Empirical Financial Economics and Efficient Markets Hypothesis

Empirical Financial Economics The Efficient Markets Hypothesis - Generalized Method of Moments

Random Walk Hypothesis Random Walk hypothesis a special case of EMH = = ( ) E r + t = 1,2, + 2 + 2 2 2 ( ) E r t Overidentification of model Provides a test of model (variance ratio criterion) Allows for estimation of parameters (GMM paradigm)

Variance ratio tests using sample quantities 1 Var( Var( ) ) r k ( ) = 1 2 = + ) ( ) + (1 VR k t r = 1 k t n 1 n = 2 2 ( ) r + 1 = 1 The variance ratio is asymptotically Normal n 1 n = 2 2 ( ) ( ) r + = 1 2 ( ) N[0,2( ( ) VR = 2 n VR ( ) 1) a ( 1)]

Overlapping observations Non-overlapping observations ln(pt) t t+T Overlapping observations ln(pt) n 1 = 2 2 ( ) r + 1 1 n unbiassed estimators = 1 n 1 m 1 n = = + 2 2 ( ) ( ) ; ( 1) 1 r m n + = (2 1)( 3 1) n VR ( ) 1) a ( N[0,2 ] Variance ratio is asymptotically Normal Lo, A. and A.C. MacKinley, 1988, Stock market prices do not follow random walks:Evidence from a simple specification test Review of Financial Studies 1(1), 41-66.

Random walk model and GMM = = + + r + 1 1 t t + 2 + 2 2 2 t j r r j j 2 t = + + 2 + 2 2 2 k k 3 t k t aggregate into moment conditions: 1 T 1 T = + r + 1 1 t t 1 T 1 T = + + 2 + 2 2 2 t j r j j 2 t 1 T 1 T = + + 2 + 2 2 2 t k r k k 3 t and express as three observations of a nonlinear regression model: = = = + + + + + + + + + 2 2 y X X X w 1 11 21 31 1 2 2 y X X X w 2 12 22 32 2 2 2 y X X X w 3 13 23 33 3

An Aside on Linear Least Squares = + = u y X u y X Min u Au = u X Au = : 0 0 FOC Au X A y = : 0 Choose X 1 1 X AX X Ay X AX X Au = = + ( ) 1 1 X AX X A Euu AX X AX = Var

An Aside on Nonlinear Least Squares = + = u ( , ) f X ( , ) f X y u y ( ) y y D 0 0 Min u Au = u D Au = : 0 0 FOC Au + D A = : ( ) 0 Choose y y D 0 0 = 1 1 D AD D A y D AD D Au = + = ( ) y 0 0 ( ) 1 1 D AD D A Euu AD D AD Var

Generalized method of moment estimators w Aw , A Choose to minimize . is referred to as the optimal weighting matrix, equal to the inverse covariance matrix of Estimators are asymptotically Normal and efficient Minimand is distributed as Chi-square with d.f. number of overidentifying information Methods of obtaining 1. Set (Ordinary Least Squares). Estimate model. Set (Generalized Least Squares). Reestimate . 2. Use analytic methods to infer w A = A I = 1 A

GMM and the Efficient Market Hypothesis ( ) = [ | , ] 0 E r E r x z 1 asset and 1 instrument: + + , , j t j t t j t 1 equation and k unknowns: ( ) , ] = [ | 0 E r E r x z m assets and 1 instrument: + + t t t t m equations and >k unknowns: ( ) , ] = [ | 0 E r E r x Z m assets and n instrument: + + t t t t mxn equations and >k unknowns: t ( ) ( ) = , ] ( ) = ( ) 1 [ | ; f E r E r x Z g T f + + t t t t t T t = 1 t Hansen, L.P. and K.J. Singleton, 1982, Generalized instrumental variables estimation of nonlinear rational expectations models Econometrica 50(5), 269-286.

Autocovariances and cross autocovariances xt ( ) k xx = ( ) ( ) k ( ) k k xy yx xy yt ( ) k yy t t+k t-k

Cross autocovariances are not symmetrical! Autocovariances are given by: = = = ( ) k [( )( )] [( )( )] ( ) E x x E x x k t k + t k xx t x x t x x xx = = = ( ) k [( )( )] [( )( )] ( ) E y y E y y k t k + t k yy t y y t y y yy Cross autocovariances are given by: = = = ( ) k [( )( )] [( )( )] ( ) E x y E y x k t k + t k xy t x y t y x yx

Cross autocovariances and the weighting function T 1 T x t = 1 t = For w 1 T T y t = 1 t T T T T x x x y 1 t t = = = = 1 1 1 1 t t = Cov( ) w 2 T T T T T y x y y t t = = = = 1 1 1 1 t t

Assuming stationarity ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) 0 0 1 2 1 0 1 2 1 0 3 2 1 4 3 2 xx xx xx xx xx xx xx xx xx xx xx xx xx xx xx = Cov( ) x 3 2 1 0 1 xx xx xx xx xx xx xx xx xx xx and so 1 T T T 1 T T k ( ) 0 ( ) k ( ) 0 ( ) k = + + 2 2 x x xx xx xx xx T t = = = = 1 1 1 1 t k k

Apply this to cross covariances 1 1 T T T T 1 T T k T k ( ) 0 ( ) k ( ) = + + k x y xy xy xy T T t = = = = 1 1 1 1 t k k 1 1 T T T k T k ( ) 0 ( ) k ( ) k = + + xy xy yx T T = = 1 1 k k and 1 1 T T T T 1 T T k T k ( ) 0 ( ) k ( ) = + + k y x yx yx yx T T t = = = = 1 1 1 1 t k k 1 1 T T T k T k ( ) 0 ( ) k ( ) k = + + yx yx xy T T = = 1 1 k k

A simple expression for the inverse weighting matrix T T T T x x x y + 1 A A t t = = = = 1 1 1 1 t t = = Cov( ) w 2 2 T T T T T y x y y t t = = = = 1 1 1 1 t t where 1 1 T T T k T k ( ) 0 ( ) k ( ) 0 ( ) k ( ) 0 ( ) k ( ) 0 ( ) k + + + + 2 2 2 2 xx xx xy xy xx x x xy xy T T = = = = 1 1 1 1 k k k k = A 1 1 T T T k T k ( ) 0 ( ) k ( ) 0 ( ) k ( ) 0 ( ) k ( ) 0 ( ) k + + + + 2 2 2 2 yx yx yy yy yx yx yy yy T T = = = = 1 1 1 1 k k k k

Some applications of GMM Fixed income securities : CIR model = ( , ) A + ( , ) ln p B i + + t t Construct moments of returns based on distribution of it+ Estimate by comparing to sample moments Derivative securities Construct moments of returns by simulating PDE given Estimate by comparing to sample moments Asset pricing with time-varying risk premia

CEV Example BS dt = + + (1 ) /2 dS AS S dz s Special cases: = 1: dS is a mean reverting process = + = = ( ) ( , / ) dS dS S dS = K S dt S dz K B A B t s t = 2: / is a standard Gaussian diffusion t + = + ( ) S dt S dz A B t s t S Method 1: Solve for Define moments of , Compare to sample moments ... t = i S i ( , , ) A B S f t

CEV Example BS dt = + + (1 ) /2 dS Special cases: AS S dz s = 1: dS is a mean reverting process = + = = ( ) ( , / ) dS dS S dS = K S dt S dz K B A B t s t = 2: / is a standard Gaussian diffusion t + = + ( ) S dt S dz A B t s t ( , , ) A B dS S Method 2: Simulate given starting value and Estimate moments of , Compare to sample moments ... 0 f = i S i ( , , ) A B S t