GMM - Generalized Method of Moments

Transcription

1 GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion Weighing marix Properies of GMM esimaors esing over-idenifying resricions Use of condiioning informaion OLS as a GMM esimaion. 6 5 Running GMM in R 7 6 Summarizing GMM 9 GMM esimaion, shor inroducion Well know ha Maximum Likelihood is generally an opimal mehod for seing up esimaors. If we know we are in a seing where ML applies, we should be using ha. However, here are many cases where ML can no be applied. Imporan cases We are no able o specify he acual probabiliy disribuion. he calculaion of he full probabiliy disribuion is no feasible. In boh cases we are no able o use neiher OLS nor ML. An alernaive way of doing esimaion is base on an old idea in saisics, ha of mahcing momens I wan o spend some ime on he analysis of he Generalized Mehod of Momens, no only because I like i, bu also because i is becoming more and more commonly used in curren research. Anoher reason ha we wan o spend some ime on GMM is ha almos all he oher mehods of analysing daa we have seen so far can be viewed as special cases of GMM. We can hus use he general resuls abou GMM problems o show resuls abou all hese special cases. his is he value of knowing he mos general cases, i is easier o specialize a general case o a special case, han o keep rack of 0 differen special cases. Anoher reason ha GMM is so popular is ha i is easy o see he mapping from he economic model o he esimaion seup.

2 2 GMM inuiion: Maching momens We will sar wih he inuiion of maching momens. Suppose x (µ 0, σ 2 ) Consider he problem of esimaing µ 0. We know ha Wha if his is all we are willing o assume? Consider By a law of large numbers, we will have hus, consider If we le increase, lim E[x ] = µ 0 x x = E[x ] = µ 0 E[x µ 0 ] = 0 x µ 0 0 as Given his, a reasonable way o esimae µ 0 is o look a he soluion of or ake as our esimaor x µ = 0, µ = x, he sample mean, which in his case is he naural esimaor. he name maching momens is coming from he fac ha e.g. he expecaion is he firs momen, and by consrucing he momen condiion x µ = 0, we are in some sense maching he firs momen, he expecaion E[x µ 0 ] = 0 2

3 3 General overview of GMM esimaion. he main ingredien of a GMM esimaion is a funcion h(θ, w ), wih θ parameers o esimae and w daa. We ofen use he erm orhogonaliy condiion abou his expecaion. By assumpion E[h(θ 0, w )] = 0 under he rue parameers θ = θ 0. Define Y = {w, w 2,..., w } g(θ, Y ) = h(θ, w ) If he number of parameers o esimae equals he number of orhogonaliy condiions, we can find ˆθ direcly as he soluion o g(ˆθ, Y ) = 0 Oherwise, if he number of parameers o esimae is less he number of orhogonaliy condiions, we can find ˆθ as ˆθ = arg min J(θ, Y ) = arg min g(θ, Y ) W g(θ, Y ) θ θ where W is some posiive definie weighing marix. o show consisency of GMM, he main seps will consis of. Show g(θ, Y ) E[h(θ, w)] for all θ 2. Assume is a unique minimum, or alernaively min E[h(θ 0, w) h(θ 0, w)] E[h(θ)] 0 θ θ 0 3. Assume coninuiy of h( ) 4. Given his, we argue ha arg min g(θ, Y) W g(θ, Y ) P θ 0 θ We will no go ino he deails of his argumen, i is similar o earlier proofs for oher cases, bu he laws of large numbers used o show his is somewha advanced. If you are ineresed, look a he references given below. 3

4 3. Weighing marix. Wha is he marix W in g(θ, Y ) W g(θ, Y ) he easies way o see wha i should be is o argue by he analogy o GLS, where we found ha b gls = arg min(y Xb) Ω (y Xb) b he marix Ω is he covariance marix of he error erms, and he inuion was ha he higher he variance of a paricular observaion, he lower weigh should i be given in he esimaion. A similar resul urns ou o be he case for GMM esimaion, he opimal W o use should be an esimae of he inverse of he covariance marix of he momen condiions. W = ( Ŝ ) S = var h(θ, w ) h(θ, w ) he weighing marix W is an imporan par of GMM, i is wha makes he mehod very robus. If we wrie ou ( ) S = var h(θ, w ) h(θ, w ) = i E[h(θ, w +i ) h(θ, w +j )], j a kind of average of he error erms. A paricular simple version is o assume independence of he error erms, which implies ha S = E[h(θ, w ) h(θ, w )], which we would esimae by Ŝ = h(θ, w ) h(θ, w ) Already we see ha o esimae he marix S you need an esimae of θ, bu o esimae θ we need an esimae of S. he way o ge around his circulariy is o proceed in seps. Esimae θ using he ideniy marix I as a weighing marix. 2. Esimae Ŝ using his θ. 3. Re esimae θ using Ŝ as a weighing marix. 3.2 Properies of GMM esimaors. he main general resuls abou GMM esimaors are ha, under he appropriae regulariy condiions, which we will no go ino, P ˆθ θ0 (ˆθ θ0 ) D N (0, V ) where V = [ DS D ] [ ] D = E g(θ, Y) θ S = covariance marix of momen condiions. 4

5 3.3 esing over-idenifying resricions Specificaion of a GMM model will consising in finding a se of momen condiions which have expecaion zero. I is ofen he case ha he model will supply more momen condiions han we have parameers o esimae. Le E[h(x, b)] = 0 be he momen condiions. Suppose we have n momen condiions and r < n parameers o esimae. If he model is correcly specified, a he rue parameers he sample equivalen of he momen condiion will go o zero for all he momen condiions. Suppose we now use only he firs r momen condiions o do he esimaion of he parameers. his will choose parameers θ o se he sample mean of h (θ, w ) h 2 (θ, w ). h r (θ, w ) = 0 where h i (θ, w ) signifies he i h elemen of h(θ, w ). If he parameers are correc, and he model correcly specified, he sample mean of he momen condiions ha are no used in he esimaion should also be close o zero. If hey are no, his is a sign ha he model is no correcly specified. he es of over-idenifying resricions measure his disance from zero of he lef over momen condiions, and will rejec he model formulaion if his saisic is large. You will ofen see his es ermed Hansen s J-es. We consruc he es saisic as J(θ) = g (θ, y)s g (θ, y) Since g (θ) is asympoically normal wih limiing covariance marix S, J(θ) is chi-square disribued wih degrees of freedom equal o he number of momen condiions less he number of parameers o esimae. he J-es is a es of he model formulaion, if J(ˆθ) > criical value, hen we may wan o hink again abou he model formulaion. In pracice his es is no very powerful, i can be hard o rejec a mis-specified model using his paricular es. 3.4 Use of condiioning informaion. he abiliy o use condiioning informaion in a meaningful way is one of he major reasons for GMM o be of very wide use. Noe a couple of ways o use condiioning informaion Use of variables in he informaion se as insrumens in he esimaion. ry o model he condiional expecaions direcly (laen variables) 5

6 4 OLS as a GMM esimaion. As anoher example of GMM seup, consider an OLS srucure y = x b + ɛ where y and x are scalars, and he following are saisfied E[ɛ ] = 0 Momen condiion Sample equivalen Solve for GMM esimaor If we define he appropriae vecors we can wrie x = E[x ɛ ] = 0 var(ɛ ) = σ 2 E[x ɛ ] = E[x (y x b)] x (y x b) = 0 ˆbGMM = x x 2. x n [ ] [ x x, y = y y 2. y n ˆbGMM = (x x) x y x y ] ɛ, ɛ = ɛ 2. ɛ n and var(ɛ) = σ 2 I Now, wha is he sandard error of he GMM esimae ˆb GMM. using he resul (ˆθ θ0 ) D N(0, V ) Mapping he OLS example ino his noaion V = ( DS D ) D = g(θ, Y) θ S = cov(momen condiions) θ = b S = σ 2 (X X) D = b x (y xb) = x x DS D = (x x)(σ 2 (x x)) (x x) = σ 2 (x x) which should be familiar. (DS D) = σ 2 (x x) 6

7 5 Running GMM in R here is a very good implemenaion of GMM esimaion in R, which covers many of he relevan applicaions for finance. he user has o do is o wrie code for calculaion of he momen condiions, and hen R akes care of he res. he momen condiions can be specified by wriing a funcion reurning he marix of momen condiions, or, in he case of a linear model, by simply wriing he linear model in he same way as a OLS regression. For documenaion of he gmm package, see?. We will use some examples o illusrae he use of he package. Exercise. Consider he model y = a + bx + ε Simulae he following model leing x be he numbers from o 0, a =, b = and ε is normally disrubued wih mean zero and variance one.. Esimae he model using OLS. 2. Esimae he model using GMM. Soluion o Exercise. he following is he R code which does his, and hen he oupu of he wo esimaions. Firs simulaing he model x <- :0 b <- a <- e <- rnorm(0) y <- a + b * x + e Running he OLS regression reg <- lm(y x) summary(reg) wih oupu lm(formula = y x) Residuals: Min Q Median 3Q Max Coefficiens: Esimae Sd. Error value Pr(> ) (Inercep) x e-05 *** --- Signif. codes: 0 âăÿ***âăź 0.00 âăÿ**âăź 0.0 âăÿ*âăź 0.05 âăÿ.âăź 0. âăÿ âăź Residual sandard error:.2 on 8 degrees of freedom Muliple R-squared: ,Adjused R-squared: F-saisic: on and 8 DF, p-value: 4.689e-05 > reg$coefficiens (Inercep) x and hen doing he same using GMM. Noe he need o load he gmm library. 7

8 library(gmm) res <- gmm(y x,x) summary(res) resuls in he oupu gmm(g = y x, x = x) Mehod: wosep Kernel: Quadraic Specral Coefficiens: Esimae Sd. Error value Pr(> ) (Inercep).3087e e e e-0 x e e e e-5 J-es: degrees of freedom is 0 J-es P-value es E(g)=0: e-29 ******* > res$coefficiens (Inercep) x

9 6 Summarizing GMM Inuiion: Maching Momens (firs momen) Also, poenially E[] = 0 E[() 2 σ 2 ] = 0 (second momen) Basis for consrucing esimaors. Allow for esimaion in many seings where esimaion oherwise impossible. Generaliy coss: Less precision when e.g. ML applies. Imporan cases: Condiional expecaions can be he basis for modelling. Robus o more general error srucure (heeroskedasiciy and auocorrelaion robus) References Pierre Chaussé. Compuing generalized mehod of momens and generalized empirical likelihood wih r. Journal of Saisical Sofware, 34(): 35,