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Nonlinear econometrics for finance
创建日期：2024-04-18 14:40:52
所属分类：金融Finance
标签： finance
Nonlinear econometrics for finance

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Nonlinear econometrics for finance 

HOMEWORK 2 (From linear econometrics to GMM) Problem 1 (Estimating the linear regression model using GMM.) (44 points.) Consider the following regression model: yt = θ1x1,t + θ2x2,t + ...+ θkxk,t + εt, (1) with t = 1, ..., T . Notice that, more compactly, we can also write yt = x > t θ + εt, where xt =  x1,t x2,t ... xk,t  and θ =  θ1 θ2 ... θk  . Notice, also, that the first regressor can be a column of ones. In this case, θ1 would correspond to the intercept. Assumptions: 1. E[εt|xt] = 0. 2. E(εt|xt,xt−1, ...εt−1, εt−2, ...) = 0. 3. E(ε2t |xt,xt−1, ...εt−1, εt−2, ...) = σ2ε . On the assumptions: (1) (3 points.) Explain why Assumption 1. is an assumption of correct specification of the model. 1 (2) (3 points.) Explain why Assumption 2. implies that the errors are uncorrelated. (Hint: the law of iterated expectations may help.) (3) (3 points.) Explain why Assumption 3. implies that the errors are homoskedastic. (Hint: the law of iterated expectations may, again, help.) GMM estimation and inference: (4) (3 points.) Prove that Assumption 1. implies the following moment condition: E[xt(yt − x>t θ)] = 0. (5) (4 points.) Using the moment condition, compute the GMM estimator for this model. What do you notice? (Hint: the typical GMM estimator requires you to minimize a criterion. Can you find the minimum value of the criterion mathemati- cally, i.e., without a computer?) (6) (4 points.) Compute the variance of the GMM estimator for this model. What do you notice? (Hint: Begin with the general HAC variance that we discussed in class and simplify it using Assumption 2. and 3.). Implementation: The file predictability.xls contains monthly returns on the market (value-weighted in column 2 and equally-weighted in column 3), on the dividend yield and on treasury bill rates (which you can assume to be risk-free rates). It also contains data on the book-to- market ratio and on the earnings-to-price ratio. (7) (4 points.) Using the value-weighted market returns, estimate by GMM the follow- ing predictive (for excess continuously-compounded market returns) regression: log(1 +RMt )− log(1 +Rft ) = θ0 + θ1 ( d p ) t−1 + εt, where ( d p ) t−1 is the previous period dividend yield. Note that this is the same model as in Eq. (1) but the number of parameters k is equal to 2 and the predictor is lagged by one period. (8) (4 points.) Compute the GMM variance in two ways: with HAC (with one lead and one lag) and without. What do you notice about the statistical significance of the slope estimate if you use HAC versus the case without leads and lags? (9) (4 points.) Now, rather than predicting one-month ahead, we predict one-year ahead. Estimate by GMM the following predictive (for excess continuously-compounded market returns) regression: 11∑ j=0 [ log(1 +RMt+j)− log(1 +Rft+j) ] = θ0 + θ1 ( d p ) t−1 + εt+11, 2 (10) (4 points.) Compute the GMM variance in two ways: with HAC (with 11 leads and lags) and without. What do you notice about the statistical significance of the slope estimate if you use HAC versus the case without leads and lags? (11) (4 points.) Now, we predict 5-years ahead. Estimate by GMM the following pre- dictive (for excess continuously-compounded market returns) regression: 59∑ j=0 [ log(1 +RMt+j)− log(1 +Rft+j) ] = θ0 + θ1 ( d p ) t−1 + εt+59 (12) (4 points.) Compute the GMM variance in two ways: with HAC (with 59 leads and lags) and without. What do you notice about the statistical significance of the slope estimate if you use HAC versus the case without leads and lags? Problem 2 (Nonlinear least squares using GMM.) (26 points.) Consider the following cross-sectional regression model (notice the index, it is not t, it is i): yi = θ1 + θ2x θ3 i + εi, (2) with i = 1, ..., N . The model is nonlinear in the parameters. Assumptions: We have E[εi|xi] = 0 (3) and the errors are cross-sectionally uncorrelated and homoskedastic. (1) (4 points.) Using Matlab, simulate x and ε first and then y so that the relation in Eq. (2) is satisfied. Set N = 1000. You can chose all of the parameters. (Notice that if you are drawing iid errors with mean zero that are independent of the x observations, the assumptions are satisfied.) In choosing the parameters, keep in mind that I am expecting something that is realistic. For example, something similar to the scatterplot below. (2) (5 points) Use the Assumption in Eq. (3) to come up with a set of moment condi- tions. (Recall, you want to have more moment conditions than parameters or the same number. In this case, you have 3 parameters. You need at least 3 moment conditions. Hint: Is E[f(xi)εi] = 0, for any f , given Eq. (3)? If so, you can choose three, or more, different fs. Two are natural (see Problem 1), the others you need to come up with but you have flexibility). (3) (4 points.) Estimate your model by GMM on your simulated data. (4) (5 points.) Compute the GMM variance and the corresponding standard errors for all parameter estimates. 3 Figure 1: Scatterplot of y versus x: yi = θ1 + θ2x θ3 i + εi (5) (4 points.) Extend the sample and show that the GMM estimator is consistent for the true parameter vector. You should show the result for all θ̂Ns. (6) (4 points.) Draw multiple samples and show that the GMM estimator is asymp- totically normal with a variance equal to the one in point (4) above. Again, you should do it for all θ̂Ns. 4 

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