Refer to the data given in Table 14.1 and the NLRM (14.2.2). Using the Eviews 4 nonlinear regression routine, which uses the linearization method,7 we obtained the following regression results; the coefficients, their standard errors, and their t values are given in a tabular form:
Variable Coefficient Std. error t value p value
Intercept 0.5089 0.0074 68.2246 0.0000
Asset -0.0059 0.00048 -12.3150 0.0000
From these results, we can write the estimated model as:
Before we discuss these results, it may be noted that if you do not supply the initial values of the parameters to start the linearization process, Eviews will do it on its own. It took Eviews five iterations to obtain the results shown in (14.5.1). However, you can supply your own initial values to start the process. To demonstrate, we chose the initial value of p1 = 0.45 and fS2 = 0.01. We obtained the same results as in (14.5.1) but it took eight iterations. It is important to note that fewer iterations will be required if your initial values are not very far from the final values. In some cases you can choose the initial values of the parameters by simply running an OLS regression of the regressand on the regressor(s), simply ignoring the non-linearities. For instance, using the data in Table 14.1, if you were to regress fee on assets, the OLS estimate of p1 is 0.5028 and that of fi2 is -0.002, which are much closer to the final values given in (14.5.1). (For the technical details, see Appendix 14A, Section 14A.3.)
Now about the properties of NLLS estimators. You may recall that, in the case of linear regression models with normally distributed error terms, we were able to develop exact inference procedures (i.e., test hypotheses) using the t, F, and x2 tests in small as well as large samples. Unfortunately, this is not the case with NLRMs, even with normally distributed error terms. The NLLS estimators are not normally distributed, are not unbiased, and do not have minimum variance in finite, or small, samples. As a result, we cannot use the t test (to test the significance of an individual coefficient) or the F test (to test the overall significance of the estimated regression) because we cannot obtain an unbiased estimate of the error variance a2 from the estimated residuals. Furthermore, the residuals (the difference between the actual Y values and the estimated Y values from the NLRM) do not necessarily sum to zero, ESS and RSS do not necessarily add up to the TSS, and therefore R2 = ESS/TSS may not be a meaningful descriptive statistic for such models. However, we can compute R2 as:
where Y = regressand and u, = Yi - Yi, where Yi are the estimated Y values from the (fitted) NLRM.
7Eviews provides three options: quadratic hill climbing, Newton-Raphson, and Berndt-Hall-Hall-Hausman. The default option is quadratic hill climbing, which is a variation of the Newton-Raphson method.
CHAPTER FOURTEEN: NONLINEAR REGRESSION MODELS 571
EXAMPLE 14.1 (Continued)
Consequently, inferences about the regression parameters in nonlinear regression are usually based on large-sample theory. This theory tells us that the least-squares and maximum likelihood estimators for nonlinear regression models with normal error terms, when the sample size is large, are approximately normally distributed and almost unbiased, and have almost minimum variance. This large-sample theory also applies when the error terms are not normally distributed.8
In short, then, all inference procedures in NLRM are large sample, or asymptotic. Returning to Example 14.1, the t statistics given in (14.5.1) are meaningful only if interpreted in the large-sample context. In that sense, we can say that estimated coefficients shown in Eq. (14.5.1) are individually statistically significant. Of course, our sample in the present instance is rather small.
Returning to Eq. (14.5.1), how do we find out the rate of change of Y ( = fee) with respect to X (asset size)? Using the basic rules of derivatives, the reader can see that the rate of change of Y with respect to X is:
As can be seen, the rate of change of fee depends on the value of the assets. For example, if X = 20 (million), the expected rate of change in the fees charged can be seen from (14.5.3) to be about -0.0031 percent. Of course, this answer will change depending on the X value used in the computation. Judged by the R2 as computed from (14.5.2), the R2 value of 0.9385 suggests that the chosen NLRM fits the data in Table 14.1 quite well. The estimated Durbin-Watson value of 0.3493 may suggest that there is autocorrelation or possibly model specification error. Although there are procedures to take care of these problems as well as the problem of heteroscedasticity in NLRM, we will not pursue these topics here. The interested reader may consult the references.
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