Chapter 19 gmm and regression based tests of linear factor model Fan longzhen
Chapter 19 GMM and regressionbased tests of linear factor model Fan Longzhen
General gmm formula Let y, be an h-vector of variables that are observed at date t, let 0 denote an unknown vector of coefficients, h(e, y,) Be an r-vector real function let 0 denote true value of 8, and suppose this true value is characterized by the property that Eh(eo, 3)=0 A sample of size T is Y=(T,yr-1m,y1) Denote sample average of he,)isg(￥Y)=∑Mn) The gmm estimator 0, is the value of 0 that minimizes the scalar Q(;Yn)=[g(62,Y7)W[8(G,Y) Where W is a sequence of positive definite matrices which may be a function of sample
General GMM formula • Let be an h-vector of variables that are observed at date t, let θ denote an unknown vector of coefficients, • Be an r-vector real function. Let denote true value of θ, and suppose this true value is characterized by the property that • A sample of size T is • Denote sample average of is • The GMM estimator is the value of that minimizes the scalar t y ( , )t h θ y θ 0 { ( , ) } 0 E h θ 0 yt = ( ', ',..., ') 1 1 y y y ΥT = T T − ( , )t h θ y ( ; ) [ ( ; )]' [ ( ; )] T T T T Q θ Υ = g θ Υ W g θ Υ ( , ) 1 ( , ) 1 ∑= Υ = T t T t h y T g θ θ θ T ˆ θ Where is a sequence of positive definite matrices which may be a function of sample WT
example sample y, is from a standard t-distributions withv degrees of freedom, so that its density is f(;)=40、1+(2112 Ifv> 2. its mean is zero. and its variance is 42=E(y2)=v/v-2) For large sample T, the sample moment should be close to the population moment o we have V7 T This is classical moment estimator
example • Sample is from a standard t-distributions with v degrees of freedom, so that its density is • If ,its mean is zero, and its variance is • For large sample T, the sample moment should be close to the population moment • So we have • This is classical moment estimator. t y 2 ( 1)/ 2 1/ 2 [1 ( / )] ( ) ( / 2 ) [( 1 ) / 2] ( ; ) − + + Γ Γ + = v Y t t y v v v v f y v t π v > 2 ( ) /( 2 ) 2 µ2 = E yt = v v − 2 1 2 2, 1 µˆ = ∑ ⎯⎯→ µ = p T t T t y T ˆ 1 2 ˆ ˆ 2, 2, − = T T T v µ µ
Example: Generalized method of moments If v>4, the population fourth moment of the t-distribution is 4=E(y) (y-2)1 If we want to choose v to match both moment, we have following minimization problem where Q(,yrs=gwg } 4T (v-2)v-4) W is 2x2 positive define symmetric matrix reflecting the importance given to matching each of the moments
Example:Generalized Method of Moments • If , the population fourth moment of the t-distribution is • If we want to choose v to match both moment, we have following minimization problem • where • W is positive define symmetric matrix reflecting the importance given to matching each of the moments ν > 4 ( 2)( 4 ) 3 ( ) 2 4 4 − − = = v v v E y µ t { } Q v y T y g Wg v ( , ,..., ) ' min 1 = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − = ( 2)( 4 ) 3 ˆ 2 ˆ 2 4, 2, v v v v v g T T µ µ 2 × 2
Number of parameter and Number of equation If the number of parameter a is the same as the number of the equations r, we simply estimate A by solving g(;2Yr)=0 Usually r>a, so we have to set estimates to minimize the Q,Y)=[g(6;,Yr)Wr[8(6,Yr)
Number of parameter and Number of equation • If the number of parameter a is the same as the number of the equations r, we simply estimate θ by solving • Usually r>a, so we have to set estimates to minimize the ; ) 0 ˆ g (θ T ΥT = ( ; ) [ ( ; )]' [ ( ; )] Q θ ΥT = g θ ΥT WT g θ ΥT
Optimal weighting matrix The optimal weighting matrix is the inverse of asymptotic variance of 8(0,Y It turn out to be S=limTE[(0 Y )1g(00: Y-1 =2I. t→) V=-00 Where r =E[h(o, y)lh(o,y-1) Ifh(0o, y,)Pe were serially uncorrelated, then the matrix S could be consistently estimated by =7∑Q,y)O2
Optimal weighting matrix • The optimal weighting matrix is the inverse of asymptotic variance of • It turn out to be • Where • If were serially uncorrelated, then the matrix S could be consistently estimated by ( , ) g θ 0 ΥT { } ∑ ∞ → ∞ =−∞ = Υ Υ = Γ v T T v t S TE [g ( ; )][ g ( ; )]' lim θ 0 θ 0 Γv = E {[ h (θ 0 , yt)][ h (θ 0 , yt− v )]' } { } ∞ t t=−∞ h ( , y ) θ 0 (){ } 1 / [ ( , )][ ( , )]' 0 0 1 * t t T t T S T ∑ h θ y h θ y = =
Optimal weighting matrix continued New-West (1987)estimate of S could be correlated, the If the vector process h(o, y)- is serially S=+∑/(q+1)kr Where =170mr Why? varu]=q9E(u)+(g-1)E(u1)+E(u1)+.+E(x-1)+E(a11 g∑B(n)
Optimal weighting matrix--- continued • If the vector process is serially correlated, the New-West (1987) estimate of S could be • Where • Why? • { } ∞ t t=−∞ h ( , y ) θ 0 { } ) ˆ ˆ 1 [ /( 1)] ( ˆ ˆ ' , , 1 0, v T v T q v T T S = Γ + ∑ − v q + Γ + Γ = ˆ 1 / {[ ( ˆ, )][ ( ˆ, )]' } 1 , t t v T t v v T T h y h y − = + Γ = ∑ θ θ ( ') var[ ] ( ) ( 1)[ ( ) ( )] ... [ ( ') ( ')] ' 1 ' 1 ' 1 t t k q v q t t t t t t t t q t q t q v v E u u q q v q u qE u u q E u u E u u E u u E u u − = − − − − − = ∑ ∑ − = = + − + + + +
Asymptotic distribution of the gmm estimates Ae the value that minimizes [g(0: Y)'S [8(8; Y)I With S- regarded fixed with respect 0 and S-PS The gmm estimates e is typically a solution to the following system of nonlinear equations ∫eg(G,Y) 6=0 xS7×g(n,Y)=0 In many situations( stationary of y, continuity of h(, and restriction on higher moments )it should be the case 7g(;Y)-→N0.S)
Asymptotic distribution of the GMM estimates • Let be the value that minimizes • With regarded fixed with respect θ and • The GMM estimates is typically a solution to the following system of nonlinear equations : • In many situations( stationary of y, continuity of h(), and restriction on higher moments) it should be the case θ T ˆ [ ( ; )] ˆ [ ( ; )]' 1 T T T g Υ S g Υ − θ θ 1 ˆ − S T S S p ˆ T ⎯⎯→ θ T ˆ [ ; ) ] 0 ˆ ( ˆ ' ' ( ; ) 1 ˆ × × Υ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ Υ − = T T T T S g g T θ θ θ θ θ ( ; ) ( 0, ) 0 T g N S L θ ΥT ⎯⎯→
proposition With suitable conditions (2)7g{(;Yr)-→N0,S) ()For any sequence (e Pa satisfying 0, 0 · It is case that plim 08(Y, IXO ·Then √7(G1-)-2→N0,) where V=DS
proposition • With suitable conditions • (1) • (2) • (3) For any sequence satisfying • It is case that • Then • where 0 ˆ θ ⎯⎯→θ P T ( ; ) (0, ) 0 T g N S L θ ΥT ⎯⎯→ { }∞=1 *T T θ 0 * θ ⎯⎯→θ P T ' 0 * ' ( ; ) lim ' ( ; ) lim r a T T D g p g p T = = = × ⎭⎬⎫ ⎩⎨⎧ ∂ ∂ Υ = ⎭⎬⎫ ⎩⎨⎧ ∂ ∂ Υ θ θ θ θ θ θ θ θ ) (0, ) ˆ ( T 0 N V L θ T −θ ⎯⎯→ { } 1 1 ' − − V = DS D
Delta method We want to estimate a quantity that is a nonlinear function of sample means b=PE(x,)]=p(u) The estimates is b=E(7∑x The sample variance is var(br)Tl du cov(X x For example, a correlation coefficient can be written as a function of sample means as corr(x, y) E(r,)-Ex, Ey Just take yB2-(Ex)2yE(y2)=(E() u=Ex, Ex, Ey, Ey! Ex, yl
Delta method • We want to estimate a quantity that is a nonlinear function of sample means • The estimates is • The sample variance is • For example, a correlation coefficient can be written as a function of sample means as • Just take = φ[ ( )] = φ( µ) t b E x )] 1 [ ( ˆ 1 ∑= = T t t x T b φ E ' cov( , ') ' 1 ) ˆ var( ∑ ∞ =−∞ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = j T t t j d d x x d d T b µ φ µ φ 2 2 2 2 ( ) ( ) ( ( ) ) ( ) ( , ) t t t t t t t t t t Ex Ex E y E y E x y Ex Ey corr x y − − − = [ ] 2 2 t t t t t t µ = Ex Ex Ey Ey Ex y
