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Empirical Financial Economics

This paper provides a comprehensive overview of the Efficient Markets Hypothesis (EMH) and its connection to the Random Walk Hypothesis (RWH). It discusses the Generalized Method of Moments (GMM) framework for testing these hypotheses, focusing on variance ratio tests and overidentifying restrictions. We delve into the applications of GMM, implications of autocovariances, and estimation techniques derived from financial modeling. Through empirical analysis and theoretical exploration, we aim to enhance understanding of market efficiency and its implications for asset pricing.

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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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