Consider a simple time series model where the explanatory variable has classical measureme...

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

yt = β0 + β1xt* + ut

xt = xt* + et,

where ut has zero mean and is uncorrelated with xt* and et. We observe yt and xt only. Assume that et has zero mean and is uncorrelated with xt* and that xt* also has a zero mean (this last assumption is only to simplify the algebra).

(i) Write xt* = xt — et and plug this into . Show that the error term in the new equation, say, βt, 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 u and e are uncorrelated with all past values of xt* and et; in particular, with xt-1* and et-1. Show that E(xt-1vt) = 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?