Simultaneous Equations

• Consider

qt = 0 + 1pt + 2yt + udt qt = 0 + 1pt + 2wt + ust

as a system of equations for supply and demand of bananas. Suggest how to

test

H₀ : ₂ = ₂ = 0 vs. H_A : ₂ =6 ₂ 6= 0.

• Consider the model

y = X + u

where some of the X variables are potentially endogenous. Let Z be a valid set of instruments for X.

1. Show that the OLS estimator of is asymptotically biased if any of the X variables are endogenous and asymptotically unbiased otherwise.
2. Show that the IV estimator of is asymptotically unbiased whether or

not the X variables are endogenous.

1. Use this to construct a test statistic forH₀ : X is exogenous vs. H_A : X is endogenous.

• Consider the model:

where y_i is output at rm i, X_i is input at rm i, and A_i is a rm-speci c productivity factor.

1. Suggest how to estimate using OLS.
2. Suggest why, if rms are pro t maximizing, X_i would be endogenous and

show the precise relationship between X_i and the error.

1. Suggest a reasonable instrument for X_i. Explain why it is a valid instru-

ment.

1