Consider a simple time series model where the explanatory variable has classical measurement error:

where ut has zero mean and is uncorrelated with x*t, and et. We observe yt and xt only. Assume that et has zero mean and is uncorrelated with x*t and that xt also has a zero mean (this last assumption is only to simplify the algebra).
(i) Write x*t = xt - et and plug this into (15.58). Show that the error term in the new equation, say, v, is negatively correlated with xt if β1, > 0. What does this imply about the OLS estimator of β1 from the regression of yt on xt?
(ii) In addition to the previous assumptions, assume that ut and et are uncorrelated with all past values of xt and et; in particular, with xt-1 and et-1 Show that E(xt-1 vt) = 0, where vt is the error term in the model from part (i).
(iii) Are xt and xt-1 likely to be correlated? Explain.
(iv) What do parts (ii) and (iii) suggest as a useful strategy for consistently estimating β0 and β1?

Transcribed Image Text:

15.58