Model selection tests for nonlinear dynamic models


  • Douglas Rivers,

    1. 1 Department of Political Science Stanford University Stanford CA 94305-60442 Department of Economics University of Southern California Los Angeles CA 90089-0253, USA
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  • and 1 Quang Vuong 1, 2

    1. 1 Department of Political Science Stanford University Stanford CA 94305-60442 Department of Economics University of Southern California Los Angeles CA 90089-0253, USA
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    • 1Corresponding author. E-mail:

  • 2As noted in Vuong (1989), the LR statistic can also be adjusted by some correction factors such as those proposed by Akaike (1973), Akaike (1974), Schwarz (1978), and Hannan and Quinn (1979) to reflect the parsimony of each competing model. For a recent contribution on penalizing the LR statistic, see Sin and White (1996).

  • 3Applications of Vuong's test, as it is called in the econometric literature, have appeared in empirical work. For instance, it has been used to test for the presence of collusion in Gasmi et al. (1992), for the presence of asymmetric information in Wolak (1994), for distributional assumptions in Paarsch (1997), and for discriminating a structural nonlinear model from linear counterparts in Caballero and Engel (1999).

  • 4It is worth noting that extensions of Cox's tests followed the lines described previously, namely extensions to time series models and incompletely specified models estimated by methods other than ML. See Walker (1967), Davidson and MacKinnnon (1981), Ericsson (1983), Godfrey (1983), Gourieroux et al. (1983) and Mizon and Richard (1986), among others.

  • 5Findley (1990) proposes an interesting graphical procedure that addresses this issue when the competing models are Gaussian ARMA or ARIMA models.

  • 6We are grateful to a referee for suggesting this example. Other model selection problems can be worked out similarly such as choosing between an AR(1) model and a MA(1) model. In particular, the latter problem has been treated differently using some Cox-type tests for nonnested hypotheses (see e.g. Walker (1967), King and McAleer (1987)).

  • 7To simplify, we assume that the sample size used for estimation is equal to the out-of-sample size used for model selection. Appropriate changes can accommodate an out-of-sample size p that increases at the same rate as n. See also West (1994), West (1996) for other situations such as limn→∞p/n = 0 or .

  • 8This assumption is stronger than necessary, but greatly facilitates the verification of the assumptions. Whenever possible, we indicate when it can be weakened.

  • 9Gaussianity can be relaxed as non-Gaussian ARMA (p, q) processes are also α-mixing of arbitrary size under appropriate conditions. See Pham and Tran (1980).

  • 10The preceding argument shows that stationarity and Gaussianity can be weakened for Assumption 15 (ii), (iii) to hold as it suffices that EYt2r be uniformly bounded for some r > 1.

  • 11The general case where Qnj (ω, γj) = dj {Mnj (ω, θj, τj), θj, τj} was not treated to economize on proofs and notations, but follows similarly. Moreover, to simplify, inline image is again assumed independent of n.

  • 12Sin and White (1996) provide conditions on the penalty functions ensuring weak or strong consistency of the adjusted likelihood criterion. Thus, combining their results with ours delivers a likelihood-based procedure that is consistent both as a model selection criterion and a model selection test of H0*.

  • 13Findley (1990) notes that comparing the (in-sample) log-likelihood values is also equivalent to comparing the one-step MSEP when the competing models are Gaussian ARMA or ARIMA models. Diebold and Mariano (1995) allow for more general losses than the MSEP, though their results require either that the parameters of the competing models be known or limn→∞p/n = 0, as noted by West (1996).

  • 14The rates n−1/4 and n−1/8 arise from the rate of mn in Assumption 27. As its proof shows, Theorem 4 actually holds for any rate of mn that guarantees the consistency of inline image for Vn, provided inline image in (i) and inline image in (ii). In particular, Andrews (1991) shows that the optimal rate of mn for the Bartlett weights w used by Newey and West (1987b) is O(n1/3), and hence does not satisfy Assumption 27. See also Andrews (1991) for optimal weights and data-dependent automatic determination of mn.

  • 15Note that inline image, where Unt is defined as in (21) but with inline image replacing inline image. Hence, from E(Unt) =μnt it can be easily shown that inline image, if inline image. The latter condition, however, is not sufficient to ensure the consistency of inline image to σn2 because the near-epoch dependence of R′n Unt on Xt does not guarantee that the raw moment E(R′n Unt R′n Un,t−τ) vanishes as τ increases, when E(R′n Unt) |= 0. On the other hand, E(R′n Unt) = 0 for all n, t trivially implies conditions (i)–(ii).

  • 16Similarly, for the in-sample MSEP studied in Section 18, the estimator inline image appearing in (17) can be taken to be given by (20), where inline image is replaced by the difference in squared prediction errors inline image.

  • 17Similar results hold when using the in-sample MSEP for choosing between the two competing AR models.


This paper generalizes Vuong (1989) asymptotically normal tests for model selection in several important directions. First, it allows for incompletely parametrized models such as econometric models defined by moment conditions. Second, it allows for a broad class of estimation methods that includes most estimators currently used in practice. Third, it considers model selection criteria other than the models’ likelihoods such as the mean squared errors of prediction. Fourth, the proposed tests are applicable to possibly misspecified nonlinear dynamic models with weakly dependent heterogeneous data. Cases where the estimation methods optimize the model selection criteria are distinguished from cases where they do not. We also consider the estimation of the asymptotic variance of the difference between the competing models’ selection criteria, which is necessary to our tests. Finally, we discuss conditions under which our tests are valid. It is seen that the competing models must be essentially nonnested.