Why It Is OK to Use the HAR-RV(1,5,21) Model

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1 Why It Is OK to Use the HAR-RV(1,5,21) Model Mihaela Craioveanu University of Central Missouri Eric Hillebrand Aarhus University August 31, 2012 Abstract The lag structure (1,5,21) is most commonly used for the HAR-RV model for realized volatility (Corsi 2009), where the terms are thought to represent a daily, a weekly, and a monthly time scale. The aggregation of the three scales approximates long memory. We explore flexible lag selection for the model on realized volatility constructed from tick-level data of the thirty constituting stocks of the Dow Jones Industrial Average between 1995 and The computational costs for flexible lag selection are substantial, and we use a parallel computing environment. We find that flexible lags do not improve in-sample or out-of-sample fit. Our results therefore confirm the standard practice in a large-scale data application. Corresponding author. 307G Dockery, Warrensburg, MO 64093, phone , fax craioveanu@ucmo.edu. Fuglesangs Allé 4, building 1326, room 225, 8210, Aarhus V, Denmark, ehillebrand@creates.au.dk. Portions of this research were conducted with high performance computational resources provided by Louisiana State University ( 1

2 1 Introduction We consider the problem of modeling and forecasting realized volatility. The objective of this study is to explore flexible lag structures in the context of time series models for realized volatility. We consider a linear and a log-linear specification of the HAR-RV model (Corsi, 2009) that can approximate long memory by including lags of realized volatility aggregated over different time horizons. The model typically employed in the literature considers daily, weekly, and monthly time scales. Allowing for a flexible lag structure is problematic in practice since a flexible lag structure is associated with high computational costs. Given this computational intensity, we set up the estimation procedure on a parallel computing framework. The contribution of this paper is to show that for a tick-level data set of the 30 constituting stocks of the Dow Jones Industrial Average, a lag structure of daily, weekly, and monthly time scales has a sufficiently good forecasting performance, eliminating the need to explore other lag specifications. A large body of literature has been developed to model and forecast volatility. Several studies in the literature document the phenomenon of volatility clustering and high persistence in volatility. Several types of models have been proposed to account for volatility clustering: the ARCH/GARCH class (Engle, 1982, Bollerslev, 1986), stochastic volatility models (Taylor, 1986, Hull and White, 1987, 1988, Harvey, 1998), and long memory models (Granger and Joyeux, 1980, Baillie et al., 1996). In all these models volatility is unobservable and modeled as latent. With increasing availability of high-frequency intra-day data, realized volatility was proposed by Andersen and Bollerslev (1998) as an alternative method of volatility estimation. In essence, realized volatility is the square root of the sum of intra-day squared returns (i.e. realized variance), which is used as a proxy for daily integrated variance. After the introduction of realized volatility the literature has evolved in two directions: how to estimate realized volatility and how to model it. The distribution of realized volatility is addressed by several studies (Andersen et al., 2001a, 2001b). Realized volatility data exhibit all the stylized facts established earlier in the case of latent model specifications, in particular high persistence. To capture this long memory property, some authors proposed ARFIMA specifications (Andersen et al. 2000b, 2003, Martens et al., 2004). An alternative to ARFIMA are models that approximate long memory by aggregation. The aggregation of different time scales induces long memory (Granger, 1980, LeBaron, 2001). A commonly employed model that uses this result is the HAR-RV model(heterogeneous AutoRegressive model of Realized Volatility) proposed by Corsi (2009). The model builds on the HARCH specification proposed by Müller et. al (1997). Volatility is modeled as a sum of different short memory processes at different time horizons: daily, weekly, and monthly. There are several studies that either employ similar specifications or modify the original model to capture various stylized facts about realized volatility (Andersen et al., 2007, Andersen et al., 2009, Corsi et al., 2008, Bollerslev et al. 2009, Barndorff-Nielsen and Shephard, 2006, McAleer and Medeiros, 2008, Hillebrand and Medeiros, 2009). While some studies model realized volatility (Corsi, 2009, 2

3 Corsi et al., 2008, Andersen et al., 2007), others specify models based on log realized volatility or realized variance (Andersen et al., 2007, Andersen et al., 2009, and Bollerslev et al., 2009). This paper investigates the forecast implications of a flexible lag structure in HAR-RV models beyond the standard specification of day, week, and month. We apply our methodology to thirty Dow Jones Industrial Average stocks and determine the optimal lag combination for each of the stocks and each of the models considered. Tick-by-tick transaction data are obtained from the NYSE TAQ dataset and cover the period between January 3, 1995 to July 31, We look at two different numbers of merit when determining the optimal lag specification: maximum likelihood (in-sample fit) and minimum mean-squared error of the one-day ahead volatility forecast (out-of-sample fit) on a forecast sample. Looking at both in-sample and out-of-sample fit, we find that for our data set and stocks considered, the HAR-RV model with daily, weekly, and monthly time scales is not significantly outperformed by models with a flexible lag structure or models with fewer lags. The outline of the rest of the paper is as follows. Section 2 describes the different models considered. The methodology used in identifying the optimal lag structure is presented in Section 3. Section 4 describes the dataset used in the empirical application. Section 5 presents the identification of the optimal lag structure. This section also includes a forecasting comparison of models with flexible chosen lags against the commonly employed model with daily, weekly, and monthly realized volatility. Section 6 concludes. 2 Models for Day-to-Day Realized Volatility Let y t be a consistent and unbiased estimator of the square root of daily integrated variance. We consider different models for daily realized volatility, depending on the measure of volatility chosen: realized volatility, log realized volatility, and realized variance. Model A (General HAR-RV). Let and consider y t,k = 1 k y t+1 = c+ k j K k y t i+1 (1) i=1 β j y t,kj +w t+1, (2) where K = (k 1, k 2,..., k N ) is a set of N indices with k 1 < k 2 < < k N, j = 1,...,N, and w t+1 WN(0,σ 2 w). By substituting y t,kj into equation (2) we can write y t+1 = c+ β 1 k 1 k 1 i=1 y t i+1 + β 2 k 2 k 2 i=1 y t i β N k N k N i=1 y t i+1 +w t+1. (3) 3

4 Let θ j = N i=j β i/k i for j = 1,2,...,N. Equation (3) becomes y t+1 = c+θ 1 y t +θ 1 y t 1 + +θ 1 y t k1+1 +θ 2 y t k1 + +θ 2 y t k2+1+ +θ N y t kn 1 + +θ N y t kn+1 +w t+1. (4) Equation (2) can thus be viewed as a restricted autoregressive model. By analogy with AR models, we can state that the model is covariance-stationary if and only if the roots of 1 θ 1 z θ 1 z k1 θ 2 z k1+1 θ 2 z k2 θ N z kn 1+1 θ N z kn = 0 lie outside the unit circle. Let φ = N j=1 k jθ j = N j=1 β j be the sum of the autoregressive coefficients. If β i > 0 for all i, the stationarity condition is φ < 1. The parameter φ can be interpreted as a measure of persistence. The closer φ is to one, the more persistent is the volatility process. High persistence means slow reversion to the mean, while low persistence means fast reversion to the mean. Provided that the time series is stationary, the mean of the process is c/(1 φ). Model A is a generalization of the HAR-RV model proposed by Corsi (2009). The HAR-RV model is inspired by the HARCH specification introduced by Müller et al. (1997) in the ARCH framework. HARCH advocates heterogeneity among market participants with respect to their time horizons and models volatility as a function of squared returns aggregated over different time horizons. HAR-RV applies the same idea in the context of realized volatility. Corsi (2009) specifies daily realized volatility as a linear combination of past daily, weekly, and monthly realized volatility. In our notation this corresponds to K = (1, 5, 21). Andersen et al. (2007) propose a similar setup but add jump components, while Corsi et al. (2008) employ the same specification in studying the volatility of realized volatility. Model B (Log linear model). The use of logs of realized volatility is very common in the literature (Andersen et al., 2001a, 2001b). Log realized volatility is approximately normally distributed (Andersen et al., 2003). This finding is confirmed for all stocks considered in our dataset. Consider Model A in equation (2). Define y t,k as in (1), but let y t be a consistent and unbiased estimator of the log of the square root of daily integrated variance. This specification is the log version of Model A. Equation (2) can be reduced to an autoregressive form and standard results for autoregressive models can be applied. The sum of the β-coefficients must be less than one for stability. A similar specification was proposed by Bollerslev et al. (2009). 3 Lag Selection The HAR-RV model employed in practice typically comprises daily, weekly, and monthly realized volatility. This is partly motivated by the simple interpretation of these terms, but also by the high computational cost associated with a flexible lag structure. We treat the maximum lag L as fixed and determine the N terms of the HAR-RV model by examining all possible combinations of lags. We always include lag one, and therefore the number of models to estimate 4

5 is ( L N 1). This demands substantial computing power, and we implement the lag selection in a parallel computing framework. Since the estimation processes are independent of each other, the problem is trivially parallelizable. We search for the best model specification according to two different criteria: in-sample fit and out-ofsample fit. For the in-sample fit, the estimation is simply maximum likelihood. For the out-of-sample fit, we minimize the mean-squared error (MSE) of the one-day ahead volatility forecast on a training sample. For each of the two criteria, the estimation algorithm delivers the optimal lag structure and parameter estimates. We apply this procedure to the thirty constituting stocks of the Dow Jones Industrial Average in the empirical part of the paper. The implementation was coded in C++ and set up on Louisiana State University s supercomputing framework, which, at the point of implementation, provided a computing power of Gflops/second. In this setup, the selection of the optimal lag structure with L = 250 and N = 3 took fifteen minutes for each single estimation. We confirm if the procedure accurately captures a specific data-generating process. We consider Model A with daily, weekly, and monthly realized volatility and set the lag structure to (1,5,21) with β 0 = 0.002, β 1 = 0.45, β 2 = 0.30, and β 3 = We simulate 10,000 days of realized volatility and use the first 5,000 observations as estimation sample and the last 5,000 as forecast sample. We repeat this process 10,000 times. On each run, we select the lag structure and estimate Model A with L = 250 and N = 3. We always include lag one, so that we have to estimate ( ) = 31,125 specifications. For each sample, the estimation procedure searches for the lag combination (1, k 2, k 3 ) that results in the highest log-likelihood for the in-sample fit and the smallest mean-squared error of the one-day-ahead forecast for the out-of-sample fit. Thus, the simulation results in 10,000 lag triples and corresponding parameter estimates. Figure 1 shows the histograms of the selected k 2 and k 3. For both, in-sample and out-of-sample fit, the histogram median is 5 for the second term and 21 for the third term, so we retrieve the data-generating lag structure. 4 Data We use high-frequency tick-by-tick trades on thirty Dow Jones Industrial Average Index stocks as listed in Table 1: Alcoa Inc. (AA), American International Group Inc. (AIG), American Express Inc. (AXP), Boeing Co. (BA), Citigroup Inc. (C), Caterpillar Inc. (CAT), Du Pont De Nemours (DD), Walt Disney Co. (DIS), General Electric (GE), General Motors (GM), Home Depot Inc. (HD), Honeywell International (HON), Hewlett Packard (HPQ), International Business Machines Co. (IBM), Intel Co. (INTC), Johnson and Johnson (JNJ), JP Morgan Chase (JPM), Coca Cola (KO), McDonald s (MCD), 3M Company (MMM), Altria Group (MO), Merck Co. (MRK), Microsoft Co. (MSFT), Pfizer Inc. (PFE), Procter and Gamble 5

6 (PG), AT&T (T), United Tech (UTX), Verizon Communications (VZ), Wal-Mart Stores (WMT), and Exxon Mobil (XOM). The data are obtained from the NYSE TAQ (Trade and Quote) database. The sample period starts in January 3, 1995 and ends in July 31, In Table 1 we report the number of days in the sample and the average number of transactions per day for each of the stocks considered. Realized volatility as an estimator of asset price volatility was originally proposed by Andersen and Bollerslev (1998). Realized variance is a consistent estimator of integrated variance in the absence of microstructure noise (Andersen et al., 2003). However, in the presence of market microstructure such as price discreteness and the bid-ask bounce, the estimator becomes inconsistent (Bandi and Russell, 2006, Hansen and Lunde, 2006, Oomen, 2006 and Zhang et al., 2005). Barndorff-Nielsen and Shephard (2002) have studied the properties of the estimation error in the presence of microstructure noise. The realized variance estimator becomes biased, and the bias is increasing with the sampling frequency. Therefore one is confronted with a trade-off: a higher sampling frequency reduces the measurement error, but because of the market microstructure effect, a higher sampling frequency induces a larger bias. In order to overcome the problem of microstructure noise, Andersen et al. (2000a, 2001a) propose sparse sampling. The sampling frequency used in the literature varies from 5 minutes to 30 minutes. Bandi and Russell (2006, 2008) and Zhang et al. (2005) propose an optimal sampling frequency. Zhang et al. (2005) and Zhang (2006) propose subsampling, while Zhou (1996), Barndorff-Nielsen et al. (2008) propose kernel-based estimators of realized volatility and Hansen et al. (2006) propose pre-filtering. Finally, Barucci and Reno (2002) and Malliavin and Mancino (2002) propose Fourier methods. There are now a number of consistent and unbiased estimators of realized volatility in the presence of microstructure noise: the two-time scales realized volatility estimator proposed by Zhang et al. (2005), the realized kernel estimator of Barndorff- Nielsen et al. (2008), the modified-ma filter of Hansen et al. (2007), and the realized quantile-based estimator of Christensen et al. (2009). For the purposes of this study, we employ the realized kernel estimator with modified Tukey-Hanning weights of Barndorff-Nielsen et al. (2008) to calculate daily realized volatility. We start by cleaning the data for outliers. We consider transactions between 9.30 am through 4.00 pm. Following Barndorff- Nielsen et al. (2008) we employ the following 60-second activity-fixed tick-time sampling scheme: fq i = 1+60n i /(t 0,i t ni,i), where fq i is the sampling frequency, n i represents the number of transactions for day i, and t 0,i, t ni,i are the times for the first and last trade for day i. This is tick-time sampling chosen such that the same number of observations is obtained each day. Figure 2 shows plots of daily realized volatility and logarithmic realized volatility series for Walmart Inc., a typical stock in our sample, from January 3, 1995 to December 31, Daily realized volatility is calculated from intraday log-returns measured in percentage (i.e., multiplied by 100). The first two panels show that both series are characterized by a high degree of volatility clustering with periods of high volatility 6

7 and low volatility. A common finding in the literature (Andersen et al., 2001a) is that logarithmic realized volatility is close to normal, while realized volatility is not normal. Panels 3 and 4 present QQ-plots for the two series. The last two panels graph the sample autocorrelation functions for the two series. Both panels exhibit slow hyperbolic decay, indicating the presence of long memory. These findings are consistent across all stocks considered in our study. 5 Main Results 5.1 Estimation: Flexible Lag Selection We divide the sample period of January 3, 1995 through July 31, 2007 into three parts: an estimation sample (January 3, 1995 through December 31, 2004), a training sample (January 3, 2005 through December 31, 2005), and a forecast sample (January 3, 2006 through July 31, 2007). We consider Models A and B as described in Section 2. The number of lagged terms is N = 3 and the maximum lag is L = 250. We always include lag one, an AR(1) term. The algorithm can choose two free lags, which results in estimation costs of ( ) = 31,125 models. We run the estimation procedure on each data set and search for the best specification of lagged terms for in-sample and out-of-sample fit. The in-sample specification is chosen by maximum likelihood on the estimation sample. The out-of-sample specification is chosen by minimizing the one-day-ahead mean-squared forecast error on the training sample with respect to the parameter vector. The forecast performance of both specifications is then evaluated on the forecast sample. Tables 2 and 3 present the results for the estimation of the best specification according to maximum likelihood(in-sample) and minimum mean-squared error of the one-day-ahead forecast on the training sample (out-of-sample) for Model A. The tables report the optimal lag structure, the corresponding parameter estimates, and standard errors according to Newey and West (1987) for each of the stocks considered. In the last column we list the estimated persistence parameter φ, calculated as the sum of the estimated coefficients. With a few exceptions, all estimated coefficients are highly significant, for both in-sample and out-ofsample fit. The last column indicates high persistence in realized volatility, as shown by the high values of φ. With the single exception of hpq, all coefficients are above 0.9 for both criteria. Furthermore, for the majority of stocks, the estimated persistence parameter is greater than Focusing on the in-sample fit, there seems to be a substantial influence of the 5-14 day lag on daily realized volatility. The long lag is estimated at or close to the boundary of 250, which we interpret as another indicator of long memory. In the case of the out-of-sample fit, for the first lag the influence of 2-16 days is found. The most frequent long lags are (6 times) and (7 times). Tables 4 and 5 present the results for the estimation of the best specification for this model according to the two criteria considered for Model B. The tables are organized in the same manner as Tables 2 and 3 7

8 and the results are very similar. The in-sample fit analysis confirms the influence of the 4-14 lag on daily realized volatility. The long lag is found close to or at the boundary of 250. For the out-of-sample analysis, the long lag is no longer found on the boundary. The most frequent lags are and The short lag is found between 2 and 24 days. 5.2 Forecast Performance We compare the forecasting performance of models with flexibly chosen lag structure against the benchmark of the common HAR-RV(1,5,21) model using Giacomini s and White s (2006) test for conditional predictive ability (CPA). Table 6 presents one-day ahead forecast errors as well as results from the CPA test for Model A. For the models estimated based on in-sample and out-of-sample fit, respectively, columns 2 and 4 report the mean-squared error (MSE) of the one-day ahead forecast as a percentage of the mean-squared error of the benchmark model HAR-RV(1,5,21). Columns 3 and 5 report the p-values for the CPA test for the model based on in-sample fit and out-of-sample fit, respectively. The null hypothesis for the CPA test is that the expected loss of the benchmark model of lags (1,5,21) and the model with flexible lags is the same. A p-value smaller than the significance level indicates that one of the models has a better forecasting performance than the other. In this case, the model selected by the CPA test is reported in parentheses, where (B) stands for the benchmark HAR-RV(1,5,21) model and (FL) stands for flexible lag chosen according to in-sample or out-of-sample fit. We can reject the null hypothesis only in a few cases. For two out of the thirty stocks considered (ge and pfe), the model based on the optimal lag combination for the in-sample fit outperforms the benchmark model at the five percent significance level. The benchmark model outperforms the optimal lag specification for two stocks (dis and utx). Turning to the flexible lag model for the out-of-sample fit, at the ten percent significance level, for six stocks the model outperforms the benchmark, and for one stock the benchmark model outperforms the flexible lag specification. At the five percent significance level, the benchmark model is outperformed in three cases and outperforms in one. We can conclude that, overall, the benchmark model is not clearly outperformed by any of the competing models. When comparing the forecast performance between the models based on in-sample and out-of-sample fit, we observe that for the model specification based on out-of-sample fit, we find MSE values smaller than one for all stocks, indicating a better fit. For Model B, the comparison of the forecast performance of a flexible lag with the benchmark model HAR-RV(1,5,21) is displayed in Table 7. The table reports the MSE and p-values from the CPA test. The results are similar to the findings for Model A. The in-sample fit is outperformed twice at ten percent significance level and once at five percent. It outperforms the benchmark once at ten percent. The key result that emerges from Tables 6 and 7 is that when we consider models with three lags, allowing for a flexible lag structure does not obviously improve the forecasting performance over the HAR-RV(1,5,21) 8

9 model. This result is found for both, Models A and B. We illustrate this result with plots of the log-likelihood and the MSE as a function of different lags in Figure 3. We present the graphs for wmt, but similar results are found for the other stocks in our sample. Panel (1) shows the likelihood surface as a function of the second and third lagged term. Lag one is always included. Panel (2) displays part of this same surface, where we zoom into a region between 1 and 15 for the second lagged term. We can see from panel (1) that the second term can be chosen at any value in this range, but once it is chosen, the third lagged term must be at the maximum value of L = 250. The results are different for the out-of-sample fit and are presented in panels (3) and (4). These plot the MSE for the one-day-ahead forecast over the training sample. We see that for a given lag of the second lagged term, the surface is flat after a value of approximately 20 for the third lagged term. There is a spike in the surface area at very small values for the third lagged term. Panel (4) zooms in and shows that, as long as the second lag is somewhere between 2 and 15, the choice of the third lagged term does not matter as long as it is bigger than 20, since the surface is flat. Therefore, while the choice of the second lagged term is crucial, the choice of the third lagged term is less important. 5.3 How Many Terms? Given the little influence of the third lagged term, in this section we study the consequences of its exclusion from the model. We compare the forecast performance of models with one lagged term against models with two and three lagged terms. Table 8 shows the results of a CPA test of models with two lagged terms against an AR(1) benchmark. The model with two lagged terms includes lag one; the second term is chosen flexibly. For Model A, with the single exception of xom, the CPA test indicates that two lagged terms outperform the AR(1)-specification at all common significance levels for both, in-sample and out-of-sample fit. Allowing for two lags leads to similar improvements in the case of Model B. With the exception of two stocks, the test indicates significant forecast improvements for the in-sample specification. For the out-ofsample specification, all stocks except aa benefit from a two term specification. Based on the results for both Models A and B, we conclude that the model specification with two terms is the better forecasting model than AR(1). Table 9 reports the lag structure chosen for the second term. We find that for the majority of stocks and both, Models A and B, the in-sample fit chooses the second lag on or close to the boundary value of 250. The results for the out-of-sample fit are very different, and the second lag is found to be somewhere between 15 and 20 in most cases. Next we compare models with two terms versus models with three terms in Table 10. We find that at the ten percent significance level, for 20 stocks the CPA test chooses the specification with three terms for the in-sample fit of Model A. The out-of-sample fit benefits less from the inclusion of the third lag; only 13 stocks show forecasts improvements for the three terms model. The results are similar for Model B. The 9

10 CPA tests selects the three term specification for 18 stocks for the in-sample fit and for eight stocks for the out-of-sample fit. While the case for the inclusion of the third term is not very strong, it does add predictive power for about half of the stocks. It is not costly in the sense that only one additional parameter is to be estimated and only one stock (dis) indicates a significantly better performance of the model with two terms. Therefore, we advocate the use of three terms. In contradistinction to Table 10, where we compare models with three flexibly chosen terms against those with two flexibly chosen terms, we now compare these two model classes with a few fixed lag structures, namely (1,5), (1,21), and (1,5,21). The results of pairwise Giacomini and White (2006) CPA tests are reported in Table 11. The table is organized such that we report the number of stocks (out of the 30) for which one model outperforms the other at the ten percent level of significance. The cell for each pair of models contains two entries, one denoted with a plus, indicating the number of stocks for which the columnmodel outperformed the row-model, the other denoted with a minus, indicating the number of stocks for which the row-model outperformed the column-model. Therefore, we only populate the lower triangle. If a cell is left blank in the lower triangle, this means that none of the two models performed better than the other. We denote the flexible specifications with two and three terms as FL2 and FL3, respectively. The cells FL2 vs. FL3 therefore contain the results from Table 10. Table 11 shows that if a model with only two terms is to be estimated, the (1,21) specification is preferred over a flexible second term: it outperforms for the in-sample fit and it does equally well for the out-of-sample fit. Extending to a third term, for the in-sample fit, the (1,5,21) specification outperforms (1,21) in seven and ten cases for Models A and B, respectively. The (1,5,21) specification also outperforms a flexible second term, in 15 and 16 cases for Models A and B, respectively. For the out-of-sample fit, the performance of the (1,5,21) specification is indistinguishable from (1,21) and a flexible second term. The comparison (1,5,21) versus two flexible terms summarizes the findings in Tables 5 and 6. In most cases, the flexible specification does not yield forecast gains, with the exception of the out-of-sample specification of Model A, where the flexible specification outperforms (1,5,21) in six cases. However, since the out-of-sample fit procedure is not very common and Model B more popular than Model A, we do not interpret this finding as a strong point for a flexible specification. 6 Conclusion We address the problem of lag selection in the HAR-RV model for daily realized volatility, motivated by the extensive use of this model in the analysis of time series of realized volatility. We focus on two specifications: one for realized volatility and one for logarithmic realized volatility. We fit both specifications by maximum 10

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16 Table 1: Data Description. The first two columns display the symbols and names of the stocks considered in the empirical investigation. The third column gives the average number of transactions per day. Column 4 shows the number of days in the sample. Symbol Stock Avg. transactions / day No days aa Alcoa Inc aig American International Group Inc axp American Express Co ba Boeing Co c Citigroup Inc cat Caterpillar Inc dd Du Pont de Nemours&Co dis Walt Disney Co ge General Electric Co gm General Motors Corp hd Home Depot Inc hon Honeywell International Inc hpq Hewlett-Packard Co ibm International Business Machines Corp intc Intel Co jnj Johnson&Johnson jpm JPMorgan Chase&Co ko Coca-Cola Co mcd McDonald s Corp mmm 3M Co mo Altria Group Inc mrk Merck&Co. Inc msft Microsoft Co pfe Pfizer Inc pg Procter&Gamble Co t AT&T Inc utx United Technologies Corp vz Verizon Communications Inc wmt Wal-Mart Stores Inc xom Exxon Mobil Corp

17 Table 2: Model A: In-Sample Fit Estimation. The first column displays the stock symbol. Column 2 presents the optimal lags according to the in-sample fit. Columns 3 to 6 give the parameter estimates. The last column is the persistence measure, calculated as the sum of the β-coefficients. The figures in parentheses are Newey-West (1987) standard errors. lags ˆβ1 ˆβ2 ˆβ3 ĉ ˆφ aa (1,5,250) 0.252(.029) 0.578(.036) 0.134(.027) 0.061(.034) aig (1,6,246) 0.330(.056) 0.514(.053) 0.103(.033) 0.078(.031) axp (1,5,249) 0.368(.041) 0.468(.045) 0.137(.035) 0.045(.032) ba (1,8,250) 0.388(.035) 0.420(.037) 0.153(.029) 0.065(.042) c (1,10,248) 0.391(.043) 0.465(.042) 0.115(.031) 0.046(.033) cat (1,13,250) 0.285(.042) 0.562(.072) 0.100(.065) 0.098(.075) dd (1,6,250) 0.338(.051) 0.498(.057) 0.132(.026) 0.051(.025) dis (1,6,248) 0.272(.029) 0.555(.029) 0.135(.029) 0.064(.031) ge (1,8,248) 0.413(.031) 0.435(.040) 0.116(.028) 0.054(.031) gm (1,10,248) 0.311(.032) 0.546(.041) 0.084(.035) 0.086(.039) hd (1,7,249) 0.375(.035) 0.480(.035) 0.105(.030) 0.069(.036) 0.96 hon (1,8,250) 0.344(.052) 0.470(.060) 0.148(.034) 0.067(.037) hpq (1,4,246) 0.242(.064) 0.300(.088) 0.219(.041) 0.248(.070) ibm (1,7,247) 0.398(.030) 0.442(.036) 0.133(.033) 0.039(.038) intc (1,9,247) 0.468(.025) 0.408(.033) 0.102(.023) 0.046(.030) jnj (1,8,249) 0.343(.032) 0.504(.039) 0.079(.035) 0.097(.038) jpm (1,9,250) 0.322(.029) 0.540(.034) 0.104(.026) 0.057(.027) ko (1,4,250) 0.314(.036) 0.500(.041) 0.142(.034) 0.062(.026) mcd (1,14,248) 0.338(.029) 0.502(.041) 0.115(.042) 0.068(.041) mmm (1,14,250) 0.406(.031) 0.473(.042) 0.086(.037) 0.047(.036) mo (1,9,250) 0.317(.030) 0.456(.037) 0.130(.043) 0.145(.053) mrk (1,8,245) 0.382(.038) 0.449(.041) 0.096(.032) 0.107(.039) msft (1,7,249) 0.469(.029) 0.382(.038) 0.133(.024) 0.024(.027) pfe (1,10,248) 0.374(.028) 0.439(.036) 0.123(.040) 0.102(.046) pg (1,9,248) 0.413(.035) 0.450(.040) 0.109(.027) 0.039(.027) t (1,9,249) 0.362(.035) 0.436(.039) 0.163(.031) 0.070(.035) utx (1,8,248) 0.358(.048) 0.487(.057) 0.119(.025) 0.057(.028) vz (1,7,250) 0.300(.070) 0.638(.057) 0.020(.038) 0.062(.038) wmt (1,9,249) 0.359(.036) 0.497(.041) 0.118(.033) 0.042(.032) xom (1,4,248) 0.265(.071) 0.605(.070) 0.110(.036) 0.016(.030)

18 Table 3: Model A: Out-Of-Sample Fit Estimation. The first column displays the stock symbol. Column 2 presents the optimal lags according to the out-of-sample fit. Columns 3 to 6 give the parameter estimates. The last column is the persistence measure, calculated as the sum of the β-coefficients. The figures in parentheses are Newey-West (1987) standard errors. lags ˆβ1 ˆβ2 ˆβ3 ĉ ˆφ aa (1,11,131) 0.344(.035) 0.519(.037) 0.089(.036) 0.080(.036) aig (1,16,117) 0.431(.056) 0.446(.056) 0.065(.040) 0.086(.039) axp (1,7,209) 0.409(.041) 0.439(.042) 0.126(.035) 0.042(.031) ba (1,8,246) 0.389(.035) 0.418(.037) 0.155(.029) 0.062(.042) c (1,10,217) 0.391(.043) 0.46(.041) 0.123(.030) 0.042(.031) cat (1,10,247) 0.284(.040) 0.529(.066) 0.136(.062) 0.097(.070) dd (1,11,250) 0.401(.047) 0.467(.052) 0.099(.029) 0.053(.028) dis (1,5,154) 0.256(.028) 0.537(.030) 0.172(.030) 0.056(.029) ge (1,10,209) 0.431(.031) 0.425(.040) 0.111(.029) 0.051(.031) gm (1,2,27) 0.21(.036) 0.301(.067) 0.397(.045) 0.131(.041) hd (1,11,207) 0.419(.037) 0.454(.037) 0.088(.032) 0.067(.037) hon (1,5,127) 0.309(.045) 0.426(.053) 0.225(.037) 0.070(.035) hpq (1,13,102) 0.317(.061) 0.369(.102) 0.203(.082) 0.138(.060) 0.89 ibm (1,6,244) 0.390(.031) 0.435(.037) 0.149(.034) 0.036(.037) intc (1,12,138) 0.485(.026) 0.384(.040) 0.103(.030) 0.057(.030) jnj (1,10,208) 0.364(.031) 0.492(.040) 0.072(.036) 0.094(.036) jpm (1,3,129) 0.229(.033) 0.494(.044) 0.24(.032) 0.06(.028) ko (1,6,247) 0.373(.040) 0.463(.041) 0.124(.033) 0.055(.027) 0.96 mcd (1,5,28) 0.275(.037) 0.296(.050) 0.354(.045) 0.113(.026) mmm (1,8,248) 0.371(.028) 0.465(.037) 0.129(.031) 0.046(.030) mo (1,2,64) 0.187(.029) 0.345(.043) 0.379(.047) 0.129(.040) mrk (1,16,56) 0.44(.037) 0.399(.068) 0.074(.063) 0.123(.039) msft (1,8,246) 0.485(.028) 0.367(.037) 0.133(.025) 0.022(.027) pfe (1,20,199) 0.42(.027) 0.425(.046) 0.089(.049) 0.103(.051) pg (1,10,244) 0.421(.035) 0.447(.040) 0.102(.027) 0.04(.027) t (1,6,63) 0.349(.044) 0.338(.058) 0.269(.038) 0.073(.029) utx (1,3,70) 0.234(.031) 0.468(.044) 0.256(.045) 0.064(.026) vz (1,2,68) 0.213(.066) 0.487(.071) 0.262(.054) 0.054(.043) wmt (1,10,212) 0.365(.036) 0.493(.043) 0.115(.034) 0.042(.031) xom (1,12,13) 0.321(.062) 0.458(.348) 0.165(.357) 0.072(.031)

19 Table 4: Model B: In-Sample Fit Estimation. The first column displays the stock symbol. Column 2 presents the optimal lags according to the in-sample fit. Columns 3 to 6 give the parameter estimates. The last column is the persistence measure, calculated as the sum of the β-coefficients. The figures in parentheses are Newey-West (1987) standard errors. lags ˆβ1 ˆβ2 ˆβ3 ĉ ˆφ aa (1,6,250) 0.279(.027) 0.535(.034) 0.152(.026) 0.014(.009) aig (1,5,247) 0.281(.033) 0.555(.038) 0.114(.027) 0.016(.007) axp (1,12,248) 0.422(.028) 0.472(.032) 0.097(.025) 0.000(.008) ba (1,13,250) 0.375(.027) 0.469(.034) 0.120(.030) 0.015(.010) c (1,10,250) 0.373(.029) 0.489(.033) 0.122(.025) 0.004(.007) cat (1,14,250) 0.278(.030) 0.566(.050) 0.119(.047) 0.026(.017) dd (1,10,229) 0.327(.028) 0.537(.035) 0.113(.025) 0.008(.007) dis (1,7,249) 0.286(.028) 0.541(.034) 0.139(.024) 0.016(.006) ge (1,10,250) 0.375(.028) 0.500(.034) 0.102(.024) 0.007(.006) gm (1,10,249) 0.281(.025) 0.568(.031) 0.100(.033) 0.015(.009) hd (1,11,249) 0.366(.027) 0.509(.030) 0.096(.030) 0.013(.009) hon (1,9,234) 0.300(.031) 0.510(.040) 0.156(.033) 0.016(.009) hpq (1,4,27) 0.282(.042) 0.430(.065) 0.266(.046) 0.005(.013) intc (1,9,247) 0.455(.024) 0.407(.030) 0.119(.022) 0.012(.009) ibm (1,7,182) 0.374(.026) 0.465(.034) 0.149(.025) 0.001(.006) jnj (1,8,250) 0.311(.026) 0.527(.034) 0.112(.029) 0.010(.006) jpm (1,10,250) 0.328(.029) 0.549(.031) 0.099(.024) 0.010(.007) ko (1,6,249) 0.306(.030) 0.514(.037) 0.146(.030) 0.008(.006) mcd (1,14,250) 0.312(.027) 0.526(.038) 0.128(.037) 0.011(.010) mmm (1,14,250) 0.379(.025) 0.500(.034) 0.094(.035) 0.005(.007) mo (1,19,250) 0.380(.024) 0.491(.037) 0.052(.042) 0.024(.012) mrk (1,8,242) 0.345(.024) 0.470(.034) 0.116(.030) 0.022(.009) msft (1,7,176) 0.399(.025) 0.438(.031) 0.148(.023) 0.002(.008) pfe (1,9,249) 0.355(.024) 0.462(.034) 0.126(.034) 0.022(.010) pg (1,10,248) 0.411(.025) 0.462(.029) 0.107(.025) 0.004(.006) t (1,9,240) 0.323(.025) 0.456(.032) 0.178(.028) 0.023(.008) utx (1,6,249) 0.261(.026) 0.568(.031) 0.131(.021) 0.016(.007) vz (1,10,231) 0.317(.037) 0.615(.046) 0.047(.036) 0.000(.011) wmt (1,8,249) 0.318(.029) 0.534(.036) 0.130(.027) 0.006(.007) xom (1,4,27) 0.210(.038) 0.554(.053) 0.192(.032) 0.006(.006)

20 Table 5: Model B: Out-Of-Sample Fit Estimation. The first column displays the stock symbol. Column 2 presents the optimal lags according to the out-of-sample fit. Columns 3 to 6 give the parameter estimates. The last column is the persistence measure, calculated as the sum of the β-coefficients. The figures in parentheses are Newey-West (1987) standard errors. lags ˆβ1 ˆβ2 ˆβ3 ĉ ˆφ aa (1,11,132) 0.325(.027) 0.511(.036) 0.122(.033) 0.017(.009) aig (1,9,15) 0.343(.030) 0.339(.064) 0.237(.069) 0.023(.007) axp (1,7,133) 0.378(.026) 0.438(.034) 0.170(.027) 0.003(.007) ba (1,4,131) 0.318(.027) 0.366(.040) 0.268(.032) 0.020(.010) c (1,8,131) 0.354(.027) 0.454(.032) (.027) 0.005(.007) cat (1,11,195) 0.283(.028) 0.523(.049) 0.149(.045) 0.029(.015) dd (1,13,250) 0.345(.028) 0.541(.034) 0.090(.027) 0.009(.008) dis (1,5,154) 0.247(.027) 0.519(.033) 0.209(.030) 0.010(.007) ge (1,5,140) 0.328(.028) 0.457(.036) 0.190(.026) 0.007(.006) gm (1,4,14) 0.234(.025) 0.208(.047) 0.456(.042) 0.028(.006) hd (1,2,15) 0.230(.034) 0.189(.049) 0.509(.035) 0.031(.007) hon (1,5,127) 0.256(.029) 0.452(.041) 0.255(.034) 0.016(.008) hpq (1,2,16) 0.231(.048) 0.318(.059) 0.420(.039) 0.011(.013) ibm (1,6,126) 0.372(.027) 0.432(.037) 0.176(.027) 0.005(.006) intc (1,16,90) 0.483(.024) 0.361(.041) 0.127(.032) 0.019(.009) jnj (1,10,247) 0.332(.025) 0.520(.033) 0.105(.030) 0.007(.006) jpm (1,3,126) 0.245(.027) 0.452(.038) 0.279(.033) 0.008(.007) ko (1,6,125) 0.311(.028) 0.471(.038) 0.181(.034) 0.009(.005) mcd (1,5,28) 0.244(.024) 0.309(.041) 0.377(.039) 0.022(.008) mmm (1,6,125) 0.309(.028) 0.465(.039) 0.183(.031) 0.009(.005) mo (1,2,40) 0.220(.031) 0.300(.042) 0.403(.030) 0.021(.006) mrk (1,16,56) 0.401(.022) 0.402(.054) 0.123(.051) 0.020(.008) msft (1,8,137) 0.415(.024) 0.417(.031) 0.148(.022) 0.006(.008) pfe (1,20,209) 0.402(.022) 0.453(.037) 0.094(.041) 0.019(.011) pg (1,8,68) 0.376(.027) 0.436(.034) 0.149(.029) 0.009(.005) t (1,6,63) 0.286(.026) 0.390(.037) 0.289(.031) 0.016(.007) utx (1,4,70) 0.222(.025) 0.458(.034) 0.282(.031) 0.013(.007) vz (1,2,21) 0.208(.044) 0.296(.058) 0.456(.038) 0.013(.010) wmt (1,24,212) 0.416(.032) 0.479(.047) 0.084(.040) 0.006(.009) xom (1,12,13) 0.376(.035) 0.878(.294) (.296) 0.009(.007)

21 Table 6: Model A: Forecasting Results. The first column displays the stock symbol. Column 2 reports the mean-squared error (MSE) for the one-day ahead forecast of realized volatility using the model chosen by the in-sample fit (IS). Column 3 reports the p-values of the Conditional Predictive Ability (CPA) test of Giacomini and White (2006) for the in-sample fit. The null hypothesis for the CPA test is that the benchmark HAR-RV(1,5,21) model and the model with flexible lag structure have the same expected loss. The last two columns present the MSE and p-values for the model specification based on the out-of-sample fit (OOS). The MSE are calculated as a percentage of the MSE of the benchmark model. p-values less than common significance levels indicate that one of the two models outperforms the other. In these cases, (B) indicates that the benchmark model outperformed the flexible lags model, (FL) indicates that the flexible lags model performed better. MSE(1) IS p val IS MSE(1) OOS p val OOS aa aig (B) axp ba c cat dd dis (B) ge (FL) (FL) gm (FL) hd hon (FL) hpq ibm intc jnj (FL) jpm ko mcd mmm mo (FL) mrk msft pfe (FL) (FL) pg t utx (B) vz wmt xom