MACROMICRO & AG FIN
MACROMICRO & AG FIN AEB 6933
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Date Created: 09/18/15
Introduction to Statistics Probability and Econometrics Lecture I I The basic question to be answered on the rst day is What are we going to study over the next fteen weeks and how does it t into my graduate studies in Food and Resource Economics A The simplest and most accurate answer to the rst question is that we are going to develop statistical reasoning using mathematical reasoning and techniques B The answer to the second part of the question requires is a little more complicated 1 Following Kmenta statistical applications can be divided into two subgroups descriptive statistics and statistical inference a Kmenta s claim is that most statistical applications in economics involve the application of techniques for statistical inference b However this position ignores the concept of decision making under risk c From a general statistical perspective mathematical statistics allows for the formalization of statistical inference 1 How do we formulate a test for quality light bulb life 2 How do we develop a test for the signi cance of an income effect in a demand equation 2 Related to the general problem of statistical inference is the study of Econometrics a quotEconometrics is concerned with the systematic study of economic phenomena using observed data Spanos p3 b quotEconometrics is concerned with the empirical determination of economic laws Theil p1 c Econometrics is the systematic study of economic phenomena using observed data and economic theory 3 Economic theory most particularly production economics relies on the implicit randomness of economic variables to develop models of decision making under risk a Expected Utility Theory b Capital Asset Pricing Models c Asymmetric Information II An Example of Inference versus Decision Making A Skipping ahead a little bit the normal distribution function depicts the AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture 1 Professor Charles B Moss Fall 2007 probability density for a given outcome x as a function of the mean and variance of the distribution fx0392 1 exp M 032 20392 Graphically the shape of the function can be depicted as o Assuming that u 0 and o2 1 B Statistical inference involves testing a sample of observations drawn from this data set against an alternative assumption for example n n A C Economic applications involve the choice between the two distribution functions 111 What is probability A Two de nitions 1 Bayesian 7 probability expresses the degree of belief a person has about an event or statement by a number between zero and one 2 Classical 7 the relative number of time that an event will occur as the number of experiments becomes very large lim PO r 0 Name N AEB 6933 Mathematical Statistics for Food and Resource Economics Professor Charles B Moss IV What i A B D s statistics Lecture I Fall 2007 De nition I Statistics is the science of assigning a probability of an event on the basis of experiments De nition II Statistics is the science of observing data and making inferences about the characteristics of a random mechanism that has generated the data By random mechanisms we are most often concerned with random variables A Discrete Random Variable is some outcome that can only take on a xed number of values 1 a b The number of dots on a die is a classic example of a discrete random variable A more abstract random variable is the number of red rice grains in a given measure of rice It is obvious that if the measure is small this is little different than the number of dots on the die However if the measure of rice becomes large a barge load of rice the discrete outcome becomes a countable in nity but the random variable is still discrete in a classical sense A Continuous Random Variable represents an outcome that cannot be technically counted a b Amemiya uses the height of an individual as an example of a continuous random variable This assumes an in nite precision of measurement The normally distributed random variable presented above is an example of a continuous random variable The exact difference between the two types of random variables has an effect on notions of probability a b c The standard notions of Bayesian or Classical probability t the discrete case well We would anticipate a probability of 16 for any face of the die In the continuous scenario the probability of any outcome is zero However the probability density function yields a measure of relative probability The concepts of discrete and continuous random variables are then uni ed under the broader concept of a probability density function De nition III Statistics is the science of estimating the probability distribution of a random variable on the basis of repeated observations drawn from the same random variable Moments of More than One Random Variable Lecture IX Covariance and Correlation A Definition 431 XY EXEY XEY EXEY EXY EXEY 1 Note that this is simply a generalization of the standard variance formulation Speci cally letting Y X yields ConX EXX EXEX 2 EX2e EX 2 From a sample perspective we have VXl quotx2 n H 2 l n Cov XY Zt1x2yt 3 Together the variance and covariance matrices are typically written as a variance matrix xx xy V X Cov XY Cov XY V Y W W Note that Cov XY ny 6yr Cov YX 4 Substituting the sample measures into the variance matrix yields AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IX Fall 2007 Professor Charles B Moss 1 n l n S S10C SW n Hxlx n tlxtyt 7 SW SW 7 l n l Zz yzxz lyzyz n n l 21x x 2 1x y n V1 V1 Z ym my The sample covariance matrix can then be written as 6 Example 432 XY 1 0 1 1 0167 0083 0167 0417 0 0083 0000 0083 0167 1 0167 0083 0167 0417 0417 0167 0417 CXY is then computed as CX B Theorem 432 V XiY V X V Y iCov XY AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IX Professor Charles B Moss Fall 2007 V XiY E XiY XiY E XX2XYYY E XX E YY r 2E X Y VXVYi2CovXY 1 Note that this result can be obtained from the variance matrix Speci cally X Y can be written as a vector operation 1 X Y12XY Given this vectorization of the problem we can de ne the variance of the sum as XIV W1 W C Theorem 433 Let X i12be pairwise independent Then V ZLXI 1V X D The simplest proofto this theorem is to use the variance matrix Note in the preceding example ifX and Y are independent we have 1 XIV W1 W Uu6yy Since independence implies 6W 0 Extending this result to three variables implies 1 011 012 013 1 v 1 012 022 023 1 11 2 12 2 13 22 2 23 33 1 013 023 033 1 AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IX Professor Charles B Moss Fall 2007 If the xsare independent the covariance terms are zero and this expression simply becomes the sum ofthe variances E Definition 432 The correlation coefficient for two variables is defined as C X y C X Y orr loi loi 1 Note that the covariance between any random variable and a constant is equal to zero Letting Y equal to zero we have 2 It stands to reason the correlation coef cient between a random variable and a constant is also zero F It now possible to derive the ordinary least squares estimator for a linear regression equation 1 We de ne the ordinary least squares estimator as that set of parameters that minimizes the squared error of the estimate mi nEszl 2EY ZEXY 2 2 EX ZEszl The rst order conditions for this minimization problem then becomes a S72EY 20 2BEX 0 001 6S 2 a B72EXY 20 EX 2BE X 0 Solving the rst equation for 0L yields ocE Y 713E X Substituting this expression into the second rst order condition yields AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IX Professor Charles B Moss Fall 2007 X ELXZJ 0 2 2 j EX 0 X 0 BCOVXY V00 2 A little razzledazzle minSZmin j l3 l3 Y 39X39X Using matrix differentiation VHS2 Y39X X39YB X XX39XB 2X Y2X XB0 B X X 71X Y G Theorem 436 The best linear predictor or more exactly the minimum meansquarederror linear predictor of Y based on X is given by of 3X where of and 5 are the least square estimates II Conditional Mean and Variance A Definition 441 Let XY be a bivariate discrete random variable taking values xly ij12 Let P yle be the conditional probability of Y yj given X Let i be an arbitrary function Then the conditional mean of d XY given X denoted E XY IX or by EY X 4 XY is defined by Eyw p XYZ1ltp Xy P y X B Definition 442 Let XY be a bivarite continuous random variable with conditional density f ylx Let I be an arbitrary function Then the conditional mean of XY given X is de ned by AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IX Professor Charles B Moss Fall 2007 EMU XaY1JltP Xay f le dy C Theorem 441 Law of Iterated Means E 1gt XY J Where the symbol EX denotes the expectation with respect to X D Theorem 442 Proof Vypr P Ein PZJ Eypz P 2 Implies EX VW P i E 21EX Ein P T By definition ofconditional variance VX Eleq XEYlXp2Ep 2 Adding these expressions yields 2 EXP E V E Theorem 443 The best predictor or the minimum meansquared error predictor of Y based on X is given by EYX Basic Axioms of Probability Lecture 11 I Basics of Probability A Using the example from Birenens Chapter 1 Assume we are interested in the game Texas lotto similar to Florida lotto 1 In this game players choose a set of 6 numbers out of the first 50 Note that the ordering does not count so that 3520151545 is the same of3551520145 2 How many different sets of numbers can be drawn a First we note that we could draw any one of 50 numbers in the first draw b However for the second draw we can only draw 49 possible numbers one of the numbers has been eliminated Thus there are 50 x 49 different ways to draw two numbers c Again for the third draw we only have 48 possible numbers left Therefore the total number of possible ways to choose 6 numbers out of 50 is 50 k 5 50 j d Finally note that there are 6 ways to draw a set of 6 numbers you could draw 35 first or 20 first Thus the total number of ways to draw an unordered set of 6 numbers out of 50 is so 15 890 700 6 6 50 6 e This is a combinatorial It also is useful for binomial arithmetic ab n n njakbnik lt gt B Definitions 1 Sample Space The set of all possible outcomes In the Texas lotto scenario the sample space is all possible 15890700 sets of6 numbers which could be drawn 2 Event A subset of the sample space In the Texas lotto scenario possible events include single draws such as 3520151545 or complex draws such as all possible lotto tickets including 352015 Note that this could be 352015123 352015 124 3 Simple Event An event which cannot be a union of other events In the Texas lotto scenario this is a single draw such as 3520151545 4 Composite Event An event which is not a simple event AEB 6933 Mathematical Statistics for Food and Resource Economics Professor Charles B Moss Lecture 11 Fall 2007 II Axiomatic Foundations A Aseta1co 03 1k of different combinations of outcomes is called an event These events could be simple events or compound events In the Texas lotto case the important aspect is that the event is something you could bet on for example you could bet on three numbers in the draw 352015 A collection of events F is called a family of subsets of sample space Q This family consists of all possible subsets of Q including Q itself and the nullset 1 Following the betting line you could bet on all possible numbers covering the board so that Q is a valid bet 2 Altematively you could bet on nothing or is a valid bet Next we will examine a variety of closure conditions These are conditions that guarantee that if one set is an contained in a family another related set must also be contained in that family 1 First we note that the family is closed under complementarity IfAeFthen171QAeF In this case 1 QA e F denotes all elements of Q that are not contained in A 2 Second we note that the family is closed under union IfAB eF thenAUB 6F 3 De nition 11 Bierens A collection F of subsets of a nonempty set Q satisfying closure under complementarity and closure under union is called an algebra 4 Adding closure under in nite union de ned as IfA eF forj 123 then UA EF 1 5 De nition 12 Bierens A collection F of subsets of a nonempty set Q satisfying closure under complementarity and in nite union is called a o algebra sigmaalgebra or a Borel Field AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture H Professor Charles B Moss Fall 2007 D Building on this foundation a probability measure is then measure which maps from the event space into real number on the 01 interval PA 1 We typically think of this as an odds function ie what are the odds ofa winning lotto ticket 115890700 2 To be mathematically precise suppose we de ne a set of events Aa1ae 2 say that we choose 11 different numbers The probability of winning the lotto is PA Pa1a nN a Our intuition would indicate that P 2 1 or the probability of winning given that you have covered the board is equal to one a certainty b Further if you don t bet the probability of winning is zeros or P 0 3 De nition 122 Given a sample space Q and an associated o algebra F a probability function is a function PA with domain F that satisfies a PA20for all AEF b 109 1 c If 1411426 Fare pairwise disjoint then PUA PA 11 AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture 11 Professor Charles B Moss Fall 2007 E Axioms of Probability 1 PA20 for any event A 2 PS 1where S is the sample space 3 If A i 12 are mutually exclusive that is A NA Q for all i7 j then PAl NA mPAlPA2 F In a little more detail from Casella and Berger De nition 111 The set S of all possible outcomes of a particular experiment is called the sample space for the experiment 2 De nition 112 An event is any collection of possible outcomes of an experiment that is any subset of S including S itself De ning the subset relationship a ACBcgtxeA2xeB b ABltgtACB and BCA c Union The union of A and B written AvB is the set of elements that belong to either A or B AUBXEXEAOI XEB Intersection The intersection of A and B written AmB is the set of elements that belong to both A and B A Bxx A andxeBgt e Complementation The complement of A written A0 is the set of all elements that are not in A A x x EA 4 Theorem 111 For any three events A B and C de ned on a sample space S a Commutativity A UB B UA A MB B MA E Q V b Associativity AvBvCAvBvC A B CA B C c Distributative Laws AmBvCAvBmAmC AuB CA BuA C d DeMorgan s Laws Ava A UBC Ame A UBC III Counting Techniques A Simple Events with Equal Probabilities PM 7104 715 the probability of event A is simply the number possible occurrences of A divided by the number of possible occurrences in the sample De nition 231 The number of permutations of taking relements from n elements is a number of distinct ordered sets consisting of r distinct elements which can be formed out of a set of n distinctive elements and is denoted Pquot 1 The rst point to consider is that of factorials For example if you DJ AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture H Professor Charles B Moss Fall 2007 have two objects A and B how many different ways are there to order the object Two AB or EA If you have three orderings how many ways are there to order the objects Six ABCACBBACBCACAB or CBA The sequence then becomes two objects can be drawn in two sequences three objects can be drawn in six sequences 2 X 3 By inductive proof four objects can be drawn in 24 sequences 6 X 4 2 The total possible number of sequences is then for n objects is n defined as nnn71n721 n n 7 r I De nition 232 The number of combinations of taking r elements from n elements is the number of distinct sets consisting of r distinct elements which can be formed out of a set of n distinct elements and is denoted Cf 0 Theorem 231 B U n n r n rr Moment Generating Functions Lecture X I Moment Generating Functions A Definition 233 Let X be a random variable with cumulative distribution function F X The moment generating function mgf of X orF X denotedMX t is MX t Ee X provided that the expectation exists for t in some neighborhood of 0 That is there is an h gt 0 such that for all t in h ltt lt h Ee X exists 1 If the expectation does not exist in a neighborhood of 0 we say that the moment generating function does not exist 2 More explicitly the moment generating function can be de ned as M X t J e C f x dx for continuous random variables and M X t Z e P X x for discrete random variables x B Theorem 232 If X has mngX t then EX M 0 X where we define dn M g 0 MX t dtquot 0 1 First note that 2 can be approximated around zero using a Taylor series expansion 1 Act x oz 03 6 2 t3 E 8 E L J6 Note for any moment 14 n dquot n Thus as I 0 AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture X Professor Charles B Moss Fall 2007 M 0 Ex 2 Leibnitz s Rule If f x9 a 9 and b 9 are differentiable with respect to 9 then d e d ELfaqfa b 17 fx dx a 3 Berger and Casella proof Assume that we can differentiate under the integral using Leibnitz s rule we have d d M t e C x dx dt X dt w f ge39xjf x dx Exetxj x dx Letting t gt 0 this integral simply becomes f xf x dxE x 4 This proof can be extended for any moment of the distribution function C Moment Generating Functions for Specific Distributions 1 Application to the Uniform Distribution DC 17 172 at a a t a t b a Following the expansion developed earlier we have 1 2 1 b3 of f 6 X b at b2 613 IS 1 2 b at 1 1 by t2 1 bid b2aba2 t3 2 b b a t 1l ab tl azabb2 t2 2 6 Letting b 1 and a 0 the last expression becomes AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture X Professor Charles B Moss Fall2007 1 1 1 MX t 1 t t2 t3 2 6 24 The rst three moments of the uniform distribution are then N H o C II N cxl NIH N H LAID I O H ID i i M O H N 1 1 2 Application to the Univariate Normal Distribution 1 Hi 2 l W 2 X t 6 2 jeb e2 quot dx 1 t dx Focusing on the term in the exponent we have 2 M 1 x 1 x L 2txlts2 2 6 2 6 1962 21962 5 2 1962 2 7 2 1962 2 E 2 The next state is to complete the square in the numerator x2 72x utoz H2 c0 2 x7 ut62 0 x2 2x uto u22t62ut2640 c 2min t2cs4 The complete expression then becomes AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture X Professor Charles B Moss Fall 2007 lx uz 1x 2r 2 tx3 Z 2 S The moment generating function then becomes x t 2 K J Taking the rst derivative with respect to t we get M t u62t exp Letting t gt 0 this becomes M g 0 p The second derivative of the moment generating function with respect to t yields 1 2 2 P l 2 l Again letting t gt 0 yields M X2 0 62 pl 3 Let X and Y be independent random variables with moment generating functions MX t and My t Consider their sum ZXYand its moment generating function M r quotM Eeb Ee y tMY t a We conclude that the moment generating function for two independent random variables is equal to the product of the moment generating functions of each variable b Skipping ahead slightly the multivariate normal distribution function can be written as AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture X Professor Charles B Moss Fall 2007 where E is the variance matrix and u is a vector of means c In order to derive the moment generating function we now need a vector t The moment generating function can then be de ned as X M f exp u39 f39ifj d Normal variables are independent if the variance matrix is a diagonal matrix e Note that if the variance matrix is diagonal the moment generating function for the normal can be written as Mg t 1 2 exp u3t3 563 MXl tMXZ tM t X
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