MIDAS Regressions: Further Results and New Directions

Abstract We explore mixed data sampling (henceforth MIDAS) regression models. The regressions involve time series data sampled at different frequencies. Volatility and related processes are our prime focus, though the regression method has wider applications in macroeconomics and finance, among other areas. The regressions combine recent developments regarding estimation of volatility and a not-so-recent literature on distributed lag models. We study various lag structures to parameterize parsimoniously the regressions and relate them to existing models. We also propose several new extensions of the MIDAS framework. The paper concludes with an empirical section where we provide further evidence and new results on the risk–return trade-off. We also report empirical evidence on microstructure noise and volatility forecasting. Keywords: Microstructure noiseNonlinear MIDASRiskTick-by-tick applicationsVolatilityJEL Classification: C22C53 ACKNOWLEDGMENTS We thank two referees and an associate editor, Alberto Plazzi, Pedro Santa-Clara, and seminar participants at City University of Hong Kong, Emory University, the Federal Reserve Board, ITAM, Korea University, New York University, Oxford University, Tsinghua University, University of Iowa, UNC, USC, participants at the Symposium on New Frontiers in Financial Volatility Modeling, Florence, the Academia Sinica Conference on Analysis of High-Frequency Financial Data and Market Microstructure, Taipei, the CIREQ-CIRANO-MITACS conference on Financial Econometrics, Montreal and the Research Triangle Conference, for helpful comments. All remaining errors are our own. Notes 1This situation is becoming more frequent now as dramatic improvements in information gathering have produced new high-frequency data sets, particularly in the area of financial econometrics. 2For further evidence on the risk–return tradeoff using MIDAS, see e.g., Angel et al. (Citation2004), Wang (Citation2004), and Charoenrook and Conrad (Citation2005). Models of idiosyncratic volatility using MIDAS appear in e.g., Jiang and Lee (Citation2004) and Brown and Ferreira (Citation2004). 3Convex shapes appear when θ1 > θ2. While those shapes are not of immediate interest in our volatility applications, they might be useful in other applications. 4The terminology of weak GARCH originated with the work of Drost and Nijman (Citation1993) and refers to volatility predictions involving only linear functionals of past returns and squared returns. Obviously, many ARCH-type models involve nonlinear functions of past (daily) returns. It would be possible to study nonlinear functions involving distributed lags of high-frequency returns. This possibility is explored later in the paper. 5The GARCH parameters of (Equation11) are related to the GARCH diffusion via formulas appearing in Corollary 3.2 of Drost and Werker (Citation1996). Likewise, Drost and Nijman (Citation1993) derive the mappings between GARCH parameters corresponding to processes with sampled with different values of m. 6See for instance Chernov et al. (Citation2002) for further discussion. Meddahi (Citation2002) derives a weak GARCH(2, 2) representation of a two-factor SV model which could be used in this particular case, but not in a more general setting. 7There is a considerable literature on the subject. See, e.g., Breitung and Swanson (Citation2000) for a recent discussion. The table shows results from estimating Equations (Equation21–Equation25) at one-, two-, three-, and four-week frequencies. The MIDAS weights are parameterized to follow the exponential Almon polynomial (Equation2). The estimation is performed by quasi-maximum likelihood using Dow Jones index return data from April 1993 to October 2003. The estimates of μ and γ are displayed in the first two columns. In column three, we show the t-statistic of γ under the null of no risk–return trade-off, and the standard errors are computed using the heteorskedasticity-robust Bollerslev and Wooldridge (Citation1992) method. We compute in column four the mean absolute deviation (MAD) as a measure of the goodness-of-fit of the MIDAS regression, because it is robust to heteroskedasticity in the data. The fraction of the weights placed on lags 1 to 5 (one week), lags 6 to 20 (one month), and higher, are shown in columns five to seven, respectively. The panels contain the results for squared daily returns , absolute daily returns (|r t |), daily ranges ([hi − lo] t ), daily realized volatility (Q t ), and daily realized power (P t ), as explained in the text. Sample April 1993 to October 2003. The table shows results from estimating Equations (Equation21–Equation25) at one-, two-, three-, and four-week frequencies. The MIDAS weights are parameterized to follow the beta polynomial (Equation3). The estimation is performed by quasi-maximum likelihood using Dow Jones index return data from April 1993 to October 2003. The estimates of μ and γ are displayed in the first two columns. In column three, we show the t-statistic of γ under the null of no risk–return trade-off, and the standard errors are computed using the heteorskedasticity-robust Bollerslev and Wooldridge (Citation1992) method. We compute in column four the mean absolute deviation (MAD) as a measure of the goodness-of-fit of the MIDAS regression, because it is robust to heteroskedasticity in the data. The fraction of the weights placed on lags 1 to 5 (one week), lags 6 to 20 (one month), and higher, are shown in columns five to seven, respectively. The panels contain the results for squared daily returns (r t 2), absolute daily returns (|r t |), daily ranges ([hi − lo] t ), daily realized volatility (Q t ), and daily realized power (P t ), as explained in the text. Sample April 1993 to October 2003. 8For additional references see O'Hara (Citation1995), Hasbrouck (Citation2004). Each entry in the table corresponds to the sample mean for the different daily volatility measure and different subsample. Subsample 2000–2002 covers January 3, 2000 to December 31, 2002. Subsample 2003–2004 covers January 3, 2003 to December 31, 2004. The names of the variables are consistent with the notation in Hansen and Lunde (Citation2005). Each entry in the table corresponds to the R 2 for different models (Equation24) and different estimation samples. The whole sample covers January 3, 2000 to December 31, 2004. Subsample 2000–2002 covers January 3, 2000 to December 31, 2002. The regressions are run on a weekly (5 days) data sampling scheme. The names of the variables are consistent with the notation in Hansen and Lunde (Citation2005). Every column corresponds to the explanatory power of the different LHS variables for the same RHS variable. Each entry in the table corresponds to the R 2 for different models (Equation24) and different estimation samples. The whole sample covers January 3, 2000 to December 31, 2004. Subsample 2000–2002 covers January 3, 2000 to December 31, 2002. The regressions are run on a weekly (5 days) data sampling scheme. The names of the variables are consistent with the notation in Hansen and Lunde (Citation2005). Every column corresponds to the explanatory power of the different LHS variables for the same RHS variable.