Nonlinear econometrics for finance
HOMEWORK 2
(LIE, NLS and GMM) Problem 1 (Law of Iterated Expectations.) (6 points) A financial analyst wants to predict the return on a portfolio. The portfolio gives either a return of 1 or 2 percent in each period. She only knows that the joint probability of returns at time t and t + 1 is
Questions:
1. (2 points) Compute the unconditional expected value E(rt+1).
2. (2 points) Compute the conditional expected value Et (rt+1) = E(rt+1jrt ).
3. (2 points) Verify the law of iterated expectations. In particular, show
- numerically - that E(rt+1) = E[E(rt+1jrt )].
Note: you can use my document “Examples for the LIE” under Lecture Notes on OneDrive to solve this problem.
Problem 2 (Nonlinear least squares, NLS.) (70 Points) Consider the model:
where qt is output/production, kt is capital and lt is labor. This is the typical specification of a Cobb-Douglas production function.The parameter θ 1 is a proportionality factor capturing “total factor pro- ductivity,” the impact on output/production of factors other than capital and labor. The parameters θ2 and θ3 capture the “output elasticity” of capital and labor, respectively. This is easy to see:
Now, if we take aderivative of, say, the logarithm of output (qt ) with respect to the logarithm of capital (kt ), we obtain
Because I am multiplying and dividing by the same object, this is, however, the same as:
Now, notice that and are just the derivative of log(qt ) with respect to qt (namely, ) and the derivative of log(kt ) with respect to kt (namely, ). Thus,which means that θ2 measures the “percentage change in output” given a “percentage change in capital” . This is “the elasticity of output with respect to capital” . It addresses the question: if capital increases by 1%, say, what is the percentage increase in output? The answer is: θ2 1%.Importantly, we expect θ2 to be smaller than 1. The idea is that a change in capital does not yield a one-to-one change in output. 代 写Nonlinear econometrics for finance HOMEWORK 2Python
If, for example, θ2 = 0.5, a 1% change in capital would translate into a 0.5% change in output. Naturally, θ3 has the exact same interpretation (for labor) and we are also expecting θ3 to be smaller than 1.One interesting assumption to test is whether θ2 + θ3 = 1. This is called a case of “constant returns to scale” . What does it mean? Assume we change all inputs by ψ multiplicatively. Then, we are also changing the output by the same amount:The case θ2 + θ3 > 1 is called “increasing returns to scale” (if we scale all inputs by ψ we are scaling output by more). The case θ2 + θ3 < 1 is called “decreasing returns to scale” (if we scale all inputs by ψ we are scaling output by less).
Questions:
1. (20 Points) Adapt my code for nonlinear least squares with two param- eters to estimate the model with three parameters in Eq. (1) using the Mizon data.
2. (20 Points) Report (1) estimates, (2) standard errors and (3) p values for the three parameters and comment on the statistical significance of your estimates.
3. (10 Points) Comment on the economic significance of your estimates as I did in the discussion above: are θ2 and θ3 positive and smaller than 1? What does it mean?
4. (10 Points) Test for “constant returns to scale” by testing H0 : θ2 +θ3 = 1.
5. (10 Points) Test for “constant returns to scale” by testing H0 : θ2 = 0.2 and θ3 = 0.8.Note: you can use my document “Asymptotic theory and testing” under Lecture Notes on OneDrive to double check the construction of your tests.
Problem 3 (Generalized Method of Moments, GMM.) (24 Points)
Use my GMM code to test the following three hypothesis (using optimal second-stage estimates):
1. (10 Points) H0 : θ1 = 50/1θ2 .
2. (10 Points) H0 : θ1 = 0.9 and θ2 = 50.
3. (4 Points) Test if the pricing errors are zero.Note: you can use my document “Asymptotic theory and testing” under Lecture Notes on OneDrive to double check the construction of your tests.
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