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Nonlinear econometrics for finance

创建日期：2024-04-17 17:16:56

所属分类：金融Finance

标签： finance

Nonlinear econometrics for finance

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Nonlinear econometrics for finance

HOMEWORK 3 (GMM and MLE) Problem 1: CCAPM and GMM (30 points) Consider, as we did in class, a representative investor who lives for two peri- ods (t and t+ 1) and has income et in period t and et+1 in period t+ 1. The utility function of the representative investor is: U(ct, ct+1) = u(ct) + βEt[u(ct+1)]. The investor can invest in an asset by buying ϑ shares at the unit price pt. The asset’s payoff xt+1 in the second period is uncertain. The investor chooses how many units (ϑ) of the asset to buy in order to maximize her/his utility function: max ϑ u(ct) + βEt[u(ct+1)], subject to the income/wealth constraints ct = et − ϑpt, ct+1 = et+1 + ϑxt+1. (1) (2 points) Assume the investor has a CRRA utility (like in class): u(ct) = c1−γt 1− γ . Derive the economy’s pricing equations both in terms of prices and in terms of returns. The return equation will give you estimable moment conditions. 1 Use the same data that we used in class. Let d be the number of parame- ters to estimate and let N be the number of assets. Modify (only when a modification is needed, of course) the GMM code to address the following questions. (2) (3 points) Estimation. Compute first-stage GMM estimates of the d model parameters using the weight matrix WT = IN . (3) (5 points) Estimation. Second stage. Using the first-stage estimates, re-estimate the parameters using the optimal weight matrix. The opti- mal weight matrix should be HAC with a number of auto-covariances (forward and backward) equal to 4. (4) (5 points) Inference. Compute the asymptotic variance of the GMM estimates. Please note: (a) The matrix Γ0 should be estimated without numerical differenti- ation. In other words, you should compute the gradient by hand (like we have done for MLE). (b) The matrix Φ0 should be HAC, like before, with a number of auto-covariances (forward and backward) equal to 4. (5) (3 points) Inference. Test whether γ = 0.8. (6) (5 points) Inference. Test whether γ = 0.8 and β = 0.9 jointly. (7) (2 points) Interpret your results in economic terms. What do you learn about the representative investor? (8) (5 points) Inference. Test for over-identifying restrictions. Problem 2: MLE (25 Points) Consider a sample (x1, x2, ..., xN) of Bernoulli random variables withN obser- vations. 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. 2 (1) (3 points) Explain briefly why the joint probability of the sample can be written as follows: L({x}, p) = N∏ i=1 pxi(1− p)(1−xi) . (2) (3 points) Use the joint probability to compute the standardized (by N) log likelihood. (3) (3 points) Compute the first derivative (with respect to p) of the stan- dardized log likelihood. (4) (5 points) Use the result in Point (3) to provide (i) an MLE estimator of the probability p and (ii) the asymptotic variance of the estimator. Interpret your results in words. (5) (11 points) Write a short Matlab code which simulates 1000 Bernoulli observations with p = 0.2 and computes an MLE estimate of p along with standard errors and t-statistics. (Hint: if you use the structure of the MLE − normal − iid− code, writing the code should be very easy and quick. Your Bernoulli code should just be a simpler version of that code in which you simulate from a Bernoulli random variable - which has only one parameter - rather than from a normal random variable - which has, instead, 2 parameters.) Problem 3: MLE for GARCH-M (15 Points) Modify the GARCH code to estimate a GARCH-M model. You only need to arrive at estimates and t−statistics.

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