Policy Research Working Paper 9018
The Moroccan New Keynesian Phillips Curve
A Structural Econometric Analysis
Vincent Belinga
Mohamed Doukali
Macroeconomics, Trade and Investment Global Practice
September 2019
Policy Research Working Paper 9018
Abstract
The Phillips curve is central to discussions of inflation estimator. Data from Morocco are used to examine the
dynamics and monetary policy. In particular, the New ability of the New Keynesian Phillips Curve to explain
Keynesian Phillips Curve is a valuable tool to describe Moroccan inflation dynamics. The analysis finds that by
how past inflation, expected future inflation, and real adding more information to the hybrid version of the New
marginal cost or an output gap drive the current inflation Keynesian Phillips Curve model by increasing the number
rate. However, economists have had difficulty applying the of moment conditions, the inflation dynamics in Morocco
New Keynesian Phillips Curve to real-world data due to can be well-described by the New Keynesian Phillips Curve.
empirical limitations. This paper overcomes these limita- This framework suggests that the New Keynesian Phillips
tions by using an identification-robust estimation method Curve would be a strong candidate for short-run inflation
called the Tikhonov Jackknife instrumental variables forecasting.
This paper is a product of the Macroeconomics, Trade and Investment Global Practice. It is part of a larger effort by the
World Bank to provide open access to its research and make a contribution to development policy discussions around the
world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may
be contacted at vtsounguibelinga@worldbank.org, and m.doukali@bkam.ma.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
The Moroccan New Keynesian Phillips Curve: A
Structural Econometric Analysis†
Vincent Belinga‡ , Mohamed Doukali §
Keywords: Inﬂation dynamics, monetary policy, New Keynesian Phillips curve, identiﬁcation
robust estimations, weak instruments.
JEL classiﬁcation: C13, C52, E31, E52, E58.
† The views expressed are those of the authors and do not reﬂect the views of the World Bank Group or Central Bank
of Morocco.
‡ The author is Economist at the World Bank Group, and can be contacted at vtsounguibelinga@worldbank.org
§ The author is Economist at the Central Bank of Morocco, and can be contacted at m.doukali@bkam.ma
1 Introduction
An important issue in monetary policy models is the short-run dynamics of inﬂation, especially
for how central banks should react to developments in the economy while maintaining inﬂation tar-
gets. In this respect, important advances have been made during the last two decades in the theoretical
modeling of inﬂation dynamics. Much of the modern analysis of inﬂation is based on the New Keyne-
sian Phillips Curve; see Galı and Gertler (1999) (GG henceforth). This model of inﬂation dynamics is
grounded in an optimizing framework where imperfectly competitive ﬁrms are constrained by costly
price adjustments.
The New Keynesian Phillips curve (NKPC) is a forward-looking model of inﬂation dynamics, where
short-run inﬂation dynamics are driven by the expected discounted stream of real marginal costs.
However, given the statistical failure of the basic NKPC formulation when confronted with data, the
curve has since evolved into its more empirically viable hybrid form. In particular, adding lagged in-
ﬂation to the model corrects the signs of estimated coefﬁcients (see Fuhrer and Moore (1995), Fuhrer
(1997) and Roberts (1997)).
GG estimate the hybrid version of this model, in which real marginal cost is the labor income share
and the parameters are functions of three key structural parameters: the fraction of backward-looking
price-setters, the average duration that an individual price is ﬁxed, and a discount factor. Using data
for the U.S, they ﬁnd that the real marginal costs are statistically signiﬁcant and the hybrid speciﬁ-
cation of the NKPC outperforms the purely forward-looking version of the NKPC (without a lag of
inﬂation in the dynamics). Indeed, the popularity of the NKPC models comes in large part from the
fact that the model is supported by empirical data.
But even as the popularity and usage of the NKPC has grown, its empirical identiﬁability has been
criticized. Many authors argue that the above results are unreliable because they are derived using
methods such as the generalized method of moments (GMM), which are not robust to identiﬁcation
problems, also known as the weak instrument problem; see Stock et al. (2002), Dufour (1997), Du-
four (2003), Kleibergen and Mavroeidis (2009) and Dufour et al. (2006). In fact, the identiﬁcation
of coefﬁcients for endogenous variables in linear structural equations is achieved using instrumental
variables that are assumed to be correlated with the right-hand-side endogenous variables (strong)
2
and uncorrelated with the structural error (valid). However, when instruments are weak, i.e., when
the correlations between an instrument and the endogenous variables is low, conventional estimation
procedures can be misleading. The problem of weak instruments or weak identiﬁcation has received
considerable attention in both the theoretical and applied econometric literature. Empirical examples
include include Angrist and Krueger (1991), who measure returns to schooling, and Eichenbaum,
Hansen, and Singleton (1988), who consider consumption asset pricing models.
Morocco has generally experienced low and stable inﬂation over the last two decades. This period
coincided with domestic macroeconomic stability and lower global inﬂation. However, these inﬂation
dynamics are expected to change, following the fact that Moroccan authorities have started in 2018
the reform of its exchange rate regime in the respective of adopting inﬂation targeting. In this paper,
we estimate the NKPC curve for the Moroccan case, in light of recent econometric ﬁndings related to
instruments. Our aim is to produce more reliable estimates based on identiﬁcation-robust methods.
The problem of weak instruments is due to a very small concentration parameter, which is a measure
of the strength of the instruments or moment conditions. To boost the concentration parameter, we
increase the number of moment conditions to a large number (from 4 to 20). However, an excessive
number of moments may create bias.
Our estimation strategy differs from the other related papers in an important way. The small-sample
bias is addressed using the Tikhonov Jackknife instrumental variable estimator (TJIVE) developed by
Hansen and Kozbur (2014), which can be viewed as a regularized version of the Jackknife estimator.
This estimator has better ﬁnite sample properties than the GMM estimator used by GG. Carrasco
and Doukali (2017) extend their work by considering Landweber-Fridman and principal components
regularization schemes. They also provide a data-driven method for selecting the regularization pa-
rameter based on an expansion of the mean-squared error.
Recently, Barnichon and Mesters (2018) estimate the NKPC for the U.S. by applying an instrument
selection method based on the least absolute shrinkage and selection operator (LASSO) method. The
regularization approach, that we consider in this paper, does not rely on variable selection (such as the
LASSO method). Therefore, it does not require ex ante knowledge about the ordering of moments or
instruments. This ensures that all available moments are used (without discarding any a priori) in an
efﬁcient way, even if there are weak instruments. In fact, the regularization method allows bias to be
3
reduced in the presence of many moments by solving the problem of the singularity of the covariance
matrix of moments. The use of many moments ﬁrst increases the bias, but when the regularization is
introduced, this bias shrinks.
An important contribution of this paper is the implementation of a new overidentifying restrictions
J test proposed by Chao et al. (2014), which is robust to the presence of many moments. The is-
sue with weak identiﬁcation is that various test statistics deteriorate. Indeed, the conventional J test
for overidentifying restrictions performs poorly when the number of moments grows (see Kunitomo,
Morimune, and Tsukuda (1983) and Burnside and Eichenbaum (1996)). To address this problem, we
use Chao et al. (2014)’s test, which is a new version of the J test that is robust to the presence of many
instruments and heteroskedasticity. Their test is based on subtracting out the diagonal terms in the
numerator of the test statistic.
Our estimation strategy leads to the following conclusions. First, our results show that the hybrid
version of the NKPC is dominant. Second, the J test proposed by Chao et al. (2014) rejects the
forward-looking speciﬁcation of the NKPC curve and generally accepts the hybrid form. Third, our
results are robust to the inclusion of additional lags of inﬂation and the output gap. In effect, the
TJIVE, which is robust to the number of instruments, allows us to reduce the bias in the many in-
struments setting and ensures that all available instruments are used efﬁciently, even if there are weak
instruments that are not discarded a priori.
This paper also assesses the usefulness of the Phillips curve for inﬂation forecasting in a small open
economy. Many studies support the use of the Phillips curve for forecasting. For instance, Stock and
Watson (1999) use generalized Phillips curve forecasts to forecast the inﬂation rate one year ahead.
Stock and Watson (2008) compare Phillips curve forecasts to several multivariate speciﬁcations of
forecasting models and ﬁnd a good Phillips curve performance for the U.S.
Moroccan monetary authorities expressed an intention to move to inﬂation targeting1 over the next
few years. This will require a formulation of a monetary policy function a la the Taylor rule, to sta-
bilize inﬂation around its target and the output gap. In this setting, forecasting inﬂation will become
1 An inﬂation-targeting regime is an institutional arrangement in which the central bank’s mandate is to target a deﬁned
medium-term inﬂation rate that is compatible with macroeconomic stability. The main policy instrument generally used
in this setup is the ofﬁcial policy interest rate, which is adjusted whenever the projected inﬂation rate over the forecast
horizon signiﬁcantly deviates from the central bank’s stated inﬂation target.
4
crucial for policymakers as well as for the public, which tries to understand and react to central banks’
decisions. An important implication of this paper is that the Moroccan central bank should consider
the Phillips curve model, including the New Keynesian feature, as a potential strong candidate for
inﬂation forecasting when it fully transitions to its inﬂation-targeting regime.
The structure of this paper is organized as follows. Section 2 presents the theoretical framework un-
derpinning the NKPC. Section 3 describes the speciﬁc model and the methodology used in this paper.
Section 4 presents the new J test. Section 5 presents our empirical results. Section 6 provides some
policy lessons. Section 7 concludes.
2 The New Keynesian Phillips Curve
2.1 Speciﬁcations
In this section, we present the hybrid version of the NKPC. This hybrid speciﬁcation can be
derived from the microfoundations framework where ﬁrms evolve in a monopolistically competitive
environment with price stickiness (ﬁrms cannot adjust their prices at certain times). Following GG,
each ﬁrm, in any given period, may change its price with a ﬁxed probability of 1 − θ , and its price
will be kept unchanged or proportional to trend inﬂation with probability θ . Those probabilities are
independent of the ﬁrm’s price history. In such an environment, proﬁle-maximization and rational
expectations lead to the following hybrid NKPC equation for inﬂation, πt :
πt = λ mct + γ f Et πt +1 + γb πt −1 (2.1)
where
(1 − ω )(1 − θ )(1 − β θ )
λ= (2.2)
θ + ω − ωθ + ωβ θ
βθ
γf = (2.3)
θ + ω − ωθ + ωβ θ
5
ω
γb = (2.4)
θ + ω − ωθ + ωβ θ
where πt is the inﬂation rate at time t , mct is the marginal cost and Et πt +1 is the expectation of
future inﬂation conditional on the information set at time t . The parameter γ f determines the forward-
looking component of inﬂation and γb determines its backward-looking part, β is the subjective dis-
count rate and ω is the proportion of ﬁrms that use a backward-looking rule of thumb. Equation 2.1
is usually referred to as the “semi-structural” speciﬁcation corresponding to a deeper microfounded
structural model. GG estimates a version of this model in which the real marginal costs are the labor
share. Using data for the U.S., they ﬁnd that the real marginal costs are statistically signiﬁcant and in-
ﬂation dynamics are predominantly forward-looking. They also ﬁnd that γb is statistically signiﬁcant
but quantitatively small relative to γ f .
2.2 Measure of marginal cost
Since the NKPC curve provides a relationship between marginal costs and inﬂation, the measure-
ment of marginal costs is important for the identiﬁcation of the NKPC parameters. In fact, when
applying the NKPC to data, a ﬁrst problem relates to the choice of an appropriate proxy for mct .
For example, Galı and Gertler (1999) emphasize the importance of using direct measures of the real
marginal cost, such as the labor income share, whereas Leith and Malley (2007) use a proxy based
on the cost of intermediate goods (which corresponds to the largest determinant of the total cost of
production). Guay et al. (2004) derive the NKPC when ﬁrms use alternative production functions.
Others consider a measure of marginal cost based on the assumption of a Cobb-Douglas technology:
Yt = Ktν (At Ht )(1−ν ) , where Yt is the output, Kt is the capital stock, At is labor-augmenting technol-
ogy, Ht is hours worked, and ν is the output elasticity of capital. Real marginal cost is then given
Wt Ht
by St /(1 − ν ), where St = Pt Yt is the labor income share, Wt is the nominal wage, and Pt is the
price level. In a log-linear deviation from the steady state, the real marginal costs are given by:
mct = st = wt + ht − pt − yt . These studies generally argue that these corrections do not affect the
qualitative nature of the results discussed below.
In this paper, we consider a second proxy of marginal costs, which is the output gap as the relevant
6
indicator of real economic activity. The output gap is deﬁned as the difference between the actual
output of the economy and its potential output. The output gap is an important variable for monetary
policy, as it is a key source of inﬂation pressure in an economy. However, measuring this key vari-
able is no easy task because unlike actual output, the level of potential output (and hence the output
gap) cannot be observed directly, and so cannot be measured precisely. We measure the output gap
using the Hodrick-Prescott ﬁlter. In this measure, the output gap is a combination of lags, leads, and
contemporaneous values of output.
3 Estimation issues
3.1 Standard GMM approach
GG use the standard GMM estimator developed by Hansen (1982) to estimate the hybrid version
of the NKPC. The reduced form can be written as:
πt = λ mct + γ f πt +1 + γb πt −1 + εt +1 (3.1)
where εt +1 = πt +1 − Et πt +1 is the error term orthogonal to the information set in period t . We can
rewrite the above NKPC model in terms of the orthogonality conditions:
Et [(πt − λ mct − γ f πt +1 − γb πt −1 )zt ] = 0
The vector of instruments zt includes variables that are orthogonal to εt +1 , allowing for GMM esti-
mation. GG choose a small number of lags for instruments to minimize the potential estimation bias
that can arise in small samples due to the number of overidentifying restrictions. It is well known that
the standard GMM estimator suffers from a small-sample bias in the presence of endogeneity, which
is increased dramatically when many instruments are used and/or when the instruments are weakly
correlated with the endogenous variables; see Yogo (2004).
In the next section, we present an estimation procedure that remains valid in the presence of many
instruments. The strategy uses the Jackknife method to estimate ﬁrst stage predictions of the endoge-
nous variables. The chief contribution of this procedure is the use of ridge regression at each iteration
7
of the Jackknife. The advantage of regularization is that all available instruments can be used without
discarding any a priori.
3.2 Estimation strategy
Several authors argue that the above results are unreliable because they are derived using methods
that are not robust to weak instrument problems (see Canova and Sala (2009), Mavroeidis (2005) and
Nason and Smith (2008)). As we explain below, the weak instrument problem arises if marginal costs
have limited dynamics or if their coefﬁcients are close to zero. In other words, a ﬂat NKPC lacks the
necessary exogenous variation in inﬂation forecasts. More precisely, the weakness of the instruments
is characterized in linear instrumental variables (IV) regression models by a unitless measure known
as the “concentration parameter” (see, for example, Phillips (1983) and Rothenberg (1984)). Hence,
adding more instruments is a way to boost the concentration parameter. Where do you ﬁnd these new
instruments? If you already have exogenous instruments, it is possible to interact them, as Angrist
and Krueger (1991) do in estimating returns from schooling. It is also possible to take higher-order
powers of the same instruments, as in Dagenais and Dagenais (1997).
In macroeconomic models, the use of lag variables is usually a source of many instruments or moment
conditions. However, in ﬁnite samples, the inclusion of an excessive number of moments may be
harmful. To address this issue, we consider the TJIVE developed by Hansen and Kozbur (2014),
because of its useful properties relative to other existing IV competing estimators in the presence of
many (possibly weak) instruments. We use this alternative estimation method for two reasons. First,
the TJIVE does not depend on the number of instruments used in the reduced equation. Second, this
leading regularized estimator performs very well (i.e. is nearly median unbiased) even in the case of
relatively weak instruments (see Carrasco and Doukali (2017) and Hansen and Kozbur (2014)).
We now present our estimation procedure. We note that, unlike the standard GMM estimator, the
number of instruments is not restricted, and instruments are allowed to be weak.
8
We can rewrite Equation 3.1 as :
yt = Xt δ + εt (3.2)
Xt = ϒt + ut (3.3)
where t = 1, . . . , T. The vector of interest is δ = [λ , γ f , γb ]. Xt = [mct , πt +1 , πt −1 ] is the vector of the
exogenous variables. yt is the inﬂation rate. The vector ϒt is the optimal instrument. The estimation
will be based on a sequence of instruments Zt = Z (τ ; νt ), where νt is a vector of exogenous variables
and τ is an index taking countable values.
First, we recall the expression of the classical Jackknife estimator (JIVE):
ˆ X )−1 (ϒ
ˆ = (ϒ
δ ˆ Y) (3.4)
n n
ˆ t Xt )−1 ∑ ϒ
= (∑ ϒ ˆ t yt (3.5)
t =1 i=t
ˆ t , is deﬁned as ϒ
The leave-one-out estimator, ϒ ˆ t = Zt µ ˆ −t = (Z Z − Zt Zt )−1 (Z X − Zt Xt )
ˆ −t , where µ
is the ordinary least squares (OLS) coefﬁcient from running a regression of X on Z using all but the
t th observation.
The JIVE can alternatively be written as:
n n
∑ ˆ −t Zt Xt )−1 ∑ µ
ˆ =( µ
δ ˆ −t Zt yt (3.6)
t =1 i=1
with
ˆ −t Zt = (X Z (Z Z )−1 Zt − Ptt Xt )/(1 − Ptt ) = ∑n
µ s=t Pts Xs /(1 − Ptt )
where P is an n × n projection matrix deﬁned as P = Z (Z Z )−1 Z , and Pts denotes the (t,s)th element
of P.
Then, the JIVE estimator is given by:
n
ˆ −1 ∑ Xt Pts (1 − Pss )−1 ys ,
ˆ =H
δ
t =s
9
ˆ = ∑n Xt Pts (1 − Pss )−1 Xs , and ∑t =s denotes the double sum ∑t ∑s=t . When the number of
where H t =s
the instruments is large, the inverse of Z’Z needs to be regularized because it is singular or nearly
singular.
ˆ , is:
The expression of the TJIVE, δ
n
δ ˆ −1 ∑ Xi Pts
ˆ =H α α −1
(1 − Pss ) ys , (3.7)
t =s
n
ˆ = ∑ Xt Pst
α α −1
H (1 − Pss ) Xs (3.8)
t =s
where Pα is an n × n matrix after regularization deﬁned as:
Pα = Z (Z Z + α I )−1 Z ,
α denotes the (t , s)th element of Pα . The TJIVE depends on a regularization term, α . We select
and Pts
α that minimizes the mean squared error (MSE) as in Carrasco and Doukali (2017).
In section 5, we present estimates of the reduced-form parameters for the hybrid and forward-looking
versions using classical GMM estimators and TJIVEs. First, we use the four moment conditions set,
before considering our proposed set of moment conditions by increasing the number of moments in a
robustness analysis. We consider the following speciﬁcation:
Et [(πt − λ mct − γ f πt +1 − γb πt −1 )zt ] = 0
As noted earlier, one important issue is the weakness of instruments used to estimate the NKPC. For
instance, GMM estimates are unreliable because they are not robust to the weak instrument problem.
To strengthen the instruments, we increase the number of instruments by allowing more lag variables.
We use the set of instruments {πt −i }k =10 k=10
i=1 and {mct −i }i=1 , plus a constant term. According to Hansen
et al. (2008, 403), the concentration parameter is a better indication of the potential weak instrument
problem than the F-statistic. Since the increase of the number of instruments improves efﬁciency and
the regularized Jackknife corrects for the bias due to the many instruments problem, we expect to
obtain reliable point estimates.
10
4 Test statistic
Another important issue in our estimation strategy is testing overidentifying restrictions. Many
empirical studies are used to implement the classical J test, which can be seen also as a speciﬁcation
test for the linear instrumental variables regression (see Hansen (1982)). If the model is correctly
speciﬁed, all the moment conditions (including the overidentifying restrictions) should be close to
zero. However, it was shown that the conventional J test for overidentifying restrictions performs
poorly when the moments condition is increased. To deal with this, we use Chao et al. (2014)’s test,
which is a new version of the J test that is robust to the many moments condition and heteroskedas-
ticity. Their test is based on subtracting out the diagonal terms in the numerator of the test statistic.
Their proposed test corrects rejection frequency as long as the number of moments increases. It is
also correct under homoskedasticity with a ﬁxed number of moments. Their test statistic takes the
form:
ε ˆ − ∑tn Ptt ε
ˆ Pε ˆt2
JCHNSW = √ +L (4.1)
Vˆ
with
ˆ (2) P(2)ε
ε ˆ (2) − ∑tn Ptt
2εˆt4 ∑tn=s ε
ˆt2 Pts
2ε 2
ˆs
ˆ=
V =
tr(P) L
ˆ, ε
ˆt = yt − Wt δ
where L is the number of moments conditions, P is the projection matrix, ε ˆ (2) =
ˆ1
(ε 2 , ...., ε
ˆn2 ), P(2) is the n-dimensional square matrix where component t , sth is P2 . Note that the nu-
ts
merator of the test statistic, ∑tn=s ε ˆs , is the numerator of the traditional Sargan test without the ob-
ˆs Pts ε
servation i. The denominator is a heteroskedastic-consistent estimator of the variance of ∑tn=s ε ˆs .
ˆt Pts ε
The test rejects the null hypothesis when JCHNSW is greater than the critical value of a chi-squared
distribution with L − p degrees of freedom.
5 Empirical results
As noted, a natural response to improve the estimation of the NKPC and deal with the identi-
ﬁcation problem is to add more variables to the instrument set, but there are limits to how many
11
instruments one can use. This is because of the so called “many instruments” problem, which biases
the GMM estimator. Andrews and Stock (2007) showed that, provided the number of instruments
is not too large relative to the sample size, the identiﬁcation-robust statistics remain size-correct in
the instrumental variables regression model with many weak instruments. But again, substantial size
distortion can arise in ﬁnite samples. To address this issue, we consider the TJIVE, which is robust to
the many instruments problem, and compare it to the GMM method.
In this section, we report the results for the forward-looking NKPC and the hybrid NKPC. We report
the estimators corresponding to the GMM approach and the TJIVE method. As a ﬁrst step, we use
a small number of instrument sets. Second, we increase the number of instruments from 4 to 20 by
allowing more lags variables in a robustness analysis. To do so, we conduct estimations with the
following sets of instruments: [1] two lags of inﬂation and two lags of the output gap (just-identiﬁed
case), [2] four lags of inﬂation and four lags of the output gap, and [3] 10 lags of inﬂation and 10 lags
of the output gap. We examine how informative additional lags of inﬂation and the output gap are in
the NKPC. Finally, as previously mentioned, the TJIVE considered in this paper depends on a tuning
parameter, α , which needs to be selected. We use a data-driven method to select the regularization
parameter α ; by minimizing an estimator of the approximate MSE (see formula (5.3) of Carrasco and
Doukali (2017)). Estimation results are reported in Tables 1 and 2.
The point estimates we obtain are comparable to those found in many other studies that use a GMM
approach when the instruments set is small. Also, the results are similar for the both the forward-
looking and hybrid NKPC in the just-identiﬁed case (small number of instruments). Nevertheless,
many authors have argued that the above results are unreliable because they are derived using meth-
ods such as GMM that are not robust to identiﬁcation problems. These point estimates (when L is
small) are biased and not informative, suggesting that the parameters of the curve are indeed not well-
identiﬁed.
To circumvent the difﬁculties associated with weak instruments, we increase the number of the instru-
ments to boost the concentration parameter, which is a measure of the strength of the instruments or
moment conditions. However, an excessive number of moments may induce a bias. The small-sample
bias is addressed by using the TJIVE, which is robust to identiﬁcation problems.
Interestingly, the results based on the TJIVE are more encouraging for the hybrid NKPC when we
12
increase the number of instruments. In fact, in accounting for inﬂation dynamics, the forward-looking
component is larger than the backward-looking component. In effect, the reduced-form coefﬁcients
γ f and γb are signiﬁcantly different from zero. We also note that the quantitative importance of the
backward-looking component for inﬂation dynamics is not negligible, even if the forward-looking
component remains dominant in the dynamics of inﬂation. The coefﬁcient of the output gap is sta-
tistically signiﬁcant and has the correct sign. The empirical evidence is found when the number of
instruments is large for the hybrid NKPC.
The ﬁnal column in Tables 1 and 2 shows Chao et al. (2014)’s test of the overidentifying restric-
tions. We ﬁnd that the J statistics is larger than chi-square critical value, which means that the null
hypothesis is rejected for the forward-looking NKPC. However, the J test statistic is smaller than the
chi-squared critical values when L = 8 and L = 20 for the hybrid NKPC, so we can conclude that the
model is correctly speciﬁed.
First, our results show that the hybrid version of the NKPC is dominant. Second, the J test suggested
by Chao et al. (2014) rejects the forward-looking speciﬁcation of the NKPC curve and generally ac-
cepts the hybrid form. Third, our results are robust to the inclusion of additional lags of inﬂation and
the output gap. In effect, the TJIVE, which is robust to the number of instruments, allows us to reduce
the bias in the presence of many instruments, and ensures that all available instruments are used in an
efﬁcient way (without discarding any a priori), even if there are weak instruments.
13
Table 1: Forward-Looking NKPC
Method Instruments λ γf JCHNSW
GMM L=4 0.24 0.883 17.09
[0.20] [0.19]
L=8 0,312 0.913 23.28
[0.25] [0.31]
L = 20 0.37 0.950 36.21
[0.28] [0.32]
T JIV E L=4 0.23 0.845 16.61
[0.14] [0.16]
L=8 0.27 0.734 20.50
[0.12] [0.09]
L = 20 0.28 0.729 32.40
[0.05] [0.06]
The p-values appear in brackets for the null hypothesis that the estimate is equal to zero.
14
Table 2: Hybrid NKPC
Method Instruments λ γf γb JCHNSW
GMM L=4 0.174 0.403 0.350 13.40
[0.31] [0.10] [0.12]
L=8 0.350 0.813 0.444 17.15*
[0.23] [0.17] [0.15]
L = 20 0.380 0.983 0.521 19.32
[0.27] [0.16] [0.12]
T JIV E L=4 0.112 0.494 0.351 12.45
[0.08] [0.17] [0.12]
L=8 0.12 0.674 0.277 10.25*
[0.05] [0.03] [0.01]
L = 20 0.151 0.656 0.320 15.2*
[0.04] [0.002] [0.001]
The p-values appear in brackets for the null hypothesis that the estimate is equal to zero. * means the
model is correctly speciﬁed (at the 5% level).
15
6 Discussion and policy implications
In 2016, Bank Al-Maghrib (the Moroccan central bank) started to use a new forecasting and pol-
icy analysis system (FPAS) called the Moroccan Quarterly Projection Model (QPM). The goal of the
QPM is to improve monetary policy decisions to align Bank Al-Maghreb with central banking best
practices. Additionally, Morocco is expected to transition to a full-ﬂedged inﬂation targeting regime
in the near future. This will require monetary policy that stabilizes inﬂation around its target and
the output gap, as outlined in the Taylor rule. Under such a regime, forecasting inﬂation is crucial
for a central bank, as well as for the public, which tries to understand and react to the central bank’s
decisions. Modeling inﬂation is therefore a core task for inﬂation-targeting central banks.
While the literature is divided on the usefulness of the Phillips curve in forecasting inﬂation, sev-
eral papers ﬁnd that different forms of the Phillips curves (including New Keynesian versions, which
include indicators for real economic activity, past inﬂation and future inﬂation) forecast inﬂation
well. For instance, Stock and Watson (2008) compare Phillips curve inﬂation forecasts to several
multivariate speciﬁcations of forecasting models and ﬁnd that the Phillips curve performs well in
forecasting U.S. inﬂation. Recently, Gabrielyan (2018) investigates the forecasting ability of the
Phillips curve for headline inﬂation for Sweden, Canada and New Zealand, three countries that used
an inﬂation-targeting regime from 1983-2016. She ﬁnds that Phillips curve models can improve inﬂa-
tion forecasts against the random walk and autoregressive models’ benchmarks when the central bank
is explicitly targeting inﬂation. However, the results from earlier periods are not homogeneous across
various econometric speciﬁcations and different sample periods. The latter ﬁnding suggests that the
Phillips curve is more appropriate for inﬂation forecasting when a central bank has already adopted
an inﬂation-targeting regime.
An important implication is that Bank Al-Maghrib should consider the Phillips curve model, includ-
ing the New Keynesian version discussed in this paper, as a potential strong candidate for inﬂation
forecasting when it fully transitions to an inﬂation-targeting regime. However, we emphasize that this
model is not a pure forecasting device. Rather, it is a tool that helps to structure monetary policy dis-
cussion, identiﬁes the impact of key activities and inﬂation drivers (e.g. the output gap), and focuses
on a forward-looking perspective.
16
7 Conclusion
This paper discusses the identiﬁcation-robust many and possibly weak moments estimation of the
parameters of the New Keynesian Phillips curve, and applies this method to Moroccan data. The
diagnostic criteria for weak identiﬁcation, such as the concentration parameter, indicate that existing
econometrics techniques perform poorly. The approach in this paper addresses several important
econometrics issues. First, we use a regularization schema (TJIVE) to solve the problem of many
weak instruments when we include more variables in the moments set. The use of many moments
increases bias, but when regularization is introduced, the bias shrinks. Second, we implement a new
J test developed by Chao et al. (2014) to test the null hypothesis that the model is correctly speciﬁed.
While our focus is on obtaining consistent coefﬁcient estimates of the New Keynesian Phillips curve
and testing the hypothesis that the model is well speciﬁed, our model provides a solid framework for
developing near-term inﬂation projections.
Nevertheless, our analysis could be improved. Because of data limitations, we are forced to take the
output gap as a proxy for the real marginal cost. An important extension would be to develop efﬁcient
algorithms for simulating the Bayesian posterior to correct for the proxy of the real marginal cost.
17
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21
Appendix
A. Data sources
All data series are quarterly, beginning in 1998:Q1 and ending in 2016:Q4. Data on gross domestic
product and inﬂation are from national statistical ofﬁce.
The output gap is the deviation of real GDP from its steady state. We give a measure of the output
gap using the Hodrick-Prescott ﬁlter. In this measure, the output gap is a combination of lags, leads,
and contemporaneous values of output.
B. Presentation of the Tikhonov regularization
We consider the general case where the estimation is based on a sequence of instruments Zi =
Z (τ ; νi ), with τ ∈ N . Assume that τ lies in a space Ξ (Ξ = {1, .., L} or Ξ = N) and let π be a positive
measure on Ξ. Let K be the covariance operator for instruments from L2 (π ) to L2 (π ) such that:
L
(Kg)(τ ) = ∑ E (Z (τ , νi)Z (τl , νi))g(τl )π (τl ).
l =1
where L2 (π ) denotes the Hilbert space of square integrable functions with respect to π . K is supposed
to be a nuclear operator, which means that its trace is ﬁnite. Let λ j and ψ j , j = 1... be the eigenvalues
(ordered in decreasing order) and the orthogonal eigenfunctions of K, respectively. The operator can
be estimated by Kn deﬁned as:
Kn : L2 (π ) → L2 (π )
L
1 n
(Kn g)(τ ) = ∑ ∑ (Z (τ , νi)Z (τl , νi))g(τl )π (τl ).
l =1 n i=1
If the number of instruments L is large relative to n, inverting the operator K is considered as an
ill-posed problem, which means that the inverse is not continuous and its sample counterpart, Kn ,
is singular or nearly singular. To solve this problem, we need to stabilize the inverse of Kn using
regularization. A regularized inverse of an operator K is deﬁned as Rα : L2 (π ) → L2 (π ), such that
22
limα →0 Rα K ρ = ρ , ∀ρ ∈ L2 (π ), where α is the regularization parameter (see Kress (1999) and Car-
rasco, Florens, and Renault (2007)).
Tikhonov regularization
We consider the Tikhonov regularization scheme
−1
(K α )−1 = (K 2 + α I ) K .
∞ λj
(K α )−1 r = ∑ 2 r, ψ j ψ j .
j=1 λ j + α
where α > 0 and I is the identity operator. For the asymptotic efﬁciency, α has to go to zero at a
certain rate. The Tikhonov regularization is related to ridge regularization. The ridge method was
ﬁrst proposed in the presence of many regressors. The aim was to stabilize the inverse of XX by
replacing XX by XX + α I . However, this was done at the expense of bias relative to the OLS es-
timator. In the IV regression, the two-stage least squares estimator is already biased, and the use of
many instruments usually increases this bias. The implementation of the Tikhonov regularization and
the selection of an appropriate ridge parameter for the ﬁrst-step regression helps to reduce this bias.
α )−1 be the regularized inverse of K and Pα be an n × n matrix as deﬁned in Carrasco (2012)
Let (Kn n
α )−1 T ∗ , where T : L2 (π ) → Rn with
by Pα = T (Kn
T g = (< Z1 , g >, < Z2 , g > , ...., < Zn , g > )
and T ∗ : Rn → L2 (π ), with
T ∗v = 1 n
n ∑ j Z jv j
ˆ 2 ≥ ..... ≥
ˆ1 ≥ λ
such that Kn = T ∗ T and T T ∗ is an n × n matrix with typical element n . ˆj, λ
Let φ
0, j = 1, 2, ... be the orthonormalized eigenfunctions and eigenvalues of Kn , and ψ j be the eigenfunc-
tions of T T ∗ .
√ √
ˆj =
We then have T φ λ j ψ j and T ∗ ψ j = λ j φ
ˆ j . For v ∈ Rn , Pα v = ∑∞ 2
j q(α , λ j ) < v, ψ j > ψ j where
λj2
q(α , λ j2 ) = 2
λ j +α
.
23
Note that the case when α = 0 corresponds to no regularization. Thus, we have q(α , λ j2 ) = 1 and
P0 = Z (Z Z )−1 Z .
24