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

July 2007 Projection Minimum Distance 2 The Paper in a Nutshell: An Efficient Limited Information Method Step 1: estimate the Wold representation of the data semiparametrically (local projections, Jord à, 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

July 2007 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

July 2007 Projection Minimum Distance 4 Motivating Example: Galí and Gertler (1999) xr t could be a predictor of  hence a valid instrument/omitted variable Let

July 2007 Projection Minimum Distance 5 Implications Substituting the Wold representation into the model

July 2007 Projection Minimum Distance 6 Remarks xr t is 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

July 2007 Projection Minimum Distance 7 1 st Step: Local Projections Suppose: with  i.i.d. and assume the Wold rep is invertible such that

July 2007 Projection Minimum Distance 8 Local Projections then, iterating the VAR(  ) with

July 2007 Projection Minimum Distance 9 Local Projections in finite samples Consider estimating a truncated version given by

July 2007 Projection Minimum Distance 10 Local Projections – Least Squares

July 2007 Projection Minimum Distance 11 Local Projections (cont.)

July 2007 Projection Minimum Distance 12 2 nd Step – Minimum Distance Notice that: is a compact way of expressing Wold conditions with Objective:

July 2007 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.

July 2007 Projection Minimum Distance 14 Min. Chi-Square – Least Squares

July 2007 Projection Minimum Distance 15 Key assumptions for Asymptotics   is stochastically equicontinuous since b is infinite-dimensional when h   as T    Instrument relevance:  Identification:

July 2007 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.

July 2007 Projection Minimum Distance 17 Misspecification Test Correct specification means the minimum distance function is zero. Hence we can test overidentifying conditions Since then

July 2007 Projection Minimum Distance 18 GMM vs. PMD: An Example Estimated Model: True Model: Instrument validity condition:

July 2007 Projection Minimum Distance 19 However… Let: Notice that: Hence: Lesson: Orthogonalize instruments w.r.t. possibly omitted variables

July 2007 Projection Minimum Distance 20 GMM

July 2007 Projection Minimum Distance 21 PMD

July 2007 Projection Minimum Distance 22 Monte Carlo Experiments 1. PMD vs MLE: ARMA(1,1) PMD vs MLE DGP: Parameter pairs (  ,  1 ): (0.25, 0.5); (0.5; 0.25); (0, 0.5); (0.5; 0) T = 50, 100, 400 Lag length determined automatically by AIC c h = 2, 5, 10

July 2007 Projection Minimum Distance 23  1 = 0.5;  1 = 0.25

July 2007 Projection Minimum Distance 24  1 = 0.5;  1 =

July 2007 Projection Minimum Distance 25 Monte Carlo Comparison: PMD vs GMM Euler equation:

July 2007 Projection Minimum Distance 26 When Model is Correctly Specified PMD GMM

July 2007 Projection Minimum Distance 27 Omitted Endogenous Dynamics

July 2007 Projection Minimum Distance 28 Omitted Exogenous Dynamics

July 2007 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

July 2007 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)

July 2007 Projection Minimum Distance 31 Results – Output Euler Equation MethodSpecification  (S.E.)  (S.E.) GMMHP0.52 (0.053) (0.0094) GMMST0.51 (0.049) (0.0093) MLEHP0.47 (0.035) (0.0037) MLEST0.42 (0.052) (0.0055) OI-GMMHP0.47 (0.062) (0.023) OI-GMMST0.41 (0.064) (0.022) PMD (h = 12)HP0.47 (0.025) (0.008) PMD (h = 12)ST0.47 (0.027) (0.009)

July 2007 Projection Minimum Distance 32 Inflation Euler Equations MethodSpecification  (S.E.)  (S.E.) GMMHP0.66 (0.13) (0.072) GMMST0.63 (0.13) (0.050) GMMRULC0.60 (0.086)0.053 (0.038) MLEHP0.17 (0.037)0.10 (0.042) MLEST0.18 (0.036)0.074 (0.034) MLERULC0.47 (0.024)0.050 (0.0081) OI-GMMHP0.23 (0.093)0.12 (0.042) OI-GMMST0.21 (0.11)0.097 (0.039) OI-GMMRULC0.45 (0.028)0.054 (0.0081) PMD (h = 16)HP0.59 (0.036) (0.019) PMD (h = 9)ST0.63 (0.050) (0.019) PMD (h = 15)RULC0.56 (0.027)0.022 (0.010)

July 2007 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