Christopher Dougherty EC220 - Introduction to econometrics (chapter 1) Slideshow: exercise 1.9 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 1). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

1.9 A researcher has international cross-section data on aggregate wages, W, aggregate profits, P, and aggregate income, Y, for a sample of n countries. By definition, Y = W + P The regressions are fitted using OLS regression analysis. Show that the regression coefficients will automatically satisfy the follow- ing equations: a 2 + b 2 = 1 a 1 + b 1 = 0 Explain intuitively why this should be so. 1 EXERCISE 1.9

2 The slope coefficients are given by the standard expressions.

3 EXERCISE 1.9 The expressions have the same denominator and a common factor in the numerator. This allows one to combine the numerators as shown.

4 EXERCISE 1.9 The sum of W i and P i is Y i, by definition. The same is true of their sample means.

5 EXERCISE 1.9 Since the numerator is just an expansion of the denominator, the ratio is equal to 1.

6 EXERCISE 1.9 The intercepts are given by the standard expressions.

7 EXERCISE 1.9 The order of the terms has been changed.

8 The first two terms are equal to the mean value of Y. The last two are equal to minus the mean value of Y. Hence a 1 + b 1 is equal to 0. EXERCISE 1.9

Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use