As in the lecture, assume independence across i and you can assume that moments exist and matrices are invertible as needed.
Suppose you observe an i.i.d. sample {yi, x1i, x2i}n , where yi is a scalar, and x1i
and x2i are each vectors of length k1 and k2 respectively.
In a first step, you regress yi on x1i, i.e.:
yi = xJ1iβ˜1 + eˆi
where β˜1 denotes the OLS estimator and eˆi the corresponding residuals.
(a) State the ordinary least squares estimator β˜1 in terms of {yi, x1i}n. (2 Points)
In a second step, you regress the residuals eˆi from the first regression on x2i. Let β˜2 denote the ordinary least squares estimator for this second regression.
(b) State the ordinary least squares estimator β˜2 of the second regression in terms of
{yi, x1i, x2i}n. (3 Points)
Next you regress yi on both x1i and x2i at the same time. Let βˆ1 and βˆ2 denote the ordinary least squares estimators from this regression, i.e.:
yi = xJ1iβˆ1 + xJ2iβˆ2 + uˆi
(c) We want to compare the estimated coefficients for x2i from these two approaches, i.e.
β˜2 and βˆ2. Write β˜2 in terms of βˆ2 and {x1i, x2i}n . (5 Points)
(d) Are the two estimates β˜2 and βˆ2 the same, i.e. does it generally hold that β˜2 = βˆ2? If so, give an intuition. If not, under which conditions are they the same? (3 Points)
As in the lecture, assume independence across i and you can assume that moments exist and matrices are invertible as needed.
Consider the following model with x1i and x2i being scalar;
yi = x1iβ1 + x2iβ2 + ei where β2 < 0
E[ei|xi] = 0
(a) Suggest a consistent estimator for the parameter θ = β1 . (1 Points)
(b) Derive the asymptotic variance Vθ of θˆ, that is, of the estimator for θ you suggested in the previous part using the delta-method and let V = Vβ,11 Vβ,12 denote the
Vβ,21 Vβ,22
asymptotic variance of (βˆ1βˆ2)J. (4 Points)
Hint: If you take the derivative of a scalar-valued function with respect to a k- dimensional column-vector, the result is a k-dimensional row-vector with the ith entry being the derivative of the function with respect to the ith entry of the column-vector:
∂f (x) = ∂f (x) , . . . , ∂f (x) for x ∈ Rk×1
Now assume yi is yearly income in 1,000 euro and x1i is age in years and x2i = x2 is age in years squared.
(c) You want to test whether the expected yearly income of a 30 year old is 40,000 euro. Write down your null hypothesis. (2 Points)
(d) Write down a test statistic for this null hypothesis, where Vˆβ denotes a consistent es- timator for the asymptotic variance of (βˆ1βˆ2)J, and state the asymptotic distribution of the test statistic. (3 Points)
(e) Derive the age x1i at which the expected yearly income is maximized as a function of β1 and β2. Write down the null hypothesis that the expected yearly income is maximized at age 45. (2 Points)
(f) Write down a test statistic for this null hypothesis and state its asymptotic distribu- tion. (4 Points)
As in the lecture, assume independence across i and you can assume that moments exist and matrices are invertible as needed.
Consider the model with a scalar xi and scalar instrumental variable zi for xi:
yi = β0 + β1xi + ui
(a) State the OLS estimator βˆOLS for β1 and derive the difference between β1 and the probability limit of βˆOLS as a function of correlations between and variances of {x, u}.(3 Points)
(b) State the IV estimator βˆIV for β1 and derive the difference between β1 and the probability limit of βˆIV as a function of correlations between and variances of {x, u, z}. (3 Points)
Now assume in the following that you know that all variables have a variance of 1 and correlations are as follows corr(x, u) = 0.8, corr(z, u) = 0.1 and corr(x, z) = 0.2.
(c) Compare your answers from (a) and (b). The probability limit of which estimator is closer to the true value? (1 Points)
(d) Derive the asymptotic variance of βˆIV . You may assume that the sample average as well as the mean of z are zero. (6 Points)
Hint: You need to subtract the asymptotic bias to apply central limit theorems. If you did not calculate the bias in previous questions, you may use ”b” as a placeholder in your derivation.
As in the lecture, assume independence across i and you can assume that moments exist and matrices are invertible as needed.
Consider the static model where xit is a scalar and you are interested in β:
yit = α + βxit + ci + uit
(a) State the assumptions needed such that the pooled OLS estimator is consistent. You may omit rank-conditions and the existence of moments. (2 Points)
(b) State the assumptions needed such that the fixed-effect estimator is consistent. You may omit rank-conditions and the existence of moments. (2 Points)
From now on consider the following model where xit is a scalar and you are interested in β, γ and δ.:
yit = α + βxit + γyit−1 + δyit−2 + ci + uit
(c) Explain if the pooled OLS estimation approach can be used for this model. Discuss the applicability of the assumptions stated in (a). (2 Points)
(d) Explain if the fixed-effect estimation approach can be used. Discuss the applicability of the assumptions stated in (b). (2 Points)
For the following suppose that uit is not serially correlated and the data set has five time periods t=1,2,3,4,5.
(e) Suppose that xit is strictly exogenous. Derive an appropriate estimation approach for this dynamic model. Derive all possible moment conditions. Which is the degree of over-identification? (6 Points)
(f) Now suppose that xit is sequentially exogenous. How do you have to modify your previous approach. Derive all possible moment conditions. Which is the degree of over-identification? (2 Points)
(g) Now suppose additionally that yit is stationary, i.e. in a steady state in t=1. Which additional moment conditions can you derive. Derive all possible moment conditions for the system GMM estimator. Which is the degree of over-identification? (2 Points)
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