1 Introductory Econometrics for Finance Chris Brooks Solutions to Review - Chapter 3 1. (a) The use of vertical rather than horizontal distances relates to the idea that the explanatory variable, x, is fixed in repeated samples, so what the model tries to do is to fit the most appropriate value of y using the model for a given value of x. Taking horizontal distances would have suggested that we had fixed the value of y and tried to find the appropriate values of x. (b) When we calculate the deviations of the points, y t, from the fitted values, t y ˆ, some points will lie above the line ( y t > t y ˆ ) and some will lie below the line ( y t < t y ˆ ). When we calculate the residuals ( t u ˆ = y t – t y ˆ those corresponding to points above the line will be positive and those below the line negative, so adding them would mean that they would largely cancel out. In fact, we could fit an infinite number of lines with a zero average residual. By squaring the residuals before summing them, we ensure that they all contribute to the measure of loss and that they do not cancel. It is then possible to define unique (ordinary least squares) estimates of the intercept and slope. (c) Taking the absolute values of the residuals and minimising their sum would certainly also get around the problem of positive and negative residuals cancelling. However, the absolute value function is much harder to work with than a square. Squared terms are easy to differentiate, so it is simple to find analytical formulae for the mean and the variance. 2. The population regression function ( PRF) is a description of the model that is thought to be generating the actual data and it represents the true relationship between the variables. The population regression function is also known as the data generating process ( DGP). The PRF embodies the true values of and , and for the bivariate.
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