Problem:

(a) Why is an error term added to a regression and explain its importance in the OLS procedure?

(b) Suppose we have a linear equation with a constant term, one explanatory variable and an error term, show in details the steps required, using the OLS procedure, to derive the estimates of the constant term and that of the coefficient of the explanatory variable.

(c) Suppose a researcher wants to test whether the returns on a company stock (y) show unit sensitivity to two factors (factor x2 and factor x3) among three considered. The regression is carried out on 144 monthly observations. The regression is: Y_t = β1 + β2x2t + β3x3t + β4x4t+ ut

i) What are the restricted and unrestricted regressions?

ii)  If the two RSS are 502.4 and 302.8  respectively, perform an F-test of the restriction, showing all the required steps.

(d) How are the t-test and the F-test related and describe in which cases they cannot be applied?