Consider the following linear model:
y = β0 + β1x1 + β2x2 + β3x3 + u
[a] Specify under what conditions we can identify the parameters of the model. Be very specific and write only the conditions needed.
[b] Assume that x3 is an endogenous variable, that is E[x3u] options of x3 being an endogenous regressor? 0. What are the implications of x3 being an endogenous regressor?
[c] One of your classmates, Ana, tells you that you can use an instrument, z, for the endogenous variable. What are the assumptions required for z to be an instrument for x3. Show mathematically, how you will conduct the IV and 2SLS procedure? Be very specific and layout the steps in a clear way.
[d] In the 2SLS approach explain how will you test the relevance assumption of the in- strument? State the null and alternative hypothesis. Also, explain how can we test endogeneity. State the null and alternative hypothesis.
Now consider the following model
y = β0 + β1x1 + β2x2 + u
[e] Set up the problem of minimizing the sum of squared residuals, derive the first order conditions and obtain the least square estimates βˆ0, βˆ1, and βˆ2. Make sure to show all the steps very clearly.
Now consider that one of your classmates tells you that x2 is an endogenous variable
and proposes an instrument z, and argues that Cov(z, u) = 0 and Cov(z, x2) 0. Your classmate proposes a two-stage least square (2SLS) procedure as follows:
x2 = γ0 + γ2z + u
y = β0 + β1x1 + β2xˆ2 + u
[f] Is the procedure proposed above correct? Explain why or why not.
Suppose that the demand function of a good is given by q = γ0 + γ1p + u, where p represents the price of the good, q represents the quantity of the good, and u represents the unobserved factors that determine the demand function. Also, let the supply of the good be, q = δ0 + δ1p + η, where η captures the unobserved factors that determine the supply function. Let E[u] = E[η] = 0, V ar(u) = σ2, and V ar(η) = σ2. In addition assume that the
unobserved components, u and η, are not correlated, that is Cov(u, η) = 0.
[a] Solve the system of equations (demand and supply) and show that p and q depend on the unobserved components u and η.
[b] Obtain the means of the price and quantity derived in part [a].
[c] Obtain the variance of the price and quantity derived in part [a].
Now let {(qi, pi) : i = 1, 2, . . . , N} be a random sample and we regress qi on pi.
(i) Use your results in part [b] and [c] to derive the estimates of the regression.
(ii) An economist uses the estimate of γ1 as the slope of the demand function. Is the estimated value of γ1 too large or too small? Show and explain in detail clearly. [Remember that the demand is downward and the supply is upward.]
Now suppose that you have the following Cobb-Douglas production function for a firm:
yi = AiLαKβ
where Li denotes labor, Ki denotes capital, and Ai is technology. Assume that Ai = eβ0+β1xi+ui , where xi denotes an observed firm characteristic and ui denote the firm’s unobserved characteristics. Our parameters of interest are β1, α and β.
[d] Rewrite the Cobb-Douglas production function in a linear form, that is the param- eters must enter the model linearly. [Hint: remember the log transformation from intermediate micro class.]
[e] Write down in a clear way the identification conditions required for the model param- eters to be identified.
[f] How would you test the joint significance of α and β? Show the steps in a clear way.
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