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Question 5. Point allocation: (a) 25%; (b) 25%; (c) 25%; (d) 25% Suppose that the solution of a representative agent's portfolio allocation problem is given by the following Euler equation 1 = BE M ( P. + D.) P, where P, is the price of a financial asset and D, is the dividend paid out by the financial asset. Suppose that the stochastic discount factor, M,+1, is given by M. = C, ICH where C, is consumption. In addition, suppose you have quarterly time series data on P, and D, for one financial asset, and on C. Finally, suppose you estimate B using GMM with the moment conditions E(Z,u,4) = 0, where Z, = (1 X,)' , 1,4 = 1- AY,., and X, = M, (P, + D,)/P._. (a) Write down Hansen's J-statistic and explain how you would use it to make inference about the validity of the moment conditions and the model implied by the above Euler equation. If you were to reject the null hypothesis of Hansen's J-test, what would you do? (b) Suppose you are unable to reject the null hypothesis of Hansen's /-test and that your estimate of B is 0.96 with a standard error of 1.5. This standard error is calculated using the usual formula, which is a function of the variance of the sample moment conditions, the first derivative of the moment conditions, and the weighting matrix. Do these results imply that the parameter B is well identified? If so, explain why. If not, explain how you would determine whether this parameter is well identified. (c) Suppose that you estimate B using the moment condition E(X, u,, ) =0 rather than E(Z,u, ) =0. Show that your estimate is inconsistent and explain in words why it is inconsistent. (d) Suppose you are unwilling to assume that stochastic discount factor is given by M. = C, /C, . Instead, you are reluctant to go beyond the general model M, = f(W.), where We denotes a large vector of macroeconomic variables. Your friend suggests that you take the first & principal components of W, and assume that f is linear, i.e., specify M,. =d, +a+...+a.V, where Viel denotes the /" principal component of W#1. Write down the sample moment conditions that you would use to obtain the GMM estimates of the unknown parameters (do, ..., a, B) and describe some advantages and disadvantages of this modeling approach.Question 4. Point allocation: (a) 15%; (b) 25%; (c) 25%; (d) 35% Suppose the data are generated by the following AR(2) y, = (1+ P)YM - Py 2 + 6,; 5, ~ (0,0'); Ipkl (a) Show that this AR(2) is not stationary. (b) Calculate the impulse response for this process. Hint: y.. - (), + 8,) = AVNs + ... + Ay, + 6, Suppose now that you incorrectly specify the process as (c) Calculate the limit in probability of the least squares estimator for B. (d) Derive the asymptotic distribution for B. You may sue the following results without proof: The Functional Central Limit Theorem: X,()-y(WO; X,()= - Zu; w(Lju, =E, The Continuous Mapping TheoremQuestion 3. Point allocation: (a) 30%; (b) 45%; (c) 25% Consider the model Y, = P +u, (1) where / is a time trend, u, is an i.i.d. sequence with mean zero and variance o'. Suppose we have a sample of size 7, where the first R observations are used to estimate the model, and P = T-R observations are reserved to test the out-of-sample forecasting performance of the model. Specifically, we are interested in evaluating the one period-ahead forecasts generated by where B, refers to the least-squares estimate of expression (1) with the first R observations, and 1 = R+1, R+2, .... T. Define the P one period-ahead forecast errors for / = R+1, ..., Tas A test of forecasting bias can be constructed by estimating the following regression with these forecast errors i, = a + 5 , and then testing the null hypothesis Ho : a = 0 with a simple t-statistic. The following results are provided for convenience: v +1 SLy, - N(0,q' /3) for v, a M.D.S. Answer the following questions: (a) Suppose R - co, derive the asymptotic distribution of Be . (b) Suppose T, R, P -> ; P/R -> > 0] = //(1-e-#) Vy"] = / Vly"ly" > 0) = He #/ (1 - e-M)2. In this question the variable y, is not completely observed. (a) Suppose we observe only 3 = 1 if y/ 2 1 3 = 0 if y, = 0 Give with justification the objective function for a consistent estimator of B. (b) Suppose we observe only yo = vi if y/ 2 1. Give with justification the objective function for the MLE of B. (c) In the same situation as in part (b), give an alternative consistent estimator for / that uses nonlinear least squares. (d) In the same situation as in part (b), give an alternative consistent estimator for / that uses GMM. (e) Suppose that now we completely observe y", but the data are panel data. We suppose that yit is Poisson distributed with density f(yit) = e-Mutual / yoel, yit = 0, 1, 2, ... Suppose we obtain the Poisson MLE of / by Poisson regression with mean /y = exp(x /). What are the properties of this estimator (consistency and efficiency) if in fact /y = 0, exp(x/) where the additional component o, is assumed to be iid [1, 62]? Give a solid explanation.Question 1.Point allocation: (a) 10%; (b) 20%; (c) 20%; (d) 20%; (e) 10%; (f) 20%. Consider the following density for the continuous positive random variable y f(y) = exp(-0/y)02(1/y)3/2; y > 0, 0>0, where it can be shown that Ely) = 0 Vyl = 00 E[1/y) = 2/0 V[1/y] = 2/03 Suppose we have a random sample (y, x,), i = 1, ...N, where x, is a k x 1 nonstochastic regressor vector and y; has the above density with 0; = exp(X,B), where A = Bo in the data generating process. For much of this question we are concerned with the properties of the MLE of S under conditions weaker than correct specification of the density. You can apply any laws of large numbers and central limit theorems without formally verifying the necessary assumptions. (a) Give the formula for the objective function Qu() equal to N-1 times the log-likelihood function. (b) Obtain plim QN(B). (c) Given your answer in (b), what assumptions are the essential assumptions to ensure consistency of A that maximizes ON(G). (d) Assuming that the density is correctly specified, give the limit distribution of B. [Your deriva- tion can be as brief as possible]. (e) Hence give a method to test at the 5% significance level Ho : 8, = 1 against Hi : B, # 1, where , is the coefficient of the joh regressor. (f) Now suppose that the conditional density is misspecified, though in such a way that the MLE retains its consistency. Give the formula for a consistent estimate of the asymptotic variance- covariance matrix of B. [Your answer and derivation can be as brief as possible]