Lag-Order Selection Criteria

Definition

Lag-order selection criteria choose the autoregressive order $p$ of a VAR model by balancing fit against parsimony. Given a maximum candidate order $p_{\max}$ , one fits vector autoregression (VAR) $(m)$ models for $m = 0, \ldots, p_{\max}$ (holding the effective sample size fixed by using the same $p_{\max}$ presample values throughout) and picks the $\hat{p}$ that minimises a penalised log-determinant criterion.

Key Ideas

The fit term $\log\det(\hat{\Sigma}(m))$ decreases (weakly) as $m$ increases — more lags always improve in-sample fit.
The penalty term offsets overfitting; the rate at which the penalty grows with $T$ determines whether the criterion is consistent.
AIC (Akaike Information Criterion) is asymptotically biased upward (overestimates $p$ with positive probability even in large samples).
HQ (Hannan-Quinn) and SC (Schwarz Criterion) are both consistent ( $\hat{p} \xrightarrow{p} p$ ) under mild conditions, including $I(1)$ /cointegrated processes (Paulsen 1984).
The finite-sample inequality $\hat{p}(\text{SC}) \leq \hat{p}(\text{HQ}) \leq \hat{p}(\text{AIC})$ holds for every sample with $T > 16$ (Lütkepohl 1991).

How It Works

General Form

All popular criteria share the structure

$C_T(m) = \log\det\!\bigl(\hat{\Sigma}(m)\bigr) + c_T \cdot \varphi(m) \tag{1}$

where $\hat{\Sigma}(m) = T^{-1}\sum_{t} \hat{u}_t \hat{u}_t'$ is the residual covariance from a VAR $(m)$ , $\varphi(m)$ is a penalty function (usually $mK^2$ , the total free parameters in $A_1,\ldots,A_m$ ), and $c_T$ is a sequence depending on $T$ .

The Three Standard Criteria

Criterion	Formula	Penalty rate $c_T$	Consistent?
AIC (Akaike 1973, 1974)	$\log\det(\hat\Sigma(m)) + \tfrac{2}{T} mK^2$	$2/T$	No
HQ (Hannan-Quinn 1979)	$\log\det(\hat\Sigma(m)) + \tfrac{2\log\log T}{T} mK^2$	$2\log\log T/T$	Yes
SC (Schwarz 1978; Rissanen 1978)	$\log\det(\hat\Sigma(m)) + \tfrac{\log T}{T} mK^2$	$\log T/T$	Yes

SC is sometimes called BIC (Bayesian Information Criterion); it imposes the heaviest penalty and therefore selects the most parsimonious model.

Finite-Sample Ordering

For every sample of size $T > 16$ :

$\hat{p}(\text{SC}) \leq \hat{p}(\text{HQ}) \leq \hat{p}(\text{AIC})$

SC can never select a higher order than HQ, and HQ can never select a higher order than AIC. This holds as a finite-sample algebraic identity, not just asymptotically.

Consistency Under Integration

The consistency results for HQ and SC extend to $I(1)$ cointegrated processes (Paulsen 1984), even though the ordinary least squares (OLS) estimator's covariance matrix is singular in that case. Consistency of $\hat{p}$ holds provided the true data-generating process (DGP) has a finite-order VAR representation and $p_{\max}$ exceeds the true order.

Practical Considerations

The sample size $T$ is held constant across candidate orders (same presample values for all $m$ ); this makes $\hat\Sigma(m)$ comparable across models.
All criteria are computed from the levels VAR form (2.1), because the cointegrating rank is typically unknown at the lag selection stage.
Sequential testing (eliminate lags one by one via $F$ -tests) is an alternative but its overall Type I error grows with $p_{\max}$ ; information criteria avoid this problem.

Multi-Step Prediction Criterion (Schorfheide 2005)

Standard information criteria choose the lag order $p$ but fix the estimator as maximum likelihood estimator (MLE). Schorfheide (2005) extends selection to jointly choose $p$ and the predictor type: the MLE predictor (iterates the one-step companion matrix, $\hat{F}_T^h Y_t$ ) or the LFE predictor (Loss Function Estimator — regresses $y_t$ directly on $y_{t-h}$ , minimizing the $h$ -step quadratic loss).

The two predictors share the same form but differ in which lag- $h$ cross-covariance they use: MLE uses $\hat{S}_{01}\hat{S}_{11}^{-1}$ (lag-1 covariances iterated $h$ times); LFE uses $\hat{S}_{0h}\hat{S}_{hh}^{-1}$ (lag- $h$ covariances estimated directly). Under correct specification they are asymptotically equivalent; under misspecification the direct LFE estimation tilts bias toward the $h$ -step loss.

The Prediction Criterion (PC) modifies Shibata's (1980) Final Prediction Error to provide an asymptotically unbiased estimate of $h$ -step prediction risk $\mathcal{R}(p, \text{est})$ :

$\text{PC}(p,\text{est}) = \hat{\mathcal{R}}(p,\text{est}) + \tfrac{2h}{T}\,\text{tr}\!\bigl(\hat{\Omega}\,\hat{\Xi}(p,\text{est})\bigr)$

where the bias-correction term penalizes estimator variance through the asymptotic covariance $\hat{\Omega}$ of the one-step innovations and $\hat{\Xi}$ depends on the predictor type and lag order.

Monte Carlo results (100,000 replications). Under local misspecification parameterized by scalar $a$ (DGP drifts toward VAR $(p^*)$ at rate $T^{-1/2}$ ):

$a = 0$ (no misspecification): SIC (Schwarz Information Criterion)+MLE(VAR(1)) is optimal; MLE(VAR(1)) reduces risk $\approx 70\%$ over baseline.
$a \geq 5$ (large misspecification): PC+LFE dominates; LFE's direct estimation absorbs the bias at the cost of higher variance.
MLE and LFE are substitutes in correcting misspecification: adding more lags to an MLE predictor and switching to LFE both reduce bias but increase variance; the PC selects which trade-off is better ex ante.

Relation to Bayesian VARs. Bayesian VARs (Doan-Litterman-Sims 1984; Litterman 1986) are asymptotically equivalent to MLE predictors but dominate in finite samples through shrinkage. SIC approximates the Bayesian marginal data density in large samples.

Bayesian Marginal Likelihood Approach (Warne 2006)

When the cointegration rank $r$ is also unknown, frequentist information criteria cannot directly handle the rank-lag interaction. Warne (2006) replaces the information-criterion framework with marginal likelihood comparison over the joint $(r,k)$ space:

$p(r,k|Y) \propto p(Y|r,k)\, p(r,k)$

where $p(Y|r,k)$ is computed by the Chib (1995) Marginal Likelihood Identity (MLI) (see Marginal Data Density). Under the informative $\Gamma|\Omega$ prior (block-diagonal Minnesota shrinkage), the marginal likelihood at full rank $r=n$ for any lag order $k$ is available analytically (Corollary 1):

$\log p(Y|k,r=n) = \text{const} + \log\frac{|\hat{S}_\Omega(k)|^{-(T+q+nk+d+n+1)/2}}{|\hat{S}_\Omega(0)|^{-(q+n+1)/2}} + \log\frac{|\Sigma_\Gamma(k)|^{-n/2}}{1}$

(exact form depends on prior hyperparameters; see Warne 2006 Corollary 1 for the full expression). This eliminates the need for Markov Chain Monte Carlo (MCMC) when comparing lag orders at full rank. For $r < n$ , MCMC via the Gibbs sampler is required.

Comparison with frequentist criteria. In the euro area M3 application (Warne 2006), the marginal likelihood strongly selects $k=2$ (posterior $\approx 98\%$ ) across ranks — consistent with SC/HQ. The advantage over AIC/HQ/SC is that (a) prior information is incorporated, (b) rank and lag are selected jointly rather than sequentially, and (c) uncertainty over $(r,k)$ is represented by a full posterior rather than a point estimate.

Why It Matters

Over-parameterised VARs produce imprecise forecasts; under-parameterised VARs are mis-specified — lag-order selection is the first specification decision in any VAR analysis.
The Bayesian approach (see Sims-Zha Prior) sidesteps the problem by shrinking rather than truncating lags, but the frequentist criteria remain the default in applied work and are needed for choosing $p_{\max}$ even in Bayesian analyses.
AIC's inconsistency is a known caveat: it tends to include extra lags, which can cause spurious dynamics in small samples.

Open Questions

Which criterion to use in small samples? SC's heavier penalty can underfit when $T$ is small; HQ is a compromise but still outperforms AIC asymptotically.
With high-dimensional VARs ( $K$ large), all criteria select short orders because the parameter count $mK^2$ grows rapidly — a motivation for Bayesian shrinkage.