VAR(p) vector autoregression by multivariate least squares

SOURCE. Lutkepohl, H. (2005), New Introduction to Multiple Time Series Analysis, Springer; doi:10.1007/978-3-540-27752-1. The model is Eq. (2.1.1)/(3.2.1), y_t = nu + A_1 y_{t-1} + ... + A_p y_{t-p} + u_t, and the estimator is the multivariate least squares estimator of Section 3.2. Because every equation carries the same regressor set Z_t = (1, y_{t-1}', ..., y_{t-p}')', the LS estimator separates equation by equation into B = (Z'Z)^{-1} Z'Y and no SUR/GLS step is needed.

Usage

Varest(Y, p = 1, intercept = TRUE)

Arguments

Y

T-by-K matrix of observations; a plain vector is one series.

p

Lag order, at least 1.

intercept

Include the constant nu.

Value

List with coef, sigma_u, sigma_ml, resid, loglik, aic, bic, hq, logdet, n, k, p.

Details

Two residual covariances are reported: the ML plug-in U'U/T and U'U/(T - Kp - 1). The Gaussian log-likelihood (Eq. 3.4.5) is evaluated at the former, and the three order-selection criteria are log|Sigma-tilde| + c(T) * (free parameters)/T.

NOT read from the book's own page images – Lutkepohl (2005) is not in the local corpus. The estimator is the standard multivariate LS form, anchored against stats::lm reached by a different route.

References

Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer. doi:10.1007/978-3-540-27752-1.

Examples

set.seed(2)
Y <- matrix(rnorm(80), 40, 2)
Y[-1, ] <- Y[-1, ] + 0.5 * Y[-40, ]   # a weak VAR(1) signal
Varest(Y, 1)$loglik
#> [1] -119.7593