Class Note for ECON 696P at UA
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Date Created: 02/06/15
Econ 696 Spring 2008 Lecture Note 1 Three Examples Example 1 Normal Model Suppose that we observe random variables Yi i 17 771 and we assume that wikd39wa 177n lt1 The notation of 1 indicates that conditional on a quantity u the Y are independent and identically distributed as normal with mean u and variance 1 This de nes for a given u a joint distribution for YhYg 7Yn7 so we can speak of a joint probability density associated with any vector of possible values for Y1 WY fy177ynl H gol em lt2 11 where the product form follows from the iid assumption7 which can be simpli ed to 7 1 1 If 2 lt2vrgtnZexpl if M 3 We will suppose that u is not known but is assumed to have a distribution with density 1901 For concreteness7 suppose 1901 is a normal density with mean a and variance 1 It will simplify some of the formulas if we work instead with the inverse of the variance 739 11 This is called the precision Then 739 2 if 7 4 pm mew W M ltgt If we regard pu as a marginal density for u and yl 7ynlp as a conditional density for the data7 this speci es a joint distribution 190173417 Wyn pMfy1 Wynn which in turn de nes the conditional distribution MbWyn plt ly17ln7yngt i pylvquot397yn pylv7yn 39 5 Here we are using to refer to a number of different density functions which one is meant can be inferred by examining its arguments Notice that the denominator of 5 is a constant in In and can be obtained by integrating the numerator with respect to a ManyWyn pM7y177yndpppfy177ynlpdp lntuitively the denominator is whatever constant c that makes 1 29042417 Wyn E p fy17 Wynn a proper density function in the sense of integrating to 1 Because 0 is a multiplicative factor which does not depend on a we can simplify the notation by using cx to mean proportional to77 and write pMly177yn 0ltPltHgtflt91779nl gt With some algebra this can be shown to be proportional to 1 2 exp T M M 7 6 where 7L i T n l u 397 rna rn The expression in 6 is proportional to equal to a constant times the normal density for a with mean if and precision r We conclude that all1 2417 7Yn yn Moi r n 1 Notice that if the conditional mean of a is a weighted average of a and the sample mean ELI y with the weight depending on r and n Suppose we desire to provide a point estimate at of a that minimizes the quadratic loss a 7 12 Since a is regarded as having a distribution rather than being xed we could choose at to minimize EMd2ly177yn7 where the expectation is with respect to the conditional distribution of a The best choice will generally depend on yl yn so the optimal policy can be regarded as a function dy1 yn Solving for the optimum dy1yn Ealy1yn p Notice that as 7 H 0 we get the usual estimator g This notion of optimality applies if the mean a does in fact have the speci ed normal distribution What if instead a is xed but unknown and allowed to take any value in R In later lectures we will study optimality in this case and show that in a certain sense the procedure just described still does pretty well The marginal density 1901 is usually called a pn39or density and puly1 yn is called a posten39or density So far we have not really discussed how to interpret or choose a prior but we will have more to say about this in later lectures Equation 5 is a special case of Bayes7 Theorem hence the term Bayesian analysis Example 2 Asset Allocation with Estimation Risk This example is based on Barberis 2000 Consider an investor trying to decide how much of her wealth to allocate among two assets whose returns are uncertain say a stock index and a bond index Denote by ytl the log return on the stock index and ya the log return on the bond index at time 25 At time T the investor has wealth WT 1 and must choose how much to place in each asset Let a E 0 1 denote the allocation to the stock index we are ruling out short sales and for simplicity suppose that the investor has a buy and hold strategy she chooses the portfolio allocation a at time T and holds on to it for H periods Wealth in period T H is thus given by WTH anpyT11 yTH1 1 a expyT12 yTH2 If the investor has isoelastic utility W1 717 then it seems reasonable for her to choose a in order to maximize expected utility UW a expltyT11 yTH1 1 a expyT12 yTH2177 7 E 17V 7 where the expectation is with respect to the distribution of future returns We will discuss expected utility theory in more detail in lecture note 3 but for now assume that this is a reasonable thing to do This raises the question which distribution to use We could assume that the vector of returns yt ytl ytg has a multivariate normal distribution ml 2 N MOLE 8 Assuming that u and E are given the investor could proceed by evaluating 7 for different values of a and choose the allocation that gives the highest expected utility What happens if instead there is uncertainty estimation risk concerning 1 and E If the investor has access to data on earlier returns yl yT this could be useful for predicting future returns By analogy with the previous section we could choose a prior pu2 for the parameters and use Bayes7 Theorem to obtain the posterior 19013341 yT In later lectures we will derive an explicit expression for the posterior density but for now assume that this can be done However we are no longer directly concerned with the parameters but instead would like to place a distribution on future returns yT1 yTH that incorporates uncertainty about the parameters Notice that Myra7yTHly177yr PZJT177yTH7M72l29177yTdM72 9 This is called a predictive density We can decompose the density inside the integral so this can be written pyr177yTHlmE7y17yTp 7Ely177yTd 7E A further simpli cation is possible since by 8 the distribution of yt conditional on u2 is inde pendent of past returns pyr17 7 yTHlM7 32901732417 WWWi 2 The rst term inside the integral is a product of multivariate normal densities and the second term is the posterior density for the parameters It is important to recognize that the distribution of 34TH yTH given uE is independent of y1 yT but it is not independent ofthe past data if we remove the conditioning on the unknown 112 From the perspective of the investor who does not know 12 the return process can display persistence even though the true process does not If this were not the case there would be no value to observing the historical return data yl yT for making forecasts about future returns Treatment Assignment Many studies have examined the impact of job training and other social programs on economic outcomes of individuals Dehejia 2005 considers data from a social experiment the GAIN pro gram in California This was a randomized experiment comparing standard AFDC welfare to a treatment which consisted of education job training and job search activities Dehejia argues that instead of focusing on simply estimating an average treatment effect it may be useful to consider the problem of a social planner who can choose to assign individuals to either of the two options based on the individual s background characteristics Let T 1 denote that individual i received the GAIN program and let T 0 denote receipt of the standard AFDC program The outcome of interest is individual earnings in various quarters after the program Since many welfare recipients had zero earnings Dehejia used a Tobit model A simpli ed version of Dehejia s model is Y max0 Othl39 6139 where the q are llD N0 02 We assume that individuals in the data are a random sample from some population so that Xi Ti are llD Dehejia estimated this model using the 71 experimental subjects and then produced predictive distributions for a hypothetical n 1th subject to assess different ways of assigning treatments To put this in a similar framework as the previous examples let the parameter vector be 9 041 042 043 02 For a given individual i the probability of observing zero earnings is 131 230 PTO1i Otgtl39 Oz ml39 ti 6139 220 E39 az39atamt Pr ltJlt7 1 l 21 3 l llml fi a a I Ozzti Oz i 45139 7 039 where I is the standard normal CDF If Y gt 0 its density is gtyila1i aztz39 a m 7513057 where gtylu 02 is the PDF function of a normal random variable with mean u and variance 02 So we can write the singleobservation conditional likelihood function as ox39042tom t 1yi0 gt X 1222 0433M ti 021yzgt0 fltyili7ti7m lt The joint conditional likelihood based on the n observations in the experimental data can be written as V L fy177ynl177n7t177tm0 H yilmti l i1 If we have a prior distribution for the parameters 0 say 190 then the posterior can be viewed as pwlylv7yn7177n7t17wtngt mp0fylvwynl177n7t177tn70 Now suppose that a social planner wants to assign a new individual 71 1 to one of the two treatments based on observing their covariates Xn1 For simplicity suppose that the social planner s objective is to choose treatment if to maximize expected income EYn1an1Tn1 25 If the social planner knew 0 then she could compare ElYn1an17 Tn1 170l VS ElYn1an1an1 Oval and pick whichever treatment gave higher expected earnings lf 9 is not known7 but the social planner has access to the data 17 771 then the planner could form a predictive distribution for Yn given Xn1 and the past data Exercise show that fyn1l17tn17y17 wym 17 7mt17 wtn fltyn1ln17tn170p0lylv wym 17 7mt17 7tn With this predictive distribution7 the planner can form two expectations for Yn1 corresponding to the two possible values of TWA7 and assign the individual to the treatment that gives higher ex pected earnings with respect to the predictive distribution Note that this predictive distribution will depend on the prior 190 because the posterior distribution Malt1 7 y m1 Wm 2517 725 depends on 190
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