Stochastic Volatility, Intraday Seasonality, and Market Microstructure Noise

Transcription

1 Stochastic Volatility, Intraday Seasonality, and Market Microstructure Noise Duy Tien Tran March 8, 26 ABSTRACT In this paper, we model the dynamics of intraday stock returns directly, taking into account the dynamics of market microstructure (MM) noise, as the sum of two components: the implicit efficient price and the MM noise term. The models are then applied to high frequency returns of several active versus inactive individual stocks to study many interesting issues. We find that the two factor stochastic volatility models or the closely similar ones are good candidate for modeling the implicit efficient price. At the fifteen minute level, the MM noise is present in the transaction prices. The estimation results are consistent with the intuition that active stocks have smaller cost of trade than those of inactive ones. The higher volatility of thinly traded stocks is the outcome of the higher volatility of the implicit efficient price as well as the MM noise. Both the number of transactions (NT) and the average trade size (Size) are crucial determinants of both volatility factors. Both NT and Size have a larger effect on the erratic volatility factor than that of the persistent one, and this result is robust to whether the stocks are actively or thinly traded. Moreover, both NT and Size play a more significant role in affecting the volatility of the relatively thinly traded stocks than that of the actively traded stocks, and this is true for both (persistent and erratic) volatility factors. Our models can mimic the unconditional dynamics of returns and perform quite well in capturing the crosscorrelation pattern between returns and return volatility at the daily, weekly and monthly levels and need some improvement in capturing the autocorrelation pattern of return volatility. The correlation between the implicit efficient price and the MM noise is positively estimated and does not help in the temporal aggregation process. Keywords: market microstructure noise, two factor stochastic volatility, intraday periodicity, determinants of volatility, temporal aggregation, effective spread, high frequency data, EMM. Economics Department, Duke University, Durham, NC dtt3@duke.edu. We thank members and participants of the Duke Monday Econometrics and Finance Lunch Group for helpful comments and suggestions.

2 I. Introduction A. Related literature Volatility plays an important role in risk measurement and management. One of the leading quantitative risk measurement techniques is the notion of Value-at-Risk (VaR), which is defined as the maximum expected loss of stock returns over a specific time interval at a given confidence level. The specific time interval is conventionally chosen to be ten days for financial banks and related institutions. Nevertheless, as pointed out by Bauwens and Giot (2), active market participants such as day traders, the specialists and market makers are interested in quantifying the stock return risk at a much shorter time period. In other words, they are interested in high frequency (intraday) VaR (see also Beltratti and Morana (999) and Giot(22)). The recent advances in the area of realized volatility (RV) (Andersen and Bollerslev (998), Barndorff-Nielsen and Shephard (22, 24) and Meddahi (22)) provide an attractive means for modeling and measuring volatility as illustrated convincingly in Andersen et al. (24). Realized volatility of a given time period, usually one trading day, is defined as the sum of the squared intraday returns for that given day. Therefore RV is not applicable to quantify high frequency (intraday) return risk. This is particularly true because the contamination by microstructure noise at very high frequency data causes RV to be a biased and inconsistent estimator for the integrated volatility (see e.g. Andreou and Ghysels (22) and Oomen (22)). Probably, the only way to determine intraday stock return risk is to model directly the intraday returns, taking into account the dynamics of microstructure noise. For daily or lower frequency return data, the (Generalized) Autoregressive Conditional Heteroskedasticity models (introduced by Engle (982) and extended in Bollerslev (986)) and Stochastic Volatility (SV) models are widely used for these purposes. It is challenging to find a model that captures most of the features of financial data at all frequency levels, especially at high frequencies. In this paper, we model the dynamics of intraday stock returns, taking into account the dynamics of market microstructure noise.

3 Researchers on financial markets now have access to stock price data at almost all frequency levels thanks to the availability of tick by tick data. When dealing with high frequency financial data, market microstructure noise is inevitable. In market microstructure literature, it is standard to assume that the observed stock price equals the implicit efficient price plus some microstructure noise. Several important structural market microstructure models such as Huang and Stoll (996) and Madhavan, Richarson and Roomans (997) generate the reduced form of this structure (see also Ait-Sahalia, Mykland, and Zhang (24) and the references therein). The efficient price of a stock at time t is defined as the conditional expectation of the end-of-trading value of that stock given all available public information at time t as well as the information that may be inferred from the trades at time t (see also Black (986), Hasbrouck (99) and O hara (995)). Roll (984) labels the difference between the signed observed transaction price and the implicit efficient price as the effective spread or the effective cost of trade. The observed mid quote is usually taken as a proxy for the implicit efficient price by many empirical researchers. Even in this case, the effective spread is still different from half of the bid ask spread because the transaction price can occur inside or outside the bid ask quotes. Because investors look for stock market with high liquidity and the effective spread is one of the intuitive measures of market liquidity, there is a huge literature devoted to research on spread and its components (see e.g. Glosten and Harris (988), Hasbrouck (988), Campbell, Lo, and MacKinlay (997, ch. 3) and Huang and Stoll (997) and the references therein). It is worthy to note that Kyle (985) defines the concept of market liquidity in terms of price resiliency as how small the effective spreads are and how fast they are rectified. The observed price tends to converge to the implicit efficient price faster when the two prices are further away from each other. Also, tight spreads signify that the market is highly liquid and hence impose lower cost to investors. That is, the market for a particular stock is considered as of high quality when the effective spread (cost of trade) does not disperse too much. These concepts of market quality can be used to study regulation impacts or the effectiveness of a given market policy, which are interested by market regulators. For example, it is interested to the NYSE regulators to 2

4 compare market qualities before and after the implementation of price decimalization in the year 2. Recent papers that also are interested in estimating the effective spreads are Bandi and Russell (24a, 24b). Those papers propose a new approach, based on the framework of realized volatility (RV) (see also Andersen and Bollerslev (998), Barndorff-Nielsen and Shephard (22)), to estimate the effective spreads. Those papers are nonparametric and attractive in the sense that they do not make any assumption on a specific expression of the latent price. Instead, we are going to use parametric models. The common assumption about the implicit efficient price is that it follows a random walk process. This assumption implies that implicit returns are independent over time. However, the enormous literature on estimating and forecasting the second moment of returns provides us with convincing evidence supporting the predictability of return volatility. (Generalized) Autoregressive Conditional Heteroskedasticity models (introduced by Engle (982) and extended in Bollerslev (986)) and Stochastic Volatility (SV) models are widely used for these purposes. Like the ARCH type models, stochastic volatility models have been enjoyed a long history of developments and extensions. The first variation of stochastic volatility is proposed by Clark (973) and refined in Tauchen and Pitts (983). Since the work of Hull and White (987), in which they extend the famous Black-Scholes options pricing method to the case where asset prices follow a continuous time SV model, there has been a renewed interest in SV models. For a more detailed description and comprehensive analysis of SV models, readers may refer to Taylor (986), Ghysels, Harvey and Renault (996), Chernov et al. (23), Shephard (24), and Tauchen (24). B. Economic questions and empirical issues Accordingly, in this paper, we model and study the dynamics of the implicit efficient price and the effective spreads using a parametric model, where a stochastic volatility (SV) process is 3

5 used to model the implicit efficient price. More particularly, the logarithm of the efficient price process is assumed to be the solution of an extension (two factors) of the logarithmic stochastic volatility model introduced by Taylor (986) and Scott (987). It is now well established that we need two factors (one factor to account for the persistence of the volatility and a second to capture the fast mean-reverting volatility) to properly model volatility. The short list of papers supports this claim include: Chacko and Viceira (999), Engle and Lee (999), Gallant, Hsu, and Tauchen (999), Alizadeh et al. (22), Bollerslev and Zhou (24), Chernov et al. (24) and Tauchen (24). Therefore, we propose the model that uses a two factor stochastic volatility model and a simple first order autoregressive process to capture the dynamics of the implicit efficient price and the market microstructure noise, respectively. By doing this, we can estimate the effective spread, the intraday variation of the market microstructure noise as well as intraday volatility of the implicit efficient price together. Empirical results from this paper reconfirm that the two factor volatility models do a much better job in capturing the dynamics of returns data than the one factor model. The model provides us with the framework to conveniently answer several interesting questions brought up from the extant empirical research. First, are the estimated results consistent with the intuition that market quality of the active stocks is higher than that of the thinly traded stocks in terms of smaller effective spreads? In other words, does the fact that active stocks are more liquid mean they have smaller cost of trade? Second, we know that the prices of relatively thinly traded stocks are more volatile than those of active stocks (Jones et al. (994), Andersen (996), Gopinath and Krishnamurti (2), Darrat et al. (23) and Huang and Masulis (23)). This feature will also be seen from our summary statistics of the observed data later. The interesting question is that does the higher volatility of thinly traded stocks come from the implicit efficient price or from the market microstructure noise or both? 4

6 From this line of research many scholars also pay attention to the question: what are the determinants of volatility? Jones et al. (994) show that activity as measured by the number of transactions is the source of generating volatility and the size of trades does not play an important role to determine volatility. Using high frequency data, Xu and Wu (999) shows that, contrary to Jones et al. s finding, average trade size has significant information content for stock volatility. As argued above, we need two factors to correctly model volatility. The justifiable and interesting questions surface: do the number of transactions and the size of trades still significantly determine each component of volatility? Moreover, how important are the contribution of these source of volatility to each component? We are going to apply the models to fifteen minute individual stock return data in an attempt to provide some understandings on these issues and also study the effects of temporal aggregation. It is appealing to see how the models perform at the daily, weekly and monthly frequency levels. We are going to study the unconditional distributions and the conditional dynamics of the returns. The dynamics include the autocorrelation of realized volatility and the crosscorrelations pattern between returns and return volatility. This study of temporal aggregation is important because ones might be interested in studying distributions and dynamics of aggregate return data but usually at this aggregated level the data are not available or not long enough. Put it simple, empirical researchers commonly apply their models to daily or lower frequency data, which contain less information than higher frequency data, may head to an imprecise view of the true models, leading to potential errors in decision making. The remainder of this paper is structured as follows. Section II sets up the model. The data and their necessary adjustment method are outlined in Section III, and the estimation method is briefly mentioned in Section IV. Section V then presents our empirical results and their implications. Finally, Section VI is the conclusion and the ongoing and future works are laid out here as well. 5

7 II. Model Stochastic volatility models have been used extensively to estimate and forecast asset return volatility. The current literature has indicated that two factor stochastic volatility models (one highly persistent factor and one quickly mean-reverting factor) capture the dynamics of stock returns data well. Alizadeh et al. (22) used ranged-based estimation to indicate the existence of two volatility factors. Specifically, the paper employed daily data on the range of currency futures contracts as the proxy for volatility and estimate stochastic volatility models using quasi-maximum likelihood method. Chernov et al. (24) applied the Efficient Method of Moments (EMM) to estimate and compare several alternative stochastic volatility models using the long data set of daily returns on the Dow Jones industrial average (DJIA) index. They argue that the two factor volatility model can separate the tail thickness feature from the volatility persistence effect in the return data dynamics. Chacko and Viceira (23) developed the spectral GMM estimation based on the characteristic function and apply it to several financial data set of different frequency: daily, weekly and monthly. They are also in favor of two factors in stochastic volatility models. Recently, by studying stochastic volatility models in the equilibrium framework, Tauchen (24) also develops a two-factor volatility model. The fundamental economic structure by Tauchen (24) assumes the recursive utility function introduced by Epstein and Zin (99) and Weil (989), as have been done by Bansal, Khatchatrian, and Yaron (23), Bansal and Yaron (24), with a more general form of the dynamics of the growth rate of consumption than that of those papers. Using this structure and applying the log-linearization methods of Campbell and Shiller (988), Tauchen( 24) finds that the stock market volatility indeed implies two factor structure. Other earlier papers supporting the appeal of two factor stochastic volatility models for stock returns are Engle and Lee (999) and Gallant, Hsu, and Tauchen (999). The above papers deal with daily or lower frequency financial data. The dynamics of high frequency financial data are certainly more complicated than the low frequency one. High frequency data are inevitably contaminated by market microstructure noise. We need more 6

8 complex models for those extra characteristics of the data. In this paper we use an extension (two factors) of the logarithmic stochastic volatility model proposed by Taylor (986) and Scott (987) to model the dynamics of the implicit efficient price and we add to it another component to account for the market microstructure noise. Noting yet again that the loglinear two-factor volatility model can capture the dynamics of daily asset returns quite well, as have been shown by Chernov et al. (23). Importantly, we need the model to be consistent with the current empirical literature of market microstructure noise, where it is standard to assume that the observed stock price equals the implicit efficient price plus some microstructure noise. The implicit efficient price is defined as the conditional expectation of the end-of-trading value of the stock, given all available information, including the information possibly inferred from trades. We assume that there is an efficient log price process m t so that at any transaction time the observed asset price equals its corresponding implicit efficient price plus the market microstructure noise term. Let y t = log(p t ) be the logarithm of the observed asset price. Then, the observed continuouslycompounded return r t is the difference between y t and y t, i.e. r t = (y t y t ). The dynamics of observed asset price are: y t m t + u t, () where u t is the stationary process with unconditional mean of zero to capture market microstructure noise or the effective spread. The implicit efficient price is modeled as the nonstationary process. Again, the unconditional expectation of the absolute value of u t or the effective spread is the natural measure of market quality, which is an interested variable to active market participants. From this literature, a basic model for the dynamics of m t are described by equations (2), (3), and (4): ( V,t m t m t = β exp + V ) 2,t ε t (2) 2 2 7

9 V,t = α V,t + β ε t (3) V 2,t = α 2 V 2,t + β 2 ε 2t. (4) Andersen et al. (23) and Barndorff-Nielsen and Sphephard (22) assume the efficient prices follow continuous semi-martingales (a local martingale plus a finite variation drift term) to estimate their quadratic variation. We choose to exclude the instantaneous expected rate of asset return from (2). This practice is common when working with high frequency data (e.g. see Bollerslev and Zhou, 22), as first pointed out by Merton (98) that when estimating the variance of asset return it is more accurate when leaving out the drift part. Put the other way, the variance part is mathematically more significant than the drift part (see also Ait- Sahalia, Mykland and Zhang, 24). In the next sections, we will apply our models to several individual stocks, which have very small sample means of returns: the average is about.2 percent. It justifies the exclusion of the drift term in (2). Equations (3) and (4) are the standard first-order autoregressive processes, capturing the dynamics of two volatility factors V,t and V 2,t, respectively. The parameters α and α 2 control the speed of mean reversion of these two volatility factors. When the rates of mean-reversion are small, V,t and V 2,t have a tendency to stay above (or below) its long run mean of zero for longer period (persistent volatility). The volatility factors V,t and V 2,t are stationary when α and α 2 are greater than minus one and less than one. Another concept that is directly related with the rate of mean reversion is half-life. The half-lives for two volatility factors equal to ln(2)/(α ) and ln(2)/(α 2 ), respectively. The half-life of volatility is the time it takes to move back halfway to the long run unconditional volatility level when there is a deviation from it. Also in the model, the positive parameters β and β 2 determine the volatility of the stochastic volatility V,t and V 2,t, respectively. Intuitively, it is expected that the volatility of the persistent volatility factor is less than that of the fast mean-reverting factor. Finally, the positive parameter β captures the long run unconditional mean of the return volatility factor. 8

10 We use two volatility factors in the dynamics of (2) as the general case. For comparison purpose, the one factor volatility model will also be estimated and tested. Equation (2) can be written in a shorter version as m t m t = σ t ε t. This equation clearly shows that the implicit return process m t m t is the martingale difference process or asset prices are unpredictable given all the currently available information. This is consistent with the definition that the efficient price of a stock at time t is the conditional expectation of the end-of-trading value of that stock given all available public information at time t as well as the information that may be inferred from the trades at time t. The error terms ε t and ε 2t are assumed independent, normally distributed with mean zero and variance one. The error term ε t also follows the standard normal distribution and is correlated with both ε t and ε 2t according to: corr(ε t,ε t ) = ρ (5) corr(ε t,ε 2t ) = ρ 2 (6) These two correlation coefficients, ρ and ρ, capture the well known leverage effect originated by Black (976). That is, when stock prices decrease or returns are negative, the firm becomes more risky due to an increase in its debt-equity ratio, leading to an increase in its stock volatility (see also Christie (982) and Nelson (99)). Meddahi and Renault(24) shows that for a general class of SV model in order to have a leverage effect at the temporal aggregation level it is necessary to model the leverage effect at the high frequency level. We are going to see that the leverage effect at the daily, weekly and monthly level is present in the observed return data. We will present the equation of the leverage effect coefficient later; here we want to focus on describing the dynamics of the microstructure noise, u t, to complete the specification of the model. 9

11 The simplest case is that we assume the noise term u t follows an independent normal distribution with mean zero and variance η 2 2. Without asymmetric information, this parameter captures the unrelated-information market microstructure noise such as price discreteness. The fact that, at the transaction frequency level, order flows may be highly correlated (the buy (sell) order often follows by a buy (sell) order as traders may split a large order to many small ones) suggests that the noise term at intraday frequency may also be autocorrelated. Therefore, we model the noise term u t as the first order autoregressive process as in (7). The noise u t is a stationary process when the absolute value of η is less than one. This parameter is associated with the resiliency of the stock price (Kyle (985) s concept of market liquidity) and it controls the speed of convergence of the noise term to its long run mean of zero. The parameter η 2 is the standard deviation of the standard normal distribution error term ε 3t. We assume that ε 3t is independent from the error terms ε t and ε 2t in the volatility equations. As noted by Ait-Sahalia, Mykland and Zhang (24), Hansen and Lunde(26) and others, if the microstructure noise arises from asymmetric information, then the error term ε 3t in (7) is likely to be correlated with the error term ε t of the implicit efficient price equation (2). u t = η u t + η 2 ε 3t (7) corr(ε t,ε 3t ) = ρ 3 (8) This completes the specification of the model. For the purpose of clarity, we combine (), (2), (5), (6), (7), and (8) to present the dynamics of the observed return as follows: Main model: ( V,t y t y t β exp 2 ε t = + V ) 2,t ε t + (η )u t + η 2 ε 3t (9) 2 ρ 2 ρ2 2 ρ2 3 ϕ t + ρ ε t + ρ 2 ε 2t + ρ 3 ε 3t, ()

12 V,t = α V,t + β ε t () V 2,t = α 2 V 2,t + β 2 ε 2t. (2) where ϕ t is an i.i.d. standard normal random variable and the two equations governing the dynamics of the volatility factors are re-written here for convenience. We use the following names: SV2N, SV2N, SV2AutoN, and SV2N2 to label the models that we are going to estimate and draw implications. Models SV2N, SV2AutoN, and SV2N2 are all two factor SV models with different specification of the noise terms. More particularly they are governed by (9), (),(), and (2) with the following restrictions: the restrictions on SV2N are ρ 3 =, η = and η 2 =, on SV2N are ρ 3 = and η =, on SV2AutoN is ρ 3 = and on SV2N2 is η =. In the next section, we describe the data used in this paper and the necessary adjustment method for the data. Then in section IV, we briefly describe the EMM methods used to estimate these models. The ease of the EMM methods is that the above structural complicated models can be estimated efficiently once we can simulate from them and the simulation task can be carried out easily from equations (9), (),(), and (2). III. Data and Adjustments A. Data To answer the mentioned questions in part I, we apply the models to the data of two groups of stocks: one represents the active stocks and other relatively thinly traded ones. The stocks chosen as active stocks are International Business Machines (IBM), WalMart (WMT) and COCA COLA (KO). These are the stocks with the highest market capitalizations (more than billions) in their corresponding industries: Computer Systems, Variety Stores and Food Industries. And in these same industries, we arbitrarily choose three stocks with much lower mar-

13 ket capitalizations (at about ten billions): Lexmark(LXK), Dollar General(DG) and Campbell Soup(CPB) to represent the relatively inactive stocks. Relatively thinly traded stocks are stocks with low market capitalizations and are traded infrequently and in low volumes. We empirically study the model by applying it to the price data for IBM, WMT, KO, LXK, DG and CPB from the New York Stock Exchange (NYSE). From the TAQ database we obtain stock transaction prices from 24 Jun, 997 to 3 DEC, 23 during official trading hours of the NYSE (9:3 AM to 4: PM). We choose 24-Jun-97, the day when the process of listing the stock prices at NYSE as one sixteenth for one tick instead of one eighth completes, as the starting day of the data. This choice is influenced by the fact that LXK starts its listing at the NYSE around the beginning of 996 and also because when traded at the tick of one eighth there are many prices occurs at the same value, which requires a more complicated method than the proposed method here to adjust the U-shape pattern in asset return volatility. Since IBM is the most active stocks here, we will also try our models to the transaction IBM price series dated back to 993. With fifteen minute frequency data, the ten year period data series of IBM, from 993 to 23, contains almost 8, observations. This choice of frequency is high enough so that the data are contaminated by microstructure noise, which we want to study their dynamics in our model, and is low enough to be empirically manageable. The fifteen minute frequency is also conventionally chosen by many authors. B. Active versus inactive stocks Table I illustrates the difference between active stocks and relatively inactive ones in terms of capitalizations, transaction durations, numbers of trades and trading volume. The pattern of the relationships among the variables is clear. The stocks with higher market values are traded more actively in term of shorter transaction durations, having higher average numbers of trades per day and higher level of trading volume (measured as the number of shares traded per day or in dollars value). For the period from 24 June997 to 3 December 23 the 2

14 average trading duration of the relatively inactive stocks LXK, DG and CPB is 3.7 seconds and is about three times more than that of IBM, WMT and KO with.49 seconds. The converse is true for the number of transactions per day. The dollar values of trading volume (and share volumes) for IBM, WMT and KO are much higher than those of the less actively traded stocks of LXK, DG and CPB. Specifically, the average volume traded per day for the active stocks equals to 68 millions of dollars and is about ten times more than that of the relatively inactive ones. Both components of the observed logarithm of the prices are latent and going to be estimated. The magnitude of the effective spread is very tiny compared to that of the implicit efficient price. This characteristic can be seen from table II, where the observed effective spread and the quote spread are reported. The first two columns are the effective spreads and the quote spreads of the six cited stocks in unit of dollars. The effective spreads are ranged from about 2 cents for DG to about 5 cents for LXK. The quotes spreads of these stocks are also very much in this range. There is no clear pattern to distinguish between the active versus the relatively inactive ones in term of the dollar spreads. However, when expressing these measures in unit of percent as in the third and fourth columns, the apparent characteristic is that the spreads (cost of trade) of the thinly traded stocks are larger than those of the active ones. The average of the effective spread and the quote spread of the three thinly traded stocks (LXK, CPB and DG) are.93 and.35 percents, meanwhile the corresponding values of IBM, WMT and KO are.65 and.68 percents, respectively. This trait is one of main appeals for investors to invest in active stocks rather than the thinly traded stocks. The original data set contains about ten millions observations for the active stocks and about one and a half million observations for the relatively thinly traded stocks from the period of June 997 to Dec 23. We then filter out prices that are not traded at the New York Stock Exchange and the trading days that are not traded for the full day. These public holidays (24 days for the 7 year period) are listed in NYSE website. For those transactions that occur at the same time, we take the average of those prices. Dacorogna et al. (2) describe two 3

15 alternative methods to convert the transaction prices into fifteen minute frequency; they are the linear interpolation and previous interpolation method. The former method is popular in the literature of realized volatility because it mitigates the discreteness microstructure noises that are widespread in high frequency data. However, in our model we want to deal with market microstructure directly so we choose the previous interpolation method. The idea is that the price at a particular fifteen minute mark takes the transaction price whose time stamp is immediately before (previous) the mark. Figure 5 is the plot of the differences between the fifteen minute marks and the time stamps of the immediate previous transaction prices. Almost all of the difference durations are less than one minute, indicating that the market is very active and the interpolation is justified. Figure is the plot of fifteen minute frequency IBM price level. We need to adjust the price data accordingly to the splits before computing the returns. For the concerned stocks, the splitting days and ratios (in the brackets) are IBM: 28-May-97 [2:], 27-May-99 [2:], WMT: 2-Apr-99 [2:], LXK: -Jun-99 [2:] and DG : 23-Sep-97 [5:4], 24-Mar-98 [5:4], 22-Sep-98 [5:4], 25-May-99 [5:4], 23-May- [5:4]. These prices are then used to construct the unadjusted returns series in percent unit. There are observations in the final data set of IBM or 2737 days and of the remaining stocks or 67 days with 27 fifteen minute frequency returns each day. We present their summary statistics in Table VI. The returns are obviously not normally distributed as the skewness is much less than zero and the raw kurtosis much greater than three. Noting that the fifteen minute standard deviation for the active stocks are.42,.45,.38 percent for IBM, KO and WMT. These are less then the fifteen minute standard deviation of the thinly traded stocks LXK, DG and CPB of.66,.63 and.43 percent. This implies that the active stocks are less volatile than the inactive ones. A fifteen minute standard deviation of.42 percent is equivalent to about 35 percent at the annual frequency, assuming there are 25 trading days per year. 4

16 C. Intraday periodicity One of the well known regularity of high frequency return data is the U-shape or J-shape pattern in asset return volatility (see e.g. McInish and Wood (99), Lockwood and Linn (99) and Andersen and Bollerslev (997)). There are several ways to deal with this intraday periodicity pattern in return volatility. Dacorogna et al. (993) use a different time-scale method. The idea is to expand (scale up) times having high mean volatility and contract (scale down) times having low volatility such that seasonal patterns almost fade out in the new time-scale. Other way is to remove the seasonal components of the data before the adjusted series can be studied by standard econometric models. Andersen and Bollerslev (997) is a typical example. In their paper, they show how to use the Flexible Fourier functional form developed by Gallant (98) to model and remove the intraday periodic volatility components before using a standard MA()-GARCH(,) model to capture the conditional variance dynamics of the adjusted return series. On the other hand, intraday periodicity can also be modeled explicitly. Baillie and Bollerslev (99) seem to be among the first to incorporate seasonal components into the standard GARCH model. In that paper, they augment the standard GARCH(,) model with twenty four hourly dummy variables, a vacation dummy and a seasonal ARCH dummy variable to study the dynamics of hourly foreign exchange rates data. In this paper, we use the simple OLS regression methods put forward by Gallant, Rossi and Tauchen (992) to adjust the returns. Following Andersen and Bollerslev (997), we use the sine and cosine functional form initiated by Gallant (98) to capture the intraday periodic volatility components. More particularly, we estimate the following two equations using OLS: r = x β + e (3) log(e 2 ) = x γ + e 2. (4) 5

17 Then, the adjusted returns are computed by: r ad j = a + b ( eˆ /exp ( x γ/2 )), (5) where a and b are chosen so that the sample means and variances of r and r ad j are the same. The components of x in the mean equation are: a constant, a dummy variable for the day of the week, a dummy for the overnight return, a dummy for the first interval of the day, a dummy for the last interval of the day, and a holiday dummy. The components of x in the variance equation are the variables from the mean equation plus the intraday seasonal component: n n 2 P ( µ j + µ 2 j + N N 2 γ p j cos 2pπt p= N + δ p j sin 2pπt ), (6) N where N is the number of returns per day, N = (N + )/2, and N 2 = (N + ) (N + 2)/6. Andersen and Bollerslev (997) show that this seasonal component is able to capture the deterministic U-shape pattern in volatility. The OLS estimates are presented in Table III, Table IV and Table V. In the mean equation of data adjustment, estimates for the parameters of dummy variable for the day of the week are statistically insignificant so these variables are left out. The sign of the coefficients of the dummy for the first interval of the day and the holiday dummy are indecisive and many are not significant whereas the coefficients of the dummy for the last interval of the day are mostly positive with many of them are statistically insignificant. Noting that the R 2 obtained from these mean equation estimations are all close to zero (less than.5 percent) implying the low explanation power of these dummy variables on the variation of returns. The intraday deterministic component in volatility is captured by the combination of (6) and the two dummy variables for the first interval and the last interval of the day. Importantly, the estimates for µ 2, the coefficient of the quadratic form, are all positive and significant as expected to generate the U-shape pattern in the volatility. The R 2 obtained from the variance equation OLS estimation are from three to six percent, indicating that only about five percent 6

18 of the variation in volatility is explained by the intraday deterministic component. The coefficients of the holiday dummy in the variance equation are mostly negative as expected that on average return volatility before the holiday is lower than the usual days. We present the plot of the unadjusted return and absolute unadjusted return in Figure 2 and the corresponding graph for adjusted series in Figure 3. Figure 4 depicts the intraday seasonality in volatility and the top graph clearly shows the well-documented inverse J-shape pattern in volatility. The bottom graph shows that the adjustment method indeed cleans out the intraday deterministic component in volatility. Finally, we present the adjusted return s summary statistics in Table VII. By construction, the adjustment method preserves the mean and variance of the data to be the same. The method change higher moments of returns such skewness and kurtosis in the way that the adjusted data are less asymmetric and less heavy tailed but still clearly not normally distributed, implying that the intraday periodicity accounts partly for the non-normality of the data. IV. Estimation Methods In general, it is difficult to estimate SV models because the conditional probability density function of the observed variables is not available in closed form. The Efficient Method of Moments (EMM) developed by Gallant & Tauchen (996, 22) is appropriate here because for the applicability of the EMM methods it only requires that we can simulate the observed variables from the structural model. Chernov et al. (23) also mention other advantages of using EMM, including () formal statistical tests of a model fit, (2) formal diagnostics of model inadequacies and most importantly (3) nonnested specifications can be compared in a meaningful way. The EMM method is briefly described here, the details can be found in the original paper. The notations used here are taken from Gallant and Tauchen (22) for convenience. Let 7

19 {y t } t= n RM be a stationary time series of interest. Gallant and Tauchen (996) assume that the conditional density of {y t } t= n RM are completely defined by a structural density p(y L,...,y,y /ρ). We are interested in estimating the unknown parameter vector ρ and testing the assumption of correct model specification. In many occasions, especially when working with continuous time model, the analytic expression for p(y L,...,y,y /ρ) is difficult (even impossible) to obtained in closed form and the usual maximum likelihood estimation (MLE) is implausible. Among many methods have been proposed to overcome this problem the efficient method of moments (EMM) by Gallant and Tauchen (996) is attractive for its closed form expression of asymptotic properties of the estimate of ρ, which are derived directly from the well known asymptotics of generalized method of moments (GMM) estimator proposed by Hansen (982). Gallant and Tauchen (996) show that EMM estimator is asymptotically efficient under mild condition. Unlike GMM, EMM provides a systematic way of choosing the moment conditions used in the estimation process. EMM method exploits the fact that an expectation of the form E ρ (g) = R... R g(y L,...,y,y )p(y L,...,y,y /ρ)dy L,...,y,y can be computed by simulation for a given value of ρ and a random function g(.). That is, for a given and a reasonable large value of N one can generate the simulation {ŷ t } t= N so that E ρ(g) is closely approximated by E ρ (g) = N N t= g(ŷ t L,...,ŷ t,ŷ t ). EMM method allocates function g(.) as the derivative of logarithm of a density function f (y t /x t,θ) of a specified auxiliary model,where x t = ( y t L,...,y t 2 t ),y is the lag state vector, and θ is a vector of parameters. It is shown in Gallant and Tauchen (996) that the EMM estimator is asymptotically efficient provided that the auxiliary model is consistent. The auxiliary model should have a tractable form and is called the scored generator. Gallant, Tauchen, and their co-workers have consistently promoted the use of semiparametric (SNP) model developed in Gallant and Nychka (987) and Gallant and Tauchen (989) as the score generator process. The SNP method will be briefly discussed later on. 8

20 After projecting the data on the auxiliary model the score vector, θ log f (y t/x t,θ), enters the minimum chi-squared criterion of the EMM estimator as moment conditions and the classical GMM is carried out. More particularly, let {y t } n t= RM be the observed data of the time series of interest and θ n the estimate of θ obtained by maximizing the log likelihood function of the score generator or by solving its first order condition. Define m(ρ,θ) = E ρ θ log f (y t/x t,θ) as an expectation of the derivative of logarithm of a density function of the score generator with respect to the structural model. Using simulation data {ŷ t } t= N and the estimate value θ n this moment vector is computed as m(ρ, θ n ) = N N t= θ log f (ŷ t/ x t, θ n ). Then the EMM estimator ρ n is obtained by minimizing: m (ρ, θ n )(Ĩ n ) m(ρ, θ n ) (7) [ ][ where, Ĩ n = n n t= θ log f (ỹ t/ x t, θ n ) θ log f (ỹ t/ x t, θ n )]. The estimator Ĩ n is appropriate when the score generator is a good statistical approximation to the data generating process. If this is not the case the heteroskedasticity and autocorrelation consistent (HAC) estimator of the variance-covariance matrix should be used. It is of the form Ĩ n = [ ][ n n t= n τ= w tτ θ log f (ỹ t/ x t, θ n ), θ log f (ỹ τ/ x τ, θ n )] where the weights wtτ are chosen to ensure positivity and consistency. Further details can be found in Andrews (99) and Gallant and Tauchen (996). As mentioned earlier the estimate ρ n is a consistent estimate of the true value ρ and asymptotically normal. In order to test the null that p(y L,...,y,y /ρ) is the correct model we use the common χ 2 test L = n m ( ρ, θ n )(Ĩ n ) m( ρ, θ n ) D χ 2 dim(θ) dim(ρ). This test is able to detect the misspecification of structural model if we use data-based SNP model as our generator score. The discussion is in Tauchen (997) and Aguires-Torres (2). In case of rejecting the structural specification the t-ratio T n = Sn n m( ρ, θ n ), where S n is the corresponding diagonal elements of the variance-covariance estimate of ρ n, can be used to detect where the structural fails to fit. Tauchen (998) emphasizes another important factor when choosing an auxiliary model: it is important that the auxiliary model is stable. Some practical ways to make SNP 9

21 more stable, including centered and scaled transformation of the original data and logistic or spline transformation of the lagged data series x t, are discussed in SNP user s guide manual. We use the C++ version of the EMM package (see Gallant and Tauchen, 25) to estimate the model. The C++ version of EMM methods avoids the classical hill climbing optimization steps of the Fortran version, owing to the use of MCMC chain for the parameter estimation method. The idea is based on the paper by Chernozhukov and Hong (23), where many other useful applications are presented. Recall that the EMM estimator ρ n is obtained by minimizing: s n (ρ) = m (ρ, θ n )(Ĩ n ) m(ρ, θ n ), or equivalent maximizing the likelihood exp( ns n (ρ)). This maximization step is carried out using the MCMC method (Chernozhukov and Hong (23), Gallant and Tauchen (24)). For a given value of ρ, its next proposal value (the candidate) is drawn using the random walk process (other proposal process, such as independent chain, can be use here). The new draw of ρ is then accepted or rejected by comparing its likelihood with that of the old ρ. To put it simple, the greater the ratio of the likelihood of the new draw with that of the old value, the higher the probability of accepting that new draw is. At the end we obtain the whole MCMC chain of parameter ρ. The parameter value that is corresponding to the mode of the objective function is the EMM estimate of ρ. V. Results We choose the SNP model according to the BIC criterion. The SNP models for all stocks are fully nonlinear nonparametric models with GARCH(2,) as the leading term with the innovation distribution depends on past values of returns and the shape of the distribution of observed returns deviates from the Gaussian density. Table VIII reports the estimated results for the structural model SV2N, where the MM noise is in its simplest form of independently normally distribution with standard deviation of η 2, for all six stocks. Figure 6 shows the 2

22 MCMC chains of the parameters of model SV2N along with the values of the objective function π. The MCMC chain mixes well and looks stationary. The estimated values of the autoregressive coefficients in all two factor volatility models suggest that one of the factors is highly persistent and the other is fast mean-reverting. Specifically, to have a sense of quantitative measures, the half-lives of the persistent volatility factor of these stocks are about two, three weeks up to three months and the half-lives of the fast mean-reversion volatility factor are about thirty minutes to one hour. Besides, comparing the estimates of β and β 2 for all stocks suggests that the standard deviation of the persistent volatility factor is much smaller than that of the fast mean-reversion factor; this is an expected result. The leverage effect coefficient is defined as the correlation between the stock return and the change in volatility. From the estimates in Table VIII, we can determine the leverage effect coefficient: ( corr m t m t, V,t V,t + V ) 2,t V 2,t 2 2 = ρ β + ρ 2 β 2 β 2 + β2 2 (8) (see also Chernov et al. (23) for the continuous time version). The leverage coefficients for IBM, KO, WMT, LXK, CPB and DG are.8,.32,.25,.73,.63 and.94, respectively. For individual stocks, these estimates of the leverage effect are all reasonably negative and quite insignificant in economic interpretation. This is consistent with the empirical result that the leverage effect is present in the stock market index rather than in individual stocks (Braun et al. (995)). Playing a part in capturing the leverage effect, the signs and magnitudes of two correlation coefficients, ρ and ρ are robustly estimated acorss all stocks. The correlation between the persistent volatility factor and the efficient price are significantly negative both in terms of economic and statistical senses across the stocks, whereas, the correlation between the erratic volatility component and the efficient price are all close to zero. This result is consistent with Durham (24), but fairly different from Chernov 2

23 et al. (23) who found that the magnitudes of both correlations are ralatively the same and statistically as well as economically significant. A. The effective spread or the cost of trade The estimates of η 2, the standard deviation of the market microstructure noise, for IBM, KO, WMT, LXK, CPB and DG are.649,.577,.628,.523,.932 and.56, respectively. These estimates imply that the effective spreads (cost of trade), E( η 2 ), are.732,.73,.655,.59,.5 and.83 percent, respectively. Comparing these values with the observed effective and quote spreads in table II, we see that they are in the same magnitudes. This is one of the successful aspects of the model, indicating that the two factor SV models or the similar ones are good candidates for the implicit efficient price. Bandi and Russell (24c) s estimate of the effective spread of S&P stocks over the month of February 22 is about.96 percent. The effective spread of LXK is in the same magnitude with those of the actively traded stocks of IBM, KO and WMT, and these values are much lower (almost half) than those of the other two relatively thinly traded stocks of CPB and DG. This is consistent with the intuition that active stocks have smaller cost of trade or narrower effective spreads than those inactive stocks. This is one of the reasons that investors prefer the active stocks than inactive ones. The variance of the market microstructure noise return accounts for about four to eleven percent of the variance of observed return. More particularly, for LXK the ratio is about four percent, and about eleven percent for IBM, and the average for these six stocks equals eight percent. For active stocks the variation of the implicit return and the market microstructure noise return are.99 and.77, respectively, whereas the values for the relatively inactive stocks are.676 and.5. The estimation results imply that the higher volatility of thinly traded stocks is the outcome of the higher volatility of the implicit efficient price as well as the market microstructure noise. 22

24 B. Temporal aggregation characteristic Now we are going to see how do the models behave at the daily, weekly and monthly frequency levels? We are going to study the unconditional distributions and the conditional dynamics of the returns. The dynamics include the autocorrelation of realized volatility and the crosscorrelations pattern between returns and return volatility. If the model represents the true dynamics of the data at the fifteen minute frequency, we believe that the dynamics of the daily, weekly, and monthly returns aggregated from the fifteen minute simulated returns should be resemble to those of the observed data. Meddahi and Renault (24) discuss some theoretical aspects of temporal aggregation of volatility models, and Bollerslev et al. (25) derive an aggregation formula for the correlations between the squared daily returns and future and past daily returns as of the underlying high-frequency correlations. From the top to the bottom of all figures from 7 through 8, we present six graphs corresponding to data of six stocks simulated from the SV2N model. Figures 7 through show the quantile-quantile plot of the observed and simulated returns at fifteen minute, daily, weekly and monthly frequencies, respectively. We see that the structural model SV2N is able to capture the unconditional distribution of the observed data at the daily, weekly and monthly levels despite some mismatch at the fifteen minute level. Given this result, it would not be surprising to see the same behavior in the quantile-quantile plot of the absolute returns (these figures are omitted to save space). We plot the pattern of the autocorrelations of observed and simulated absolute returns, realized volatility at various frequencies in Figures 2 through 5. At the fifteen minute level, the model is doing well in capturing the autocorrelations of the absolute of returns for KO, DG and CPB, and they deviate from the same statistical aspects of the observed data at up to fifty fifteen minute lags for LXK, IBM and WMT. At the daily level, the model is doing well at the first few lags and not as good as before for the longer lags. For the weekly and monthly level, we see that the autocorrelations of the realized volatility of the simulated return data obtained from model SV2N deviate from the same statistical aspects of the observed data. 23