14.30 Introduction to Statistical Methods in Economics Spring 2009

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1 MIT OpeCourseWare Itroductio to Statistical Methods i Ecoomics Sprig 009 For iformatio about citig these materials or our Terms of Use, visit:

2 430 Itroductio to Statistical Methods i Ecoomics Lecture Notes 8 Korad Mezel April 3, 009 Properties of Estimators (cotiued) Stadard Error Ofte we also wat to make statemets about the precisio of the estimator - we ca always state the value of the estimate, but how cofidet are we that it is actually close to the true parameter? Defiitio The stadard error σ(θˆ) of a estimate is the stadard deviatio (or estimated stadard deviatio) of the estimator, SE(θˆ) = Var(θˆ(X,,X )) Should recall that a estimator is a fuctio of the radom variables, ad therefore a radom variable for which we ca calculate expectatio, variace ad other momets Example The mea X of a iid sample X,,X where Var(X i ) = σ σ has variace Therefore, the stadard error is σ SE(X ) = If we do t kow σ, we calculate the estimated stadard error SÊ(X ) = σˆ The stadard error is a way of comparig the precisio of estimators, ad we d obviously favor the estimator which has the smaller variace/stadard error Defiitio If θˆa ad θˆb are ubiased estimators for θ, ie E θ0 [θˆa] = E θ0 [θˆb] = θ 0, the we say that θˆa is efficiet relative to θˆb if Var(θˆB) Var(θˆA) Sometimes we look at a etire class of estimators Θ = {θˆ, θˆ, }, ad we say that θˆa is efficiet i that class if it has the lowest variace of all members of Θ Example Suppose that X ad Y are scores from two differet Math tests You are iterested i some uderlyig math ability, ad the two scores are oisy (ad possibly correlated) measuremets with E[X] = E[Y ] = µ, Var(X) = σx, Var(Y ) = σ Y, ad Cov(X, Y ) = σ XY Istead of usig oly oe of the measuremets, you decide to combie them ito a weighted average px + ( p)y istead What is the

3 expectatio of this weighted average? Which value of p miimizes the variace of the weighted average? We ca iterpret this as a estimatio problem i which we wat to estimate µ usig a sample of oly two observatios Sice all weighted averages of X ad Y have mea µ, we ll try to fid the efficiet estimator From the formula of the variace of a sum of radom variables, σ σ Var(pX + ( p)y ) = p X + p( p)σ XY + ( p) Y I order to fid the optimal p, we set the first derivative equal to zero, ie 0 = pσx + ( p)σ XY ( p)σy Solvig for p, we get, assumig that σx + σy > σ XY (otice that this is also the sufficiet coditio for a local miimum) σ Y σ XY Var(Y ) Cov(X, Y ) Cov(Y X, Y ) p = = = σx σ XY + σy Var(Y X) Var(Y X) Note that if X ad Y are ucorrelated, the efficiet estimator puts weight p greater the lower the variace of X is relative to that of Y σ = Y σ X +σ Y o X which is Methods for Costructig Estimators Method of Momets This method was proposed by the British statisticia Karl Pearso i 894: suppose we have to estimate k parameters of a distributio the we ca look at the first k sample momets of the data, X = X i X = X k = ad equate them to the correspodig populatio momets for a give parameter value, calculated uder the distributio, µ (θ) = E θ [X i ] xf X (x; θ)dx µ k (θ) = E θ [X k i ] X i X k i x k f X (x θ)dx ()

4 The the method of momets (MoM) estimator θˆ ca be obtaied by solvig the equatios µ j (θˆ) = X j j =,,k for θ Example 3 Suppose X,, X is a iid sample from a Poisso distributio with ukow parameter λ, ie X P(λ) The distributio has oly oe ukow parameter, ad the first populatio momet is give by µ (λ) = E λ [X] = λ Therefore, the MoM estimator is give by λˆ = µ (λˆ) X = X i What if we used more momets tha ecessary to estimate the parameter? - We also kow that for the Poisso distributio E λ [X ] = Var λ (X) + E λ [X] = λ + λ Example 4 A double expoetial radom variable has pdf f Y (y) = λe λ y µ so we have to estimate two parameters (λ, µ) We ca look up i a statistics book that so the method of momets estimator solves E[Y ] = µ E[Y ] = Var(Y ) + E[Y ] = + µ λ Ȳ = µˆ Y = + ˆµ λˆ so that, solvig for (ˆ λ, µ), ˆ µˆ = Y, λ ˆ = (Y (Ȳ ) ) / Maximum Likelihood Estimatio While the method of momets oly tries to match a selected umber of momets of the populatio to their sample couterparts, we might alteratively costruct a estimator which makes the populatio distributio as a whole match the sample distributio as closely as possible This is what the maximum likelihood estimator of a parameter θ does, which is loosely speakig, the value which most likely would have geerated the observed sample: Suppose we have a iid sample Y,,Y where the pdf of Y is give by f Y (y θ), which is kow up to the parameter θ The Maximum Likelihood estimator (MLE) is a fuctio θˆ of the data maximizig the joit pdf of the data uder θ More specifically, we defie the likelihood of the sample as L(θ) = f(y,,y θ) = f(y i θ) 3

5 Usually it is much easier to maximize the logarithm of the likelihood fuctio, L(θ) = log(l(θ)) = log f(y i θ) Note that sice the logarithm is a strictly icreasig fuctio, L(θ) ad L(θ) will be maximized at the same value Propositio The expectatio of the log-likelihood uder the parameter θ 0, is maximized at the true parameter θ 0 Proof: Sice the true desity over which we take the expectatio is f(y θ 0 ), we ca show that E θ0 [L(Y θ) L(Y θ 0 )] 0 for all values of θ usig Jese s Iequality ad the fact that log( ) is cocave [ ( f(y θ) )] E θ0 [L(Y θ) L(Y θ 0 )] = E θ0 [log f(y θ) log f(y θ 0 )] = E θ0 log f(y θ 0 ) ( [ ) f(y θ) ]) ( f(y θ) log E θ0 = log f(y θ 0 )dy f(y θ 0 ) f(y θ 0 ) ( ) = log f(y θ)dy = log() = 0 p log f(y i θ) E[log f(y θ)] It turs out that it s much easier to maximize the log-likelihood, { } (X i µ) log L(θ) = log e σ πσ { } ( Xi µ ) = log σ πσ E θ0 [L(θ)] = E[log f(y θ)] sice f(y θ) is a desity ad therefore itegrates to Therefore E θ0 [L(Y θ 0 )] E θ0 [L(Y θ)] for all values of θ, so that θ 0 maximizes the fuctio Sice by the Law of Large Numbers, the log likelihood for ad iid sample we d thik that maximizig the log likelihood for a large iid sample should therefore give us a parameter close to θ 0 Example 5 Suppose X N(µ 0, σ 0 ), ad we wat to estimate the parameters µ ad σ from a iid sample X,, X The likelihood fuctio is L(θ) = e σ πσ (X i µ) = log(πσ ) σ (X i µ) 4

6 I order to fid the maximum, we take the derivatives with respect to µ ad σ, ad set them equal to zero: 0 = (Xi µˆ) µˆ = Xi σ Similarly, π µ) 0 = + ( ) X ) (Xi µˆ) σ = (X i ˆ = (X i πσ σ Recall that we already showed that this estimator is ot ubiased for σ 0, so i geeral, Maximum Likelihood Estimators eed ot be ubiased Example 6 Goig back to the example with the uiform distributio, suppose X U[0, θ], ad we are iterested i estimatig θ For the method of momets estimator, you ca see that so equatig this with the sample mea, we obtai µ (θ) = E θ [X] = θ θˆmom = X What is the maximum likelihood estimator? Clearly, we would t pick ay θˆ max{x,,x } because a sample with realizatios greater tha θˆ has zero probability uder θˆ Formally, the likelihood is { ( ) L(θ) = θ if 0 X i θ for all i =,, 0 otherwise We ca see that ay value of θ max{x,,x } ca t be a maximum because L(θ) is zero for all those poits Also, for θ max{x,,x } the likelihood fuctio is strictly decreasig i θ, ad therefore, it is maximized at θˆmle = max{x,,x } Note that sice X i < θ 0 with probability, the Maximum Likelihood estimator is also goig to be less tha θ 0 with probability oe, so it s ot ubiased More specifically, the pdf of X () is give by so that { fx() (y) = [F X (y)] f X (y) = θ 0 θ 0 θ 0 ) if 0 y θ 0 0 otherwise ( y θ0 ( ) y E[X () ] = yf X() (y)dy = dy = θ 0 0 θ 0 + We could easily costruct a ubiased estimator θ = + X () 3 Properties of the MLE The followig is just a summary of mai theoretical results o MLE (wo t do proofs for ow) If there is a efficiet estimator i the class of cosistet estimators, MLE will produce it 5

7 Uder certai regularity coditios, MLE s will have a asymptotically ormal distributio (this comes essetially from a applicatio of the Cetral Limit Theorem) Is Maximum Likelihood always the best thig to do? - ot ecessarily may be biased ofte hard to compute might be sesitive to icorrect assumptios o uderlyig distributio 6

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