Simultaneous bandwidths determination for DK-HAC estimators and long-run variance estimation in nonparametric settings

Abstract

We consider the derivation of data-dependent simultaneous bandwidths for double kernel heteroscedasticity and autocorrelation consistent (DK-HAC) estimators. In addition to the usual smoothing over lagged autocovariances for classical HAC estimators, the DK-HAC estimator also applies smoothing over the time direction. We obtain the optimal bandwidths that jointly minimize the global asymptotic MSE criterion and discuss the tradeoff between bias and variance with respect to smoothing over lagged autocovariances and over time. Unlike the MSE results of Andrews, we establish how nonstationarity affects the bias-variance tradeoff. We use the plug-in approach to construct data-dependent bandwidths for the DK-HAC estimators and compare them with the DK-HAC estimators from Casini that use data-dependent bandwidths obtained from a sequential MSE criterion. The former performs better in terms of size control, especially with stationary and close to stationary data. Finally, we consider long-run variance (LRV) estimation under the assumption that the series is a function of a nonparametric estimator rather than of a semiparametric estimator that enjoys the usual $\sqrt{T}$ rate of convergence. Thus, we also establish the validity of consistent LRV estimation in nonparametric parameter estimation settings.

Keywords:

• HAC standard errors
• long-run variance
• nonstationarity
• outliers
• segmented locally stationary

JEL Classification:

• C12
• C13
• C18
• C22
• C32
• C51

Notes

1 The fixed-b literature is extensive. Pioneering contribution of Kiefer et al. (2000) and Kiefer and Vogelsang (2002; 2005) introduced the fixed-b LRV estimators. Additional contributions can be found in Dou (2019), Lazarus et al. (2020), Lazarus et al. (2018), Gonçalves and Vogelsang (2011), de Jong and Davidson (2000), Ibragimov and Müller (2010), Jansson (2004), Müller (2007, 2014), Phillips (2005), Politis (2011), Preinerstorfer and Pötscher (2016), Pötscher and Preinerstorfer (2018; 2019), Robinson (1998), Sun (2013; 2014a; 2014c), Sun et al. (2008), Velasco and Robinson (2001) and Zhang and Shao (2013). We refer to Casini (2019, 2022c) and Casini et al. (2022) for discussions and comparisons between our approach based on DK-HAC and the fixed-b approach.

2 Note that the MSE bounds in Section 8 in Andrews (1991) are not correct. See Casini (2022a) for details.

3 Some authors have used alternative notions of local stationarity that allow for discontinuities and have established some results in other contexts [cf. Dahlhaus (2009) and Zhou (2013)].

4 That is, ρ_t varies smoothly between 0 and 0.8071.

5 We have excluded Andrews’s (1991) HAC estimator since its performance is similar to that of the Newey-West estimator.

6 The method of Sun (2014b) suffers more from this problem than KVB’s fixed-b since the LRV estimator is the same but the critical values of Sun’s (2014b) are in practice larger than the KVB’s fixed-b critical values.

7 For the test normalized by the latter estimator, we use the fixed-b critical values.

8 The results for $q_{\pi }=0$ and the other combinations of q_u and $q_{\pi }$ are similar and not reported.