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
Topic: Investigation. Level of competency of the IO and transparency of the role of investigation officer must be increased. Investigation police should be brought under the control of operation police in a way that there should be two additional SHOs.
Naturalistic observation Participant observation Survey & interviews Archival data Systematic (contrived) observation Observational Methods Naturalistic Observation: Observation and description of behaviors within a natural setting Jane Goodall Dian Fossey Good for behaviors that don't occur (as well) in more controlled...
This marks the two ends of the process. Be sure this aligns with your . defined scope! M. Current State Map. Current State Map. Start/End. Routine. Time. Current State Map Symbols . This marks any step in the process. Record...
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Notes for each lecture should begin on a new page. Date your lecture notes and number all pages. Never use a sentence when you can use a phrase, or a phrase when you can use a word. Notetaking Tips, Cont....