The Millionaires Brain

Note: GPDIBL = gross private domestic investment, billions of 1992 dollars. GPDIM = gross private domestic investments, millions of 1992 dollars. GDPB = gross domestic product, billions of 1992 dollars. GDPM = gross domestic product, millions of 1992 dollars. Source: Economic Report of the President, 1999, Table B-2, p. 328.

Note: GPDIBL = gross private domestic investment, billions of 1992 dollars. GPDIM = gross private domestic investments, millions of 1992 dollars. GDPB = gross domestic product, billions of 1992 dollars. GDPM = gross domestic product, millions of 1992 dollars. Source: Economic Report of the President, 1999, Table B-2, p. 328.

170 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Suppose in the regression of GPDI on GDP one researcher uses data in billions of dollars but another expresses data in millions of dollars. Will the regression results be the same in both cases? If not, which results should one use? In short, do the units in which the regressand and regressor(s) are measured make any difference in the regression results? If so, what is the sensible course to follow in choosing units of measurement for regression analysis? To answer these questions, let us proceed systematically. Let

where w1 and w2 are constants, called the scale factors;w1 may equal w2 or be different.

From (6.2.2) and (6.2.3) it is clear that Y* and X* are rescaled Yi and Xi. Thus, if Yi and Xi are measured in billions of dollars and one wants to express them in millions of dollars, we will have Y* = 1000 Yi and X* = 1000 Xi; here wi = W2 = 1000.

Now consider the regression using Y* and X* variables:

We want to find out the relationships between the following pairs:

From least-squares theory we know (see Chapter 3) that fa = Y fa X (6.2.5)

T,xiy Ex2

CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 171

Ex2 Eu2

Applying the OLS method to (6.2.4), we obtain similarly

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