Heteroskedasticity and Autocorrelation Robust Estimation of Variance Covariance Matrices

Inspection of the asymptotic normality results for least mean distance and generalized method of moments estimators given in, e.g., Theorems 11.2(a) and 11.5(a) shows that in both cases a matrix of the form $$C_n^{ - 1}{D_n}{D'_n}C_n^{ - 1'}$$ acts as an asymptotic variance covariance matrix of $${n^{1/2}}\left( {{{\hat \beta }_n} - {{\hat \beta }_n}} \right)$$ , where C _n and D _n are given in those theorems. For purposes of inference we need estimators of C _n and D _n . Inspection of the matrices C _n reveals that these matrices are essentially composed of terms of the form $${n^{ - 1}}\sum\nolimits_{t = 1}^n {E{w_{t,n}}} $$ , where $${w_{t,n}} = {w_t}\left( {{{\bar \tau }_n},{{\bar \beta }_n}} \right)$$ equals $${\nabla _{\beta \beta qt}}\left( {{z_t},{{\bar \tau }_n},{{\bar \beta }_n}} \right)$$ in the case of least mean distance estimators or equals $${\nabla _{\beta qt}}\left( {{z_t},{{\bar \tau }_n},{{\bar \beta }_n}} \right)$$ in the case of generalized method of moments estimators. The matrices D _n — apart from containing similar terms — also contain an expression of the form $${n^{ - 1}}E\left[ {\left( {\sum\limits_{t = 1}^n {{V_{t,n}}} } \right)\left( {\sum\limits_{t = 1}^n {{{v'}_{i,n}}} } \right)} \right],$$ where $${v_{t,n}} = {v_t}\left( {{{\bar \tau }_n},{{\bar \beta }_n}} \right)$$ equals $$\nabla \beta 'qt\left( {{z_t},{{\overline \tau }_n},{{\overline \beta }_n}} \right)$$ in the case of least mean distance estimators or equals $${q_t}\left( {{z_t},{{\bar \tau }_n},{{\bar \beta }_n}} \right)$$ in the case of generalized method of moments estimators. The expressions of the form $${n^{ - 1}}\sum\nolimits_{t = 1}^n {E{w_{t,n}}} $$ will typically be estimated by $${n^{ - 1}}\sum\nolimits_{t = 1}^n {{{\hat w}_{t,n}}} $$ where $${\hat w_{t,n}} = {w_t}\left( {{{\hat \tau }_n},{{\hat \beta }_n}} \right)$$ . Consistency of such estimators can be derived from ULLNs and from consistency of $$\left( {{{\hat \tau }_n},{{\hat \beta }_n}} \right)$$ in a rather straightforward manner via Lemma 3.2.1 The estimation of $${\Psi _n} = {n^{ - 1}}\sum\limits_{t = 1}^n {E{v_{t,n}}{{v'}_{t,n}}j = \sum\limits_{j = 1}^{n - 1} {{n^{ - 1}}} \sum\limits_{t = 1}^{n - j} {\left[ {E{v_{t,n}}{{v'}_{t + j,n}} + E{v_{t = j,n}}v't,n} \right]} } $$ reduces to a similar problem in the important special case where (v_t,_n) is a martingale difference array (or, more generally is uncorrelated and has mean zero), since then the expression for Ψ _n reduces to $${n^{ - 1}}\sum\nolimits_{t = 1}^n {E{v_{t,n}}{{v'}_{t,n}}} $$ . As discussed above (v _t,n ) will have a martingale difference structure in certain correctly specified cases. In the general case, however, (v _t,n ) will typically be autocorrelated (with autocorrelation of unknown form) and hence the estimation of Ψ _n is more involved.