Estimation and Inference by the Method of Projection Minimum Distance

Estimation and Inference by the Method of Projection Minimum Distance

Òscar Jordà Sharon KozickiU.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, 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


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


Motivating Example: Galí and Gertler (1999)

xrt could be a predictor of hence

a valid instrument/omitted variable Let


Implications

Substituting the Wold representation into the model


Remarks

xrt 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


1st Step: Local Projections

Suppose:

with i.i.d. and

assume the Wold rep is invertible such that


Local Projections

then, iterating the VAR()

with


Local Projections in finite samples

Consider estimating a truncated version given by


Local Projections – Least Squares


Local Projections (cont.)

rh rh B

Ir 0r . . . 0r

B1 Ir . . . 0r

. . .

Bh 1 Bh 2 . . . Ir


2nd Step – Minimum Distance

Notice that:

is a compact way of expressing Wold conditions with

Objective:


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.


Min. Chi-Square – Least Squares


Key assumptions for Asymptotics

1.

2. is stochastically equicontinuous since b is infinite-dimensional when h as T

3. Instrument relevance:4. Identification:


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.


Misspecification Test

Correct specification means the minimum distance function is zero.

Hence we can test overidentifying conditions

Since then


GMM vs. PMD: An Example

Estimated Model: True Model:

Instrument validity condition:


However…

Let:

Notice that: Hence: Lesson: Orthogonalize instruments

w.r.t. possibly omitted variables

ytM t 1 1Etyt 1M t 1 tM t 1

E tM t 1yt h 0;h 1, . . . ,H


GMM

min vec 0 h 1,0

1 h,0

WT

GMM

vec 0 h 1,0

1 h,0

WGMM 0 j 1

j j

j T k h 1 t kT h Yt,hYt j,h j

t kT h u t

u t j

E h

GMM 1 2 h 1 1 h 1

h 0 as h


PMD

min vec 0 h 1|1 k

1 h|1 k

WT

PMD

vec 0 h 1|1 k

1 h|1 k

WT

PMD

0|1 k 1

v

1

v B Ih

B

E h

PMD 1 2 h 1|k 1 h 1|k

h|k 0 when either h 1 (at h 1 it is exactly zero) or h


Monte Carlo Experiments1. 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

AICc

h = 2, 5, 10


1 = 0.5; 1 = 0.25


1 = 0.5; 1 = 0.25

0.20.255


Monte Carlo Comparison: PMD vs GMM

Euler equation:


When Model is Correctly Specified

PMD

GMM


Omitted Endogenous Dynamics


Omitted Exogenous Dynamics


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


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)


Results – Output Euler Equation

Method Specification (S.E.) (S.E.)

GMM HP 0.52 (0.053) 0.0024 (0.0094)

GMM ST 0.51 (0.049) 0.0029 (0.0093)

MLE HP 0.47 (0.035) -0.0056 (0.0037)

MLE ST 0.42 (0.052) -0.0084 (0.0055)

OI-GMM HP 0.47 (0.062) -0.0010 (0.023)

OI-GMM ST 0.41 (0.064) -0.0010 (0.022)

PMD (h = 12) HP 0.47 (0.025) -0.014 (0.008)

PMD (h = 12) ST 0.47 (0.027) -0.016 (0.009)


Inflation Euler EquationsMethod Specification (S.E.) (S.E.)

GMM HP 0.66 (0.13) -0.055 (0.072)

GMM ST 0.63 (0.13) -0.030 (0.050)

GMM RULC 0.60 (0.086) 0.053 (0.038)

MLE HP 0.17 (0.037) 0.10 (0.042)

MLE ST 0.18 (0.036) 0.074 (0.034)

MLE RULC 0.47 (0.024) 0.050 (0.0081)

OI-GMM HP 0.23 (0.093) 0.12 (0.042)

OI-GMM ST 0.21 (0.11) 0.097 (0.039)

OI-GMM RULC 0.45 (0.028) 0.054 (0.0081)

PMD (h = 16) HP 0.59 (0.036) -0.018 (0.019)

PMD (h = 9) ST 0.63 (0.050) -0.040 (0.019)

PMD (h = 15) RULC 0.56 (0.027) 0.022 (0.010)


Summary

1. Models that require MLE + numerical techniques can be estimated by LS with PMD and nearly as efficiently (e.g. VARMA models)

2. PMD is asymptotically MLE3. PMD accounts for serial correlation

parametrically – hence it is more efficient than GMM

4. PMD does appropriate, unsupervised conditioning of instruments, solving some cases of instrument invalidity

5. PMD provides natural statistics to evaluate model fit: J-test + plots of parameter variation as a function of h

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Estimation and Inference by the Method of Projection Minimum Distance