Nonlinear econometrics for finance HOMEWORK 3
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Nonlinear econometrics for finance
HOMEWORK 3
GMM, MLE and Volatility
Problem 1. (40 points) Return to the first problem in Assignment 2. Consider, again, the Cobb-Douglas production function:
qt = 91 kt(9)2 lt(9)3 + et , (1)
where qt is output/production, kt is capital and lt is labor. Assume E(et lkt , lt ) = 0.
Questions:
1. (20 Points) You have already estimated the model with NLS. Now, you will estimate it with GMM. Adapt the code “gmm-vs-nls” to es- timate the model in Eq. (1) using GMM and, again, the Mizon data. Report (1) estimates, (2) standard errors and (3) t-statistics for the three parameters and comment on the statistical signi cance of your estimates.
Note: You can choose the moment conditions (at least three, since there are three parameters) freely. If you wish, you can estimate the three parameters with only three moment conditions, i.e., you can es- timate an exactly-identi ed model.
2. (10 Points) Test for “constant returns to scale”, i.e., H0 : 92 + 93 = 1 using the GMM estimates.
3. (10 Points) Test the null H0 : 92 = 0.5 and 93 = 0.5 using the GMM estimates.
Problem 2. (30 points) Consider a sample (z1 , z2 , ..., zT ) of Bernoulli random variables with T observations. As you know from your statistics classes, these are random variables which take on the value 1 with probability p and the value 0 with probability 1 - p. Hence,
L({z}, p) = p(zT , zT_1 , . . . , z1 , p)
= p(zT , p)p(zT_1 , p) . . . p(z1 , p)
T
= p(zt , p)
t=1
T
= p北t (1 - p)(1_北t) .
t=1
Note that p(zt , p) = p北t (1 -p)(1_北t) because, if zt = 1, we obtain p. If zt = 0, we obtain (1 - p).
Questions:
1. (10 Points) Write the standardized log-likelihood for this model.
2. (20 Points) Adapt the code “mle-Normal” to (1) define a Python func- tion for the standardized log-likelihood, (2) estimate the single param- eter p and (3) compute standard errors for your estimate with the two methods discussed in class (i.e., with Ω0 and with B0 ).
Problem 3. (30 points) Estimate a GARCH(1,1)-M by ML:
rt = 8ht + et ,
et = ^htut with Et_1 (ut ) = 0 and Et_1 (ut(2)) = 1,
ht = u* + 6* ht_1 + o*et(2)_1 ,
Assume the errors (ut ) are normal.
Questions:
1. (25 points) Modify the code “mle-GARCH” to estimate this model using the data in S&P500daily-level.xlsx. Compute (1) parameter es- timates, (2) standard errors and (3) t-statistics. Report all figures in a table.
2. (5 points) Plot the time series of the conditional variances. Do you see any interesting event?
2023-03-04
GMM, MLE and Volatility