2. WEAK CONSISTENCY OF INFORMATION CRITERIA

We assume that the multivariate system may be written as an m-dimensional VAR(k) process given by

where the error term ε_t is a zero mean independent and identically distributed (i.i.d.) vector with finite positive definite covariance matrix. This VAR(k) process will be referred to as cointegrated of rank r if Π = I − Φ₁ − ··· − Φ_k has rank r. In this case the matrix Π may be decomposed as Π = αβ′ where α and β are matrices of dimension m × r. The error correction representation¹

of the system is given by

where Ψ_i, i = 1,…,k − 1 are functions of Φ_i, i = 1,…,k.

The general form of the loss function minimized by information criteria is given by

where l_T(θ) is the log likelihood of the model, s is the number of free parameters, and c_T(s) is a penalty term promoting model parsimony depending on s and the sample size. For three common information criteria the penalty terms are as follows: s (Akaike's information criterion) (Akaike, 1973), (s/2)ln(T) (Bayesian information criterion) (Schwarz, 1978), and s ln(ln(T)) (Hannan–Quinn information criterion [HQ]) (Hannan and Quinn, 1979). The model specification chosen is that for which IC(k) is minimized. In lag order selection it is well known that the estimated lag order for stationary and unit root nonstationary vector autoregressive (VAR) models will be weakly consistent iff c_T(k)

as T → ∞ and c_T(k) is bounded in k where k is the lag order. For a proof of the latter case, for deterministic penalty terms, see Paulsen (1984). Clearly, whereas BIC and HQ are weakly consistent for lag order selection, AIC is not. This is a well-known result for AIC (see, e.g., Shibata, 1976). Note that we choose to have a general expression for the penalty term to accommodate other less widely used criteria such as, e.g., the generalized information criterion (GIC) (see Konishi and Kitagawa, 1996) and the PIC (see Phillips, 1996; Phillips and Ploberger, 1994; Phillips and Ploberger, 1996). Note that whereas AIC, BIC, and HQ have deterministic penalty terms, GIC and PIC have stochastic penalty terms (hence the notation concerning the conditions on the asymptotic behavior of the generic penalty term). These results have been shown by various authors to extend to more general model selection frameworks (see, e.g., Sin and White, 1996; Kapetanios, 2001).

Aznar and Salvador (2000) show that the standard conditions on the penalty terms for weak consistency in lag selection extend to the cointegration framework. In particular, they show that the cointegration rank and the lag order may be jointly weakly consistently estimated iff c_T(k) → ∞ and c_T(k)/T → 0 as T → ∞. This result holds only for information criteria whose penalty terms are deterministic, and therefore criteria such as the GIC are not covered. We also note that the asymptotic properties of PIC have been discussed in Chao and Phillips (1999) where weak consistency of PIC in jointly estimating cointegration rank and VAR lag order is established. Note that this is the first paper to give a consistency result for estimating cointegration rank via an information criterion.

3. THE ASYMPTOTIC DISTRIBUTION OF THE RANK ESTIMATE USING AKAIKE'S INFORMATION CRITERION

Following Pesaran and Pesaran (1997) we specify the number of free parameters for a multivariate cointegrated model with no intercept or time trend to be equal to s = m²(k⁰ − 1) + 2mr − r² where k⁰ is the true lag order of the system. The penalty terms for AIC, BIC, and HQ are then given, respectively, by s, (s/2)ln(T), and s ln(ln(T)) or equivalently by

where

.

It is clear that AIC is not consistent in rank determination. Nevertheless it is also clear that the probability of picking a rank that is lower than the true rank goes to zero asymptotically, as we also show in the Appendix. The following theorem provides the means for determining the asymptotic probabilities that AIC will pick a higher rank than the true one.

THEOREM. Consider the VAR model of (1) with μ = 0 and known k⁰. The asymptotic distribution of the rank estimate obtained through AIC is given by

where p_r are given by expression (A.7) in the Appendix and r⁰ is the true rank of the model.

Note that we assume a known true lag order for the derivation of the asymptotic distribution of AIC. Unlike the result of Aznar and Salvador (2000) on the joint determination of lag order and cointegration rank, allowing for an unknown lag order in this context would obviously change the asymptotic distribution of the cointegration rank estimated using AIC.

The asymptotic distribution of the estimate of the cointegration rank depends only on d = m − r⁰. Tables 1 and 2 show the distribution of the cointegration rank estimate for the case of no deterministic terms and the case of an unrestricted constant obtained through simulation. Brownian motion is simulated using a random walk of 1,000 observations. Five thousand replications have been used.

In the standard case of lag order selection, the asymptotic probability of AIC picking a lag order larger than the true one is quite small and declines rapidly for higher lag orders. The results for the cointegration rank estimate do not follow this pattern. The probabilities of overestimation are quite large and depend crucially on the nature of the deterministic terms included in the model. In most cases the true rank is not even the mode of the asymptotic distribution of the estimate. As m − r⁰ rises, the problem is further accentuated. In the extreme case considered in the tables, when m − r⁰ = 9 and a constant is included in the model, the probability of picking the right rank is equal to just 1.9%.

The motivation behind the derivation of AIC is not consistency in the selection of the true model but optimization in terms of goodness of the selected model as measured by the Kullback–Leibler information metric.²

Therefore, our result does not necessarily imply that the criterion is in general “bad” in selecting cointegration rank because such a judgment would have to be related to a particular modeling purpose. Nevertheless, the optimality properties of AIC hold for stationary models. Currently, there is no compelling theoretical reason motivating the use of AIC for rank determination in cointegration models. Furthermore, we can provide some evidence in favor of methods that are parsimonious in cointegration rank selection such as BIC and PIC. In terms of forecasting and over long horizons, error correction models have been in general shown to have an advantage. However, Christoffersen and Diebold (1998) cast doubt on the notion that error correction models are better forecasting tools even at long horizons, at least with respect to the standard root mean square forecasting error criterion. They also argue that although unit roots are estimated consistently, modeling nonstationary series in (log) levels is likely to produce forecasts that are suboptimal in finite samples relative to a procedure that imposes unit roots, a phenomenon exacerbated by small sample estimation bias. Developing this argument, they suggest that for cointegrated series it is better to overestimate rather than underestimate the number of common trends, or in other words, underestimate the cointegrating rank.