Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Abstract

International audienceLinear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations

Adaptive Wild Bootstrap Tests for a Unit Root With Non‐Stationary Volatility

“…For the empirically more relevant case where the volatility function is not observed, we show that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that non-parametric estimation of the volatility process leads to the same asymptotic power envelope. Similar adaptivity results were obtained for stable (auto-)regressions by Hansen (1995), Xu and Phillips (2008) and Patilea and Raïssi (2012). The test statistics that come out of this analysis have an asymptotic null distribution that depends on the realization of the volatility process.…”

“…In practice, the window width can be chosen by a leave-one-out cross-validation procedure, which involves minimizing Wasserman (2006). 3 As discussed by Patilea and Raïssi (2012), a formal analysis of post-selection consistency requires the result of Lemma 4.1 to hold uniformly over N ∈ [N l , N u ] with N l and N u satisfying the rate requirement of the lemma. In non-parametric estimation problems, leave-one-out cross-validation has been shown to lead to a window width that 3 Note that {w tt (N )} n t=1 are the diagonal elements of the smoothing matrix that maps the vector ( ε 2 1 , .…”

Testing normality for unconditionally heteroscedastic macroeconomic variables

“…Straightforward computations give √ n(θ(ω)−θ(ω 0 )) = o p (1), so that using the results of Patilea and Raïssi (2012) we have…”

Testing Second-Order Dynamics for Autoregressive Processes in Presence of Time-Varying Variance

“…In practice, the autoregressive order p in (2.1) is usually unknown. In this case, the order p can be selected using the adapted tools proposed in Patilea andRaïssi (2012, 2013). We will thus assume in the sequel that the order p is given.…”

“…Xu and Phillips (2008) proposed adaptive estimators for autoregressive parameters of stable univariate processes using kernel smoothing of the unconditional variance of the innovations. Patilea andRaïssi (2012, 2013) extended the work of Xu and Phillips and proposed adapted tools to estimate and validate multivariate autoregressive stable processes with time-varying unconditional variance of the innovations. Kim and Park (2010) investigated cointegrated systems assuming smooth changes for the unconditional variance.…”