Question

2. Consider the same model in a time series context, namely, yt = β0 + β1xt + ut, t = 1, . . . , T where ut = ρut−1 + vt, |ρ| < 1, vt is i.i.d. with E(vt) = 0 and Var(vt) = σ 2 v . (a) What is the problem in using OLS to estimate the model? Is there any problem in hypothesis testing? (b) Show that Cov(ut, ut−τ ) = ρ τVar(ut−τ ) for τ = 0, 1, 2, . . . . (c) How would you test the hypothesis that ρ = 0 using an alternative of your choice? (d) How would you estimate the model correcting for autocorrelation if you know that ρ = .55? Show all the steps. (e) Can you do part (d) above if ρ is unknown? Explain the steps.

Answer 1

Let's consider the following model :

     where t=1,2,,,T

and    

,

• OLS estimate cant be used for this model. Because here , random errors are correlated among themselves

,

• which violates the OLS assumption of non-autocorrelation (that errors are uncorrelated among them).
• During hypothesis testing if we use the OLS estimates here, in this model, then the estimates will n't be BLUE (Best Linear Unbiassed Estimator).
• Also the accepted regions of the hypothesis tests will become wider, showing the regression coefficients to be insignificant and Null hypothesis will be accepted readily.

(c) To test for the presence of autocorrelation using the hypothesis Ho: rho =0 vs say

H1: rho not equal to 0

• we can develope a test statistic called Durbin Watson d statistic, given

by

where observed residual at t th time point of the dataset

• If d = large(around 4 ) => Negative correlation is present (Reject Ho : rho =0)
• d= small (around 0 )=> Positive correlation is present (Reject Ho: rho= 0 )
• d=intermediate (around 2 ) => No autocorrelation( Accept Ho: rho =0 )

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