课程名称︰经济学与计量经济学专题 课程性质︰经济系、所选修 课程教师︰郭汉豪 开课学院：社科院 开课系所︰经济系 考试日期（年月日）︰111.06.06~111.06.08 考试时限（分钟）：4320 试题 : 部分数学式以TeX语法编写。 1. Bayesian Point Estimation (30 points) This question is about estimating functions of parameters by Bayesian methods. The random variable under consideration is denoted by x. Image that our potent- ial data will consist of realization(s) of x. The conditional density of condi- tional likelihood of x is f(x|μ), where μ us a scalar parameter. The prior density of μ is g(μ). The marginal density (unconditional density) of x is denoted by f_G(x) = \int_μ f(x|μ)g(μ)dμ. The posterior density of μ us denoted by f(μ|x) = f(x|μ)g(μ)/f_G(x). The parameter of interest is denoted by λ=h(μ), which is a function of μ. In our classes, we considered the cases where λ=μ. The Bayes desicion is δ(x). The loss function for point estimation is L(δ(x),λ)=(δ(x)-λ)^2. The Bayes risk is defined as in the reference book that W(δ) = E[E(L|x)]. The Bayes dec- ision is the minimizer of the Bayes risk. (i.) Bayes Point Estimate Prove that the Bayes point estimate of λ\equiv h(μ) is the posterior mean of λ, E(λ|x). Write your arguments clearly. (ii.) Normal Random Variable Suppose x is normally distributed with mean μ and variance σ^2. The variance σ^2 is a known constant so that we do not need to have a prior distribution of it. Our parameter of interest is λ=exp{μ/σ}. The prior density of μ is not specified. Derive the Bayes point estimate of λ, in terms of the mariginal de- nsity. Note that this question is not example 1.3.6 in the reference book. The functional form of λ is different. 2. James-Stein Theorem (50 points) This question is about proving a general version of James-Stein theorem in whi- ch the random variables and the parameters of interest are independently norma- lly distributed. Precisely, we have the assumption (1.32) in Efron (2010, p.6). The observed ra- ndom variables are z_i, where i=1,...,N. The conditional distribution of z_i is N(μ_i, σ_0^2). The prior distribution of μ_i is N(M,A). (i.) Marginal and Posterior Distributions Prove the results in (1.33) in Efron (2010, p.6). That is, prove that the marg- inal distribution of z_i is N(M,A+σ_0^2) and that the posterior distribution of μ_i is N(M+B(z_i-M), Bσ_0^2), where B=A/(A+σ_0^2). (ii.) Empirical Bayes Estimator If the piror distribution is known, the Bayesian estimator of μ_i is the post- erior mean M+B(z_i-M). Suppose the parameters in the prior are unknown. Show that the James-Stein empirical Bayes estimator is μ_i in (1.35) in Efron (2010 , p.7). Hint: you need to derive an unbiased estimator of B by using the prope- rties of chi-squared distribution and gamma function. (iii.) James-Stein Theorem Prove the James-Stein theorem under this setting. Precisely, prove that (1.26) in Efron (2010, p.5) holds if N\geq4. Note that you do not need to use the pri- or distribution when you are proving the James-Stein theorem. The expectations in (1.26) are conditional on μ_i's. 3. Shrinkage (20 points) This question is about shrinkage estimation. Suppose y is a p*1 column of rand- om variables. Its i-th entry is y_i. They have heterogeneous first and second moments across i. For all i and j, E_μ(y_i)=μ_i, E_μ(y_i-μ_i)^2=σ_i^2, and E_μ(y_i-μ_i)(y_j-μ_j)=σ_{ij}. Note that σ_i^2 = σ_{ii}. The expectat- ions are conditional on μ. The situation is as follows. We have one observation of y_i for each i. Theref- ore, an obvious unbiased estimator for μ_i is \hat{μ_i}\equiv y_i; that is, \hat{μ}=y. Obviously, for all i and j, E_μ(\hat{μ_i})=μ_i, E_μ(\hat{μ}- μ_i)^2 = σ_i^2, and E_μ(\hat{μ}_i-μ_i)(\hat{μ_j}-μ_j)=σ_{ij}. Our main purpose is to estimate μ_i with small expected total mean squared er- rors. Thus we consider the following shrinkage estimatorL \hat{μ_i^s}\equiv\xi_i\hat{μ_i}, where \xi_i is a nonrandom number between 0 and 1. Please derive the conditions under which we have E_μMSE(\hat{μ^s},μ) < E_μMSE(\hat{μ},μ). --

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