Linear Regression

Equation 12: Var[3] = =1 AT Exercise 4: One doesn't always have a choice about where to take observations of the response variable; but it one can choose the @ values at which the response variable y is observed, what does this theorem imply about that choice? Everything else being equal, the smaller the variance of an estimator the better. The error variance, o2, is what it is, and isn't going to change no matter where you choose to make observations. But the choice of where observations are made can change the denominator of Equation 12, and so you want to choose to take observations that make the denominator larger. If, in addition, the error term is assumed to be normally distributed then one can make all sorts of inferences about the parameter estimates, e.g. confidence intervals, tests of hypothesis, etc. Theorem 1: The least squares regression contains the centroid of the data, i.e. the point (z, y). Note that Theorem 1 extends in the natural way to the general case of p parameters. It even extends to the case of weighted least squares regression, if the notion of a centroid is properly interpreted. To be a little more precise about the assumptions on the error term, write: Equation11: Y, = a + f(z; 7) + &amp