Analysis of Taylor Rule Deviations

Cheng-CheHsu*

Department of Economics, National Taiwan University, No. 1, Sector 4, Roosevelt Road, Taipei, Taiwan

Abstract

ais 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 oers a better explanation of the interac-

tions among interest rates, the output gap, and ination. In addition, we use dierent

datasets and an alternative estimation approach to check the empirical validity of theNK

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 Classication: C32,E12,E52

*Tel.: +886 (2)3366 3366 ext 55753. E-mail: d01323005@ntu.edu.tw.

1

1 Introduction

Taylor rule deviation is the dierence between the nominal interest rate and the level

prescribed by the Taylor rule, which is proposed by Taylor (1993). ae Taylor rule is

expressed as

it 2 t 0.5t 2 0.5 xt

where it denotes the nominal interest rate and t denotes the annual ination rate. ae

term xt represents the output gap or the deviation of the log of the real GDP from that

of the potential GDP.1 ae 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 nancial imbalances.

In fact, nomonetary authority announces a standard benchmark for determining the

interest rate structure; thus, the observed deviation depends on the reaction coecients

in the rule. ae Taylor rule parameters play an important role in the evaluation of the

interest rate policy, but it is not normative to calibrate (estimate) the coecients. More-

over, Cochrane (2011) questioned the aspects of the identication 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 identied from the residuals of or-

dinary least square regressions (e.g., Rudebusch (2002)) or vector autoregression (e.g.,1In 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 reects the shis in preferences or the responses to unexpected disturbances of

the monetary authority. ae monetary authority might decide a signicantly negative

deviation, as seen in the era following the global nancial crisis. ae persistently loose

monetary policymakes estimation of the coecient of the interest rate rule unreliable in

a single-equation approach. aerefore, as indicated byCochrane (2007), it is a promising

possibility to obtain more convincing results from a full-system approach; for example,

see Smets andWouters (2003) and Ireland (2007).

Since the variables are determined simultaneously in the system, the interaction be-

tween the variableswill 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-identied). aat is, there exists a

dierent set of parameters that generate similar observational implications, as shown by

Onatski andWilliams (2004). To avoid the under-identication problem, it is desirable to

obtain reasonable estimates from a relatively simple system, for instance, thewell-known,

fundamental three-equation NK model.

In this study, with the inclusion of Taylor rule deviation, we show that theNK 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 ndings reveal that the NK model, which includes the deviation, depicts real

economic dynamics. First, the pure forward-looking NK model generates high ination

and a persistent output gap. We show that the one-step-ahead forecast of ination and

the output gap is fairly accurate, and the forecast errors are unpredictable, which implies

that the forecasts are rational. In addition, we use dierent datasets and an alternative2ais literature refers deviation as amonetary policy shock.

3

estimation approach (the generalizedmethod ofmoments, 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 exploreways inwhich possible factors impact the de-

viation. Taylor rule deviation can be explained by factors containing information about

future economic paths. ae 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 themodel and

the robustness of the estimation approach, and investigates the possible factors that aect

the deviation. Section 5 summarizes the research ndings.

2 Baseline Model

We consider a simple, well-known NK model in our analysis. ae economic environ-

ment is described by two key log-linear equations. aese functions are derived under

several assumptions such as nominal rigidities and monetary policy non-neutralities.

SeeWoodford (2003a) andGal (2009) for detailed information on a standard derivation

under rst principles.

2.1 Simple NK Model

An intertemporal IS curve (aggregate demand) and aNewKeynesianPhilips curve (NKPC,

aggregate supply) take the form3

xt Etxt1 1it Ett1 ret , (1)

t xt Ett1 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, t is the ination rate, it is the nominal interest3Modelswith similar components are described byWoodford (2003b), Iskrev (2010),Giannoni (2014),

andmany others.4Et E St

4

rate, is the discount factor, and and are positive coecients. Equation (1) is derived

from a representative consumers intertemporal Euler equation,where is the coecient

of relative risk aversion, denoting the reciprocal of the elasticity of intertemporal substi-

tution. Equation (2) is obtained by optimal pricing-setting for amonopolistic rm under

the Calvo (1983) framework, where denotes the rate of price adjustment. ae term ret

denotes the ecient rate of interest and ut represents the cost-push disturbance. aese

are the real exogenous disturbances, where the word "disturbance," rather than "shock,"

is used to remind us that ret and ut can be serially correlated.

ae cost-push disturbance depends on several exogenous disturbances such as tech-

nology shocks, shis in labor supply, and variations in material costs. ae ecient rate

of interest varies across time, whereas the response to preference shocks or uctuations

in government expenditures occur in the short run. Please refer to Woodford (2003a)

and Gal (2009) to see how disturbances arise from rst principles. From the above, the

exogenous disturbances ret and ut comprise various potential disturbances with dierent

degrees of persistence. For simplicity, both disturbances are assumed to follow a station-

ary AR(1) process as dened below:

ret r ret1 rt , (3)

ut u ut1 ut . (4)

ae exogenous disturbances here are more like the gaps between current aggregate de-

mand or supply levels and equilibrium levels. ae real exogenous shocks at period t are rt

and ut . ae current aggregate demand and supply levels are the sum of past shocks with

declining weights. An oil or nancial crisis can be considered to be a signicant shock

at specic periods, thereby aecting the aggregate supply or demand levels and causing

uctuations in the economy. ae size of the eect depends on the persistence of the ex-

ogenous disturbances, i.e., the AR1 coecient. Please note that the AR1 structure is

5

the key assumption to derive the unique stationary solution of the forward-looking linear

rational expectations model.

2.2 Taylor Rule

ae interest rate rule assumes that themonetary authority adopts a simple Taylor rule:

it t xxt t . (5)

ae interest rate policy is decomposed into two parts: a systematic component (described

by the Taylor rule) and a nonsystematic component (called Taylor rule deviation). ae

coecients of policy rule ( , x) are committed by themonetary authority at the begin-

ning. For each period, themonetary authority decides the level of t to adjust the interest

rate from the systematic policy.

ais setting allows the monetary authority to retain discretion in response to ma-

jor unexpected disturbances. ae term t 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 arehighly autocorrelated. From a single-

equation perspective, an inertial Taylor rule (i.e., a Taylor rulewith a partial adjustment)

is widely used in empirical studies since this rule appears to t 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. ae deviation of the interest rate rule implies that the mone-

tary authority responds to exogenous inuences aggressively, which is intuitively more

consistent with a central banks actual behavior.

Taylor rule deviation comprises many distinct components. ae original Taylor rule

contains an intercept,which implies an interest rateunder zero ination and full-employment5For example, it it1 1 xxt t t .

6

output in the long run, i.e., the so-called natural interest rate. Woodford (2001) noted

that the natural interest rate is aected by real disturbances; thus, the intercept should

be time-varying. However, the sources of the stochastic intercept are dicult to identify.

ae deviation can be considered to be a policy term considering stochastic intercept,

while uctuation in Taylor deviation reects the fact that the intercept varies over time.

Moreover, an interest policy would not consider only ination and the output gap

t , but may also involve the monetary authoritys response to other persistent shocks,

time-varying rules, etc. Alternatively, current ination 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 specication of (5), the mea-

surement error will enter into Taylor rule deviation. aere 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 coecients are determined, t is observed

by the residuals of (5). ais setting may aid identication, and thus, avoid the under-

identication problem.

Deviations reect shis in preferences or responses to unexpected disturbances of

the monetary authority and the measurement error faced by the monetary authority. It

is also aected by various potential disturbanceswith dierent degrees of persistence. For

simplicity, Taylor rule deviation t is also assumed to follow a stationary AR1 process,

as given by

t t1 t . (6)

In particular, Taylor rule deviation is correlated with the ination and the output gap.

For instance, there existed a signicant reduction in output and a large forecast error on

7

ination and the output gap during the global nancial crisis. When a single-equation

approach (either OLS or GMM) is adopted, this endogeneity causes the estimates of the

Taylor rule coecients 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 dierence equations of the following form:

Et

xt1

t1

=AAAAAA? A

xt

t

=AAAAAA? B et , (7)

where et ret , ut , t and

A

1 1x 1 1

1

=AAAAAA?, B

1 1 1

0 1 0

=AAAAAA?. (8)

ais system has a unique equilibrium only if both eigenvalues ofmatrixA lie outside the

unit circle. When the coecients in the policy rule are restricted to being non-negative

( , x A 0),Woodford (2003a) showed that the well-known condition of unique equi-

librium holds only if

1

x A 1. (9)

Woodford (2001) pointed to a simple implication of this condition. Equation (2) shows

that a permanent 1 percent increase in ination will raise the long-term average output

gap by ~1 percent. Plugging this fact into (5), suggesting that the interest rate

should increase by 1 x~, the Taylor principle stipulates that amonetary au-

thority should raise the nominal interest ratemore than the increase in ination,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 satised. Given , A 0, the nonlinear constraint in (9) is satised when the param-

eters are restricted to A 1 and x A 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 rst assume that competitive

equilibrium is a function of exogenous shocks and Taylor deviation. ae solution of this

system then takes the following form:

xt

t

=AAAAAA?

cxr cxu cx

cr cu c

=AAAAAA?

ret

ut

t

=AAAAAAAAA?

C et , (10)

where C is a 23matrix. Using themethod of undetermined coecients, the coecients

in matrix C can be shown as6

C

1 rr 1 rx

u u 1 ux

1 1 x

r 1 rx

x 1 uu 1 ux

1 x

=AAAAAAAAAA?

, (11)

where j 1 j1 j j, j > r, u, are terms unaected by the Taylor

rule coecients. We assume j 1 jx A 0 for j > r, u, . Given the

absolute value of the coecient of a stationary AR1 process is less than unity, , A 0,

and A 1 , x A 0, we can determine the sign of the coecients in matrix C. A positive

Taylor rule deviationwill lower both ination and the output gap. Both ination and the

output gap will increase due to a positive eciency rate shock, whereas a positive cost-

push shock will raise ination but lower the output gap; thus, we treat the eciency 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 ination rate in (5), the interest rate

also becomes a function of exogenous shocks and Taylor deviation. Hence, all endoge-6See Appendix 1 for details.

9

nous variables can be described as

zt

xt

t

it

=AAAAAAAAA?

cxr cxu cx

cr cu c

cir ciu ci

=AAAAAAAAA?

ret

ut

t

=AAAAAAAAA?

H et , (12)

with a 3 3 matrix H equal to

H

1 rr 1 rx

u u 1 ux

1 1 x

r 1 rx

x 1 uu 1 ux

1 x

1 rx yr 1 rx

ux 1 uu 1 ux

1 1 x y

1 x

=AAAAAAAAAAAAAAAAAAAA?

. (13)

3 Estimation

In this section, we check our progress in the estimation.

3.1 Parameter Estimation

Nason and Smith (2008) noted thepossible identication problemswith the single-equation

method. aerefore,we disregard the single-equation approach. Due to thepossible 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

ofmonetary policy aer themid-1980s. Even if we consider the Bayesian approach (see

Smets and Wouters (2003) and Rabanal and Rubio-Ramrez (2005)), the empirical re-

sult will depend on the tight prior distribution in small samples and may not provide a

reliable estimation with diuse prior distributions (see Cochrane (2007) for a detailed

discussion). Dierent prior distribution is needed if another dataset is used.

Previous studies have oen used the vector autoregression (VAR) approach to esti-

mate theNKmodel (Rudebusch and Svensson (1999), Del Negro et al. (2007), andKolasa,

10

Rubaszek, and Skrzypczyski (2012)). To increase the tness, the empirical model tends

to includemore lagged dependent variables but yields the over-parameterized problem.

If the cost-push shocks are signicantly serially correlated, then this could lead to biased

estimators (see Kuester, Mller, and Stlting (2009) and Zhang and Clovis (2010) for

more detailed discussions).

However, thehighdegreeof autocorrelation in time seriesdata, includingmore lagged

variables, can make the residualswhite noise and render the estimation easier. Although

the empirical model generates better performance from out-of-sample forecasts, the pol-

icy analysis becomesmore complicated under such a hybridNKmodel. For convenience

of analysis, it is desirable to obtain reliable estimates in a relatively simplemodel. Lind

(2005) indicated that FIML is a usefulway to obtain better estimates in the simultaneous

system.7 Considering the linear constraints of the coecients,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. ae long-run level of the output gap should be zero, but

the steady state of the ination rate is dicult to identify. Because we assume all shocks

have a zero mean, all variables are previously demeaned; thus, the sample mean of the

identied shock would also be zero. Actual data show that ination, the output gap,

and interest rates are highly autocorrelated but stationary. To investigate the explanatory

power of themodel, we retain any trends in the original data.

We estimateonly theparameters that appear in (12),which are , , , x , , , r , u.

Since the discount factor cannot be observed directly andNason and Smith (2008) sug-7We 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 may aid identication, we simply calibrate

= 0.99 for quarterly data.8 In the simultaneous equation system, the Taylor rule relates

only current ination, the output gap, and interest rates. ae series t can be observed

by the residuals from (5) once the parameters x , are determined. ae parameter

can be estimated directly using a simple ordinary least square (OLS) estimator once

x , are given.9 aerefore, we need to estimate six parameters simultaneously using

FIML; the estimation method is similar to that of Lind (2005). aemodel equilibrium

is given by (12) and the shock structure is

et et1 t , (14)

where

r 0 0

0 u 0

0 0

=AAAAAAAAA?

, t

rt

ut

t

=AAAAAAAAA?

. (15)

We assume t i .i .d . N0, . From (12), the exogenous shocks can be recovered by the

data zt using

et H1zt . (16)

aus, (14) can be rewritten as

Ht Het Het1 zt HH1zt1. (17)

aen, we have

zt Szt1 i .i .d . NHH1zt1,HH1. (18)8We also tried to estimate the discount factor instead of using the calibrated value. ae estimated value

is 0.933 and coecients in IS and NKPC are changed slightly. However, the Taylor rule coecients arealmost unchanged and the observational implications of themodel are very similar.

9Our results show that the OLS andMLE estimators are rather close.

12

Let t zt HH1zt1 and HH1, ae conditional log-likelihood function is

thus

ln , , x , , r , u T~2 ln2 T~2 ln SS 1~2T

Qt1

t1 t . (19)

Because we do not have any prior information about the shock structure, there are no

restrictions imposed on the covariancematrix in the conditional log-likelihood func-

tion. However, while we determine the parameters, the exogenous shocks are identied

and the covariance matrix can be estimated directly from the identied shocks. ae

estimator is obtained by the sample covariancematrix of H1zt H1zt1. ae con-

ditional log-likelihood function then becomes

ln , , x , , r , u T~2 ln2 T~2 ln T T 1~2T

Qt1

t 1t , (20)

where H H1. ae FIML estimator is obtained by maximizing (20) with linear

constraints, including , A 0, 0 @ r , u @ 1, A 1, and A 0.

3.2 Estimation on US data

ae 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 desirablemeasures of ination and the output gap is important. Other-

wise, the observedTaylor deviationmay deviate from the situation faced by themonetary

authority.

It iswidely known that theTaylor rulewas seen as oering an appropriate description

of the interest rate policy regime aer the mid-1980s. ae interest rates suggested by

the rule were substantially consistent with the federal fund rates during periods of low

ination and low macroeconomic volatility. ae second oil crisis led to stagation in

the 1980s; thus, selecting it as the basis would make ination and interest rates show

13

a signicant downward trend. ae Taylor rule may deviate from the actual interest rate

policy during periods of high ination,whichwouldmake the estimatedTaylor 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 ination rate to be

measured as the annual change in the consumer price index (CPI).11to the GDP dea-

tor that Taylor (1993) originally used. However, policymakers may look at various CPI

measures. aemost common ination measure for policymakers is the core CPI, which

excludes food and energy items. ae core CPI excludes items that tend to uctuate dra-

matically; thus, using an ination rate dened by the core CPI could avoid excessive

volatility in interest rates caused by severe uctuations in ination. Hence, we calculate

the ination rate using the annual change in the core CPI. For the output gap,we consider

the potential output estimated by the Congressional Budget Oce (CBO).12 ae output

gap is measured by 100log yt log yt , where yt 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-

ation increases distinguish the third oil crisis. We see that when the economy is in re-

cession, there is a signicant decline in the output gap and themonetary authority tends

to cut interest rates substantially. ais may support the validity of the Taylor rule. Note

that aer the nancial crisis, the interest rate fell to exceptionally low levels. ae starting

value in the estimations for , , x , , r , u are 0.0238, 0.1567, 1.5,0.5, 0.5, 0.5, re-10We also consider dierent sample periods in the later robustness analysis.11See http://www.federalreserve.gov/newsevents/press/monetary/20120125c.htm.12ae estimates prepared by the Federal Reserve sta are discovered aer a ve-year lag.13ae interest rate is measured by the quarterly average federal funds rate (FEDFUNDS). ae series IDs

of core CPI, real GDP, and real potential output are CPILFESL, GDPC1, and GDPPOT, respectively.14ae NBER-based recession indicators are also obtained from FRED.

14

https://research.stlouisfed.org/fred2/http://www.federalreserve.gov/newsevents/press/monetary/20120125c.htm

Table 1: Estimation results for U.S. data

x r u0.99 0.073

0.0291.9811.182

1.5490.199

0.6330.120

0.9740.008

0.9470.013

ru r u 2r 2u 20.863 -0.712 0.324 -0.215 4.51 0.012 1.481

Notes Standard errors in parentheses under the estimator. ae hat denotes the estimated value by FIML,

the tilde denotes the sample counterparts of identied shocks, and is the calibrated value.

spectively, where 0.0238 and 0.1567 as suggested by Rotemberg andWoodford

(1997) for theUS data, and 1.5 and x 0.5 are taken from Taylor (1993).15 Because

we have no prior information for the exogenous shocks, the initial values for r and u

are simply set to 0.5. ae estimation results are presented in Table (1) and the identied

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

is relatively larger than that of Rotemberg andWoodford (1997), indicating that the e-

ciency of monetary policy is lower. ae estimated values of , , and x are very close

to the estimations by Rotemberg andWoodford (1997) and Taylor (1993), although we

use dierent datasets. In particular, Gal, Gertler, and Lopez-Salido (2001) obtained a

signicantly negative from an output-gap-basedNKPC, which is inconsistent with the

theory. With a full-system approach, our estimation results suggest theoretically consis-

tent estimates of theNKPC coecients. Even though the linear constraints of coecients

are disregarded,we still obtain the same estimates. ais feature implies that the estimated15We also try using alternative initial values 0.34, 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.16ae identied shocks are obtained by et H1zt , where the corresponding values in H1 with the

estimated parameters are

0.24 0.54 0.720.06 0.14 0.060.63 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, ination, and interest rate. When the output gap is divided by two, ,

, and x become twice and others remain the same. When the ination is divided by

two, and become half and x becomes twice,while the others remain the same. If the

interest rate is multiplied by two, then only x and x become twice. From the above,

denotes the relationship between the expected ination and the current output gap in the

IS curve. ae measures the ination-output trade-o, and and x are parameters to

identify the nonsystematic component in the interest rate rule. ae parameters r and u

determine the persistence of the exogenous shocks.

Moreover, the identied demand and supply shocks uctuated markedly in 1990,

2001, and 2008. aese dates correspond to the 1990 oil price shock, the dot-com bub-

ble, and the nancial crisis, respectively. ae identied shocks reect the external dis-

turbances encountered by the real economy. aus, the simple NK model oers a good

empirical description of the output gap, ination, 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 ination. 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 AR1 structure of exogenous shocks is the key assumption to formulate the rational

expectations and characterize the equilibrium. aerefore, we will discuss the empirical

16

performance of the pure forward-looking NK model with AR1 shocks.

ae persistence of exogenous shocks is signicant, 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 cwrret cwuut cwt ,

for w > x , , i. aerefore, the variable is also stationary and the rst-order autocor-

relation coecient of themodel is

1 Covwt ,wt1Varwt

. (21)

Many current models use the Phillips curve, which includes lagged variables, to gener-

ate high ination 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 rst-order autocorrelation coecient of the estimatedNK

model is very similar to the sample counterpart.18 ais suggests that persistent ination

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 ae one-step-

ahead forecast is constructed by

Etzt1 EtHet1 H et H H1zt . (22)17We also compared estimated parameters r and u with the values estimated directly from the iden-

tied shocks. ae results show that they are very close.18ae sample autocorrelation coecients of the output gap, ination, and interest rate are 0.969, 0.983,

and 0.985, respectively, whereas the values implied by themodel are 0.973, 0.976, and 0.982, respectively.19Although the one-step-ahead forecast is dependent on current variables only, the coecients inmatrix

H are obtained when the full sample is used.

17

Figure (3) plots the comparison of expected and actual values. ae 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, ination, and interest rate. For highly autocorrelated

data, the prediction generates small forecast errors. We assume that the shock structure

follows a simple AR1 process. Hence, the forecast errors can be represented as

zt1 Etzt1 H et1 H et H t1. (23)

Forecast errors are determined by the real disturbance term t1.

According to the denition of rational expectation, the prediction error is themean

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. aemean independent condition becomes

E zt1 Etzt1St E H t1 S ret , ut , t 0 (24)

We conduct a simple inspection to checkwhether this condition holds. At rst, 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. ae estimation equations are

wt1 Etwt1 w ,r ret w ,u ut w , t . (25)

for w > x , , i . Table (2) reports the estimation results.

ae empirical results show that the variance in the prediction error is small, especially

for ination. Compared with the single NKPC, considering the simultaneous equations

oers a better explanation of the ination dynamics. aere is strong evidence that exoge-

nous shocks provide no information on disturbance terms; thus, themean 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 Etxt1 t1 Ett1 it1 Etit1

x ,r x ,u x , R2 ,r ,u , R

2 i ,r i ,u i , R2

0.020.07

0.380.66

0.000.04

0.010.27

0.000.03

0.040.27

0.000.02

0.000.05

0.040.05

0.510.59

0.010.03

0.010.24

Notes aeNewey-West robust standard errors in parentheses under the estimator. Mean square error in paren-

theses under R2.

ers (SPF) to compute the out-of-sample forecast errors.20 ae recursive scheme is used

to evaluate the out-of-sample one-step-ahead forecasts implied by the NK model. ae

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, ination, 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

ae comparison charts are presented in Figure (4). Interestingly, forecast errors of

both output and ination demonstrate the same tendency and suered signicant fore-

cast errors during the global nancial crisis. ae SPF forecasts of interest rates aremore

accurate than those of the NK model, but the forecast errors are similar in the zero-rate

era. However, the forecast errors reect the exogenous shocks faced by the economy.

aat the forecast errors of theNK model are similar to the real-time forecast, in practice,

implies that the identied exogenous shocks reect 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.20See https://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-

forecasters.21ae forecast errors are calculated by computing the one-step-ahead forecast minus the last vintage of

realization.

19

https://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-forecastershttps://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-forecasters

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 dierent sample period is employed, the estimated coecients may be dierent.

To check the robustness of the estimation procedure, we run the same algorithm with

a dierent dataset. ae stability of the estimated parameters reects the validity of the

NKTM model; therefore, we can check whether the validity is tied to a specic dataset

or sample period.

In the literature, in addition to measuring ination by CPI, the price index for per-

sonal consumption expenditures (PCE) is alsooenused tomeasure ination (e.g.,Rude-

busch (2002) and Cogley, Primiceri, and Sargent (2008)). Aer 2000 and for several rea-

sons, the Fed switched its focus from CPI to PCE when measuring ination.22 Although

bothmeasures draw on similar components, each uses very dierentweights. Compared

with CPI ination, PCE ination is a better predictor of ination faced by the general

population. aerefore, we replace CPI ination with PCE ination 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) l-

ter.23 Moreover, because Taylor deviation plays a very important role in the estimation,

selecting a measure of interest rate policy instrument aects the estimation results. Al-

though the federal funds rate is well known as a key policy instrument in theUS, we still

substitute the treasury bill (T-bill) rate for the federal funds rate as a robustness check.

Because the nancial crisis caused a signicant reduction in output, the quadratic

trendmodel makes the estimates of potential output unreliable,which results in a signif-22See http://www.federalreserve.gov/newsevents/press/monetary/20120125c.htm.23For quarterly data, the common smoothing parameter 1600 is used.

20

http://www.federalreserve.gov/newsevents/press/monetary/20120125c.htm

icant dierence 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 comparedwith theQT and CBO

output gaps, the HP output gap has a relatively small uctuation. Before 2004, PCE in-

ation was signicantly lower than CPI ination 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. ae rst sub-sample period is motivated by Taylor (1993)

and the starting period is 1987. ae second sub-sample period ends at 2008:Q2, so we

can determine whether the estimated coecients were signicantly dierent before and

aer the global nancial crisis. Table (3) reports the estimation results. Except for ,

the results are quantitatively similar. ae response coecients in the Taylor rule are also

not far from 1.5-2.0 and 0.5-1.0. ae estimated value of depends on the scale of the

variables. Although the estimators of seem very dierent, the economic implication

of in themodel is that it denotes the transmission eciency of the interest rate policy.

ae larger the , the lower the eciency of the interest rate policy. If we take the inverse

of the estimated , then the dierence becomes insignicant. We also examine various

exogenous shocks recovered from dierent datasets and found that the movements are

similar. Even if we use both the T-bill rates and theHP 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 Lind

(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-24ae estimated parameters are 0.037, 1.601, 1.955, 1.135, 0.994, 0.927, 0.834.25In fact, all identied shocks t reject the null hypothesis for the test of normality.

21

Table 3: Estimation results from a dierent dataset

Quadratic TrendOutput Gap (1983Q3:2008Q2) x r u

0.0620.034

3.2041.794

1.6240.241

0.5180.179

0.9840.013

0.9430.014

0.845

H-P Filter Output Gap x r u

0.0350.007

1.4301.213

2.1070.219

1.3960.367

0.9960.001

0.9240.014

0.829

Core PCE x r u

0.0080.003

13.76010.836

2.0350.433

0.7700.177

0.9880.001

0.9380.011

0.893

Treasury Bill Rates x r u

0.0680.025

1.7730.897

1.4450.173

0.5480.101

0.9730.008

0.9470.013

0.811

Sub-sample (1987:Q1 - 2015:Q3) x r u

0.0260.008

11.0418.110

1.5890.390

0.6470.274

0.9830.001

0.9430.014

0.904

Sub-sample (1983:Q3 - 2008:Q2) x r u

0.0790.036

2.0510.998

1.7310.165

0.9890.148

0.9890.010

0.9400.014

0.753

GMM x r u

0.0870.043

1.9381.512

1.4950.257

0.6720.165

0.9830.018

0.9670.027

0.815

Notes Standard errors in parentheses under the estimator.

equation GMM is that the former provides richer knowledge about the three-equations

model. aerefore, 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 dierent estimation approach is used, the estimation results

may vary greatly if the model is misspecied. If the estimators of these two approaches

22

were quite similar, then it would prove that the NK model is a good specication for the

actual economy. aus, 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 Gal and Gertler (1999) and Gal, Gertler, and Lopez-Salido (2001), we

use the lagged variables as instruments.

ae orthogonality conditions are 26

EH1zt H1zt1Szt1 0.

Sincewehave three exogenousdisturbances (r , u , ) and three instruments (xt1, t1, it1),

there are ninemoment 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 Hansens J-test statistic (0.6817, p = 0.8775),

which supports the models validity. ae results are consistent with previous estimates.

Although the two estimation methods suggest similar estimates, we nd that the object

function of the GMM approach is highly nonlinear. aat 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 dierent

initial values.

We have provided strong evidence that our estimation results are quite robust. Even

with dierent materials, estimation methods, and sub-sample periods, we obtain consis-

tent results.26ae orthogonality conditions are equal to Et Szt1 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 ination-targeting countries such as Canada and the

UK. It is not appropriate to apply the instrument rule to ination-targeting central banks.

Although the simple Taylor rule is not suited to an ination-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, ina-

tion, and interest rate, it should not be valid for a specic country only. In this subsection,

wewill explorewhether this model can help explain the dynamics of variables inCanada

and the UK.

ae T-bill rate serves as the operating target for the nominal interest rate. It is well

known that the ination measure of the retail price index, excluding mortgage interest

payments (RPIX), was the UKs target rate of ination 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 ameasure of ination in theUK. In Canada, the ination-

control target is to keep the total CPI ination 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 ination is monitored as an operational guide to achieve the total

ination (ination measured by CPI) target. aerefore, the interest rate policy is more

likely to respond to core ination (ination measured by core CPI) due to the relatively

large volatility in total ination. aus,weuse the inationmeasured by the annual change

in the core CPI for Canada. ae output gap announced by the Bank of Canada is used as

the output gap measure.27

27ae output gap obtained from the Bank of Canada is very similar to that implied by the real detrendedGDP based on theHP lter.

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 lter is

close to the ocial output gap in the US and Canada. aus, the output gap is measured

by the HP lter detrended real output.28 ae sample period runs from 1983:Q3 through

2015:Q1.29 Figures (6) and (7) present the time series plots, and the identied shocks are

plotted in Figures (8) and (9).30

Aer the 1990 oil crisis, both countries controlled ination at about 2%. As can be

seen, the interest rates disregard the output gap andmatch the ination in the UK while

the central bank cut interest rates signicantly during the Canadian recession in the early

2000s. ais reects the fact that the central bank moved the real interest rate in response

to ination. Table (4) reveals that the structural parameters are quite similar to those for

theUS,which is reasonable in that both are developed countries. Comparedwith theUS,

the estimates on are relatively large and the output gap does not play an important role

in the interest rate policy rules in eitherCanada or theUK.aese resultsmay characterize

the behavior of ination-targeting central banks. Although not reported, the variance in

the prediction error is small, and current variables are not signicant predictive factors

in prediction errors in these two countries; the autocorrelations suggested by themodel

are also very close to the sample counterparts.

ais section has provided evidence that the estimation approach is not empirically

valid for only one country. We use this model to identify the coecients in the Taylor

rule for countries that do not explicitly follow the Taylor rule. We observe the dierent

degrees of response to ination in dierent countries. ae coecient estimates show a

stronger response to ination from an ination-targeting central bank. From the em-28We get similar estimates from the quadratic detrended real output over the period 1983:Q1-2008:Q2.29ae ination data in Canada and the UK are not available from the FRED aer 2015:Q1.30ae ination data are obtained from FRED. ae series IDs are CANCPICORMINMEI for Canada

and CPRPTT02GBQ661N for the UK. ae data of real GDP and T-bill rates are from the InternationalFinancial Statistics (IFS) published by the International Monetary Fund (IMF).

25

Table 4: Estimation results in dierent countries

Canada (1983Q3:2015Q1) x r u

0.2380.240

15.64010.120

2.7430.442

0.0000.753

0.9790.011

0.9060.024

0.829

United Kingdom (1983Q3:2015Q1) x r u

0.0590.069

20.28613.242

3.1640.738

0.0002.452

0.9990.010

0.9350.007

0.918

Notes Standard errors in parentheses under the estimator.

pirical results, we nd 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 aect deviation. To investi-

gate the decision process of the deviation, we estimate the following regression model:

t a t1 3Qj0b jAt j t ,

where At is the economic indicator in period t and t represents the regression residual.

ae additional lags of the dependent variable are considered since the reaction rate for

future information is dierent among indicators.

ae deviation in the monetary policy rule represents the monetary authoritys re-

sponse to transitory shocks. In general, themonetary authoritymight respond to uctu-

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 twomore indicators: the unem-

26

ployment rate and consumer sentiment. One of the objectives of themonetary 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 ination 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 ination should help explain the deviation from theTaylor rule.

Consumer sentiment is an indicator that reects consumer optimism and expectations

about the overall state of the economy, which may also explain the deviation.

ae exchange rate is measured by the real eective exchange rate (REER) index and

the commodity price ismeasured by the producer price index (PPI).aeUS stockmarket

is measured by the S&P500 stock price index, the Canadian stock market is measured by

the S&P/TSX stock price index, and the FTSE 100 index is used for the UK.31 Consumer

sentiment is measured by the consumer condence index. ae data source is described

in Appendix 2 and all data are plotted in Figures (10)-(12). All economic indicators are

measured by the rst dierences of the logarithmic (seasonally adjusted) index (1983:Q3

= 100) except unemployment. Change in employment ismeasured by the rst dierences

of the (seasonally adjusted) unemployment rates.

If the indicator can help explain the deviation, then the coecients on At j should

be jointly signicantly dierent from zero. Table (5) examines whether these indicators

help explain the deviation. Unemployment and consumer condence 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 indicators31Since 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 CommodityPrice Stock ExchangeRate Unemployment Rate ConsumerCondence

US 1.190.32

0.240.92

1.150.34

1.220.31

0.470.76

1.320.27

CA 0.970.43

1.650.17

0.470.76

1.240.30

2.280.06

2.06*0.09

GB 0.120.97

5.84***0.00

0.970.43

0.810.52

1.150.34

0.470.76

Notes P-values in parentheses under the test statistic. Asterisks ***, **, and * denote signicance 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. ae indicator may aect

the monetary policy only when the uctuations are relatively intense, for example, the

1990 oil price spike and the dot-com bubble. aerefore,we apply quantile regression (see

Koenker and Bassett Jr (1978)) to reveal information on the nonlinear eect of economic

indicators on the deviation.

ae quantile regression model is represented as follows:

Qt St1,At j a t1 3Qj0b jAt j,

where Qt St1,At j is the conditional th quantile of t . ae notation b j stresses

that the marginal eect of the economic indicator may be dierent for each respective

quantile . We focus on the deviation dynamics across two specic quantiles, 0.25

and 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 mina,,b j

3j0

Qi

t

a t1

3Qj0b jAt j

=AAAA?(26)

28

Table 6: F-test to check the nonlinear eect

House CommodityPrice Stock ExchangeRate Unemployment Rate ConsumerCondence

US

0.25 2.56**0.04

1.000.41

2.03*0.10

2.05*0.09

0.440.78

9.45***0.00

0.75 13.35***0.00

0.750.56

7.62***0.00

0.930.45

2.03*0.09

2.25*0.07

CA

0.25 3.31***0.01

2.03*0.09

0.550.70

11.07***0.00

3.31**0.01

1.280.28

0.75 0.150.96

1.570.19

7.94***0.00

0.620.65

0.320.86

0.890.47

GB

0.25 0.100.98

2.80**0.03

3.65***0.01

1.010.04

0.610.65

0.920.45

0.75 1.090.37

4.84***0.00

0.370.83

3.83***0.01

0.190.95

0.850.40

Notes P-values in parentheses under the test statistic. Asterisks ***, **, and * denote signicance at 1%,5%, and 10%, respectively.

where is the check function (dened as z z for z C 0) and z 1z if

z @ 0. To explore whether the economic indicator aects the accommodative (contrac-

tionary) monetary policy, we test the hypothesis that the coecients on At j should be

jointly signicantly dierent from zero given 0.25 ( 0.75).

ae estimation results are reported in Table (6). Results aremixed. ae housing mar-

ket andunemployment aect themonetarypolicy in theUS andCanada. ae commodity

price aects 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 ination. ae 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 themon-

etary policy in the US. In summary, the economic indicators that contain information

about consumer expectations and condence help explain the Taylor rule deviations in

29

the US. Factors that may contribute to price volatility and inuence 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. ae unrestricted

model involves all economic indicators and the nal model is selected by Akaikes infor-

mation criterion. ae estimated equations are as follows:

US

t 0.0080.055

0.8980.037

t1 0.0760.039

STOCKt3 0.1220.074

REERt 0.0550.026

CONFt2 , R2 0.82.

CA

t 0.1420.188

0.8680.037

t1 0.2050.136

PPIt1 0.2740.141

PPIt3 0.0560.027

STOCKt

1.3620.822

UNEMt1 0.5040.314

CONFt 0.4140.355

CONFt1 , R2 0.72.

GB

t 0.5700.193

0.9070.022

t1 0.9390.177

PPIt 0.4550.237

PPIt2 0.4160.201

PPIt3

0.0810.049

REERt 0.0980.056

REERt3 1.2740.750

UNEMt2 1.3481.059

UNEMt3

0.4570.331

CONFt1 0.3800.322

CONFt3 , R2 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 awide variety of exogenous disturbanceswith dierent degrees of persistence in

either linear or nonlinear format. ae deviation does not follow a stochastic process but

is rather decided by themonetary authority. However, if the deviation is a reaction func-

tion of exogenous disturbances, itwill approximate the dynamics of a stochastic process.

30

Furthermore, suppose themodel selection criteria are changed to a Bayesian information

criterion, then themodel that only contains lagged deviation is dominant in the US and

Canada. ais 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 AR1

process.

5 Conclusion

ae 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 suer from identication problems as described byCanova 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 themodels 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 nd quite robust evidence that the pure forward-

looking version of theNKmodel has performed outstandingly well under empirical test-

ing. Nevertheless, there is room for improvement. For instance, parameter estimation is

based on themodel 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-

ed shocks are slightly serially correlated, suggesting that there is scope for improvement

in the assumptions of the error term structure.

ae study investigates an alternativeway 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 identied

deviation and real economic activity is stronger. Further research may focus on the out-

of-sample exchange rate and interest rate predictability of the identied 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 themonetary authority as following

a simple instrument rulemechanically. 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. ae 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 ndings may provide useful

recommendations for further research on the specications of interest rate policy.

32See http://www.brookings.edu/blogs/ben-bernanke/posts/2015/04/28-taylor-rule-monetary-policyfor the detailed discussion.

32

http://www.brookings.edu/blogs/ben-bernanke/posts/2015/04/28-taylor-rule-monetary-policy

Appendix 1

Using (5) to eliminate interest rate in (1) and (2) gives the equations:

xt Etxt1 1t xxt t Ett1 ret (A1)

t xt Ett1 ut (A2)

We rst guess the form of solution is

xt cxr ret cxu ut cx t (A3)

t cr ret cu ut c t (A4)

By AR1 structure, the conditional expectation for t 1 evaluated at time t is

Etxt1 Etcxr rt1 cxu ut1 cx t1 cxrrret cxuuut cxt (A5)

Ett1 Etcr rt1 cu ut1 c t1 crrret cuuut ct (A6)

Using (A5) and (A6) to substitute the shocks in (A1) and (A2), we then have

r 1 xcxr r cr 1ret u 1 xcxu u cuut

1 xcx c 1t 0 (A7)

cxr r 1crret cxu u 1 1ut cx 1ct 0 (A8)

We assume all shocks have zero mean; then, expectation on both sides in (A7) and (A8)

gives

r 1 xcxr r cr 1 0

u 1 xcxu u cu 0

1 xcx c 1 0

cxr r 1cr 0

cxu u 1cu 1 0

cx 1c 0

33

Given the realization of parameters in , , , , x , r , u , , then solving the six

equations and six unknowns cxr , cxu , cx , cr , cu , c 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 S&P 500 S&P/TSX FTSE 100

Source IFS Yahoo Finance Yahoo Finance

Real Eective Exchange Rate

Source IFS IFS IFS

Unemployment Rate

Source FRED FRED FRED

Code UNRATE LRUNTTTTCAQ156S LMUNRRTTGBQ156S

Consumer Condent Index

Source FRED Datastream Datastream

Code UMCSENT CNOCS005Q UKOCS005Q

34

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37

1985 1990 1995 2000 2005 2010 2015

64

20

2Output Gap

1985 1990 1995 2000 2005 2010 2015

12

34

5

Inflation Rate

1985 1990 1995 2000 2005 2010 2015

02

46

810

Interest Rate

Figure 1: Output gap, ination, and interest rates (shaded areas indicate the period fol-lowing the peak through the trough)

1985 1990 1995 2000 2005 2010 2015

42

02

4 Demand Shock

1985 1990 1995 2000 2005 2010 2015

0.4

0.2

0.00.2

0.4

Supply Shock

1985 1990 1995 2000 2005 2010 2015

20

24

6

Taylor Rule Deviation

Figure 2: Identied shocks (shaded areas indicate the period following the peak throughthe trough)

38

Time

1985 1990 1995 2000 2005 2010 2015

62

2

Time

1985 1990 1995 2000 2005 2010 2015

62

2

Time

1985 1990 1995 2000 2005 2010 2015

62

2

Output Gap

Time

1985 1990 1995 2000 2005 2010 2015

02

4

Time

1985 1990 1995 2000 2005 2010 2015

02

4

Time

1985 1990 1995 2000 2005 2010 2015

02

4

Inflation Rate

1985 1990 1995 2000 2005 2010 2015

04

8

1985 1990 1995 2000 2005 2010 2015

04

8

1985 1990 1995 2000 2005 2010 2015

04

8

Interest Rate

Figure 3: Rational expectations and forecast errors (the dashed-line denotes expectedvalues and the dotted-line denotes forecast errors)

Time

2005 2010 2015

20

24

Time

2005 2010 2015

20

24

GreenbookNK model

Real GDP

Time

2008 2010 2012 2014

0.5

0.5

1.0

Time

2008 2010 2012 2014

0.5

0.5

1.0 Greenbook

NK model

Core CPI

2005 2010 2015

0.5

0.5

2005 2010 2015

0.5

0.5

GreenbookNK model

Interest rates

Figure 4: Real-time forecast errors comparison

39

Time

1985 1990 1995 2000 2005 2010 2015

64

20

24 CBO

QT

HP

Time

1985 1990 1995 2000 2005 2010 2015

12

34

5 CPIPCE

1985 1990 1995 2000 2005 2010 2015

02

46

810 Fed

Tbill

Figure 5: Data comparison

1985 1990 1995 2000 2005 2010 2015

32

10

12

3

Output Gap

1985 1990 1995 2000 2005 2010 2015

01

23

45

6

Inflation Rate

1985 1990 1995 2000 2005 2010 2015

02

46

810

1214 Interest Rate

Figure 6: Output gap, ination, and interest rates in Canada (shaded areas indicate theOECD-based recession)

40

1985 1990 1995 2000 2005 2010 2015

32

10

12

3Output Gap

1985 1990 1995 2000 2005 2010 2015

24

68

Inflation Rate

1985 1990 1995 2000 2005 2010 2015

02

46

810

1214

Interest Rate

Figure 7: Output gap, ination, and interest rates in the UK (shaded areas indicate theOECD-based recession)

1985 1990 1995 2000 2005 2010 2015

50

5

Demand Shock

1985 1990 1995 2000 2005 2010 2015

0.5

0.00.5

1.0

Supply Shock

1985 1990 1995 2000 2005 2010 2015

50

5

Taylor Rule Deviation

Figure 8: Identied shocks forCanada (shaded areas indicate theOECD-based recession)

41

1985 1990 1995 2000 2005 2010 2015

50

5

Demand Shock

1985 1990 1995 2000 2005 2010 2015

0.2

0.00.2

0.4

Supply Shock

1985 1990 1995 2000 2005 2010 2015

105

05

Taylor Rule Deviation

Figure 9: Identied shocks for the UK (OECD-based recession)

Time

1985 1995 2005 2015

4.6

5.0

5.4

5.8 House Price

Timets(U

Sppi

, sta

rt =

c(19

83, 3

), fre

q =

4)

1985 1995 2005 2015

4.6

5.0

Commodity Price

Time

1985 1995 2005 2015

4.5

5.5

6.5

Stock Price

Timets(U

Sree

r, st

art =

c(1

983,

3),

freq

= 4)

1985 1995 2005 2015

4.3

4.5

4.7

REER

1985 1995 2005 2015

46

810

Unemployment Rate

ts(U

Scfi,

sta

rt =

c(19

83, 3

), fre

q =

4)

1985 1995 2005 2015

4.2

4.4

4.6

4.8 Consumer Confidence Index

Figure 10: Economic indicators in the US

42

Time

1985 1995 2005 2015

4.6

5.0

5.4

House Price

Timets(C

Appi

, sta

rt =

c(19

83, 3

), fre

q =

4)

1985 1995 2005 2015

4.6

4.8

5.0

5.2 Commodity Price

Time

1985 1995 2005 2015

4.5

4.7

4.9

5.1

Stock Price

Timets(C

Aree

r, st

art =

c(1

983,

3),

freq

= 4)

1985 1995 2005 2015

4.2

4.4

4.6 REER

1985 1995 2005 2015

68

1012 Unemployment Rate

ts(C

Acfi,

sta

rt =

c(19

83, 3

), fre

q =

4)

1985 1995 2005 2015

4.56

4.59

Consumer Confidence Index

Figure 11: Economic indicators in Canada

Time

1985 1995 2005 2015

5.0

6.0

House Price

Timets(G

Bppi

, sta

rt =

c(19

83, 3

), fre

q =

4)

1985 1995 2005 2015

4.6

5.0

5.4 Commodity Price

Time

1985 1990 1995 2000 2005 2010 2015

5.0

5.5

6.0

6.5 Stock Price

Timets(G

Bree

r, st

art =

c(1

983,

3),

freq

= 4)

1985 1995 2005 2015

4.45

4.60

4.75

REER

1985 1995 2005 2015

57

911

Unemployment Rate

ts(G

Bcfi,

sta

rt =

c(19

83, 3

), fre

q =

4)

1985 1995 2005 2015

4.56

4.58

4.60

4.62 Consumer Confidence Index

Figure 12: Economic indicators in the UK

43