Define the 2sls estimator of beta using zi as an instrument

Question

Econometrics 710 Final Exam 1999

1. Take a linear regression model

Y = Xβ + e

where X is n × k aud β is h × 1. Assume E (e | X) = 0. Suppose the parameters β are known to satisfy the restristions

β = Qθ

where Q is k × m and θ is m × 1, m < k. Q is known but θ is unknown.

(a) Is there a simple way to estimate θ by least squares? Find this estimator (θ^˜).

(b) Let β^ˆ denote the OLS estimator for β.  One can also estimate θ from β^ˆ using the "minimum chi-square criterion." Specifically, for some positive definate k × k matrix W, define

C(θ) = (β^ˆ - Qθ)' W (β^ˆ - Qθ)

and define θ^ˆ  as the θ which minimizes C(θ):

θ^ˆ = Argmin C(θ).

Find this θ^ˆ.

(c) Is θ^ˆ consistent for θ?

(d) Find the asymptotic distribution of θ^ˆ.

(e) Is there a choice of W so that θ^˜ = θ^ˆ?  [θ^˜ is defined in part (a)]

2. Take the linear model

Y = Xβ + e.

Let the OLS estimator for β be β^ˆ and the OLS residual be e^ˆ = Y - Xθ^ˆ.

Let the 2SLS estimator for β using some instrument Z be β^˜ and the 2SLS residual be e^˜ = Y - Xβ^˜. If X is indeed endogeneous, will 2SLS "fit" better than OLS, in the sense that e^˜' e^˜ < e^ˆ' e^ˆ, at least in large samples?

3. Consider the single equation model

y_i = x_iβ + e_i,

where y_i and x_i are both real-valued (1 × 1).  Let β^ˆ denote the 2SLS estimator of β using as an instrument a dummy variable d_i (takes only the values 0 and 1). Find a simple expression for the 2SLS estimator in this context.

4. Suppose that y_i is generated by

y­_i = αy_t-2 + e_t

with e_t  iid (0, σ²) and 0 < α < 1.

(a) Is y_t stationary and ergodic?

(b) Find the asymptotic distribution of the OLS estimator αˆ.

5. A latent variable y^*_i is generated by

y^*_i  = x_iβ + e_i

The distribution of e_i, conditional on x_i, is N (O, σ_i²), where σ_i² = γ0 + x_i²γ₁ with γ₀ > 0 and γ₁ > 0. The binary variable y_i equals 1 if y_i^* ≥ 0, else y_i = 0. Find the log-likelihood function for the conditional distribution of y_i given x_i (the parameters are β, γ₀, γ₁).

6. The equation of interest is

y­_i = g(x_i, β)+ e_i

E (e_i | z_i) = 0.

The observed data is (y_i, x_i, z_i).  z_i is k × 1 and β is m × 1. Show how to construct the efficient GMM estimator for β.

 7. In the linear model

y_i = x_iβ + e_i

Suppose σ_i² = E(e_i² | x_i) is known. Show that the GLS estimator of β can be written as au instrumental variables estimator using some instrument z_i. (Find an expression for z_i.)

8. Take the linear model

y_i = x_iβ + e_i

E(e_i|x_i) = 0.

where x_i and β are 1 × 1.

(a) Show that E(x_ie_i) = 0 and E(x_i²e_i) = 0. Is z_i = (x_i  x_i²) a valid instrumental variable for estimation of β?

(b) Define the 2SLS estimator of β, using z_i as an instrument for x_i. How does this differ from OLS?

(c) Find the efficient GMM estimator of β, based on the moment condition E(z_i (y_i - x_iβ)) = 0. Does this differ from 2SLS and/or OLS?