Simultaneous Equations
1) Consider
qt = β0 + β1pt + β2yt + udt qt = α0 + α1pt + α2wt + ust
as a system of equations for supply and demand of bananas. Suggest how to
test
H0 : β2 = α2 = 0 vs. HA : β2 =6 α2 6= 0.
2) Consider the model
y = Xβ + u
where some of the X variables are potentially endogenous. Let Z be a valid set of instruments for X.
a) Show that the OLS estimator of β is asymptotically biased if any of the X variables are endogenous and asymptotically unbiased otherwise.
b) Show that the IV estimator of β is asymptotically unbiased whether or
not the X variables are endogenous.
c) Use this to construct a test statistic forH0 : X is exogenous vs. HA : X is endogenous.
3) Consider the model:
where yi is output at rm i, Xi is input at rm i, and Ai is a rm-speci c productivity factor.
a) Suggest how to estimate α using OLS.
b) Suggest why, if rms are pro t maximizing, Xi would be endogenous and
show the precise relationship between Xi and the error.
c) Suggest a reasonable instrument for Xi. Explain why it is a valid instru-
ment.
