HAR Inference: Recommendations for Practice

The classic papers by Newey and West (1987) and Andrews (1991) spurred a large body of work on how to improve heteroscedasticity- and autocorrelation-robust (HAR) inference in time series regression. This literature finds that using a larger-than-usual truncation parameter to estimate the long-run variance, combined with Kiefer-Vogelsang (2002, 2005) fixed-b critical values, can substantially reduce size distortions, at only a modest cost in (size-adjusted) power. Empirical practice, however, has not kept up. This article therefore draws on the post-Newey West/Andrews literature to make concrete recommendations for HAR inference. We derive truncation parameter rules that choose a point on the size-power tradeoff to minimize a loss function. If Newey-West tests are used, we recommend the truncation parameter rule S = 1.3T1/2 and (nonstandard) fixed-b critical values. For tests of a single restriction, we find advantages to using the equal-weighted cosine (EWC) test, where the long run variance is estimated by projections onto Type II cosines, using ν = 0.4T2/3 cosine terms; for this test, fixed-b critical values are, conveniently, tν or F. We assess these rules using first an ARMA/GARCH Monte Carlo design, then a dynamic factor model design estimated using a 207 quarterly U.S. macroeconomic time series.纽威与韦斯特（Newey and West，1987）以及安德鲁斯（Andrews，1991）的经典论文，催生了大量关于如何优化时间序列回归中异方差和自相关稳健（heteroscedasticity- and autocorrelation-robust, HAR）推断的研究。该领域文献表明，采用大于常规取值的截断参数估计长期方差，并结合基弗-沃格尔桑（Kiefer-Vogelsang，2002、2005）提出的固定b临界值，可大幅降低检验水平扭曲，仅在调整检验水平后的功效上付出微小代价。然而，实证实践并未跟上这一理论进展。因此，本文借鉴纽威-韦斯特/安德鲁斯之后的相关文献，为HAR推断提出具体可行的建议。我们推导了截断参数选择规则，通过在检验水平与功效的权衡中选取最优解以最小化损失函数。若采用纽威-韦斯特检验，我们建议使用截断参数规则S = 1.3T^(1/2)，并配合（非标准的）固定b临界值。针对单一约束检验，我们发现采用等权重余弦（equal-weighted cosine, EWC）检验具有优势：该检验通过对第二类余弦基进行投影来估计长期方差，使用ν = 0.4T^(2/3)个余弦项；对此类检验而言，固定b临界值恰好为t_ν或F分布临界值，使用起来十分便捷。我们首先通过ARMA/GARCH蒙特卡洛设计，随后采用基于207组美国宏观经济季度时间序列估计的动态因子模型设计，对上述规则进行了评估。