2. Consider the model; yt = βxt + ut, t= 1,..,T, and u∼(0,σ2IT ).
a) Suppose that xt is measured with error and that what is observed is
x∗
t = xt + vt,
where v ∼(0,σ2
v IT ) and xt and vt are independent. Show that the OLS estimator for
β from a regression of x∗
t on yt is inconsistent, detailing any additional assumptions you
make.
b) Suppose instead that yt is measured with error and
y∗
t = yt + vt,
with vt and yt independent. Show that the OLS estimator for β from a regression of xt
on y∗
t is consistent, detailing any additional assumptions you make.
3. Consider the regression model; yt = βxt + ut, t = 1,..,T, with E(u) = 0 and
ˆ
Var[u] = σ2IT and the OLS estimator of β,
β=
T
t=1 xtyt / T
t=1 x2
t.
ˆ
a) Is
b) Is
this case.
β consistent when xt = λt and 0 <λ<1?
ˆ
β consistent when xt = t1/2? If so determine the asymptotic distribution of
ˆ
β in