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Empirical Financial Economics

Empirical Financial Economics. The Efficient Markets Hypothesis - Generalized Method of Moments. Random Walk Hypothesis. Random Walk hypothesis a special case of EMH Overidentification of model Provides a test of model (variance ratio criterion)

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Empirical Financial Economics

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

  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 Non-overlapping observations ln(pt) t t+T Overlapping observations ln(pt) unbiassed estimators Variance ratio is asymptotically Normal Lo, A. and A.C. MacKinley, 1988, Stock market prices do not follow random walks:Evidencefrom a simple specification test Review of Financial Studies 1(1), 41-66.

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

  6. An Aside on Linear Least Squares

  7. An Aside on Nonlinear Least Squares

  8. 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

  9. 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: Hansen, L.P. and K.J. Singleton, 1982, Generalized instrumental variables estimation of nonlinear rational expectations models Econometrica 50(5), 269-286.

  10. Autocovariances and cross autocovariances xt yt t t+k t-k

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

  12. Cross autocovariances and the weighting function

  13. Assuming stationarity

  14. Apply this to cross covariances

  15. A simple expression for the inverse weighting matrix

  16. Some applications of GMM • Fixed income securities • 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

  17. CEV Example Special cases: is a mean reverting process is a standard Gaussian diffusion Method 1: Solve for Define moments of , Compare to sample moments ...

  18. CEV Example Special cases: is a mean reverting process is a standard Gaussian diffusion Method 2: Simulate given starting value and Estimate moments of , Compare to sample moments ...

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