Estimation and Inference by the Method of Projection Minimum Distance

1. Estimation and Inference by the Method of Projection Minimum Distance Òscar Jordà Sharon Kozicki U.C. Davis Bank of Canada

2. The Paper in a Nutshell: An Efficient Limited Information Method • Step 1: estimate the Wold representation of the data semiparametrically (local projections, Jordà, 2005) • Step 2: Replace the variables in the model by their Wold representation • Minimize the distance function relating the model’s parameters and the semiparametric estimates of the Wold coefficients Projection Minimum Distance

3. Preview of Results • Local projections are consistent and asymptotically normal (and only require least-squares) • Minimum chi-square step produces consistent and asymptotically normal estimates of the parameters (often only requires least-squares) • A 2 test of the distance in the second step is a model misspecification test. • PMD is asymptotically MLE/fully efficient • GMM is a special case of PMD but PMD addresses some invalid/weak instrument problems + efficient Projection Minimum Distance

4. Motivating Example: Galí and Gertler (1999) • xrt could be a predictor of , hence a valid instrument/omitted variable • Let Projection Minimum Distance

5. Implications • Substituting the Wold representation into the model Projection Minimum Distance

6. Remarks • xrtis a natural predictor of inflation and fulfills two roles: • As an instrument: the impulse responses of the included variables with respect to xr are used to estimate the parameters • As a possibly omitted variable: even if we do not use the previous impulse responses, the responses of the included variables are calculated, orthogonal to xr Projection Minimum Distance

7. 1st Step: Local Projections • Suppose: • with i.i.d. and • assume the Wold rep is invertible such that Projection Minimum Distance

8. Local Projections • then, iterating the VAR() • with Projection Minimum Distance

9. Local Projections in finite samples • Consider estimating a truncated version given by Projection Minimum Distance

10. Local Projections – Least Squares Projection Minimum Distance

11. Local Projections (cont.) Projection Minimum Distance

12. 2nd Step – Minimum Distance • Notice that: is a compact way of expressing Wold conditions with • Objective: Projection Minimum Distance

13. Minimum Chi-Square • Objective function: • Relative to classical minimum distance, the key is that first stage estimates appear both in the left and right hand sides, e.g. Projection Minimum Distance

14. Min. Chi-Square – Least Squares Projection Minimum Distance

15. Key assumptions for Asymptotics • is stochastically equicontinuous since b is infinite-dimensional when h as T  • Instrument relevance: • Identification: Projection Minimum Distance

16. Asymptotic Normality - Remarks • Consistency and asymptotic normality is based on omitted lags vanishing asymptotically with the sample • becomes infinite-dimensional with the sample: • need stochastic equicontinuity condition as moment conditions go to infinity with the sample • need condition that ensures enough explanatory power in the first stage estimates as the sample grows. In practice, use Hall et al. (2007) information criterion • W is a function of nuisance parameters. Use equal weights estimator first to obtain consistent estimates and then plug into W and iterate. Projection Minimum Distance

17. Misspecification Test • Correct specification means the minimum distance function is zero. • Hence we can test overidentifying conditions • Since • then Projection Minimum Distance

18. GMM vs. PMD: An Example • Estimated Model: • True Model: • Instrument validity condition: Projection Minimum Distance

19. However… • Let: • Notice that: • Hence: • Lesson: Orthogonalize instruments w.r.t. possibly omitted variables Projection Minimum Distance

20. GMM Projection Minimum Distance

21. PMD Projection Minimum Distance

22. Monte Carlo Experiments1. PMD vs MLE: ARMA(1,1) PMD vs MLE • DGP: • Parameter pairs (1, 1): (0.25, 0.5); (0.5; 0.25); (0, 0.5); (0.5; 0) • T = 50, 100, 400 • Lag length determined automatically by AICc • h = 2, 5, 10 Projection Minimum Distance

23. 1 = 0.5; 1 = 0.25 Projection Minimum Distance

24. p1 = 0.5; 1 = 0.25 0.25 Projection Minimum Distance

25. Monte Carlo Comparison: PMD vs GMM • Euler equation: Projection Minimum Distance

26. When Model is Correctly Specified PMD GMM Projection Minimum Distance

27. Omitted Endogenous Dynamics Projection Minimum Distance

28. Omitted Exogenous Dynamics Projection Minimum Distance

29. PMD in Practice: PMD, MLE, GMM • Fuhrer and Olivei (2005) • Output Euler:z is the output gap and x is real interest rates • Inflation Euler:z is inflation, x is the output gap Projection Minimum Distance

30. Fuhrer and Olivei (2005) • Sample: 1966:Q1 – 2001:Q4 • Output gap: log deviation of GDP from (1) HP trend; (2) Segmented linear trend (ST) • Inflation: log change in GDP chain-weighted index • Real interest rate: fed funds rate – next quarter’s inflation • Real Unit Labor Costs (RULC) Projection Minimum Distance

31. Results – Output Euler Equation Projection Minimum Distance

32. Inflation Euler Equations Projection Minimum Distance

33. Summary • Models that require MLE + numerical techniques can be estimated by LS with PMD and nearly as efficiently (e.g. VARMA models) • PMD is asymptotically MLE • PMD accounts for serial correlation parametrically – hence it is more efficient than GMM • PMD does appropriate, unsupervised conditioning of instruments, solving some cases of instrument invalidity • PMD provides natural statistics to evaluate model fit: J-test + plots of parameter variation as a function of h Projection Minimum Distance