Lag order selection for long-run variance estimation in econometrics

Abstract.

Estimating the long-run variance (LRV) is crucial for several econometric issues. Constructing reliable heteroskedasticity autocorrelation consistent (HAC) variance-covariance matrices and implementing efficient generalized method of moments (GMM) estimation procedures require a consistent LRV estimate. A good VARHAC estimator (HAC matrix with the spectral density at frequency zero constructed using a VAR spectral estimation) requires accurately estimating the sum of autoregressive (AR) coefficients; however, a criterion that minimizes the innovation variance does not necessarily yield the best spectral estimate. This article implements an optimal VARHAC estimator using an alternative information criterion, considering the bias in the sum of the parameters for the AR estimator of the spectral density at frequency zero.

Keywords:

• Long-run Variance
• HAC
• VAR
• Spectral Density at Frequency Zero
• Information Criteria

JEL classification::

• C12
• C22
• C32
• C51
• C52
• C58

Notes

1. For a complete survey of LRV estimation methods in time series, see Hirukawa (2023).

2. Sections 2.1, 2.2, and 5.2.1 are based on Morales (2010).

3. As we see later in Section 2.1, for the univariate case, to minimize the MAE is equivalent to minimize the MSE.

4. Given that ln(x) is just a monotonic transformation of x, minimizing any of them gives the same minimizer.

5. Remember that for lag orders below the true order, the negative bias in the AR coefficients is higher the larger the true lag order (Shaman and Stine, 1988). This way, the bias component of ASIC is decreasing to a minimum value around the true order of the model.

6. See Newey and West (1994) for the determination of the bandwidth M.

7. Given that an invertible MA(1) process can be written as an infinitive AR series, this is the most important case to consider for the selection of the correct number of lags for a good finite order approximation.

8. If the same autocorrelation of the disturbance term is assumed for the regressors, the negative MA case will be similar to the positive MA case. This is because the autocorrelation for U_tX_t is given by the product of the autocorrelations of U_t and X_t, which would be independent of the sign of the MA coefficient.

9. Following Andrews and Monahan (1992), we consider multiplicative heteroskedasticity as in HET1, where the variance of the error term is proportional to the first non constant regressor.

10. The two instrumental variables are both AR(1) processes plus $i.i.d.N(0,1)$noise and not correlated with the error terms of the endogenous variables.