The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
Deep Habits in the New Keynesian
Phillips Curve
WP 11-08
Thomas A. Lubik
Federal Reserve Bank of Richmond
Wing Leong Teo
University of Nottingham
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Deep Habits in the New Keynesian Phillips Curve
Thomas A. Lubik
Federal Reserve Bank of Richmondy
Wing Leong Teo
University of Nottinghamz
November 2011
Working Paper No. 11-08
Abstract
We derive and estimate a New Keynesian Phillips curve (NKPC) in a model where
consumers are assumed to have deep habits. Habits are deep in the sense that they
apply to individual consumption goods instead of aggregate consumption. This alters
the NKPC in a fundamental manner as it introduces expected and contemporaneous
consumption growth as well as the expected marginal value of future demand as additional driving forces for in‡ation dynamics. We construct the driving process in the
deep habits NKPC by using the model’s optimality conditions to impute time series
for unobservable variables. The resulting series is considerably more volatile than unit
labor cost. General Methods of Moments (GMM) estimation of the NKPC shows an
improved …t and a much lower degree of indexation than in the standard NKPC. Our
analysis also reveals that the crucial parameters for the performance of the deep habit
NKPC are the habit parameter and the substitution elasticity between di¤erentiated
products. The results are broadly robust to alternative speci…cations.
JEL Classification: E31; E32.
Key Words: Phillips curve; GMM; marginal costs; deep habits.
The views expressed in this paper are those of the authors and should not be interpreted as those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.
y
Research Department, P.O. Box 27622, Richmond, VA 23261. Tel.: +1-804-697-8246. Email:
thomas.lubik@rich.frb.org.
z
School of Economics, University of Nottingham - Malaysia Campus, Jalan Broga, 43500 Semenyih,
Selangor, Malaysia. Tel.: +603-8924-8698. Email: WingLeong.Teo@nottingham.edu.my
1
1
Introduction
The New Keynesian Phillips curve (NKPC) is the centerpiece of modern macroeconomic
models used for monetary policy analysis. It is derived from the optimal price-setting
problem of a monopolistically competitive …rm that operates in an environment where …rms
face downward-sloping demand curves. The NKPC, in contrast to earlier accelerationist
Phillips curves, is explicitly forward-looking and imposes theoretical restrictions on the
comovement of its components. Speci…cally, theory identi…es marginal cost as the main
driver of in‡ation dynamics. However, the NKPC faced the early criticism that marginal
cost is not observable to the empirical researcher and that the stochastic properties of
various proxies do not line up with the properties of the in‡ation process they claim to
explain.
In a seminal contribution, Galí and Gertler (1999) show that the performance of the
NKPC can be improved by introducing backward-looking price-setting or indexation to the
e¤ect that lagged in‡ation enters the structural speci…cation. Moreover, they demonstrate
that marginal cost is well proxied by unit labor cost under the assumption of perfectly
competitive factor markets. Their results established a benchmark in the literature, namely
that in‡ation dynamics are explained both by intrinsic factors, such as in‡ation indexation
in price-setting, and by external driving forces, such as marginal cost movements. Much
of the follow-up research con…rmed their initial …ndings and established their robustness
under the chosen modeling environment (for instance, Galí et al., 2005).
This paper follows in the steps of more recent research that modi…es the environment
in which …rms operate. We introduce deep habits in the preferences of the consumer and
derive the corresponding NKPC. Habit formation is deep in the sense that it extends to
each individual good of the consumption bundle available to consumers, and not only to
the consumption composite. This seemingly simple modi…cation, however, has far-reaching
consequences. It implies a downward-sloping demand function that depends on the lagged
level of the consumer’s purchases. Since …rms take this demand function as a constraint
in their optimal price-setting problem, the time dependence carries over to the NKPC and
results in the introduction of future, current, and lagged consumption in this relationship.
We thus show that deep habits fundamentally a¤ect the analytical form and interpretation
of the driving process and the interaction of marginal cost with in‡ation.
We estimate the NKPC under deep habits using generalized methods of moments
(GMM) techniques. More speci…cally, our empirical approach is a mixture of calibration
2
and structural estimation. In our benchmark speci…cation, we combine the additional explanatory variables introduced by the deep habits environment with marginal cost into a
single driving process. We then impute this unobservable series using data on consumption
and real unit labor cost. The weights on the various elements are functions of the model’s
structural parameters, which we calibrate. This procedure allows us to compare the driving
process of the standard NKPC, namely marginal cost, with the one implied by deep habits.
We show that the latter is considerably more volatile than real unit labor cost. This observation is re‡ected in the NKPC estimates for the coe¢ cients on the driving process and
the weight on the intrinsic contribution to in‡ation dynamics, which is much smaller than
in the standard speci…cation. Moreover, the …t of the deep habits NKPC is much improved
over the standard NKPC per typical speci…cation measures in the GMM literature.
The representation of the driving process under deep habits involves expectations of
both observable variables, such as consumption, and unobservables, speci…cally the marginal
value of demand. In order to back out processes for these variables we pursue a parametric
approach in that we use the optimality conditions of the model to link the marginal value
of demand to observable consumption. In the deep habits environment, this involves an
expectational di¤erence equation which we solve forward to express the marginal value of
demand as the present value of future consumption growth and marginal cost. We then
pursue alternatively a univariate and a multivariate approach to produce forecasts for the
latter variables, which are used to construct a synthetic series for these unobservables. Our
approach is therefore of a partial equilibrium and limited information nature in that we do
not use all potential information available within the general equilibrium context of the full
theoretical model.
We also study an alternative representation of the deep habits NKPC that uses analytical representations for the unobserved and expectational terms in the driving process.
This yields a more reduced-form representation in terms of marginal cost and current consumption growth, which we can estimate directly. The coe¢ cients in this representation are
functions of the structural parameters. This representation then allows us to identify the
crucial elements for the improved performance of the deep habits model. Not surprisingly,
the size of the deep habits parameter is the central element. We show that performance
notably improves for values above 0:6. Our GMM-estimate of this parameter is 0:85, which
is identical to previous estimates in the literature (see Ravn et al., 2010). The second important element is the degree of substitutability of di¤erentiated products in the consumer’s
preferences, which is inversely related to the …rm’s markup. We estimate this to be consis-
3
tent with a markup of 74%, which may be considered high. A robustness analysis shows,
however, that the performance of the deep habits NKPC is still very satisfactory for more
typical markups of 10%.
The modeling environment in our paper draws from, and contributes to, an emerging literature on deep habits, starting with the original contribution by Ravn et al. (2006). Their
key insight was that deep habits impart additional internal propagation on a model, on top
of what an external habit formation would produce, while at the same time being arguably
more plausible. We extend this insight to the speci…cation of the NKPC by showing that
at a purely empirical level deep habits add additional regressors to the empirical in‡ation
equation. Moreover, Ravn et al. (2006) show that deep habits give rise to countercyclical
markups, which translates into procyclical marginal cost in the context of the NKPC. We
utilize this insight in the construction of a modi…ed driving process. Ravn et al. (2010)
build on their earlier paper and extend it to a New Keynesian framework. They estimate the
model using impulse-response function matching for identi…ed monetary policy shocks. Our
estimates of the key structural parameters using a GMM approach are virtually identical
to theirs, which suggests robustness of the derived insights.
Our paper also touches upon a host of other research on the NKPC that is concerned
with modifying the notion of the driving process. In the wake of Galí and Gertler (1999),
researchers at …rst assessed the robustness of their conclusions with respect to proxies for
marginal costs, the instrument set used in the estimation, and the empirical approach being
taken.1 Our paper follows the spirit of this research in that we do not modify or speci…cally
address the nature of marginal cost. We thus continue to proxy marginal cost with unit
labor cost, which is justi…ed under the assumption of competitive factor markets and a
standard neoclassical production structure. More recent contributions, however, deviate
from this assumption. The key insight is that modi…cations to the production structure
change the nature of the driving process in the NKPC either by introducing additional
elements, that is, regressors, or by altering the responsiveness of in‡ation to marginal cost
movements
For instance, this literature introduces search and matching frictions in the labor market
(e.g. Krause et al., 2008), or incorporates models with …nished goods inventories that
highlight the distinction between marginal costs of production and marginal costs of sales
(e.g. Lubik and Teo, 2011). In both approaches, modi…cation of the production side
does not improve the performance of the NKPC. The reason is that the driving process in
1
Galí et al. (2005) is an example of the former, Kurmann (2007) an example of the latter. Nason and
Smith (2008) give a comprehensive survey of the developments in this literature.
4
either speci…cation relies solely on present-value relationships of the cost of, respectively,
maintaining long-run employment relationships and of keeping an inventory. By their very
nature, present discounted values tend to smooth out the movements imparted by volatile
driving processes. In the two cases this virtually negates the e¤ect of additional regressors.
This e¤ect is present in our paper, too, but it is compensated by the presence of current
consumption growth, as we show in the reduced-form representation of the NKPC.
Guerrieri et al. (2010) provide a bridge between these di¤erent aspects of the literature.
In an otherwise standard model of a small open economy, they introduce both variable
demand elasticities on the preference side as well as a production structure that relies
on intermediate inputs from domestic and foreign sources. They derive an NKPC, whose
driving process is a function of marginal cost, relative prices and exogenous variations in the
aggregate elasticity of …nal products, and …nd that the estimated NKPC provides a better
…t than a standard speci…cation, mainly due to the addition of the relative price term. Our
paper similarly improves the …t of the NKPC via changes to the demand structure in the
form of persistence-inducing deep habits.
We now proceed as follows. In the next section, we present our theoretical model and
show how the introduction of deep habits modi…es the speci…cation of the NKPC. In Section
3, we discuss our empirical approach, the data being used, and the various speci…cations of
the deep habits NKPC that we estimate. Section 4 presents the GMM estimation results for
a standard NKPC and our deep habits model. In Section 5, we conduct various robustness
checks and consider alternative speci…cations. The …nal section concludes.
2
The Model
Our theoretical model is based on Ravn et al. (2010). It describes a New Keynesian monetary economy with utility-maximizing households and pro…t-maximizing …rms. Households
consume a bundle of consumption goods, each of which individually is subject to habit
formation. Firms are monopolistically competitive and hire labor from the households
as input in the production process. They set their prices subject to a downward-sloping
demand schedule, which is derived from household preferences, and they are subject to
quadratic costs of adjusting nominal prices. It is the latter element that gives rise to an
NKPC. In what follows, we describe the full model of the economy, but focus mainly on the
relationships that are needed to derive the NKPC.
5
2.1
Households
The household sector is described by the decisions of a representative household. Its intertemporal utility function is given by:
V 0 = E0
1
X
1
t
t=0
0<
xt1
1
< 1 is the discount factor, and
ht :
(1)
> 0 is the coe¢ cient of relative risk aversion. ht
is the amount of labor supplied by the household to the …rm sector. The core of the deep
habits formulation is captured by the term xt . It is a sub-utility function that depends on
a CES aggregate of consumption goods i, cit , relative to the previous period’s consumption
of goods i:
xt =
Z
1=(1 1=")
1
(cit
1
1)
cit
1="
di
:
(2)
0
" > 1 is the substitution elasticity between the di¤erentiated goods, while 0
< 1 is the
deep habits parameter. In contrast to habits at the aggregate level of consumption, deep
habits apply to each individual consumption good and are thus deeply embedded into the
utility function.
Solving for the household’s optimality conditions is slightly more involved than in the
standard case. We note that we can rewrite the household’s problem as an expenditure
minimization problem:
min Xt =
cit
Z
1
Pit cit di;
0
subject to the de…nition of the habit stock (2). The …rst-order condition implies the following
demand function:
cit =
Pit
Pt
Pt =
Z
where
0
1
"
xt + cit
1;
(3)
1=(1 ")
Pit1 " di
(4)
is the associated aggregate price index. The demand function has the typical feature that
it is downward-sloping in its own relative price Pit =Pt . In addition, deep habits render
consumption demand persistent through lagged consumption cit
1.
It is speci…cally this
feature which changes the nature of the driving process in the NKPC.
We can now describe the household’s intertemporal utility optimization problem. The
budget constraint of the representative household is given by:
Pt xt + {t + Bt = Rt
6
1 Bt 1
+ W t ht +
t;
(5)
where
{t =
Z
1
Pit cit
1 di
0
is the current consumption expenditure required to maintain the habit consumption level
from the previous period. Wt is the nominal wage payment for labor services ht .
t
is
the residual pro…t accruing to the household from its ownership of the …rms. Finally, we
assume that households have access to a risk-free one period nominal government bond Bt
that pays a gross interest rate of Rt .
The household maximizes the utility function (1) by choosing sequences of xt , ht , and
Bt . The resulting …rst-order conditions are:
xt
xt
=
=
Wt
;
Pt
R t Et
(6)
Pt
x
Pt+1 t+1
(7)
The …rst equation is the standard labor-leisure trade-o¤ a household faces, where the variable that determines the overall level of economic activity is the habit stock xt instead of
the usual consumption aggregate. Similarly, the second optimality condition is a standard
Euler-equation for intertemporal smoothing with xt instead of consumption. This relationship also de…nes a stochastic discount factor which …rms use to evaluate future pro…t
streams.
2.2
Firms
The …rm sector is composed of a continuum of monopolistically competitive …rms that face
a downward-sloping demand schedule for their product. Demand for …rm i0 s output is
given by equation (3), where we associate each good i with a speci…c …rm. Firms choose
an optimal price, but they are subject to quadratic costs of price adjustment. We assume
that …rms have access to a linear production technology that uses labor as its only input.2
Production is subject to aggregate productivity disturbances At . We write the production
function as yit = At hit . Each …rm hires labor input from the representative household for
a competitive wage Wt .
The …rm has to solve an intertemporal pro…t maximization problem:
"
#
1
2
X
Pit
max E0
qt Pit cit Wt hit
Pt ct ;
t
2 Pit 1
fPit ;cit ;hit g
(8)
t=0
2
The assumption of constant returns to scale is immaterial for the NKPC as it is independent of the curvature of the production function. Derivations and results for alternative production functions are available
from the authors upon request.
7
by choosing sequences of prices Pit , output cit , and labor input hit , subject to the demand
function (3) and the production function. The discount factor qt =
the household’s ownership of the …rm.
t
xt =Pt re‡ects
> 0 is the price adjustment cost parameter. We
assume that …rms only incur this cost when the chosen price path deviates from the weighted
in‡ation rate
t
1
=
(
t 1)
, where
is steady state in‡ation and 0
1 is the degree
of indexation in the targeted in‡ation rate.
The …rst order conditions of …rm i are given by:
Wt
=
Pt
c
t
Pt
Pit 1
Pit
Pit 1
t
ct +"
y
t
+
Pt
(cit
Pit
y
t At ;
(9)
Pit
qt+1 Pt+1 c
= Et
;
Pt
qt Pt t+1
qt+1 Pit+1
cit 1 ) ct cit = Et
qt
Pit
(10)
Pit+1
Pit
t+1
Pt+1
ct+1 :
Pit
(11)
The …rst condition equates the real wage to the marginal product of the worker, which is
simply productivity with linear production .
y
t
is the Lagrange-multiplier on the production
function. It can be interpreted as the real marginal cost. To see this, denote the real wage
wt = Wt =Pt . Total cost of production is wt hit = wt
y
t,
to yit , to get mct = wt =At =
can now be used to eliminate
yit
At
. Take the derivative with respect
where the last equality follows from (9). This expression
y
t
from the optimality condition (10), which becomes an
expectational di¤erence equation in
c
t,
with the driving variable being marginal cost.
The second …rst-order condition connects marginal cost
the demand function (3).
c
t
y
t
with
c
t,
the multiplier on
can be interpreted as the marginal value of demand. In the
absence of deep habits (when
= 0), it equals relative prices minus marginal cost. Under
deep habits, however, the persistence of demand for individual goods a¤ects …rms’demand
for labor intertemporally. Finally, the third …rst-order condition captures the optimal pricesetting problem of the …rm. We now derive the NKPC from this equation.
2.3
Deriving the NKPC
The …rst step is to impose a symmetric equilibrium. That is, we assume in line with the
literature that all …rms behave identically and are ex-post homogeneous. This amounts
to erasing the …rm-speci…c subscripts i, which simpli…es the above expressions considerably. We now de…ne the aggregate, consumption-based (gross) in‡ation rate
t
= Pt =Pt
1.
Substituting in the stochastic discount factor qt results in the following expression:
" ct xt +
t( t
t ) ct
= ct +
Et
8
xt+1
xt
t+1 ( t+1
t+1 ) ct+1 :
(12)
This is an expectational di¤erence equation in cross-products of in‡ation
t
and consump-
tion ct . We identify as driving forces the terms involving xt and the marginal value of
demand
c
t.
Our goal is now to re-write this expression in terms of marginal cost and
potentially other variables.
As is common in the literature, we consider a linearized version in terms of deviations
from the steady state. Denote the (log-) deviation of a variable zt from its steady state z
as zet = log zt
log z. We can now linearize (12) around its steady state, whereby we note
that the resulting relationship is independent of the steady-state in‡ation rate up to …rst
order, since …rms face price adjustment cost only to the extent that their prices deviate
from the aggregate price path
t.
The expression for marginal value of demand
c
t
can also
be linearized around its steady state. We substitute these into the linearized form of (12)
and collect terms.3
The NKPC in the model with deep habits is:
et =
Et et+1 +
1+
et
1+
1
ft
1 mc
+
where the coe¢ cients are given by:
1
=
"(1
) (1
(1+ )
)
;
2
=
It is straightforward to verify that
itive if " > (1
(1
) =(1
) = (1
1
);
(1+
2,
2 Et
3
4
e
ct+1
(1+
)1
(1+ )
=
and
3
3
;
ec
4 Et t+1 ;
e
ct
4
=
(1+
(13)
):
are strictly non-negative.
1
is pos-
). We impose this condition henceforth. The critical value
) increases monotonically with .4 For
2 (0; 0:99), the critical value is below 2 for
changes as
+
= 0, the critical value equals 1. For
= 0:99. Figure 1 shows how (1
varies between 0 and 0.99 with
) = (1
set at 0.99.
There are a few observations to make. First, in the absence of deep habits, when
it can be easily veri…ed that
2
=
3
=
)
4
= 0,
= 0. The relationship thus reduces to the
standard NKPC derived in Galí and Gertler (1999):
et =
1+
Et et+1 +
1+
et
1
+
" 1
mc
f t;
(1 + )
(14)
so that the speci…cation with deep habits cleanly nests the standard speci…cation. Second,
the introduction of deep habits a¤ects the conditional responsiveness of in‡ation to marginal
cost. It is straightforward to show that
1
<
" 1
(1+ ) ,
the standard NKPC-coe¢ cient.
Ceteris paribus in‡ation under the deep habits formulation is less reliant on marginal cost
as driving process.
3
4
Details of the derivation are given in the Appendix.
It is easy to show that the derivative of (1
) =(1
9
) with respect to
is positive.
The third observation is that deep habits add additional terms to the NKPC, above and
beyond marginal cost, to wit, expected and current consumption growth and the expected
marginal value of future demand. They stem from the fact that …rms have to consider
the e¤ect of current pricing decisions on future demand through its feedback via the persistence of demand. Contemporaneous consumption growth
e
ct engenders future consumption
growth via deep habits formation, which …rms encourage by lowering their prices. Higher
c
expected marginal value of future demand Et et+1 reduces current in‡ation since it creates
an incentive for …rms to lower their prices in order to capture future market share (Ravn et
al., 2010). In contrast, higher expected future consumption growth raises current in‡ation
as …rms do not have to lower their prices to generate an increase in future demand.
We …nd it useful for comparison with the standard NKPC to rewrite equation (13) in
a slightly di¤erent way by factoring out a common coe¢ cient and by grouping the relevant
terms together:
et =
1+
We refer to
Et et+1 +
=
coe¢ cients are ~ i
" 1
(1+ )
1
1+
et
1
+
h
~ mc
~
ct+1
1 f t + 2 Et e
~
3
e
ct
i
~ Et e c
4
t+1 : (15)
> 0 as the NKPC-coe¢ cient. The expressions for the responsei,
i = 1; :::; 4 which preserves the sign restrictions that we impose
on the original coe¢ cients.
Factoring out the coe¢ cient
Note that if
thus allows us to de…ne the driving process det as:
det = ~ 1 mc
f t + ~ 2 Et e
ct+1
~
3
e
ct
~ Et e c :
4
t+1
(16)
= 0, we have det = mc
f t , and the speci…cation reduces to the standard NKPC.
We treat (15) as our benchmark speci…cation for the NKPC under deep habits. Summarizing
the additional regressors in terms of a driving process allows us to compare it directly to
the driving process in the standard NKPC, namely marginal cost. The remainder of our
paper is concerned with computing this driving process. The key challenge is to determine
c
the behavior of the unobserved term Et e . Once we derive a series for det we can then
t+1
assess the performance of the NKPC under deep habits in a limited information setting.
3
In‡ation Dynamics and Deep Habits: A Limited Information Approach
We now proceed to a formal empirical analysis of the NKPC under deep habits. We pursue
a limited information approach in that we do not use all the information available in the full
10
general equilibrium model that embeds the NKPC. To be more precise, we do not impose
the cross-equation and cross-coe¢ cient restrictions on the comovement of the endogenous
variables that the full model would prescribe. We thus treat the NKPC as a moment
condition which we estimate with a GMM approach. We begin with a short description
of the data and our estimation method. We then describe how to deal with unobservable
variables in the formulation of the driving process by backing them out of intertemporal
optimality conditions. We apply two methods: First, our benchmark speci…cation, which
treats observable, but exogenous processes as separate univariate processes, and, second, a
VAR-based method.
3.1
Data and Empirical Approach
We extract quarterly data from the FRED database at the Federal Reserve Bank of St.
Louis. Our sample period ranges from 1955:1 to 2011:2, but we also consider a sub-sample
from 1984:1 onwards, which covers the Great Moderation during which the behavior of many
macroeconomic time series changed. Output and consumption are constructed by dividing
real GDP in chained dollars (GDPC96 in FRED) and real consumption in chained dollar
(PCECC96) by the civilian non-institutional population aged 16 and over (CNP16OV).
The GDP implicit price de‡ator is our measure of Pt (GDPDEF). Real unit labor cost is
constructed by dividing nominal unit labor cost of the nonfarm business sector (ULCNFB)
by the price de‡ator. We remove a linear trend from GDP and consumption. We also use
compensation per hour in the nonfarm business sector (COMPNFB) as a measure of the
nominal wage to construct wage in‡ation, which we then use as an instrument in the GMM
estimation.
Table 1 reports some moments of the data series. Over the full sample period, GDP is
more volatile than real unit labor cost, which has been used in the literature as a proxy
for marginal cost. Concentrating on the sub-sample from 1984 on, we …nd, however, that
the volatility of GDP drops, while that of the marginal cost proxy increases, to the e¤ect
that the latter becomes now more volatile than the former. In‡ation and marginal cost are
mildly positively correlated, both for the full sample and for the sub-sample. This pattern is
almost a requirement for the validity of the NKPC as the logic of the relationship suggests
that increases in marginal cost should drive up in‡ation. We also note that the correlation
pattern between GDP and real unit labor costs is negative for the full sample but quite
positive for the period of the Great Moderation. Finally, we report second moments for
consumption growth as it appears in the driving process of the deep habits speci…cation. It
11
comoves negatively with in‡ation over both the full sample and sub-sample. Since current
consumption growth a¤ects in‡ation negatively per equation (16), this suggests that deep
habits have a role to play in explaining in‡ation dynamics.
Let zt denote a vector of variables observed at time t. The NKPC then de…nes a
h
i
e
set of orthogonality conditions: Et et
e
e
d
zt = 0. Given these
t
f t+1
b t 1
conditions, we can estimate the model using a GMM approach. To aid comparison with the
recent literature, we use the same set of instruments as Galí et al. (2005). Speci…cally, we
use 4 lags of in‡ation and 2 lags of the regressors and wage in‡ation as instruments. The
weighting matrix is computed from the estimated heteroskedasticity- and autocorrelationadjusted (HAC) covariance matrix, where the number of lags in the HAC estimation is
chosen based on the criterion in Andrews (1991). We consider two empirical speci…cations:
…rst, a reduced-form version which estimates only the coe¢ cients in the moment conditions.
The focus here is on the relative importance of the backward and forward in‡ation terms,
and thus the degree of intrinsic price dynamics, and on the NKPC-coe¢ cient
, which
captures the strength of the transmission mechanism between the real and nominal side
and indicates the presumed degree of price stickiness. The second version attempts to
estimate the underlying structural parameters of the model embedded in the reduced-form
coe¢ cients.
3.2
Computing the Driving Process (I): Baseline
The baseline speci…cation we intend to estimate is:
et =
f Et e t+1
+
b et 1
+ det ;
(17)
where the driving process det is given by equation (16). The advantage of this speci…cation
is that the coe¢ cient estimates are immediately comparable to those from the standard
NKPC. The only di¤erence is that allowing for deep habits a¤ects the nature of the driving
process, which is no longer marginal costs alone, but a composite of marginal cost, expected
and current consumption and the expected marginal value of future demand. Once we have
constructed a times series for det , estimating this speci…cation is straightforward.
However, this approach presents a few challenges. First, marginal cost is unobservable
to the econometrician. We proxy mc
f t with real unit labor cost in line with most of the
NKPC literature. Second, while we can compute current consumption growth straight from
the data, expected consumption growth is unobservable. This can be obtained in several
ways. One possibility is to proxy expected consumption using survey data from sources
12
such as the Survey of Consumer Finances. This approach has numerous drawbacks, such as
the potential inconsistency of forecast horizons and forecast object between the model and
the survey respondents. Second, we can try to back out expected consumption from other
equilibrium conditions. However, as it turns out, all possible relationships involve unobservable variables, speci…cally, Lagrange multipliers that also would have to be proxied. We
therefore choose to specify a parametric model for consumption, which we use to compute
conditional expectations. In this section, we specify univariate processes for the observable
variables, while the subsequent section assumes a multi-variate relationship that allows for
richer interactions.
The third challenge to computing the driving process det is the presence of the term
c
Et e , which involves the conditional expectation of an unobservable quantity, namely the
t+1
marginal value of future demand. We follow the approach of Lubik and Teo (2011) and
use an intertemporal equilibrium condition to relate the unobservable expected marginal
value of future demand to observables. We …nd such a relationship in the …rm’s …rst-order
condition (10). Using the relationship mct =
y
t,
and linearizing (10) around the steady
state yields:5
ec =
t
c
Et et+1
Et
1
( e
ct+1
e
ct )
[" (1
)
)] mc
f t:
(1
(18)
c
This is an expectational di¤erence equation in et which can be solved forward.6 We …nd
that:
ec =
t
=
1
X
(
)j Et
(
)j Et
j=0
1
X
j=0
1
1
( e
ct+j+1
1
e
ct+j )
2
e
ct+j+1
2
[" (1
[" (1
)
)
(1
(1
)] mc
f t+j
)] mc
f t+j
e
ct :
(19)
1
The last equality follows from collecting terms in consumption growth. The marginal value
+
of demand can now be expressed as a function of observable
e
ct and as the present dis-
counted value of future consumption growth and marginal cost. Proxying the latter by real
c
unit labor cost, we can now back out et from a parametric model for the two series, e
ct
and mc
f t.
As a …rst pass, we assume that both variables follow AR(1)-processes:
5
6
mc
ft =
ft 1
mc mc
The derivation is shown in the Appendix.
Since 0 <
< 1, the equation has a unique solution.
13
+ "mc;t ;
(20)
where j
mc ;
cj
that Et mc
f t+j =
e
ct =
< 1 and "mc;t and "
j
f t,
mc mc
c;t
e
ct
c
1
+"
c;t ;
(21)
are i:i:d: random variables with zero mean. Noting
we can substitute this into the above expression and solve out the
in…nite discounted sum. This results in the following expression:
ec =
t
2
1
1
1
c
1
2
1
c
e
ct
" (1
)
1
(1
)
mc
mc
f t;
(22)
which is a weighted average of consumption growth and marginal cost. Since the latter
can be proxied by unit labor cost, this expression allows us to impute a time series for the
c
unobservable et =
ct
f t.
c e
mc mc
The conditional one period-ahead forecast can then be computed by iterating forward
one more time:
c
Et et+1 =
c
c
e
ct
f t:
mc mc mc
(23)
The sign of the coe¢ cients depends on the size of the deep habits parameter.
since we impose " > (1
if
>
mc
is positive
) = (1
). Furthermore, it is easy to show that
c > 0
e
c . Whether movements in Et t+1 reinforce or dampen movements of the other
variables in the driving process thus depends on the size of the habit parameter and whether
consumption growth and marginal cost comove positively or negatively.7 Before we can
make further empirical progress, however, our …nal step assigns numerical values to the
structural parameters. In our benchmark exercise, we calibrate all parameters required
to impute the unobservable series since we focus on the impact of changes in the driving
process only. In a robustness exercise below, we show how to use the previous expression
to compute a reduced-form representation of the NKPC, which allows us to estimate some
of these parameters.
The calibrated parameter values are detailed in Table 2. We base our calibration on the
estimates in Ravn et al. (2010), which is to the best of our knowledge the …rst empirical
study of a deep habits model. We …x the discount factor
= 0:99 to be consistent with
an annual real interest rate of 4%. We set the coe¢ cient of relative risk aversion to
which implies log utility. As an alternative, we consider a value of
= 1,
= 3, which implies
much more risk averse households. We follow again Ravn et al. (2010) in choosing the
habit parameter
= 0:85. The substitution elasticity between di¤erentiated products ",
which can be interpreted as a demand elasticity is set at " = 2:48, based on the empirical
estimates in Ravn et al. (2010). In standard models without deep habits this parameter is
Recall that the coe¢ cient on the expected marginal value of future demand in (15), ~ 4 , is strictly
positive.
7
14
usually …xed at " = 11 to imply a markup of 10% over marginal cost. In our model with
deep habits, the steady state markup is given by
"(1
"(1 )
) (1
8
).
Figure 2 plots the steady state markup as " varies for the standard model without deep
habits and our deep habits model.
and
are …xed at 0.99 and 0.85, respectively. The
markup is slightly higher for a given value of " in our deep habits model compared to the
standard model without deep habits. In our deep habits model, " = 11 implies a steady
state markup of 10.6% instead of 10%. The di¤erence arises because for a given value of
" the presence of deep habits makes demand less elastic, giving …rms incentive to charge a
higher markup. Our benchmark " = 2:48 imposes a steady state markup of 74:2%, which
may seem excessive. We discuss this assumption further in the robustness section, where we
also investigate alternative values. Finally, we …t separate AR(1)-processes to consumption
growth and real unit labor costs. This results in estimates of
which satis…es the restriction
>
c
for the coe¢ cient
c
mc
= 0:98 and
c
= 0:31,
to be positive.
Figure 3 depicts the constructed driving process and marginal cost. The former has a
standard deviation of 5.68%, which is higher than marginal cost. The correlation of the two
series is 0:57. This con…rms that the introduction of the additional elements into the NKPC
via deep habits renders the driving process more volatile. This is also re‡ected in the less
than perfect comovement, since some elements of det enter the driving process with negative
signs, as the previous discussion has shown. Nevertheless, what we cannot distinguish at
this stage is whether the changed properties of the driving process are simply due to the
increased number of regressors or to the changed responsiveness of the coe¢ cients. We
attempt to disentangle this further below.
3.3
Computing the Driving Process (II): A VAR Approach
In the previous section we used independent AR(1) processes for marginal cost and conc
sumption growth as predictors for the behavior of Et et+1 . In order to capture potential
additional information in the data, we alternatively pursue a VAR-based approach. Con-
sider a generic data vector vt , which contains consumption growth, marginal cost and other
variables that we judge useful for forecasting. Assume that vt is described by a VAR:
vt = Avt
1
+ t .9 The conditional forecast is then given by Et vt+j = Aj vt , j
note the extraction vector for some element at of the vector vt as
Et ( e
ct+j ) =
8
cA
jv
t
and Et (mc
f t+j ) =
mc A
jv .
t
a,
1,8t. De-
so that, for instance,
Estimating the VAR and the coe¢ cient
See the Appendix for a derivation.
The …rst-order speci…cation is without loss of generality since any higher-order VAR can be written in
…rst-order companion form. We discuss this speci…cation for expositional expediency.
9
15
b therefore allows us to construct a time series for conditional expectations of the
matrix A
variables of interest.
We can use these expressions in the equation for the expected marginal value of future
demand. Following the same steps as above, this yields:
c
Et et+1 =
1
1
2
+
1
b
c Avt :
2
b2
cA
[" (1
)
(1
)]
b
mc A
h
I
i
b
A
1
vt
(24)
Given this expression, observed consumption growth e
ct , expected consumption growth
b t , and our proxy for marginal cost mc
f t , we can now construct an imEt e
ct+1 = c Av
e
puted time series for the driving process dt from equation (16). As before, we impose our
benchmark calibration, that is,
= 0:85, " = 2:48, and
= 1. We estimate a VAR(4) in
consumption growth, real unit labor cost and output growth to construct expectations.
The constructed driving process from the deep habits speci…cation is depicted in Figure
4, together with the marginal cost proxy. The standard deviation of deV AR is 4:82%, whereas
t
that of real unit labor cost is 3:44%. The correlation of the two series is 0:45. Both numbers
are lower than the corresponding values from the baseline speci…cation with independent
AR(1) processes for expected marginal cost and consumption growth.10 Nevertheless, the
imputed driving process exhibits substantial volatility. Figure 5 depicts the imputed series
c
for Et et+1 against marginal cost. The contemporaneous correlation of both series is 0:94,
while the standard deviation of the expected marginal value of future demand term is
2:27%. Since the latter term enters the driving process (16) with a negative sign, and its
own coe¢ cient is positive, the negative correlation thus imparts positive comovement with
marginal cost and reinforces its contribution to the driving process.
4
Estimating the NKPC
We now provide formal estimates of the NKPC using a GMM approach. Our benchmark
speci…cation relies on the use of constructed driving processes. We …rst estimate a standard
NKPC, where we use real unit labor cost as a proxy for marginal cost. We then estimate the
corresponding NKPC, where the driving process is imputed from the …rst-order conditions
of a deep habits model, using the two methods described in Section 3.
10
This is reminiscent of the …nding in Lubik and Teo (2011), where the use of a VAR-based imputation
process tends to smooth out the present discounted value much more than simple univariate processes.
16
4.1
The Standard NKPC
In order to provide a benchmark for our deep habits speci…cation, we …rst estimate both
unrestricted and restricted versions of the standard NKPC with a proxy for marginal cost
as in the original model of Galí and Gertler (1999). Speci…cally, we estimate the following
standard NKPC speci…cation:
et =
f Et e t+1
b et 1
+
+ mc
f t:
(25)
The GMM-estimation results for the standard NKPC are reported in Table 3. The estimates
are quite similar to those found in the literature and statistically signi…cant throughout.
In the fully unrestricted speci…cation, the coe¢ cient
the coe¢ cient
b
f
on expected in‡ation is 0:79 while
on lagged in‡ation is 1=5, which is consistent with the …ndings of Galí
and Gertler (1999) and subsequent work. The coe¢ cient on marginal cost
= 0:004. The
J-test for overidentifying restrictions does not reject the speci…cation, as evidenced by a
high p-value. When
of
b
is restricted to zero,
is estimated to be 0:989, while the estimate
f
increases by 50 percent to 0:006. At the same time, the p-value for the J-test increases,
which suggests that the speci…cation without indexation is preferred.
Next, we estimate the structural parameters of the NKPC. From (14), the coe¢ cient on
expected in‡ation
f
= =(1 +
Note that when
= 0 the speci…cation reduces to the purely forward looking NKPC.
The slope coe¢ cient
= ("
), while the coe¢ cient on past in‡ation
1)=[ (1 +
)]. We impose
b
= =(1 +
).
= 0:99 on the estimation,
which is consistent with the implied value from the restricted NKPC estimation. We also
note the parameters in the coe¢ cient
0
= ("
1)= are not separately identi…able in this
speci…cation as the coe¢ cient simply scales the marginal cost term and appears nowhere
0
else. We therefore only report estimates for
and the indexation parameter . The results
are in the last line of Table 3. We note that the high p-value of the J-statistic suggests that
the cross-coe¢ cient restrictions are informative in the estimation. The estimate
corresponds to an implied backward-coe¢ cient
b
= 0:27
= 0:21, which is consistent with the
reduced-form estimate.
4.2
The NKPC with Deep Habits
We now estimate the NKPC speci…cation with deep habits:
et =
f Et e t+1
+
b et 1
+ det ;
(26)
where the driving process det is either imputed using independent AR-processes for the observables or from a VAR-based approach. We note again that the speci…cation of the NKPC
17
is such that we only vary the term det . Estimates for the forward-looking coe¢ cient
backward-looking coe¢ cient
b
and the NKPC-coe¢ cient
f,
the
thus allow us to make direct
comparisons between the models. We also report results from the structural speci…cation:
et =
1+
Et et+1 +
1+
et
1
+
" 1 e
dt :
(1 + )
(27)
Not all parameters in this speci…cation are identi…able, however. We therefore focus on the
indexation parameter
and the price adjustment cost parameter , and …x the remaining
parameters. Speci…cally, we set
= 0:99 and " = 2:48, following the estimates reported
in Ravn et al. (2010). The calibrated parameter values that go into the imputed driving
process det are as reported in the previous section. We will consider alternative calibrations
in our robustness analysis.
Table 4 contains the GMM estimates for the deep habits NKPC when the marginal
cost and consumption growth processes are assumed to be independent AR(1)-processes.
Compared to the standard NKPC, two observations stand out. First, the degree of indexation and thus the weight on the lagged in‡ation term is much lower for the deep habits
speci…cation. In the unrestricted version,
NKPC. Restricting
b
b
= 0:10 is barely half as big as in the standard
to zero results in an estimate for
implied value for the discount factor
f
= 0:984, which is identical to the
= 0:99 in the restricted speci…cation. Second, the
p-value for the J-test rises substantially from the unrestricted to the restricted speci…cation.
The respective p-values are also much higher than the corresponding values for the standard
NKPC. When interpreted as a speci…cation test, this suggests that the deep habits NKPC
captures in‡ation dynamics exceedingly well with only a minor degree of intrinsic in‡ation
persistence. This is also re‡ected in the structural estimates. The fraction of price-indexing
…rms is estimated at a highly signi…cant
coe¢ cient of
b
= 0:11, which translates into a backward-looking
= 0:099. Moreover the p-value of the J-statistic is almost one, which sug-
gests an excellent …t, based on the information content in the cross-coe¢ cient restrictions.
Finally, the estimated NKPC-coe¢ cients
are statistically signi…cant and almost twice as
large as those for the standard NKPC.
We now vary the speci…cation for the driving process and use deVt AR which has been
constructed from the VAR-based forecasts. The estimation results are reported in Table 5.
Conceptually, the estimates do not di¤er from those based on AR-forecasts. In fact, the …t
of the model is improved in the case of the unrestricted speci…cation where the coe¢ cient
on lagged in‡ation comes in at a statistically insigni…cant
b
= 0:034. The estimates for the
restricted and structural speci…cation are essentially unchanged from before. This suggests
18
that the performance of the deep habits NKPC does not rest solely on the speci…cation of the
forecasting model for variables that are extraneous to the in‡ation dynamics equation.11
The more important aspect is the fact that introducing deep habits imparts additional
regressors into the driving process. We will take up this issue again in the robustness
section.
The conclusion we can draw from our benchmark analysis is straightforward. Deep
habits dramatically improve the performance of the NKPC in describing in‡ation dynamics.
We demonstrated in the previous section that the implied driving process generated from our
model is more volatile than marginal cost. We …nd that this improves the …t of the model in
terms of a standard J-type speci…cation test, but it also much reduces, even negates, the role
of indexation in price-setting for explaining in‡ation dynamics. This device was introduced
by Galí and Gertler (1999) in order to better capture in‡ation persistence through an
intrinsic, that is, built-in, mechanism. We show that this role is played by the process for
consumption growth in the driving process above and beyond indexation in price-setting.
5
Robustness
We assess the robustness of our conclusion in three directions. First, we consider the role
that calibration plays in generating the desired stochastic properties of the driving process.
In the second exercise, we look at an alternative speci…cation for the driving process. Instead
of …rst imputing a time series for the driving process, which is then used as a single regressor,
we use the theoretical model restrictions to derive an alternative representation for a second
set of observable driving forces. This allows us to decompose the e¤ects on the NKPC into
two underlying forces, namely marginal cost and consumption growth. Speci…cally, we use
the imputed representations for expected consumption growth and the expected marginal
value of future demand, substitute them into the driving process, and thus generate a
reduced-form in marginal cost and consumption growth. We then use these as independent
regressors in the NKPC. Finally, we also look at the performance of the model for a subsample of our full data set that considers only the period of the Great Moderation from
1984 on.
11
This stands in contrast to the results in Lubik and Teo (2011), where the speci…cation of the forecasting
model matters.
19
5.1
Alternative Calibration
Our …rst robustness check simply looks at the implications of di¤erent parameterizations on
the behavior of the driving process. We focus on three parameters, namely the coe¢ cient
of relative risk aversion
, the demand elasticity ", and the habit parameter . Table 6
contains the GMM estimation results for the imputed driving processes under di¤erent
calibrations. We report only estimates for the unrestricted speci…cation where we estimate
reduced-form coe¢ cients and where the driving processes are constructed using AR-based
forecasts. Estimation results for the restricted version, the structural NKPC which estimates
the model’s parameters and the VAR-based driving process o¤er overall consistent results.
The …rst experiment documents the sensitivity of the model to the size of the habit
parameter. When
= 0:15, the NKPC estimates are in between those of the benchmark
calibration in Table 4 and the standard NKPC in Table 3. The estimates of
f
and
are larger than those of the standard NKPC but smaller than those of the benchmark
calibration of
= 0:85. The reverse is true for the estimate of
b.
Moreover, the standard
errors are wider, and the J-test statistic has a lower p-value compared to the results of the
benchmark calibration. Going to the other end of the parameter range, when
estimate of
f
= 0:95, the
is somewhat larger than that of the benchmark calibration but the J-test
statistic has a lower p-value. We also experimented with intermediate values of
that the performance of the deep habits NKPC improves notably for values of
and …nd
above 0:6,
the intuition of which we discussed in Section 2. The highest p-value is in fact attained for
our benchmark calibration.
The second experiment varies , but keeps other parameters at their benchmark values.
For
= 3, when agents are more risk-averse, the e¤ect is to increase the weight on the
forward-looking coe¢ cient. However, the J-test statistic has a lower p-value compared to
the benchmark calibration. Finally, we also consider increasing the demand elasticity " to
11, which is the value most commonly used in the calibration literature implying a markup
of 10 percent in standard model without deep habits. In this case, the estimates of
and
f
are slightly smaller and the …t of the model worsens compared to the case of the
benchmark calibration. Nonetheless, even with " = 11, the J-test statistic for the deep
habits models still has a higher p-value than in the case of the standard NKPC. When
considering joint variations of the parameters, the strongest role is played, unsurprisingly,
by the habit parameter, while small values of " improve performance. This suggest that the
benchmark calibration, which has been chosen based on the empirical estimates in Ravn et
al. (2010), does provide good estimates of the underlying parameters.
20
5.2
An Alternative Reduced-Form Speci…cation
Our benchmark speci…cation relies on an imputed series for the driving process, which we
treat as a single regressor. However, the NKPC speci…cation in equation (13) highlights the
fact that the introduction of deep habits changes the standard NKPC in two fundamental
ways. First, it a¤ects the responsiveness of in‡ation to marginal cost as captured by the
coe¢ cient
1,
which is di¤erent from the standard NKPC coe¢ cient. Second, it adds
additional regressors to the in‡ation equation, namely current and expected consumption
growth and the expected marginal value of future demand.
Consider the structural representation for the NKPC from equation (13), which we
reproduce here for convenience:
et =
1+
Et et+1 +
1+
et
+
1
ft
1 mc
+
2 Et
e
ct+1
3
c
ec
4 Et t+1 ;
e
ct
(28)
What prevents direct estimation of this relationship is that Et et+1 is not observable to
the econometrician.12 We can, however, use the tools developed in Section 3 to provide
analytical expressions for these two components of the driving process. For purposes of
exposition, we focus on the univariate representation for the driving forces. These expressions depend on marginal costs and current consumption growth. Substituting them into
the NKPC results in a reduced-form speci…cation that only depends on observables. The
exact derivations can be found in the Appendix.
We thus have:
where we treat
mc
et =
and
c
f Et e t+1
+
b et 1
+
ft
mc mc
c
e
ct ;
(29)
as reduced-form coe¢ cients. We also consider a representa-
tion that factors out the standard NKPC-coe¢ cient and thus imposes the cross-coe¢ cient
restrictions implied by theory:
12
et =
1+
Et et+1 +
1+
et
1+
" 1 h~
mc
ft
(1 + ) 5
~
6
i
e
ct :
(30)
As we discussed before, there are alternatives to our approach, one of which involves combining future
in‡ation and consumption treating them as joint elements in the moment condition. This raises issues
of normalization in the estimation, which is well known to be problematic in empirical NKPC models.
Moreover, it does not solve the problem with fundamentally unobservable marginal value of demand, for
which we would have to use a parametric model in any case. We therefore chose to be fully parsimonious in
that we treat all non-in‡ation variables in the NKPC as pure elements of the driving process. A comparison
of these additional alternative approaches would be a worthwhile exercise.
21
The coe¢ cients on mc
f t and
~
~
5
6
e
ct are, respectively, given by:
=
=
" (1
)
"
1
"
11
(1
1
1
)
1+
1
(
;
mc
c)
1
:
c
We also show in the Appendix that both coe¢ cients are positive for plausible calibrations.
Since marginal cost and current consumption growth are negatively correlated, this implies
that the latter reinforces the impact of marginal cost on in‡ation. Both speci…cations show
that the main e¤ect of deep habits is via introducing an additional regressor in the NKPC,
namely consumption growth.
We can now estimate (29) without further modi…cations. For a preliminary assessment,
Figure 6 depicts the marginal cost series and consumption growth. Clearly, the latter is
less volatile (see also Table 1). Any reinforcement of marginal cost on in‡ation dynamics
would therefore have to be generated by the relative size of the coe¢ cient
c.
This is, in
fact, borne out by the estimation results in Table 7. The consumption growth coe¢ cient in
the unrestricted and the restricted speci…cations (where
larger than
mc ,
b
= 0) is an order of magnitude
which in turn is in line with the estimates from the standard NKPC. This
con…rms that capturing in‡ation dynamics is thus a simple matter of adding the correct
additional regressor.13 However, the …t of the model as measured by the J-test declines
relative to both the standard NKPC as well as our benchmark speci…cation.
We now turn to estimating the alternative speci…cation (30), which imposes the crosscoe¢ cient restrictions. We …x
benchmark calibration that sets
and ", and estimate
= 0:99,
and . As before, we consider a
= 1 and " = 2:48, following the estimates
reported in Ravn et al. (2010). We report the estimation results in Table 7, where we also
consider variations in key parameters.
The estimates for the indexation parameter are identical across the three speci…cations
where we vary the …xed parameters. A value of
= 0:198 corresponds to
b
= 0:166, and is
thus consistent with the reduced-form estimate. There are minor di¤erences for the other
parameters in the two speci…cations that set " = 2:48. In that case, the value of the deep
habits parameter
is close to 0:85, which is the value estimated by Ravn et al. (2010)
with a di¤erent empirical methodology and within the context of a fully-speci…ed general
equilibrium model. For the speci…cation with
13
= 3 and " = 11, we note that the habit
That this need not be the case is demonstrated by empirical studies of the NKPC that modify factor
inputs (e.g. by introducing labor market search and matching frictions as in Krause et al., 2008) or change
the structure of product markets (e.g. by introducing …nished goods inventories as in Lubik and Teo, 2011).
22
parameter is close to one and that the price adjustment cost parameter
a very high value.14 The latter estimate re‡ects the fact that " and
is estimated at
are not separately
identi…able. The higher calibrated value of the elasticity parameter " then translates into
higher implied adjustment cost in order to generate the same implied NKPC-coe¢ cient.
At the same time, the increase in
is related to the increase in ", which also suggest
identi…cation issues regarding these two preference parameters.
We investigate this issue further by looking at how the driving process changes with
respect to parameters. Figure 7 plots the marginal cost coe¢ cient ~ 5 as well as the relative
weight on consumption growth ~ = ~ against the habit parameter over the range [0; 1].
6
5
The graphs are conditional on " = 2:48. Analytically, the coe¢ cient on marginal cost is not
a¤ected by , which only enters the weight on consumption growth (see equation 30). The
bottom graph therefore contains two lines for di¤erent values of
the weight on marginal cost decreases. For
obtains. ~ is fairly inelastic to changes in
5
= 1; 3. As
increases,
= 0, the weight is one and the standard NKPC
until about 0.6, after which it declines rapidly.
Nevertheless, the weight on marginal cost drops below 0:5 only for extremely high values
of the habit parameter. The relative weight on consumption growth is the ‡ip-side of this.
For > 0:6, ~ = ~ exceeds 1 and increases exponentially afterwards. For values larger
6
5
than 0.9, the weight on consumption is 10 times that on marginal cost. Higher values of
increase the relative weight on consumption growth, but the two curves are close enough to
not make this the dominant e¤ect.
The previous graphs were plotted conditional on " = 2:48. We now study how the
response coe¢ cients change with variations in that parameter for a given . The respective
graphs are in Figure 8. We …x
= 0:85, and also plot variations to . As before, the weight
on marginal cost is not a¤ected by . As " rises over the range [2; 11], the weight on marginal
cost increases only slightly. At the same time, the relative weight on consumption growth
decreases by an order of 10 over the range. A higher value of
increases the relative weight
on consumption growth as in Figure 7. Note that the relative weights on consumption
growth are all larger than 1 in Figure 8.
This analysis shows where the improvement in …t over the standard NKPC is coming
from. It is not simply the addition of another regressor, but the fact that deep habits
increase the responsiveness of in‡ation to movements in consumption growth by an order of
magnitude without a large countervailing e¤ect from a reduced importance of marginal cost.
This result is, however, predicated on two requirements. First, the demand elasticity " has
14
We do not report the case of
impose the parameter restriction.
= 1 and " = 11 since the estimate of
23
would exceed 1 if we did not
to be small enough, while the degree of deep habits has to be large enough, as is the case for
the estimates of Ravn et al. (2010) and for our estimates using the alternative speci…cation
of this section. Although a high degree of habits reduces the weight on marginal costs, this is
more than compensated by the increase in the relative weight of consumption growth. The
second requirement is that consumption growth has to have the right statistical properties.
Although it is markedly less volatile than marginal costs, this is more than compensated
by the response coe¢ cient in terms of its overall impact. However, for the ampli…cation
e¤ect on marginal cost dynamics, consumption growth has to comove negatively with the
marginal cost proxy since consumption growth shows up in the NKPC with a negative sign
as per equation (29). Otherwise, deep habits would dampen the e¤ects of marginal cost.
Our …nal robustness check investigates such a case.
5.3
Sub-Sample Analysis
We conclude our robustness analysis by estimating the benchmark speci…cation for a subsample period that starts in 1984, which covers the period of the Great Moderation. Table 1
shows that the behavior of the series we use in this paper has in fact changed. The volatility
of all variables is smaller in the sub-sample than in the full sample with the exception of
real unit labor cost. This need not have a dramatic e¤ect on our estimates since the decline
in consumption growth volatility is compensated by increased volatility of the marginal cost
proxy. More detrimental is the change in comovement pattern between these two series; to
wit, the contemporaneous correlation between mct and
ct is a positive 0:23 over the sub-
sample. Since we show above that the latter enters the theoretical NKPC with a negative
coe¢ cient, this is likely to counter the e¤ects of marginal cost.
Our concerns are only partially borne out by the estimates reported in Table 8. The
standard NKPC estimates in Panel A show a shift towards a stronger weight on forwardlooking behavior compared to the full sample and an overall better …t, although the structural estimates impart a value for the indexation parameter that is much higher than in
the benchmark sample. Panel B shows the estimates of the deep habits NKPC using the
benchmark calibration with VAR-based forecasts. The estimates do not show dramatic
di¤erences to those in Table 5. The relative …t of the sub-sample estimation is worse, but
still much improved over the standard NKPC. However, standard errors of the estimates
are surprisingly large, technically rendering the NKPC coe¢ cient statistically insigni…cant
at the 10%-level. This arguably re‡ects the changing pattern of the comovement between
the regressors. Moreover, it may also re‡ect that unit labor cost may not be the best proxy
24
for marginal cost for this sub-sample period.15 Nevertheless, even the sub-sample analysis
shows that deep habits are a central component to explaining in‡ation dynamics per the
NKPC.
6
Conclusion
We show in this paper that deep habits in preferences are an essential element in understanding in‡ation dynamics. Compared to a standard version of the NKPC, a deep habits
speci…cation is an improvement in terms of …t and in terms of smaller standard errors of the
estimated parameters. The estimated NKPC under deep habits also puts much less weight
on lagged in‡ation. This suggests a lower degree of intrinsic in‡ation persistence, where
the required propagation is derived from the properties of the imputed driving process.
The impact of deep habits on the latter stems from two in‡uences. First, the model implies additional regressors, speci…cally consumption growth and a marginal value of demand
term. This in and of itself produces a better …t, but we also show that a large part of the
improved performance is due to the altered responsiveness of in‡ation to the coe¢ cients
in the NKPC. Deep habits therefore preserve the standard transmission mechanism from
marginal cost movements to in‡ation, but reinforce this through additional feedback.
The main concern about the validity of our results stems from the partial equilibrium
nature of our analysis, that is, we only estimate a single equation that should naturally
be seen as a part of a larger general equilibrium model. This is re‡ected in two aspects.
First, the elements of the driving process are treated as exogenous regressors, albeit ones
that still su¤er from endogeneity problems. This requires the use of instrumental variables
in the estimation, that may themselves be of dubious quality. Estimation of the full equilibrium model with, for instance, likelihood-based methods obviates this problem since the
likelihood function and the application of the Kalman-…lter automatically constructs the
optimal instruments. The drawback of a systems-approach, however, is the possibility of
misspeci…cation. Resolving this issue is outside the scope of this paper. We take comfort,
however, from the empirical estimates of Ravn et al. (2010), which are close to ours, despite
a di¤erent empirical method that does utilize more information. The second problematic
aspect lies in the way we impute the unobservable variables. We rely on present-value computations that are known to impose weak restrictions on the imputed variables. It would
therefore be a useful exercise to consider alternative methods for backing out unobservables.
15
See the discussion in Galí et al. (2005) and Nason and Smith (2008).
25
References
[1] Andrews, Donald W.K. (1991): “Heteroskedasticity and Autocorrelation Consistent
Covariance Matrix Estimation”. Econometrica, 59(3), 817-858.
[2] Galí, Jordi and Mark Gertler (1999): “In‡ation Dynamics: A Structural Econometric
Analysis”. Journal of Monetary Economics, 44(2), 195-222.
[3] Galí, Jordi, Mark Gertler, and J. David Lopez-Salido (2005): “Robustness of Estimates
of the Hybrid New Keynesian Phillips Curve”. Journal of Monetary Economics, 52,
1107-1118.
[4] Guerrieri, Luca, Christopher Gust, and J. David Lopez-Salido (2010): “International
Competition and In‡ation: A New Keynesian Perspective”. American Economic Journal: Macroeconomics, 2, 247-280.
[5] Krause, Michael U., J. David Lopez-Salido, and Thomas A. Lubik (2008): “Do Search
Frictions Matter for In‡ation Dynamics?” European Economic Review, 52(8), 14641479.
[6] Kurmann, André (2007): “VAR-based Estimation of Euler Equations with An Application to New Keynesian Pricing”. Journal of Economic Dynamics and Control, 31(3),
767-796.
[7] Lubik, Thomas A., and Wing Leong Teo (2011): “Inventories, In‡ation Dynamics, and
the New Keynesian Phillips Curve”. Forthcoming, European Economic Review.
[8] Nason, James M., and Gregor W. Smith (2008): “The New Keynesian Phillips Curve:
Lessons from Single-Equation Econometric Estimation”. Federal Reserve Bank of Richmond Economic Quarterly, 94(4), 361-395.
[9] Ravn, Morten O., Stephanie Schmitt-Grohé, and Martín Uribe (2006): “Deep Habits”.
Review of Economic Studies, 73(1), 195-218.
[10] Ravn, Morten O., Stephanie Schmitt-Grohé, Martín Uribe, and Lenno Uuskula (2010):
“Deep Habits and the Dynamic E¤ects of Monetary Policy Shocks”. Journal of the
Japanese and the International Economies, 24, 236-258.
26
Table 1. Business Cycle Statistics
Sample P eriod : 1955 : 1 2011 : 2
Variable
s.d.(%)
Cross-Correlation
GDP
Inf lation
RU LC
Cons:Growth
y
1
3.97
0.58
3.44
0.71
0.15
1
rulc
-0.20
0.24
1
c
0.14
-0.28
-0.09
1
Sample P eriod : 1984 : 1 2011 : 2
Variable
s.d.(%)
Cross-Correlation
GDP
Inf lation
RU LC
Cons:Growth
y
1
3.31
0.25
3.71
0.56
0.16
1
rulc
0.42
0.27
1
c
0.28
-0.11
0.23
1
Table 2. Benchmark Calibrated Parameter Values
Parameter
c
mc
De…nition
Value
Source
Discount Factor
Risk Aversion
Deep Habits
Elasticity of Demand
AR(1)-coe¢ cient
AR(1)-coe¢ cient
0.99
1
0.85
2.48
0.31
0.98
Annual Real Interest Rate
Log-utility
Ravn et al. (2010)
Ravn et al. (2010)
Authors’Estimates
Authors’Estimates
27
Table 3. GMM Estimates: Standard NKPC
Speci…cation
Unrestricted NKPC
f
Restricted NKPC
b =0
0.004
(0.001)
J(7)
5.428
(0.608)
0.006
(0.002)
J(8)
5.813
(0.668)
b
0.791
(0.055)
0.197
(0.058)
f
0.989
(0.014)
0
Structural NKPC
= 0:99
0.271
(0.092)
0.004
(0.001)
J(8)
5.333
(0.722)
Note: The numbers in parentheses are standard errors. For J-statistics, the numbers in parentheses are
p-values. For the structural NKPC,
0
("
1)= : The instrument set includes 4 lags of in‡ation and 2
lags of marginal cost, output and wage in‡ation. The adjusted sample period for the estimation is 1956Q3
to 2011Q2.
Table 4. GMM Estimates: Deep Habits NKPC, AR-based
Speci…cation
Unrestricted NKPC
0.008
(0.002)
J(7)
4.236
(0.752)
0.984
(0.008)
0.010
(0.001)
J(8)
3.133
(0.926)
0.110
(0.011)
203.455
(7.377)
J(10)
1.068
(0.998)
f
b
0.857
(0.041)
Restricted NKPC
b =0
Structural NKPC
= 0:99, = 1
" = 2:48, = 0:85
0.105
(0.045)
f
Note: The numbers in parentheses are standard errors. For J-statistics, the numbers in parentheses are
p-values. The instrument set includes 4 lags of in‡ation and 2 lags of constructed driving process, output
and wage in‡ation. The adjusted sample period for the estimation is 1956Q3 to 2011Q2.
28
Table 5. GMM Estimates: Deep Habits NKPC, VAR-based
Speci…cation
Unrestricted NKPC
0.017
(0.002)
J(7)
2.251
(0.945)
0.990
(0.011)
0.010
(0.003)
J(8)
3.133
(0.926)
0.138
(0.004)
334.825
(10.165)
J(8)
1.007
(0.998)
f
b
0.929
(0.028)
Restricted NKPC
b =0
Structural NKPC
= 0:99
0.034
(0.031)
f
Note: The numbers in parentheses are standard errors. For J-statistics, the numbers in parentheses are
p-values. The instrument set includes 4 lags of in‡ation and 2 lags of constructed driving process, output
and wage in‡ation. The adjusted sample period for the estimation is 1956Q3 to 2011Q2.
Table 6. Robustness: Alternative Calibration
Speci…cation
Unrestricted NKPC
= 0:15
0.825
(0.059)
0.157
(0.059)
0.005
(0.002)
J(7)
6.942
(0.543)
Unrestricted NKPC
= 0:95
0.902
(0.051)
0.064
(0.053)
0.004
(0.001)
4.615
(0.707)
Unrestricted NKPC
= 0:60
0.845
(0.058)
0.126
(0.066)
0.006
(0.002)
4.921
(0.670)
Unrestricted NKPC
=3
0.884
(0.047)
0.071
(0.051)
0.008
(0.002)
4.565
(0.713)
Unrestricted NKPC
" = 11
0.841
(0.058)
0.130
(0.066)
0.006
(0.002)
4.859
(0.677)
f
b
Note: The numbers in parentheses are standard errors. For J-statistics, the numbers in parentheses are
p-values. The instrument set includes 4 lags of in‡ation and 2 lags of constructed driving process, output
and wage in‡ation. The adjusted sample period for the estimation is 1956Q3 to 2011Q2.
29
Table 7. Robustness: Alternative Speci…cations
Speci…cation
Unrestricted NKPC
f
0.825
(0.059)
Restricted NKPC
b =0
b
0.157
(0.059)
mc
0.005
(0.002)
mc
f
1.002
(0.022)
0.004
(0.002)
c
0.069
(0.030)
J(8)
6.942
(0.543)
0.047
(0.042)
J(9)
7.871
(0.547)
c
Structural NKPC
= 0:99
= 1, " = 2:48
0.198
(0.084)
230.699
(62.834)
0.894
(0.038)
J(9)
6.710
(0.667)
Structural NKPC
= 0:99
= 3, " = 2:48
0.198
(0.084)
254.654
(74.851)
0.843
(0.052)
J(9)
6.710
(0.667)
0.956
(0.015)
J(9)
6.709
(0.667)
Structural NKPC
= 0:99
= 3, " = 11
0.198
(0.084)
1228.934
306.754
Note: The numbers in parentheses are standard errors. For J-statistics, the numbers in parentheses are
p-values. The instrument set includes 4 lags of in‡ation and 2 lags of marginal cost, consumption growth,
output and wage in‡ation. The adjusted sample period for the estimation is 1956Q3 to 2011Q2.
30
Table 8. Robustness: Sub-Sample 1984:1-2011:2
Panel A: Standard NKPC
Unrestricted NKPC
f
0.833
(0.081)
Restricted NKPC
b =0
Structural NKPC
= 0:99
0.004
(0.001)
J(7)
4.209
(0.755)
0.005
(0.002)
J(8)
5.051
(0.752)
b
0.234
(0.052)
f
1.146
(0.057)
0
0.003
(0.001)
J(8)
2.984
(0.935)
0.009
(0.012)
J(7)
3.830
(0.799)
1.073
(0.118)
0.013
(0.009)
J(8)
3.881
(0.868)
0.131
(0.159)
139.677
(128.87)
J(8)
4.420
(0.817)
0.402
(0.056)
Panel B: Deep Habits NKPC
Unrestricted NKPC
f
b
0.988
(0.196)
Restricted NKPC
b =0
Structural NKPC
= 0:99
0.074
(0.140)
f
Note: The numbers in parentheses are standard errors. For J-statistics, the numbers in parentheses are
p-values. The instrument set includes 4 lags of in‡ation and 2 lags of constructed driving process, output
and wage in‡ation. The sample period for the estimation is 1984Q1 to 2011Q2.
31
2
1.9
1.8
Critical value
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0
0.1
0.2
0.3
0.4
0.5
0.6
Degree of deep habits -
Figure 1: Critical value of " for
0.7
θ
1
0.8
0.9
1
to be positive
120
Deep habit
S tandard model
Steady state markup (in %)
100
80
60
40
20
0
2
3
4
5
6
7
Elasticity of substitution -
ε
8
Figure 2: Steady-state markup as " varies
32
9
10
11
30
AR-based driving process
Marginal cost
Percentage deviation from steady state
25
20
15
10
5
0
-5
-10
-15
-20
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Figure 3: AR-based constructed driving process and marginal cost
30
VAR-based driving process
Marginal cost
Percentage deviation from steady state
25
20
15
10
5
0
-5
-10
-15
-20
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Figure 4: VAR-based constructed driving process and marginal cost
33
2010
c
t t+1
Eλ
10
Percentage deviation from steady state
Marginal cost
5
0
-5
-10
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Figure 5: Expected marginal value of future demand and marginal cost
10
Percentage deviation from steady state
Consumption growth
Marginal cost
5
0
-5
-10
1960
1965
1970
1975
1980
1985
1990
1995
2000
Figure 6: Consumption growth and marginal cost
34
2005
2010
Weight on marginal cost
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
Relative weight on consumption growth
0.55
0
0.1
0.2
0.3
0.4
0.5
Degree of deep habits -
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
θ
90
σ =1
σ =3
80
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
Degree of deep habits -
θ
Figure 7: Weights on marginal cost and consumption growth as
varies
Weight on marginal cost
0.87
0.86
0.85
0.84
0.83
0.82
0.81
Relative weight on consumption growth
0.8
2
3
4
5
6
7
Elasticity of substitution -
ε
8
9
10
11
25
σ =1
σ =3
20
15
10
5
0
2
3
4
5
6
7
Elasticity of substitution -
ε
8
9
10
Figure 8: Weights on marginal cost and consumption growth as " varies
35
11
A
Appendix: Analytical Derivations
A.1
Derivation of Equation (18) and the Deep Habits NKPC
Substituting the de…nitions of the habit stock xt = ct
rate
"
t
1
=
c
t (ct
ct
(
t 1)
1 )+
ct
1
and the weighted in‡ation
into equation (12), we have:
t
1
t
(
t 1)
ct = ct +
xt+1
xt
Et
t+1
1
t+1
( t)
ct+1 :
(A1)
Log-linearization of the equation above gives:
c
) ~t +
" c c (1
1
c~t
1
c~t
1
1
+ c (~ t
"
c
~t
1)
= c~
ct +
cEt (~ t+1
~ t ) : (A2)
Using the steady-state relation:
(1
) = 1;
(A2b)
which is obtained from equation (A1), we can simplify equation (A2) as:
~t =
1+
~t
1
+
1
Et ~ t+1
1+
(1 +
~c +
t
)
1
(~
ct
Imposing symmetry and substituting the discount factor qt , xt = ct
c~t
ct
1,
1)
:
and
(A3)
y
t
= mct
into equation (10), we have:
mct +
c
t
=1+
ct+1
ct
ct
ct 1
Et
c
t+1 :
(A4)
Log-linearization of the equation above gives:
mcmc
ft +
c ~c
t
c
=
Et
1
(~
ct+1
c~t
c~t + c~t
1)
c
+ ~ t+1 :
(A5)
Using the steady-state relation:
mc = 1
(1
)
c
;
(A5b)
which is obtained from equation (A4) and (A2b), we can simplify equation (A5) as:
~c =
t
c
Et ~ t+1
Et
1
( c~t+1
c~t )
(" (1
)
(1
)) mc
f t;
(A6)
which is equation (18) in the main text. Substituting equation (A6) into equation (A3) and
rearranging, we then obtain equation (13) in the main text.
36
A.2
Derivation of Equation (22)
Substituting (20) and (21) into (19), we get:
ec =
t
=
1
X
)j Et
(
j=0
+
1
1
X
j=0
1
X
2
1
1
2
(
e
ct :
(
j
mc ) (" (1
c~t
f t;
mc mc
j
c)
2
1
1
c
where:
c
)) mc
ft +
(1
1
2 (0; 1) and j
mc j
1
c
A.3
0 is
>
1
2
c
;
c
1
=
1
2
(A8)
c
" (1
)
(1
1
)
:
(A9)
mc
2 [0; 1), the condition for
is the same condition as for
)] mc
f t+j
(A7)
1
c
1
mc
Since
(1
c~t
1
1
1
=
)
2
)
2
=
[" (1
c~t
c
j=0
=
e
ct+j+1
mc
> 0 is " > (1
) = (1
), which
> 0. It is then easy to verify that a su¢ cient condition for
c.
Steady-State Markup
Combining equations (A2b) and (A5b), we obtain:
mc =
" (1
) (1
" (1
)
)
:
(A10)
The steady-state markup is the inverse of steady-state real marginal cost. We therefore
have:
Steady-state markup =
37
" (1
)
" (1
) (1
)
:
(A11)
A.4
Derivation of Equations (29) and (30)
Substituting (23) and (21) into (13), we have:
et =
=
Et et+1 +
1+
e
ct
3
1+
(
3
4(
Et et+1 +
+
c
e
ct
c
1+
c
4
et
1+
c
2
et
ft
1 mc
+
1
+
f t)
mc mc mc
+(
1
+
1
4
e
ct ;
c)
e
ct
c
2
:::
ft
mc mc ) mc
(A12)
" 1
(1+ ) ,
which is in the form of equation (29) in the main text. Factoring out
we can write
equation (A12) as equation (30), where:
~
5
=
=
=
~
6
=
=
=
=
~ +~
1
4
" (1
)
"
" (1
)
"
~ +~
3
4
1+
" 1 1
1
"
11
1
"
11
~ is positive if " > (1
5
~ to be non-negative is
6
~
c
mc mc
"
)
" (1
"
1
1
)
(1
)
mc
1
mc
:
1
mc
11
+
(
1
) = (1
>
+
1
1+
1+
)
c
2
+
(1
1
(1
1
c,
2
1
1
c)
c
c
c
1
c
1
2
1
c
"
1
c
2
c
c
:
c
), which we have imposed. A su¢ cient condition for
which is the same condition for
38
c
0.
@FedFRASER