d = 0.716 R2 = 0.9284 R2 = 0.9079 d = 1.038 R2 = 0.9983 R2 = 0.9975 d = 2.70

* Yj = 141.767 + 63.478X; -(6.375) (4.778) (22.238) (13.285)

d = 0.716 R2 = 0.9284 R2 = 0.9079 d = 1.038 R2 = 0.9983 R2 = 0.9975 d = 2.70

520 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

critical d values are dL = 0.879 and du = 1.320. Likewise, the computed d value for the quadratic cost function is 1.038, whereas the 5 percent critical values are dL = 0.697 and du = 1.641, indicating indecision. But if we use the modified d test (see Chapter 12), we can say that there is positive "correlation" in the residuals, for the computed d is less than du. For the cubic cost function, the true specification, the estimated d value does not indicate any positive "correlation" in the residuals.22

The observed positive "correlation" in the residuals when we fit the linear or quadratic model is not a measure of (first-order) serial correlation but of (model) specification error(s). The observed correlation simply reflects the fact that some variable(s) that belong in the model are included in the error term and need to be culled out from it and introduced in their own right as explanatory variables: If we exclude the X3 from the cost function, then as (13.2.3) shows, the error term in the mis-specified model (13.2.2) is in fact (u\i + p4X3) and it will exhibit a systematic pattern (e.g., positive autocorrelation) if X3 in fact affects Y significantly.

To use the Durbin-Watson test for detecting model specification error(s), we proceed as follows:

1. From the assumed model, obtain the OLS residuals.

2. If it is believed that the assumed model is mis-specified because it excludes a relevant explanatory variable, say, Z from the model, order the residuals obtained in Step 1 according to increasing values of Z. Note: The Z variable could be one of the X variables included in the assumed model or it could be some function of that variable, such as X2 or X3.

3. Compute the d statistic from the residuals thus ordered by the usual d formula, namely,

Note: The subscript t is the index of observation here and does not necessarily mean that the data are time series.

4. From the Durbin-Watson tables, if the estimated d value is significant, then one can accept the hypothesis of model mis-specification. If that turns out to be the case, the remedial measures will naturally suggest themselves.

In our cost example, the Z (= X) variable (output) was already ordered.23 Therefore, we do not have to compute the d statistic afresh. As we have seen, the d statistic for both the linear and quadratic cost functions suggests d

22In the present context, a value of d = 2 will mean no specification error. (Why?)

23It does not matter if we order Ui according to X2 or X3 since these are functions of Xi, which is already ordered.

CHAPTER THIRTEEN: ECONOMETRIC MODELING 521

specification errors. The remedies are clear: Introduce the quadratic and cubic terms in the linear cost function and the cubic term in the quadratic cost function. In short, run the cubic cost model.

Ramsey's RESET Test. Ramsey has proposed a general test of specification error called RESET (regression specification error test).24 Here we will illustrate only the simplest version of the test. To fix ideas, let us continue with our cost-output example and assume that the cost function is linear in output as

where Y = total cost and X = output. Now if we plot the residuals U obtained from this regression against Y, the estimated Yi from this model, we get the picture shown in Figure 13.2. Although J] U and E UY are necessarily zero

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