Assessing the severity and consequences of collinearity in nonlinear models is more complicated than in linear models. To illustrate, we first discuss its detection in nonlinear regression, and then in the context of maximum likelihood estimation.
Consider the nonlinear regression model
y = f (X, P) + e, (12.19)
where e ~ (0, о 2I) and f(X, P) is some nonlinear function that relates the independent variables and parameters to form the systematic portion of the model. The nonlinear least squares estimator chooses S to minimize
S(P) = [y – f(X, P)]'[y – f(X, P)j.
The first order conditions yield the least squares solution,
Z(p)'[y – f(X, p)j = 0, (12.20)
where the T x K matrix Z(P) = cf(X, P)/3P’. Since equation (12.20) is nonlinear, the least squares estimates S must be obtained using numerical methods.
A useful algorithm for finding the minimum of S(P) is the Gauss-Newton. The Gauss-Newton algorithm is based on a first order Taylor’s series expansion of f(X, P) around a starting value p1. From that we obtain the linearized model
y(P0 = Z(P0P + e, (12.21)
where y(P1) = y – f(X, p1) + Z(P1)P1. In (12.21) the dependent variable and the "regressors" Z(P1) are completely determined given p1. The next round estimate is obtained by applying least squares to (12.21), and in general the iterations are
pn+1 = [Z(P„ )’Z(P„ )]-1 Z(P„ )’y(P„).
The iterations continue until a convergence criterion is met, perhaps that P„ ~ P„+1 = S, which defines the nonlinear least squares estimates of p. Given that f(X, P) is a nice function, then, asymptotically,
S ~ N(P, o2[Z(p)’Z(p)]-1) (12.23)
and the asymptotic covariance matrix of S is estimated as
acov(S) = 62[Z(S),Z(S)]-1, (12.24)
where 62 = S(S)/(T – K). Equations (12.21)-(12.23) show that Z(P) in nonlinear regression plays the role of X in the linear regression model. Consequently, it is the columns of Z(P), which we examine via the BKW diagnostics in Section 3, that we must consider when diagnosing collinearity in the nonlinear regression model.