Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T21:16:12.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of Deterministic and Stochastic Predictors in Nonlinear Systems When the Disturbances Are Small

Published online by Cambridge University Press:  11 February 2009

Hassan Arvin-Rad
Hassan Arvin-Rad
Affiliation:
Florida International University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper compares the deterministic and stochastic predictors of nonlinear models when the disturbances are small. Large-sample properties of these predictors have been analyzed extensively in the econometric literature. While the deterministic predictors are asymptotically biased, there are some Monte Carlo experiments that suggest the magnitude of this bias is rather insignificant. Here, we offer a possible explanation of the smallness of the deterministic bias. It is shown that when the error terms have small standard deviation, the deterministic predictor turns out to be asymptotically unbiased. The results are based on deriving asymptotic expansions for alternative predictors. The asymptotic expansions carried out here are similar to the large-sample asymptotic expansions; however, the expansions here are in terms of the standard deviation of the disturbance terms. The results are then used to obtain the asymptotic bias and asymptotic mean squared prediction errors of the deterministic and stochastic predictors of a model containing the Box-Cox transformation.

Type
Articles
Information
Econometric Theory , Volume 13 , Issue 3 , June 1997 , pp. 368 - 391
Copyright
Copyright © Cambridge University Press 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCE

Amemiya, T. & Powell, J. (1981) A comparison of the Box-Cox maximum likelihood estimator and the non-linear two-stage least squares estimator. Journal of Econometrics 17, 351–381.CrossRefGoogle Scholar
Arvin-Rad, H. (1996) A Note on Deriving Asymptotic Expansions for Some Estimators. Mimeo, Florida International University, Miami.Google Scholar
Bemdt, E.R., Showalter, M.H., & Wooldridge, J.M. (1993) An empirical investigation of the Box-Cox model and a nonlinear least squares alternative. Econometric Reviews 12, 6S–102.Google Scholar
Bianchi, C. & Calzolari, G. (1980) The one-period forecast errors in nonlinear econometric models. International Economic Review 21, 201–208.CrossRefGoogle Scholar
Bianchi, C., Calzolari, G., & Corsi, P. (1976) Divergence in the results of stochastic and deterministic simulation of an Italian nonlinear econometric model. In Dekker, L. (ed.), Simulation of Systems, pp. 653–661. Amsterdam: North-Holland.Google Scholar
Bickel, P.J. & Doksum, K.A. (1981) An analysis of transformations revisited. Journal of the American Statistical Association 76, 296–311.CrossRefGoogle Scholar
Box, G.E.P. & Cox, D.R. (1964) An analysis of transformations. Journal of the Royal Statistical Society. Series B 26, 211–252.Google Scholar
Brown, B.W. & Mariano, R.S. (1984) Residual-based stochastic prediction and estimation in a nonlinear simultaneous system. Econometrica 52, 321–343.CrossRefGoogle Scholar
Brown, B.W. & Mariano, R.S. (1989a) Measures of deterministic prediction bias in nonlinear models. International Economic Review 30, 667–684.CrossRefGoogle Scholar
Brown, B.W. & (Mariano, R.S. (1989b) Predictors in dynamic nonlinear models: Large-sample behavior. Econometric Theory 5, 430–452.Google Scholar
Calzolari, G. (1979) Antithetic variates to estimate the simulation bias in nonlinear models. Economics Letters 4, 323–338.CrossRefGoogle Scholar
Carroll, R.J. & Ruppert, D. (1981) On prediction and the power transformation family. Biometrika 68, 609–615.CrossRefGoogle Scholar
Chibisov, D.M. (1973) An asymptotic expansion for a class of estimators containing maximum likelihood estimators. Theory of Probability and Its Applications 16, 295–303.Google Scholar
Collins, S. (1991) Prediction techniques for Box-Cox regression models. Journal of Business & Economic Statistics 9, 267–277.Google Scholar
Draper, N.R. & Cox, D.R. (1969) On distributions and their transformation to normality. Journal of the Royal Statistical Society, Series B 31, 472–476.Google Scholar
Duan, N. (1983) Smearing estimate:-A nonparametric ^transformation method. Journal of the American Statistical Association 78, 605–610.CrossRefGoogle Scholar
Fair, R. (1980) Estimating the expected predictive accuracy of econometric models. International Economic Review 21, 355–378.CrossRefGoogle Scholar
Fair, R. (1984) Specification, Estimation, and Analysis of Macroeconometric Models. Cambridge, Massachusetts: Harvard University Press.Google Scholar
Fisher, P. & Salmon, M. (1986) On evaluating the importance of nonlinearity in large macroeconometric models. International Economic Review 27, 625–646.CrossRefGoogle Scholar
Hall, S.G. & S.G.B. Henry (1988) Macroeconomic Modelling. Amsterdam: North-Holland.Google Scholar
Howrey, E. & Kelejian, H. (1971) Simulation versus analytical solution: The case of econometric models. In Naylor, T. (ed.), Computer Simulation Experiments with Models of Economic Systems, pp. 299–319. New York: Wiley.Google Scholar
Kadane, J. (1970) Testing overidentifying restrictions when the disturbances are small. Journal of the American Statistical Association 65, 182–185.CrossRefGoogle Scholar
Kadane, J. (1971) Comparison of k-class estimators when the disturbances are small. Econometrica 39, 723–737.CrossRefGoogle Scholar
Kadane, J. (1974) Testing a subset of the overidentifying restrictions. Econometrica 42, 853–867.CrossRefGoogle Scholar
Klein, L.R. (1971) An Essay on the Theory of Economic Prediction. Chicago: Markham Publishing.Google Scholar
Kobayashi, M. (1994) Power of tests for nonlinear transformation in regression analysis. Econometric Theory 10, 357–371.CrossRefGoogle Scholar
Mariano, R.S. & Brown, B.W. (1983) Asymptotic behavior of predictors in a nonlinear simultaneous system. International Economic Review 24, 523–536.CrossRefGoogle Scholar
Mariano, R.S. & Brown, B.W. (1985) Finite sample properties of predictors in nonlinear systems: Some initial results. Mimeo, University of Pennsylvania.Google Scholar
Nagar, A.L. (1959) The bias and moment matrix of the general k-class estimators of the parameters in structural equations. Econometrica 27, 575–595.CrossRefGoogle Scholar
Nagar, A.L. (1969) Stochastic simulation of the Brookings model. In Duesenberry, J.S., Klein, G. Fromm L.R. & Kuh, E. (eds.), The Brookings Model: Some Further Results, pp. 423–456. Amsterdam: North-Holland.Google Scholar
Nelson, H. & C.W.J. Granger (1979) Experience with using the Box-Cox transformation when forecasting economic time series. Journal of Econometrics 10, 57–69.CrossRefGoogle Scholar
Sargan, J.D. (1974) On the validity of Nagar's expansion for the moments of econometric estimators. Econometrica 42, 169–176.CrossRefGoogle Scholar
Spitzer, J. J. (1978) A Monte Carlo investigation of the Box-Cox transformation in small samples. Journal of the American Statistical Association 73, 488–495.Google Scholar
Spitzer, J. J. (1982a) A fast and efficient algorithm for the estimation of parameters in models with the Box-and-Cox transformation. Journal of the American Statistical Association 77, 760–766.CrossRefGoogle Scholar
Spitzer, J.J. (1982b) A primer on Box-Cox estimation. The Review of Economics and Statistics 64, 307–313.CrossRefGoogle Scholar
Spitzer, J.J. (1984) Variance estimates in models with Box-Cox transformation: Implications for estimation and hypothesis testing. The Review of Economics and Statistics 66, 645–652.CrossRefGoogle Scholar
Srinivasan, T.N. (1970) Approximations to finite sample moments of estimators whose exact sampling distributions are unknown. Econometrica 38, 533–541.CrossRefGoogle Scholar
Toedter, K. (1992) Structural estimation and stochastic simulation of large non-linear models. Economic Modelling 9, 121–128.Google Scholar
Wooldridge, J.M. (1992) Some alternatives to the Box-Cox regression model. International Economic Review 33, 935–955.CrossRefGoogle Scholar
Zarembka, P. (1974) Transformation of variables in econometrics. In Zarembka, P. (ed.), Frontiers in Econometrics, pp. 81–104. New York: Academic Press.Google Scholar