1. 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 2006

2. Random Walk Hypothesis  Random Walk hypothesis a special case of EMH  Overidentification of model  Provides a test of model (variance ratio criterion)  Allows for estimation of parameters (GMM paradigm)

3. Variance ratio tests using sample quantities The variance ratio is asymptotically Normal

4. Overlapping observations ln(p t ) t Non-overlapping observations Overlapping observations t+T unbiassed estimators Variance ratio is asymptotically Normal

5. Random walk model and GMM aggregate into moment conditions: and express as three observations of a nonlinear regression model:

6. Generalized method of moment estimators 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

7. GMM and the Efficient Market Hypothesis 1 asset and 1 instrument: … 1 equation and k unknowns: m assets and 1 instrument: … m equations and >k unknowns: m assets and n instrument: … mxn equations and >k unknowns:

8. Autocovariances and cross autocovariances t xtxt ytyt t+k t-k

9. Cross autocovariances are not symmetrical! Autocovariances are given by: Cross autocovariances are given by:

10. Cross autocovariances and the weighting function

11. Assuming stationarity

12. Apply this to cross covariances

13. A simple expression for the inverse weighting matrix

14. Some applications of GMM  Fixed income securities  Construct moments of returns based on distribution of i t+  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