Question

1. The following figure shows three (n = 3) observations of variables X1 and 22 along with a line 8 that represents the regression of xı on X2 (xi1 = do +822i2 + êii) and a line 1 that represents the regression of x2 on X1 (xi2 = 10 + 11Xi1 + fi2). (Note that residuals for the first regression are ê, while for the second regression they are î). 6 + - - - - - - - -+ 0.0 0.5 10 i5 20 25 30 35 40 45 5.0 In the figure, label êil and fiz for each of the three observations. Also label regression line 8 and regression line 1. Provide a clear explanation of your reasoning. It should not be necessary to calculate any ols estimators to answer this question. (Note that: īj = 3, 72 = 6)

Answer 1

é i3 @iz=ria 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

The residuals and the regression lines are labeled as shown in the diagram above.

When x₁ = 0, x₂ = 3 and when x₁ = 3, x₂ =6. Given the positive slope of the line, the intercept has to be positive if we believe (x₁ = 0, x₂ = 3). Hence, the bold line refers to the regression of x₂ on x₁.

On the other hand, when x₂ = 0, x₁ = 1.5 is possible with a possible slope line if the regression is of x₁ on x₂.