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77

"G. E. P. Box and D. R. Cox, "An Analysis of Transformations" (with discussion). Journal of the Royal Statistical Society, Series B, 1962, pp. 211-243.

( ) = exp(9.932) = 20,577. Further analysis of these data is left as an exercise. (What is the for the equation in the deflated variables? What can you conclude?)

*5.6 Testing tlie Linear Versus Log-Linear Functional Form

As we mentioned in the introduction, sometimes equations are estimated in log form to take care of the heteroskedasticity problem. In many cases the choice of the functional form is dictated by other considerations like convenience in interpretation and some economic reasoning. For instance, if we are estimating a production function, the linear form

• = a + p,£ + p2*

where X is the output, L the labor, and the capital, implies perfect substitutability among the inputs of production. On the other hand, the logarithmic form

log = a + , log L + p2 log

implies a Cobb-Douglas production function with unit elasticity of substitution. Both these formulations are special cases of the CES (constant elasticity of substitution) production function.

For the estimation of demand functions the log form is often preferred because it is easy to interpret the coefficients as elasticities. For instance,

log e = a -b , log P + log Y

where Q is the quantity demanded, P the price, and Fthe income, implies that Pi is the price elasticity and P2 is the income elasticity. A linear demand function implies that these elasticities depend on the particular point along the demand curve that we are at. In this case we have to consider some methods of choosing statistically between the two functional forms.

When comparing the linear with the log-linear forms, we cannot compare the R-s because R~ is the ratio of explained variance to the total variance and the variances of and log are different. Comparing Rs in this case is Uke comparing two individuals A and B, where A eats 65% of a carrot cake and eats 70% of a strawberry cake. The comparison does not make sense because there are two different cakes.

The Box-Cox Test

One solution to this problem is to consider a more general model of which both the linear and log-linear forms are special cases. Box and Cox" consider the transformation



5.6 TESTING THE LINEAR VERSUS LOG-LINEAR FUNCTIONAL FORM 221

for \ 0

(5.6)

for \ = 0

This transformation is well defined for all > 0. Also, the transformation is continuous since

y> - 1

lim- = log \

Box and Cox consider the regression model

y,{\) = , + , (5.7)

where , ~ 1N(0, ). For the sake of simplicity of exposition we are considering only one explanatory variable. Also, instead of considering x, we can consider x,(\). For X = 0 this is a log-linear model, and for X = 1 this is a linear model.

There are two main problems with the specification in equation (5.7):

1. The assumption that the errors w, in (5.7) are 1N(0, cr) for all values of X is not a reasonable assumption.

2. Since > 0, unless X = 0 the definition of y(X) in (5.6) imposes some constraints on y(X) that depend on the unknown X. Since > 0, we have, from equation (5.6),

y(X) > - - if X > 0 and y(X) < - i if X < 0 X X

However, we will ignore these problems and describe the Box-Cox method.

Based on the specification given by (5.7) Box and Cox suggest estimating X by the maximum likelihood (ML) method. We can then test the hypotheses: X = 0 and X = 1. If the hypothesis X = 0 is accepted, we use log as the explained variable. If the hypothesis X = 1 is accepted, we use as the explained variable. A problem arises only if both hypotheses are rejected or both accepted. In this case we have to use the estimated X, and work with y(X).

The ML method suggested by Box and Cox amounts to the following procedure:"

1. Divide each by the geometric mean of the ys.

2. Now compute y(X) for different values of X and regress it on x. Compute the residual sum of squares and denote it by uHk).

3. Choose the value of X for which (x) is minimum. This value of X is the ML estimator of X.

As a special case, consider the problem of choosing between the linear and log-Unear models:

•G. S. Maddala, Econometrics (New York: McGraw-Hill. 1977), pp. 316-317.



= a + +

logy = a + x +

What we do is first divide each y, by the geometric mean of the ys. Then we estimate the two regressions and choose the one with the smaller residual sum of squares. This is the Box-Cox procedure.

We will now describe the two other tests that are based on artificial regressions.

The BM Test

This is the test suggested by Bera and McAleer." Suppose the log-linear and linear models to be tested are given by

: log y, = Po + + «0, ~ IN(cTg) The BM test involves three steps.

Step 1. Obtain the predicted values log Si and y, from the two equations, respectively. The predicted value of y, from the log-linear equation is exp(log y,). The predicted value of log y, from the linear equation is log y,.

Step 2. Compute the artificial regressions:

exp{log:P,) = Po + + v„

log y, = Po + P, X, + Vo,

Let the estimated residuals from these two regression equations be v„ and 0,, respectively. Step 3. The tests for Ho and , are based on and 6, in the artificial regressions:

logy, = Po + + % , + e,

y, = Po + + 0,Vo, -I- e,

We use the usual /-tests to test these hypotheses. If = 0 is accepted, we choose the log-linear model. If 6, =0 is accepted, we choose the linear model. A problem arises if both these hypotheses are rejected or both are accepted.

"A. K. Bera and M. McAleer, "Further Results on Testing Linear and Log-Linear Regression Models." paper presented at the SSRC Econometric Group Conference on Model Specification and Testing, Warwick, England, 1982.



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