Analysis of Taylor Rule Deviations

Analysis of Taylor Rule Deviations
Cheng-Che Hsu*
Department of Economics, National Taiwan University, No. 1, Sector 4, Roosevelt Road, Taipei, Taiwan
Abstract
This study provides a resolution to identify the parameters of the Taylor rule. In partic-
ular, we introduce a deviation from the Taylor rule into a standard new Keynesian (NK)
trinity model. We estimate the parameters using a canonical, pure forward-looking NK
model with a full information maximum likelihood approach. All structural shocks are
assumed to follow an AR(1) process. With inclusion of the deviation, our results show
strong evidence that the estimated NK model offers a better explanation of the interac-
tions among interest rates, the output gap, and inflation. In addition, we use different
datasets and an alternative estimation approach to check the empirical validity of the NK
trinity model. We provide strong evidence that the interest rate policy can be decom-
posed into a systematic component, described by the Taylor rule, and a nonsystematic
component, which is known as Taylor rule deviations.
Keywords : Monetary policy rule; Taylor rule; New Keynesian model; Forward-looking;
Full information maximum likelihood
JEL Classification: C32,E12,E52
*Tel.: +886 (2)3366 3366 ext 55753. E-mail: d01323005@ntu.edu.tw.
1

1 Introduction
Taylor rule deviation is the difference between the nominal interest rate and the level
prescribed by the Taylor rule, which is proposed by Taylor (1993). The Taylor rule is
expressed as
it 2 πt 0.5ˆπt 2 0.5 xt
where it denotes the nominal interest rate and πt denotes the annual inflation rate. The
term xt represents the output gap or the deviation of the log of the real GDP from that
of the potential GDP.1 The Taylor rule is considered a useful benchmark for monetary
authorities (see Peersman and Smets (1999) and Kozicki (1999)). A negative (positive)
deviation is associated with an accommodative (contractionary) monetary policy. How-
ever, Taylor (2009) argued that when the interest rate deviates from the level suggested
by the Taylor rule, it results in asset market bubbles. Kahn et al. (2010) also pointed out
that the deviation contributes to a buildup of financial imbalances.
In fact, no monetary authority announces a standard benchmark for determining the
interest rate structure; thus, the observed deviation depends on the reaction coefficients
in the rule. The Taylor rule parameters play an important role in the evaluation of the
interest rate policy, but it is not normative to calibrate (estimate) the coefficients. More-
over, Cochrane (2011) questioned the aspects of the identification problem of the Taylor
rule parameters. In this study, we attempt to provide a resolution to identify the Taylor
rule parameters. To address this question, we introduce a deviation from the Taylor rule
into a standard new Keynesian (NK) trinity model.
In previous literature, Taylor rule deviation is identified from the residuals of or-
dinary least square regressions (e.g., Rudebusch (2002)) or vector autoregression (e.g.,
1
In the original paper, the potential output is measured by linear trend regression.
2

Christiano, Eichenbaum, and Evans (1999)).2 Rudebusch (1998) emphasized that the
least square approach appears too structurally fragile to identify the deviations. Devi-
ation reflects the shifts in preferences or the responses to unexpected disturbances of
the monetary authority. The monetary authority might decide a significantly negative
deviation, as seen in the era following the global financial crisis. The persistently loose
monetary policy makes estimation of the coefficient of the interest rate rule unreliable in
a single-equation approach. Therefore, as indicated by Cochrane (2007), it is a promising
possibility to obtain more convincing results from a full-system approach; for example,
see Smets and Wouters (2003) and Ireland (2007).
Since the variables are determined simultaneously in the system, the interaction be-
tween the variables will help distinguish the structure of the interest rate policy. However,
a large system implies a large number of parameters; thus, the model tends to be over-
parameterized (the structural parameters are under-identified). That is, there exists a
different set of parameters that generate similar observational implications, as shown by
Onatski and Williams (2004). To avoid the under-identification problem, it is desirable to
obtain reasonable estimates from a relatively simple system, for instance, the well-known,
fundamental three-equation NK model.
In this study, with the inclusion of Taylor rule deviation, we show that the NK model
is empirically valid. We estimate the parameters in a canonical, pure forward-looking
NK model by adopting the full information maximum likelihood (FIML) method. Our
empirical findings reveal that the NK model, which includes the deviation, depicts real
economic dynamics. First, the pure forward-looking NK model generates high inflation
and a persistent output gap. We show that the one-step-ahead forecast of inflation and
the output gap is fairly accurate, and the forecast errors are unpredictable, which implies
that the forecasts are rational. In addition, we use different datasets and an alternative
2
This literature refers deviation as a monetary policy shock.
3

estimation approach (the generalized method of moments, GMM) to show that our em-
pirical result is quite robust. We also use data from Canada and the UK to investigate
the external validity. Moreover, we explore ways in which possible factors impact the de-
viation. Taylor rule deviation can be explained by factors containing information about
future economic paths. The rest of the paper is organized as follows. Section 2 intro-
duces the structural model and the equilibrium. Section 3 describes the way in which
the parameter estimation is conducted. Section 4 explores the validity of the model and
the robustness of the estimation approach, and investigates the possible factors that affect
the deviation. Section 5 summarizes the research findings.
2 Baseline Model
We consider a simple, well-known NK model in our analysis. The economic environ-
ment is described by two key log-linear equations. These functions are derived under
several assumptions such as nominal rigidities and monetary policy non-neutralities.
See Woodford (2003a) and Galí (2009) for detailed information on a standard derivation
under first principles.
2.1 Simple NK Model
An intertemporal IS curve (aggregate demand) and a New Keynesian Philips curve (NKPC,
aggregate supply) take the form3
xt Etˆxt1 σ1
it Etˆπt1 re
t ¥, (1)
πt κxt βEtˆπt1 ut, (2)
where Et is the conditional expectation operator evaluated for the information set in pe-
riod t,4 yt denotes the log of output, πt is the inflation rate, it is the nominal interest
3
Models with similar components are described by Woodford (2003b), Iskrev (2010), Giannoni (2014),
and many others.
4
Etˆ ¡ Eˆ ƒΩt
4

rate, β is the discount factor, and σ and κ are positive coefficients. Equation (1) is derived
from a representative consumer’s intertemporal Euler equation, where σ is the coefficient
of relative risk aversion, denoting the reciprocal of the elasticity of intertemporal substi-
tution. Equation (2) is obtained by optimal pricing-setting for a monopolistic firm under
the Calvo (1983) framework, where κ denotes the rate of price adjustment. The term re
t
denotes the efficient rate of interest and ut represents the cost-push disturbance. These
are the real exogenous disturbances, where the word disturbance, rather than shock,
is used to remind us that re
t and ut can be serially correlated.
The cost-push disturbance depends on several exogenous disturbances such as tech-
nology shocks, shifts in labor supply, and variations in material costs. The efficient rate
of interest varies across time, whereas the response to preference shocks or fluctuations
in government expenditures occur in the short run. Please refer to Woodford (2003a)
and Galí (2009) to see how disturbances arise from first principles. From the above, the
exogenous disturbances re
t and ut comprise various potential disturbances with different
degrees of persistence. For simplicity, both disturbances are assumed to follow a station-
ary AR(1) process as defined below:
re
t ρr re
t1 εr
t , (3)
ut ρu ut1 εu
t . (4)
The exogenous disturbances here are more like the gaps between current aggregate de-
mand or supply levels and equilibrium levels. The real exogenous shocks at period t are εr
t
and εu
t . The current aggregate demand and supply levels are the sum of past shocks with
declining weights. An oil or financial crisis can be considered to be a significant shock
at specific periods, thereby affecting the aggregate supply or demand levels and causing
fluctuations in the economy. The size of the effect depends on the persistence of the ex-
ogenous disturbances, i.e., the ARˆ1 coefficient. Please note that the ARˆ1 structure is
5

the key assumption to derive the unique stationary solution of the forward-looking linear
rational expectations model.
2.2 Taylor Rule
The interest rate rule assumes that the monetary authority adopts a simple Taylor rule:
it ϕππt ϕx xt ηt. (5)
The interest rate policy is decomposed into two parts: a systematic component (described
by the Taylor rule) and a nonsystematic component (called Taylor rule deviation). The
coefficients of policy rule (ϕπ, ϕx) are committed by the monetary authority at the begin-
ning. For each period, the monetary authority decides the level of ηt to adjust the interest
rate from the systematic policy.
This setting allows the monetary authority to retain discretion in response to ma-
jor unexpected disturbances. The term ηt denotes Taylor rule deviation or refers to the
monetary policy disturbance, i.e., the deviation of the nominal interest rate from the sys-
tematic rule. Actual data show that interest rates are highly autocorrelated. From a single-
equation perspective, an inertial Taylor rule (i.e., a Taylor rule with a partial adjustment)
is widely used in empirical studies since this rule appears to fit the data as well.5 On the
contrary, Rudebusch (2002) indicated that a Taylor rule with autocorrelated monetary
policy shocks (we prefer calling them deviations rather than shocks) is a better setting
for the interest rate rule. The deviation of the interest rate rule implies that the mone-
tary authority responds to exogenous influences aggressively, which is intuitively more
consistent with a central bank’s actual behavior.
Taylor rule deviation comprises many distinct components. The original Taylor rule
contains an intercept, which implies an interest rate under zero inflation and full-employment
5
For example, it ρit1 ˆ1 ρˆϕx xt ϕππt εt.
6

output in the long run, i.e., the so-called natural interest rate. Woodford (2001) noted
that the natural interest rate is affected by real disturbances; thus, the intercept should
be time-varying. However, the sources of the stochastic intercept are difficult to identify.
The deviation can be considered to be a policy term considering stochastic intercept,
while fluctuation in Taylor deviation reflects the fact that the intercept varies over time.
Moreover, an interest policy would not consider only inflation and the output gap
ηt, but may also involve the monetary authority’s response to other persistent shocks,
time-varying rules, etc. Alternatively, current inflation and the output gap are not pol-
icy instruments for the monetary authority since they not observable at the beginning.
Instead, an interest rate policy based on measured variables using real-time forecast es-
timates (see Orphanides (2001) and Bernanke (2010)) is a reasonable approximation in
practice. If such a rule were adopted, then based on the specification of (5), the mea-
surement error will enter into Taylor rule deviation. There is an important advantage
if we consider interest rate with a current variable instead of adopting an interest rate
rule based on real-time data. Once the policy coefficients are determined, ηt is observed
by the residuals of (5). This setting may aid identification, and thus, avoid the under-
identification problem.
Deviations reflect shifts in preferences or responses to unexpected disturbances of
the monetary authority and the measurement error faced by the monetary authority. It
is also affected by various potential disturbances with different degrees of persistence. For
simplicity, Taylor rule deviation ηt is also assumed to follow a stationary ARˆ1 process,
as given by
ηt ρηηt1 ε
η
t . (6)
In particular, Taylor rule deviation is correlated with the inflation and the output gap.
For instance, there existed a significant reduction in output and a large forecast error on
7

inflation and the output gap during the global financial crisis. When a single-equation
approach (either OLS or GMM) is adopted, this endogeneity causes the estimates of the
Taylor rule coefficients to be unreliable.
2.3 Equilibrium
Using (5) to eliminate the interest rate in (1) and (2), the economic dynamics can be
written as a system of difference equations of the following form:
Et
`
d
d
d
d
d
d
b
xt1
πt1
a
e
e
e
e
e
e
c
A
`
d
d
d
d
d
d
b
xt
πt
a
e
e
e
e
e
e
c
B et, (7)
where et ¡ re
t , ut, ηt¥œ and
A
`
d
d
d
d
d
d
b
1 σ1ˆϕx
κ
β σ1ˆϕπ
1
β

κ
β
1
β
a
e
e
e
e
e
e
c
, B
`
d
d
d
d
d
d
b
σ1 1
βσ σ1
0
1
β 0
a
e
e
e
e
e
e
c
. (8)
This system has a unique equilibrium only if both eigenvalues of matrix A lie outside the
unit circle. When the coefficients in the policy rule are restricted to being non-negative
(ϕπ, ϕx e 0), Woodford (2003a) showed that the well-known condition of unique equi-
librium holds only if
ϕπ
1 β
κ
ϕx e 1. (9)
Woodford (2001) pointed to a simple implication of this condition. Equation (2) shows
that a permanent 1 percent increase in inflation will raise the long-term average output
gap by κ~ˆ1 β percent. Plugging this fact into (5), suggesting that the interest rate
should increase by ϕπ ˆ1 βϕx~κ, the Taylor principle stipulates that a monetary au-
thority should raise the nominal interest rate more than the increase in inflation, which is
consistent with (9). Our estimation approach is based on the equilibrium of this model,
to ensure the determinacy of equilibrium since the unique equilibrium condition must
8

be satisfied. Given κ, σ e 0, the nonlinear constraint in (9) is satisfied when the param-
eters are restricted to ϕπ e 1 and ϕx e 0. Imposing a linear constraint is much simpler
than imposing a nonlinear constraint during estimation. For a system involving (7) with
stationary shock structures involving (3), (4), and (6), we first assume that competitive
equilibrium is a function of exogenous shocks and Taylor deviation. The solution of this
system then takes the following form:
`
d
d
d
d
d
d
b
xt
πt
a
e
e
e
e
e
e
c

`
d
d
d
d
d
d
b
cxr cxu cxη
cπr cπu cπη
a
e
e
e
e
e
e
c
`
d
d
d
d
d
d
d
d
d
b
re
t
ut
ηt
a
e
e
e
e
e
e
e
e
e
c
C et, (10)
where C is a 2! 3 matrix. Using the method of undetermined coefficients, the coefficients
in matrix C can be shown as6
C
`
d
d
d
d
d
d
d
d
d
d
b
1 βρr
Ωr ˆ1 βρrϕx κ ϕπ
ρu ϕπ
Ωu ˆ1 βρuϕx κ ϕπ
ˆ1 βρη
Ωη ˆ1 βρηϕx κ ϕπ
κ
ϕx σˆ1 ρu
κ
a
e
e
e
e
e
e
e
e
e
e
c
, (11)
where Ωj σˆ1 ρjˆ1 βρj κρj, j b ˜r, u, η are terms unaffected by the Taylor
rule coefficients. We assume Ωj ˆ1 βρjϕx κϕπ e 0 for j b ˜r, u, η. Given the
absolute value of the coefficient of a stationary ARˆ1 process is less than unity, κ, σ e 0,
and ϕπ e 1 , ϕx e 0, we can determine the sign of the coefficients in matrix C. A positive
Taylor rule deviation will lower both inflation and the output gap. Both inflation and the
output gap will increase due to a positive efficiency rate shock, whereas a positive cost-
push shock will raise inflation but lower the output gap; thus, we treat the efficiency rate
shock as a demand-side shock and cost-push as a negative supply-side shock.
When we substitute (10) for the output gap and inflation rate in (5), the interest rate
also becomes a function of exogenous shocks and Taylor deviation. Hence, all endoge-
6
See Appendix 1 for details.
9

nous variables can be described as
zt ¡
`
d
d
d
d
d
d
d
d
d
b
xt
πt
it
a
e
e
e
e
e
e
e
e
e
c

`
d
d
d
d
d
d
d
d
d
b
cxr cxu cxη
cπr cπu cπη
cir ciu ciη
a
e
e
e
e
e
e
e
e
e
c
`
d
d
d
d
d
d
d
d
d
b
re
t
ut
ηt
a
e
e
e
e
e
e
e
e
e
c
H et, (12)
with a 3 ! 3 matrix H equal to
H
`
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
b
1 βρr
ρu ϕπ
ˆ1 βρη
κ
ϕx σˆ1 ρu
κ
ˆ1 βρrϕx κϕy
ρuϕx σˆ1 ρuϕπ
1
ˆ1 βρηϕx κϕy¥
a
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
c
. (13)
3 Estimation
In this section, we check our progress in the estimation.
3.1 Parameter Estimation
Nason and Smith (2008) noted the possible identification problems with the single-equation
method. Therefore, we disregard the single-equation approach. Due to the possible struc-
tural changes, the empirical macroeconomic time series, although appropriate, are typ-
ically short. For instance, the Taylor rule was seen as providing a suitable description
of monetary policy after the mid-1980s. Even if we consider the Bayesian approach (see
Smets and Wouters (2003) and Rabanal and Rubio-Ramírez (2005)), the empirical re-
sult will depend on the tight prior distribution in small samples and may not provide a
reliable estimation with diffuse prior distributions (see Cochrane (2007) for a detailed
discussion). Different prior distribution is needed if another dataset is used.
Previous studies have often used the vector autoregression (VAR) approach to esti-
mate the NK model (Rudebusch and Svensson (1999), Del Negro et al. (2007), and Kolasa,
10

Rubaszek, and Skrzypczyński (2012)). To increase the fitness, the empirical model tends
to include more lagged dependent variables but yields the over-parameterized problem.
If the cost-push shocks are significantly serially correlated, then this could lead to biased
estimators (see Kuester, Müller, and Stölting (2009) and Zhang and Clovis (2010) for
more detailed discussions).
However, the high degree of autocorrelation in time series data, including more lagged
variables, can make the residuals white noise and render the estimation easier. Although
the empirical model generates better performance from out-of-sample forecasts, the pol-
icy analysis becomes more complicated under such a hybrid NK model. For convenience
of analysis, it is desirable to obtain reliable estimates in a relatively simple model. Lindé
(2005) indicated that FIML is a useful way to obtain better estimates in the simultaneous
system.7 Considering the linear constraints of the coefficients, we use the FIML approach
in this study.
Because all our variables are functions of exogenous shocks and Taylor deviation,
taking the expected value of both sides in (12) shows that the average for all variables
is zero. In the original NK model, the value of the endogenous variable represents the
deviation from the steady state. The long-run level of the output gap should be zero, but
the steady state of the inflation rate is difficult to identify. Because we assume all shocks
have a zero mean, all variables are previously demeaned; thus, the sample mean of the
identified shock would also be zero. Actual data show that inflation, the output gap,
and interest rates are highly autocorrelated but stationary. To investigate the explanatory
power of the model, we retain any trends in the original data.
We estimate only the parameters that appear in (12), which are ˜β, κ, σ, ϕx , ϕπ, ρη, ρr, ρu.
Since the discount factor β cannot be observed directly and Nason and Smith (2008) sug-
7
We will use a GMM approach as suggested by Gali, Gertler, and Lopez-Salido (2005) in the later
robustness check.
11

gested that calibrating a discount factor β may aid identification, we simply calibrate β
= 0.99 for quarterly data.8 In the simultaneous equation system, the Taylor rule relates
only current inflation, the output gap, and interest rates. The series ηt can be observed
by the residuals from (5) once the parameters ˆ Âϕx , Ãϕπ are determined. The parameter
ρη can be estimated directly using a simple ordinary least square (OLS) estimator once
ˆ Âϕx , Ãϕπ are given.9 Therefore, we need to estimate six parameters simultaneously using
FIML; the estimation method is similar to that of Lindé (2005). The model equilibrium
is given by (12) and the shock structure is
et ρet1 εt, (14)
where
ρ
`
d
d
d
d
d
d
d
d
d
b
ρr 0 0
0 ρu 0
0 0 ρη
a
e
e
e
e
e
e
e
e
e
c
, εt
`
d
d
d
d
d
d
d
d
d
b
εr
t
εu
t
ε
η
t
a
e
e
e
e
e
e
e
e
e
c
. (15)
We assume εt ¢i.i.d. Nˆ0, Σε. From (12), the exogenous shocks can be recovered by the
data zt using
et H1
zt. (16)
Thus, (14) can be rewritten as
Hεt Het Hρet1 zt HρH1
zt1. (17)
Then, we have
ztƒzt1 ¢i.i.d.
Nˆ HρH1
zt1, HΣεH1
. (18)
8
We also tried to estimate the discount factor instead of using the calibrated value. The estimated value
is 0.933 and coefficients in IS and NKPC are changed slightly. However, the Taylor rule coefficients are
almost unchanged and the observational implications of the model are very similar.
9
Our results show that the OLS and MLE estimators are rather close.
12

Let νt zt HρH1zt1 and Σν HΣεH1, The conditional log-likelihood function is
thus
ln ℓˆκ, σ, ϕx , ϕπ, ρr, ρu ˆT~2 lnˆ2π ˆT~2 lnƒΣνƒ ˆ1~2
T
Qt 1
νœ
tΣ1
ν νt. (19)
Because we do not have any prior information about the shock structure, there are no
restrictions imposed on the covariance matrix Σε in the conditional log-likelihood func-
tion. However, while we determine the parameters, the exogenous shocks are identified
and the covariance matrix Σε can be estimated directly from the identified shocks. The
estimator Σε is obtained by the sample covariance matrix of H1zt ρH1zt1. The con-
ditional log-likelihood function then becomes
ln ℓˆκ, σ, ϕx , ϕπ, ρr, ρu ˆT~2 lnˆ2π ˆT~2 ln„ ÂΣν„ ˆ1~2
T
Qt 1
νœ
t
ÂΣν
1
νt, (20)
where ÂΣν H ÂΣεH1. The FIML estimator is obtained by maximizing (20) with linear
constraints, including κ, σ e 0, 0 d ρr, ρu d 1, ϕπ e 1, and ϕπ e 0.
3.2 Estimation on US data
The main objective of this study is to identify Taylor rule deviation. Since we consider
a Taylor-type instrument rule for interest rate policy, selecting an appropriate sample
period and using desirable measures of inflation and the output gap is important. Other-
wise, the observed Taylor deviation may deviate from the situation faced by the monetary
authority.
It is widely known that the Taylor rule was seen as offering an appropriate description
of the interest rate policy regime after the mid-1980s. The interest rates suggested by
the rule were substantially consistent with the federal fund rates during periods of low
inflation and low macroeconomic volatility. The second oil crisis led to stagflation in
the 1980s; thus, selecting it as the basis would make inflation and interest rates show
13

a significant downward trend. The Taylor rule may deviate from the actual interest rate
policy during periods of high inflation, which would make the estimated Taylor deviation
unreliable. To make the appropriate sample period as long as possible, we chose the
sample span from 1983:Q3to 2015:Q3.10
In the formulation of interest rate policy, the FOMC prefers the inflation rate to be
measured as the annual change in the consumer price index (CPI).11to the GDP defla-
tor that Taylor (1993) originally used. However, policymakers may look at various CPI
measures. The most common inflation measure for policymakers is the core CPI, which
excludes food and energy items. The core CPI excludes items that tend to fluctuate dra-
matically; thus, using an inflation rate defined by the core CPI could avoid excessive
volatility in interest rates caused by severe fluctuations in inflation. Hence, we calculate
the inflation rate using the annual change in the core CPI. For the output gap, we consider
the potential output estimated by the Congressional Budget Office (CBO).12 The output
gap is measured by 100ˆlog yt log y‡
t , where y‡
t denotes the real potential output and
yt is the real GDP.
We use data from FRED.13 Figure (1) shows the time series plot of variables; the vari-
ables with shaded areas indicate the period following the peak through the trough.14 In-
flation increases distinguish the third oil crisis. We see that when the economy is in re-
cession, there is a significant decline in the output gap and the monetary authority tends
to cut interest rates substantially. This may support the validity of the Taylor rule. Note
that after the financial crisis, the interest rate fell to exceptionally low levels. The starting
value in the estimations for ˆκ, σ, ϕx , ϕπ, ρr, ρu are ˆ0.0238, 0.1567, 1.5,0.5, 0.5, 0.5, re-
10
We also consider different sample periods in the later robustness analysis.
11
See http://www.federalreserve.gov/newsevents/press/monetary/20120125c.htm.
12
The estimates prepared by the Federal Reserve staff are discovered after a five-year lag.
13
The interest rate is measured by the quarterly average federal funds rate (FEDFUNDS). The series IDs
of core CPI, real GDP, and real potential output are CPILFESL, GDPC1, and GDPPOT, respectively.
14
The NBER-based recession indicators are also obtained from FRED.
14

Table 1: Estimation results for U.S. data
β Âκ Âσ Ãϕπ
Âϕx Âρr Âρu
0.99 0.073
ˆ0.029
1.981
ˆ1.182
1.549
ˆ0.199
0.633
ˆ0.120
0.974
ˆ0.008
0.947
ˆ0.013
Çρη Èρru Èρrη Èρuη
Çσ2
r
Çσ2
u
Çσ2
η
0.863 -0.712 0.324 -0.215 4.51 0.012 1.481
Notes ¢ Standard errors in parentheses under the estimator. The hat denotes the estimated value by FIML,
the tilde denotes the sample counterparts of identified shocks, and β is the calibrated value.
spectively, where κ 0.0238 and σ 0.1567 as suggested by Rotemberg and Woodford
(1997) for the US data, and ϕπ 1.5 and ϕx 0.5 are taken from Taylor (1993).15 Because
we have no prior information for the exogenous shocks, the initial values for ρr and ρu
are simply set to 0.5. The estimation results are presented in Table (1) and the identified
shocks are shown in Figure (2).16 Standard errors are obtained by computing the square
roots of the diagonal elements of the inverted Hessian matrix. Our results show that σ
is relatively larger than that of Rotemberg and Woodford (1997), indicating that the effi-
ciency of monetary policy is lower. The estimated values of κ, ϕπ, and ϕx are very close
to the estimations by Rotemberg and Woodford (1997) and Taylor (1993), although we
use different datasets. In particular, Galı, Gertler, and Lopez-Salido (2001) obtained a
significantly negative λ from an output-gap-based NKPC, which is inconsistent with the
theory. With a full-system approach, our estimation results suggest theoretically consis-
tent estimates of the NKPC coefficients. Even though the linear constraints of coefficients
are disregarded, we still obtain the same estimates. This feature implies that the estimated
15
We also try using alternative initial values ˆ0.34, 1, 2, 1, 0.5, 0.5, but the result remains almost un-
changed. In fact, the estimation result by the FIML approach is not sensitive to the chosen initial value.
16
The identified shocks are obtained by et H1
zt, where the corresponding values in H1
with the
estimated parameters are
@@@@@
0.24 0.54 0.72
0.06 0.14 0.06
0.63 1.55 1.00
=AAAAA?
.
15

parameters achieved the global maximum in the parameters’ space.
We further discuss the implication of the structural parameters by changing the scale
of the output gap, inflation, and interest rate. When the output gap is divided by two, Âκ,
Âσ, and Âϕx become twice and others remain the same. When the inflation is divided by
two, Âκ and Âσ become half and Âϕx becomes twice, while the others remain the same. If the
interest rate is multiplied by two, then only Âϕx and Âϕx become twice. From the above, σ
denotes the relationship between the expected inflation and the current output gap in the
IS curve. The κ measures the inflation-output trade-off, and Ãϕπ and Âϕx are parameters to
identify the nonsystematic component in the interest rate rule. The parameters ρr and ρu
determine the persistence of the exogenous shocks.
Moreover, the identified demand and supply shocks fluctuated markedly in 1990,
2001, and 2008. These dates correspond to the 1990 oil price shock, the dot-com bub-
ble, and the financial crisis, respectively. The identified shocks reflect the external dis-
turbances encountered by the real economy. Thus, the simple NK model offers a good
empirical description of the output gap, inflation, and interest rate dynamics. We also
provide a resolution to bridge the substantial gap between the theoretical work version
and empirical model in the NK framework.
4 Validity of the NK Model
4.1 Rational Expectation
Cochrane (2007) strongly questioned the NK model for implying rational expectation
paths with explosive inflation. Chari, Kehoe, and McGrattan (2009) also pointed out
that the NK model is not an accurate structural model for quarterly data. In this model,
the ARˆ1 structure of exogenous shocks is the key assumption to formulate the rational
expectations and characterize the equilibrium. Therefore, we will discuss the empirical
16

performance of the pure forward-looking NK model with ARˆ1 shocks.
The persistence of exogenous shocks is significant, which may be a possible source
of the high autocorrelation in variables.17 In this model, the variable is represented as a
function of stationary shocks:
wt cwrre
t cwuut cwηηt,
for w b ˜ x, π, i. Therefore, the variable is also stationary and the first-order autocor-
relation coefficient of the model is
ρˆ1
Covˆwt, wt1
Varˆwt
. (21)
Many current models use the Phillips curve, which includes lagged variables, to gener-
ate high inflation persistence in empirical studies (e.g., Smets and Wouters (2003) and
Christiano, Eichenbaum, and Evans (2005)). However, using this model for policy anal-
ysis is undesirable because the analytical solution of the hybrid NK model is quite com-
plex. Our results show that the first-order autocorrelation coefficient of the estimated NK
model is very similar to the sample counterpart.18 This suggests that persistent inflation
can be generated by the pure forward-looking Philips curve in the simultaneous system
without lagged variables, which is widely used in policy analysis.
To ensure that our model captures the dynamics of the economy, we examine in-
sample predictability to check whether the rational expectation operator provides accu-
rate predictions for the next period and whether the forecast is rational.19 The one-step-
ahead forecast is constructed by
Etˆzt1 EtˆHet1 H ρ et H ρ H1
zt. (22)
17
We also compared estimated parameters Âρr and Âρu with the values estimated directly from the iden-
tified shocks. The results show that they are very close.
18
The sample autocorrelation coefficients of the output gap, inflation, and interest rate are 0.969, 0.983,
and 0.985, respectively, whereas the values implied by the model are 0.973, 0.976, and 0.982, respectively.
19
Although the one-step-ahead forecast is dependent on current variables only, the coefficients in matrix
H are obtained when the full sample is used.
17

Figure (3) plots the comparison of expected and actual values. The result indicates that
the predicted value for the next period is similar to the current value. Actual data show
the persistence of the output gap, inflation, and interest rate. For highly autocorrelated
data, the prediction generates small forecast errors. We assume that the shock structure
follows a simple ARˆ1 process. Hence, the forecast errors can be represented as
zt1 Etˆzt1 H et1 H ρ et H εt1. (23)
Forecast errors are determined by the real disturbance term εt1.
According to the definition of rational expectation, the prediction error is the mean
independent of every variable contained in the information set. In this model, the com-
petitive equilibrium of a variable is a function of current exogenous shocks and Taylor
deviation. The mean independent condition becomes
E zt1 Etˆzt1ƒΩt¥ E ˆH εt1 ƒ re
t , ut, ηt 0 (24)
We conduct a simple inspection to check whether this condition holds. At first, all sample
means of prediction errors are close to zero. Next, we regress the prediction error of each
variable on exogenous shocks and Taylor deviation. The estimation equations are
wt1 Etˆwt1 βw,r re
t βw,u ut βw,η ηt. (25)
for w b ˜ x, π, i . Table (2) reports the estimation results.
The empirical results show that the variance in the prediction error is small, especially
for inflation. Compared with the single NKPC, considering the simultaneous equations
offers a better explanation of the inflation dynamics. There is strong evidence that exoge-
nous shocks provide no information on disturbance terms; thus, the mean independence
condition of the rational expectation holds in this model.
We further compare the out-of-sample predictability of the NK model and the real-
time forecast in practice. We use real-time data from the Survey of Professional Forecast-
18

Table 2: Forecast error exogeneity
xt1 Etˆxt1 πt1 Etˆπt1 it1 Etˆit1
Ãβx,r
Äβx,u
Äβx,η R
2
Ãβπ,r
Äβπ,u
Äβπ,η R
2
Ãβi,r
Ãβi,u
Ãβi,η R
2
0.02
ˆ0.07
0.38
ˆ0.66
0.00
ˆ0.04
0.01
ˆ0.27
0.00
ˆ0.03
0.04
ˆ0.27
0.00
ˆ0.02
0.00
ˆ0.05
0.04
ˆ0.05
0.51
ˆ0.59
0.01
ˆ0.03
0.01
ˆ0.24
Notes ¢ The Newey-West robust standard errors in parentheses under the estimator. Mean square error in paren-
theses under R
2
.
ers (SPF) to compute the out-of-sample forecast errors.20 The recursive scheme is used
to evaluate the out-of-sample one-step-ahead forecasts implied by the NK model. The
full sample has been split into two sub-periods: the in-sample period 1983:Q3-2000:Q4
and the out-of-sample period 2001:Q1-2015:Q3. For real-time data, the forecast errors of
the output gap, inflation, and interest rates are, respectively, measured by the real-time
one-step-ahead forecast errors of real GDP, CPI, and 3-month treasury bill rate.21
The comparison charts are presented in Figure (4). Interestingly, forecast errors of
both output and inflation demonstrate the same tendency and suffered significant fore-
cast errors during the global financial crisis. The SPF forecasts of interest rates are more
accurate than those of the NK model, but the forecast errors are similar in the zero-rate
era. However, the forecast errors reflect the exogenous shocks faced by the economy.
That the forecast errors of the NK model are similar to the real-time forecast, in practice,
implies that the identified exogenous shocks reflect the current state of the economy.
Hence, we provide some evidence to show that the economic environment described by
the simple NK model is close to the actual economy.
20
See https://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-
forecasters.
21
The forecast errors are calculated by computing the one-step-ahead forecast minus the last vintage of
realization.
19

4.2 Robustness Analysis
In this subsection, we investigate the robustness of the estimation approach used in this
study. Because the likelihood function is associated with the dataset, if another dataset
or a different sample period is employed, the estimated coefficients may be different.
To check the robustness of the estimation procedure, we run the same algorithm with
a different dataset. The stability of the estimated parameters reflects the validity of the
NKTM model; therefore, we can check whether the validity is tied to a specific dataset
or sample period.
In the literature, in addition to measuring inflation by CPI, the price index for per-
sonal consumption expenditures (PCE) is also often used to measure inflation (e.g., Rude-
busch (2002) and Cogley, Primiceri, and Sargent (2008)). After 2000 and for several rea-
sons, the Fed switched its focus from CPI to PCE when measuring inflation.22 Although
both measures draw on similar components, each uses very different weights. Compared
with CPI inflation, PCE inflation is a better predictor of inflation faced by the general
population. Therefore, we replace CPI inflation with PCE inflation in our estimations
for the robustness check. Apart from the potential output estimated from CBO, we also
consider the potential output suggested by the other two commonly used methods in
empirical studies: quadratic trend (QT) regression and the Hodrick-Prescott (HP) fil-
ter.23 Moreover, because Taylor deviation plays a very important role in the estimation,
selecting a measure of interest rate policy instrument affects the estimation results. Al-
though the federal funds rate is well known as a key policy instrument in the US, we still
substitute the treasury bill (T-bill) rate for the federal funds rate as a robustness check.
Because the financial crisis caused a significant reduction in output, the quadratic
trend model makes the estimates of potential output unreliable, which results in a signif-
22
See http://www.federalreserve.gov/newsevents/press/monetary/20120125c.htm.
23
For quarterly data, the common smoothing parameter λ 1600 is used.
20

icant difference from the CBO output gap. For this reason, we select a sub-sample from
1983:Q3 to 2008:Q2 while using quadratic de-trended output data. Comparison charts
of the various materials are plotted in Figure (5). When compared with the QT and CBO
output gaps, the HP output gap has a relatively small fluctuation. Before 2004, PCE in-
flation was significantly lower than CPI inflation and the T-bill rates were slightly lower
than the federal fund rate.
We consider two more sample periods, 1987:Q1-2015:Q3 and 1983:Q3-2008:Q2, for
additional robustness checks. The first sub-sample period is motivated by Taylor (1993)
and the starting period is 1987. The second sub-sample period ends at 2008:Q2, so we
can determine whether the estimated coefficients were significantly different before and
after the global financial crisis. Table (3) reports the estimation results. Except for σ,
the results are quantitatively similar. The response coefficients in the Taylor rule are also
not far from 1.5-2.0 and 0.5-1.0. The estimated value of σ depends on the scale of the
variables. Although the estimators of σ seem very different, the economic implication
of σ in the model is that it denotes the transmission efficiency of the interest rate policy.
The larger the σ, the lower the efficiency of the interest rate policy. If we take the inverse
of the estimated σ, then the difference becomes insignificant. We also examine various
exogenous shocks recovered from different datasets and found that the movements are
similar. Even if we use both the T-bill rates and the HP de-trended output to replace the
original data, the results are still similar.24
In this paper, we estimate the model using the FIML approach. Although Lindé
(2005) believed that FIML is useful for obtaining better estimates, the normality as-
sumption of residuals may be a potential threat.25 Gali, Gertler, and Lopez-Salido (2005)
pointed out the reason that the FIML approach generates better estimates than the single-
24
The estimated parameters are ˜0.037, 1.601, 1.955, 1.135, 0.994, 0.927, 0.834.
25
In fact, all identified shocks εt reject the null hypothesis for the test of normality.
21

Table 3: Estimation results from a different dataset
Quadratic Trend Output Gap (1983Q3:2008Q2)
Âκ Âσ Ãϕπ
Âϕx Âρr Âρu Çρη
0.062
ˆ0.034
3.204
ˆ1.794
1.624
ˆ0.241
0.518
ˆ0.179
0.984
ˆ0.013
0.943
ˆ0.014
0.845
H-P Filter Output Gap
Âκ Âσ Ãϕπ
0.035
ˆ0.007
1.430
ˆ1.213
2.107
ˆ0.219
1.396
ˆ0.367
0.996
ˆ0.001
0.924
ˆ0.014
0.829
Core PCE
Âκ Âσ Ãϕπ
0.008
ˆ0.003
13.760
ˆ10.836
2.035
ˆ0.433
0.770
ˆ0.177
0.988
ˆ0.001
0.938
ˆ0.011
0.893
Treasury Bill Rates
Âκ Âσ Ãϕπ
0.068
ˆ0.025
1.773
ˆ0.897
1.445
ˆ0.173
0.548
ˆ0.101
0.973
ˆ0.008
0.947
ˆ0.013
0.811
Sub-sample (1987:Q1 - 2015:Q3)
Âκ Âσ Ãϕπ
0.026
ˆ0.008
11.041
ˆ8.110
1.589
ˆ0.390
0.647
ˆ0.274
0.983
ˆ0.001
0.943
ˆ0.014
0.904
Sub-sample (1983:Q3 - 2008:Q2)
Âκ Âσ Ãϕπ
0.079
ˆ0.036
2.051
ˆ0.998
1.731
ˆ0.165
0.989
ˆ0.148
0.989
ˆ0.010
0.940
ˆ0.014
0.753
GMM
Âκ Âσ Ãϕπ
0.087
ˆ0.043
1.938
ˆ1.512
1.495
ˆ0.257
0.672
ˆ0.165
0.983
ˆ0.018
0.967
ˆ0.027
0.815
Notes ¢ Standard errors in parentheses under the estimator.
equation GMM is that the former provides richer knowledge about the three-equations
model. Therefore, compared with the single-equation GMM method, which relies only
on the NKPC, the GMM method also generates reliable estimations while the model
structure is considered. If a different estimation approach is used, the estimation results
may vary greatly if the model is misspecified. If the estimators of these two approaches
22

were quite similar, then it would prove that the NK model is a good specification for the
actual economy. Thus, we perform another estimation using the GMM method.
Given the full realization in the model structure, the FIML estimator is obtained by
assuming that the disturbances are normally distributed; however, the GMM estimator
is obtained by assuming that the disturbances are orthogonal to the instruments. In this
case, following Galı and Gertler (1999) and Galı, Gertler, and Lopez-Salido (2001), we
use the lagged variables as instruments.
The orthogonality conditions are 26
EˆH1
zt ρH1
zt1ƒzt1 0.
Since we have three exogenous disturbances (εr, εu, εη) and three instruments (xt1, πt1, it1),
there are nine moment conditions for solving for six parameters. We can also perform a
test of over-identifying restrictions to check whether the moment conditions hold. Ta-
ble (3) presents the estimation results and Hansen’s J-test statistic (0.6817, p = 0.8775),
which supports the model’s validity. The results are consistent with previous estimates.
Although the two estimation methods suggest similar estimates, we find that the object
function of the GMM approach is highly nonlinear. That is, its curvature is large and a
good guess of initial values is required. Unlike the FIML approach, the estimation result
of GMM is sensitive to the chosen initial value, and it is time-consuming to try different
initial values.
We have provided strong evidence that our estimation results are quite robust. Even
with different materials, estimation methods, and sub-sample periods, we obtain consis-
tent results.
26
The orthogonality conditions are equal to Eˆεtƒzt1 0 when the optimal weighting matrix is con-
sidered.
23

4.3 External Validity
Svensson (2003) indicated that commitment to a simple instrument rule does not cap-
ture the interest rate dynamics in inflation-targeting countries such as Canada and the
UK. It is not appropriate to apply the instrument rule to inflation-targeting central banks.
Although the simple Taylor rule is not suited to an inflation-targeting interest rate policy
regime, in this model, the deviation contains the information about the behavior of the
monetary authority. If this model explains the interactions among the output gap, infla-
tion, and interest rate, it should not be valid for a specific country only. In this subsection,
we will explore whether this model can help explain the dynamics of variables in Canada
and the UK.
The T-bill rate serves as the operating target for the nominal interest rate. It is well
known that the inflation measure of the retail price index, excluding mortgage interest
payments (RPIX), was the UK’s target rate of inflation before 2003 and prior to being
changed to CPI. Because most of the samples are drawn from this period, we use the
annual change in the RPIX as a measure of inflation in the UK. In Canada, the inflation-
control target is to keep the total CPI inflation within the range of 1-3%. Because this
study focuses on the deviation, it is very important to select an appropriate variable in
the operational guidelines of the interest rate rule. However, the Bank of Canada has
emphasized that core inflation is monitored as an operational guide to achieve the total
inflation (inflation measured by CPI) target. Therefore, the interest rate policy is more
likely to respond to core inflation (inflation measured by core CPI) due to the relatively
large volatility in total inflation. Thus, we use the inflation measured by the annual change
in the core CPI for Canada. The output gap announced by the Bank of Canada is used as
the output gap measure.27
27
The output gap obtained from the Bank of Canada is very similar to that implied by the real detrended
GDP based on the HP filter.
24

Unlike the US and Canada, the Bank of England does not release point estimates
on the output gap. From the above, the detrended real output based on the HP filter is
close to the official output gap in the US and Canada. Thus, the output gap is measured
by the HP filter detrended real output.28 The sample period runs from 1983:Q3 through
2015:Q1.29 Figures (6) and (7) present the time series plots, and the identified shocks are
plotted in Figures (8) and (9).30
After the 1990 oil crisis, both countries controlled inflation at about 2%. As can be
seen, the interest rates disregard the output gap and match the inflation in the UK while
the central bank cut interest rates significantly during the Canadian recession in the early
2000s. This reflects the fact that the central bank moved the real interest rate in response
to inflation. Table (4) reveals that the structural parameters are quite similar to those for
the US, which is reasonable in that both are developed countries. Compared with the US,
the estimates on ϕπ are relatively large and the output gap does not play an important role
in the interest rate policy rules in either Canada or the UK. These results may characterize
the behavior of inflation-targeting central banks. Although not reported, the variance in
the prediction error is small, and current variables are not significant predictive factors
in prediction errors in these two countries; the autocorrelations suggested by the model
are also very close to the sample counterparts.
This section has provided evidence that the estimation approach is not empirically
valid for only one country. We use this model to identify the coefficients in the Taylor
rule for countries that do not explicitly follow the Taylor rule. We observe the different
degrees of response to inflation in different countries. The coefficient estimates show a
stronger response to inflation from an inflation-targeting central bank. From the em-
28
We get similar estimates from the quadratic detrended real output over the period 1983:Q1-2008:Q2.
29
The inflation data in Canada and the UK are not available from the FRED after 2015:Q1.
30
The inflation data are obtained from FRED. The series IDs are CANCPICORMINMEI for Canada
and CPRPTT02GBQ661N for the UK. The data of real GDP and T-bill rates are from the International
Financial Statistics (IFS) published by the International Monetary Fund (IMF).
25

Table 4: Estimation results in different countries
Canada (1983Q3:2015Q1)
Âκ Âσ Ãϕπ
Âϕx Çρη Âρr Âρu
0.238
ˆ0.240
15.640
ˆ10.120
2.743
ˆ0.442
0.000
ˆ0.753
0.979
ˆ0.011
0.906
ˆ0.024
0.829
United Kingdom (1983Q3:2015Q1)
Âκ Âσ Ãϕπ
Âϕx Çρη Âρr Âρu
0.059
ˆ0.069
20.286
ˆ13.242
3.164
ˆ0.738
0.000
ˆ2.452
0.999
ˆ0.010
0.935
ˆ0.007
0.918
Notes ¢ Standard errors in parentheses under the estimator.
pirical results, we find that even though the interest rate policy is not conducted by the
Taylor rule in practice, it is still desirable to set the interest rate rule as a Taylor rule with
an autocorrelated deviation in the NK model.
4.4 Source of Deviation
In this subsection, we explore the economic indicators that affect deviation. To investi-
gate the decision process of the deviation, we estimate the following regression model:
ηt a ρηt1
3
Qj 0
bjAtj ζt,
where At is the economic indicator in period t and ζt represents the regression residual.
The additional lags of the dependent variable are considered since the reaction rate for
future information is different among indicators.
The deviation in the monetary policy rule represents the monetary authority’s re-
sponse to transitory shocks. In general, the monetary authority might respond to fluctu-
ations in stock market (see Rigobon and Sack (2001)), exchange rates (see Taylor (2001)),
and so on. Moreover, Kahn et al. (2010) demonstrated that deviation can be predicted by
changes in housing and commodity prices. We consider two more indicators: the unem-
26

ployment rate and consumer sentiment. One of the objectives of the monetary authority
is to achieve full employment; thus, unemployment can be considered as a proxy for the
output gap in the interest rate rule (see Clarida, Gali, and Gertler (2000)). Accordingly,
changes in unemployment may explain certain parts of the monetary policy. On the
other hand, Bernanke (2010) argued that the deviation declines when the real-time fore-
casts of output gap and inflation are used as target variables. If the monetary authority
adopts a forward-looking framework, then the variables that provide information about
the future output gap and inflation should help explain the deviation from the Taylor rule.
Consumer sentiment is an indicator that reflects consumer optimism and expectations
about the overall state of the economy, which may also explain the deviation.
The exchange rate is measured by the real effective exchange rate (REER) index and
the commodity price is measured by the producer price index (PPI). The US stock market
is measured by the SP500 stock price index, the Canadian stock market is measured by
the SP/TSX stock price index, and the FTSE 100 index is used for the UK.31 Consumer
sentiment is measured by the consumer confidence index. The data source is described
in Appendix 2 and all data are plotted in Figures (10)-(12). All economic indicators are
measured by the first differences of the logarithmic (seasonally adjusted) index (1983:Q3
= 100) except unemployment. Change in employment is measured by the first differences
of the (seasonally adjusted) unemployment rates.
If the indicator can help explain the deviation, then the coefficients on Atj should
be jointly significantly different from zero. Table (5) examines whether these indicators
help explain the deviation. Unemployment and consumer confidence can help explain
the deviation in Canada, whereas deviation in the UK can be explained by commodity
price and stock market. Contrary to expectations, the result shows that all the indicators
31
Since the FTSE starts from 1984:Q1, the sample period will also be adjusted to 1984:Q1-2008:Q2 for
the UK.
27

Table 5: F-test to check whether the economic indicator can explain the deviation
House Commodity Price Stock Exchange Rate Unemployment Rate Consumer Confidence
US 1.19
ˆ0.32
0.24
ˆ0.92
1.15
ˆ0.34
1.22
ˆ0.31
0.47
ˆ0.76
1.32
ˆ0.27
CA 0.97
ˆ0.43
1.65
ˆ0.17
0.47
ˆ0.76
1.24
ˆ0.30
2.28‡
ˆ0.06
2.06*
ˆ0.09
GB 0.12
ˆ0.97
5.84***
ˆ0.00
0.97
ˆ0.43
0.81
ˆ0.52
1.15
ˆ0.34
0.47
ˆ0.76
Notes ¢ P-values in parentheses under the test statistic. Asterisks ***, **, and * denote significance at 1%,
5%, and 10%, respectively.
do not explain the deviation in the US. One possible explanation is that the relationship
between the deviation and the economic indicator is nonlinear. The indicator may affect
the monetary policy only when the fluctuations are relatively intense, for example, the
1990 oil price spike and the dot-com bubble. Therefore, we apply quantile regression (see
Koenker and Bassett Jr (1978)) to reveal information on the nonlinear effect of economic
indicators on the deviation.
The quantile regression model is represented as follows:
Qτˆηtƒηt1, Atj aˆτ γˆτηt1
3
Qj 0
bˆτjAtj,
where Qτˆηtƒηt1, Atj is the conditional τth quantile of ηt. The notation bˆτj stresses
that the marginal effect of the economic indicator may be different for each respective
quantile τ. We focus on the deviation dynamics across two specific quantiles, τ 0.25
and τ 0.75, which, respectively, represent accommodative and contractionary mone-
tary policy. According to Koenker and Bassett Jr (1978), the estimator of parameters is
obtained by solving the following problem:
arg min
aˆτ,γˆτ,™bˆτjž
3
j 0
Qi
ρτ
`
d
d
d
d
b
ηt
’
”
aˆτ γˆτηt1
3
Qj 0
bˆτjAtj
“
•
a
e
e
e
e
c
(26)
28

Table 6: F-test to check the nonlinear effect
House Commodity Price Stock Exchange Rate Unemployment Rate Consumer Confidence
US
τ 0.25 2.56**
ˆ0.04
1.00
ˆ0.41
2.03*
ˆ0.10
2.05*
ˆ0.09
0.44
ˆ0.78
9.45***
ˆ0.00
τ 0.75 13.35***
ˆ0.00
0.75
ˆ0.56
7.62***
ˆ0.00
0.93
ˆ0.45
2.03*
ˆ0.09
2.25*
ˆ0.07
CA
τ 0.25 3.31***
ˆ0.01
2.03*
ˆ0.09
0.55
ˆ0.70
11.07***
ˆ0.00
3.31**
ˆ0.01
1.28
ˆ0.28
τ 0.75 0.15
ˆ0.96
1.57
ˆ0.19
7.94***
ˆ0.00
0.62
ˆ0.65
0.32
ˆ0.86
0.89
ˆ0.47
GB
τ 0.25 0.10
ˆ0.98
2.80**
ˆ0.03
3.65***
ˆ0.01
1.01
ˆ0.04
0.61
ˆ0.65
0.92
ˆ0.45
τ 0.75 1.09
ˆ0.37
4.84***
ˆ0.00
0.37
ˆ0.83
3.83***
ˆ0.01
0.19
ˆ0.95
0.85
ˆ0.40
Notes ¢ P-values in parentheses under the test statistic. Asterisks ***, **, and * denote significance at 1%,
5%, and 10%, respectively.
where ρτ is the check function (defined as ρτˆz τz for z g 0) and ρτˆz ˆτ 1z if
z d 0. To explore whether the economic indicator affects the accommodative (contrac-
tionary) monetary policy, we test the hypothesis that the coefficients on Atj should be
jointly significantly different from zero given τ 0.25 (τ 0.75).
The estimation results are reported in Table (6). Results are mixed. The housing mar-
ket and unemployment affect the monetary policy in the US and Canada. The commodity
price affects the monetary policy in Canada and the UK, perhaps due to the changes in
commodity price, which are argued to be leading indicators of future inflation. The ef-
fects of the stock market and exchange rate on monetary policy are consistent across all
three countries. In addition, consumer sentiment provides information about the mon-
etary policy in the US. In summary, the economic indicators that contain information
about consumer expectations and confidence help explain the Taylor rule deviations in
29

the US. Factors that may contribute to price volatility and influence monetary policy in
Canada and the UK are commodity price and exchange rate.
Moreover, to distinguish the key variables in the decision process of deviation, we
estimate a regression model to explain the dynamics of the deviation. The unrestricted
model involves all economic indicators and the final model is selected by Akaike’s infor-
mation criterion. The estimated equations are as follows:
US ¢
Âηt 0.008
ˆ0.055
0.898
ˆ0.037
ηt1 0.076
ˆ0.039
∆STOCKt3 0.122
ˆ0.074
∆REERt 0.055
ˆ0.026
∆CONFt2, R
2
0.82.
CA ¢
Âηt 0.142
ˆ0.188
0.868
ˆ0.037
ηt1 0.205
ˆ0.136
∆PPIt1 0.274
ˆ0.141
∆PPIt3 0.056
ˆ0.027
∆STOCKt
1.362
ˆ0.822
∆UNEMt1 0.504
ˆ0.314
∆CONFt 0.414
ˆ0.355
∆CONFt1, R
2
0.72.
GB ¢
Âηt 0.570
ˆ0.193
0.907
ˆ0.022
ηt1 0.939
ˆ0.177
∆PPIt 0.455
ˆ0.237
∆PPIt2 0.416
ˆ0.201
∆PPIt3
0.081
ˆ0.049
∆REERt 0.098
ˆ0.056
∆REERt3 1.274
ˆ0.750
∆UNEMt2 1.348
ˆ1.059
∆UNEMt3
0.457
ˆ0.331
∆CONFt1 0.380
ˆ0.322
∆CONFt3, R
2
0.88.
, where the numbers in parenthesis are Newey-West robust standard errors.
In this subsection, we provide evidence to show that the Taylor rule deviation is af-
fected by a wide variety of exogenous disturbances with different degrees of persistence in
either linear or nonlinear format. The deviation does not follow a stochastic process but
is rather decided by the monetary authority. However, if the deviation is a reaction func-
tion of exogenous disturbances, it will approximate the dynamics of a stochastic process.
30

Furthermore, suppose the model selection criteria are changed to a Bayesian information
criterion, then the model that only contains lagged deviation is dominant in the US and
Canada. This result implies that the explanatory power of economic indicators is limited.
In conclusion, it is not too critical to assume that the deviation follows a stationary ARˆ1
process.
5 Conclusion
The simple NK model and the Taylor rule are popular due to their simplicity, but have
been criticized for their inability to characterize the real economy. Moreover, the param-
eter estimation may suffer from identification problems as described by Canova and Sala
(2009) and Cochrane (2011). In this study, we provide resolution to identify the struc-
ture parameters in the NK model. We show that if interest rate rules are set as a Taylor
rule with autocorrelated deviations in the NK model, then this model provides a good
representation of reality. In this manner, the inconsistency problems of the models used
in theoretical and empirical analyses have also been resolved since the exogenous distur-
bances in the IS and Phillips curve are serially correlated. To make the error term white
noise, previous studies tend to use a hybrid NK trinity model to make the estimators
consistent with theory. However, we find quite robust evidence that the pure forward-
looking version of the NK model has performed outstandingly well under empirical test-
ing. Nevertheless, there is room for improvement. For instance, parameter estimation is
based on the model equilibrium, which is dependent on the shock structure of the error
terms. We assumed that the shocks are i.i.d. in NK modeling, but found that the identi-
fied shocks are slightly serially correlated, suggesting that there is scope for improvement
in the assumptions of the error term structure.
The study investigates an alternative way to identify Taylor rule deviation. Compared
with a single-equation estimation or a calibration scheme, the empirical results from a
31

full-system approach are more convincing and the connection between the identified
deviation and real economic activity is stronger. Further research may focus on the out-
of-sample exchange rate and interest rate predictability of the identified Taylor rule de-
viations.
Ben S. Bernanke stressed that the interest rate policy should be systematic, not auto-
matic.32 It is too arbitrary to interpret the behavior of the monetary authority as following
a simple instrument rule mechanically. We provide evidence that the interest rate policy
can be decomposed into two parts: a systematic part, described by the Taylor rule, and a
nonsystematic component. The nonsystematic component can be explained by the eco-
nomic indicators that contain the information about the future path of the economy, for
example, consumer sentiment and commodity prices. Our findings may provide useful
recommendations for further research on the specifications of interest rate policy.
32
See http://www.brookings.edu/blogs/ben-bernanke/posts/2015/04/28-taylor-rule-monetary-policy
for the detailed discussion.
32

Appendix 1
Using (5) to eliminate interest rate in (1) and (2) gives the equations:
xt Etˆxt1 σ1
ϕππt ϕx xt ηt Etˆπt1 re
t ¥ (A1)
πt κxt βEtˆπt1 ut (A2)
We first guess the form of solution is
xt cxr re
t cxu ut cxη ηt (A3)
πt cπr re
t cπu ut cπη ηt (A4)
By ARˆ1 structure, the conditional expectation for t 1 evaluated at time t is
Etˆxt1 Etˆcxr rt1 cxu ut1 cxη ηt1 cxrρrre
t cxuρuut cxηρηηt (A5)
Etˆπt1 Etˆcπr rt1 cπu ut1 cπη ηt1 cπrρrre
t cπuρuut cπηρηηt (A6)
Using (A5) and (A6) to substitute the shocks in (A1) and (A2), we then have
˜ σˆρr 1 ϕx¥cxr ˆρr ϕπcπr 1re
t ˜ σˆρu 1 ϕx¥cxu ˆρu ϕπcπu ut
˜ σˆρη 1 ϕx¥cxη ˆρη ϕπcπη 1ηt 0 (A7)
κcxr ˆβρr 1cπr¥re
t κcxu ˆβρu 1 1¥ut κcxη ˆβρη 1cπη¥ηt 0 (A8)
We assume all shocks have zero mean; then, expectation on both sides in (A7) and (A8)
gives
σˆρr 1 ϕx¥cxr ˆρr ϕπcπr 1 0
σˆρu 1 ϕx¥cxu ˆρu ϕπcπu 0
σˆρη 1 ϕx¥cxη ˆρη ϕπcπη 1 0
κcxr ˆβρr 1cπr 0
κcxu ˆβρu 1cπu 1 0
κcxη ˆβρη 1cπη 0
33

Given the realization of parameters in ˜β, κ, σ, ϕπ, ϕx , ρr, ρu, ρη, then solving the six
equations and six unknowns ˜cxr, cxu, cxη, cπr , cπu , cπη yields the result shown in (11).
Appendix 2
Data sources are described in the following table.
US CA GB
House Price Index
Source Datastream Datastream Datastream
Code USXPHI..E CNXPHI..F UKXPHI..E
Producer Price Index
Source IFS IFS IFS
Stock Price Index
Market SP 500 SP/TSX FTSE 100
Source IFS Yahoo Finance Yahoo Finance
Real Effective Exchange Rate
Source IFS IFS IFS
Unemployment Rate
Source FRED FRED FRED
Code UNRATE LRUNTTTTCAQ156S LMUNRRTTGBQ156S
Consumer Confident Index
Source FRED Datastream Datastream
Code UMCSENT CNOCS005Q UKOCS005Q
34

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37

1985 1990 1995 2000 2005 2010 2015
−6−4−202
Output Gap
1985 1990 1995 2000 2005 2010 2015
12345
Inflation Rate
1985 1990 1995 2000 2005 2010 2015
0246810
Interest Rate
Figure 1: Output gap, inflation, and interest rates (shaded areas indicate the period fol-
lowing the peak through the trough)
1985 1990 1995 2000 2005 2010 2015
−4−2024
Demand Shock
1985 1990 1995 2000 2005 2010 2015
−0.4−0.20.00.20.4
Supply Shock
1985 1990 1995 2000 2005 2010 2015
−20246
Taylor Rule Deviation
Figure 2: Identified shocks (shaded areas indicate the period following the peak through
the trough)
38

Time
1985 1990 1995 2000 2005 2010 2015
−6−22
Time
1985 1990 1995 2000 2005 2010 2015
−6−22
Time
1985 1990 1995 2000 2005 2010 2015
−6−22
Output Gap
Time
1985 1990 1995 2000 2005 2010 2015
024
Time
1985 1990 1995 2000 2005 2010 2015
024
Time
1985 1990 1995 2000 2005 2010 2015
024
Inflation Rate
1985 1990 1995 2000 2005 2010 2015
048
1985 1990 1995 2000 2005 2010 2015
048
1985 1990 1995 2000 2005 2010 2015
048
Interest Rate
Figure 3: Rational expectations and forecast errors (the dashed-line denotes expected
values and the dotted-line denotes forecast errors)
Time
2005 2010 2015
−2024
Time
2005 2010 2015
−2024
Greenbook
NK model
Real GDP
Time
2008 2010 2012 2014
−0.50.51.0
Time
2008 2010 2012 2014
−0.50.51.0
Greenbook
NK model
Core CPI
2005 2010 2015
−0.50.5
2005 2010 2015
−0.50.5
Greenbook
NK model
Interest rates
Figure 4: Real-time forecast errors comparison
39

Time
1985 1990 1995 2000 2005 2010 2015
−6−4−2024
CBO
QT
HP
Time
1985 1990 1995 2000 2005 2010 2015
12345
CPI
PCE
1985 1990 1995 2000 2005 2010 2015
0246810
Fed
T−bill
Figure 5: Data comparison
1985 1990 1995 2000 2005 2010 2015
−3−2−10123
Output Gap
1985 1990 1995 2000 2005 2010 2015
0123456
Inflation Rate
1985 1990 1995 2000 2005 2010 2015
02468101214
Interest Rate
Figure 6: Output gap, inflation, and interest rates in Canada (shaded areas indicate the
OECD-based recession)
40

1985 1990 1995 2000 2005 2010 2015
−3−2−10123
Output Gap
1985 1990 1995 2000 2005 2010 2015
2468
Inflation Rate
1985 1990 1995 2000 2005 2010 2015
02468101214
Interest Rate
Figure 7: Output gap, inflation, and interest rates in the UK (shaded areas indicate the
OECD-based recession)
1985 1990 1995 2000 2005 2010 2015
−505
Demand Shock
1985 1990 1995 2000 2005 2010 2015
−0.50.00.51.0
Supply Shock
1985 1990 1995 2000 2005 2010 2015
−505
Figure 8: Identified shocks for Canada (shaded areas indicate the OECD-based recession)
41

1985 1990 1995 2000 2005 2010 2015
−505
Demand Shock
1985 1990 1995 2000 2005 2010 2015
−0.20.00.20.4
Supply Shock
1985 1990 1995 2000 2005 2010 2015
−10−505
Figure 9: Identified shocks for the UK (OECD-based recession)
Time
1985 1995 2005 2015
4.65.05.45.8
House Price
Time
ts(USppi,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.65.0
Commodity Price
Time
1985 1995 2005 2015
4.55.56.5
Stock Price
Time
ts(USreer,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.34.54.7
REER
1985 1995 2005 2015
46810
Unemployment Rate
ts(UScfi,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.24.44.64.8
Consumer Confidence Index
Figure 10: Economic indicators in the US
42

Time
1985 1995 2005 2015
4.65.05.4
House Price
Time
ts(CAppi,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.64.85.05.2
Commodity Price
Time
1985 1995 2005 2015
4.54.74.95.1
Stock Price
Time
ts(CAreer,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.24.44.6
REER
1985 1995 2005 2015
681012
Unemployment Rate
ts(CAcfi,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.564.59
Figure 11: Economic indicators in Canada
Time
1985 1995 2005 2015
5.06.0
House Price
Time
ts(GBppi,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.65.05.4
Commodity Price
Time
1985 1990 1995 2000 2005 2010 2015
5.05.56.06.5
Stock Price
Time
ts(GBreer,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.454.604.75
REER
1985 1995 2005 2015
57911
Unemployment Rate
ts(GBcfi,start=c(1983,3),freq=4)
1985 1995 2005 2015
4.564.584.604.62
Figure 12: Economic indicators in the UK
43

Analysis of Taylor Rule Deviations

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