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            pdfauthor={Christopher Carroll <ccarroll@jhu.edu>, Jirka Slacalek <jiri.slacalek@ecb.int>, Martin Sommer <msommer@imf.org>},
            pdftitle={International Evidence on Sticky Consumption Growth},
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\begin{verbatimwrite}{\jobname.title}
International Evidence On Sticky Consumption Growth
\end{verbatimwrite}

\title{\LARGE International Evidence On \\ Sticky Consumption Growth%
\footnote{\small Carroll: Department of Economics, Johns Hopkins University, Baltimore, MD, \url{http://econ.jhu.edu/people/ccarroll/}, \href{mailto:ccarroll@jhu.edu}{\texttt{ccarroll@jhu.edu}}; Slacalek: European Central Bank, Frankfurt am Main, Germany, \url{http://www.slacalek.com/}, \href{mailto:jiri.slacalek@ecb.int}{\texttt{jiri.slacalek@ecb.int}}; Sommer: International Monetary Fund, Washington, DC, \url{http://martinsommeronline.googlepages.com/}, \href{mailto:msommer@imf.org}{\texttt{msommer@imf.org}}.
We thank Angela Espiritu and Stephanie Denis for excellent research assistance and the participants at the NBER Summer Institute for insightful comments. We are grateful to Ray Barrell, Carol Bertaut, Amanda Choy, Nathalie Girouard, Roberto Golinelli and Robert Metz for help in constructing the dataset.
Data and econometric programs that generated all of the results in the paper are available from the first author's web page,
\url{http://econ.jhu.edu/people/ccarroll/}.
The views presented in this paper are those of the authors, and should not be attributed to the International Monetary Fund, its Executive Board, or management, or to the European Central Bank.%
}}

\author{Christopher D. Carroll\\ JHU
\and
Jirka Slacalek \\  ECB
\and
Martin Sommer \\ IMF
}

\date{\today}

\maketitle

\begin{abstract}\large

We estimate the degree of `stickiness' in aggregate
consumption growth (sometimes interpreted as reflecting consumption
habits) for thirteen advanced economies.
We find that, after controlling for measurement error, consumption growth has a high degree of autocorrelation, with a stickiness parameter of about 0.7 on average across countries.
The sticky-consumption-growth model
outperforms the random walk model of \cite{hal78}, and typically fits
the data better than the popular \cite{cm89} model. In several countries, the sticky-consumption-growth and
Campbell--Mankiw models work about equally well.

\end{abstract}

\noindent\textbf{Keywords}: Sticky Expectations, Consumption Dynamics, Habit Formation
\\[1mm]
\textbf{JEL classification}: C6, D9, E2

\medskip\medskip

\begin{tabular}{rc}
   Paper:   & {\footnotesize \url{http://econ.jhu.edu/people/ccarroll/papers/cssIntlStickyC.pdf}}
\\ Archive: & {\footnotesize \url{http://econ.jhu.edu/people/ccarroll/papers/cssIntlStickyC.zip}}
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\section{Introduction}

A large literature ranging across macroeconomics, finance, and
international economics has argued that `habit formation' can explain
many empirical facts related to consumption
dynamics, including stickiness in aggregate consumption growth.\footnote{Facts that have been interpreted using habit formation models
include the equity premium puzzle (\citet{con90} and
  \citet{cc99}), Granger causality from
  growth rates to saving rates (\citet{cow00}), the hump-shaped
  response of consumption to income shocks (\citet{fuh00}), the
  dynamic effects of fiscal policy (\citet{lu00}), persistence
  in current account balances (\citet{gru04}), and the home bias
  puzzle (\citet{sw06}).
} Other explanations
for the persistence of aggregate spending growth, or `excess
smoothness' (in \citet{campbell&deaton:smooth}'s terminology),
include imperfect attentiveness to macroeconomic news on the part of consumers (\citet{cs06}; \citet{rei06}),
or consumers' inability to distinguish micro- from macro-economic shocks~(\cite{pischke:micromacro}).
Further explanations could undoubtedly be imagined.

But a full consensus has not yet emerged on whether empirical data
are irreconcilable with \cite{hal78}'s benchmark random walk model
of consumption. Hall's model implies that consumption growth is
unpredictable (excess smoothness is zero). However, standard
extensions of the \citeauthor{hal78} model can generate some degree
of stickiness in consumption growth. For example, excess smoothness
might merely reflect the fact that spending decisions are made more
frequently than consumption data are measured (\cite{wor60}; this
viewpoint has recently been advocated in well known papers by
\citet{ll01,ll04}). Also, in the presence of uncertainty, the
precautionary saving motive slows down consumers' response to
shocks, which could also explain part (though not all) of the excess
smoothness (\citet{ludvigson&michaelides:excesses}).  Another
possibility, not often mentioned but nevertheless
worth serious consideration, is that the smoothness of
measured spending reflects data construction methods (e.g.\ for
components of spending for which quarterly observations are imputed
using annual data sources) rather than actual spending smoothness.
Finally, many of the papers in the habit formation literature have
not carefully examined the possibility that their results might
reflect the presence of some `rule-of-thumb' consumers, who simply
set consumption equal to income in each period, as proposed in
influential papers by \citet{cm89,cm91}.

Motivated by this debate and by the fact that much of the empirical
evidence on excess smoothness has come from a single country (the U.S.), this paper provides systematic estimates of three
simple canonical models of consumption dynamics using data for
all advanced economies for which we were able to construct
appropriate datasets (thirteen countries in all).  We compare the random walk model of
\cite{hal78} with two alternatives: the \cite{cm89} model,
and a model that permits (but does not require) excess smoothness. (We do not take a stand here on whether
such smoothness reflects habits, inattention, or other factors.)

Using both instrumental variables (IV) (section \ref{ssIV}) and Kalman
filter structural (section \ref{ssKalman}) estimation methods, we
find strong evidence of excess smoothness (`stickiness') in
consumption growth in every country in our sample.\footnote{Section \ref{subsubsec:DSGE} shows
how our Kalman filter technique can be interpreted as a particularly simple example
of structural estimation of a DSGE model.  Embedding our 
framework in a larger macroeconomic structure would be relatively straightforward.}   Although there is
some variation across countries in the degree of stickiness, in every
country we can reject the hypothesis that the stickiness coefficient is zero (the random walk theory), while in no country can we reject
the hypothesis that it is 0.7.  Furthermore, wherever there is a clear
distinction between the two non-random-walk models, the consumption
stickiness model outperforms the rule-of-thumb model, usually by a
decisive statistical margin.  (In a few cases, the two non-random-walk
models are not statistically distinguishable from each
other.)\footnote{To our knowledge, the only comparable
  paper is \citet{bcf93} (henceforth BCF), who estimate a habit formation model
  using data on total personal consumption expenditures for six
  countries. BCF find evidence for stickiness in aggregate consumption
  growth data in most countries. Their estimates of the habit
  persistence coefficient range between 0.57 and 0.93, but
  are often insignificant. Their paper also does not test the assumption
  of habit formation against alternative models of consumption
  dynamics, such as the Campbell--Mankiw model.  \citet{fc91} report in a
  framework closely related to BCF that the evidence for habit
  formation seems stronger in the U.S. data than in their international
  dataset.  However, both papers use GMM to estimate a nonlinear Euler
  equation, a method which is not robust to the presence of
  substantial measurement error in consumption data.  }

The large size of our estimated stickiness parameter may come as a
surprise to some readers, because the serial correlation
coefficient for spending growth in the raw data is much lower than 0.7
(for instance, it is about 0.35 in U.S. data).  The discrepancy reflects our use of
econometric methods that are robust to the presence of measurement
error.  Consistent with \citet{som07}'s findings for the United
States, our estimates suggest that in most countries at least half of
the quarterly variation in consumption growth reflects temporary
variation that can be interpreted either as measurement error or as
truly transitory spending disturbances unrelated to the theoretical
consumption model (caused, for example, by unseasonal weather, which can have a nontrivial effect at the quarterly frequency, cf.\ \citet{som07}).\footnote{Interestingly, \cite{friedman:atheory}'s original
statement of the permanent income hypothesis gave almost equal billing
to transitory consumption shocks and transitory income shocks.}

The remainder of the paper is organized as follows. Section \ref{sTheory} outlines
two theoretical frameworks that generate sticky consumption growth
and provide the conceptual framework for our estimation strategy. Section \ref{sEmpirics}
presents the main empirical results and Section \ref{sConcl} concludes.


\section{Two Theories of Stickiness} \label{sTheory}

This section sketches the two most popular theoretical frameworks---habit formation and
sticky expectations---that can generate serial correlation in
aggregate consumption growth.  In the habit formation model, the serial
correlation coefficient $\chi$ reflects the strength of
habits (if $\chi=0$, the model collapses to the Hall random
walk model); in the sticky information model, $\chi$ is the fraction of
aggregate expenditure by households that have not fully updated their
information set about the latest macroeconomic developments (again,
$\chi=0$ corresponds to the Hall model). Because
the implications of the two frameworks are indistinguishable in aggregate data, our
empirical evidence is consistent with either model.\footnote{\citet{cs06} argue that the models can be distinguished using microeconomic data.}

\subsection{Habit Formation}        \label{ssHabits}
\citet{muellbauerAES88} proposed a simple model of habit persistence, in which the representative consumer maximizes time-nonseparable utility
\be
\max \Exp\sum_{t=s}^{\infty} \beta^{t-s} \mathbf{u}(C_t-\chi C_{t-1}) \label{eq:Objective}
\ee
subject to the usual transversality condition and the dynamic budget constraint:
\be
  B_{t+1}  =  (B_t-C_t)R+Y_{t+1},  \label{dbc}
\ee
where $\beta$ is the discount factor, $C$ is the consumption level, $B$ is beginning-of-period net assets, $R$ is the constant interest factor, and $Y$ is noncapital income. $C_{t-1}$ in \eqref{eq:Objective} represents the `habit stock,' i.e., the reference level of consumption to which the consumer compares the current consumption level. The parameter $\chi$ captures the strength of habits. After rewriting the utility function as $\mathbf{u}(C_t-\chi C_{t-1})=\mathbf{u}\big((1-\chi)C_t+\chi \Delta C_t\big)$, one can see that, for $\chi\in(0,1)$, the consumer derives utility from both the level and the change in consumption.

\citet{muellbauerAES88} and \citet{dyn00} have shown that for a habit-forming consumer with a  Constant Relative Risk Aversion (CRRA) outer utility $\mathbf{u}(X)=X^{1-\rho}/(1-\rho)$ and $R\beta=1$, optimal consumption growth approximately follows an AR(1) process:
\be
\Delta \log C_t=\chi\Delta \log C_{t-1}+\epsilon_t,     \label{cHabMicro}
\ee
where $\epsilon_t$ denotes innovations to lifetime resources. Hence, in contrast to the standard intertemporally separable utility specification, some of the period $t$ consumption growth is predetermined at time $t-1$. The strength of habits $\chi$ can be estimated as the autocorrelation coefficient in the equation for aggregate consumption growth.


\subsection{Sticky Expectations} \label{ssStickyE}

\cite{cs06} present an alternative model of consumer behavior that
also generates sticky aggregate consumption growth, but without departing
from the standard intertemporally separable utility specification. The key assumption  is that consumers are mildly inattentive to macro
developments---for example, they do not immediately and fully take
into account information contained in aggregate macroeconomic
indicators such as productivity growth or the unemployment rate.\footnote{For evidence on this kind of inattention in the context
of inflation expectations, see \cite{carroll:epidemicInflQJE}.}

Assume for a moment that consumers maximize the discounted sum of time separable quadratic utility streams $-\sum_{t=s}^{\infty} \beta^{t-s} (C_t-\bar C)^2$ subject to the budget constraint \eqref{dbc}. In the standard \cite{hal78} model, in which households use all available information, the optimal consumption level follows a random walk and consumption growth is a white noise: $\Delta C_t=\epsilon_t$.

Suppose now instead that the economy consists of a continuum of
\emph{inattentive} consumers, each of whom updates the information
about his permanent income with probability $\Pi$ in each period. For
each consumer, this probability is assumed to be independent of the
date when the consumer last updated his information set and
independent of his income or wealth. The model therefore is similar to
the \citet{calvo:prices} model of price setting frequently used in the
monetary economics literature. \cite{cs06} show that the change in
aggregate consumption, $\Delta C_t$,
approximately follows an AR(1) process, whose autocorrelation roughly
equals the share of consumers ($1-\Pi$) who do not have up-to-date
information about macroeconomic developments.  If utility is of
the CRRA form, consumption growth is
well approximated by:\footnote{Similar dynamics of aggregate consumption growth are also implied by the `rational inattention'
  models of \citet{rei06} and potentially also \cite{sims:inattention}.}

\be
\Delta\log C_t= \underbrace{(1-\Pi)}_{\equiv \chi} \Delta\log C_{t-1}+\epsilon_t. \label{eq:stickyCrec}
\ee

In addition, in the spirit of \citet{akerlof&yellen:nearrational}
and \citet{cochrane:nearrational}, \cite{cs06} show that the utility loss from the
infrequent updating of expectations is very small.

As noted above, since the two theories of stickiness generate identical implications for aggregate consumption
growth dynamics, evidence of a positive serial correlation coefficient will
be consistent with either theory.  (The theories can be distinguished in other
ways, e.g.\ using microeconomic data; see \citeauthor{cs06} for details and
evidence.)


\section{Empirical Results} \label{sEmpirics}


This section tests the model of sticky consumption growth \eqref{cHabMicro} and \eqref{eq:stickyCrec} against the alternatives of rule-of-thumb behavior and the random walk hypothesis. The organizing framework for our empirical analysis is a specification for consumption growth adopted in the excess sensitivity literature,%
\footnote{Early contributions include \cite{fla81}, \cite{campbell&deaton:smooth}, and \cite{cm89}.  } which has been expanded here to include a term capturing stickiness of consumption growth:
\be
\Delta \log C_t = \varsigma+\chi \Exp_{t-2}[\Delta \log C_{t-1}]+\eta \Exp_{t-2}[\Delta \log Y_{t}]+\alpha \Exp_{t-2}[A_{t-1}]+\epsilon_t,  \label{cDynMacroPrime}
\ee
where $Y$ denotes household income and $A$ denotes household (net) assets. The first two right-hand side regressors correspond to two of the tested theories of consumption behavior: inattentiveness or habit formation ($\Delta \log C_{t-1}$) and rule-of-thumb consumers ($\Delta \log Y_{t}$). Under the third tested theory---the random walk hypothesis---the coefficients $\chi$ and $\eta$ should both be zero. The third term in the equation above ($A_{t-1}$) is included as a control---any of the three theories allow for some direct effect of asset holdings on consumption growth, either due to effects related to uncertainty (which induces a precautionary saving motive) or due to time variation in interest rates (which
we assume is captured by time variation in the capital-to-income ratio $A$).\footnote{By including the assets in the estimated equation, we follow the literature on precautionary savings and liquidity constraints. Calibrated theoretical models imply that the relationship between $A_t$ and $\Exp_t[R_{t+1}]$ is almost linear. However, empirical estimates of Euler equations using macro data generally produce insignificant (or even implausible) coefficients on expected interest rates (see, e.g., \citet{hal88} and table 3 of \citet{cm91}; and \citet{vis02} for evidence in micro consumption data).
}

There are at least three reasons to expect the OLS estimates of
coefficients in \eqref{cDynMacroPrime} to be biased and
inconsistent.  First, as argued by \citet{wil92} and \citet{som07},
quarterly consumption data may be contaminated with substantial
measurement error. Second is the undoubted existence of transitory spending disturbances such as
those related to unseasonal weather (or even, for some smaller
countries, one-time events such as the hosting of the
Olympics).  None of the theoretical models include these kinds of shocks,
yet back-of-the-envelope calculations
suggest their effects could be substantial in quarterly data.
Our final reason for expecting OLS to be biased is the well-known problem
of time aggregation.\footnote{\citet{wor60}'s analysis shows that if consumers with time separable
  preferences make purchase decisions more often than consumption data
  are observed, time aggregation generates an MA(1) process in
  observed consumption growth even when preferences are otherwise
  standard as in \citet{hal78}. In a simple habit formation or sticky
  information model of the type presented in this paper, time
  aggregation generates an MA(2) process in consumption growth
  (\citet{muellbauerAES88}, \citet{som07}), but the MA(2)
  coefficient is generally small.}

We develop these points using the United States as an example.  The
clearest source of measurement error in \emph{quarterly} aggregate
consumption data is the services sector, because many components of
the quarterly services data are calculated by interpolating from the
underlying annual or biennial data sources (\citet{bea06}).
However, sampling and nonsampling errors introduce significant
measurement error even into other categories of consumption data
(see \citet{som07} for details). \citeauthor{som07} also argues
that weather-related events can significantly influence aggregate
consumption expenditures. For example, under some plausible
assumptions, Hurricane Katrina may have reduced quarterly personal
consumption expenditure (PCE) growth by about 1 percentage point on
an annualized basis in Q3:2005. However, even a much more benign
event such as mild winter can reduce annualized quarterly
consumption growth significantly---for instance, by about 1/4
percentage point in the United States in Q1:2006---through lower
outlays on energy. All in all, \citeauthor{som07} estimates that
measurement error and transitory consumption together account for
about 50 percent of the quarterly U.S. nondurables and services
consumption volatility, consistent with his empirical finding that
the IV estimates of consumption persistence are about twice as high
as the OLS estimates.

To address these three estimation issues (measurement error, transitory consumption, and time aggregation) in quarterly consumption data, we
use two econometric methods developed in \citet{som07}. The first
technique attempts to correct for the estimation issues using
instrumental variables regressions. As with any IV
method, validity of the estimation results depends on the ability to
find suitable instruments.  As an alternative, the second technique
therefore uses the Kalman filter to separate `true' consumption growth
from its transitory components and measurement error.\footnote{Aficionados of Bayesian estimation of DSGE models may wish to reinterpret our estimates as a maximum likelihood estimator of a particularly simple structural model with measurement error and a weak prior. See Section 3.2 for details.}  In this case,
the usual caveat applies:  The validity of this maximum likelihood method
hinges on the assumed structure of the stochastic processes for measurement error and
`true' consumption dynamics.



\subsection{Sticky Consumption Growth in IV Regressions}        \label{ssIV}

\subsubsection{Dataset}

Equation \eqref{cDynMacroPrime} is estimated using aggregate quarterly data from thirteen advanced economies
ranging roughly over the past forty years (table \ref{tCsource} provides data details). Our preferred measure
of consumption is the sum of expenditures on nondurable goods and services. However, this measure is available
only for six countries in our sample (Canada, France, Germany, Italy, the U.K. and the U.S.); total personal consumption
expenditures are therefore used for the other sample countries.%
\footnote{For the six countries for which nondurables and services
data are readily available, regression results using total PCE are
similar to those reported in the paper for nondurables and services. Since durable consumption
growth is generally mildly negatively autocorrelated
(\citet{mankiw:durgoods}), the estimates of consumption
persistence $\chi$ for the other countries for which we use data on
the total PCE (see the bottom panel of table \ref{tAllC}) may be
biased \emph{downward}, making our evidence in favor of strong
consumption stickiness likely to be conservative. Japan is not included in
our sample as creating a quarterly dataset with consumption data
prior to 1980 would involve splicing consumption series based
on three very different methodologies. Adjustments to the Japanese
national accounts methodology in 2002 and 2004 have significantly
improved the reliability of quarterly consumption series but the
current-methodology data are only available since Q1:1994
(\citet{imf06}).  For the U.S., it is possible to perform similar
experiments using data on purely nondurable goods spending and on
retail sales spending, with results similar to those reported here
for PCE excluding durables.} Finally, $Y$ and $A$ are measured as
household disposable income and the ratio of financial wealth to
disposable income, respectively.

\subsubsection{Instruments}

The main advantage of IV estimation is that with appropriate
instruments, there is no need to make assumptions about the stochastic
structure of measurement error and other transitory fluctuations in
quarterly consumption growth.  The only requirements are that the
instruments are uncorrelated with measurement error and temporary
consumption fluctuations, but correlated with the instrumented
variables.

Under habit formation or sticky expectations, \citet{som07} shows that
time aggregation makes ``true'' consumption growth $\Delta\log C_t^*$
(i.e., consumption growth without measurement error and transitory
consumption) follow an ARMA(1,2) process:
\be \Delta\log
C_t^*=c_0+\chi\Delta\log
C_{t-1}^*+v_t+\lambda_1(\chi)v_{t-1}+\lambda_2(\chi)v_{t-2}, \label{trueC}
\ee
where the $\lambda$s are complicated functions of $\chi$. In
addition, \citeauthor{som07} verifies that the MA(2) coefficient
$\lambda_2$ is close to zero for all reasonable values of
$\chi\in(0,1)$, so that $\Delta\log C_t^*$ is approximately
ARMA(1,1). Given these considerations, equation \eqref{cDynMacroPrime}
can be estimated using the IV estimator with instruments lagged at
least twice (e.g., dated as of time $t-2$ and earlier).%
\footnote{Ideally, it would be desirable to use instruments dated $t-3$ or earlier, but for some countries
the $t-3$ instruments did not have sufficient predictive power for the instrumented
variables.}

The baseline instrument set for the IV regressions consists of
variables that are strongly correlated with consumption growth and yet
unlikely to be correlated with measurement error: the unemployment
rate, a long-term interest rate, and an index of price
volatility.\footnote{Consumer price volatility is robustly negatively
  correlated with real consumption growth in all sample
  countries---this relationship is known among business cycle
  forecasters as the `Katona Effect'; see, e.g., \citet{oku81}, p.\
  216. In economic terms, periods of above-average price volatility
  tend to be associated with shocks that may also have an impact on
  permanent income. This instrument is attractive because it can be
  readily calculated for any country and it is unlikely to be
  correlated with measurement error in consumption growth. The variable appears
  to be widely used in the professional forecasting community but is not
  as common in academic work.  Price volatility at time $t$,
  $V^P_t\!\!$, is calculated as the coefficient of variation over the
  past four quarters: $V^P_t=\sigma^P_{t-4,t}/\mu^P_{t-4,t},$ where
  $\sigma^P_{t-4,t}$ is the standard deviation of price level between
  quarters $t-4$ and $t$ and $\mu^P_{t-4,t}$ denotes the mean of price
  level between quarters $t-4$ and $t$.}  Consumer sentiment is also
used as an instrument whenever available (the G-7 countries and
Australia), as in \cite{cfw94} and others.


\subsubsection{Estimation Results}

Table \ref{tAllC} summarizes the baseline estimation results for four
alternative econometric specifications nested in equation
\eqref{cDynMacroPrime}.\footnote{An advantage of our reduced-form
  estimates of the consumption function over the estimated dynamic
  stochastic general equilibrium models (DSGE, which started with the
  influential work of \citet{sw03}; see \citet{as07} for a review)
  is that we do not use informative priors. Our parameters could thus
  be used as an input for prior distributions (as in \citet{ds04}).}
The left panel reports the results from univariate regressions in
which each right-hand side variable enters the estimated specification
as the only regressor. The first column extends \citetalias{som07}
findings for the United States to the international data: the IV
estimates of consumption persistence $\chi$ are for all countries much
higher than would be the OLS estimates and are highly statistically
significant. The IV estimates of consumption persistence in table
\ref{tAllC} are on average about 0.7---a strong rejection of the
random walk proposition which implies a coefficient of zero.
\begin{comment}
This points to substantial
sluggishness in consumption growth, which can be interpreted either as
evidence that consumers have strong habits or that households
typically update their information about aggregate shocks only roughly
once every three quarters, i.e., $\Pi\approx0.3$ in the context of
quarterly data in equation \eqref{eq:stickyCrec}.
\end{comment}

The second column estimates the Campbell--Mankiw model. Our results are
broadly consistent with the evidence presented in
\cite{cm91}:  Rule-of-thumb consumers (for whom, by
assumption, consumption equals current income) are on average
estimated to earn about $\eta\approx0.4$ of aggregate
income. Interestingly, the estimates of $\eta$ in the left panel are
often less significant than those of consumption persistence $\chi$
and are in four cases insignificant. This means that---aside from the
question of how the Campbell--Mankiw model stands up against the
alternative of habit formation or sticky expectations---rule-of-thumb
spending behavior cannot be reliably detected in about a third of our
sample countries.

The third column investigates the relative importance of wealth (expressed as the ratio of net financial assets to income) in aggregate consumption dynamics. The coefficient on the wealth--to--income ratio, $\alpha$, turns out to be statistically significant only in four countries, where, in addition, the coefficient $\alpha$ has the opposite sign to that predicted by either precautionary saving theory or intertemporal substitution as channelled through the interest rate.  This is unsurprising for at least two reasons. First, the overwhelming significance of consumption (and also income) in the previous regressions implies a severe omitted-variable bias problem with the univariate regression that only includes wealth. Second, the previous literature generally finds little evidence of interest rate or precautionary saving effects in aggregate consumption data.%
\footnote{Microeconomic evidence suggests that the precautionary saving motive may be an important determinant of household-level consumption decisions, see for example \citet{cs97}, \citet{gp02}, and \citet{fss05}.
}

The fourth column displays the adjusted $R^2$s from the first-stage
regressions of consumption growth on instruments (denoted $\bar
R_c^2$). This measure of the strength of instruments ranges between
0.1 and 0.2 for most countries.%
\footnote{%
Ideally, one would prefer first stage $\bar R_c^{2}$ coefficients
larger than those generated by our instrument set for some countries.  For each individual country
it is possible to find a country-specific instrument set that performs considerably
better than our universal instrument set.  We preferred to run the well-understood
risks of weak instruments (coefficients biased toward the OLS value) rather than the much
more difficult to quantify risks associated with cherry picking a different instrument
set for each country.
}$^{,}$\footnote{The adjusted $R^2$s from
  the first-stage regressions on income growth are comparable with the
  $R^2$s from the first-stage regressions on consumption growth. The
  $R^2$s are much higher for the wealth--to--income ratio, about 0.8.
}
\begin{comment}
---an indication of significant
predictability in consumption growth. The finding is confirmed by the
first-stage F statistics, which are overwhelmingly significant
and---interestingly---are, as shown in figure \ref{fFstats}, in
general positively related to the estimates of $\chi$ across
countries: the better the instruments, the higher the estimated
persistence of consumption growth.
\end{comment}

The right panel of table \ref{tAllC} reports estimation results when
all three regressors are included in equation
\eqref{cDynMacroPrime}. The results strongly suggest that past
consumption growth is by far the strongest predictor of current
consumption growth. The average persistence parameter in the country
regressions falls only very slightly compared with the average
estimates from univariate regressions reported in the left panel (from
$\chi\approx0.7$ to $\chi\approx0.6$) and remains statistically
significant at the five percent level in ten of our thirteen countries. The predicted
income growth term dominates the lagged consumption term only in one country,
Germany.\footnote{Germany tends to be an outlier in all our IV regressions (reported and unreported).  This may
reflect data difficulties associated with comparing pre- and post-reunification
German data, or the unpredictable movements in consumption growth during the years
following reunification.} The last column of the right panel reports the p-values of
the Hansen's overidentification test---results of which imply that the null
of instrument exogeneity cannot be rejected.

Table \ref{tAllCG} averages the coefficient estimates from table
\ref{tAllC} across various country groups. As in table \ref{tAllC},
while the average consumption persistence $\chi$ falls relatively
little after income and wealth are added to the estimated equations
(compare the right and left panels of the table), the income and
wealth coefficients become essentially zero. The result holds for all
five groups of countries reported in the table which suggests
considerable homogeneity in $\chi$ among advanced economies, a fact
already apparent in the previous table with the results for individual
countries.


Table \ref{tAllCAlt}, whose format is identical to table \ref{tAllC},
estimates aggregate consumption dynamics with an alternative
instrument set, in which long-run interest rates and price volatility
have been replaced with income growth and the interest-rate spread. The
estimation results are broadly consistent with our baseline: (i) the
coefficient on lagged consumption growth in univariate regressions is
large and significant for ten countries, (ii) in the regressions that include all three
regressors, the coefficients on instrumented income growth and wealth
tend to be small and less often statistically significant compared
with univariate regressions (iii) lagged consumption growth beats
lagged income in nine horse-race regressions (but gets badly beaten in
German data).


\subsection{Kalman Filter/Maximum Likelihood Evidence on Sticky Consumption Growth}     \label{ssKalman}

As a more efficient alternative to IV, we also estimate the dynamics of consumption growth using the Kalman filter.  To proceed, it is necessary to specify an assumption about the stochastic process of measurement error. We follow the methodology of \citet{som07} and assume that measurement error in the log-level of consumption follows an MA(1) process.\footnote{Taking a classical approach with white noise measurement error in the level of consumption is \textit{a priori} not justifiable because all three main measurement error types are likely to be serially correlated. The measurement error is therefore allowed to be serially correlated in our model but the impact of error on the serial correlation
properties of the consumption data is limited. See \citet{som07} for a more detailed discussion.
} Observed consumption growth, $\Delta\log C_t$, can be written as the sum of `true' consumption growth, $\Delta\log C_t^*$, and a measurement error, $u_t$, as follows:
\begin{eqnarray}
\Delta\log C_t&=&\Delta\log C_t^*+u_t+(\theta-1)u_{t-1}-\theta u_{t-2}, \label{observedC}\\
\Delta\log C_t^*&=&c_0+\chi\Delta\log C_{t-1}^*+v_t+\lambda_1(\chi)v_{t-1}+\lambda_2(\chi)v_{t-2}. \label{trueC1}
\end{eqnarray}

As noted above, $\lambda$s are not free parameters but are complicated functions of $\chi$. The Kalman filter jointly estimates the sticky expectations coefficient $\chi$ and the degree of the first autocorrelation in measurement errors, $\theta$. The filter also generates separate estimates of `true' consumption growth, $\Delta \log C_t^*$, and the measurement error component, $u_t$. For the purposes of this subsection, we assume that the correlation structure of measurement error remains unchanged over the sample period.

The model described in equations \eqref{observedC} and \eqref{trueC1}
has been rewritten in a state-space form (see appendix B) and
estimated using consumption data for the countries in our dataset (listed
in table \ref{tCsource}). Table \ref{tKalman} presents the
estimation results. As in the case of the IV estimation, the
coefficient reflecting consumption growth stickiness, $\chi$, is large
and highly statistically significant in almost all sample
countries. The value of $\chi$ typically ranges between 0.6 and 0.8,
with only Denmark and the United Kingdom coefficients estimated
below 0.4. For the United States, the estimated consumption
persistence is about 0.7, which is consistent with previous studies.

It is encouraging that the Kalman filter estimates of consumption
persistence tend to be close to the IV estimates. This indicates that
even if instruments such as the lags of consumer sentiment and
interest rates happened to be contaminated with some measurement error
from the published consumption data, the practical impact on the IV
estimates reported in the previous subsection is likely not large. The
estimation results also suggest that measurement error in the level of
consumption is positively and significantly autocorrelated in about
half of our sample countries---a fact that is not surprising given the
interpolation techniques that are often used by statistical agencies when
constructing quarterly consumption data.

The Kalman filter's estimate of
``true'' consumption growth, $\Delta \log C_t^*$, is presented, along with the
raw data, in figures
\ref{fG7} and \ref{fOther}. The Kalman filter
estimation suggests that the share of transitory components in
published quarterly consumption data is large (about 50 percent for
the United States and even more for some countries), as a result of
the combination of measurement error and transitory components.%
\footnote{There is an
  interesting link between the signal-to-noise ratio from the
  estimated Kalman filter models, $\var (\Delta\log C^*_t)/\var
    (\Delta\log C_t)$, in Table \ref{tKalman} and the first-stage $R^2$ for consumption growth from the IV
  regressions in Table \ref{tAllC}. The correlation between the two statistics is about 80
  percent across countries, confirming that consumption growth can be
  predicted better in the countries with smaller measurement error and
  transitory fluctuations.  } The interesting
question is whether the Campbell--Mankiw predicted income term carries
any information about `true' consumption growth beyond the information
already contained in consumption persistence. In other words, the
question is whether the sticky expectations model is a better model of
consumption growth than the rule-of-thumb model after measurement
error and transitory consumption have been Kalman-filtered from the data.\footnote{The
figures are interesting in ways that are independent of our analysis in this 
paper; measurement errors seem to be systematically much larger in some countries
than in others, and for several countries appear to have declined markedly over time.  These
would be interesting topics for future research, but are beyond the scope of our analysis.}


Table \ref{tAllCMLE} presents the second-stage IV regression
results. ``True'' consumption growth (as measured by the Kalman smoother) displays little correlation with
predicted income, especially when the regressions include a term
reflecting sticky expectations (or habit formation). All
point estimates are much smaller than the 0.5 estimated by
\citet{cm89} and other authors. The estimates of $\eta$ are mostly
statistically insignificant, and none exceeds
0.15. Interestingly, the coefficient on lagged consumption growth changes
very little after adding predicted income and wealth to the univariate
regression with only consumption growth (compare the left and right panels
of the table).


\subsubsection{Relationship with the Structural Estimation Literature}\label{subsubsec:DSGE}

The state-space representation \eqref{observedC}--\eqref{trueC1}  fits nicely into the structural DSGE framework  recently proposed by \citet{ire04}, who estimates a small (three-equation) log-linearized model with the Kalman filter. Control variables $\mathbf{f}_t$ in his model can be solved in terms of state variables $\mathbf{s}_t$ and residuals $\mathbf{u}_t$:
\be
\mathbf{f}_t=\mathbf{Cs}_t+\mathbf{u}_t. \label{ireland}
\ee
\citeauthor{ire04}, p.\ 1210 views the disturbances $\mathbf{u}_t$ as follows: ``the residuals [$\mathbf{u}_t$] may \dots\ soak up  both measurement errors, but they can be interpreted more liberally as capturing all of the movements and co-movements in the data that the real business cycle model, because of its elegance and simplicity, cannot explain.'' Once we plug our transition equation for consumption growth \eqref{trueC1} into the measurement equation \eqref{observedC} the Kalman filter model we estimate above has exactly the structure \eqref{ireland} with $\mathbf{f}_t=\Delta\log C_t$, $\mathbf{s}_t=\Delta\log C^*_{t-1}$,  $\mathbf{u}_t=u_t+(\theta-1) u_{t-1}-\theta u_{t-2}+v_t+\lambda_1(\chi)v_{t-1}+\lambda_2(\chi)v_{t-2}$ and $\mathbf{C}=\chi$.

Thus the state-space representation \eqref{observedC}--\eqref{trueC1} can be interpreted as a stripped-down version of \citeauthor{ire04}'s model with consumption habits in which measured consumption is affected by a combination of measurement errors $u_t$ and shocks $v_t$ to ``true'' consumption $C_t^*$. As our main goal is to estimate consumption stickiness $\chi$, we do not take a stand on where the consumption shocks $v_t$ come from (be it news about income, wealth, interest rates, fiscal policy or something else).

Our model is simple enough to be estimable using the classical techniques, including the maximum likelihood estimator. From one point of view, this approach allows the data to have a complete control over the estimates of $\chi$. From another perspective, our estimates of $\chi$ could be used as an extra (out-of-sample) information to calibrate priors about consumption sluggishness (or habit persistence) in larger-scale Bayesian DSGE models. In fact, our Kalman filter estimation could also be seen as a special case of the Bayesian framework with uninformative priors. 



\section{Conclusions} \label{sConcl}

\citet{hal78} provided macroeconomists with a clean theoretical benchmark against which actual consumption data could be compared: Consumption growth should be essentially unpredictable.  In contrast with this benchmark, we find that, when econometric techniques that
account for measurement error are used, consumption growth exhibits a high degree of persistence or
``momentum.''  The stickiness of aggregate consumption growth can be
interpreted as reflecting the behavior of fully informed households
with a strong consumption habit, or the behavior of an aggregate
economy in which households are not always perfectly up to date
in their knowledge of macroeconomic developments. Fitting the model
to data from thirteen countries, we estimate that consumption growth
persistence is always significantly above the random-walk benchmark of 0
and is never robustly different from about 0.7. Our
analysis also suggests that, on balance, the model of sticky
consumption growth describes aggregate consumption data better than
the rule-of-thumb model of \citet{cm89}, although our point
estimates do typically indicate that a modest proportion of households
(in the range of 10--20 percent) may simply consume their current
income every quarter.

Our findings imply that the large literature claiming to find
evidence of sticky consumption growth in the U.S. probably cannot be
explained away as reflecting time aggregation problems or other mistreatment
of the data, suggesting that many of
the insights gleaned from that literature are likely applicable to
other countries as well.  (However, it is worth bearing in mind that
analyses that rely heavily on the literal interpretation of the
habits-in-the-utility-function framework, such as calculations of
the welfare cost of aggregate fluctuations, may not
hold up under alternative interpretations of consumption growth
stickiness.)

Our analysis also strengthens a key policy message about the sluggish average response
of consumption to monetary and fiscal policy innovations highlighted earlier in the context
of the habit formation literature---an important policy consideration at the current
cyclical juncture in many countries, including in the United States.



\newpage

% Tables and Figures
\clearpage

\input../tables/tAllCountriesFinBase.tex
\input../tables/tAllCgroupsBase.tex
\input../tables/tAllCountriesFinAlt.tex
\input../tables/t1stStage.tex
\input../tables/tAllCountriesFinMLE.tex

\clearpage

\begin{figure}
\caption{Measured and ``True'' Consumption Growth---G7 Countries} \label{fG7}
\begin{center}
\includegraphics[height=5in,width=\textwidth]{../figures/fG7}
\end{center}
\end{figure}

\begin{figure}
\caption{Measured and ``True'' Consumption Growth---Other Countries} \label{fOther}
\begin{center}
\includegraphics[height=6.7in,width=\textwidth]{../figures/fOther}
\end{center}
\end{figure}

\clearpage\vfill\eject
\pagebreak

\appendix

\section*{Appendix A: Description of Data}\small
Data for the G-7 economies are from the Haver Analytics database. Data for other countries are from the database of the NiGEM model of the NIESR Institute, London. The original sources for most of these data are OECD, Eurostat, national statistical offices and central banks. Income is measured as personal disposable income. Wealth is approximated using data on the net financial wealth. All series  were deflated with consumption deflators and expressed in per capita terms. The population series are from  DRI International and were interpolated from annual data to quarterly observations. Japan is not included  in our sample as creating a quarterly dataset with consumption data going prior to 1980 would involve  splicing consumption series based on three very different methodologies. Adjustments to the Japanese  national accounts methodology in 2002 and 2004 have significantly improved the reliability of quarterly  consumption series but the current-methodology data are only available since Q1:1994 (\citet{imf06}).

We thank Roberto Golinelli for consumer sentiment series for G7 countries and Australia used (and described in detail) in \cite{gp04}. (We have not used consumer sentiment series for the remaining countries, because the data are not available before 1985.) We are grateful to Carol Bertaut and Nathalie Girouard for providing us with the data used in \cite{ber02} and \cite{cgp04}, respectively. Ray Barrell, Amanda Choy and Robert Metz answered our questions about the NiGEM's database.

\begin{table} \large\small
\caption{Consumption Data, Its Sources, and Samples for IV Regressions \label{tCsource}}
\begin{center}
\begin{tabular}{@{}lccc@{}}
\toprule
 Country & Time Frame  & Consumption Series  &  Source \\
\midrule
\multicolumn{4}{l}{\!\!\!G7 Countries} \\
Canada & Q4:1970--Q3:2002 & Nondurables${}+{}$Services & Haver Analytics\\
France & Q1:1985--Q4:2003 & Nondurables${}+{}$Services & Haver Analytics\\
Germany${}^{\dagger}$ & Q4:1975--Q4:2002 & Nondurables${}+{}$Services & Haver Analytics\\
Italy & Q1:1981--Q4:2003 & Nondurables${}+{}$Services & Haver Analytics\\
United Kingdom & Q1:1974--Q4:2003 & Nondurables${}+{}$Services & Haver Analytics\\
United States & Q3:1962--Q2:2004 & Nondurables${}+{}$Services & Haver Analytics\\
\midrule
\multicolumn{4}{l}{\!\!\!Other Countries}\\
Australia & Q4:1975--Q4:1999 & Total PCE & Haver Analytics\\
Belgium & Q2:1980--Q4:2002 & Total PCE & NiGEM/MEI \\
Denmark & Q1:1977--Q2:2003 & Total PCE & NiGEM/MEI \\
Finland & Q3:1973--Q2:2003 & Total PCE & NiGEM/MEI \\
Netherlands & Q1:1975--Q4:2002 & Total PCE & NiGEM/MEI \\
Spain & Q1:1978--Q4:1999 & Total PCE & NiGEM/MEI \\
Sweden & Q1:1977--Q4:2002 & Total PCE & NiGEM/MEI \\
\bottomrule\end{tabular}
\end{center}
 {\footnotesize Notes: ${}^{\dagger}$: Regressions for Germany were estimated with a reunification dummy in Q1:1991; Source: NiGEM---Database of the NiGEM model of the NIESR Institute, London, MEI---Main Economic Indicators of OECD.}
\end{table}


\section*{Appendix B:  Details of the Kalman Filter Estimation}

Following \citet{som07}, equations \eqref{observedC} and \eqref{trueC1} can be rewritten in the state-space
form with the measurement equation:

\begin{equation*}
\Delta \log C_{t}=c_{0}+\big[1\,\,0\,\,0\,\,1\,\,0\,\,0\big] \left[
\begin{array}{c}
\Delta \log C_t^* \\
u_t \\
-u_t+\theta\Delta u_t \\
\Delta u_t+\theta\Delta{u}_{t-1} \\
v_t \\
v_{t-1}%
\end{array}%
\right] +0,
\end{equation*}
and the state-evolution equation:
\begin{equation*}
\left[
\begin{array}{c}
\Delta \log C_t^* \\
u_t \\
-u_t+\theta\Delta u_t \\
\Delta u_t+\theta\Delta u_{t-1} \\
v_t \\
v_{t-1}%
\end{array}%
\right] =\left[
\begin{array}{cccccc}
\chi & 0 & 0 & 0 & \lambda _{1} & \lambda _{2} \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & -\theta & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0
\end{array}%
\right]
\left[
\begin{array}{c}
\Delta \log C_{t-1}^* \\
u_{t-1} \\
-u_{t-1}+\theta\Delta u_{t-1} \\
\Delta{u}_{t-1}+\theta\Delta u_{t-2}\\
v_{t-1} \\
v_{t-2}%
\end{array}%
\right]
+\left[
\begin{array}{c}
v_t \\
u_t \\
(\theta-1)u_t \\
u_t \\
v_t \\
0%
\end{array}%
\right] ,
\end{equation*}%
and with the associated covariance matrices $H=0$ and
\begin{equation*}
Q=\left[
\begin{array}{cccccc}
\sigma _v^2 & 0 & 0 & 0 & \sigma _v^2 & 0 \\
0 & \sigma_u^2 & (\theta-1)\sigma_u^2  & \sigma_u^2 & 0
& 0 \\
0 & (\theta-1)\sigma_u^2 & (\theta-1)^{2}\sigma_u^2
& (\theta-1)\sigma_u^2 & 0 & 0 \\
0 & \sigma_u^2 & (\theta-1)\sigma_u^2 & \sigma_u^2 & 0
& 0 \\
\sigma_v^2 & 0 & 0 & 0 & \sigma_v^2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0%
\end{array}%
\right],
\end{equation*}
respectively.

The state-space form is estimated with the Kalman filter using the consumption series described in table \ref{tCsource}. The coefficients $\lambda _{1}$ and $\lambda _{2}$ are not free parameters but instead depend on the consumption persistence coefficient $\chi$: $\lambda _{1}=f(\chi),\lambda _{2}=g(\chi )$ as detailed in the appendix to \citet{som07}. Our Kalman filter estimation incorporates this relationship between $\chi $, $\lambda _{1}$, and $\lambda _{2}$.

Figures \ref{fG7} and \ref{fOther} display the measured consumption growth  $\Delta\log C_t$  and true consumption $\Delta\log C_t^*$ estimated using the Kalman smoother based on the above state-space model.

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\end{document}
