Abstract
Constrained Maximum Likelihood (CML), developed at Aptech Systems, generates maximum likelihood estimates with general parametric constraints (linear or nonlinear, equality or inequality), using the sequential quadratic programming method. CML computes two classes of confidence intervals, by inversion of the Wald and likelihood ratio statistics, and by simulation. The inversion techniques can produce misleading test sizes, but Monte Carlo evidence suggests this problem can be corrected under certain circumstances.
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References
Amemiya, Takeshi (1985). Advanced Econometrics, (Cambridge, MA: Harvard University Press).
Bates, Douglas M. and Watts, Donald G. (1988). Nonlinear Regression Analysis and Its Applications, (New York: John Wiley and Sons).
Berndt, E., Hall, B., Hall, R., and Hausman, J. (1974). ‘Estimation and inference in nonlinear structural models’. Annals of Economic and Social Measurement, 3, 653-665.
Brent, R.P. (1972). Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice-Hall.
Browne, Michael W. and Arminger, Gerhard (1995). ‘Specification and estimation of mean-and covariance-structure models’, in Handbook of Statistical Modeling for the Social and Behavioral Sciences, Gerhard Arminger, Clifford C. Clogg, and Michael E. Sobel (eds.), New York: Plenum.
Cox, D.R. and Hinkley, D.V. (1974). Theoretical Statistics. London: Chapman and Hall.
Cook, R.D. and Weisberg, S. (1990). ‘Confidence curves in nonlinear regression’, Journal of the American Statistical Association, 85, 544-551.
Meeker, W.Q. and Escobar, L.A. (1995). ‘Teaching about approximate confidence regions based on maximum likelihood estimation’, The American Statistician, 49, 48-53.
Dennis, Jr., J.E., and Schnabel, R.B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall.
Efron, Gradley, Tibshirani, Robert, J. (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.
Fletcher, R. (1987). Practical Methods of Optimization. New York: Wiley.
Gallant, A.R. (1987). Nonlinear Statistical Models. New York: Wiley.
Gill, P.E. and Murray, W. (1972). ‘Quasi-Newton methods for unconstrained optimization’. J. Inst. Math. Appl., 9, 91-108.
Gourieroux, Christian, Holly, Alberto, and Monfort, Alain (1982). ‘Likelihood ratio test, Wald Test, and Kuhn-Tucker test in linear models with inequality constraints on the regression parameters’, Econometrica, 50, 63-80.
Greene, William H. (1990). Econometric Analysis. New York: Macmillan.
Han, S.P. (1977). ‘A globally convergent method for nonlinear programming.’ Journal of Optimization Theory and Applications, 22, 297-309.
Hartmann, Wolfgang M. and Hartwig, Robert E. (1995). ‘Computing the Moore-Penrose inverse for the covariance matrix in constrained nonlinear estimation’, SAS Institute, Inc., Cary, NC.
Hock, Willi and Schittkowski, Klaus (1981). Lecture Notes in Economics and Mathematical Systems. New York: Springer-Verlag.
Jamshidian, Mortaza and Bentler, P.M. (1993). ‘A modified Newton method for constrained estimation in covariance structure analysis.’ Computational Statistics & Data Analysis, 15 133-146.
Newton, M.A. and Raftery, A.E. (1994). ‘Approximate Bayesian inference with the weighted likelihood bootstrap’, J.R. Statist. Soc. B, 56 3-48.
O'Leary, Dianne P. and Rust, Bert W. (1986). ‘Confidence intervals for inequality-constrained least squares problems, with applications to ill-posed problems’. American Journal for Scientific and Statistical Computing, 7(2), 473-489.
Rubin, D.B. (1988). ‘Using the SIR algorithm to simulate posterior distributions’, in Bayesian Statistics 3, Bernardo, J.M., DeGroot, M.H., Lindley, D.V. and Smith, A.F.M. (eds.), pp. 395-402.
Rust, Bert W., and Burrus, Walter R. (1972). Mathematical Programming and the Numerical Solution of Linear Equations. New York: American Elsevier.
Schoenberg, Ronald (1995). CML Users Guide. Maple Valley, WA, USA: Aptech Systems, Inc.
Self, Steven G. and Liang, Kung-Yee (1987). ‘Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions’, Journal of the American Statistical Association, 82, 605-610.
Terrell, G.R. (1990). ‘The maximal smoothing principle in density estimation’, Journal of the American Statistical Association, 85, 470-477.
White, H. (1981). ‘Consequences and detection of misspecified nonlinear regression models.’ Journal of the American Statistical Association, 76, 419-433.
White, H. (1982). ‘Maximum likelihood estimation of misspecified models.’ Econometrica, 50, 1-25.
Wolak, Frank (1991). ‘The local nature of hypothesis tests involving inequality constraints in nonlinear models’, Econometrica, 59, 981-995.
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Schoenberg, R. Constrained Maximum Likelihood. Computational Economics 10, 251–266 (1997). https://doi.org/10.1023/A:1008669208700
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DOI: https://doi.org/10.1023/A:1008669208700