## The Exponential Function Given In 1422

The function under consideration is:

Note: For ease of manipulation, we have dropped the observation subscript.

Remember that in this function the unknowns are the fa coefficients. Let us linearize this function at fai = fa* and fa2 = fa2, where the starred quantities are given fixed values. To linearize this, we proceed as follows:

Y = f(fai, fa2) = f(fa*, fa*) + ffai(fa*, fa2)(fai - fa*) + ffa2(fa*, fa2)(fa2 - fa*) (2)

where fgi and fp2 are the partial derivatives of the function (i) with respect to the unknowns and these derivatives will be evaluated at the (assumed) starred values of the unknown parameters. Note that we are using only the first derivatives in the preceding expression, since we are linearizing the function. Now assume that fa* = 0.45 andfa2 = 0.01, which are pure guess-estimates of the true coefficients. Now f (fa* = 0.45, fa2* = 0.01) = 0.45e0 01Xi ffai = efa2 Xi and ^ = fai Xefa2 Xi

578 PART THREE: TOPICS IN ECONOMETRICS

by the standard rules of differentiation. Evaluating these derivatives at the given values and reverting to (2), we obtain:

Y = 0.45eomXi + eomXi (01 - 0.45) + (0.45)Xie0 01X- (02 - 0.01) (4) which we write as:

Now let Y* = (Yi - 0.45e0 01Xi), X1 = e0 01 Xi, and X2i = 0.45Xie001X<. Using these definitions and adding the error term ui, we can finally write (5) as:

Lo and behold, we now have a linear regression model. Since Y*, X1i, and X2i can be readily computed from the data, we can easily estimate (7) by OLS and obtain the values of a1 and a2. Then, from (6), we obtain:

Call these values 0*~ and 02\ respectively. Using these (revised) values, we can start the iterative process given in (2), obtaining yet another set of values of the 0 coefficients. We can go on iterating (or linearizing) in this fashion until there is no substantial change in the values of the 0 coefficients. In Example 14.1, it took five iterations, but for the Mexican Cobb-Douglas example it took 32 iterations. But the underlying logic behind these iterations is the procedure just illustrated.

For the mutual fund fee structure, the Y* X1, and X2 as given in (6) are as shown in Table 14.4; the basic data are given in Table 14.1. From these values, the regression results corresponding to (7) are:

 Variable Coefficient Std. error t statistic Probability 