Heteroskedasticity Autocorrelation Robust Inference In Time Series Regressions With Missing Data

In this article, we investigate the properties of heteroskedasticity and autocorrelation robust (HAR) test statistics in time series regression settings when observations are missing. We primarily focus on the nonrandom missing process case where we treat the missing locations to be fixed as T → ∞ by mapping the missing and observed cutoff dates into points on [0,1] based on the proportion of time periods in the sample that occur up to those cutoff dates. We consider two models, the amplitude modulated series (Parzen, 1963) regression model, which amounts to plugging in zeros for missing observations, and the equal space regression model, which simply ignores the missing observations. When the amplitude modulated series regression model is used, the fixed-b limits of the HAR test statistics depend on the locations of missing observations but are otherwise pivotal. When the equal space regression model is used, the fixed-b limits of the HAR test statistics have the standard fixed-b limits as in Kiefer and Vogelsang (2005). We discuss methods for obtaining fixed-b critical values with a focus on bootstrap methods and find the naive i.i.d. bootstrap with missing dates fixed to be an effective and practical way to obtain the fixed-b critical values.