Question

Consider the following regression: uneme = Bo + B infe + Bzinft-1 + Bsyear + u where unem is the civilian unemployment rate and inf is the CPI inflation rate. Assume ut = put,itet with p70. Answer the following questions. 1. What is serial correlation? Explain. 2. Write the transformed model that is estimated by GLS. 3. Outline the intuition behind the BG test for serial correlation. 4. If = P19-1 + P2U-2+ et, the GLS model described in class will not fix the problems with OLS. How can we get around this problem?

Answer 1

(1) When a variable is dependent on its own lag, it is called serial correlation. In the above regression, the error term is dependent on itself with lag 1. (2) The above model can be transformed as follows: unem = Bo + B.inft + B inft-1 + Bayear: + Ut ... (1) => unemt-1 = Bo + B, inft-1 + B inft-2 + Bayeart-1 + 47-1 ... (2) (considering lag) => punemt-1 = PB. + pß.inf-1 + PB, inft-2 +pBayeart-1 + put-1 ... (3) (multiplying p both sides) Now subtract both equation (3) from equation (1). Hence, unem-1-punemt-1 = (1-P) B. + ... + ut – put-1 => unem,* = B.* + B,*inft* + B *inf-1* + Ba year,* + € ... (4) since Ep = uz – put-1 Equation 4 represents the transformed model. It can be observed that the error term of equation 4 is er which follows all OLS assumptions. Hence, OLS can be applied to the modified model. (3) The above example considers first order-serial correlation. Breusch-Godfrey (BG) test helps to test the higher order serial correlation. Suppose up = Plut-1 + Poup-2 + + pput-p + Et be the p-th order serial correlation scheme where E N (0,0%). BG test conducts the hypothesis test with Ho: P1 = P2 = ... = Pp = 0. The test statistic (n - p)R follows chi-square distribution with p degree of freedom. Based on the test statistic, if the null is not rejected, we conclude that there is no serial correlation and vice versa.