'Millions of 1960 pesos; "•Thousands of people; ^Millions of 1960 pesos.

Source: Victor J. Elias, Sources of Growth: A Study of Seven Latin American Economies, International Center for Economic Growth, ICS Press, San Francisco, 1992. Data from Tables E5, E12, and E14.

'Millions of 1960 pesos; "•Thousands of people; ^Millions of 1960 pesos.

Source: Victor J. Elias, Sources of Growth: A Study of Seven Latin American Economies, International Center for Economic Growth, ICS Press, San Francisco, 1992. Data from Tables E5, E12, and E14.

CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 271

The F test given in (8.7.10) or its equivalent (8.7.9) provides a general method of testing hypotheses about one or more parameters of the k-variable regression model:

Yi = 01 + 02 X2i + ft X3i + ... + l3kXki + ui (8.7.15)

The F test of (8.5.16) or the t test of (8.6.3) is but a specific application of (8.7.10). Thus, hypotheses such as

which involve some linear restrictions on the parameters of the k-variable model, or hypotheses such as

which imply that some regressors are absent from the model, can all be tested by the F test of (8.7.10).

From the discussion in Sections 8.5 and 8.7, the reader will have noticed that the general strategy of F testing is this: There is a larger model, the unconstrained model (8.7.15), and then there is a smaller model, the constrained or restricted model, which is obtained from the larger model by deleting some variables from it, e.g., (8.7.18), or by putting some linear restrictions on one or more coefficients of the larger model, e.g., (8.7.16) or (8.7.17).

We then fit the unconstrained and constrained models to the data and obtain the respective coefficients of determination, namely, Rjr and RR. We note the df in the unconstrained model ( = n — k) and also note the df in the constrained model ( = m), m being the number of linear restriction [e.g., 1 in (8.7.16) or (8.7.18)] or the number of regressors omitted from the model [e.g., m = 4 if (8.7.18) holds, since four regressors are assumed to be absent from the model]. We then compute the F ratio as indicated in (8.7.9) or (8.7.10) and use this Decision Rule: If the computed F exceeds Fa(m, n — k), where Fa (m, n — k) is the critical F at the a level of significance, we reject the null hypothesis: otherwise we do not reject it.

214If one is using the maximum likelihood approach to estimation, then a test similar to the one discussed shortly is the likelihood ratio test, which is slightly involved and is therefore discussed in the appendix to the chapter. For further discussion, see Theil, op. cit., pp. 179-184.

272 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Let us illustrate:

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