Nonlinear Regression Analysis and Nonlinear Simulation Models

This paper is a survey of SAS System features for nonlinear models, with emphasis on new features for nonlinear regression. Topics include automatic calculation of analytic derivatives, estimation with nonlinear parameter restrictions, tests of nonlinear hypotheses, maximum likelihood and generalized method of moments (GMM) estimation, estimation of simultaneous systems of nonlinear regression equations, and distributed lags and time series error processes for nonlinear models. In addition, this paper will briefly discuss solving nonlinear equation systems, dynamic simulation of nonlinear systems, and optimization of nonlinear functions. The MODEL, NLIN, NLP, and GENMOD procedures are discussed. Introduction For simplicity, many researchers assume that their problems can be represented by linear models. This assumption is valid if the problem is truly linear or if you are restricted to studying only a small area of the problem space. Other researchers transform their problems appropriately to obtain linear models. While linear models are useful for much research, nonlinearity pervades our every day life and should not be ignored. This paper will concentrate on the estimation and simulation of nonlinear models. Estimation of nonlinear models usually requires finding the minimum (or maximum) of a nonlinear function. Solving for unknown variables in nonlinear equations requires finding zeros of the equations as a function of the unknown variables. Nonlinear Estimation A model can be nonlinear in its parameters, nonlinear in its observed variables, or nonlinear in both its parameters and variables. Nonlinear in the parameters means that the mathematical relationship between the variables and parameters is not required to have a linear form. (A linear model is a special case of a nonlinear model.) Example of Nonlinear Estimation Consider a simple exponential model for the decay of a radioactive isotope: conc = conc0 exp(rate t) (1) where conc0 is the initial concentration, t is time, and rate is the rate of decay. This model can also be written as a linear model with a log link function, the function that associates the regressors with the response variable. log(conc) = logconc0 + rate t (2) Because the model can be written as a generalized linear model, the GENMOD procedure can be used to estimate the model parameters using the following SAS code: proc genmod data=decay; model conc = t / dist = normal link = log noscale; run; The output is shown in Output 1. The reported INTERCEPT value of 1.3756 is the log of the parameter conc0. Output 1. PROC GENMOD Estimation Results