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

# Model selection tests for nonlinear dynamic models

Version of Record online: 4 NOV 2002

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

Additional Information

#### 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 online: 4 NOV 2002
- Version of Record online: 4 NOV 2002
- Received on November 1999

### References

- 11973); Information theory and an extension of the likelihood ratio principle. . In(
*Proceedings of the Second International Symposium of Information Theory*, (PetrovB.N.& CsakiF.. ed.) , pp. 257–81. Akademiai Kiado, Budapest. - 21974); A new look at the statistical model identification. IEEE Transactions and Automatic Control AC-19: 716–23.(
- 3 (
- 41985);(
*Advanced Econometrics*, Harvard University Press, Cambridge. - 51987); Consistency in nonlinear econometric models: a generic uniform law of large numbers. Econometrica 55: 1465–71.(
- 61991); Heteroskedasticity and autocorrelation consistent matrix estimation. Econometrica 59: 817–58.(
- 71994); Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62: 43–72.(
- 81988); Inferences in econometric models with structural change. Review of Economic Studies 55: 615–40.& (
- 91985); A unified theory of consistent estimation for parametric models. Econometric Theory 1: 151–78.& (
- 101999); Explaining investment dynamics in U.S. manufacturing: a generalized (S, s) approach. Econometrica 67: 783–826.& (Direct Link:
- 111961); Tests of separate families of hypotheses. . In(
*In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability*, vol. 1. , pp. 105–23. - 121962); Further results on tests of separate families of hypotheses. Journal of the Royal Statistical Society, Series B 24: 406–24.(
- 131981); Several tests for model specification in the presence of alternative hypotheses. Econometrica 49: 781–93.& (
- 141995); Comparing predictive accuracy. Journal of Business and Economic Statistics 13: 253–63.& (
- 151982); Misspecified models with dependent observations. Journal of Econometrics 20: 35–58.& (
- 161983); Asymptotic properties of instrumental variables statistics for testing non-nested hypotheses. Review of Economic Studies 50: 287–304.(
- 171990); Comparing information in forecasts from econometric models. American Economic Review 80: 375–89.& (
- 181990); Making Difficult Model Comparisons, mimeo, U.S. Bureau of the Census.(
- 191991); Convergence of finite multistep predictors from incorrect models and its role in model selection. Note di Matematica XI: 145–55.(
- 201998); New capabilities and methods of the X-12-ARIMA seasonal adjustment program. Journal of Business and Economic Statistics 16: 127–77., , , & (
- 211993); Moment bound for deriving time series CLT's and model selection procedures. Statistica Sinica 3: 453–80.& (
- 221976);(
*Introduction to Statistical Time Series*, Wiley, New York. - 231988);& (
*A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models*, Basil Blackwell, New York. - 241992); Econometric analysis of collusive behavior in a soft drink industry. Journal of Economics and Management Strategy 1: 277–311., & (Direct Link:
- 251990); Testing nonnested Euler conditions with quadrature-based methods of approximation. Journal of Econometrics 46: 273–308.& (
- 261983); Testing non-nested models after estimation by instrumental variables or least squares. Econometrica 51: 355–65.(
- 272000); Discrepancy Risk Model Selection Test Theory for Comparing Possibly Misspecified or Nonnested Models, mimeo, University of Texas, Dallas.(
- 281983); Testing nested or non-nested hypotheses. Journal of Econometrics 21: 83–115., & (
- 292000); Economic and statistical measures of forecast accuracy. Journal of Forecasting 19: 537–60.& (
- 301986);, , & (
*Robust Statistics: The Approach Based on Influence Functions*, Wiley, New York. - 311979); The determination of the order of an autoregression. Journal of the Royal Statistical Society, Series B 41: 190–5.& (
- 321982); Large sample properties of generalized method of moments estimators. Econometrica 50: 1029–54.(
- 332000);(
*Econometrics*, Princeton University Press, Princeton. - 341981); Heterogeneity and state dependence. . In(
*Studies in Labor Markets*, (RosenS.. ed.) , pp. 91–139. University of Chicago Press, Chicago. - 35 (
- 361971);& (
*Independent and Stationary Sequences of Random Variables*, Wolters-Noordhoff, Groningen. - 371969); Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40: 633–43.(
- 381987); Further results on testing AR(1) against MA(1) disturbances in the linear regression model. Review of Economic Studies 54: 649–63.& (
- 391996); Generalized information criterion in model selection. Biometrika 83: 875–90.& (
- 401951); On information and sufficiency. Annals of Mathematical Statistics 22: 79–86.& (
- 411998); Selection of regressors in econometrics: parametric and nonparametric methods. Econometric Reviews 17: 227–73.(
- 421996); Nonparametric selection of regressors: the nonnested case. Econometrica 64: 207–19.& (
- 43 & (
- 441987); Selecting the best linear regression model: a classical approach. Journal of Econometrics, Annals 35: 3–23.& (
- 451988); A test whether two AIC's differ significantly. South African Statistical Journal 22: 153–61.(
- 461986);& (
*Model Selection*, Wiley, New York. - 471993); Robust model selection and M-estimation. Econometric Theory 9: 478–93.(
- 481983); Model specification tests against non-nested alternatives. Econometric Reviews 2: 85–110.(
- 49 (
- 502000); Model Selection for Non-linear Dynamic Models, mimeo, Universita Bocconi.(
- 511980); Robust estimation of autoregressive models. . In(
*Directions in Time Series*, (BrillingerD.R.& TiaoG.C.. ed.) , pp. 228–62. Institute of Mathematical Statistics, Hayward. - 521987); Specification tests for separate models: a survey. . In(
*Specification Analysis in the Linear Model*, (KingM.L.& GilesD.E. A.. ed.) , pp. 146–95. Routledge and Kegan Paul, London. - 531983); Empirical exchange rate models of the seventies: do they fit out of sample?Journal of International Economics 14: 3–24.& (
- 541986); The encompassing principle and its applications to testing non-nested hypotheses. Econometrica 54: 657–78.& (
- 551978); Chi-square tests. . In(
*Studies in Statistics*, vol. 19. (HoggR.V.. ed.) The Mathematical Association of America, - 561990); Alternative approaches to model choice. Journal of Economic Behavior and Organization 14: 97–125., & (
- 571994); Large sample estimation and hypothesis testing. . In& (
*Handbook of Econometrics*, vol. 4. (EngleR.F.& McFaddenD.. ed.) , pp. 2111–245. North Holland, Amsterdam. - 581987a); Hypothesis testing with efficient method of moments estimators. International Economic Review 28: 777–87.& (
- 591987b); A simple positive semi-definite heteroskedasticty and autocorrelation consistent covariance matrix. Econometrica 55: 703–8.& (
- 601997); Deriving an estimate of the optimal reserve price: an application to British Columbian timber sales. Journal of Econometrics 78: 333–57.(
- 611974); On the general problem of model selection. Review of Economic Studies 41: 153–71.(
- 621994); A generalized R& (
^{2}criterion for regression models estimated by the instrumental variables method. Econometrica 62: 705–10. - 631980); The Strong Mixing Properties of the Autoregressive Moving Average Time Series Models, Seminaire de Statistique, Grenoble.& (
- 641994); Estimation of semiparametric models. . In(
*Handbook of Econometrics*, vol. 4. (EngleR.F.& McFaddenD.. ed.) , pp. 2443–521. North-Holland, Amsterdam. - 651991); Model selection tests for nonlinear dynamic models, Working Paper 9108, INRA-ESR, Toulouse.& (
- 661985); Robust model selection in regression. Statistics and Probability Letters 3: 21–3.(
- 671958); The estimation of economic relationships using instrumental variables. Econometrica 26: 393–415.(
- 68 (
- 691998); An application of multiple comparison techniques to model selection. Annals of the Institute of Statistical Mathematics 50: 1–15.(
- 701996); Information criteria for selecting possibly misspecified parametric models. Journal of Econometrics 71: 207–25.& (
- 711992); Non-nested tests for competing models estimated by generalized method of moments. Econometrica 60: 973–80.(
- 721968); A test of the mean square error criterion for restrictions in linear regression. Journal of the American Statistical Association 63: 558–72.& (
- 731989); Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57: 307–33.(
- 741991); Tests for model selection using power divergency statistics, Working Paper 9106, INRA–ESR, Toulouse.& (
- 751993a); Minimum chi-square estimation and tests for model selection. Journal of Econometrics 56: 141–68.& (
- 761993b); Selecting estimates using chi-square statistics. Annales d’Economie et de Statistique 30: 143–64.& (
- 77 (
- 781994); Asymptotic Inference about predictive ability, mimeo, University of Wisconsin.(
- 791996); Asymptotic inference about predictive ability. Econometrica 64: 1067–84.(
- 801982); Maximum likelihood estimation of misspecified models. Econometrica 50: 1–25.(
- 811984);(
*Asymptotic Theory for Econometricians*, Academic, New York. - 822000); A reality check for data snooping. Econometrica 68: 1097–126.(
- 831994); An econometric analysis of the asymmetric information regulator utility interaction. Annales d’Economie et de Statistique 34: 13–69.(