 te on a constant and q lagged values. Under the null hypothesis of no ARCH effects, the statistic has a limiting chi-squared distribution with q degrees of freedom. Values larg- er than the critical table value give evidence of the presence of ARCH (or GARCH) effects (Greene, 2002, p. 244). 2. Empirical Results 2.1. The Fisher’s Effect on the Warsaw Stock Exchange The Fisher’s effect on the Warsaw Stock Exchange has been detected by Olbryś (2011b). The empirical results show a pronounced Fisher’s effect in the case of the WIG, mWIG40 and sWIG80 series. We observe the most clear ef- fect for the sWIG80 series. We have no reason to reject the null hypothesis (1) only in the case of the WIG20 series. As mentioned above, this evidence is con- sistent with most of the literature on friction in the trading process because the observed correlation is higher in those indexes that give greater weight to the securities of smaller firms. For the Fisher’s effect reason, we can use Dimson’s (1979) correction and include lagged values of the market factor (i.e. the main index of WSE companies – WIG) as an additional independent variable in the regressions of market-timing models of Polish equity open-end mutual funds to accommodate infrequent trading (Olbryś, 2011a). 2.2. Data The period investigated is January 2, 2003 – June 30, 2011 (T=2137 obser- vations). To detect for the ARCH effect in market-timing models of Polish Joanna Olbryś 192 funds in subsamples of various length, the entire sample has been divided into eight subsamples: P1, P2, P3, P4, P5, P6, P7, P8 (Table 1). Table 1. Subsamples in the period from Jan 2, 2003 to June 30, 2011 Subsample T P1 Jan 2, 2003–June 30, 2011 2137 P2 Jan 2, 2004–June 30, 2011 1886 P3 Jan 3, 2005–June 30, 2011 1631 P4 Jan 2, 2006–June 30, 2011 1380 P5 Jan 2, 2007–June 30, 2011 1129 P6 Jan 2, 2008–June 30, 2011 880 P7 Jan 5, 2009–June 30, 2011 629 P8 Jan 4, 2010–June 30, 2011 377 Note: T is the number of data points. Table 2. Equity open-end mutual funds in Poland by the end of 2002 Equity fund (current name) Short Name Year of creation 1 Arka BZ WBK FIO Subfundusz Arka Akcji Arka 1998 2 Aviva Investors FIO Subfundusz Aviva Investors Polskich Akcji Aviva 2002 3 BPH FIO Parasolowy BPH Subfundusz Akcji BPH 1999 4 ING Parasol FIO ING Subfundusz Akcji ING 1998 5 Investor Top 25 Małych Spółek FIO Investor 25 2002 6 Investor Akcji Dużych Spółek FIO Investor ADS 1998 7 Investor Akcji FIO Investor 1998 8 Legg Mason Akcji FIO Legg Mason 1999 9 Millennium FIO Subfundusz Akcji Millennium 2002 10 Novo FIO Subfundusz Novo Akcji Novo 1998 11 Pioneer FIO Subfundusz Pioneer Akcji Polskich Pioneer 1995 12 PKO Akcji – FIO PKO 1998 13 PZU FIO Parasolowy Subfundusz PZU Akcji Krakowiak PZU 1999 14 Skarbiec FIO Subfundusz Akcji Skarbiec – Akcji Skarbiec 1998 15 UniFundusze FIO Subfundusz UniKorona Akcje UniKorona 1997 Note: The source of this Table is the Polish Financial Supervision Authority http://www.knf.gov.pl (Sept 8, 2011). We have examined the performance of 15 selected equity open-end Polish mutual funds which were created up to the end of 2002. Our dataset includes returns on all the equity funds in existence in Poland from 2002 to 2011, there- fore our results are free of survivorship bias (Table 2). We have studied daily logarithmic excess returns from Jan 2003 to June 2011. Daily returns on the main index of WSE companies are used as the returns on the market portfolio. The daily average of returns on 52-week Treasury bills are used as the returns on riskless assets. All calculations were done using Gretl 1.9.5. 2.3. ARCH Effect in Market-Timing Models with Lagged Market Variable Volatility clustering, which is a common cause of heteroskedasticity, is more likely to be present in financial models built using higher-frequency data, such as daily data (Brzeszczyński et al., 2011). To detect for the ARCH effect in ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 193 market-timing models of Polish equity open-end mutual fund portfolios in the period investigated Jan 2, 2003 – June 30, 2011, the LM (Lagrange Multiplier) and the LB (Ljung-Box) tests have been applied. The empirical results present- ed in Table 3 show strong ARCH effect in the case of all of the funds. The null hypothesis (7) is rejected in these cases. Because we are using daily logarithmic excess returns on fund portfolios, the LM test at the lag 5q has been applied. On the other hand, the LB test at the lag 8)2137ln( q has been used (Tsay, 2010). The p -values of all statistics are very close to zero. Table 3. The ARCH effect in market-timing models (2) and (3) of Polish equity mutual funds in the entire sample P1 (period from Jan 2, 2003 to June 30, 2011) Equity fund (short name) T-M model H-M model LM p-value LB p-value LM p-value LB p-value 1 Arka 326.8 110-68 157.6 410-30 346.9 710-73 159.5 210-30 2 Aviva 257.1 110-53 299.9 410-60 258.3 910-54 306.7 110-61 3 BPH 424.6 110-89 434.2 910-89 427.0 410-90 436.5 210-89 4 ING 443.9 110-93 442.4 110-90 445.2 510-94 444.5 510-91 5 Investor 25 404.4 310-85 145.1 210-27 390.4 310-82 142.1 810-27 6 Investor ADS 524.4 410-111 474.7 110-97 531.8 110-112 475.1 110-97 7 Investor 460.3 210-97 498.0 110-102 459.4 410-97 497.2 210-102 8 Legg Mason 402.1 110-84 333.1 310-67 408.6 410-86 334.2 210-67 9 Millennium 437.6 210-92 371.2 210-75 439.8 710-93 374.7 410-76 10 Novo 622.2 310-132 489.3 110-100 609.5 110-129 485.7 810-100 11 Pioneer 423.7 210-89 372.1 110-75 426.6 510-90 374.4 510-76 12 PKO 485.9 810-103 379.1 510-77 477.1 710-101 379.0 510-77 13 PZU 402.0 110-84 387.0 110-78 404.9 210-85 391.8 110-79 14 Skarbiec 384.4 610-81 427.0 310-87 385.5 410-81 426.7 310-87 15 UniKorona 371.4 410-78 519.3 510-107 376.8 210-79 519.7 410-107 Note: The table is based on the entire sample P1; T-M (2) is the classical Treynor-Mazuy model with the lagged excess return on market portfolio M as additional factor; H-M (3) is the classical Henriksson-Merton model with the lagged excess return on market portfolio M as additional factor; LM is the Engle (1982) statistic at the lag equal to five, which should be distributed as chi-squared; LB is the Ljung-Box (1978) statistic at the lag equal to eight, which should be distributed as chi-squared. Tables 4a–4b present further analysis, including more details about empiri- cal results of testing the ARCH effect in the T-M and the H-M market-timing models. The ARCH effect has been tested in the case of all funds and in all subsamples. Several results in Tables 4a–4b are worth special notice. The ARCH effect disappears as the interval is shortened and only in the case of 5 out of 15 funds (i.e. Arka, Investor 25, Novo, Skarbiec and UniKorona) it persists in all samples P1–P8. Furthermore, if the ARCH effects are not present in the model, simple OLS regression is quite sufficient (Brzeszczyński et al., 2011). Table 4a. Result summary of the ARCH effect in the T-M and H-M models; subsamples P1–P4 Fu nd . N o. P1 P2 P3 P4 T-M H-M T-M H-M T-M H-M T-M H-M LM LB LM LB LM LB LM LB LM LB LM LB LM LB LM LB 1 + + + + + + + + + + + + + + + + 2 + + + + + + + + + + + + + + + + 3 + + + + + + + + + - + - + - + - 4 + + + + + + + + + + + + + + + + 5 + + + + + + + + + + + + + + + + 6 + + + + + + + + + + + + + + + + 7 + + + + + + + + + + + + + + + + 8 + + + + + + + + + - + - + - + - 9 + + + + + + + + + - + - + - + - 10 + + + + + + + + + + + + + + + + 11 + + + + + + + + + - + - + - + - 12 + + + + + + + + + + + + + + + + 13 + + + + + + + + + - + - + - + - 14 + + + + + + + + + + + + + + + + 15 + + + + + + + + + + + + + + + + Table 4b. Result summary of the ARCH effect in the T-M and H-M models; subsamples P5–P8 Fu nd . N o. P5 P6 P7 P8 T-M H-M T-M H-M T-M H-M T-M H-M LM LB LM LB LM LB LM LB LM LB LM LB LM LB LM LB 1 + + + + + + + + + + + + + + + + 2 + + + + - + - + - + - + + + + + 3 + - + - + - + - - - - - + - + - 4 + + + + + - + - + - + - - + - + 5 + + + + + + + + + + + + + + + + 6 + + + + + + + + + + + + - - - - 7 + + + + + + + + + + + + + - + - 8 + - + - + - + - + + + + - - - - 9 + - + - + - + - + - + - + - + - 10 + + + + + + + + + + + + + + + + 11 + - + - + - + - + - + - - - - - 12 + + + + + + + + + - + - - - - - 13 + - + - + - + - + + + + + + + + 14 + + + + + + + + + + + + + + + + 15 + + + + + + + + + + + + + + + + Note: Table 4a is based on the samples P1– P4 and Table 4b is based on the samples P5– P8 (Table 1); T-M (2) is the classical Treynor-Mazuy model with the lagged excess return on market portfolio M as additional factor; H-M (3) is the classical Henriksson-Merton model with the lagged excess return on market portfolio M as additional factor; LM is the Engle (1982) statistic at the lag q, which should be distributed as chi-squared; LB is the Ljung-Box (1978) statistic at the lag q, which should be distributed as chi-squared; + denotes that statistic value is larger than the critical table value of chi-squared and gives evidence of the presence of ARCH effect; – denotes that statistic value is smaller than the critical table value of chi-squared. ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 195 2.4. The GARCH Versions of Market-Timing Models of Polish Equity Mutual Funds Hamilton (2008) stresses that even if the researcher’s primary interest is in es- timating the conditional mean, having a correct description of the conditional variance can still be quite important. By incorporating the observed features of the heteroskedasticity into the estimation of the conditional mean, substantially more efficient estimates of the conditional mean can be obtained. The most popular White or Newey–West corrections may not fully correct for the influ- ence problems introduced by ARCH. The testing results from the Polish equity Table 5a. The HAC estimates of the T-M market-timing models of Polish equity mutual funds in the entire period from Jan 2, 2003 to June 30, 2011 Equity fund P̂ 1P̂ 2 P P̂ 2R 1 Arka 410-4 ** (110-4) 0.71*** (0.03) 0.14*** (0.03) -1.86** (0.83) 0.66 2 Aviva 310-4 ** (110-4) 0.75*** (0.03) 0.15*** (0.02) -1.45** (0.69) 0.73 3 BPH 210-5 (110-4) 0.72*** (0.02) 0.12*** (0.02) -0.56 (0.48) 0.76 4 ING -210-6 (110-4) 0.76*** (0.02) 0.13*** (0.02) -0.48 (0.43) 0.74 5 Investor 25 210-4 (210-4) 0.39*** (0.02) 0.31*** (0.03) -1.76** (0.85) 0.39 6 Investor ADS -110-4 (210-4) 0.63*** (0.03) 0.38*** (0.03) -0.55 (1.26) 0.55 7 Investor -410-5 (110-4) 0.54*** (0.03) 0.37*** (0.03) -0.75 (0.86) 0.55 8 Legg Mason 210-4 *(910-5) 0.69*** (0.02) 0.12*** (0.02) -0.68 (0.51) 0.74 9 Millennium -410-6 (110-4) 0.69*** (0.02) 0.13*** (0.02) -0.76 (0.57) 0.72 10 Novo 810-5 (210-4) 0.48*** (0.03) 0.45*** (0.02) -1.20 (1.14) 0.55 11 Pioneer -910-5 (110-4) 0.80*** (0.03) 0.16*** (0.02) -1.13 (0.81) 0.74 12 PKO 610-5 (210-4) 0.55*** (0.03) 0.28*** (0.03) -1.66 (1.19) 0.56 13 PZU 310-5 (110-4) 0.71*** (0.02) 0.12*** (0.02) -0.98** (0.49) 0.73 14 Skarbiec -410-6 (110-4) 0.46*** (0.03) 0.38*** (0.03) 0.15 (0.73) 0.51 15 UniKorona 110-4 (110-4) 0.48*** (0.03) 0.45***(0.03) -0.54 (0.89) 0.55 Note: The table is based on the entire sample P1; T-M (2) is the classical Treynor-Mazuy model with the lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors are in parentheses next to the coefficient estimates; the values of the adjusted determination coefficient are in the last column; * significant at the 10 per cent level; ** significant at the 5 per cent level; *** significant at the 1 per cent level. mutual funds dataset show pronounced ARCH effect in market-timing models (Tables 3–4). For this reason, the estimation of the market-timing models as the GARCH(p, q) models is well-founded. Although the ARCH(q) model (6) is simple, it often requires many parameters to adequately describe the volatility process. The modeling procedure of the ARCH(q) model can also be used to build a GARCH(p, q) model (8). However, specifying the order of a GARCH(p, q) model is not easy. Only the lower order GARCH models are used in most applications, i.e. GARCH(1,1), GARCH(1,2), GARCH(2,1), and GARCH(2,2) models (Tsay, 2010). According to the literature, GARCH(p, q) models are usually compared and selected by the information criterion of Akaike (AIC) and the information criterion of Schwartz (SC). Lower values of Joanna Olbryś 196 the AIC and SC indexes indicate the preferred model, that is, the one with the fewest parameters that still provides an adequate fit to the data2. As an example, we present the comparison of the estimation results of mar- ket-timing models T-M and H-M of Polish equity mutual funds in the entire period from Jan 2, 2003 to June 30, 2011. We use the Newey-West robust esti- mates (HAC) as well as the robust quasi-maximum likelihood estimates (QML) of the parameters of the suitable GARCH(p, q) version of the market-timing model. Tables 5a–5b provide details on the robust HAC estimates of the T-M and H-M market-timing models, respectively. Table 5b. The HAC estimates of the H-M market-timing models of Polish equity mutual funds in the entire period from Jan 2, 2003 to June 30, 2011 Equity fund P̂ 1P̂ 2 P P̂ 2R 1 Arka 710-4 ***(210-4) 0.65*** (0.04) 0.14*** (0.03) -0.14** (0.06) 0.66 2 Aviva 610-4 **(210-4) 0.70*** (0.04) 0.15*** (0.02) -0.11** (0.05) 0.73 3 BPH 110-4 (110-4) 0.70*** (0.03) 0.12*** (0.02) -0.04 (0.04) 0.76 4 ING 110-4 (110-4) 0.74*** (0.03) 0.13*** (0.02) -0.04 (0.04) 0.74 5 Investor 25 410-4 (310-4) 0.33*** (0.03) 0.31*** (0.03) -0.13* (0.06) 0.39 6 Investor ADS -210-4 (310-4) 0.63*** (0.04) 0.38*** (0.03) -0.01 (0.08) 0.55 7 Investor 210-5 (310-4) 0.52***(0.03) 0.37*** (0.03) -0.04 (0.06) 0.55 8 Legg Mason 310-4 *(110-4) 0.67***(0.03) 0.12*** (0.02) -0.05 (0.04) 0.74 9 Millennium 210-4 (210-4) 0.65*** (0.03) 0.13*** (0.02) -0.07 (0.04) 0.72 10 Novo 110-4 (310-4) 0.45*** (0.03) 0.45*** (0.03) -0.06 (0.07) 0.55 11 Pioneer 910-5 (210-4) 0.76*** (0.04) 0.16*** (0.02) -0.08 (0.06) 0.74 12 PKO 310-4 (310-4) 0.50*** (0.04) 0.28*** (0.03) -0.12 (0.07) 0.56 13 PZU 210-4 (210-4) 0.67*** (0.03) 0.12*** (0.02) -0.08* (0.04) 0.73 14 Skarbiec -810-5 (310-4) 0.47*** (0.03) 0.38*** (0.03) 0.02 (0.06) 0.51 15 UniKorona 110-4 (310-4) 0.47*** (0.03) 0.45*** (0.03) -0.02 (0.06) 0.55 Note: The table is based on the entire sample P1; H-M (3) is the classical Henriksson-Merton model with the lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors are in parentheses next to the coefficient estimates; the values of the adjusted determination coefficient are in the last column; * significant at the 10 per cent level; ** significant at the 5 per cent level; *** significant at the 1 per cent level. The robust QML estimates of the parameters of the suitable GARCH(p, q) version of market-timing models are presented in Tables 6a–6b, respectively. It is worth stressing that some restrictions for the parameters in the GARCH(p, q) models (8) can be relaxed. For example, it is not necessary for the 2 parameter in the conditional variance equation in the GARCH(1,2) model to be nonnegative (Fiszeder, 2009). Note that in the case of all funds, both for the T-M model and for the H-M model the same variant of the GARCH(p, q) model has been chosen (Tables 6a–6b). 2 When the values of the information criterions AIC or SC for different variants of the GARCH(p, q) models are almost equal, the statistical significance of the parameters in the condi- tional mean and conditional variance equations of the GARCH(p, q) model has been analyzed to choose the appropriate model. ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 197 In summary, the results in Tables 5a–5b and 6a–6b clearly show that despite the strong ARCH effect in all models built based on the sample P1, the simpler robust HAC method is quite sufficient. Therefore, in our opinion, the GARCH(p, q) model is suitable but not necessary for such applications. Table 6a. The GARCH(p, q) versions of the T-M market-timing models of Polish equity mutual funds in the entire period from Jan 2, 2003 to June 30, 2011 Fu nd . N o. T-M model – conditional mean equation (p, q) conditional variance equation P̂ 1P̂ 2 P P̂ 0̂ 1 2̂ 1̂ 2̂ 1 310-4 (110-4) 0.80 (0.01) 0.04 (0.01) -1.60 (0.52) (1,1) 210-7 (110-7) 0.09 (0.02) - 0.90 (0.02) - 2 310-4 (710-5) 0.86 (0.01) 0.01 (0.005) -1.95 (0.33) (1,2) 210-7 (610-8) 0.47 (0.12) -0.30 (0.12) 0.85 (0.03) - 3 -710-5 (510-5) 0.84 (0.005) 0.005 (0.004) 0.06 (0.18) (1,1) 710-8 (210-8) 0.08 (0.01) - 0.91 (0.01) - 4 -110-5 (510-5) 0.89 (0.005) 0.003 (0.005) 0.06 (0.21) (1,1) 710-8 (210-8) 0.07 (0.01) - 0.91 (0.01) - 5 710-5 (210-4) 0.34 (0.02) 0.25 (0.02) -1.06 (1.07) (2,2) 410-6 (110-6) 0.17 (0.04) 0.20 (0.04) -0.16 (0.05) 0.74 (0.05) 6 -210-4 (910-5) 0.94 (0.03) 0.07 (0.02) 0.64 (0.69) (1,2) 410-8 (310-8) 0.28 (0.04) -0.21 (0.04) 0.93 (0.01) - 7 -110-4 (710-5) 0.79 (0.01) 0.08 (0.01) -0.28 (0.40) (1,2) 510-8 (310-8) 0.35 (0.04) -0.26 (0.04) 0.91 (0.01) - 8 910-5 (610-5) 0.83 (0.007) 0.02 (0.005) -0.009 (0.29) (1,2) 510-8 (210-8) 0.14 (0.03) -0.07 (0.03) 0.93 (0.01) - 9 -210-4 (610-5) 0.81 (0.008) 0.009 (0.005) -210-4 (0.31) (2,1) 810-8 (410-8) 0.13 (0.03) - 0.33 (0.18) 0.54 (0.18) 10 210-5 (110-4) 0.14 (0.08) 0.64 (0.04) -0.89 (0.79) (1,2) 910-8 (610-8) 0.40 (0.07) -0.30 (0.07) 0.91 (0.02) - 11 -210-4 (610-5) 0.89 (0.006) 0.02 (0.005) -0.38 (0.28) (2,1) 910-8 (410-8) 0.13 (0.02) - 0.39 (0.10) 0.47 (0.09) 12 -410-5 (910-5) 0.73 (0.02) 0.06 (0.01) -0.34 (0.52) (1,2) 610-8 (310-8) 0.27 (0.04) -0.17 (0.03) 0.91 (0.01) - 13 -110-4 (510-5) 0.84 (0.005) 0.01 (0.004) -0.17 (0.18) (1,1) 610-8 (210-8) 0.10 (0.02) - 0.90 (0.02) - 14 210-5 (110-4) 0.42 (0.05) 0.41 (0.05) 0.35 (0.88) (1,2) 910-7 (410-7) 0.24 (0.04) -0.16 (0.04) 0.91 (0.02) - 15 -610-5 (110-4) 0.40 (0.04) 0.53 (0.03) 0.80 (1.08) (1,2) 810-7 (510-7) 0.29 (0.03) -0.20 (0.04) 0.90 (0.03) - Note: The table is based on the entire sample P1; T-M (2) is the classical Treynor-Mazuy model with the lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors are in parentheses below the coefficient estimates; the variance-covariance matrix of the estimated parameters is based on the QML algorithm; the distribution for the innovations is supposed to be normal. Joanna Olbryś 198 Table 6b. The GARCH(p, q) versions of the H-M market-timing models of Polish equi- ty mutual funds in the entire period from Jan 2, 2003 to June 30, 2011 Fu nd . N o. H-M model – conditional mean equation (p, q) conditional variance equation P̂ 1P̂ 2 P P̂ 0̂ 1 2̂ 1̂ 2̂ 1 510-4 (110-4) 0.75 (0.02) 0.04 (0.01) -0.10 (0.03) (1,1) 210-7 (110-7) 0.09 (0.02) - 0.90 (0.02) - 2 810-4 (710-5) 0.75 (0.03) 0.02 (0.005) -0.18 (0.03) (1,2) 210-7 (610-8) 0.52 (0.17) -0.37 (0.16) 0.86 (0.03) - 3 -610-5 (710-5) 0.83 (0.008) 0.005 (0.004) -410-4 (0.01) (1,1) 710-8 (210-8) 0.08 (0.01) - 0.91 (0.01) - 4 -410-5 (710-5) 0.89 (0.009) 0.002 (0.005) -0.01 (0.01) (1,1) 710-8 (210-8) 0.08 (0.01) - 0.91 (0.01) - 5 210-4 (210-4) 0.33 (0.03) 0.25 (0.02) -0.06 (0.06) (2,2) 310-8 (110-8) 0.20 (0.04) -0.20 (0.04) 1.70 (0.06) -0.70 (0.05) 6 -310-4 (110-4) 0.97 (0.03) 0.07 (0.02) 0.05 (0.04) (1,2) 410-8 (310-8) 0.29 (0.04) -0.21 (0.04) 0.93 (0.01) - 7 -110-4 (110-4) 0.78 (0.02) 0.08 (0.01) -0.02 (0.02) (1,2) 510-8 (310-8) 0.35 (0.04) -0.25 (0.04) 0.91 (0.01) - 8 910-5 (910-5) 0.83 (0.01) 0.02 (0.005) -810-4 (0.02) (1,2) 610-8 (210-8) 0.14 (0.03) -0.07 (0.03) 0.93 (0.01) - 9 -710-5 (910-5) 0.81 (0.01) 0.009 (0.005) -0.02 (0.02) (2,1) 810-8 (410-8) 0.13 (0.03) - 0.32 (0.18) 0.54 (0.18) 10 -210-4 (110-4) 0.14 (0.06) 0.65 (0.05) 0.02 (0.04) (1,2) 910-8 (510-8) 0.39 (0.07) -0.29 (0.08) 0.91 (0.02) - 11 -210-4 (810-5) 0.88 (0.01) 0.02 (0.005) -0.02 (0.02) (2,1) 910-8 (410-8) 0.13 (0.02) - 0.39 (0.10) 0.46 (0.09) 12 510-5 (110-4) 0.71 (0.03) 0.06 (0.01) -0.03 (0.03) (1,2) 610-8 (310-8) 0.27 (0.03) -0.18 (0.03) 0.91 (0.01) - 13 -210-5 (810-5) 0.83 (0.008) 0.01 (0.004) -0.02 (0.01) (1,1) 610-8 (310-8) 0.10 (0.02) - 0.90 (0.02) - 14 -110-4 (210-4) 0.44 (0.05) 0.41 (0.05) 0.04 (0.05) (1,2) 910-7 (410-7) 0.24 (0.04) -0.16 (0.04) 0.91 (0.02) - 15 -210-4 (210-4) 0.43 (0.04) 0.53 (0.03) 0.07 (0.05) (1,2) 910-7 (510-7) 0.29 (0.03) -0.20 (0.04) 0.90 (0.03) - Note: The table is based on the entire sample P1; H-M (3) is the classical Henriksson-Merton model with the lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors are in parentheses below the coefficient estimates; the variance-covariance matrix of the estimated parameters is based on the QML algorithm; the distribution for the innovations is supposed to be normal. Conclusions Our research provides evidence of pronounced ARCH effects in the classical market-timing models of Polish open-end mutual funds. We detect for the ARCH effects in the entire period from Jan 2, 2003 to June 30, 2011, as well as for the 7 subperiods. For comparison, we estimate the market-timing models using two methods. Results on both the HAC and the GARCH estimates are qualitatively similar, and even better in the case of the simpler HAC method. For this reason, it is not necessary to estimate the GARCH versions of market- ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 199 timing models in the case of Polish mutual funds, even despite the strong ARCH effects that exist in these models. As for the interpretation of the esti- mated coefficients, our empirical results can be summarized as follows: 1. There is no evidence that equity fund managers are successful in selecti- vity ( ˆ P ). 2. 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The Journal of Finance, 55, 1655–1703. Efekt ARCH w klasycznych modelach market-timing z opóźnioną zmienną rynkową: przypadek rynku polskiego Z a r y s t r e ś c i. W artykule przedstawiono badania dokumentujące występowanie efektu ARCH w klasycznych modelach market-timing z opóźnioną zmienną rynkową w przypadku polskich funduszy akcji, w okresie styczeń 2003-czerwiec 2011. Dokonano estymacji wersji GARCH odpowiednich modeli oraz porównano jakość modeli GARCH i modeli uzyskanych metodą HAC. Wyniki wskazują, że modele GARCH są odpowiednie, ale metoda HAC jest wy- starczająca, pomimo występowania efektu ARCH. Podano również interpretacje parametrów otrzymanych modeli w badanej grupie funduszy. S ł o w a k l u c z o w e: model market-timing, niesynchroniczne transakcje, efekt ARCH, model GARCH.