Lag Selection and Parsimonious VARs

Why Lag Selection Is a Model-Size Decision

A VAR with more lags can represent richer dynamics, but every additional lag adds coefficients to every equation. In a multivariate system, this cost grows quickly: adding one lag to a VAR with \(m\) variables adds \(m^2\) slope coefficients before deterministic terms are counted. If the available sample is short, an unrestricted high-order VAR can use a large share of the data estimating parameters.

Information criteria provide a systematic way to compare candidate lag orders. They reward improved fit while penalizing model size. Their scores are relative: among the candidates being compared, the criterion with the lowest value is preferred. A criterion is not a test that establishes the true lag order, and its chosen order depends on the candidate set, sample, and specification.

Model and Parameter Count

Write a VAR(\(p\)) with deterministic regressors as

where \(y_t\) is an \(m\times1\) vector, each \(A_j\) is an \(m\times m\) coefficient matrix, \(d_t\) collects deterministic terms, and \(u_t\) is the innovation vector. The sample size \(n\) here is the number of usable observations after forming lags. All candidate models must use the same \(n\) for a direct information-criterion comparison.

If every equation has \(q\) common deterministic regressors, there are \(mq\) deterministic coefficients and \(pm^2\) lag coefficients. Thus the number of free regression coefficients in an unrestricted VAR(\(p\)) is

For an intercept-only specification, \(q=1\), so \(k_p=m+pm^2\). The innovation covariance matrix is also estimated, but its number of distinct entries is \(m(m+1)/2\), the same for every candidate lag order in this comparison. We omit that common count from \(k_p\); including it in every model's penalty adds the same amount to all scores and does not change their ranking. Under restrictions, count the actual free regression coefficients rather than using the unrestricted formula.

AIC, BIC, and HQ

Let \(\widehat{\Sigma}_p\) be the estimated innovation covariance matrix for a candidate VAR(\(p\)). With Gaussian innovations, the concentrated log likelihood, after dropping constants common to all candidates, is proportional to \(-n\log|\widehat{\Sigma}_p|/2\). The following per-observation criteria use that fit term and penalize the number of free regression coefficients:

The logarithmic determinant measures residual covariance jointly across the system; smaller values indicate a better-fitting covariance estimate under this criterion. The penalty per free coefficient is \(2/n\) for AIC, \(\log(n)/n\) for BIC, and \(2\log(\log n)/n\) for HQ. In the usual settings where these criteria are compared, HQ's penalty lies between AIC's and BIC's. The penalty changes with \(n\), so a penalty calculated for one sample size must not be reused for another.

These expressions use the maximum-likelihood covariance estimate, conventionally formed with divisor \(n\), and the same convention for every candidate. Software may use alternative covariance corrections or report scores on a different scale. Rankings are meaningful only when models are evaluated on a common sample and with compatible definitions.

Why the Criteria Penalize Complexity

The fit term cannot be used by itself: adding regressors generally cannot worsen the in-sample least-squares fit, so selecting solely by fit tends to favor unnecessarily large models. The penalty makes additional parameters worthwhile only when the reduction in the fit term is sufficiently large.

One justification for the BIC penalty comes from approximating a model's marginal likelihood. Under regularity conditions, with a well-behaved likelihood near an interior maximum and a prior that is positive there, a quadratic expansion of the log likelihood around its maximum gives a Gaussian integral over \(K\) parameters. Each parameter contributes a factor of order \(n^{-1/2}\), so the log marginal likelihood is approximately

Multiplying by \(-2\) yields \(-2\log L(\widehat{\theta})+K\log n\), the BIC form up to terms that do not vary across candidates. AIC instead uses a penalty motivated by estimating expected predictive information loss. The criteria therefore embody different goals and asymptotic reasoning; they need not select the same order.

Short Samples and Restricted Lag Structures

In a short sample, a restricted structure can reduce the number of coefficients and may make estimation more stable. A restriction might exclude selected cross-variable lags, impose a common lag order only on certain variables, or omit all coefficients at a higher lag. Such restrictions should have a substantive or design-based rationale. They change the model being fitted, so the resulting estimate is conditional on those restrictions being correct enough for the intended use.

Restrictions must be reflected in the penalty. For a restricted model, \(k\) is the number of regression coefficients left free after imposing the restrictions. Do not penalize it as if every coefficient in the unrestricted VAR were estimated. Conversely, do not count each equation's regressors as though they were the entire system: the total coefficient count is across all \(m\) equations.

Practical Choice and Common Pitfalls

A useful comparison begins by stating the candidate set and checking that all models use the same observations and deterministic terms. Compute the free-parameter count for each model, apply the criterion with the actual effective sample size, and compare the resulting scores. If restrictions are considered, describe precisely which coefficients are fixed or excluded and count only those left free.

For a system with many variables, the \(pm^2\) term can become large even at modest \(p\). A high-order unrestricted VAR may therefore be difficult to estimate precisely in a short sample. AIC often favors richer models than BIC because its penalty per parameter does not grow with \(\log n\); this is a tendency, not a guarantee for any particular data set. HQ provides a different balance. The preferred order should be understood alongside the specification and the purpose of the analysis, not as a universal property of the data.

• Wrong: “VAR(3) fits best in sample, so it must be preferred.” Correction: In-sample fit alone rewards added regressors; compare fit and complexity using a stated criterion.
• Wrong: “A VAR(2) has two parameters.” Correction: Lag order is not the parameter count. With \(m\) variables, each lag adds \(m^2\) unrestricted slope coefficients.
• Wrong: “The lowest BIC means the model is true.” Correction: It is the lowest score among the candidates under this definition, not a truth probability or a guarantee against misspecification.
• Wrong: “A restricted model has the same penalty as the unrestricted model with the same maximum lag.” Correction: Count its actual free coefficients; restrictions change the parameter count.