In economics we are often faced with causal problems where the endogeneity arises because of simultaneity. A classic example is that if you are interested in estimating a demand curve, the issue is what you observe in the data are equli- birum and prices and quantity which is not only a function of demand but also a function of supply. In this excercise we will generalize the problem of simultaneity bias. Consider a situation where we are interested in the effect of Ti on Yi, i.e. obtaining a consistent estimate of α. But Yi also effects Ti. Let Xi be some exogenous covariates affecting both Yi and Ti. Let us imagine you have an exogenous variable Z which only effects Ti but does not directly effect Yi. In particular, consider the following structural equations: Yi = αTi+Xi′β+ui (1) Ti = ρYi+Xi′γ+Ziδ+νi (2) where E(ui | Xi,Zi) = 0 and E(vi | Xi,Zi) = 0 (a) Show why you cannot you use OLS to estimate α consistently in model 1? (b) Solve for the reduced form equation for Ti i.e. use (1) to plug in for Yi in (2). Simplify and denote the coefficients in front of Xi, Zi as β∗2 and δ2∗ and denote the composite error term as νi∗. Show why can you get a consistent estimate of β∗2 and δ2∗ by regressing T on X and Z. [Hint: Use the assumptions on the conditional expectations above to show that exogeneity will hold in this case with νi∗.] (c) Solve for the reduced form equation fo Yi i.e. use the reduced for equation for Ti derived above and plug in (1). Simplify and denote the the coefficients in front of Xi, Zi as β∗1 and δ1∗ and denote the composite error term as ui∗. Show why can you get a consistent estimate of β∗1 and δ1∗ by regressing Y on X and Z. [Hint: Use the assumptions on the conditional expectations above to show that exogeneity will hold in this case with ui∗.] (d) Now with consistent estimates obtained in steps (b) and (c), show that you can get a consistent estimate of α. (e) Would this have been possible if you did not have Zi? Explain in words what is the role of Zi in solving the simultaneity bias problem.
In economics we are often faced with causal problems where the endogeneity arises because of simultaneity. A classic example is that if you are interested in estimating a demand curve, the issue is what you observe in the data are equli- birum and prices and quantity which is not only a
Consider a situation where we are interested in the effect of Ti on Yi, i.e. obtaining a consistent estimate of α. But Yi also effects Ti. Let Xi be some exogenous
In particular, consider the following structural equations:
Yi = αTi+Xi′β+ui (1)
Ti = ρYi+Xi′γ+Ziδ+νi (2)
where E(ui | Xi,Zi) = 0 and E(vi | Xi,Zi) = 0
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(a) Show why you cannot you use OLS to estimate α consistently in model 1?
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(b) Solve for the reduced form equation for Ti i.e. use (1) to plug in for Yi in (2). Simplify and denote the coefficients in front of Xi, Zi as β∗2 and δ2∗ and denote the composite error term as νi∗. Show why can you get a consistent estimate of β∗2 and δ2∗ by regressing T on X and Z. [Hint: Use the assumptions on the conditional expectations above to show that exogeneity will hold in this case with νi∗.]
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(c) Solve for the reduced form equation fo Yi i.e. use the reduced for equation for Ti derived above and plug in (1). Simplify and denote the the coefficients in front of Xi, Zi as β∗1 and δ1∗ and denote the composite error term as ui∗. Show why can you get a consistent estimate of β∗1 and δ1∗ by regressing Y on X and Z. [Hint: Use the assumptions on the conditional expectations above to show that exogeneity will hold in this case with ui∗.]
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(d) Now with consistent estimates obtained in steps (b) and (c), show that you can get a consistent estimate of α.
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(e) Would this have been possible if you did not have Zi? Explain in words what is the role of Zi in solving the simultaneity bias problem.
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