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Estimation and Inference by the Method of Projection Minimum Distance. Òscar Jordà Sharon Kozicki U.C. Davis Bank of Canada. The Paper in a Nutshell: An Efficient Limited Information Method .
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Estimation and Inference by the Method of Projection Minimum Distance Òscar Jordà Sharon Kozicki U.C. Davis Bank of Canada
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
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
Motivating Example: Galí and Gertler (1999) • xrt could be a predictor of , hence a valid instrument/omitted variable • Let Projection Minimum Distance
Implications • Substituting the Wold representation into the model Projection Minimum Distance
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
1st Step: Local Projections • Suppose: • with i.i.d. and • assume the Wold rep is invertible such that Projection Minimum Distance
Local Projections • then, iterating the VAR() • with Projection Minimum Distance
Local Projections in finite samples • Consider estimating a truncated version given by Projection Minimum Distance
Local Projections – Least Squares Projection Minimum Distance
Local Projections (cont.) Projection Minimum Distance
2nd Step – Minimum Distance • Notice that: is a compact way of expressing Wold conditions with • Objective: Projection Minimum Distance
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
Min. Chi-Square – Least Squares Projection Minimum Distance
Key assumptions for Asymptotics • is stochastically equicontinuous since b is infinite-dimensional when h as T • Instrument relevance: • Identification: Projection Minimum Distance
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
Misspecification Test • Correct specification means the minimum distance function is zero. • Hence we can test overidentifying conditions • Since • then Projection Minimum Distance
GMM vs. PMD: An Example • Estimated Model: • True Model: • Instrument validity condition: Projection Minimum Distance
However… • Let: • Notice that: • Hence: • Lesson: Orthogonalize instruments w.r.t. possibly omitted variables Projection Minimum Distance
GMM Projection Minimum Distance
PMD Projection Minimum Distance
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
1 = 0.5; 1 = 0.25 Projection Minimum Distance
p1 = 0.5; 1 = 0.25 0.25 Projection Minimum Distance
Monte Carlo Comparison: PMD vs GMM • Euler equation: Projection Minimum Distance
When Model is Correctly Specified PMD GMM Projection Minimum Distance
Omitted Endogenous Dynamics Projection Minimum Distance
Omitted Exogenous Dynamics Projection Minimum Distance
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
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
Results – Output Euler Equation Projection Minimum Distance
Inflation Euler Equations Projection Minimum Distance
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