Econ 583 Final Exam Fall 2008 Eric Zivot December 11, 2008 Exam is due at 9:00 am in my office on Friday, December 12. 1 Maximum Likelihood Estimation and Asymptotic Theory Let X 1,...,X n be iid random variables representing monthly continuously compounded returns on an asset (i.e., X i =ln(p i /P i 1 ) where P i denotes the price at the end of of month i) withx 1 N(μ, σ 2 ). 1. Show that the maximum likelihood estimators of μ and σ 2 are ˆμ mle = X = 1 n nx X i, ˆσ 2 mle = 1 n i=1 nx (X i X) 2. i=1 2. Show that ˆμ mle is unbiased and that ˆσ 2 mle is biased. 3. What is the MLE for σ? Justify your result. 4. Let θ =(μ, σ 2 ) 0 and ˆθ mle =(ˆμ mle, ˆσ 2 mle) 0. What is the asymptotic distribution of n(ˆθ mle θ) and what is the asymptotic distribution of ˆθ mle? How would you consistently estimate avar(ˆθ mle )? (Note: I am not asking you to prove this result - just give me the result along with a justification). 5. For α (0, 1), show that the α 100% quantile of the N(μ, σ 2 ) distribution can be expressed as q X α = μ + σ q Z α, where q Z α is the α 100% quantile of the N(0, 1) distribution (i.e, Pr(Z q Z α )= α). 1
6. The ML estimate of the α 100% quantile is ˆq X α =ˆμ mle +ˆσ mle q Z α Using the asymptotic distribution of ˆθ mle and the delta method compute an asymptotic standard error for ˆq X α. 7. In risk management, lower quantiles of return distributions are used to compute so-called value-at-risk (VaR) measures. VaR represents the dollar loss over an investment horizon with a stated probability. For example, with an initial investment of $W the 5% VaR for a monthly investment with continuously compounded return X N(μ, σ 2 ) is An estimate of VaR 0.05 is VaR 0.05 =$W exp(q X 0.05) 1. [VaR 0.05 =$W exp(ˆq X 0.05) 1. Use the results from question 6. error for [VaR 0.05. above to compute an asymptotic standard 8. A common performance measure is the so-called Sharpe ratio (SR) SR = μ r f, σ where r f is a fixed (non-random) risk-free rate. An estimate of the SR is csr = ˆμ mle r f. ˆσ mle Use the results from question 4. above to compute an asymptotic standard error for SR. c 2 Asymptotics for Nonlinear Regression Consider the nonlinear regression model y t = f(x t,β 0 )+ε t,,...,t where y t is a scalar observable random variable, x t is a vector of exogenous variables, β 0 is a K vector of unknown parameters and ε t is and iid random error term with E[ε t ]=0and var(ε t )=σ 2 0. An example of such a nonlinear regression is the Cobb- Douglas production function with additive noise y t = β 1 x β 2 1t x β 3 2t + ε t, 2
where y t,x 1t and x 2t denote output, capital input and labor input, respectively. The nonlinear least squares (NLS) estimator solves ˆβ NLS =argmin β SSR(β), where SSR(β) = (y t f(x t,β)) 2 (1) Then, the NLS estimator of β satisfies the K first order conditions 0= SSR(ˆβ NLS ) = 2 [y t f(x t,β)] f(x t,β). (2) 1. Derive the Gauss-Newton iteration scheme to solve for the NLS estimator of β. Recall, the GN iteration starts from a first order Taylor series expansion of f(x t,β) about some starting value β 1 and then the iteration scheme results from minimization of the SSR for the linearized model with respect to β. 2. Recognizing that the NLS estimator is an extremum estimator, state the assumptions required for an extremum estimator to be consistent. 3. For the NLS estimator, the following conditions are often assumed to prove consistency of ˆβ NLS : a) f(x t,β)/ exists and is continuous. b) f(x t,β) is continuous in β uniformly in t. c) T P 1 T f(x t,β 1 )f(x t,β 2 ) converges uniformly in β 1 and β 2. d) lim T T P 1 T [f(x t,β 0 ) f(x t,β)] 2 6=0if β 6= β 0. Show that T 1 SSR(β) =A 1 + A 2 + A 3 where A 1 = T 1 ε 2 t,a 2 = T 1 [f(x t,β 0 ) f(x t,β)] 2 A 3 = 2T 1 [f(x t,β 0 ) f(x t,β)] ε t 4. Assuming that A 3 p 0 uniformly in β, show that lim T 1 SSR(β) is uniquely minimized at β 0. 3
5. Asymptotic normality of ˆβ NLS starts from a first order Taylor series expansion of the first order conditions (2) about the true value β 0 : 0= SSR(ˆβ NLS ) = SSR(β 0) + 2 SSR(β 0 ) 0 (ˆβ NLS β 0 )+o p (1). Derive the limit distribution of T (ˆβ NLS β 0 ) assuming that where 1 SSR(β 0 ) d N(0, 4σ 2 T 0C), (3) 1 T 1 C = lim T T 2 SSR(β 0 ) 0 p 2C, (4) f(x t,β 0 ) f(x t,β 0 ) 0. 6. Using (1), show that SSR(β 0 ) = SSR(β) = 2 β=β0 2 SSR(β 0 ) 0 = 2 SSR(β) 0 = 2 β=β0 f(x t,β 0 ) f(x t,β 0 ) 0 2 f(x t,β ε 0 ) t, and give arguments to support the results in (3) and (4). 3 Stylized Wage Equation Consider the stylized cross sectional wage equation ln wage i = δ 0 + δ 1 educ i + δ 2 exper i + δ 3 age i + δ 4 age 2 i + u i ε t 2 f(x t,β 0 ) 0, where educ represents years of schooling, and exper represents work experience (e.g. age - educ - 6). In matrix notation, the wage equation may be represented as y i = z 0 iδ + u i,i=1,...,n where z i and δ are L 1 vectors, with L =5. 1. Briefly discuss why educ is a potentially endogenous variable in the wage equation. 4
Let x i denote a K 1 vector of instruments, with K L, satisfying E[x i u i ] = 0 E[x i z 0 i] = Σ xz,rank(σ xz )=L Assume that (y i, z i, x i ) is stationary and ergodic, and that g i = x i u i is a martingale difference sequence with K K covariance matrix E[g i g 0 i]=s 2. Give an example of suitable instruments for educ. 3. Based on the instruments x i, derive the efficient two-step GMM estimator for δ. Make sure to describe how to consistently estimate the efficient weight matrix (note: you do not need to prove that this weight matrix is consistent). 4. Does iterating the two-step GMM estimator improve its asymptotic efficiency? 5. Suppose that K>L.Briefly describe how you would test the overidentifying restrictions. Give the test statistic you would use and state its asymptotic distribution under the null hypothesis that the overidentifying conditions are valid (you do not have to derive the asymptotic distribution of the test). What does it mean if you reject null hypothesis? 6. Let x 1 =(1,age i,age 2 i,med i,fed i ) 0 where med i and fed i represent mother s education and father s education, respectively. Assume that x i represents a set of instruments that are assumed to be valid. Suppose you are unsure if exper i is exogenous (uncorrelated with ε i ) inthewageequation. Describehowyoucan test the null hypothesis that exper i is exogenous against the alternative that exper i is endogenous. Be sure to give the asymptotic distribution of your test statistic. 7. Suppose you have two potentially valid instruments for educ, say x 1 and x 2 (e.g. med and fed). Assume that all other variables are exogenous. The asymptotic theory for the GMM estimator requires that the instruments be relevant for the endogenous variable, educ. Briefly describe how you can test for the relevance of these two instruments? 8. Continuing with the previous question, suppose you determine that x 1 and x 2 are only weakly correlated with educ. Briefly discuss the main implications of weak instruments for estimation and inference with respect to the coefficient on educ. 5
4 Seemingly unrelated regressions Consider a SUR model with 2 regression equations expressed as the system y1 X1 0 β1 ε1 = + y 2 0 X 2 β 2 ε 2 (5) where y 1 and y 2 are n 1 vectors, X 1 is a n k 1 matrix, X 2 is a n k 2 matrix, β 1 is a k 1 1 vector, β 2 is a k 2 1 vector and ε 1 and ε 2 are n 1 vectors. Assume that X 1 and X 2 contain exogenous variables independent of the error terms. The error terms satisfy E[ε 1 ε 0 1]=σ 11 I n,e[ε 2 ε 0 2]=σ 22 I n and E[ε 1 ε 0 2]=E[ε 2 ε 0 1]=σ 12 I n where I n is the n dimensional identity matrix. The SUR model can be written as the giant regression y = Xβ + ε (6) where y =(y 0 1, y 0 2) 0 and X and ε are obtained similarly from the SUR model. y and ε are 2n 1, X is 2n k and β is k 1 where k = k 1 + k 2. The stacked errors ε have covariance matrix E[εε 0 ]=Σ I n = V which is of dimension 2n 2n and where Σ is a 2 2 matrix with elements σ ij (i, j =1, 2). 1. Show that OLS on the giant regression (6) is the same as OLS on each equation in (5) taken separately. 2. Compare the estimated variance-covariance matrix of the coefficients computed from the giant regression (6) and from OLS on each equation in (5). Are they the same? 3. What is the (infeasible) GLS estimator of β in the giant regression (6)? Note: the GLS estimator is equivalent to the efficient GMM estimator. 4. Show that β b GLS is numerically µ equivalent to β b OLS when Σ is a diagonal matrix. σ (Hint: write Σ 1 1 11 0 = 0 σ 1 and crank away.) 22 5. Show that b β GLS is numerically equivalent to b β OLS when X 1 = X 2 = X. 6