复旦大学：《投资学讲义》（英文版） Chapter 19 GMM and regression- based tests of linear factor model

Chapter 19 gmm and regression based tests of linear factor model Fan longzhen

Chapter 19 GMM and regression￾based 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