Evaluating Forecast Uncertainty in Econometric Models: The Effect of Alternative Estimators of Maximum Likelihood Covariance Matrix

Calzolari, Giorgio and Panattoni, Lorenzo (1984): Evaluating Forecast Uncertainty in Econometric Models: The Effect of Alternative Estimators of Maximum Likelihood Covariance Matrix. Published in: paper presented at The Fourth International Symposium on Forecasting. London Business School, July 8-11 (8 July 1984): pp. 1-33.

Abstract

Most of the methods proposed in the literature for evaluating forecast uncertainty in econometric models need an estimate of the structural coefficiencs covariance matrix among input data. When estimation is performed with full information maximum likelihood, alternative estimators of such a covariance matrix (Hessian, outer product, generalized least squares type matrix, quasi maximum likelihood type matrix), although asymptotically equ1valent, often produce large differences in practical applications. Experimental results will be given for some econometric models well known in the literature, both with hiscorical data and with data generated by Monte Carlo.

• Item Type: MPRA Paper
• Language: English
• Keywords: Econometric models; simultaneous equations; maximum likelihood; covariance matrix; standard error of forecast
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables

References

Amemiya, T. (1977), "The Maximum Likelihood and the Nonlinear Three-Stage Least Squares in the General Nonlinear Simultaneous Equation Model", Econometrica 45, 955-968.

Artus, P., G. Laroque, and G.Michel (1982), "Estimation of a Quarterly Model with Quantity Rationing". Paris: INSEE, discussion paper presented at the European Meeting of the Econometric Society, Dublin.

Berndt, E. K, Hall. B. H.. Hall, R.E., Hausman, J. A.: Estimation and inference in nonlinear structural models. Annals of Economic and Social Measurement 3. 653-665 (1974).

Bianchi, C., and G. Calzolari (1980), "The One-Period Forecast Errors in Nonlinear Econometric Models", International Economic Review 21, 201-208.

Bianchi, C., and G. Calzolari (1982), "Evaluating Forecast Uncertainty Due to Errors in Estimated Coefficients: Empirical Comparison of Alternative Methods", in Evaluating the Reliability of Macro-Economic Models, ed. by G. C. Chow and P. Corsi. New York: John Wiley & Sons, Inc., 251-277.

Brundy,J.M. and D.W.Jorgenson (1971), "Efficient Estimation of Simultaneous Equations by Instrumental Variables", The Review of Economics and Statistics 53, 207-224.

Calzolari, G. (1981), "A Note on the Variance of Ex-Post Forecasts in Econometric Models", Econometrica 49, 1593-1595.

Calzolari,G., and L.Panattoni (1983), "Hessian and Approximated Hessian Matrices in Maximum Likelihood Estimation : A Monte Carlo Study". Pisa: Centro Scientifico IBM, discussion paper presented at the European Meeting of the Econometric Society, Pisa.

Calzolari, G., and F. P. Sterbenz (1983), "Efficient Computation of Reduced Form Variances in Nonlinear Econometric Models", presented at the European Meeting of the Econometric Society, Pisa.

Chernoff, H., and N. Divinsky (1953), "The Computation of Maximum-Likelihood Estimates of Linear Structural Equations", in Studies in Econometric Method, ed. by W. C. Hood and T. C. Koopmans. New York: John Wiley & Sons, Inc., Cowles Commission Monograph No. 14, 236-302.

Dagenais, M. G. (1978), The computation of FIML-estimates as iterative generalized least-squares estimates in linear and nonlinear simultaneous equation models. Econometrica 46, 1351-1362.

Dhrymes, P. J. (1970), Econometrics: Statistical Foundations and Applications. New York: Harper & Row.

Eisenpress, H. and J. Greenstadt (1966), "The Estimation of Nonlinear Econometric Systems". Econometrica 34, 851-86l.

Fair, R. C. (1980), "Estimating the Expected Predictive Accuracy of Econometric Models", International Economic Review 21, 355-378.

Girshick, M. A., and T. Haavelmo (1953), "Statistical Analysis of the Demand for Food: Examples of Simultaneous Estimation of Structural Equations", in Studies in Econometric Method, ed. by W. C. Hood and T. C. Koopmans. New York: John Wiley & Sons, Inc., Cowles Commission Monograph No. 14, 92-111.

Goldberger, A. S., A. L. Nagar and H. S. Odeh (1961), "The Covariance Matrices of Reduced-Form Coefficients and of Forecasts for a Structural Econometric Model", Econometrica 29, 556-573.

Gourieroux, C., A. Monfort, and A. Trognon (1984), "Pseudo Maximum Likelihood Methods: Theory". Econometrica 52, 681-700.

Hatanaka, M. (1978), "On the Efficient Estimation Methods for the Macro-Economic Models Nonlinear in Variables", Journal of Econometrics 8, 323-356.

Hausman, J. A. (1974), "Full Information Instrumental Variables Estimation of Simultaneous Equations Systems", Annals of Economic and Social Measurement 3, 641-652.

Hendry, D. F. (1971), "Maximum Likelihood Estimation of Systems of Simultaneous Regression Equations with Errors Generated by a Vector Autoregressive Process", International Economic Review 12, 257-272.

Klein, L. R. (1969), "Estimation of lnterdependent Systems in Macroeconometrics", Econometrica 37, 171-192.

Kmenta, J. (1971), Elements of Econometrics. New York: The Macmillan Company.

McCarthy, M. D. (1972), "Some Notes on the Generation of Pseudo-Structural Errors for Use in Stochastic Simulation Studies", in Econometric Models of Cyclical Behavior, ed. by B. G. Hickman. New York: National Bureau of Economic Research, Studies in Income and Wealth No. 36, Columbia University Press, 185-191.

Nissen, D. H. (1968), "A Note on the Variance of a Matrix", Econometrica 36, 603-604.

Parke, W. R.: An algorithm for FIML and 3SLS estimation of large nonlinear models. Econometrica 50, 81-95 (1982).

Rothenberg,T.J. and C.T.Leenders (1964), "Efficient Estimation of Simultaneous Equation Systems", Econometrica 32, 57-16.

Rothenberg.T.J. (1973). Efficient Estimation with A Priori Information, Cowles Foundation Monograph 23. New Haven: Yale Unlversity Press.

Schink, G. R. (1971), "Small Sample Estimates of the Variance Covariance Matrix of Forecast Error for Large Econometric Models: The Stochastic Simulation Technique". University of Pennsylvania: Graduate School of Arts and Sciences, Ph. D. Dissertation.

Sitzia, B., and M. Tivegna (1975), "Un Modello Aggregato dell'Economia Italiana 1952-1971", in Contributi alla Ricerca Economica No.4. Roma: Banca d'ltalia, 195-223.

Trivellato, U., and E. Rettore (1986), "Preliminary Data Errors and Their Impact on the Forecast Error of Simultaneous Equations Models". University of Padova: Dipartimento di Scienze Statistiche, discussion paper presented at the Fourth International Symposium on Forecasting, London.

White, H. (1982), Maximum Likelihood Estimation of Misspecified Models, Econometrica 50, 1-25.