# Empirical Financial Economics - New York University

Empirical Financial Economics 2. The Efficient Markets Hypothesis - Generalized Method of Moments Stephen Brown NYU Stern School of Business UNSW PhD Seminar, June 19-21 Random Walk Hypothesis Random Walk hypothesis a special case of EMH E (rt ) 1, 2, 2 2 2 2 E (rt ) Overidentification of model Provides a test of model (variance ratio criterion) Allows for estimation of parameters (GMM paradigm) Variance ratio tests 1 Var(rt ) k VR( ) 1 2 (1 ) ( k ) Var(rt ) k 1 using sample quantities n 1

2 (r1 ) 2 n 1 n 1 2 ( ) (r ) 2 n 1 2 ( ) VR ( ) The variance ratio 2 asymptotically Normal a ( ) 1) n (VR N[0, 2( 1)] is Overlapping observations Non-overlapping observations Overlapping observations 1 n 2 ( r ) 1

n 1 1 t t+T ln(p t) ln(p t) unbiassed n 1 1 2 ( ) (r ) 2 ; m (n 1) 1 estimators m n (2 1)( 1) a n ( VR ( ) 1) N[0,

2 ] Variance ratio is 3 asymptotically Normal 2 Random walk model and GMM rt 1 1t rt 2 j j 2 j 2 2 2t rt 2k k 2 k 2 2 3t aggregate into moment conditions: 1 rt 1 T 1 rt 2 j T 1 rt 2k T 1 1t T

1 j 2 j 2 2 2t T 1 k 2 k 2 2 3t T and express as three observations of a nonlinear regression model: y X X 2 X 2 w 1 11 21 31 1 y2 X 12 X 22 2 X 32 2 w2 y3 X 13 X 23 2 X 33 2 w3 Generalized method of moment estimators wAw A . Choose to minimize is , referred to as the optimal weighting matrix, equal to the inverse w 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 A 1. Set A I (Ordinary Least Squares). Estimate model. Set 1 (Generalized Least A Squares). Reestimate . 2. Use analytic methods to infer GMM and the Efficient Market Hypothesis E rj ,t E[rj ,t | xt , j ] zt 0 1 asset and 1 instrument: 1 equation and k unknow E rt E[rt | xt , ] zt 0 m assets and 1 instrument: m equations and >k unknowns: E rt E[rt | xt , ] Z t 0 m assets and n instrument: mxn equations and >k unknowns: t

ft ( ) E rt E[rt | xt , ] Z t ; gT ( ) T 1 f t ( ) t 1 Autocovariances and cross autocovariances xt xx (k ) xy ( k ) yx ( k ) xy ( k ) yt yy (k ) t-k t t+ k Cross autocovariances are not symmetrical! Autocovariances are given by: xx (k ) E[( xt x )( xt k x )] E[( xt x )( xt k x )] xx ( k ) yy (k ) E[( yt y )( yt k y )] E[( yt y )( yt k y )] yy ( k ) Cross autocovariances are given

by: xy (k ) E[( xt x )( yt k y )] E[( yt y )( xt k x )] yx ( k ) Cross autocovariances and the weighting function 1 T T xt t 1 For w 1 T T xt t 1 T T xt x 1 t 1 1 Cov( w) 2 T T T yt x t 1 1 T T xt y t 1 1 T T yt y t 1 1

Assuming stationarity xx 0 xx 1 xx 2 xx 3 xx 4 xx 1 xx 0 xx 1 xx 2 xx 3 xx 2 xx 1 xx 0 xx 1 xx 2 Cov( x) xx 3 xx 2 xx 1 xx 0 xx 1 4 3 2 1 0 xx xx xx xx xx and so T1 1 T T

T k xt x xx 0 2 xx k xx 0 2 xx k T t 1 1 T k 1 k 1 Apply this to cross covariances T1 T1 1 T T T k T k xt y xy 0 xy k xy k T t 1 1 T T k 1 k 1 T1 T1 T k T k xy 0 xy k yx k T T k 1 k 1 and

T1 T1 1 T T T k T k yt x yx 0 yx k yx k T t 1 1 T T k 1 k 1 T1 T1 T k T k yx 0 yx k xy k T T k 1 k 1 A simple expression for the inverse weighting matrix T T x x 1 t 1 1 t Cov( w) 2 T T T yt x t 1 1 T

T xt y t 1 1 A A T T 2 yt y t 1 1 where T1 T 0 2 xx T k 1 A T1 T

0 2 yx T k 1 k k T1 xx k yx k T k xy 0 2 xy k xx 0 2 xx k xy 0 2 xy k T k 1 k 1 k 1 T1

T k yy 0 2 yy k yx 0 2 yx k yy 0 2 yy k T k 1 k 1 k 1 Some applications of GMM Fixed income securities CIR Construct model : moments ln pt ofAreturns ( , ) based B ( ,on )it 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

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