Consider a simple time series model where the explanatory variable has classical measurement error: YtXt=0+1Xt+Ut,=Xt+t, where Ut has zero mean and is uncorrelated with Xt and t. You observe Yt and Xt only. Assume that t has zero mean and is uncorrelated with Xt and that Xt also has a zero mean (this last assumption is only to simplify the calculations). 1. (8 marks) Write Xt=Xtt and plug this into (3). Show that the error term in the new equation, say Vt, is negatively correlated with Xt if 1>0. What does this imply about the OLS estimator of 1 from the regression of Yt on Xt ? 2. ( 4 marks) In addition to the previous assumptions, assume that Ut and t are uncorrelated with all past values of Xt and t; in particular, with Xt1 and t1. Show that E(Xt1Vt)= 0 , where Vt is the error term in the model from part 1 . 3. (8 marks) Taking into account that Xt and Xt1 are likely to be correlated, what does part 2 suggest as a useful strategy for consistently estimating 0 and 1 ? Consider a simple time series model where the explanatory variable has classical measurement error: YtXt=0+1Xt+Ut,=Xt+t, where Ut has zero mean and is uncorrelated with Xt and t. You observe Yt and Xt only. Assume that t has zero mean and is uncorrelated with Xt and that Xt also has a zero mean (this last assumption is only to simplify the calculations). 1. (8 marks) Write Xt=Xtt and plug this into (3). Show that the error term in the new equation, say Vt, is negatively correlated with Xt if 1>0. What does this imply about the OLS estimator of 1 from the regression of Yt on Xt ? 2. ( 4 marks) In addition to the previous assumptions, assume that Ut and t are uncorrelated with all past values of Xt and t; in particular, with Xt1 and t1. Show that E(Xt1Vt)= 0 , where Vt is the error term in the model from part 1 . 3. (8 marks) Taking into account that Xt and Xt1 are likely to be correlated, what does part 2 suggest as a useful strategy for consistently estimating 0 and 1