Assessing GMM Estimates of the Federal Reserve
Reaction Function
Clémentine Florens, Eric Jondeau and Hervé Le Bihan∗
December 2000
Abstract
Estimating a forward-looking monetary policy rule by the Generalized Method
of Moments (GMM) has become a popular approach since the influencial paper by Clarida, Gali, and Gertler (1998). However an abundant econometric
literature, motivated by asset-pricing models, points to the unappealing small
samples properties of GMM. Focusing on the Federal Reserve reaction function, this paper assesses GMM estimators in the context of monetary policy
rules. First, we show that three usual alternative GMM estimators yield substantially different results in the case of forward-looking Taylor rules. Then, we
contrast the GMM estimators to two Maximum-Likelihood (ML) estimators.
A small structural model of the economy is used to perform ML estimation.
Results suggest that the inflation coefficient in the Taylor rule is lower than
that produced by GMM. We perform Monte-Carlo simulations, which confirm
that GMM small-sample bias leads to overstatement of the inflation coefficient.
The simulations also underline the wide dispersion of GMM estimates in small
samples, especially in the case of continuous-updating GMM.
Keywords: Forward-looking model, monetary policy reaction function, GMM
estimator, FIML estimator.
JEL classification: E52, E58, F41.
∗ GREMAQ, Université de Toulouse 1, Manufacture des Tabacs, 21 allée de Brienne, 31000
Toulouse, France. Banque de France, 41-1391 Centre de recherche, 31 rue Croix des Petits Champs,
75049 Paris, France. E-mail : [email protected] ; [email protected] ;
[email protected]. We are grateful to Simon Gilchrist for making his AIM algorithm
code available to us.
1
1
Introduction
The benchmark Taylor rule states that the central bank should set the short-term
interest rate in proportion of the inflation rate and the output gap. However, since
Taylor’s (1993) prominent contribution, a number of empirical as well as theoretical
studies have claimed that central banks may have a forward-looking behavior (Clarida and Gertler, 1997, Clarida, Gali, and Gertler, 1998, 2000, Haldane and Batini,
1998). The econometric implication of this assumption is that, to deal with the presence of expected inflation in the policy rule, an adequate estimation method such
as the Generalized Method of Moments, must be used. Estimating forward-looking
monetary policy rules by GMM has become a popular approach since the influential
work by Clarida, Gali, and Gertler (1998).1
The purpose of the present paper is to assess the robustness of GMM estimates in
the context of forward-looking reaction functions. This investigation is motivated by
the growing literature on the small-sample properties of the GMM procedure. Using
Monte-Carlo experiments, Tauchen (1986), Fuhrer, Moore, and Schuh (1995) and Andersen and Sørensen (1996) provided evidence that GMM estimators can be strongly
biased and widely dispersed in small samples. These studies have investigated small
sample properties of the GMM mainly in the context of asset-pricing, consumption
or inventories models. Forward-looking monetary policy rules offer an original field
for investigating GMM properties.
The methodology we use is to contrast the GMM estimators with an alternative estimation procedure for forward-looking models: the maximum likelihood (ML)
method. The ML approach involves the estimation of a structural model of the economy. We use a version of the Rudebusch and Svensson (1998) model and solve the
whole rational-expectations model using the Anderson and Moore (1985) procedure.
The properties of ML and GMM estimates of the reaction function are then compared
using Monte Carlo simulations.
While we focus on the widely studied Federal Reserve reaction function over the
period 1979 to 1998, we present some original empirical results. In the literature
on the reaction function, estimation has been typically performed using the twostep GMM estimator. We also estimate the rule using two alternative versions of the
GMM estimator, as well as the ML and FIML. ML estimates are found to provide a
clearer support than GMM in favor of the forward-looking specification. In addition,
Monte-Carlo simulations point to a small sample bias of the GMM, in particular
towards overstating the response of interest rate to expected inflation. The bias is
limited, so that our overall assessment of GMM in the case of a monetary policy
rule is less critical than that of Fuhrer, Moore, and Schuh (1995) in the case of
inventories. However GMM parameters estimates are imprecise, especially in the case
of continuous-updating GMM.
The paper is organized as follows. The benchmark specification of a forwardlooking reaction function is presented in section 2. Section 3 summarizes GMM esExamples include Clarida, Gali, and Gertler (2000), Mehra (1999), Orphanides (2000) for US
data, or Verdelhan (1999), Peersman and Smets (1998), Gerlach and Schnabel (1999), Nelson (2000),
Angeloni and Dedola (1999) for European data.
1
2
timation procedures and presents empirical results. In section 4 ML estimation is
introduced and implemented. Section 5 provides some interpretation of empirical results, using Monte-Carlo simulations. Section 6 concludes.
2
The monetary-policy reaction function: specification
In the baseline Taylor rule, the central bank is assumed to set the level of the nominal
short-term interest rate as a function of the rate of inflation and output gap:
it
= i∗ + β (π − π ∗) + γy
t
(1)
t
where i denotes the short-term nominal interest rate, π the inflation rate, y the
output gap. The inflation rate is the four-quarter moving average of inflation in the
implicit GDP deflator. The output gap is defined as the percent deviation of real GDP
from its deterministic trend. The constant i∗ is the long-run equilibrium nominal
interest rate and π∗ is the inflation target. The output-gap target is assumed to be
zero. The coefficient on inflation β is a crucial parameter since in most macroeconomic
models based on the Phillips curve and the I-S curve, β > 1 is a relevant condition
for stability (Taylor, 1999, and in the context of forward-looking models, Kerr and
King, 1996, Clarida, Gali, and Gertler, 2000).
The Taylor rule has received a widespread attention in the empirical literature.
In particular, it has be shown to provide a rough description of US monetary policy
during the Greenspan and Volcker tenures (Taylor, 1993, 1998 Judd and Rudebusch,
1998). However, most estimate consider ”modified” Taylor rules. First, central banks
often appear to smooth changes in interest rates. Many motivations for such an
interest-rate smoothing have been proposed (Sack and Wieland, 1999, Woodford,
1999). For instance, in case of uncertainty concerning the model’s parameters, it is
optimal for the central bank to adjust interest rates only gradually. Therefore, the
short-term rate is allowed to adjust gradually to its target given by equation (1).
A second issue, central for analyzing monetary policy, is whether the central bank
sets the level of the interest rate as a function of observed or expected inflation.
Though the terminology is not uncontroversial, we will refer to the policy rule as being ”forward-looking” if the central bank reacts to some expectation of inflation. A
number of authors have claimed that the forward-looking reaction function is consistent with the observed behavior of central banks (Clarida and Gertler, 1997, Clarida,
Gali, and Gertler, 1998, 2000, Orphanides, 1998). Most central banks explicitly
claim that they do not only consider past or current economic conditions, but they
also include economic forecasts in their macroeconomic conditions statement. In addition, from the normative point of view, that policy rules should be forward-looking
has been advocated by Haldane and Batini (1998) and Svensson (1997). Therefore,
a large number of recent studies estimate the following more general specification,
which incorporates the expected inflation rate:
t
t
it
t
= ρi −1 + (1 − ρ) (i∗ + β (E π + − π∗) + γy −1) + ε
t
t
3
t
h
t
t
(2)
where ε denotes an exogenous random policy shock, h is the horizon of the central
bank with respect to inflation and E denotes the mathematical expectation conditional to the information set containing all variables dated t − . The parameter ρ
represents the degree of interest-rate smoothing. We do not assume that the central
bank reacts to expected output gap.
Note that this specification encompasses both backward- and forward-looking
rules. The backward-looking reaction function corresponds to the case h − .
Arguably, equation (2) with h equal to 0 is already a forward-looking specification
since inflation is not observed contemporaneously (Orphanides, 1998). We consider
the baseline forward-looking rule to be with h , but we also examine horizons h
ranging from 0 to 4 quarters ahead. An important element is the information set
available to the central bank when formulating its expectation. Current inflation and
output gap are not assumed to be observed in real time by the central bank. Rather,
we assume here that, at date t, only inflation and output gap at date t − are in the
information set of the central bank.2
In our estimation specification, we simply rewrite equation (2) as
i
ρi −1
− ρ α βE π + γy −1 ε
(3)
where α i∗ βπ∗. Estimating equation (3) provides estimates of the weights on
inflation and output gap in the monetary policy rule and the speed of adjustment to
the rule. The long-run inflation target π∗ is not identified but can be computed if we
assume a value for the long-run equilibrium real interest rate rr∗. For instance in the
case one chooses the sample average real rate, denoted rr, one obtains
t
t
1
=
1
= 4
1
t =
=
t
+ (1
)(
+
t
t
h +
t
) +
t
+
π
∗
=
1
(β
−
1)
(rr
−α
)
.
(4)
3 GMM estimation
Given that expectations are unobserved, the standard approach is to substitute
3
E π + with the actual value π + in equation (3).
Since the current interest rate
shock is likely to affect future inflation, the OLS procedure provides biased estimators, whereas GMM provides a consistent estimation procedure. GMM estimators
have been shown to be strongly consistent, asymptotically normal (Hansen, 1982) and
have been applied to rational expectation models since Cumby, Huizinga, and Obstfeld (1983) and Hansen and Singleton (1982). In the rational expectation context,
the GMM approach is very appealing, because it does not require strong assumptions
on the underlying model. In fact, it only requires identifying relevant instrument
variables, strongly correlated with RHS variables, but uncorrelated with innovations.
t
t
h
t
h
2
To some extent, even this assumption is questionable. The output gap is measured precisely
after many quarters. See Orphanides (2000) who claims that, if monetary policy during Chairman
Burns tenure appears a posteriori non-optimal, this is because the output-gap measure has been
dramatically corrected since the end of the 70s.
3
An alternative approach for models with expectations, not pursued here, is to use actual inflation
forecasts. McNees (1985, 1992) and Orphanides (1998) used the Board of Governors’ staff forecasts
presented at each FOMC meeting.
4
The central bank’s information set at time t is here assumed to include four lagged
values of interest rate, inflation and output gap. (We reduce intentionally the information set to the variables used in the structural model discussed in section 3.)
These instruments are plausibly correlated with future inflation.
3.1 GMM estimators
Let equation (3) be expressed in standard regression notation as
y = Xθ + ε
x1 , ..., x
is a vector of
with y a T × vector and X a T × n matrix. X
observation and θ is the n × vector of unknown parameters. Let Z be the T × q
matrix of instrumental variables, in which each of the q instruments (q > n) are
assumed to be orthogonal to ε, so that the orthogonality conditions Eg θ
are
satisfied, where g θ Z y − X θ Z ε .
The GMM estimator, denoted θ̂ , is the value of θ that minimizes the scalar
(
1)
(
(
t = (
)
nt )
t
1)
(
)
t ( ) = 0
t ( ) =
t ( t
t ) =
t t
GM M
QT (θ) = ḡT (θ) WT ḡT (θ)
where the q × vector ḡ θ denotes the sample average of g θ . W denotes
the q × q GMM weighting matrix. An asymptotically efficient estimator is obtained bychoosing
as a weighting
matrix a consistent estimator of W V −1, where
V
is the long-run covariance matrix of g θ . The GMM
− E g θ g− θ
→∞
estimator is then defined by:
(
(
T ( )
1)
t ( )
T
)
=
= lim
j
t( )
j
j
j ( )
t
t ( )
θ̂T = (X ZWT Z X )
−1
X Z WT Z y
.
(5)
When innovations are heteroskedastic and serially correlated, the optimal value of
the weighting matrix W is given by the inverse of the Newey and West’s (1987)
estimator. For a given value of θ̂ , the weighting matrix is estimated as
T
WT =
T
−1
where
ŜT
L
ŜT = S0 +
l=1
w ( l ) ( Sl + Sl )
(6)
with S 1 = +1 ε̂ ε̂ − Z Z − and w l
− +1 . The asymptotic covariance
−1
matrix of θ̂ is estimated by V̂ X Z W Z X .
Note that we need an estimate of W before estimating θ̂ and we need an estimate
of θ̂ before estimating W . In the following, we implement three alternative versions
of the GMM estimator often considered in the literature on GMM.
In the first approach, the parameter vector is estimated with the two-step twostage least squares, or ”two-step GMM”, initially proposed by Cumby, Huizinga, and
Obstfeld (1983) and Hansen and Singleton (1982).4 An initial estimate of the parameters is obtained with two-stage least squares to construct an estimate of the optimal
l = T
T
t l
t t
l
t
t
( ) = 1
l
= (
T
T
T
T
l
L
)
T
T
4
This approach has been adopted, for instance, by Clarida, Gali, and Gertler (1998, 2000) and
Peersman and Smets (1998) in the context of the reaction function.
5
weighting matrix W (0) using equation (6). Then, the two-step GMM estimate, denoted θ̂(1), is obtained from equation (5) with W (0) as weighting matrix.
The second approach, suggested by Hansen (1982), Ferson and Foester (1994) or
Hansen, Heaton, and Yaron (1996), relies on estimating parameters and the weighting matrix recursively. Thus, equations (5) and (6) are estimated
repeatedly, until
()
( −1)
convergence of parameter θ̂ is reached, such that θ̂ − θ̂
is less than a convergence criterion (chosen here to be equal to −5 ). θ̂( ) denotes the GMM estimator
at step i. This procedure is called ”iterative GMM” in the following.
In the last approach, called ”continuous-updating GMM”, developed by Hansen,
Heaton, and Yaron (1996) and studied in Stock and Wright (2000), the weighting
matrix is continuously altered as θ is changed in the minimization. Therefore, the
continuous-updating GMM estimator is the minimizer of
T
T
T
T
i
T
i
10
ḡT (θ) WT (θ) ḡT (θ)
i
T
T
.
GMM estimates are justified on asymptotic grounds. Small-samples properties of
the GMM procedure have been studied in an abundant work. The general result is
that the asymptotic theory provides a poor approximation in finite samples. Using
Monte-Carlo experiments, Tauchen (1986), Kocherlakota (1990), and Andersen and
Sørensen (1996) provided evidence that GMM estimates can be strongly biased in
small samples. More specifically, there is a strong variance/bias trade-off as the number of instruments is increased. Including more moment restrictions (more information) improves the estimation performance, but, in small samples, it also deteriorates
the precision of the estimated weighting matrix. Hansen, Heaton, and Yaron (1996)
compared the small-sample properties of alternative GMM estimators. The two-step
estimator and the iterative estimator are shown to be more widely median biased than
the continuous-updating estimator. But, the distribution of the continuous-updating
estimator has fatter tails.
The alternative GMM estimators have the same asymptotic distribution. Nevertheless, the continuous-updating GMM offers the advantage over the two-step and
iterative GMM approaches that estimates are invariant with respect to the scale of the
data and to the initial weighting matrix for W . Moreover, in an asset-pricing context, the iterative GMM approach is found to have superior small-sample properties
as compared with the two-step GMM (Ferson and Foerster, 1994).
As stressed by Nelson and Startz (1990) and Maddala and Jeong (1992) in the
context of the Instrumental Variable estimator or by Stock and Wright (2000) in
the context of the GMM estimator, the poor performance of these estimators can
be related to the weak correlation between instruments and the relevant first-order
conditions. We will consider this point in the last section of the paper.
Another important feature of GMM estimation is that the information set contains
more instruments than parameters to be estimated. Therefore, when the model is
true, there are q − n linearly independent combinations of the orthogonality conditions
that ought to be close to 0, but are not actually set to 0. This provides us with a test
of the over-identifying restrictions. A rejection of these restrictions would indicate
T
6
that some variables in the information set do not satisfy the orthogonality conditions.
The Hansen’s (1982) overidentifying restriction test statistic is
JT = T ḡT
θ̂ T
WT ḡT
θ̂T
,
which is distributed as a χ2 with q − n degrees of freedom.
3.2 Empirical results
We now present estimation of the monetary policy reaction function based on equation (3). We consider monetary policy for the Federal Reserve over the period from
1979:Q3 to 1998:Q4.5 Our sample period covers P. Volcker (1979:Q3-1987:Q2) and
A. Greenspan (1987:Q3 up to now) tenures. The assumption that a unique monetary policy regime has prevailed over this period is controversial. However statistical
evidence in favor of reaction function instability within this sample is at most mixed
(Judd and Rudebusch, 1998, Estrella and Fuhrer, 1999). We use quarterly data,
drawn from the OECD databases BSDB and MEI. Consistently with the quarterly
frequency of the data, we use the three-month interest rate as a proxy for the central
bank’s intervention rate. The output gap is defined by the deviation of (log) real
GDP from (log) potential GDP. Potential GDP is computed using a deterministic
trend with a break in trend growth rate in 1974.
Table 1 reports parameter estimates of the forward-looking reaction function for
different horizons h (from h to ). Estimates obtained using the two-step GMM,
the iterative GMM, and the continuous-updating GMM are reported in Panel A,
B, and C respectively. The lag length in the weighting matrix is chosen equal to
the conventional value L
on quarterly data or determined using the optimal
bandwidth as suggested by Andrews (1991), in which case we have L L∗.
First, we consider the two-step GMM approach with L . For the contemporaneous specification (h ), the estimated inflation parameter is remarkably close
to the . value suggested by Taylor, and the output-gap parameter (γ . ) is
also quite close to the . value suggested by Taylor. For distant horizon, the point
estimate of β increases significantly : it ranges between . for h
to . for
h
. The output-gap parameter γ becomes insignificant for models which assume
a forward-looking inflation (h > ). This result indicates that current output gap
appears in the reaction function as a forecast of future inflation only. The standard
error of estimate is the smallest for the model with contemporaneous inflation, and
it increases with the horizon of inflation.
The iterated GMM and the continuous updating GMM yield a higher estimate
for β for the current inflation specification. Furthermore, as the horizon becomes
forward, β estimates increase strongly and reach very high values. For horizon h ,
parameter β is 3.1 for the iterative GMM and 3.5 for the continuous-updating GMM.
The output gap has a significant weight in the reaction function for models with
current inflation and with one-lead inflation (h
and 1). Standard errors of
parameters also increase significantly, so that the t-stat reduces for distant horizons.
= 0
4
= 4
=
= 4
= 0
1 5
= 0 38
0 5
1 56
= 0
1 92
= 4
0
= 4
=
5
0
Estimates end in 1998:Q4, because our GMM estimates use data until the end of 1999.
7
The continuous-updating GMM estimator produces similar estimates to the Iterative
GMM estimator, but standard errors are much larger.
Computing the weighting matrix with L
or L L∗ does not materially
alter the estimated parameters for the iterative and continuous-updating GMM approaches. In the case of Two-step GMM, the optimal bandwidth in the weighting
matrix is estimated to be L∗ . In this case, the β estimate is much larger than the
estimate obtained with L . Moreover, it is closer to the estimates obtained with
the iterative GMM and continuous-updating approaches.
One major difference between the three GMM procedures concerns standard errors
of parameter estimates. Considering the inflation parameter β , we find that the
standard error is much larger using the iterative GMM as compared with the twostep GMM, and even larger using the continuous-updating GMM. For instance, for
horizon h , the standard error of β is 0.29 for two-step GMM, 0.46 for iterative
GMM and as high as 7 for continuous-updating GMM.
To sum up, several empirical results are worth emphasizing. First, the weight of
inflation in the reaction function is larger for forward-looking specifications than for
the contemporaneous specification. The standard error of β also increases with the
horizon of inflation. Second, the weight of output gap becomes insignificant for more
distant horizons. Third, the forward-looking specifications do not fit the data as
well as the contemporaneous specification. This deterioration is strengthened by the
increase in the smoothing parameter ρ with the horizon. The smoothing is reinforced
as the horizon lengthens, because other determinants become less significant. Most
of these results are in line with those obtained by Orphanides (1998) using ex-post
revised data and the IV estimator. In particular, for horizon h
, he finds an
inflation parameter equal to . , with a very large standard error of 3. Fourth,
results provided by iterative GMM and continuous updating GMM are somewhat at
variance with those provided by the usual Two-step GMM estimator.
= 4
=
= 1
= 4
= 4
= 4
3 7
3.3 Testing the robustness of the forward-looking reaction
function
As a robustness check, we wish to test the hypothesis that the central bank is forwardlooking against the alternative hypothesis that it is backward-looking. This issue is
particularly relevant for our purpose, since the use of the GMM is motivated by the
forward-looking nature of the specification. Since both hypotheses are not nested,
we estimate an encompassing model, which incorporates past inflation and expected
future inflation. To limit the problem of multicolinearity, we contrast the h −
horizon with the h horizon. Therefore, we estimate6
=
= 4
rt = ρrt−1 + α + (1
−ρ
)
β b π t−1 + β f Et π t+4 + γyt−1
+
εt
.
1
(7)
A perhaps more appealing specification for inflation would be: β (λπt−1 + (1 − λ) Etπt+4 ), in
which λ is constrained to range between 0 and 1. But, as appears in empirical results, in most cases,
free estimation of λ provides a point estimate either lower than zero or larger than 1. Therefore,
constraining the value of λ would boil down to estimate either the forward-looking specifications
presented in the tables or a backward-looking one.
6
8
Table 2 reports parameter estimates of equation (7), using the three GMM procedures with L
and L L∗. First, using the two-step GMM, we obtain a positive
parameter β and a negative β whatever the lag length in the weighting matrix.
Therefore, we may conclude that lagged inflation does not matter when expected
inflation is already introduced. We thus obtain a conclusion similar to the result of
Clarida, Gali, and Gertler (1998) who used the same GMM approach.
However, the iterative and continuous-updating GMM procedures provide the
opposite result. We obtain a positive estimate of the past inflation coefficient β ,
but a negative estimate of the expected future inflation coefficient β . With the
iterative procedure, both parameters are statistically significant, whereas they are
not significant with the continuous-updating GMM. These estimates point toward a
backward-looking reaction function.
Surprisingly, the different approaches based on the GMM provide a diverging
assessment of the forward-lookingness of the reaction function. Yet, these estimators
are supposed to be consistent and unbiased. These contrasting results suggest that
the GMM approach may lack robustness in our sample.
= 4
=
f
b
b
f
4 ML estimation
4.1 The ML approach
In this section, we contrast GMM estimation with an alternative approach, maximum
likelihood (ML) estimation. The ML approach requires that an auxiliary model is
estimated for the forcing variables (here, output gap and the inflation rate). The
auxiliary model, together with the equation of interest is then solved for forward
looking variables, yielding cross equation restrictions. An appealing advantage of
ML over GMM, in forward-looking models, is that expectations obtained with ML
estimation are fully model-consistent. Thus, the expected inflation which appears
in the reaction function (2) is consistent with the inflation equation of the model.
The ML approach is of course demanding since it requires that a structural model
is estimated for variables other than interest rates. However in the present case,
the widely used IS curve - Phillips curve framework provides in our view a reliable
benchmark model of the output-inflation joint dynamics.
For the problem at hand, the estimation can be performed in two ways: First, the
FIML procedure involves the joint estimation of the Phillips curve, the I-S curve and
the reaction function. Second, the two-step ML procedure is based on a preliminary
estimation of the Phillips curve and the I-S curve. The latter implies a loss of efficiency, but reduces the computational burden dramatically. We will provide evidence
on both procedures in the following.
Both ML estimations of the forward-looking reaction function are implemented
using the procedure developed by Anderson and Moore (1985). This procedure computes the reduced form of a forward-looking model. The forward-looking model can
9
be written in the format
0
θ
i=−τ
Hi xt+i +
(8)
Hi Et (xt+i ) = εt
i=1
where x π , y , r and H are conformable square matrices containing the model’s
coefficients. The random innovations ε are iid N , . τ and θ denote leads and
lags respectively (τ θ
in our empirical application). In the present context,
inflation leads are the only forward-looking terms.
Using the generalized saddlepath procedure of Anderson and Moore (1985), the
expectation of future terms in equation (8) is expressed as a function of expectations
of lagged terms:
−1
E x+
B E x+ +
k > .
(9)
t = (
t
t
t)
i
(0 Σ)
t
=
= 4
t (
k) =
t
t (
i
i=−τ
t
i)
k
0
Then, equation (9) is used to derive the expectation of future terms as a function
of the present and past terms. Substituting expectations into equation (8) gives the
so-called observable structure
0 S x ε .
(10)
+
i
i=−τ
t
i =
t
This procedure is very efficient and can be applied to a wide range of applications.
It has been widely used, see, e.g., Fuhrer, Moore, and Schuh (1995), Fuhrer and Moore
(1995a,b).
Finally, the concentrated log-likelihood function is computed using the observable
structure (10):
ln
L =
−
1
2
nT ln (2π )
−
1
2
T
ln Σ̂ +
t=1
ε̂t Σ̂
−1 ε̂t
1 where
=1 ε̂ ε̂ is the estimated covariance matrix.
The log-likelihood function is maximized using the BFGS algorithm. The parameter covariance matrix is computed as the inverse of the Hessian of the log-likelihood
function.
Σ̂ =
T
t
T
t t
4.2 Estimation results
4.2.1 The structural model
As a structural model, we estimate a version of the Rudebusch and Svensson model
(1998), which includes a Phillips curve (PC) and an I-S curve. This model embodies the main features of the standard macroeconomic paradigm and has proved to
be a robust representation of the US economy. The backward-looking nature of this
model can be pointed as one potential source of mis-specification. However, no compelling empirical forward-looking counterpart of the RS model has so far emerged
(see Estrella and Fuhrer, 1998). We estimate the following PC/I-S model:
π
α 1 π −1
α 2 π −2
α 3 π −3
α 4 π −4
α y −1
u
(11)
t
=
π
t
+
π
t
+
π
10
t
+
π
t
+
y
t
+
t
yt
β y1 yt−1 + β y2 yt−2 + β r (it−1
=
− π −1 − β0
t
) +
vt
.
(12)
The Phillips curve relates quarterly inflation π to lags of inflation and to lagged
output gap. Since we do not reject the assumption that the four autoregressive
parameters freely estimated sum to one, we impose this restriction in equation (11),
so that α 4 − α 1 − α 2 − a 3. This is consistent with an accelerationist form of
the Phillips curve. Moreover, we do not include a constant term in the equation, so
that the output gap is null in the long run. The I-S curve relates the output gap to its
own lags and to the difference between the lagged short nominal rate and the lagged
inflation rate. This last term is a proxy of the short real interest rate. Note that
unlike Rudebusch and Svensson, we include the lagged short real rate rather than a
4-quarter moving average of the short real rate.
The empirical model is very close to the model estimated by Rudebusch and
Svensson (1998) over the period 1961:Q1 to 1996:Q2. To be consistent with our
estimate of the reaction function, we use the sample period 1979:Q3 to 1998:Q4. OLS
estimation is reported below. Standard errors are shown in parentheses. Adjusted
2
R , standard error of residuals and the Durbin-Watson statistic are also reported.
t
π
πt
see
yt
see
= 1
π
=
0.499
=
0.869
=
1.309
=
0.704
(0.117)
(0.104)
π
π
π t−1 + 0.212 π t−2 + 0.268 π t−3 + 0.021 π t−4 + 0.100 yt−1 + ût
, R̄2
yt−1
, R̄2
(0.127)
= 0.854
, DW
(0.126)
.
= 0.907
t
, DW
(0.048)
= 1.944
− (0. 103) y −2− (0. 037)
0 393
(−)
it−1
0 071
.
.
− π −1− (1. 155)
t
3 731
.
+
v̂t
(13)
(14)
= 2.176
The sensitivity of inflation to output gap and the responsiveness of output gap to the
short real interest rate are slightly lower than in Rudebusch and Svensson (1998), but
they have right signs and are significant.
4.2.2 The two-step ML procedure
Two-step ML is obtained by using the estimated parameters of equations (13) and (14)
to solve for inflation expectations and estimate the monetary policy rules. Parameter
estimates are reported in Panel A of Table 3 for several inflation horizons h. The
inflation parameter β is noticeably stable when the horizon becomes more forward.
Parameter β equal to 1.60 for the contemporaneous model, whereas it is 1.668 for
the model with E π +4. Except for the contemporaneous specification, the two-step
ML estimate is much lower than the estimates obtained by GMM. Standard error of
β does not significantly increases with the inflation horizon. It ranges between 0.16
and 0.18. Therefore, for large inflation horizon (h ), the standard error is much
smaller than those obtained with GMM procedures. Remind that it is 0.29 for the
two-step GMM with lag length L .
The output-gap parameter γ is essentially zero and non significant. Note that it
decreases with the horizon of inflation expectations. The autoregressive parameter
t
t
= 4
= 4
11
is 0.7 only, whatever the horizon. This is much lower than estimates obtained
with GMM procedures. The standard error of residuals ranges between 1.077 for
the contemporaneous model and 1.071 for the model with E π +4. The log-likelihood
is maximized for the reaction function with E π +4 . The empirical fit appears to be
better than with other procedures.
ρ
t
t
t
t
4.2.3 The FIML procedure
We turn now to the FIML estimation. Thus, the Phillips curve, the I-S curve and the
reaction function are simultaneously estimated, assuming a non-diagonal covariance
matrix of parameters. In Panel B of Table 3, we report parameter estimates for the
reaction function. Parameter estimates for the Phillips curve and the I-S curve remain
essentially unchanged whatever the horizon. Therefore, we only indicate the FIML
estimates of the Phillips curve and the I-S curve for the horizon h = 4:
π
= 0(0.482
π −1 + 0.187 π −2 + 0.261 π −3 + 0.070 π −4 + 0.116 y −1 + û (15)
111)
(0 119)
(0 119)
(−)
(0 047)
see = 0.910
y
= 1(0.196
y −1 − 0.271 y −2 − 0.058 i −1 − π −1 − 4.078 + v̂
(16)
102)
(0 099)
(0 038)
(1 159)
see = 0.753.
As compared with the individually OLS estimated Phillips curve reported above,
we notice that parameter estimates remains unaltered, when the model is estimated
using the FIML procedure. Concerning the I-S curve, the output-gap persistence
parameters β 1 and β 2 change slightly. More importantly, the sensitivity of output
gap to the short-term real interest rate decreases: the point estimate is −0.058 with
a standard error equal to 0.039.
Considering now the parameter estimates for the reaction function, we obtain the
following results. As compared to the two-step ML estimation, the inflation parameter
β in the reaction function is slightly larger whatever the horizon. For instance, for
h = 4, we obtain β = 1.71 to be compared with the two-step ML estimate β = 1.67.
The output-gap parameter γ and the autoregressive parameter ρ are essentially the
same as the parameters obtained with two-step ML.
t
.
t
.
y
t
t
.
.
t
t
.
t
t
.
t
t
t
.
.
t
t
y
4.2.4 Testing the robustness of the forward-looking reaction function
We now estimate the encompassing model, which incorporates the lagged inflation
and the expected future inflation. Results for the two-step ML and FIML procedures
are reported in Table 4. Considering first the two-step ML, we obtain β and β
parameters which are similar to those found with the two-step GMM procedure. The
parameter of the lagged inflation is negative, whereas the parameter of the expected
future inflation is positive, although both of them are non-significant. ML estimation
thus support the forward-looking specification of Federal Reserve reaction function.
Estimating the encompassing model with the FIML procedure provides qualitatively the same results as for the two-step ML estimation. It is noteworthy that these
results contrast with those obtained by Fuhrer and Moore (1995a), who concluded
b
12
f
that the Federal Reserve reaction function is backward-looking. However, past and
future inflation as well as output gap are introduced in their encompassing model.
Moreover, they adopted a quarterly increase specification for inflation, whereas we
consider a four-quarters increase specification.
5
Interpreting empirical evidence
Previous sections have provided contrasting estimates of the forward-looking reaction
function: First, the GMM parameter estimates are substantially different from each
other and from ML estimates. Estimates seem to be less precise with the GMM
approaches than with the ML approaches. Second, the expectation horizon clearly
matters. Indeed, estimates of the inflation parameter β systematically increase when
the horizon becomes more distant. In this section, we investigate these results, with
a special focus on the expected inflation parameter. First, we conduct Monte-Carlo
experiments to illustrate small-sample properties of the various estimation procedures.
Then, we analyze the influence of the expectation horizon.
5.1 The distribution of GMM and ML estimators
5.1.1 Monte-Carlo design
In this section, we conduct Monte-Carlo experiments to assess the small- and largesample properties of estimators obtained using the GMM and ML estimators. The
design of the experiment is the following.
The data generating process (DGP) is given by the complete macroeconomic
model estimated by FIML. The DGP is thus constituted of equations (15), (16)
together with the reaction function estimated by FIML (Table 3), reported below
for convenience:
rt
=0(0.708
r −1 + 0.292 1.712 E π +4 − 0.090 y −1 + 1.48
+ ε̂ .
061)
(−)
(0 190)
(0 210)
(0 793)
.
t
t
.
t
.
t
.
t
(17)
The innovation covariance matrix, Σ̂, used in the Monte-Carlo experiments is the
residual covariance matrix of (û , v̂ , ε̂ ).
Each Monte-Carlo experiment is based on N = 500 replications for each simulation of the model.7 For a given sample size T , a sequence
of T +50 random innovations
are drawn from the normal distribution N 0, Σ̂ . In the simulations, two sample sizes
are used: T = 78, and 1500 observations. Our estimation sample contains 78 observations. The sample size T = 1500 illustrates whether biases are likely to disappear
in large samples. The random innovations are used to simulate the macroeconomic
model (15), (16) and (17). Initial conditions are set equal to the average values over
the sample. The first 50 periods are discarded to reduce the effects of initial conditions on the solution path. For each simulated dataset, the reaction function is then
t
t
t
An upper bound on the Monte-Carlo standard error of parameter, denoted σθ , is N −1/2 σθ .
Therefore, for N = 500, the upper bound on the Monte-Carlo standard error is 0.045 times the
computed standard error.
7
13
estimated using the three GMM estimators and the two ML estimators. We also report results for the OLS estimators, expected to be inconsistent in our set-up, in order
to evaluate the magnitude of the OLS bias. For each sample size and each estimator,
we have 500 parameters estimates, and the distribution of parameter estimates can
then be analyzed.
The simulations were performed using GAUSS version 3.2 on a Pentium III platform. We used the BFGS algorithm of the OPTMUM procedure for optimization.
We found no discrepancies when we used different algorithms. All experiments were
performed using numerical derivatives.
In some experiments, for T = 78 observations, the continuous-updating GMM
estimator failed to converge. The number of crashes is 2 percent of our samples.
Hansen, Heaton, and Yaron (1996) already reported some difficulties to obtain reasonable parameter estimates with the continuous-updating GMM. First, the numerical search for the minimizer sometimes fails. In addition, even when convergence
is reached, the distribution of estimates can be severely distorted, because of a few
unrealistic samples. For this reason, in Table 6, for T = 78, two rows are devoted
to the continuous-updating GMM estimator distribution. In the first row, we report
distribution statistic after we discarded only estimations which reached the maximum
number of iterations (here, 150). In the second row, we selected estimations which
satisfied the additional criterion that ρ lies inside the reasonable interval [−1; 1]. In
our Monte-Carlo experiments, 5.4 percent of estimations felt outside of this parameter
space.
The two-step ML and FIML estimators are implemented as follows in the MonteCarlo experiments. For the two-step ML estimator, for each replication, we estimate
the Phillips curve and the I-S curve with simulated data, and then we estimate the
reaction function using the procedure described in section 4.1. For the FIML, all
macroeconomic parameters are estimated jointly with the reaction function. Therefore, FIML is expected to perform better in this context.
5.1.2 Monte-Carlo evidence
The distribution of the alternative reaction-function parameter estimators is summarized in Table 5 for the small-sample case (T = 78),and in Table 6 for the large-sample
case (T = 1500). Fig. 1 and 2 also display the parameter β distribution for the various estimation approaches. In the following, the assumed true value of the parameter
is β = 1.712 (from equation (17)).
Table 5 reveals that, in small samples, the two-step ML procedure is unbiased. The
median β parameter is equal to 1.704, whereas the true parameter is β = 1.711. This
estimator is precisely estimated, since the standard deviation is equal to 0.213 and
the 90 percent confidence interval is [1.425; 1.930]. As regards the FIML estimators,
even for small samples, the estimator of β is as well unbiased. The median value is
1.694.
Turning to GMM procedures, we find that estimators are somewhat biased and
very imprecise in small samples. The GMM bias on β is positive. The smallest bias
is obtained for iterative GMM, with an estimator mean equal to 1.817, and a me14
dian equal to 1.767. The bias is more pronounced in the case of the two-step GMM
estimator (with a 1.842 mean and a 1.767 median). The continuous-updating estimator provides very imprecise estimates, even though the median is equal to 1.875.
When ”unreasonable” outcomes are excluded, the mean is equal to 1.95 and median
to 1.795. While the magnitude of the bias is not very large, one notices that it is in
fact as large as the bias obtained with the inconsistent OLS estimator.
The dispersion of GMM estimator is at least twice as large as that of ML estimators. The iterative GMM estimator has a standard deviation as high as 0.586, giving
a 90 percent confidence interval of [1.325; 2.334]. The distribution of GMM estimators
is markedly asymmetric, since the upward boundary is much more distant from the
median than the downward boundary. This provides a rationale for the very large β
estimates obtained with iterative and continuous-updating GMM on the actual data.
This finding is consistent with some previous results, which indicate that the GMM
estimator may have very unusual properties (see, e.g., Tauchen, 1986, Fuhrer, Moore,
and Schuh, 1995, and Nelson and Startz, 1990, in the case of the Instrumental Variable estimator). Noticeably, the continuous-updating GMM estimator has very fat
tails and yields a non-negligible proportion of implausible estimate.
For large samples (Table 6), the bias obtained with GMM procedures disappears,
as expected. Note that the difference between the alternative GMM estimators vanishes, even for intermediate sample sizes (T = 300, not reported here). The bias
of GMM is substantially reduced in our large-sample size, but the standard error of
GMM estimators remains about 25 percent higher than the standard error of ML
estimators. This result is illustrated in Fig. 2. The OLS estimator is asymptotically
biased. Interestingly, in our set-up, the bias on the inflation parameter is rather small.
The above results point in favor of ML estimate. However, a usual and important caveat is in order. Since we suppose that the true specification is known, the
ML approach is undoubtedly favored in the experiments. On the other hand, in the
Monte-Carlo experiments, we only use relevant instruments, and lags of relevant instruments. In actual GMM empirical estimate of reaction function, it is a common
practice to include a large number of instruments (such as the exchange rate or raw
material prices), some of which may be irrelevant, thus accentuating the small-sample
undesirable properties of GMM.
5.2 Influence of the expectation horizon
The previous section has focused on the benchmark case of one-year ahead inflation
expectation, found to be the most plausible. However, the expectation has a strong
influence on estimation results. A general result, also exhibited by Orphanides (1998)
estimates, is that the estimated inflation parameter rises when the expectation horizon
h increases (see Fig. 3). This occurs for all estimation procedures, so we investigate
for a specification bias using a simplified model. However, the rise in parameter β is
much more pronounced for the GMM estimates than for the ML estimates. This calls
for analyzing the GMM bias specifically.
15
5.2.1 The case for an horizon bias: a simple macroeconomic example
To illustrate the impact of the expectation horizon on the reaction function parameters, consider the very stylized I-S curve together with a Phillips curve model. The
output gap depends on the lagged real rate, while acceleration in inflation is related
to current output gap:
y
= −θ(i −1 − E −1π ) + v
(18)
π
= π −1 + µy + u
(19)
where v and u are random shocks. A semi-reduced form for the I-S curve and
Phillips curve is:
π = π −1 − γ (i −1 − E −1 π ) + η
(20)
where γ = θµ and η = u + µv .
Consider the case in which the true reaction function includes the contemporaneous inflation:
i = βπ + ε .
(21)
Replacing the reaction function in equation (20) and taking expectations at time
t − 1 yields
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
Et 1 π t
so that
t
t
t
t
−
t
t
t
= π −1 − γ(βπ −1 + ε −1 − E −1π )
t
−
t
Et 1 π t
t
t
= 11−−γβ
π −1 − γε −1 .
γ
t
t
t
Thus, the reaction function can be also expressed in a forward-looking version as
i = βE π +1 + ε̃
(22)
where ε̃ = 1−1 ε and β
= 11−− β , with β
> β . Therefore, an econometrician
estimating the forward-looking specification will recover a coefficient β
higher than
the true value β in the correct specification.
Note that this reaction function can also be expressed as a backward-looking
Taylor rule where the interest rate reacts both to the lagged inflation rate and to the
output gap, since combining equations (19) and (21), we obtain
i = βπ −1 + βµy + βu + ε .
(23)
Rules (21), (23), and (22) are equivalent in this simple model. Exact equivalence
arises because of the simple structure of the model, but this example rationalizes
our finding that the inflation parameter obtained when estimating a forward-looking
specification with horizon h is systematically higher than the parameter obtained
when using a less forward-looking specification with horizon h < h. The explanation
is that the monetary-policy rule, through its counter-cyclical effect on the economy,
makes inflation a mean-reverting process. Therefore, a stronger coefficient on expected
inflation is required to yield the same impact on interest rate. The simple example
also reminds that, due to the logic of Phillips curve, the output-gap coefficient shrinks
when the inflation expectation horizon rises (consistent with Tables 1 and 3).
t
t
γβ
t
t
t
γ
t
γβ
t
t
t
16
t
t
Note that these patterns for parameters emerge whatever the true specification.
Consider the case when the true reaction function is the forward-looking equation
(22), but the econometrician estimates a backward-looking rule. The same manipulation as above yields
1
E −1 π =
π −1 .
t
t
1 + γ(β − 1)
t
Therefore, the parameter recovered from estimating the contemporaneous specification (21) will be 1+ (
−1) , and thus will be smaller than the true value β
(because
β > 1).
A subsequent message is that the rules might be difficult to distinguish empirically.
In the above example, forward-looking and backward-looking rules are observationally
equivalent. This provides some explanations for the contrasting results obtained in
section 3.3.
β
γ β
5.2.2 Weak instrument correlation: the case for a GMM bias
In the previous section, we rationalize the impact of the horizon on the inflation
parameter pattern. However, in our estimations, the parameters vary much more
importantly in the GMM case than in the ML case, which should not be expected.
Therefore, we investigate now for the presence of an additional GMM specific bias
related to a poor instrument-regressor correlation for longer horizons. As pointed out
by Nelson and Startz (1990), poor instruments can lead to biased estimates.
This issue has been addressed in a number of studies. The instrument relevance
is often measured by the standard R2 from the regression of RHS variables X on
instrument variables Z . (see, for instance, Miron and Zeldes, 1988, and Campbell
and Mankiw, 1990). However, Nelson and Startz (1990) have shown that such an
approach may be misleading, when X has more than one variable. This is because all
RHS variables can be highly correlated with one of the instruments only. In this case,
only one of the parameters can be identified. Shea (1997) proposed an alternative
measure of instrument relevance based on partial correlations between RHS variables
and instruments.
In the case where X contains
only one endogenous (forward-looking) variable
X1 X2
X1 , with X =
, Shea’s measure of instrument relevance is the squared
correlation between the components of X1 and X̂1 orthogonal to X2, where X̂1 denotes
the projection of X1 on the instrument set. This statistic is also named ”partial R2”,
and denoted R2. To correct partial R2 for degrees of freedomwhen instruments
are
−
1
2
2
2
added, one defines the corrected partial R as: R̄ = 1 − − 1 − R .
Table 7 reports standard R2and Shea’s partial R2 instrument-relevance measure
for each horizon. As expected, the uncorrected standard R2 is large for the contemporaneous inflation decreases slightly for more distant horizons. It is as high as 0.99
for h = 0 and 0.87 for h = 4. Partial R2 display a somewhat different pattern. For
distant horizons, the instruments appear to be less relevant, with a partial R2 equal
to 0.64 and a corrected partial R2 equal to 0.57 for h = 4.
p
p
17
T
T
q
p
To sum up, instruments appear to be quite relevant, but the instrument relevance
decreases with the inflation horizon. This provides an explanation for an increasing
bias and a increasing standard error of β parameter GMM estimates in Table 1.
The asymptotic standard error of the GMM estimator, as shown by Shea (1997),
can be written as
ASE
β̂ GM M
= SE
β̂ OLS
2
/Rp
where β̂ is the OLS estimator when future inflation is assumed to be known.
Therefore, the discrepancy between both standard errors is expected to increase with
the horizon. For the horizon h = 4, the GMM standard error of parameter β may be
as high as twice the OLS standard error. This rationalizes GMM and OLS parameter
standard errors found in large sample Table 6.
OLS
6 Conclusion
This paper has investigated the robustness of GMM estimates of the Federal Reserve reaction function, relying on a dynamic forward-looking Taylor rule. Our main
findings are the following.
First, the various GMM (two-step GMM, iterative GMM, continuous-updating
GMM) estimators provide contrasting estimates for crucial parameters of the Taylor
rule. Iterative and continuous-updating GMM, not often considered in the reactionfunction literature, produce particularly high inflation parameters in our sample. The
test of a backward-looking vs. forward-looking specification for the policy rule also
produces contrasting results depending on the GMM estimator used.
Second, ML is a feasible alternative to GMM for estimating a forward-looking
reaction function. A traditional difficulty with FIML is that it requires a structural
model for forcing variables. But in the present context, a Phillips Curve / I-S curve
model, such as the Rudebusch and Svensson’s model, provides a fairly reliable simple
model of the economy. In our sample, inflation parameters are lower than those
estimated using GMM, and more in line with the Taylor rule. Moreover, ML gives
more precise parameter estimates, and supports the forward-looking specification.
Monte-Carlo experiments support these outcomes. We find evidence that, in small
sample, GMM estimates tend to overstate the degree to which interest rates respond
to future inflation. The size of this bias is limited however. Our assessment of GMM
in the case of a reaction function is therefore less critical than that of Fuhrer, Moore,
and Schuh (1995) in the case of inventories. However, GMM estimates, particularly
iterative and continuous-updating GMM, have a large variance.
18
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[46] Verdelhan, A. (1999), Taux de Taylor et taux de marché de la zone euro, Bulletin
de la Banque de France, 61, 85-93.
[47] Woodford, M. (1999), Optimal Monetary Policy Inertia, Manchester School, 67,
supplement 1999, 1—35.
22
Table 1: Parameter estimates of contemporaneous and forward-looking models using GMM procedures
Inflation
specification Parameter
πt
πt+1
πt+2
πt+3
πt+4
β
γ
ρ
α
Panel A: Two-step GMM
L =4
L =L *
Estimate Std Err.
Estimate Std Err.
1.560
0.127
1.851
0.234
0.382
0.160
0.348
0.172
0.739
0.064
0.764
0.067
2.023
0.646
1.155
0.813
see
JT
1.063
7.817
0.553
1.083
12.901
β
γ
ρ
α
1.616
0.325
0.774
1.909
0.154
0.179
0.056
0.730
1.985
0.252
0.803
0.842
see
JT
1.065
8.157
β
γ
ρ
α
Panel B: Iterative GMM
L =4
L =L *
Estimate Std Err.
Estimate Std Err.
1.939
0.183
2.061
0.231
0.681
0.157
0.328
0.150
0.461
0.064
0.753
0.065
1.104
0.730
0.682
0.787
0.167
1.132
5.932
0.747
1.117
7.073
0.298
0.202
0.057
0.992
2.213
0.351
0.741
0.628
0.251
0.182
0.057
0.890
2.278
0.211
0.788
0.189
0.518
1.086
13.458
0.143
1.156
6.817
1.718
0.238
0.788
1.630
0.183
0.191
0.053
0.826
2.107
0.135
0.543
0.815
0.337
0.217
0.054
1.106
see
JT
1.086
8.183
0.516
1.106
13.085
β
γ
ρ
α
1.829
0.195
0.795
1.332
0.221
0.197
0.051
0.925
see
JT
1.094
7.295
β
γ
ρ
α
see
JT
Panel C: Continuous-updated GMM
L =4
L =L *
Estimate Std Err.
Estimate Std Err.
2.006
2.616
2.082
1.534
0.412
2.099
0.343
1.115
0.660
1.104
0.744
0.478
1.075
8.143
0.623
5.028
0.630
1.173
5.971
0.743
1.124
7.018
0.635
0.287
0.166
0.058
0.942
2.477
0.328
0.708
0.236
2.166
3.987
1.328
3.101
2.311
0.232
0.779
0.128
2.983
1.309
0.595
10.279
0.656
1.127
7.751
0.560
1.311
6.387
0.701
1.138
7.690
0.566
2.473
0.334
0.726
0.030
0.306
0.192
0.061
1.009
2.544
0.063
0.788
-0.452
0.339
0.170
0.059
1.082
2.470
0.319
0.710
0.246
3.170
2.175
0.915
7.378
2.309
0.236
0.780
0.142
1.752
1.272
0.427
5.449
0.159
1.268
6.959
0.641
1.183
8.320
0.502
1.294
6.382
0.701
1.130
7.697
0.566
2.338
0.054
0.821
-0.052
0.416
0.226
0.052
1.307
2.809
0.186
0.783
-0.872
0.380
0.214
0.056
1.286
2.842
-0.049
0.823
-1.247
0.426
0.198
0.056
1.350
2.486
0.328
0.711
0.230
2.114
2.100
0.797
3.433
2.315
0.248
0.782
0.147
2.059
1.255
0.503
6.567
0.606
1.121
12.212
0.202
1.244
7.146
0.622
1.1830
7.596
0.575
1.293
6.403
0.699
1.123
7.700
0.565
1.918
0.143
0.808
1.108
0.289
0.210
0.048
1.078
2.510
0.006
0.832
-0.464
0.536
0.257
0.049
1.595
3.077
-0.023
0.833
-1.642
0.456
0.232
0.052
1.615
3.143
-0.118
0.844
-2.007
0.526
0.236
0.055
1.640
3.468
0.094
0.744
-2.644
7.041
5.784
0.900
22.877
3.113
-0.127
-1.909
0.829
3.620
1.706
0.565
10.884
1.114
6.900
0.647
1.141
11.135
0.267
1.212
7.001
0.637
1.202
7.444
0.591
1.534
6.524
0.686
1.217
7.405
0.595
Table 2: Parameter estimates of the cncompassing model using GMM procedures
Parameter
β b (π t -1)
Panel A: Two-step GMM
L =4
L =L *
Estimate Std Err.
Estimate Std Err.
-0,983
3,379
-0,181
3,059
Panel B: Iterative GMM
L =4
L =L *
Estimate Std Err.
Estimate Std Err.
2,875
0,696
2,194
0,923
Panel C: Continuous-updated GMM
L =4
L =L *
Estimate Std Err.
Estimate Std Err.
5,558
23,355
2,277
5,612
β f (E t π t+ 1)
γ
ρ
α
3,119
-0,042
0,840
0,637
4,237
0,716
0,103
2,265
2,735
-0,021
0,831
-0,552
4,137
0,682
0,122
2,918
-2,173
1,006
0,711
3,721
0,948
0,240
0,106
0,877
-0,415
0,461
0,685
1,321
1,351
0,256
0,120
1,269
-3,422
0,808
0,754
0,764
22,184
3,696
0,682
9,932
-0,517
0,467
0,670
1,390
8,550
1,586
0,745
8,974
see
JT
1,146
6,650
0,575
1,155
9,965
0,268
1,290
6,885
0,549
1,174
6,787
0,560
1,586
5,330
0,722
1,187
6,704
0,569
Table 3: Parameter estimates of contemporaneous and forward-looking models using ML p
Inflation
specification Parameter
πt
β
γ
ρ
α
πt+1
πt+2
πt+3
πt+4
Panel A: Two-step ML
Estimate
Std Err.
1.596
0.165
0.251
0.158
0.690
0.061
1.824
0.704
Panel B: FIML
Estimate
Std Err.
1.662
0.180
0.255
0.176
0.693
0.066
1.633
0.766
see
lnL
β
γ
ρ
α
1.077
-282.950
1.622
0.164
0.701
1.762
0.172
0.161
0.058
0.729
1.148
-281.436
1.682
0.164
0.703
1.591
0.187
0.178
0.063
0.790
see
lnL
β
γ
ρ
α
1.074
-282.342
1.635
0.089
0.712
1.726
0.182
0.177
0.057
0.770
1.145
-280.959
1.691
0.084
0.715
1.570
0.196
0.186
0.062
0.826
see
lnL
β
γ
ρ
α
1.081
-282.837
1.641
0.009
0.701
1.697
0.175
0.118
0.057
0.736
1.152
-281.638
1.692
-0.007
0.704
1.547
0.187
0.029
0.610
0.782
see
lnL
β
γ
ρ
α
1.074
-282.283
1.668
-0.059
0.705
1.607
0.180
0.162
0.056
0.758
1.146
-281.166
1.712
-0.090
0.708
1.480
0.190
0.210
0.061
0.793
see
lnL
1.071
-281.908
1.142
-280.888
Table 4: Parameter estimates of the cncompassing model using ML procedures
Parameter
β b (π t -1)
β f (E t π t+ 1)
γ
ρ
α
see
lnL
Panel A: Two-step ML
Estimate
Std Err.
-2.021
2.073
3.781
-0.545
0.759
1.389
1.065
-281.145
2.200
0.530
0.072
0.971
Panel B: FIML
Estimate
Std Err.
-2.970
3.712
4.806
-0.965
0.768
1.133
1.125
-280.184
3.900
1.220
0.079
1.125
Table 5: Parameter distribution statistics when the true model is forward-looking
T =78 observations, N =500 draws
Parameter
Method
Mean
Median
Std dev.
β=1,712
OLS
1.651
1.658
0.285
Two-step GMM
1.842
1.790
0.445
Iterative GMM
1.817
1.767
0.586
Continuous-updating GMM
9.400
1.785
166.441
Continuous-updating GMM (truncated)
1.949
1.795
1.521
Two-step ML
1.699
1.704
0.213
FIML
1.708
1.694
0.217
γ=-0,090
OLS
-0.066
-0.058
0.271
Two-step GMM
-0.179
-0.166
0.314
Iterative GMM
-0.196
-0.178
0.519
Continuous-updating GMM
8.329
-0.190
195.404
Continuous-updating GMM (truncated)
-0.292
-0.197
0.989
Two-step ML
-0.130
-0.137
0.238
FIML
-0.124
-0.125
0.226
ρ=0,708
OLS
0.750
0.758
0.061
Two-step GMM
0.659
0.672
0.104
Iterative GMM
0.628
0.647
0.136
Continuous-updating GMM
-76850.3
0.578
1721461.4
Continuous-updating GMM (truncated)
0.478
0.579
0.333
Two-step ML
0.642
0.648
0.083
FIML
0.639
0.648
0.087
α=1,480
OLS
1.644
1.636
1.418
Two-step GMM
2.491
2.036
3.783
Iterative GMM
2.213
1.804
5.080
Continuous-updating GMM
68.757
1.993
1449
Continuous-updating GMM (truncated)
3.526
2.021
15.175
Two-step ML
1.499
1.503
1.089
FIML
1.449
1.483
1.085
5% CI
1.133
1.308
1.133
0.790
1.040
1.336
1.377
-0.496
-0.685
-0.893
-1.302
-1.276
-0.510
-0.479
0.646
0.463
0.355
-0.492
-0.257
0.490
0.476
-0.588
-2.111
-3.989
-6.955
-4.344
-0.199
-0.381
10% CI
1.321
1.457
1.325
1.217
1.244
1.425
1.452
-0.390
-0.549
-0.642
-0.904
-0.904
-0.420
-0.410
0.671
0.522
0.445
-0.077
0.037
0.530
0.521
0.097
-0.918
-1.750
-2.618
-1.974
0.313
0.180
90% CI
1.988
2.284
2.334
2.704
2.685
1.930
1.967
0.265
0.165
0.238
0.395
0.290
0.178
0.159
0.820
0.769
0.783
0.783
0.777
0.744
0.749
3.144
6.507
6.955
10.434
9.576
2.705
2.568
95% CI
2.067
2.525
2.623
3.320
3.124
2.007
2.049
0.351
0.270
0.390
0.759
0.516
0.253
0.246
0.835
0.791
0.816
0.840
0.810
0.771
0.768
4.057
8.980
9.503
16.104
13.597
3.209
3.069
Table 6: Parameter distribution statistics when the true model is forward-looking
T =1500 observations, N =500 draws
Parameter
Method
Mean
Median
Std dev.
β=1,712
OLS
1.676
1.675
0.029
Two-step GMM
1.718
1.716
0.034
Iterative GMM
1.718
1.716
0.034
Continuous-updating GMM
1.718
1.716
0.034
Two-step ML
1.713
1.711
0.027
FIML
1.711
1.711
0.027
γ=-0,090
OLS
-0.002
-0.001
0.050
Two-step GMM
-0.096
-0.096
0.051
Iterative GMM
-0.096
-0.096
0.051
Continuous-updating GMM
-0.098
-0.097
0.051
Two-step ML
-0.089
-0.090
0.045
FIML
-0.092
-0.093
0.046
ρ=0,708
OLS
0.788
0.788
0.011
Two-step GMM
0.706
0.707
0.020
Iterative GMM
0.706
0.707
0.020
Continuous-updating GMM
0.705
0.706
0.021
Two-step ML
0.706
0.706
0.015
FIML
0.705
0.705
0.015
α=1,480
OLS
1.187
1.178
0.260
Two-step GMM
1.531
1.511
0.294
Iterative GMM
1.531
1.510
0.294
Continuous-updating GMM
1.535
1.510
0.296
Two-step ML
1.474
1.482
0.139
FIML
1.478
1.480
0.243
5% CI
1.629
1.665
1.665
1.665
1.668
1.667
-0.083
-0.176
-0.176
-0.177
-0.165
-0.163
0.772
0.672
0.672
0.670
0.681
0.679
0.772
1.097
1.098
1.094
1.241
1.097
10% CI
1.639
1.675
1.675
1.675
1.678
1.676
-0.067
-0.162
-0.163
-0.164
-0.148
-0.148
0.775
0.681
0.680
0.678
0.686
0.685
0.852
1.183
1.181
1.180
1.302
1.170
90% CI
1.712
1.761
1.761
1.762
1.747
1.746
0.061
-0.030
-0.030
-0.031
-0.034
-0.033
0.802
0.732
0.732
0.731
0.726
0.724
1.527
1.914
1.914
1.919
1.647
1.787
95% CI
1.725
1.779
1.779
1.779
1.758
1.758
0.082
-0.014
-0.013
-0.017
-0.016
-0.015
0.807
0.736
0.736
0.735
0.730
0.728
1.644
2.039
2.044
2.053
1.731
1.909
Table 7: Standard R 2 and Shea's partial R 2 instrument relevance measures
Inflation
specification
R2
corrected R
πt
0.987
0.984
0.920
0.905
πt+1
0.969
0.963
0.880
0.858
πt+2
0.941
0.931
0.805
0.769
πt+3
0.900
0.881
0.702
0.647
πt+4
0.870
0.846
0.636
0.569
0.870
0.846
0.003
-0.181
Encompassing model
2
R 2p
corrected R
2
p
Fig. 1: T=78 observations
Fig. 2: T=1500 observations
Figure 3: β parameter estimate as a function of the expectation horizon
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0
1
2
3
4
Expectation horizon h
Two-step GMM
Iterative GMM
Continuous-updating GMM
FIML