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Nonlinear econometrics for finance HOMEWOR
创建日期：2024-07-20 16:24:06
所属分类：金融Finance
标签： finance
Nonlinear econometrics for finance HOMEWOR

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Nonlinear econometrics for finance HOMEWORK 2

(C-CAPM and GMM)

In this homework you will evaluate the Consumption CAPM model using two different utility functions and asset returns data. Each should submit:

  1. Theanswers to the questions in a pdf  file.
  2. TheMatlab code(s) used for  You should combine answers and code in a single file.

Please note: it will be useful for you to read the 2014 paper by John Campbell (under Lecture Notes in Blackboard)  before you tackle the assignment.

Problem 1 (50 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 u(ct) + βEt[u(ct+1],ϑ

subject to the income/wealth constraints  

ct =   et − ϑpt,

ct+1 =   et+1 + ϑxt+1.

1.(2point) Assume the investor has a CRRA utility:

Derive the economy’s pricing equations both in terms of prices and in terms of returns. Use the return equation to derive estimable moment conditions.

  1. Use the quarterly data on 15 portfolios provided in class to estimate the parameters of the CCAPM using the GMM estimator. Use the consumption data provided in class. Let N denote the number of as- sets/portfolios and let d be the number of parameters to estimate.
  2. (6points) Compute first-stage GMM estimates of the d model parame- ters using the weight matrix WT = IN . For this step, assume that your data are an i.i.d.
  3. (10points) Second  Using the first-stage estimates, re-estimate the parameters using the optimal weight matrix. Assume, again, that your data are i.i.d.
  4. (10points) Next, use the HAC estimator to compute the optimal weight matrix in the second stage as well as the variance-covariance matrix of the GMM  In this case you are assuming that your data are stationary but not i.i.d. Provide some justification for your choice of the number of lags (k in our notation in class) in the construction of the HAC estimator. How do your estimates change?
  5. (4points) Interpret your estimation results in economic  What do you learn about the representative investor?
  6. (6 points) Test whether γ =1.
  7. (6 points) Test whether γ = 1 and β = 0.95 jointly.
  8. (6 points) Test for over-identifying restrictions.

Problem 2 (50 Points) 

Consider, again, the Consumption-CAPM model used in Problem 1. Let the preferences of the representative investor be given by the log-utility

u(ct) = ln(ct).

Using the same data you used for Problem 1, please answer the following questions:

  1. (2point) Derive the economy’s pricing equations both in terms of prices and in terms of returns. Use the return equation to derive estimable moment conditions.
  2. (6 points) Provide first-stage GMM estimates of the model parame- ter(s) using WT= IN . For this step, assume your data are an i.i.d. sample.
  3. (10points) Second  Using the first-stage estimates, re-estimate the parameters using the optimal weight matrix. Assume, again, that your data are i.i.d.
  4. (14points) Compute standard errors for the estimate(s) and test β =1.

Note: for this question you need to compute the gradient by hand. In other words, you can not use the command “gradi- ent” in Matlab 

5.(10points) Next, use the HAC estimator to compute the optimal weight matrix in the second stage as well as the variance-covariance matrix of the GMM  In this case you are assuming that your data are stationary but not i.i.d. Provide some justification for your choice of the number of lags (k in our notation in class) in the construction of the HAC estimator. How do your estimates change? Note: for this question you need to compute the gradient by hand. In other words, you can not use the command “gradient” in Matlab.

6.(4points) Interpret your estimation results in economic  What do you learn about the representative investor?

7.(4 points) Test for over-identifying restrictions.

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