You are presented with the following time series model:
Yt = Xtβ + ut
,
where Xt
is a 1 × q row vector,β is a q × 1 column vector. You have been told that the
error term has the following AR(1) structure
ut = ρut−1 + vt
Where |ρ| < 1 and vt ∼ N(0, σ2 v ) (a) Show, using recursive substitution, that the expectation of the error term, E[ut ], is zero. [4 MARKS] (b) Derive an expression for the variance of the error term ut from the information above. [10 MARKS] (c) Explain what will be the effect on the OLS estimator of β given the AR(1) structure of the error term. [2 MARKS] (d) Assume you have been told that there may be a longer AR(q) process governing the behaviour of the error term ut of the form ut = ρ1ut−1 + ρ2ut−2 + . . . + ρqut−q + vt another investigator thinks there is no autocorrelation. Outline how you would construct a statistical test to help resolve this debate. Be sure to state any assumptions you make. [9 MARKS]