HIAS-E-64

Radial Basis Functions Neural Networks for Nonlinear Time
Series Analysis and Time-Varying Eﬀects of Supply Shocks
Nobuyuki Kanazawa
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

March 2018

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Radial Basis Functions Neural Networks for
Nonlinear Time Series Analysis
and Time-Varying E↵ects of Supply Shocks
Nobuyuki Kanazawa⇤
March 8, 2018
Abstract
I propose a flexible nonlinear method for studying the time series properties ofmacroeconomic variables. In particular, I focus on a class of Artificial Neural
Networks (ANN) called the Radial Basis Functions (RBF). To assess the validity of the RBF approach in the macroeconomic time series analysis, I conduct a
Monte Carlo experiment using the data generated from a nonlinear New Keynesian (NK) model. I find that the RBF estimator can uncover the structureof the nonlinear NK model from the simulated data whose length is as small
as 300 periods. Finally, I apply the RBF estimator to the quarterly US dataand show that the response of the macroeconomic variables to a positive supply
shock exhibits a substantial time variation. In particular, the positive supply
shocks are found to have significantly weaker expansionary e↵ects during thezero lower bound periods as well as periods between 2003 and 2004. The finding is consistent with a basic NK model, which predicts that the higher realinterest rate due to the monetary policy inaction weakens the e↵ects of supply
shocks.

Keywords: Neural Networks; Radial Basis Functions; Zero Lower Bound; Supply
Shocks.
JEL Classification: C45; E31.
⇤
Hitotsubashi Institute for Advanced Study (HIAS), Hitotsubashi University (email:n.kanazawa@r.hit-u.ac.jp): I would especially like to thank Karel Mertens, Levon Barseghyan, andChristopher Huckfeldt for their great guidance and support. All remaining errors are mine.

11Introduction

In this paper, I propose an Artificial Neural Network (ANN)-based nonlinear VAR
estimator with a particular focus on the class of network called the Radial Basis Functions (RBF). The use of RBF is motivated by its ability to flexibly estimate a complex
function. Despite its flexibility, however, the RBF can be efficiently estimated as if
it is a linear estimator by following a simple two-step estimation procedure that iscommonly adopted in the neural network community.
The RBF has been studied extensively in the field of computer science and neural networks, and it has been shown that the RBF can approximate any continuousfunctions on a compact domain (known as the Universal Approximation Property).1
Unlike typical regime-switching VARs, the network-based model can flexibly estimate
a potentially state-dependent and nonlinear data-generating process as a whole, making it possible to produce nonlinear impulse responses without specifying potentiallylarge and unknown number of state. This feature is especially useful when econometricians do not have a clear understanding of the underlying data-generating process.
The strengths of the RBF time series are similar to the models that admit time variation in multivariate linear structures or the Time-Varying Coefficient VARs (TVCVAR).2 Both RBF and TVC-VAR estimators are capable of capturing very general
nonlinear dynamics without specifying the state of an economy. An additional benefit of the RBF estimator, however, is its low computational costs. As pointed out
by Kilian and Lütkepohl (2017), the Bayesian estimation of the TVC-VAR is only
possible, in practice, for small dimensional models with a small number of lags due
to its high computational costs. The curse of dimensionality is greatly reduced for
the RBF estimator because its estimation is conducted by the two-step procedurethat essentially breaks down a costly nonlinear estimation into (1) a fast clustering
method and (2) a linear estimation.3 The benefit of low computational costs allows
researchers to estimate nonlinear models that can produce time-varying impulse response functions with a number of dimension as large as what is typically used in a
linear VAR estimation.
In order to validate the use of the RBF estimator in the macroeconomic analysis,1

See Hartman et al. (1990) and Park and Sandberg (1991).
Examples include Primiceri (2005), Canova (1993), Sims (1993), Stock and Watson (1996) andCogley and Sargent (2001)
3
This procedure is called the ‘unsupervised learning’ in the neural nets field.
22one needs to test if the estimator can correctly capture the structure of aggregate economy with the number of observations that is usually available to a macroeconomist.
This is because, despite the universal approximation property of the RBF, the rate ofconvergence generally depends on the smoothness of the function to be approximatedand the number of available observations. To this end, I conduct a Monte-Carlo experiment using the simulated time series data generated from a medium-scale, nonlinear
New Keynesian (NK) model, which is frequently viewed as a representation of aggregate economy among economists and policy makers. Nonlinearity in this NK model
comes from a kink in the central bank’s monetary policy rule where the interest
rates is bounded below at 0%. I then calculate the within-sample Mean-SquaredErrors (MSEs) as defined by the distance between the true impulse responses fromthe model and the impulse responses estimated by the RBF estimator. For comparison, I also conduct the same exercise using the linear VARs. I found that the
RBF estimator outperforms the linear VARs even when the sample size is limited toas small as 300 periods. The result suggests that the RBF time series can correctly
capture the nonlinear structure of the medium-scale NK model and that the use of
the RBF estimator for macroeconomic time series analysis may be appropriate.
Finally, I apply the RBF estimator to the US quarterly data from 1978 to 2016.
Using Fernald (2014)’s utilization-adjusted TFP shocks, I estimate impulse responsesof the output, consumption, investment, hours worked, expected inflation rate, andnominal interest rate. The aim of this exercise is to test the prediction of the textbook
NK model, which states that the expansionary e↵ects of a positive supply shock is
weaker under the passive monetary policy regimes such as the zero lower bound (zlb)
periods. Consistent with the NK model’s prediction, I found that the e↵ects of thepositive TFP shock is significantly weaker during the periods of zlb.
Furthermore, the output responses to the supply shocks are found to be similarly
small between 2003 and 2004. During these periods, the federal funds rate was very
low at around 1%. In fact, the estimated nominal interest rate response suggests that
the Fed was not responding actively to the supply shocks during these periods. As
a consequence of the unresponsive nominal interest rate and the deflationary e↵ects,
the supply shocks are estimated to raise the real interest rate during these periods,
which discourages consumption, investment, and output. The finding highlights thecritical role that the response of real interest rate plays in shaping the responses ofother macroeconomic variables. The study also suggests that the estimated responses
of macroeconomic variables around 2003-2004 is remarkably similar to the responses
3during the zlb periods around 2008-2015.
These results are in contrast with the existing studies (Garı́n et al. (2016) and
Wieland (2016)) that found the strong expansionary e↵ects of a positive supply shock
during the periods of zlb. I briefly discuss potential explanations for the di↵erencein the findings between the current paper and the existing studies and show that
it is likely to be a consequence of di↵erent data specifications rather than the different methodologies. Specifically, I estimated the output response using the statedependent local projection method and show that the e↵ects of the supply shocks
are significantly weaker during the zlb periods once the growth rate specifications isused and an outlier (2008Q4) is excluded. The exercise reiterates the point that themacroeconomic dynamics during the passive monetary policy regimes are markedlydi↵erent from the usual macroeconomic dynamics under the active monetary policy
regimes as prescribed by the basic NK model.
The rest of the paper is organized as follows: Section 2 reviews literature on
the Radial Basis Functions. Section 3 goes over literature on time series analysis.
Section 4 describes the Monte Carlo simulation exercise, Section 5 illustrates empiricalapplication to the U.S. data, Section 6 briefly discusses the results, and Section 7
concludes.

2

Radial Basis Functions

In this paper, I focus on a particular class of the ANN called the Radial Basis Functions. The ANN typically consists of several layers between input and output variables. Each layer is made of multiple units, and a unit in one layer receives inputs
from the preceding layers and produces outputs for the succeeding layer. The RBF
consists of three layers with a single hidden layer between the input and output layers.4 Figure 1 shows the graphical description of the RBF. The three layers depictedin the figure are: (1) an input layer, where the independent variables enter the system; (2) a hidden layer, where the independent variables are transformed; and (3) alinear output layer, where the dependent variables are predicted.
The RBF was first introduced as a solution technique for interpolation problems,and in the late 1980s, the RBF formulation was extended to perform more general4

The word “hidden” is used to distinguish the layer in the middle from the input and output
layers4

Figure 1: Graphical description of RBF
task of approximation.5 Since then, a number of researchers including Park andSandberg (1991), Xu et al. (1994), and Girosi and Poggio (1990) showed that any
continuous function on a compact domain can be approximated arbitrarily well by
the RBF, making the RBF one of the popular choices among various ANNs.
Consider some unknown function y = f (x), which is to be approximated given adataset (xt , yt ). f (·) is a nonlinear function or the conditional expectation of yt given
xt , hence:yt = f (xt ) + ✏t ,E[✏t |xt ] = 0(1)

In the case of the RBF, f (xt ) takes the following functional form:
5

See Broomhead and Lowe (1988), Moody and Darken (1989), and Poggio and Girosi (1990).

5

f (xt ) =MXK (⇠j , xt ) j

(2)

j=1

=

M
X

D

j=1

✓

||xt

⇠j ||

◆

j

where each basis element is indexed by a location parameter or a centroid, ⇠j , anda scale parameter . M is the number of centroids in the model, and has thesame number of dimension as the input, x. I choose the gaussian density function
for the kernel, D, since it is the most popular choice of the base function for the
RBF. Roughly speaking, when a new input enters the system, the RBF computes
the Euclidean distance between the input and each of the centroids, ⇠j , and assigns
densities using the gaussian density function. The RBF then computes the predictedvalue, ybt , based on linear combinations of the densities and the weights, which is given
by j . I estimate the three sets of parameters, ⇠j , , and j by solving the following
minimization problem:min{ ,⇠j , j }M
1

T
X

yt

0

t=1

M
X

j exp

j=1

⇢

(xt

⇠j )0 (xt
2

⇠j )

!2

(3)

The weights between the input layer and the hidden layer in Figure 1 is represented
as 0 s, the units in the hidden layer are represented by ⇠j , and the linear weights
between the hidden layer and the output layer is represented as 0 s in the equation
3. The problem looks complex and highly nonlinear at first. However, I show in
Section 2.1 that this minimization problem can be solved quickly as if it is a linear
optimization problem.
Extending the RBF formulation to a lag-dependent model is straightforward byaugmenting the RBF model with lags, denoted by p. Thus, the lag-dependent RBF
model is as follows:
M
⇣
⌘
X
yt =
K ⇠j , {yt p }Pp=1 j + ✏t(4)
j=1

where ✏t is a vector of error terms, and the independent variables are now the lagged
dependent variables.

6

2.1

Parameter Estimation Methods and Renormalization

As specified in Section 2, there are three types of parameters to estimate: centroids⇠j , scale parameters , and weights j . Rate of convergence and accuracy of solution
generally depend on the method of estimation. Since this paper is not intended
for finding the optimal method of estimation, I simply choose one of the popular
estimation methods that breaks down the complex nonlinear minimization problem
into two simple steps.6 In other words, the parameters in the hidden layer and theweights ( ) are estimated separately. Some argue that the global solution method7
is ideal in order to ensure that the equation 2 is globally minimized. However, sucha method exponentially increases the computational costs, and, consequently, theglobal solution method is rarely used in practice. The two-step estimation procedure
is as follows. First, I employ a clustering method to fix the value of centroids, ⇠j .
A common choice is the K-means clustering method, in which n observations are
assigned into k clusters with each observation belonging to the cluster with nearest
mean. The “mean” of each cluster constitutes a prototype of that cluster, and for thisreason, I set the centroids to be the “means” of the clusters. Figure 2 illustrates the Kmeans clustering method. In this illustration, 1000 observations are partitioned into
four clusters, with yellow stars representing the “means” of each cluster. Essentially,
the RBF estimator constructs kernels (the gaussian density functions) centering thesecentriods, and, therefore, the centroids need to be a good representation of the entire
sample so that the RBF kernels have a comprehensive coverage over the sample.
In general, centroids determined by the K-means clustering are not unique, andthe location of centriods depends on the initial values. For this reason, I conduct a
primitive, random search optimization by simply repeating the K-means clustering
algorithm with 1,000 di↵erent initial values and select centroids that minimize the error. Other centroid-selection methods were also considered including the Orthogonal
Least Square Algorithm by Chen et al. (1991). However, the crude optimization of
the random search over di↵erent initial values repeatedly produced lower error andwas more stable.
In addition to determining the centroids, the scale parameters, , need to be
estimated as well. I fix to be the standard deviation of x so that the kernel of
6This method of training is also called the “unsupervised training,”
Or, in the language of neural networks literature, “supervised training,” in which the backpropagation is activated7

7

Figure 2: Illustration of K-means clustering(a) Observations and “Means”(b) Clusters

the RBF in equation (2) is akin to the standard normal density function. Fixing
is convienient for computational efficiency, but it has a side-e↵ect of leaving holeregions in the input space where no kernel has appreciable support. For illustration,
the hole-regions as well as the support from the RBF kernels are depicted in the top
panel of Figure 3. With kernel’s width being fixed, there is a region where no kernel
support can be reached. To avoid this problem, Friedman et al. (2008) suggested touse the normalized RBF, which is specified as follows:

y=

M
X

hj (x) j + ✏

(5)

j=1

where

D(||x ⇠j ||)hj (x) = PM
⇠k ||)
k=1 D(||x(6)

The bottom panel of Figure 3 illustrates the kernel support using the normalized
RBF, which is now extended to cover the entire region. I use the normalized RBF in
the remainder of the paper.
A convenient property of the RBF is that once ⇠j and are fixed, the function
is linear in ’s. Thus, j can be estimated in the usual OLS manner. I choose the8Figure 3: Top panel: Radial Basis Functions. Bottom panel: Nomalized RBF(Reprinted from Friedman et al. (2008)).

targeted number of centroids, M , by the Bayesian Information Criteria (BIC).83

Literature Review

This paper is related to the literature on nonlinear time series analysis. The literature
can be generally classified into three.
The first approach to capture nonlinearities relies on the use of regime-switching
VAR models including the threshold VARs (e.g., Hubrich and Teräsvirta (2013)) andthe Markov-switching VARs (Hamilton (1989)). One caveat of this approach is that,while regime-switching VARs can capture state dependence of macroeconomic variables, it assumes that the economy can be described by a finite number of regimes,
within which the structures of the economy are assumed to be the same. This approach becomes computationally challenging as the number of regimes increases or if
the true data generating process features asymmetric impulse responses within eachstate. In such a case, a new set of VAR coefficients need to be estimated for eachnew regime. In contrast, the RBF estimator can efficiently estimate a possibly state8
I also compute the Akaike Information Criteria, but BIC gives a clearer indication of the optimal
number of centroids.

9

dependent and asymmetric structure of the economy as a whole without actuallyspecifying the state or the structure of the economy.
The second approach consists of directly estimating the impulse response functions. This strand of approach was pioneered by Jordà (2005) who proposed the
Local Projection method. While Jorda’s method can easily capture nonlinearities in
the response functions, his method poses serious difficulty when its efficiency is considered as pointed out by Barnichon and Matthes (2016). Indeed, drawing inferences
on a rich set of nonlinearities including sign- and state-dependence using the local
projection method is often difficult.
Relatedly, new approach is recently proposed by Barnichon and Matthes (2016)
who uses few numbers of gaussian functions to approximate the impulse functions.
Like Jordà (2005), this approach is robust to a functional-form assumtion error but has
a great advantage of reduced efficiency costs because, instead of model-free estimationof Jordà (2005), Barnichon and Matthes (2016) imposes a flexible parametric assumption based on the mixture of gaussian base functions. The current paper is similar
to Barnichon and Matthes (2016)’s in that both papers utilize the gaussian functions
to approximate macroeconomic dynamics. However, our papers are di↵erent in thesense that the current paper employs the neural network structure that makes use of
the hidden layer. This multiple-layer structure of the RBF provides extra flexibility,
which makes it possible to estimate a nonlinear VAR system without specifying thestate of an economy. In contrast, estimating asymmetric or state-dependent impulse
responses using Barnichon and Matthes (2016)’s method still requires researchers tospecify the states of the economy often in the form of indicator variables. Thus, theRBF method can be an useful alternative when there is little information about thestates or the structure of the true data-generating process. At the same time, one
caveat for using the RBF estimator is that researchers need to assume the existence ofa VAR representation in the form of the equation 4. This assumption is not neccesaryfor Barnichon and Matthes (2016) and Jordà (2005) because they directly estimate
the impulse response functions.
The third approach is the TVC-VAR models (e.g. Primiceri (2005)). As suggested
by Kilian and Lütkepohl (2017), even though the TVC-VAR can capture very general
nonlinear dynamics without specifying the states of an economy, its expensive computational cost often prohibits researchers from estimating large dimensional models
and, thus, limiting the use of TVC-VAR in practice. Unlike the TVC-VAR, the RBF

10estimator can be estimated with a relatively low computational cost. Therefore, estimation of a larger dimensional model with a larger number of autoregressive lags canbe relatively easily handled by the RBF.9
In addition to the nonlinear time series literature listed above, the current paper
is obviously related to the literature that employs the ANN in economic and financial
time series analysis. The majority of the research that applied the ANN in economics
so far has been focused on forecasting the financial variables. In those studies, theANN models are often found to outperform the traditional time series models.10 . Few
studies tested the ANN’s ability to forecast macroeconomic variables. Swanson andWhite (1997) applied ANN models to forecast nine US macroeconomic series andconcluded that even though their results were mixed, the ANN models were promising even when no explicit nonlinearity was found in the macroeconomic dynamics.
Moshiri and Cameron (1999) investigated if the ANN can correctly forecast inflation
and Tkacz (2001) used the ANN for forecasting the Canadian GDP. An extensive
review of the use of the ANN in the context of economic analysis was written by
Kuan and White (1994).

4Monte Carlo simulation

To study the validity of the RBF estimator in the macroeconomic time series analysis, I conduct the Monte Carlo simulation in this section. I ask: “if the true datagenerating process of an economy was a medium-sized nonlinear New Keynesian (NK)model, can the RBF networks uncover the structure of the NK model from the simulated dataset?” To answer this question, I first solve a nonlinear NK model globally,and simulate data using this NK model. The nonlinearity in this NK model stems
from the kink in the central bank’s Taylor rule, in which the central bank cannot
lower the interest rate below zero. Except for the habit persistence in the household
sector, the NK model is standard. In Appendix 8.1, I describe the NK model used
for simulation in detail. After I solved the model, I simulated time series data for 1
million periods. The nominal interest rate hits zero for about 1.85% of the simulated
periods.
9

In comparing with the TVC-VAR, one limitation of the RBF estimator is that it does not admit
time-varying variance covariance matrix.
10
See Hutchinson et al. (1994), Lachtermacher and Fuller (1995), Zhang and Wan (2007), Zhang
and Hu (1998), and Guresen et al. (2011)

11

4.1

Mean Squared Error Comparison

Using the time series data generated by the NK model, I calculate the within-sample
MSEs of the impulse responses estimated by the RBF estimator and linear VARs.
The MSEs are measured by the deviation of the estimated impulse responses from thetrue impulse responses. I present the MSEs under two di↵erent scenarios. In the firstscenario, the economy is in the “normal” state, in which the nominal interest rates
is strictly positive. I make sure that the economy is far away from the zlb state byensuring that the nominal interest rates is greater than 1% for at least five consecutive
periods prior to and after the shock hits the economy. The second scenario considers
the economy under the “zlb” state, in which the nominal interest rate is binding at
zero. To ensure that the economy is distinctively di↵erent from the normal state,
I define the zlb state as the periods where the nominal interest rate is zero for atleast three consecutive periods. When calculating the MSEs, I produce the impulse
responses by giving the shock to the second of the three consecutive zlb periods.
The exogenous state variable for the NK economy is the productivity shock, at ,and the endogenous state variables are (1) price dispersion, (2) consumption, and(3) interest rates. Since econometricians typically do not have a good measure ofprice dispersion, I instead assume that I observe the inflation rate instead of the pricedispersion.
Let Xt = [⇡t+1 , Ct , Rt ]. Our RBF estimator can be written as follows:⇣⌘
Xt = RBF at , {at p , Xt p }Pp=1 + ✏t

(7)

Since at is orthogonal to the other variables, I impose the assumption that the
productivity shock can a↵ect the macroeconomic variables contemporaneously, but
not vice versa. I impose the same assumption when I estimate the VAR model byordering at at the very first.11
One complication arises as I estimate nonlinear impulse response functions, which
generally depend on future shocks and current state of the economy (history) as wellas the sign and magnitude of the current shock. To accomodate the impulse responses
to various future shocks, I use the generalized impulse response functions following
11
I also added the measurement error to the simulated inflation. The error follows the normal
distribution with standard deviation equal to the 3% of the standard deviation of ⇡.

12

Koop et al. (1996).12 Let GI denote a generalized impulse response function. Then,the impulse response of a variable, Yt+n , in n-period ahead is:GI(n, vt , !t 1 ) = E[Yt+n |vt , !t 1 ]E[Yt+n |!t 1 ] for n = 0, 1, . . .

(8)where vt is current shock and !t 1 is the history.
Examples of the generalized impulse responses are illustrated in Figure 5. For
this illustration, I selected a sample of 300 consecutive periods from the simulated
data and estimated the VAR(4) and RBF(1). To draw impulse responses, I pick two
random histories, !t 1 , from the sample and ensure that one of the selected histories
satisfies the criterion for the normal state and the other satisfies the criterion for thezlb state. I then calculate the impulse responses with the shock occurring at these
two histories.
In Figure 5, the blue solid line shows the true impulse response under the normal
state, and the red solid line shows the true impulse response under the zlb state.
Together with the true impulse responses, the left panels show the estimated responses
from the RBF(1) estimator under the normal state in the blue dotted line and under
the zlb state in the red dotted line. Shaded bands represent the 2.5th and 97.5th
percentiles.13 The right panels show the estimated impulse responses from the linear
VAR(4) with 95 percent confidence bands in black lines.
The top panels of Figure 5 show the responses of inflation rate following a positive
productivity shock. The positive productivity shock is deflationary. The figure showsthat the initial impact of the shock is slightly larger in magnitude during the normal
state, reducing the inflation rate by 0.7% on impact. The inflation rate under thenormal state then quickly returns to zero after three periods. The decline of theinflation rate is smaller under the zlb. The inflation rate goes down by 0.42% on
12Some earlier work includes Beaudry and Koop (1993), Potter (1995), Pesaran and Shin (1996),
and Potter (2000). In the generalized impulse response framework, the problem of future shockdependence is handled by averaging out the impulse responses with many di↵erent future shocks. I
compute the conditional expectation by the Monte Carlo integration, and the expectation is calculated by averaging out 1,000 di↵erent paths of simulated future shocks. See Koop et al. (1996) for
more detail.
13
The confidence bands are estimated by the residual-based block bootstrap method. WhenI conduct the bootstrap, the centroids and scaling parameters are re-estimated by repeating theK-means cluster method for 50 times with di↵erent initial values and by selecting the centroids
that minimize the errors. My baseline preference is block bootstrap method rather than the wildbootstrap because the residuals exhibit auto-correlation. The results are almost identical when thewild bootstrap is employed.

13

impact, and the response is slightly more persistent under the zlb. The RBF(1)estimator appears to be generally successful at capturing the di↵erent responses of
the inflation rate under the two di↵erent states. In addition, the confidence bands
are reasonably narrow even compared with the linear VAR(4) model. As expected,
the linear VAR(4) only captures the response during the normal state.
The next panels in Figure 5 show the responses of interest rate. When the positive
productivity shock creates a deflationary pressure in the economy, the central banklowers the nominal interest rate during the normal state following the Taylor rule. The
response of the interest rate is smaller during the zlb periods because of the central
bank’s inability to push down the interest rate any further.14 The left panel showsthat the RBF(1) estimator is, again, successful at capturing the di↵erent interest
rates responses under the two di↵erent regimes. Furthermore, the RBF(1) estimatoris able to distinguish the two di↵erent interest rate responses under the zlb state andnormal state in a statistically significant manner.
Finally, the last panels show the responses of consumption. Due to the positive
productivity shock, the consumption goes up in both normal and zlb states. The
consumption responses peak at the third quarter, reaching at 1.2% in the normal
state and 1% in the zlb state. The weak response during the zlb periods stems fromthe higher real interest rate, which is caused by the absence of central bank’s reaction
to the deflationary pressure. As the real interest rate goes up during the zlb periods,
the consumption is discouraged. The RBF(1) estimator seems to perform reasonably
well at capturing the di↵erent consumption dynamics following the positive supply
shock. In addition, the responses under the two regimes can be distinguished in astatistically significant manner.
The examples above demonstrate the potential usefulness of the RBF estimator in
uncovering the nonlinear structure of the NK economy. However, since the generalized
impulse responses are history-dependent, the performance of the RBF estimator could
be history-dependent as well. In order to assess the overall performance and accuracyof the RBF estimator, I present below the within-sample MSEs with various histories.
The MSEs are computed based on 1,000 Monte Carlo simulations. For each Monte
Carlo simulation, I take the following procedures: (1) randomly pick a new sample of
14

The nominal interest rates still appears to go down in the figure, but the responses shown here
are relative to the counterfactual where the shock did not occur. Thus, the negative interest rates
response during the zlb state is indicative that the interest rates would have increased from zero
faster if the shock did not arrive.

14

300 observations, (2) estimate the RBF and linear VARs, (3) select a history withinthat sample, (4) give a shock, and (5) compute the MSEs of the impulse responses.
In Table 1, I present the sum of the MSEs over horizons from 1 to 30 for the normal
state and zlb state, which are classified based on the level of interest rate when theshock hits the economy. In addition, I also present the MSEs for the general casewhere I simply calculate the MSEs from the randomly picked histories, paying noattention to whether or not the history belongs to the normal or zlb state. Top row
in Table 1 shows the MSEs for the normal state, the middle row shows the MSEs forthe zlb state, and the bottom row presents the MSEs for the general case.
Table 1: Average MSE Performance (estimation periods = 300)

Normal state
ZLB state
General case

Inflation
VAR(4) RBF(1)Consumption
VAR(4) RBF(1)Interest rates
VAR(4) RBF(1)18.5690
15.5559
18.427730.053437.074330.8573

6.1412
10.28996.701415.2598
10.4868
15.4047

17.5437
20.9026
18.6588

3.55494.42613.9047

Notes: Summary statistics over 1000 Monte-Carlo replications. MSE is the meansquared error of the estimated impulse response function over horizons 1 to 30.

Surprisingly, the top row of Table 1 indicates that the MSEs are smaller for theRBF(1) estimator for all the variables even under the normal state. This result isstriking given that the NK model can be well approximated by linear functions in
the normal state, and it confirms the prediction from the earlier impulse response
illustrations that the RBF estimator can identify nonlinear- and history-dependent
dynamic responses of the macroeconomic variables from a small sample of simulated
data. Furthermore, the di↵erence in the MSEs between the RBF(1) and VAR(4)becomes even larger for the zlb case, with errors from the RBF(1) as small as twothird of the MSEs of the VAR(4) at maximum. The result in this section indicates that
the RBF estimator can be a useful alternative to the traditional linear VAR model
when the form of underlying data generating process is unknown to econometricians
or when it is possibly nonlinear. The result also suggests that the use of the RBF
estimator for macroeconomic time series analysis may be appropriate.

15

5Application to the US Data

Finally, I apply the RBF estimator to the quarterly US data. Following Wieland(2016) and Garı́n et al. (2016), I ask if a positive supply shock is less expansionary
when the nominal interest rates is at the zlb. The question is motivated by the
textbook NK model that predicts a possibility of contractionary supply shock at
the zlb. This somewhat counter-intuitive prediction is the extreme version of the
responses studied in the previous section. I assume that a positive supply shock to
be an increase in neutral productivity. An increase in productivity decreases inflation
rate and the equilibrium real interest rate. When such a shock hits the economy, thecentral bank would react by lowering nominal interest rates according to the Taylor
rule in the normal state. However, if the economy is at the zlb, the central bank isunable to push down the nominal interest rate any further. Consequently, the realinterest rate will go up and the demand will su↵er. In the extreme case where thezlb is expected to last for a long period, the increase in the real interest rate can bepotentially so large that the negative demand response dominates the positive e↵ectof the productivity shock. When that happens, the GDP decreases.
In contrast to the predictions of the NK model, Wieland (2016) found that the effects of a supply shock remain the same at the zlb by examining the experiences of theGreat East Japan Earthquakes and the oil-supply shocks.15 Garı́n et al. (2016) used
the utilization-adjusted TFP shock estimated by Fernald (2014) and found that a positive supply shock is even more expansionary at the zlb. As their findings potentiallycast doubts on the assumptions that underlie the DSGE models, I also investigate
this important question by employing the RBF estimator with the utilization-adjusted
TFP shock estimated by Fernald (2014). Unlike the previous studies, I found someevidence of weaker expansionary e↵ects during the passive monetary policy regimes.

5.1

Data

The dataset includes the following six variables: the log di↵erence of (1) real GDP,ln Yt , (2) personal consumption, ln Ct , (3) private investment, ln It , as wellas (4) log of hours worked, ln Nt (5) expected inflation rate from the University of
15
Wieland (2016) also suggested that the expansionary e↵ects of a supply shock may be evenlarger at the zlb.

16

Michigan’s Survey of Consumers, ⇡te , and (6) the e↵ective federal funds rate, rt .16 In
addition to these variables, I use the utilization-adjusted TFP process estimated by
Fernald (2014).
Garı́n et al. (2016) illustrated that the utilization-adjusted TFP series by Fernald(2014) can be considered an exogenous shock process. Garı́n et al. (2016) used four
popular measures of exogenous macroeconomic shocks17 and showed that these fourmeasures do not Granger cause the utilization-adjusted TFP shock. They, therefore,
argued that the evidence is suggestive that the utilization-adjusted TFP process betreated as an exogenous shock process. Based on their evidence, I also treat theutilization-adjusted TFP process as an exogenous shock process.
The dataset begins in 1978Q1, when the quarterly data of inflation expectation
became available, and ends in 2016Q4. The US economy was in the “zlb state” fromthe last quarter of 2008 to the last quarter of 2015.

5.2

Generalized Impulse Responses

I estimate the generalized impulse responses to a positive TFP shock. The estimatedmodel is as follows:
Xt = RBF⇣⌘
ln At , { ln At p , Xt p }Pp=1 + ut

where Xt = [ ln Yt , ln Ct , ln It , ln Nt , ⇡te , rt ]. I assume that the utilization-adjusted
TFP shock can a↵ect the other variables within the same period, but not vice versa.
I set the lags to be four.
As in Section 4.1, I define generalized impulse response of a variable, Yt+n , inn-period ahead to be:
GI(n, vt , !t 1 ) = E[Yt+n |vt , !t 1 ]16

E[Yt+n |!t 1 ] for n = 0, 1, . . .

(9)Real GDP (GDPC09), personal consumption (PCEC), private investment (FPI) are from the
NIPA table. Hours are obtained by multiplying civilian employment (CE16OV) and average weekly
hours duration (PRS85006023). The GDP, consuption, investment, and hours are expressed as per
capita by dividing the variables by civilian noninstitutional population (CNP16OV).
17
The four measures are Romer and Romer (2004)’s monetary policy shocks, Romer and Romer(2010)’s tax shocks, Ramey (2011)’s defense news shock, and Kilian (2008)’s exogenous oil priceshock.

17where vt is current shock, !t 1 is the history, and t spans from 1978Q1 to 2016Q4.
I fix vt to be the standard deviation of ln At for all t. Since ln At exhibits astrong fluctuation and because the estimated impulse responses depend on the current
value of ln At , the resulting impulse responses also exhibit significant fluctuations.
To avoid this issue, the impulse responses shown below are averaged out over [t2, . . . , t + 2] for all t whenever possible.

5.3

Results

Figures 7-8 plot the point estimates of the responses after 1 quarter, 3 quarters and 10
quarters between 1980 and 2016. In Figures 10-16, the impulse responses are shownwith confidence bands for the horizons of (1) 1st quarter, (2) 2nd quarter, (3) 5th
quarter, and (4) 9th quarter. In those figures, the point estimates are drawn in blue
solid lines, and the 68 percent confidence bands are expressed as red dotted lines.
In general, the impulse responses from the RBF estimator look reasonably sensible.
The positive supply shocks have positive and persistent e↵ects on output, consumption, and investment. Expected inflation declines immediately after the shock, but
the e↵ect is short-lived. The Fed funds rate also decreases after the shock in mostperiods, which is consistent with the Taylor rule. Hour worked goes down in thefirst few quarters in most periods, but it rises in the longer horizon. Overall, these
responses are consistent with previous studies that investigated the e↵ects of supply
shocks on macroeconomic variables. Yet, the figures also illustrate that there aresubstantial time-variations in the responses of the macroeconomic variables.
Figure 7(a) show the response of the cumulative GDP. 18 As summarized above,
the TFP shock is expansionary in all periods but exhibits a substantial time-variationin terms of the magnitude of its e↵ects. The supply shock’s after-10-quarters-e↵ects
varies from approximately 0% to 0.7% depending on the timing of the shock. The
figure also suggests that the e↵ect of productivity shock is procyclical. The 2008
Financial Crisis had a particularly powerful impact of weakening the e↵ects of the
productivity shock, reducing its after-10-quarters-e↵ects from 0.57% in 2007Q2 to
0.05% in 2009Q4.
Surprisingly, the expansionary e↵ects of the productivity shocks are considerably
18
The result shown here is not GDP di↵erence as I calculated the cumulative sum of the estimatedimpulse responses.

18

weaker during the zlb periods even after the recession from the 2008 Financial Crisis
was over. These weak expansionary e↵ects are characterized by the weak on-impacte↵ects as well as the lack of persistence. Interestingly, the similarly weak output
responses are observed between 2003 and 2004, in which the persistence of the e↵ects
is also absent. The decline of the output responses is unlikely to be explained by theprocyclical e↵ects of the supply shocks since these periods do not correspond to therecessionary periods. Rather, they correspond to the recovery phase from the 2001
recession when the fed funds rate was kept very low at around 1% and also to theperiods right before the rate started to rise successively in 2005 and 2006.19
Figure 7(c) plots the estimated response of the fed funds rate. While the interest
rate is estimated to go down in most periods (consistent with the Taylor rule), it isestimated to either go up or stay the same between 2003 and 2004 and after the 2008
Financial Crisis.20 Furthermore, Figure 7(d) shows the response of real interest rate
calculated from the simple Fisher equation21 , which indicates that the real interest
rate would have been higher in response to the supply shock around 2003-2004 andafter 2008. The results suggest that the response of real interest rate plays a crucialrole in forming the responses of output. The fact that the supply shocks are foundto have weaker expansionary e↵ects not only during the zlb periods but also between
2003 and 2004 makes the prediction of the NK model, in which the monetary policy
impotency reduce the e↵ects of supply shocks, more convincing.
Figure 7(e) shows the response of cumulative consumption. Similar to the output
response, the consumption also tend to increase in response to the positive productivity shock throughout the sample. However, the response is particularly weak during
2003-2004 periods and after the 2008 Financial Crisis. Moreover, the cumulative
investment response shown in Figure 7(g) also exhibits particularly weak responses
during 2003-2004 and after 2008 periods. To reiterate, these findings are in accordance with the textbook NK model and the findings from the output and interest
rates responses above. The decline of inflation expectation, combined with the unresponsive monetary policy, raises the real interest rate and discourages consumption
and investment.
19

see Figure 9.
The responses during these periods are not statistically di↵erent from zero.
21
The real interest rate = nominal interest rate - expected inflation. Negative real interest rate
response means that the fed funds rate decreased for more than one-for-one in response the thedecline in the expected inflation.
20

19

Finally, Figure 7(f) plots the response of the hours worked. The previous literaturefound that the hours worked decreases in the short-run in response to a positive
productivity shock, which is often referred to as the evidence for price-stickiness in themacro models. My finding is consistent with the existing literature: the hours worked
decreases in the short-run, but it rises in the longer horizon. This pattern is stable
for majority of the sample periods, with the exception of the recessionary periods inthe early 1980s, 2001-2005 periods, and after 2008. These periods correspond to theperiods of a large slack in the labor market as well as the periods of relatively smallor positive real interest rate responses. Peculiarly, the positive productivity shocks
are estimated to increase the hours worked both in the short horizon and in the longhorizon after 2008.
In summary, the estimated impulse responses from the RBF estimator suggestthat the response of macroeconomic variables to a positive supply shock is substantially time varying. It also hints at the fact that the e↵ects of productivity shocks
might be quantitatively and qualitatively di↵erent after the recent Financial Crisis
as well as 2003-2004 periods when the monetary policy is possibly less sensitive tothe deflationary pressure caused by the supply shocks. In particular, the e↵ects ofthe supply shocks during the periods of monetary policy inaction are found to beconsiderably less expansionary, which is consistent with the textbook NK model but
is inconsistent with what Wieland (2016) and Garı́n et al. (2016) reported. The findings indicate that the economic responses to the supply shocks around 2003-2004 ismarkedly similar to the responses during the zlb periods and highlight the criticalrole that the response of real interest rate plays in determining the responses of othermacroeconomic variables. Importantly, all of the findings above are based on the RBF
estimator that was estimated without imposing a priori- functional form assumption.

6

Discussion

Why is the estimates in Section 5 starkly di↵erent from the existing literature that
concludes that the supply shocks are even more expansionary under the zlb? Is thedi↵erence an artefact of the untraditional methodology of the current paper? To
answer these questions, I briefly investigate the relationship between the TFP shocks
and the output using the state-dependent local projection method of Jordà (2005).
Using the same data that I used in Section 5, I estimated the following regression:
20

h
X

ln(Yt+j ) =(1Zt ) (↵hn +nhln At +nh (L)xt 1 )

(10)

j=0

+ Zt (↵hz +zhln At +zh (L)xt 1 ) + ut+h

where Zt is the indicator variable for the zlb periods and xt is the control variables.
Following Garı́n et al. (2016), I define the zlb as the periods between 2008Q4 and2015Q4. The control variables are lagged ln Yt and ln At . The number of lags are
four. Since I do not need to use the expected inflation, I now extend the dataset toperiods between 1948Q1 and 2016Q4. Additionally, I excluded the 2008Q4 because it
nontrivially a↵ects the estimated on-impact response of output during the zlb periods,
which is likely due to the large impact of the Lehman failure and not driven by the
movement in the TFP.
Figure 17 shows the estimated output responses during the normal periods in theblue line and during the zlb periods in the red line. The shaded bands represent the
90 percent confidence interval. The output response during the normal periods is
initially small but grows larger over time, culminating at around 0.2% in the 10thquarter. In contrast, the estimated response is weaker during the zlb periods and is
not statistically di↵erent from zero for all horizons. The result is consistent with thefinding from Section 5 that found the weaker output response when the monetary
policy is not reacting actively to the shocks. As a robustness check, I also regressed
the equation 10 with the dependent variable specified as (ln Yt+h ln Yt 1 ). The resultis shown in Figure 18, and the main conclusion stays the same.
In contrast to the previous literature that concluded that the expansionary e↵ectsof the supply shocks are even larger at the zlb, I found that the supply shocks becomeless expansionary during the periods of zlb. The di↵erence is likely to be a productof di↵erent data specifications rather than the methodologies. In particular, while
Garı́n et al. (2016) used aggregate GDP and log-level specification, I used the GDP
per capita and the growth rate specification. In addition, I also excluded the outlier(2008Q4). Once I treated the data as described, the supply shocks are found to have
weaker expansionary e↵ects during the passive monetary regimes even when I use thestate-dependent local projection method.

21

7Conclusion

In this paper, I investigated the applicability of the Radial Basis Functions to themacroeconomic time series analysis. The RBF estimator can be a useful alternative
to traditional estimators because of its low computational costs that allow researchers
to flexibly estimate the time-varying response functions using relatively large dimensional models.
Based on a medium-scale nonlinear New Keynesian model, I performed a series ofMonte Carlo experiments to study the small sample properties of the proposed RBF
estimator. I found that the RBF estimator outperforms the linear VARs in terms of
the within-sample MSEs. The generalized impulse responses from the RBF estimatorsuggests that it can learn the structure of the nonlinear New Keynesian model fromthe sample of simulate data whose length is as small as 300 periods.
Finally, I applied the RBF estimator to the quarterly US data and found that the
responses of macroeconomic variables to a positive supply shock exhibit substantial
time variations. The result also suggests that the expansionary e↵ects of a supply
shock became significantly weaker during the periods of monetary policy inaction,
which is consistent with the prediction of a textbook NK model.
The results shown in this paper highlights the potential benefit and validity of
the RBF estimator in analyzing the macroeconomic time series data. The estimator
is especially useful when econometricians do not have a clear understanding of theunderlying data-generating process.

22

8

Appendix

8.1

NK Model for Monte Carlo simulation

The economy is inhabited by four types of agents: Households, Final good producer,
Intermediate goods producers, and Monetary authority. Except for the habit persistence in the household sector, the following NK economy is standard.
8.1.1

Households with Habit Persistence

There is a continuum of households who consume a composite good, Ct , supply labor,
Nt , and purchase bond Bt . The representative household maximizes the expected
lifetime utility given by:
Et

"1
X
t=1

where

t 1

⇢

Ct 1 )1

(Ct

+

1

(1

controls the degree of habit persistence, andNt ) 1 
1 

#

(11)

is the discount factor.

The household is subject to the following budget constraint each period:
Pt C t +1
B t = W t Nt + B t 1 + P t ⇧t
Rt

(12)

where Pt , Wt , and Rt are commodity good price, nominal wage, and nominal interest
rates, respectively. In addition, ⇧t is the profit from the intermediate-good firms.
We can maximize utility subject to the budget constraint to obtain the optimal
allocation of consumption across time:t =where ⇡t+1 = Pt+1 /Pt andt = (Ct

Et [ t+1 Rt /⇡t+1 ]Ct 1 ).

The first order condition concerning labor supply decision is
Wt = ((1Nt )  / t )23

8.1.2

Final Good Producer

There are perfectly competitive final good producers who use intermediate goods,
Yt (i), for i 2 [0, 1] as inputs and produce final good, Yt , at a price Pt to maximize theprofit given by:Z
1

max Pt Yt
Yt (i)

Pt (i)Yt (i)di

(13)

0

The technology of the final good producer is given by the following CES aggregator:
Yt =

✓Z 1

Yt (i)

(⌘ 1)
⌘

di

0

◆ (⌘ ⌘ 1)(14)

where Yt (i) and Pt (i) are quantity and price of an intermediate good i, respectively.
8.1.3

Intermediate Goods Producers

There is a continuum of monopolistically competitive intermediate goods producers
who uses labor, Nt (i), as an input and solve the following cost minimization problem:
min T C(Yt (i)) = Wt Nt (i)(15)

Nt (i)

where T C is nominal total cost. The production technology of the intermediate goods
producers are the following:
Yt (i) = At Nt (i)(16)
where At is a productivity shock that follows the AR(1) process given by:log(At ) = ⇢a log(At 1 ) + ✏at ,✏at v N (0,

2
a)

(17)

The cost minimization problem of firm i implies
mct =

Wt
Pt At

where mct is the Lagrange multiplier and also the real marginal cost of production.
The intermediate goods producers are subject to Calvo-type price setting friction.
In this environment, only a 1 ⇠ fraction of the firms set prices optimally each period:
Pt (i) = Pt⇤ , and the remaining fraction ⇠ of the firms are not allowed to change the24price, Pt (i) = Pt 1 (i). The profit maximization problem of a re-optimizing firm i, whotakes into account the probability of adjusting its price next period and onwards, isgiven by the following:
Pt⇤

1
X
j=0

⇠ Et {⇤t+j [Pt⇤ Yt+j (i)j j

Pt+j mct+j Yt+j (i)]}(18)

where ⇤t is the household’s marginal utility of wealth at period t. I assume thatthe intermediate good producers are owned by the household, and all the profits aretransferred to the households. The intermediate goods producers solve the optimization problem described above subject to the demand curve for their own goods, which
is given by the following:✓◆ ⌘Pt (i)Yt (i) = Yt
(19)
Pt
8.1.4

Monetary Authority

Lastly, a monetary authority sets nominal interest rate according to the Taylor rule
as follows:23✓◆ R ⇣ ⌘ ✓ ◆ y !1 R⇡
¯ Rt 1⇡ t ⇡ Yt
Rt = max 4
, 15
(20)
⇡
¯/
⇡
¯
Ȳ

where ⇡¯ is inflation target, ⇡t is inflation rate between t 1 and t, and Ȳ is the output
target. This monetary authority is subject to the zero lower bound, meaning thatthe monetary authority cannot set nominal interest rate below zero (or equivalently,
R 1).
8.1.5

Aggregate conditions

The aggregate resource constraint is simply given by
C t = Yt
In the Calvo pricing setting, firms that change prices in di↵erent periods will havedi↵erent prices. Therefore, the economy needs to track price dispersion. When firms
have di↵erent relative prices, there are distortions that create a wedge between the aggregate output measured in terms of production factor inputs and aggregate demand25measured in terms of composite goods. Specifically,

At Nt (i) = Yt (i) =✓Pt (i)
Pt

◆ ⌘

Yt

◆ ⌘

di =

Yt v t
At

✓

Pt⇤
Pt

which implies, in aggregate,
Z 1

Yt
Nt =
Nt (i)di =
At
0

Z 1✓
0

Pt (i)
Pt

where price dispersion, vt , can be described as:
vt ⌘

8.2

Z 1✓
0

Pt (i)Pt◆ ⌘di = ⇠⇡t⌘ vt 1 + (1

⇠)

◆ ⌘

Equilibrium conditions

I summarize below the first order conditions that characterize the equilibrium of oureconomy. Let p⇤t = Pt⇤ /Pt and at = log(At ). Then:t =(Ctt =Ct 1 )(21)

Et [ t+1 Rt /(⇡t+1 )](22)

vt Yt =at Nt(23)
Nt )  / t )/at

mct =((1Ct =Yt2

Rt = max 4p⇤t =((1

⇡
¯

✓

Rt 1
⇡
¯/

⇠⇡ ⌘ 1 )/(1vt =⇠⇡t⌘ vt 1 + (1
St =
Ft =

(24)

◆ R

⇣⇡ ⌘ ⇡t⇡¯⇠))1/(1 ⌘)⇠)p⇤t ⌘⌘
t mct Yt + ⇠Et [⇡t+1 St+1 ]⌘ 1
t Yt + ⇠Et [⇡t+1 Ft+1 ]p⇤t =St /Ft

✓

Yt
Ȳ

◆ y !1

R

3

, 15

(25)
(26)
(27)
(28)
(29)
(30)
(31)

26

8.3

Numerical solution of nonlinear NK model

The zero lower bound in the Taylor rule introduces the nonlinearity in the NK model.
Thus, the solution to the equilibrium condition must be obtained using a global
solution method. I use the projection method. Let S = [vt , Ct 1 , Rt 1 , at ] be the statevariables of the model. There are three future variables, [⇡t+1 (S), Yt+1 (S), Ft+1 (S)],that need to be interpolated. I approximate each of future variables with the Radial
Basis Functions, [⇡t+1 (S̃), Yt+1 (S̃), Ft+1 (S̃)], in such a way that equilibrium conditionsof the model are satisfied at a set of collocation points, S̃. These collocation points,
S̃, are selected based on the Maliar and Maliar (2015)’s Epsilon-Distinguishable Set
algorithm. The solution’s accuracy is presented in the Appendix 8.5. The model iscalibrated to a set of standard parameter values in the NK literature. The calibrated
parameter values are listed in the Appendix 8.4.
The algorithm for the numerical solution of the model is given below. There are
two loops in the algorithm. The outer loop iterates over the grid, and the inner loop
iterates over policy functions.
Step 0a: Solve the log-linear version of the model and simulate data. This initial step isrequired for the clustering methods in Step 1.
Step 0b: Defining the grid and the polynomials of the RBF. Given the simulateddata, construct a grid following Maliar and Maliar (2015) and estimate the
RBF coefficients for the policy functions.
Step 1: Computing integrals. Compute the integrals using the Maliar and Maliar(2015)
Step 2: Equilibrium conditions. For each grid points, use the polynomials obtainedin Step 1 to compute the values of future variables, [⇡t+1 (S̃), Yt+1 (S̃), Ft+1 (S̃)].
Given the future variables, solve for the endogenous state variables next periodusing the model’s equilibrium conditions.
Step 3: Evaluate conditional expectations. Using the integrals computed in Step 1,
evaluate the conditional expectations in equations 23, 29, 30.
Step 4: Evaluate new policy functions. Given the conditional expectations, obtainnew values of future variables in the current period, [⇡t0 (S̃), Yt0 (S̃), Ft0 (S̃)], using
equations 23, 29, 30. Given these new values, compute the new policy functions,27

and compute the di↵erence between the polynomials of newly obtained policy
functions and those of old policy functions. Denote the percentage di↵erence asr.
Step 5: Iteration. If r < 10 8 , go to Step 6. Otherwise, update the guess and repeat
Step 1-5.
Step 6: Compute new grid. Using the solution obtained in the previous steps, simulate new data. Using these simulated data, choose a new grid using Maliar
and Maliar (2015). Compute the di↵erence between the old grid and the newgrid. Specifically, for each newly computed grid point, find the nearest point in
the old grid and compute the Euclidean distance. This forms a vector, D, that
contains the distances between each new grid point to its nearest point in theold grid. Find the maximum of D and call it rg .
Step 7: Iteration for grid. If rg is smaller than the Euclidean distance between thefarthest two points in the old grid, stop the algorithm. Otherwise, go back to
Step 2 with the new grid obtained in Step 6.

28

8.4

Calibration

The parameter values for calibration of the New Keynesian model are summarized asfollows:
Table 2: Calibration
parameter


⇠
⌘
⇡
¯
⇡
y
R

⇢a
a

8.5

value

description

0.5
0.99
1
2.65
0.60
9
1
2.21
0.07
0.82
0.80
0.019

Habit Persistence
Discount factor
Utility Curvature: Consumption
Utility Curvature: LeisureCalvo: (1 ⇠)% adjust prices each period
Price Elasticity of Demand
Inflation Target
Taylor Rule inflation coefficient
Taylor Rule output gap coefficientInterest Rate Smoothing
Prod. shock persistence
Prod. shock std.

Accuracy of numerical solution

I check the accuracy of the numerical solution by computing the errors of the residual
equations. Specifically, I proceed as follows. First, I simulate the model forward for
10,000 periods. This gives a simulation for both the state and control variables ofthe model for 10,000 periods. Second, compute the residuals from the intertemporalequations 23, 29, 30 for the 10,000 periods. I report the decimal log of the absolutevalue of these residual errors.
On average, residual equation errors are on order of -3.53 for equation 23, -4.94
for equation 29, and -2.58 for equation 30. These numbers are comparable to theother studies whose models have similar degrees of complexity.

29References
Regis Barnichon and Christian Matthes. Gaussian mixture approximations of impulse
responses and the non-linear e↵ects of monetary shocks. 2016. 10
Paul Beaudry and Gary Koop. Do recessions permanently change output? Journalof Monetary economics, 31(2):149–163, 1993. 13
David S Broomhead and David Lowe. Radial basis functions, multi-variable functional
interpolation and adaptive networks. Complex System, 2:321–355, 1988. 5
Fabio Canova. Modelling and forecasting exchange rates with a bayesian time-varying
coefficient model. Journal of Economic Dynamics and Control, 17(1-2):233–261,
1993. 2
Sheng Chen, Colin FN Cowan, and Peter M Grant. Orthogonal least squares learning algorithm for radial basis function networks. IEEE Transactions on neuralnetworks, 2(2):302–309, 1991. 7
Timothy Cogley and Thomas J Sargent. Evolving post-world war ii us inflation
dynamics. NBER macroeconomics annual, 16:331–373, 2001. 2
John G Fernald. A quarterly, utilization-adjusted series on total factor productivity.
Federal Reserve Bank of San Francisco, 2014. 3, 16, 17
Jerome Friedman, Trevor Hastie, and Robert Tibshirani. The elements of statisticallearning. Springer series in statistics New York, second edition, 2008. 8, 9
Julio Garı́n, Robert Lester, and Eric Sims. Are supply shocks contractionary at thezlb? evidence from utilization-adjusted tfp data. Technical report, National Bureau
of Economic Research, 2016. 4, 16, 17, 20, 21
Federico Girosi and Tomaso Poggio. Networks and the best approximation property.
Biological cybernetics, 63(3):169–176, 1990. 5
Erkam Guresen, Gulgun Kayakutlu, and Tugrul U Daim. Using artificial neuralnetwork models in stock market index prediction. Expert Systems with Applications,38(8):10389–10397, 2011. 11

30

James D Hamilton. A new approach to the economic analysis of nonstationary timeseries and the business cycle. Econometrica: Journal of the Econometric Society,pages 357–384, 1989. 9
Eric J Hartman, James D Keeler, and Jacek M Kowalski. Layered neural networks
with gaussian hidden units as universal approximations. Neural computation, 2(2):210–215, 1990. 2
Kirstin Hubrich and Timo Teräsvirta. Thresholds and smooth transitions in vector
autoregressive models. In VAR Models in Macroeconomics–New Developments andApplications: Essays in Honor of Christopher A. Sims, pages 273–326. Emerald
Group Publishing Limited, 2013. 9James M Hutchinson, Andrew W Lo, and Tomaso Poggio. A nonparametric approachto pricing and hedging derivative securities via learning networks. The Journal of
Finance, 49(3):851–889, 1994. 11
Òscar Jordà. Estimation and inference of impulse responses by local projections. The
American Economic Review, 95(1):161–182, 2005. 10, 20
Lutz Kilian. Exogenous oil supply shocks: how big are they and how much do theymatter for the us economy? The Review of Economics and Statistics, 90(2):216–
240, 2008. 17
Lutz Kilian and Helmut Lütkepohl. Structural vector autoregressive analysis. Cambridge University Press, 2017. 2, 10
Gary Koop, M Hashem Pesaran, and Simon M Potter. Impulse response analysis in
nonlinear multivariate models. Journal of econometrics, 74(1):119–147, 1996. 13
Chung-Ming Kuan and Halbert White. Artificial neural networks: An econometricperspective. Econometric reviews, 13(1):1–91, 1994. 11
Gerson Lachtermacher and J David Fuller. Back propagation in time-series forecasting. Journal of forecasting, 14(4):381–393, 1995. 11
Lilia Maliar and Serguei Maliar. Merging simulation and projection approaches tosolve high-dimensional problems with an application to a new keynesian model.
Quantitative Economics, 6(1):1–47, 2015. 27, 28

31

John Moody and Christian J Darken. Fast learning in networks of locally-tuned
processing units. Neural computation, 1(2):281–294, 1989. 5
Saeed Moshiri and Norman Cameron. Neural network versus econometric models in
forecasting inflation. 1999. 11
Jooyoung Park and Irwin W Sandberg. Universal approximation using radial-basisfunction networks. Neural computation, 3(2):246–257, 1991. 2, 5
M Hashem Pesaran and Yongcheol Shin. Cointegration and speed of convergence toequilibrium. Journal of econometrics, 71(1):117–143, 1996. 13
Tomaso Poggio and Federico Girosi. Networks for approximation and learning. Proceedings of the IEEE, 78(9):1481–1497, 1990. 5
Simon M Potter. A nonlinear approach to us gnp. Journal of applied econometrics,
10(2):109–125, 1995. 13
Simon M Potter. Nonlinear impulse response functions. Journal of Economic Dynamics and Control, 24(10):1425–1446, 2000. 13
Giorgio E Primiceri. Time varying structural vector autoregressions and monetary
policy. The Review of Economic Studies, 72(3):821–852, 2005. 2, 10
Valerie A Ramey. Identifying government spending shocks: it’s all in the timing. The
Quarterly Journal of Economics, 126(1):1–50, 2011. 17
Christina D Romer and David H Romer. A new measure of monetary shocks: Derivation and implications. American Economic Review, pages 1055–1084, 2004. 17
Christina D Romer and David H Romer. The macroeconomic e↵ects of tax changes:
Estimates based on a new measure of fiscal shocks. American Economic Review,
100:763–801, 2010. 17
Christopher A Sims. A nine-variable probabilistic macroeconomic forecasting model.
In Business Cycles, Indicators and Forecasting, pages 179–212. University of
Chicago Press, 1993. 2
James H Stock and Mark W Watson. Evidence on structural instability in macroeconomic time series relations. Journal of Business & Economic Statistics, 14(1):11–30, 1996. 2
32

Norman R Swanson and Halbert White. A model selection approach to real-time
macroeconomic forecasting using linear models and artificial neural networks. Review of Economics and Statistics, 79(4):540–550, 1997. 11
Greg Tkacz. Neural network forecasting of canadian gdp growth. International Journal of Forecasting, 17(1):57–69, 2001. 11
Johannes Wieland. Are negative supply shocks expansionary at the zero lower bound?
2016. 4, 16, 20
Lei Xu, Adam Krzyżak, and Alan Yuille. On radial basis function nets and kernel
regression: statistical consistency, convergence rates, and receptive field size. Neural
Networks, 7(4):609–628, 1994. 5
Gioqinang Zhang and Michael Y Hu. Neural network forecasting of the british
pound/us dollar exchange rate. Omega, 26(4):495–506, 1998. 11
Yan-Qing Zhang and Xuhui Wan. Statistical fuzzy interval neural networks for currency exchange rate time series prediction. Applied Soft Computing, 7(4):1149–
1156, 2007. 1133

Figure 5: Examples of Generalized Impulse Responses

Inflation: RBF(1)Inflation: VAR(4)Interest Rate: RBF(1)Interest Rate: VAR(4)34Figure 6: Examples of Generalized Impulse Responses

Consumption: RBF(1)Consumption: VAR(4)Notes: Impulse response functions of the inflation rate, interest rate, and consumption (in percent)to a one standard-deviation positive productivity shock. True structural impulse responses from theNK model in normal state (blue solid line) or zlb state (red solid line). Estimation from a RBF(1)in normal state (blue dashed line) or zlb state (red dashed line). Estimated impulse response froma VAR(4) (black dashed line). Shaded bands denote the 2.5th and 97.5th percentiles estimated by
the residual-based block bootstrap method.

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Figure 7: Responses at di↵erent horizons(a) cumulative GDP(b) Expected Inflation(c) Fed funds rate(d) real interest rate

Notes: Impulse response functions (in percent) to a one standard-deviation positive TFP shock
after 1st, 3rd, and 10th quarters. Black shaded areas corresponds to NBER-recession dates. Red
shaded areas represent the periods of weak output responses outside the recession dates.

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Figure 8: Responses at di↵erent horizons(e) cumulative Consumption(f) Hours worked(g) cumulative Investment
Notes: Impulse response functions (in percent) to a one standard-deviation positive TFP shock
after 1st, 3rd, and 10th quarters. Black shaded areas corresponds to NBER-recession dates. Red
shaded areas represent the periods of weak output responses outside the recession dates.

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Figure 9: Federal funds rate

Notes: Federal funds rate between 1980 and 2016. Red shaded areas represent the periods of weakoutput responses outside the recession dates.

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Figure 10: Responses of cumulative GDP(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of the cumulative real GDP (in percent) to a one standarddeviation positive TFP shock. Shaded bands denote the 68% confidence interval estimated by theresidual-based block bootstrap method.

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Figure 11: Responses of Expected Inflation(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of the expected inflation (in percent) to a one standard-deviation
positive TFP shock. Dotted lines show the 68% confidence bands estimated by the residual-based
block bootstrap method.

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Figure 12: Responses of Fed Funds rate(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of the fed funds rate (in percent) to a one standard-deviation
positive TFP shock. Dotted lines show the 68% confidence bands estimated by the residual-based
block bootstrap method.

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Figure 13: Responses of real interest rate(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of real interest rate (in percent) to a one standard-deviation
positive TFP shock. Dotted lines show the 68% confidence bands estimated by the residual-based
block bootstrap method.

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Figure 14: Responses of cumulative Consumption(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of consumption (in percent) to a one standard-deviation positive
TFP shock. Dotted lines show the 68% confidence bands estimated by the residual-based blockbootstrap method.

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Figure 15: Responses of Hours Worked(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of hours worked (in percent) to a one standard-deviation positive
TFP shock. Dotted lines show the 68% confidence bands estimated by the residual-based blockbootstrap method.

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Figure 16: Responses of cumulative Investment(a) Response after 1st quarter(b) Response after 2nd quarter(c) Response after 5th quarter(d) Response after 9th quarter

Notes: Impulse response functions of investment (in percent) to a one standard-deviation positive
TFP shock. Dotted lines show the 68% confidence bands estimated by the residual-based blockbootstrap method.

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Figure 17: Response of cumulative GDP using Local Projection method
Dependent variable:

Ph

j=0

ln Yt+j

Notes: Estimated impulse response of output to a one unit TFP shock at various horizons. The
solid blue line shows the response under the normal periods (i.e. when Zt = 0). The solid red line is
the response when the ZLB binds (Zt = 1). The shaded bands represent the 90 percent confidence
interval.

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Figure 18: Response of GDP using Local Projection method
Dependent variable: ln Yt+hln Yt 1

Notes: Estimated impulse response of output to a one unit TFP shock at various horizons. The
solid blue line shows the response under the normal periods (i.e. when Zt = 0). The solid red line is
the response when the ZLB binds (Zt = 1). The shaded bands represent the 90 percent confidence
interval.

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Figure 19: Residual equation errors(a) Equation 23(b) Equation 29(c) Equation 30
Notes: The histograms report the residual equation errors in decimal log basis. The dotted red lines
mark the mean residual equation errors.

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