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HAR Testing for Spurious Regression in Trend

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Cowles Foundation for Research in Economics, Yale University, Box 208281, Yale Station, New Haven, CT 06520, USA
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Department of Economics, University of Auckland, Auckland CBD, Auckland 1010, New Zealand
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School of Economics, Singapore Management University, 81 Victoria St, Singapore 188065, Singapore
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Department of Economics, University of Southampton, Southampton SO14 0DA, UK
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Department of Economics, The Chinese University of Hong Kong, Hong Kong 999077, China
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School of Economics, Renmin University of China, Beijing 100872, China
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Author to whom correspondence should be addressed.
Econometrics 2019, 7(4), 50; https://doi.org/10.3390/econometrics7040050
Received: 7 December 2018 / Revised: 6 October 2019 / Accepted: 28 November 2019 / Published: 16 December 2019
(This article belongs to the Special Issue Celebrated Econometricians: David Hendry)
The usual t test, the t test based on heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators, and the heteroskedasticity and autocorrelation robust (HAR) test are three statistics that are widely used in applied econometric work. The use of these significance tests in trend regression is of particular interest given the potential for spurious relationships in trend formulations. Following a longstanding tradition in the spurious regression literature, this paper investigates the asymptotic and finite sample properties of these test statistics in several spurious regression contexts, including regression of stochastic trends on time polynomials and regressions among independent random walks. Concordant with existing theory (Phillips 1986, 1998; Sun 2004, 2014b) the usual t test and HAC standardized test fail to control size as the sample size n in these spurious formulations, whereas HAR tests converge to well-defined limit distributions in each case and therefore have the capacity to be consistent and control size. However, it is shown that when the number of trend regressors K , all three statistics, including the HAR test, diverge and fail to control size as n . These findings are relevant to high-dimensional nonstationary time series regressions where machine learning methods may be employed. View Full-Text
Keywords: HAR inference; Karhunen–Loève representation; spurious regression; t-statistics HAR inference; Karhunen–Loève representation; spurious regression; t-statistics
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Phillips, P.C.B.; Wang, X.; Zhang, Y. HAR Testing for Spurious Regression in Trend. Econometrics 2019, 7, 50.

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