ARCH Effects and Trading Volume

ARCH Eects and Trading Volume Je Fleming a , Chris Kirby b , Barbara Ostdiek a a Jesse H. Jones Graduate School of Management, Rice University, Houston, TX 77005 b John E. Walker Department of Economics, Clemson, SC 29631 Abstract Studies that t volume-augmented GARCH models often nd support for the hypothesis that
trading volume explains ARCH eects in daily stock returns. We show that this nding is due
to an unrecognized constraint imposed by the GARCH specication used for the analysis. Using
a more exible specication, we nd no evidence that inserting volume into the conditional
variance function of the model reduces the importance of lagged squared returns in capturing
volatility dynamics. Volume is strongly correlated with contemporaneous return volatility, but
the correlation is driven largely by transitory volatility shocks that have little to do with the
highly persistent component of volatility captured by standard volatility models. Key words: volume-volatility relation, information ow, two-component GARCH, bivariate
mixture models, mixture of distributions hypothesis Comments welcome. We thank Scott Baggett for helping us construct the price and volume datasets from TAQ data and Joel Hasbrouck and James Weston for providing useful insights and
advice regarding TAQ data. Address correspondence to: Chris Kirby, John E. Walker Department
of Economics, Clemson University, P.O. Box 341309, Clemson, SC 29634-1309. Email addresses: jfleming@rice.edu (Je Fleming), cmkirby@clemson.edu (Chris Kirby), ostdiek@rice.edu (Barbara Ostdiek). Rice University and Clemson University Working Paper 12 September 2005 ARCH Eects and Trading Volume ABSTRACT Studies that t volume-augmented GARCH models often nd support for the hypothesis that trading volume explains ARCH eects in daily stock returns. We show that this nding is due to an unrecognized constraint imposed by the GARCH specication used for the analysis. Using a more exible specication, we nd no evidence that inserting volume into the conditional variance function of the model reduces the importance of lagged squared returns in capturing volatility dynamics. Volume is strongly correlated with contemporaneous return volatility, but the correlation is driven largely by transitory volatility shocks that have little to do with the highly persistent component of volatility captured by standard volatility models. ARCH Eects and Trading Volume This paper investigates the degree to which autoregressive conditional heteroscedasticity (ARCH) in stock returns is explained by the dynamics of trading volume. Our investigation is based on a new volume-augmented generalized ARCH (VA-GARCH) model in which return volatility is allowed to contain both short- and long-term components. In contrast to prior studies, we nd no support for the hypothesis that inserting volume into the conditional variance function of the model reduces the importance of lagged squared returns in capturing volatility dynamics. Previous conclusions that trading volume largely explains ARCH eects appear to stem from an unrecognized constraint imposed by the econometric methodology. Like most studies in the area, we use the mixture of distributions hypothesis (MDH) to guide our analysis. In one of the rst studies to use VA-GARCH models in an MDH context, Lamoureux and Lastrapes (1990) report that ARCH eects tend to disappear when contemporaneous trading volume is added to the conditional variance function of a GARCH(1,1) specication. Although subsequent studies report a less dramatic atten- uation of ARCH eects in such models, they generally nd that incorporating trading volume produces a substantial drop in volatility persistence and at least some reduction in the signicance of ARCH eects (see, e.g., Fujihara and Mougoue, 1997; Girma and Mougoue, 2002; Marsh and Wagner, 2003). Fleming, Kirby, and Ostdiek (2006), on the other hand, use linear state-space meth- ods to investigate the MDH. They t a number of MDH-based specications and nd that accounting for the dynamics of trading volume has no impact on the signicance of ARCH eects. Moreover, their analysis indicates that trading volume is primarily related to the nonpersistent component of return volatility. These ndings, which are clearly at odds with the conclusions drawn by Lamoureux and Lastrapes (1990), call into question the robustness of the VA-GARCH methodology. Our paper attempts to identify the source of these conicting results. In general, there are two specication issues that may aect the results obtained using the VA-GARCH methodology. The rst issue is the impact of simultaneity bias on the VA-GARCH coecient estimates. Some researchers, such as Liesenfeld (1998), use this issue to help motivate econometric methods that are much more computationally intensive than tting volume-augmented GARCH models. However, we show that the 1 MDH implies that the impact of simultaneity bias becomes negligible as the number of traders in the market and/or the number of daily information events becomes large. Hence, by focusing on widely held and frequently traded stocks that are often in the news, we can retain the tractability and intuitive appeal of the GARCH framework while minimizing concerns about the impact of simultaneity bias. Since Lamoureux and Lastrapes (1990) also examine actively traded stocks, it is unlikely that simultaneity bias plays a large role in their ndings. The GARCH(1,1) methodology used by Lamoureux and Lastrapes (1990), however, is vulnerable to a second specication problem that has escaped attention in the litera- ture. Although it is natural to consider a GARCH(1,1) model given its success in other applications, a VA-GARCH(1,1) model imposes an important restriction that makes it dicult to interpret the model tting results. Specically, the coecients on lagged vol- ume and lagged squared returns are constrained to decline with the lag length at the same rate. Thus, if volume provides little information about future volatility, it might be necessary to downweight lagged volume (and hence lagged squared returns) to keep the tted volatilities from becoming too noisy. This restriction may explain why ARCH eects tend to vanish when volume is added to the GARCH(1,1) model. We overcome this problem by tting a volume-augmented exponential GARCH (VA- EGARCH) model that allows for both short- and long-term volatility components. Our VA-EGARCH(2,2) model nests the VA-EGARCH(1,1) model as a special case, which allows us to directly assess the impact of relaxing the implicit restriction. We t the model to daily returns on the 20 stocks in the major market index (MMI). We nd no support for the hypothesis that volume explains ARCH eects. The results conrm that volume is strongly correlated with contemporaneous return volatility, but the correlation is driven by transitory shocks to the volatility process. Nothing in the model tting results suggests that volume explains the highly persistent component of volatility that is captured by standard volatility models. We provide further evidence on this issue by examining the relative performance of the EGARCH models in explaining realized volatility. Our approach, which follows An- dersen and Bollerslev (1998), is to regress the realized variances on the tted variances produced by each model. The regressions conrm that the VA-EGARCH(1,1) model pro- vides misleading evidence regarding of the relation between volume dynamics and ARCH eects. The tted variances from the VA-EGARCH(1,1) model produce a much lower 2 R-squared than the tted variances from the basic EGARCH(1,1) model. In contrast, the tted variances from the VA-EGARCH(2,2) model substantially outperform the tted variances from both of these models as well as those from the basic EGARCH(2,2) model. Since the superior performance of the VA-EGARCH(2,2) model is primarily attributable to the undiminished role of ARCH eects, our results suggest that we need to look beyond volume in order to identify the features of the trading process that give rise to ARCH eects in daily stock returns. The remainder of the paper is organized as follows. Section 1 covers general back- ground issues, analyzes the large-market implications of the most widely studied bivariate mixture model, and introduces the two-component EGARCH model that we use for the empirical analysis. Section 2 describes the dataset and reports the model tting results. Section 3 oers some concluding remarks. 1 Background and Methodology Lamoureux and Lastrapes (1990) is arguably the most inuential study of the relation between ARCH eects and trading volume. Using a sample of daily data for 20 U.S. rms, they t a volume-augmented GARCH(1,1) model of the form R t = + h 1/2 t z t , (1) h t = + h t1 + r 2t1 + V t , (2) where R t is the daily stock return, V t is the daily trading volume, z t is an i.i.d. N(0, 1) standardized innovation, and r t = R t is the demeaned return. They nd that they cannot reject = = 0 for 16 of the rms and the parameter estimates for the remaining
four rms suggest much lower levels of volatility persistence than those obtained under the constraint = 0. As a result, they conclude that lagged squared residuals contribute
little if any additional information about the variance of the stock return process after accounting for the rate of information ow, as measured by contemporaneous volume. There are two aspects of the methodology that raise concerns about the robustness of the results. First, it treats volume as exogenous. This can give rise to an undetermined simultaneity bias if R t and V t are jointly determined. Second, it relies on a model that constrains volume eects to decay at the same rate as ARCH eects, i.e., the coecients on V ts and r 2ts1 are both proportional to s for all s > 0. This lack of exibility can aect 3 the coecient estimates. Although simultaneity bias is generally viewed as the primary robustness issue, we argue that the lack of exibility exhibited by the GARCH(1,1) model is likely to be a more serious concern in most applications. 1 The criticism of simultaneity bias in the VA-GARCH methodology arises because most market microstructure models imply that trading volume is endogenous. It may be reasonable, however, to treat V t as exogenous under conditions in which the magnitude of the resulting bias is likely to be small. The lack of exibility in the VA-GARCH(1,1) model, on the other hand, is dicult to overcome. Suppose, for example, that V t is strongly correlated with the volatility of R t , but contains no information beyond that contained in R t about the volatility of R t+s for any s > 0. Obtaining an estimate of close to zero might be evidence that volume largely subsumes ARCH eects, or it could simply indicate that putting large weights on lagged volume (and hence lagged squared returns) makes the tted volatilities too noisy, thereby reducing the likelihood. To avoid this kind of ambiguity, we propose a more exible approach for investigating the extent to which volume explains ARCH eects. We begin by examining the issue of simultaneity bias in the context of bivariate mixture models. By analyzing the asymptotic properties of these models, we identify circumstances under which a strategy of tting VA-GARCH models should pose minimal concerns about simultaneity bias. Once this is established, we show how to overcome the structural constraints of the VA-GARCH(1,1) model without sacricing the tractability of the GARCH methodology. 1.1 Bivariate mixture models Let v t denote detrended trading volume. Much of the recent empirical work on the re- lation between volume and volatility, such as Liesenfeld (1998) and Watanabe (2000), is motivated in large part by Andersens (1996) modied MDH. The modied MDH implies a bivariate mixture model of the form r t = (IJK t ) 1/2 z rt , (3) v t = ( + IJK t ) + ( + IJK t ) 1/2 z vt , (4) 1 The results of Fleming, Kirby, and Ostdiek (2006), which are robust to simultaneity bias, indirectly support this argument. 4 where I is the number of informed (i.e., non-liquidity) traders in the market, J is the
number of information arrivals on the day used as a benchmark, K t is the intensity of information arrivals on day t relative to the benchmark day, z rt is an i.i.d. N(0, 1) standard- ized innovation, and z vt is an i.i.d. standardized innovation distributed independently of z rt+s for all t and s such that v t |K t

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