Read on Twitter
1/N
Do you like Local Projections together with IV to study dynamic responses to structural shocks? Then you need to go through this paper. There is an interesting estimator that you might want to look at. Here is how it works (thread). https://www.sciencedirect.com/science/article/pii/S0304393218302587?via%3Dihub#sec0018
Do you like Local Projections together with IV to study dynamic responses to structural shocks? Then you need to go through this paper. There is an interesting estimator that you might want to look at. Here is how it works (thread). https://www.sciencedirect.com/science/article/pii/S0304393218302587?via%3Dihub#sec0018
2/N
You have a model
Y=Xb +u
And want to estimate b. But E(Xu)~=0. Damn. But you are smart, you have an instrument Z. Cool. You run your first stage regression
X=Zd +e
and find that the instrument is relevant, i.e. d is significantly different from 0. You are happy. But wait...
You have a model
Y=Xb +u
And want to estimate b. But E(Xu)~=0. Damn. But you are smart, you have an instrument Z. Cool. You run your first stage regression
X=Zd +e
and find that the instrument is relevant, i.e. d is significantly different from 0. You are happy. But wait...
3/N
Are you sure that Z can be excluded from the main regression? For instance, US monetary policy shocks (Z) could be an instrument for interest rates (X) in economies pegged to the USD. But US Mon. Pol. could affect Y other than through local rates (X). Asset prices. Trade.
Are you sure that Z can be excluded from the main regression? For instance, US monetary policy shocks (Z) could be an instrument for interest rates (X) in economies pegged to the USD. But US Mon. Pol. could affect Y other than through local rates (X). Asset prices. Trade.
4/N
In this case the exclusion restriction does not hold. The true regression model is
Y=Xb +Zc +u = Xb +u*
Where c is the “spillover coefficient”. Now you are back to square 1. The error u* is correlated with Z and your estimate of b is biased.
In this case the exclusion restriction does not hold. The true regression model is
Y=Xb +Zc +u = Xb +u*
Where c is the “spillover coefficient”. Now you are back to square 1. The error u* is correlated with Z and your estimate of b is biased.
5/N
Here is the solution proposed in the paper.
1.Rewrite “b” as a scaled version of “c”, b= lambda*c
2.Estimate with OLS the model Y=Xb +Zc +u
3.Estimate with OLS the first stage regression X=Zd +e
4.Collect b_ols, c_ols, d_ols
Here is the solution proposed in the paper.
1.Rewrite “b” as a scaled version of “c”, b= lambda*c
2.Estimate with OLS the model Y=Xb +Zc +u
3.Estimate with OLS the first stage regression X=Zd +e
4.Collect b_ols, c_ols, d_ols
6/N
Now with your three coefficients, b_ols, c_ols, d_ols, and with a value of “lambda” that you need to calibrate (but you can try many reasonable ones and check robustness) compute a “spillover term”:
c(lambda) = (b_ols+c_ols)/(lambda*d_ols)
Good, you are almost done.
Now with your three coefficients, b_ols, c_ols, d_ols, and with a value of “lambda” that you need to calibrate (but you can try many reasonable ones and check robustness) compute a “spillover term”:
c(lambda) = (b_ols+c_ols)/(lambda*d_ols)
Good, you are almost done.
7/N
Re-run your original IV regression, but now on the left hand side correct Y with the spillover term c(lambda), i.e. do IV on the model
(Y-c(lambda)) =Xb +eta
Upside: b this time around is unbiased
Downside: b is a function of lambda, it is a b(lambda)
Try various lambdas
Re-run your original IV regression, but now on the left hand side correct Y with the spillover term c(lambda), i.e. do IV on the model
(Y-c(lambda)) =Xb +eta
Upside: b this time around is unbiased
Downside: b is a function of lambda, it is a b(lambda)
Try various lambdas
N/N
Intuition: the new error term eta is uncorrelated with the instrument Z (the proof is the methodological contribution in the paper, it is not hard to follow, if I did it you can do it!). You can now be really fancy and report spillovers corrected local projections!
Intuition: the new error term eta is uncorrelated with the instrument Z (the proof is the methodological contribution in the paper, it is not hard to follow, if I did it you can do it!). You can now be really fancy and report spillovers corrected local projections!
