^{1}Corresponding author. E-mail: qvuong@usc.edu

# Model selection tests for nonlinear dynamic models

Article first published online: 4 NOV 2002

DOI: 10.1111/1368-423X.t01-1-00071

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#### How to Cite

Rivers, D. and Vuong, Q. (2002), Model selection tests for nonlinear dynamic models. The Econometrics Journal, 5: 1–39. doi: 10.1111/1368-423X.t01-1-00071

^{2}As noted in Vuong (1989), the LR statistic can also be adjusted by some correction factors such as those proposed by , 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).^{3}Applications 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).^{4}It 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.^{5}Findley (1990) proposes an interesting graphical procedure that addresses this issue when the competing models are Gaussian ARMA or ARIMA models.^{6}We 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)).^{7}To 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 for other situations such as lim_{n∞}p/n = 0 or ∞.^{8}This assumption is stronger than necessary, but greatly facilitates the verification of the assumptions. Whenever possible, we indicate when it can be weakened.^{9}Gaussianity 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).^{10}The preceding argument shows that stationarity and Gaussianity can be weakened for Assumption 15 (ii), (iii) to hold as it suffices that EY_{t}^{2r}be uniformly bounded for some r > 1.^{11}The general case where Q_{n}^{j}(ω, γ^{j}) = d_{j}{M_{n}^{j}(ω, θ^{j}, τ^{j}), θ^{j}, τ^{j}} was not treated to economize on proofs and notations, but follows similarly. Moreover, to simplify, is again assumed independent of n.^{12}Sin 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 H_{0}^{*}.^{13}Findley (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 lim_{n∞}p/n = 0, as noted by West (1996).^{14}The rates n^{−1/4}and n^{−1/8}arise from the rate of m_{n}in Assumption 27. As its proof shows, Theorem 4 actually holds for any rate of m_{n}that guarantees the consistency of for V_{n}, provided in (i) and in (ii). In particular, Andrews (1991) shows that the optimal rate of m_{n}for the Bartlett weights w_{nτ}used by Newey and West (1987b) is O(n^{1/3}), and hence does not satisfy Assumption 27. See also Andrews (1991) for optimal weights and data-dependent automatic determination of m_{n}.^{15}Note that , where U_{nt}is defined as in (21) but with replacing . Hence, from E(U_{nt}) =μ_{nt}it can be easily shown that , if . The latter condition, however, is not sufficient to ensure the consistency of to σ_{n}^{2}because the near-epoch dependence of R′_{n}U_{nt}on X_{t}does not guarantee that the*raw*moment E(R′_{n}U_{nt}R′_{n}U_{n,t−τ}) vanishes as τ increases, when E(R′_{n}U_{nt}) |= 0. On the other hand, E(R′_{n}U_{nt}) = 0 for all n, t trivially implies conditions (i)–(ii).^{16}Similarly, for the in-sample MSEP studied in Section 18, the estimator appearing in (17) can be taken to be given by (20), where is replaced by the difference in squared prediction errors .^{17}Similar results hold when using the in-sample MSEP for choosing between the two competing AR models.

#### Publication History

- Issue published online: 4 NOV 2002
- Article first published online: 4 NOV 2002
- Received on November 1999

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