Identifying the New Keynesian Phillips Curve
James M. Nason and Gregor W. Smith†
May 2003: Preliminary. Please Quote.
Abstract
New Keynesian Phillips curves describe how past inflation, expected future inflation,
and a measure of marginal cost or an output gap drive the current inflation rate. This
paper demonstrates that typical, GMM estimation of these curves may feature parameters that are near-identified (in which case non-standard distribution theory applies)
or unidentified. This problem may supercede the usual tradeoff between robustness
and efficiency in comparing GMM estimators with full-information estimators, for we
show that FIML may make identification easier. In a full-information environment,
we illustrate the identification problem analytically. We then describe two pre-tests
than can be used to study identification, and apply these to data for the United States,
the United Kingdom, and Canada. In these countries, Phillips curves are central to
discussions of monetary policy and inflation targetting.
JEL classification: E31; C32
Keywords: Phillips curve, identification, inflation
†Nason: University of British Columbia and Research Department, Federal Reserve
Bank of Atlanta; nason@econ.ubc.ca. Smith: Queen’s University; smithgw@qed.econ.
queensu.ca. We thank the Social Sciences and Humanities Research Council of Canada
for support of this research. Smith thanks the Research Department of the Federal
Reserve Bank of Atlanta for providing the environment for this research. Richard
Luger and Katharine Neiss provided data for Canada and the U.K. respectively, while
Nikolay Gospodinov and Amir Yaron shared their code.
1. Introduction
Recent years have seen an efflorescence of work on the Phillips curve. This
work features debates on the role of backward and forward-looking components, on
whether marginal cost or an output gap best explains inflation, and on the implications for policy, including the costs of disinflation. Attempting to account for the
instability of the Phillips curve over time also has led to a great deal of research.
The specification of the curve has important implications for inflation targetting, and
specifically for how central banks may react to real events while maintaining inflation
targets. Contributions to this research are too numerous to list. Fuhrer and Moore
(1995), Galí and Gertler (1999), Roberts (1995,1997) and Sbordone (2002) make important empirical contributions. The theory and evidence is outlined and assessed by
Woodford (2002).
Most empirical work on the Phillips curve has estimated it using instrumental
variables methods. Generally, its parameters have proven to be difficult to pin down
without large instrument sets. In section 2 we study identification analytically in a
solved version of the Phillips curve difference equation. This analysis suggests two
sources of identification problems. First, higher-order dynamics in marginal cost or
the output gap are necessary for identification and testing. Second, and more importantly, if inflation Granger causes marginal cost or the output gap, then the coefficient
on lagged inflation in the hybrid Phillips curve cannot be identified. Since Grangercausality by inflation of future marginal costs is often used as an informal, weak test
of the theory, this is an unfortunate conclusion.
In section 3 we then pre-test for higher-order dynamics and for Granger-causality,
and use the results to reinterpret estimates of Phillips curves for the U.S., U.K., and
Canada using both limited information (GMM/IVE) and full information (MLE) methods. The results provide some support for the standard estimation method in the
U.S., but not in the other two countries. Section 4 concludes with some suggestions
of alternative approaches to identification.
1
2. Statistical Environment and Identification Results
Galí and Gertler (1999) proposed a variation of Calvo’s contract model, in which
firms set prices with reference to past prices. Each firm can reset its price with probability 1 − θ. Of those, a fraction 1 − ω set prices with reference to expected future
marginal costs, while the remainder set prices equal to the the average of recently
altered prices plus an adjustment based on lagged inflation. The firms have a common discount factor, β. From the three structural parameters, ω, θ, and β, define
reduced-form parameters:
φ = θ + ω[1 − θ(1 − β)]
λ = (1 − ω)(1 − θ)(1 − βθ)φ−1
(1)
γf = βθφ−1
γb = ωφ−1
The hybrid, new-Keynesian, Phillips curve then is given by:
πt = γf Et πt+1 + γb πt−1 + λxt ,
(2)
where we use xt to denote either marginal cost or an output gap.
The hybrid, new Keynesian Phillips curve (2) is a linear, second-order difference
equation, just like the first-order condition from any linear-quadratic planning problem. Our study draws on tools for formulating these problems under rational expectations developed by Hansen and Sargent (1980) and Sargent (1987). We also draw
on studies of estimation in the linear-quadratic model by Gregory, Pagan, and Smith
(1993), West and Wilcox (1994) and Fuhrer, Moore, and Schuh (1995).
We begin by illustrating the identification of the Phillips curve parameters in several different statistical frameworks. Estimation generally is by instrumental variables
(GMM) and so we focus on those methods. First, we adopt a linear, statistical model
for xt , and then we solve for inflation, πt . Using the solved (full-information) model
we then describe several different GMM (limited-information) estimators.
Our focus is first on the identification of the three parameters, {γf , γb , λ}. The
second stage of identification – finding {ω, θ, β} – is then straightforward from (1),
2
which has a unique solution if these three parameters are constrained to be positive
fractions. We consider the two classic properties of instrument sets. First, and most
obviously, identifying the three parameters of the Phillips curve requires at least three
instruments or, more generally, three pieces of identifying information which could
include restrictions on the parameters or covariance restrictions in a system setting. A
test based on over-identification requires at least four instruments or four such pieces
of information. The instruments must be uncorrelated with the GMM residuals, which
are essentially forecast errors. This is the order condition. Second, the matrix of crossproducts of the instruments and the right-hand-side variables in the Phillips curve (2)
cannot be singular. This is the rank or ‘relevance’ condition.
We illustrate each property using our model of {xt }. Our environment is linear,
so there is no distinction between local and global identification. We first show the
obvious fact that higher-order dynamics in xt often are necessary for identification.
We also show analytically how weak identification can arise. In GMM estimation, we
show that an Engle-Granger-style two-step estimator may not aid identification when
x and π are cointegrated. We show that partly solving the Phillips curve forward does
not improve identification, and we derive an expression for the loss of precision in
the Phillips curve caused by using only lagged instruments. Finally, we also show that
the hybrid Phillips curve can be identified only if πt does not Granger-cause xt . These
analytical results provide guidance for empirical work in section 3.
To see properties of Phillips curve estimators, begin by solving the difference
equation (2) using methods popularized by Sargent (1987):
∞ 1 k
λ
πt = δ1 πt−1 +
Et xt+k ,
δ2 γf k=0 δ2
(3)
where δ1 and δ2 are the stable and unstable roots, respectively, of the characteristic
equation in (2), given by:
δ1 =
1 − 1 − 4γb γf
δ2 =
2γf
1 + 1 − 4γb γf
2γf
.
(4)
We assume that {xt } is of exponential order less than δ2 so that the infinite sum in
(3) is finite, and that the roots yield a unique solution to the difference equation.
3
Suppose that xt evolves autonomously according to a J-th order autoregression:
xt =
J
ρj xt−j + t ,
(5)
j=1
where ρj = 0 ∀ j and t is an innovation with respect to the σ -field generated by the
history of x. This process can be rewritten in companion form as:
x̃t = ρ̃ x̃t−1 + ˜t ,
where x̃t = (xt xt−1 . . . xt−J−1 ) and the transition matrix is:


ρ1 ρ2 . . . ρJ


0 ... 0 
 1


1 ... 0 
ρ̃ = 
.
 0.
..
..
.. 
 .

 .
.
.
. 
0
0 ... 1
(6)
(7)
Next, define sJ as a selection row vector of length J with 1 in the first position and
zeros thereafter. It will select the first element of x̃t . Define IJ as the J × J identity
matrix. The solution for inflation then is:
−1
λ
πt = δ1 πt−1 +
x̃t + ηt .
sJ IJ − ρ̃δ−1
2
γf δ2
(8)
We assume that
|IJ − ρ̃δ−1
2 | = 0.
(9)
The stochastic singularity is avoided – so that ηt appears in the solution (8) –
by assuming that the econometrician’s information set lies strictly within that of the
price-setting agents, as originally proposed by Hansen and Sargent (1980). Call the
agents information set Gt and the econometrician’s information set Ft ⊂ Gt . The
inflation rate can always be decomposed this way:
πt = E[πt |Ft ] + πt − E[πt |Ft ]
= E[πt |Ft ] + ηt .
(10)
Thus, ηt is uncorrelated with information available to the econometrician at time t.
In particular, if the econometrician has access to current and past values of x then:
cov(ηt , t ) = 0.
4
(11)
We shall study identification in this environment. This quest excludes other potential sources of identification, such as structural breaks, varying conditional covariances, or the use of survey data on inflation expectations. We have omitted constant
terms, as if the data have been demeaned. Of course if in applications a constant term
is included in the Phillips curve then a vector of ones can be used as an instrument,
but this step adds no net identifying information.
Result 1. The number of regressors in (8) is J + 1. The parameters in ρ̃ can be
identified from the estimation of the law of motion for xt , (5). With three behavioural
parameters {γf , γb , λ} to identify, J ≥ 1 is necessary for identification in the solved
model (8). J ≥ 2 is necessary for overidentification.§
The key logic behind this result is that FIML estimation of the bivariate system
allows the econometrician to impose the covariance restriction (11). Thus only two
additional pieces of information are required from the solution for inflation (8), and
there are two regressors as long as J ≥ 1. In general, identification is possible if the
present value in the solved model (2) has a non-null projection on at least one variable
known by price-setters at time t. In our case, these variables will be elements in x̃t ,
but other variables might contribute as well.
Studies that use the system estimator include those of Fuhrer and Moore (1995),
Sbordone (2002), Kurmann (2003a), and Lindé (2002). But typically, the Phillips curve
is estimated by GMM, using the sample versions of:
Et γf πt+1 − πt + γb πt−1 + λxt = 0,
(12)
and instruments zt . A necessary condition for identification of {γb , γf , λ} using the
moment conditions (12) is that there be as many valid instruments as parameters (or
variables explaining inflation in this linear model). Of course, being dated t − 1 or
earlier is not sufficient for an instrument to be valid: it must contain incremental
information about πt+1 . This is the ‘relevance’ condition of instrumental variables
estimation.
Result 2. If zt = {πt−1 , xt , xt−1 , xt−2 , . . . , xt−J−1 } then the parameters of the Phillips
curve are identified by GMM if J ≥ 2 and overidentified if J ≥ 3.§
5
The solution (8) shows that further lags of inflation contain no identifying information, so zt is the maximal instrument set in this environment. Observe that
dim zt = J + 1 and the result follows. For example, if J = 2 then zt = {πt−1 , xt , xt−1 },
because xt−2 contains no additional information. Moving from estimating the solved
model (5) and (8) to the difference equation (12), excluding information on the properties of xt , cannot ease the conditions for identification. Result 2 shows that identification is more onerous than in the full information environment of result 1. That
is because the error-covariance restriction (11) is no longer available. Prior to considering the usual trade-off between efficiency and robustness in deciding between
FIML and GMM, the conditions for identification in GMM estimation will be stricter.
In particular, the parameters of the second-order difference equation in inflation (2)
cannot be identified by GMM if xt follows a first-order Markov process.
Pesaran (1987, Propositions 6.1 and 6.2) derived similar results in a more general
setting. He observed that identifying information is available when the lag length in
the process for xt is longer than that in the difference equation.
A number of studies of the difference equation impose a value for β (or γf ). For
example, β is sometimes set to be 0.99 in quarterly data, which implies a quarterly
discount rate of about 1 percent. Result 2 can rationalize this procedure: the Phillips
curve may not be identified without this step, if {xt } approximately follows a Markov
process.
A number of researchers have used only lagged instruments in estimating (9).
Galí and Gertler (1999), used up to four lags of various instruments. Let us denote
this information set by zt−1 .
Result 3. If zt−1 = {πt−1 , xt−1 , xt−2 , . . . , xt−J }, so that only lagged information is
used, then the parameters of the Phillips curve again are identified by GMM if J ≥ 2
and overidentified if J ≥ 3.§
The intuition for Result 3 is that the moment conditions (12) involve forecasts of
πt+1 , πt , and xt based on information at time t − 1. Notice that zt−1 is not a subset
of zt . Again dim zt−1 = J + 1 and the identification result follows.
6
As an example, suppose that xt follows a second-order autoregression, so J = 2.
Then zt = {πt−1 , xt , xt−1 } and zt−1 = {πt−1 , xt−1 , xt−2 }. Omitting the current value
of xt as an instrument means that an additional, lagged value must be used and be
relevant. If instead zt is the instrument set, then including xt−J (xt−2 in this example)
provides no overidentifying information.
In some circumstances, the investigator may know the value of λ, either from
theory or from some auxiliary statistical work. For example, if J = 1 and ρ1 = 1 then
xt and πt will be cointegrated with parameter λ, which could be estimated from a
static regression, as originally proposed by Granger and Engle. This information can
potentially aid identification of the remaining parameters, γf and γb .
Result 4. If a consistent estimate λ̂ is available, then γf and γb are overidentified
in the full information environment if J ≥ 1. They are identified in the limited information environment with zt if J ≥ 1, and overidentified if J ≥ 2. In the limitedinformation environment, with zt−1 , however, J ≥ 2 remains necessary for identification and J ≥ 3 remains necessary for overidentification.§
To see this result, consider J = 1. Then the solved model yields two coefficients
and a covariance restriction, which overidentify the two remaining parameter estimates (with ρ1 estimated in the auxiliary model). Similarly, with λ̂xt known in the
difference equation (12), the instruments xt and πt−1 can be used to identify γf and
γb . But with instruments zt−1 three variables in (12) still must be forecasted – πt+1 ,
πt , and xt – even given an estimate λ̂. Thus a two-step procedure cannot identify the
two remaining parameters unless J ≥ 2 continues to hold.
The last part of Result 4 is a generalization of an example by Pagan, Gregory, and
Smith (1993), who considered the case with ρ1 = 1. They argued that lagged instruments could not identify the parameters of the difference equation without higherorder dynamics in the x-process. Result 4 also is relevant to price-setting rules that
are written in terms of the level of prices, rather than the inflation rate, because the
price level is more likely to be nonstationary yet cointegrated with the fundamental.
7
Result 5. The conditions for identification do not change if the investigator imposes
γb = 0, so that the Phillips curve is purely forward-looking.§
This result can be checked by specializing the solution in (8), with δ1 = 0 and
δ2 = γg−1 which follow from the roots (4). Again we assume that the remaining two
parameters yield a unique solution to the difference equation. Intuitively, the investigator has dropped a parameter, γb but also a variable πt−1 . Mavroeidis (2001) also
provides a discussion of this case.
Rudd and Whelan (2001), Galí, Gertler, and López-Salido (2001), and Guay, Luger,
and Zhu (2002) solve the Phillips curve difference equation forward, as in (2), but
truncate after K leads. They then estimate by instrumental variables in:
Et−1
K
λ −k
πt − δ1 πt−1 +
δ2 xt+k .
δ2 γf k=0
(13)
Result 6. Solving forward and truncating provides no additional information to aid
identification (or improve efficiency).§
This result is obvious given Result 2. The difference equation – solved forward
and truncated – still involves the three parameters {γf , γb , λ}. Were there valid instruments for each future xt+k in (13) then these parameters would be overidentified
because (13) contains more variables than parameters when K ≥ 1. But the number of
relevant instruments remains J +1, so the conditions for identification are unchanged.
Result 7. Whether zt or zt−1 is adopted, the GMM residual follows an MA(1) process.
Both of these instrument sets are valid, but any instrument set must exclude lagged
GMM residuals. In addition, the loss of precision from excluding xt from the instrument set depends both on parameters in its law of motion and on the Phillips curve
parameters.§
The GMM residual is given by:
νt+1 ≡ γf πt+1 − πt + γb πt−1 + λxt − Et γf πt+1 − πt + γb πt−1 + λxt .
8
(14)
With zt , the residual is:
νt+1|t = γf ηt+1 + (δ1 γf − 1)ηt +
−1
λ
s̃J IJ − ρ̃δ−1
˜t+1 .
2
γf δ2
(15)
This moving average can be controlled for in constructing the weighting matrix in
GMM estimation. If zt−1 is adopted then the residual is:
νt+1|t−1 = νt+1|t + s̃J
λ −1 −1
γf (γb + ρ̃) − 1 IJ − ρ̃δ2
− λ ˜t .
γf δ2
(16)
so that the variance of the additional term depends on the parameters of the Phillips
curve in addition to those of the {xt } process.
The final identification result makes estimating or testing for hybrid versions of
the new Keynesian Phillips curve particularly difficult.
Result 8. If πt Granger causes xt then the coefficient γb is not identified in either
FIML or GMM estimation.§
Finding that inflation Granger-causes marginal cost is sometimes treated as a
weak implication and test of the new Keynesian Phillips curve. Ironically, Result 8
implies that the curve cannot be identified under these circumstances. To see this result heuristically, imagine a purely forward-looking, solved Phillips curve. We project
the present-value term in marginal cost onto the econometrician’s information set,
leaving a residual ηt :
πt =
λ
δ 2 γf
∞ 1 k
E[xt+k |Ft ] + ηt ,
δ2
k=0
(16)
where ηt reflects the part of the present-value that cannot be forecast by the econometrician. But recall that agents have more information than econometricians, so ηt
will be correlated with Gt . Moreover, it may be correlated with Gt−1 . Therefore, it will
be correlated with πt−1 for, after all, πt−1 is the optimal predictor of the present value
lagged one-period, which shares many terms with the present value in (16). Thus if
πt−1 has some additional predictive power for the present value in (16), then it will
enter the equation significantly. But its presence will reflect forecasting information,
9
and not backward-looking price-setting. More generally, the coefficient on πt−1 will
not be a consistent estimate of δ1 even if the true curve is a hybrid.
A formal proof of Result 8 was given by Sargent (1987, chapter XI, part 24), who
showed that xt must be strictly exogenous (in the classic terminology of Engle, Hendry
and Richard (1983)) for δ1 to be identified. The idea that the endogenous variable
reflects superior information has been applied by Campbell and Shiller (1987) and
Boileau and Normandin (2002), among others. For the U.S., King and Watson (1994)
report evidence that inflation does not Granger-cause the unemployment rate, which
suggests the Phillips curve in inflation and unemployment can be identified for the
U.S.. Rudd and Whelan (2001) and Mavroeidis (2001) have noted that forward and
backward dynamics may not be separately identified. To their warnings we add an
interpretation and test based on Granger causality.
Summary The key, analytical findings of this section are that (a) identification may
be easier in the FIML context than in the GMM context; (b) in either case, higher-order
dynamics in marginal cost or unemployment are necessary in order to test the theory;
and (c) the backward portion of the Phillips curve is generally not identified, when
price-setters have superior information to econometricians.
These results suggest two pre-tests that should accompany estimation. First, we
should study the dynamics of the {xt } process. Second we should study whether
{πt } Granger causes {xt }. We next implement these tests, followed by GMM and FIML
estimation, for the U.S., the U.K., and Canada.
3. Applications
In these applications, xt will denote the logarithm of real marginal cost. Following Result 7, GMM estimators will allow for a first-order moving average in the GMM
residual. The weighting matrix will be the continuous-updating version introduced
by Hansen, Heaton, and Yaron (1996), which has good finite-sample properties and is
invariant to the normalization of the Phillips curve (2). Data sources are described in
the appendix.
10
Each empirical study begins with two pre-tests. First, we test for Granger noncausality. Second, we project xt on J = 6 lags, and test the lag length using the AIC,
BIC, and likelihood ratio tests. Granger causality test results are given in table 1, while
we report Jˆ in the text.
3.1 United States
For the U.S., we find that x Granger causes π but that π does not Granger cause
x. Strictly speaking, the null hypothesis that π does not Granger cause x cannot be
rejected, with p-value 0.18. Thus, the backward-looking part of the Phillips curve may
be identified. The AIC and a likelihood ratio test suggested a lag length of 3, while
the BIC suggested a lag length of 1. In addition, the coefficient ρ̂2 was insignificantly
different from zero.
The results of these pre-tests may be seen from two perspectives. First, Result 2
implies that identifying the Phillips curve using GMM requires higher-order dynamics.
These dynamics are not overwhelming here, so that identification may be difficult.
Second, however, precisely because the Phillips curve depends on those dynamics,
measuring them might best be done in a system including the curve.
Table 2 contains estimation results, with GMM estimates in the top panel and
FIML estimates in the bottom panel. In the GMM rows, most of the work is done by
the instruments {πt−1 , xt , xt−2 }, as is suggested by the pre-test evidence that only
xt and xt−2 help forecast xt+1 . Adding further instruments increases the precision
slightly (especially for β̂) but does not lead to significant changes in the estimates.
The J-test clearly does not reject the over-identifying restrictions.
Depending on the instrument set, the point estimate of the proportion of firms
changing prices each quarter, θ̂, varies from 0.89 to 0.96. The proportion of firms
looking backward while changing prices, ω̂, ranges from 0.33 to 0.41. The discount
factor also takes plausible values. These results are comparable to those of Galí and
Gertler (1999, table 2), though we find a larger share of firms looking backward to
recent price changes and we use much smaller instrument sets.
11
The bottom panel of table 2 contains the system estimates. Now taking account
of higher-order dynamics, in the form of ρ3 , makes a greater difference, because both
equations use that information. The main economic finding is that the FIML estimators
find a larger role for backward-looking behaviour. However, the restrictions on the
joint series {πt , xt } implied by the theory are rejected at the 1 percent level.
3.2 United Kingdom
For the U.K, table 1 provides strong evidence of Granger causality in both directions, so that γ̂b will not measure the inertia in price-setting. In this case the second
set of pre-tests suggests that the lag length is J = 5. Moreover, ρ̂3 is insignificantly
different from zero.
Table 3 contains estimates of the U.K. Phillips curve. In the top panel, the GMM
estimates are very sensitive to instrument choice. Once lags up to xt−4 are included,
the estimates accord with theory and are estimated with some precision. But the
overidentifying restrictions are rejected when xt is an instrument.
In the bottom panel, the restrictions implied by the most general model, with
J = 5, are rejected. Parameter estimates are consistent with theory, but they are
estimated with little precision. For example, the estimated effect of marginal cost
on inflation, λ̂, is insignificant. For the U.K. data, the information in higher-order
dynamics makes little difference to the FIML estimates, which suggests that the current
value of xt is doing most of the explanatory work.
Our estimates of the forward-looking element in inflation are lower than those
found by Neiss and Nelson (2002). Those authors use dummy variables to control for
a variety of price-level shocks, though. Like us, Balakrishnan and López-Salido (2001)
do not find a significant, stable effect of marginal cost on inflation.
3.3 Canada
For Canada, table 1 shows that inflation does Granger cause marginal cost, so
that identification will be a problem. The pre-tests for lag length showed a similar
12
persistence pattern to that in U.S. marginal costs. Once-lagged costs play a large
predictive role and thrice-lagged costs play a significant, additional role.
Canadian results are shown in table 4. For Canada the Phillips curve is poorly
identified in both GMM and FIML estimation. Guay, Luger, and Zhu (2003) estimate the
hybrid, new-Keynesian Phillips curve using a wider range of instruments. When they
use much larger instrument sets they increase precision (and reject over-identifying
restrictions). But we reproduce their finding that λ̂ is insignificant, so that one cannot
find a role for marginal cost in inflation dynamics.
3.4 Summary
For all three countries, it is difficult to find evidence of a significant, positive role
for marginal cost in the hybrid Phillips curve. For all three countries, marginal cost
has some higher-order dynamics, but perhaps not enough to greatly improve identification. Clearly, one possibility is that the Phillips curve is a useful tool but a broader
set of instruments is needed to forecast marginal costs. Kurmann (2003b) carefully
explores the evidence on this possibility, beyond the bivariate environment considered here. A theoretical study of this possibility would require a general equilibrium
model. Another possibility is that the Calvo-style pricing problem is simply not applicable. And a final possibility is that inference can be improved even in this bivariate
environment.
Stock and Wright (2000) and Stock, Wright, and Yogo (2002) provide tools for
GMM estimation and inference with weak instruments. Ma (2002) shows using the
S-sets developed by Stock and Wright (2000) that ω is weakly identified in the GalíGertler data, so that one cannot conclude from small point estimates – as Galí and
Gertler do – that backward-looking price-setting is unimportant. We view our work
as an analytical complement to these studies. The Phillips curve (2) is linear and so
it can be solved for any {xt } that is multi-step forecastable. In the next version of
this paper, we hope to include analytical solutions for the concentration parameter,
µ, used to test the relevance condition.
13
7. Conclusion
Section 2 showed two fundamental sources of non-identification: weak, higherorder dynamics and superior information. We suggested two pre-tests corresponding
to these two syndromes: a test of the lag length J in the fundamental process {xt },
and a test of Granger non-causality. Empirical application of these tests suggest that
the Phillips curve may be better identified in the U.S. than in the U.K. or Canada. We
also showed that FIML has an identification advantage over GMM, and can identify the
Phillips curve even if xt follows a first-order Markov process.
Estimating the new Keynesian Phillips curve involves the interaction of number
of statistical issues. In FIML one might be concerned with the effects of the pre-test,
with misspecification of the x-process, and with the small-sample bias in estimates of
persistence, such as ρ̂1 . In GMM one might be concerned with the choice of weighting
matrix, the loss of test power from using irrelevant instruments, and the properties
of the J-test of overidentifying restrictions. To both of these must be added the
problem of weak or near identification. The interaction of these issues can be studied
by Monte Carlo methods, as recent studies by Lindé (2002) and Mavroeidis (2001)
fruitfully show.
A failure of identification is a property of a statistical approach, and not of a
theory. Price-setting dynamics have important implications for monetary policy, so
alternative sources of identifying information – such as regime changes – are certainly
worthy of study.
14
Data Sources
United States
The price level P(t) is the GDP implicit price deflator. The GDP deflator is available
in chain weight form and in implicit form (all the U.S. results so far are based on the
implicit GDP deflator).
Nominal unit labor cost (ULC) is the ratio of the index of hourly compensation in
the non-farm business sector, labeled COMPNFB, to output per hour of all persons in
the non-farm business sector, labeled OPHNFB. COMPNFB is an index of the nominal
wage. OPHNFB is an index of the average product of labor. These can be found in the
Federal Reserve Bank of St. Louis’ FRED databank. Thus, ULC is a measure of labor’s
share.
Real ULC equals nominal ULC deflated by P(t). Inflation is 100*ln[P(t)/P(t-1)] and
real ULC is 100(1 + a)*ln[COMPNFB(t)/OPHNFB(t)] - 100*ln[P(t)], where a is a function of
the steady-state markup and labor’s share parameter in the firm’s production function.
This adjustment renders real ULC stationary and a = 1.08.
The estimation sample period is 1947Q1-2002Q4, T = 224.
United Kingdom
The inflation rate is measured with the GDP deflator, and x is a measure of the log
of real marginal cost. Data sources are given by Katharine Neiss and Edward Nelson
(2002), who kindly provided the data. The estimation period is 1961Q1 to 2000Q4, so
T = 168.
Canada
The inflation rate is measured with the GDP deflator, while x is the log of the
labour share in the non-farm, business sector. Data sources are given by Guay, Luger,
and Zhu (2003), who kindly provided the data. The estimation period is 1963Q1 to
2000Q4.
References
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business cycles, and superior information. Journal of Monetary Economics 49,
495-520.
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Galí, Jordi, Gertler, Mark, and López-Salido, J. David (2001) Notes on estimating the
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Table 1
Granger Non-Causality Tests
Country
Lag length (d.f.)
p π → x
p x → π
U.S.
3
0.18
0.05
U.S.
4
0.24
0.08
U.K.
4
0.01
0.00
U.K.
5
0.01
0.00
Canada
3
0.00
0.73
Canada
4
0.00
0.63
Notes: The lag lengths are the same as those selected by information criteria, Jˆ. Entries are p -values for
the null hypothesis that the first variable does not Granger cause the second variable. Data sources and
sample sizes are given in the data appendix.
Table 2
U.S. Phillips Curve
πt = γf Et πt+1 + γb πt−1 + λxt
xt =
J
ρj xt−j + t
j=1
T = 224
1947Q1 – 2002Q4
Estimator
{Information}
γ̂f
γ̂b
λ̂
ω̂
θ̂
β̂
ρ̂1
(se)
(se)
(se)
(se)
(se)
(se)
(se)
χ 2 (df )
(p)
GMM
{πt−1 , xt }
—
—
—
—
—
—
—
—
GMM
{πt−1 , xt , xt−2 }
0.685
0.299
0.001
0.406
0.961
0.965
—
—
(0.356)
(0.247)
(0.006)
(0.465)
(0.162)
(0.306)
GMM
0.491
{πt−1 , xt , ...xt−2 } (0.301)
0.440
0.008
0.567
(0.206)
(0.005)
(0.321)
0.902
(0.094)
(0.480)
GMM
0.677
{πt−1 , xt , ...xt−4 } (0.219)
0.294
0.008
0.333
0.891
0.961
(0.154)
(0.005)
(0.263)
(0.064)
(0.168)
FIML
{πt−1 , xt }
0.358
0.561
-0.001
0.782
1.022
0.488
0.923 15.1(2)
(11.9)
(7.20)
(0.091)
(0.236)
(0.208)
(22.6)
(0.026) (0.00)
FIML
{πt−1 , xt , xt−2 }
0.581
0.426
-0.001
0.758
0.980
1.055
0.971 8.89(1)
(0.331)
(0.200)
(0.002)
(2.48)
(2.73)
(1.60)
(0.052) (0.00)
FIML
0.509
{πt−1 , xt , ...xt−2 } (0.497)
0.469
-0.001
0.904
1.410
0.695
0.871 8.95(2)
(0.299)
(0.003)
(1.53)
(11.32)
(5.524)
(0.068) (0.01)
0.797
—
2.11(1)
(0.34)
—
3.48(3)
(0.48)
Notes: The data are quarterly from 1949Q1 to 2001Q4. The underlying parameters ω, θ and β are
nonlinear functions of γf , γb , and λ, given in (1). The test statistic is the J -statistic in the upper panel
and the likelihood ratio statistic in the lower panel. The first two likelihood ratio statistics test the
restrictions relative to the more general specification in the last row. The last statistic tests against an
unrestricted system.
Table 3
U.K. Phillips Curve
πt = γf Et πt+1 + γb πt−1 + λxt
xt =
J
ρj xt−j + t
j=1
T = 168
1961Q1 – 2000Q4
Estimator
{Information}
γ̂f
γ̂b
λ̂
ω̂
θ̂
β̂
ρ̂1
(se)
(se)
(se)
(se)
(se)
(se)
(se)
χ 2 (df )
(p)
GMM
{πt−1 , xt }
—
—
—
—
—
—
—
—
GMM
{πt−1 , xt , xt−1 }
-2.70
2.39
0.924
0.702
0.492
-1.61
—
—
(4.78)
(3.04)
(1.53)
(0.113)
(0.337)
(1.41)
0.801
(0.302)
(0.287)
GMM
0.134
{πt−1 , xt , ...xt−4 } (0.224)
0.601
0.094
0.562
(0.135)
(0.160)
(0.139)
0.307
—
9.81(3)
(0.04)
GMM
{πt−1 , xt−1 , ...
xt−4 }
0.505
0.309
0.294
0.011
0.570
0.953
(0.263)
(0.197)
(0.153)
(0.113)
(0.065)
(0.113)
FIML
{πt−1 , xt }
0.598
0.399
0.029
0.491
0.739
0.994
0.633 0.18(1)
(0.177)
(0.079)
(0.047)
(0.086)
(0.097)
(0.325)
(0.073) (0.67)
FIML
{πt−1 , xt , xt−1 ,
xt−3 , xt−4 }
0.602
0.398
0.028
0.489
0.740
1.00
0.629 14.9(3)
(0.166)
(0.662)
(0.044)
(0.085)
(0.093)
(0.304)
(0.074) (0.00)
FIML
0.540
{πt−1 , xt , ..., xt−4 } (11.7)
—
4.40(2)
(0.22)
0.426
0.048
0.494
0.703
0.896
0.770 36.9(4)
(5.17)
(3.61)
(1.42)
(4.91)
(21.3)
(0.050) (0.00)
Notes: The data are quarterly from 1961Q1 to 2000Q4. The underlying parameters ω, θ and β are
nonlinear functions of γf , γb , and λ, given in (1). The test statistic is the J -statistic in the upper panel
and the likelihood ratio statistic in the lower panel. The first two likelihood ratio statistics test the
restrictions relative to the more general specification in the last row. The last statistic tests against an
unrestricted system.
Table 4
Canadian Phillips Curve
πt = γf Et πt+1 + γb πt−1 + λxt
xt =
J
ρj xt−j + t
j=1
T = 152
1963Q1 – 2000Q4
Estimator
{Information}
γ̂f
γ̂b
λ̂
ω̂
θ̂
β̂
ρ̂1
(se)
(se)
(se)
(se)
(se)
(se)
(se)
χ 2 (df )
(p)
GMM
{πt−1 , xt }
—
—
—
—
—
—
—
—
GMM
{πt−1 , xt , xt−2 }
-0.197
0.868
0.039
0.736
0.891
-0.187 —
—
(2.08)
(1.37)
(0.07)
(0.185)
(0.040)
(1.72)
0.891
(0.032)
(1.20)
GMM
0.337
{πt−1 , xt , ...xt−2 } (0.741)
0.520
0.019
0.662
(0.495)
(0.026)
(0.287)
0.366
—
0.29(1)
(0.86)
GMM
{πt−1 , xt−1 , ...
xt−4 }
-0.134
0.847
0.032
0.785
0.901
-0.625 —
1.24(3)
(0.818)
(0.555)
(0.029)
(0.060)
(0.039)
(0.404)
(0.87)
FIML
{πt−1 , xt }
0.566
0.428
0.131
0.406
0.545
0.986
0.958
(0.613)
(0.331)
(3.40)
(2.18)
(3.49)
(0.784)
(0.021)
Notes: The data are quarterly from 1961Q1 to 2000Q4. The underlying parameters ω, θ and β are
nonlinear functions of γf , γb , and λ, given in (1). The test statistic is the J -statistic in the upper panel
and the likelihood ratio statistic in the lower panel. The first-order FIML model is just identified. The
restrictions on the third-order FIML to reduce to this case were easily accepted.