Skip to main content


  • Seung-Hwa Rho (a1) and Timothy J. Vogelsang (a2)

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.

Corresponding author
*Address correspondence to Timothy J. Vogelsang, Department of Economics, Michigan State University, East Lansing, MI, USA; e-mail:
Seung-Hwa Rho, Amazon, Berlin, Germany; e-mail:
Hide All

We are grateful for helpful suggestions and comments from the editor, a co-editor, anonymous referees, and seminar participants at Columbia University, the Joint Montreal Econometrics Seminar and participants at the 2017 NSF/NBER Time Series Conference.

Hide All
Andrews, D.W. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817858.
Bester, C.A., Conley, T.G., Hansen, C.B., & Vogelsang, T.J. (2016) Fixed-b asymptotics for spatially dependent robust nonparametric covariance matrix estimators. Econometric Theory 32, 154186.
Bloomfield, P. (1970) Spectral analysis with randomly missing observations. Journal of the Royal Statistical Society 32, 369380.
Datta, D.D. & Du, W. (2012) Nonparametric HAC Estimation for Time Series Data With Missing Observations. Working paper.
Dunsmuir, W. & Robinson, P. (1981) Asymptotic theory for time series containing missing and amplitude modulated observations. The Indian Journal of Statistics 43, 260281.
Gonçalves, S. & Vogelsang, T.J. (2011) Block bootstrap HAC robust tests: The sophistication of the naive bootstrap. Econometric Theory 27, 745791.
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 14491459.
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21, 11301164.
Neave, H.R. (1970) Spectral analysis of a stationary time series using initially scarce data. Biometrika 57, 111122.
Newey, W.K. & West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703708.
Parzen, E. (1963) On spectral analysis with missing observations and amplitude modulation. The Indian Journal of Statistics 25(4), 383392.
Perron, P. (1991) A continuous time approximation to the unstable first-order autoregressive process: The case without an intercept. Econometrica 59(1), 211236.
Phillips, P.C. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74(3), 535547.
Scheinok, P.A. (1965) Spectral analysis with randomly missed observations: The binomial case. The Annals of Mathematical Statistics 36(3), 971977.
Sun, Y. (2014a) Fixed-smoothing asymptotics in a two-step generalized method of moments framework. Econometrica 82(6), 23272370.
Sun, Y. (2014b) Lets fix it: Fixed-b asymptotics versus small-b asymptotics in heteroskedasticity and autocorrelation robust inference. Journal of Econometrics 178, 659677.
Sun, Y., Phillips, P.C.B., & Jin, S. (2008) Optimal bandwidth selection in heteroskedasticity-autocorrelation robust testing. Econometrica 76, 175194.
Vogelsang, T.J. (2012) Heteroskedasticity, autocorrelation, and spatial correlation robust inference in linear panel models with fixed-effects. Journal of Econometrics 166, 303319.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed