MACROMICRO & AG FIN
MACROMICRO & AG FIN AEB 6933
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This 10 page Class Notes was uploaded by Weldon Rau I on Friday September 18, 2015. The Class Notes belongs to AEB 6933 at University of Florida taught by Charles Moss in Fall. Since its upload, it has received 21 views. For similar materials see /class/206590/aeb-6933-university-of-florida in Agricultural Economics And Business at University of Florida.
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Date Created: 09/18/15
Bayesian Estimation and Con dence Intervals Lecture XXII I Bayesian Estimation A Implicitly in our previous discussions about estimation we adopted a classical viewpoint 1 We had some process generating random observations 2 This random process was a function of xed but unknown 3 We then designed procedures to estimate these unknown parameters based on observed data Speci cally if we assumed that a random process such as students admitted to the University of Florida generated heights This height process can be characterized by a normal distribution We can estimate the parameters of this distribution using maximum likelihood 2 The likelihood of a particular sample can be expressed as l 2 2 L XXz 2 X 2 2 n 2 3 Our estimates of u and o 2 are then based on the value of each parameter that maximizes the likelihood of drawing that sample Turning this process around slightly Bayesian analysis assumes that we can make some kind of probability statement about parameters before we start The sample is then used to update our prior distribution 1 First assume that our prior beliefs about the distribution function can be expressed as a probability density function if 639 where 9 is the J wearequot Jin quot quot 2 Based on a sample the likelihood function we can update our knowledge of the distribution using Bayes rule L Xe 11 e 7 SIX 1 L xe 739 6 d9 Departing from the book s example assume that we have a prior of a Bernoulli distribution Our prior is that P in the Bernoulli distribution is distributed B a l The beta distribution is de ned similar to the gamma distribution 1 f PaB PM 1713 B 0H3 AEB 6933 7 Mathematical Statistics for Food and Resource Economics Lecture XXII Professor Charles Moss Fall 2007 B 05 is de ned as lH B 0L3 Jxquotquot1 l x dx Thus the beta distribution is de ned as F on 3 M f Pa P l P 1 a 1 3 1 2 Assume that we are interested in forming the posterior distribution after a single draw n PX PX 11 P up 0t 13qu 1 X 7 EPXW I 1 XdP Following the original speci cation of the beta function JZPM I 11 P ldP where 1 XX and 3 B Xl X 1 1 The posterior distribution the distribution of P after the observation is then 09 3 The Bayesian estimate of P is then the value that minimizes a loss function Several loss functions can be used but we will focus on the quadratic loss function consistent with mean square errors 2 minE15 ZEUS PJ 0 P EP AEB 6933 7 Mathematical Statistics for Food and Resource Economics Lecture XXII Professor Charles Moss Fall 2007 Taking the expectation of the posterior distribution yields E P 1 P X dP 01 1 P X dP 01 7 As before we solve the integral by creating of aX 1 and l X 1 The integral then becomes 1 1312671 1 X 1 X 1 F x32 Hence 1 X 1 EP X 1 Which can be simpli ed using thefact r a 1 ar 6 Therefore 1 X X i X 01 X 1 4 To make this estimation process operational assume that we have a prior distribution with parameters a l 4968 that yields a beta distribution with a mean P of 05 and a variance of the estimate of 00625 Next assume that we ip a coin and it comes up heads X 1 The new estimate of P becomes 06252 If on the other hand the outcome is a tail X 0 the new estimate of P is 03747 5 Extending the results to n Bernoulli trials yields PY 1 1 P Y 1 01 n where Y is the sum of the individual XS or the number of heads in the sample The estimated value of P then becomes Y 01 AEB 6933 7 Mathematical Statistics for Food and Resource Economics Lecture XXII Professor Charles Moss Fall 2007 6 Going back to the example in the last lecture in the rst draw Y 15 and n 50 This yields an estimated value of P of 03112 This value compares with the maximum likelihood estimate of 03000 Since the maximum likelihood estimator in this case is unbaised the results imply that the Bayesian estimator is baised II Bayesian Con dence Intervals A Apart from providing an alternative procedure for estimation the Bayesian approach provides a direct procedure for the formulation of parameter con dence intervals B Returning to the simple case of a single coin toss the probability density function of the estimator becomes PX 1 1 P X 0c 7 As previously discussed we know that given 0 14968 and a head the Bayesian estimator of P is 06252 However using the posterior distribution function we can also compute the probability that the value of P is less than 05 given a head PPlt5 f 1X lPX 1 Hence we have a very formal statement of con dence intervals 1 P X dP 22976 Composite Tests and The Likelihood Ratio Test Lecture XXIV I Simpl A e Tests Against 3 Composite Mathematically we now can express the tests as testing between H0 660 against H1He 1 where 1 is a subset of the parameter space Given this speci cation we must modify our de nition of the power of the test because the 5 value the probability of accepting the null hypothesis when it is false is not unique In this regard it is useful to develop the power function 1 De nition 94 1 If the distribution of the sample X depends on a vector of parameters 9 we define the power function of the test based on the critical region R by Qe PX eRe 2 De nition 942 Let Q16 and Q2 639 be the power functions of two tests respectively Then we say that the rst test is uniformly better or uniformly most power ll than the second in testing H0 660 against H1He 1 ifQ160QZ60 and Q19Z Q2 9 for all 9 e 1 and Q19gt Q2 9 for at least one 9 e 1 De nition 943 A test R is the uniformly most power UMP test of size level a for testing H0 6 60 against H1 6 e 1 if PR60 Sa and any other test Rl such that PR160 Sa we have PR6 2PR16 for any 6 e 1 The likelihood ratio test usually gives the UMP test if an UMP test exists Even if the UMP does not exist the likelihood ratio test has good properties AEB 6933 7 Mathematical Statistics for Food and Resource Economics Lecture XXIV Professor Charles Moss Fall 2005 1 De nition 944 Let Lx16 be the likelihood function and let the null and alternative hypotheses be H0 660 and H1 6681 where 1 is a subset of the parameter space 8 Then the likelihood ratio test of H0 against H1 is de ned by the critical region L90x A lt c es 1L1Jp L9x where c is chosen to satisfy PA lt clHO a for a certain value of a 2 Example 943 Let the sample be X1 Nu0392i 1 2n where o2 is assumed to be known Let x be the observed value of X1 Testing H0 u 0 against H1 u gt M The likelihood ratio test is to reject H0 if 1 exp 262 20 way 1 n ltc ew ZLZZWHY 11 3 Assume that we had the sample X678910 from the preceding example and wanted to construct a likelihood ratio test for u gt 75 eXp22 6 752 7 752 10 752 exp6 8Z 7 82 10 82 AEB 6933 7 Mathematical Statistics for Food and Resource Economics Lecture XXIV Professor Charles Moss Fall 2005 where 8 is the maximum likelihood estimate of u assuming a standard deviation of 15 yields a likelihood ratio of 1284 4 Theorem 941 Let A be the likelihood ratio test statistic Then 2A is asymptotically distributed as chisquared with the degrees of freedom equal to the number of exact restrictions implied by H 0 11 Composite Against Composite A De nition 951 A test R is the uniformly most power test of size level a if sung PR6 Sa and for any other test R1 such that sungGPR1 6 Sa we have PR6 2PltRI6 for any 6 e 8 B De nition 952 Let Lx6 be the likelihood function Then the likelihood ratio test of H 0 against H1 is de ned by the critical region su L 9 x sup L9x enUel where c is chosen to satisfy sup PA ltc6gt 0 for a certain speci ed value of a C Example 952 Let the sample X1Nu0392 with unknown 0392 i12n We want to test H0Ltu0 and 0ltUZltOO against H1Ltgtu0 and 0ltUZltOO L6 2ni 62T exp 22 21xl u2 Using the concentrated likelihood function at the null hypothesis supL 9 2n7 62 7 exp Z on 2 2 1 n 2 6 Zxlc H0 This likelihood value can be compared with the maximum likelihood value of AEB 6933 7 Mathematical Statistics for Food and Resource Economics Professor Charles Moss sg pL 9 2n7 62 7 exp 62 2ch w 1 53271161 n The critical region then becomes 62 A Z lt C 6 Lecture XXIV Fall 2005 D Turning back to the Bayesian model the Bayesian would solve the problem testing H06S60 against H16gt60 Let L209 the loss incurred by choosing H0 and by choosing H1 The Bayesian rejects H0 if jL1efexde lt 7L2efexde where f6x is the posterior distribution of 9 Definition of Estimator and Choosing among Estimators Lecture XVII I What is An Estimator A In the next several lectures we will be discussing statistical estimators and estimation The book divides this discussion into the estimation of a single number such as a mean or standard deviation or the estimation of a range such as a con dence interval At the most basic level the de nition of an estimator involves the distinction between a sample and a population 1 In general we assume that we have a random variable X with some distribution anction Next we assume that we want to estimate something about that population for example we may be interested in estimating the mean of the population or probability that the outcome will lie between two numbers a For example in a farmplanning model we may be interested in estimating the expected return for a particular crop b b In a regression context we may be interested in estimating the average effect of price or income on the quantity of goods consumed This estimation is typically based on a sample of outcomes drawn from the population instead of the population itself Common point estimators are the sample moments 1 Sample Mean 2 l X n 11 1 Sample Variance l n 7w 1 n 2 2 2 SX Zz1 1 XZerl X Sample km moment around zero 1 n X116 n 11 Sample km moment around the mean gzm 2Z
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