Download Estimating Monthly GDP for a Small Open Economy: Structural

Estimating Monthly GDP for a Small Open Economy:
Structural versus Reduced-Form Mix-Frequency Models
Ruey Yau
Department of Economics
National Central University
Taoyuan, Taiwan
February 2015
Abstract
This paper compares monthly GDP estimates for a small open economy from two classes
of mixed frequency models. One class is model-based and the other class is a reduced-form
model. Our small open economy structural model is a New Keynesian dynamic stochastic
general equilibrium (DSGE). DSGE modeling has become a popular framework by central
banks for policy and forecasting in recent years, for such a model delivers strong theoretical
coherence and is appropriate for policy evaluation. In addition, higher frequency observations may arrive for policy makers to include these timely information into their forecast
framework. This paper evaluates how much a structural mixed-frequency model can help in
generating more accurate estimates of monthly GDP and timely forecasting. The empirical
model has a state space representation and is estimated by Kalman filter technique.
JEL Classification: C5, E1
Keywords: DSGE model, mixed frequency, state-space representation, Kalman filter
1
Introduction
Central banks or institutional analysts are often eager to get access to economic status
for policy considerations or timely economic forecasts. Real GDP is considered as one of
the most important measures of the aggregate state of an economy; however, it is only
available on a quarterly basis. As an alternative, popular coincident indices of business
cycles are estimated. Examples include the composite index of coincident indicators released
by the U.S. Conference Board, the coincident indicators developed by Stock and Watson
(1989, 1991), and the business condition indicator computed by Acruoba et al. (2009) are
monthly series. The main criticism of such coincident indices is that they lack direct economic
interpretation.
To overcome such a criticism, a number of economists estimate monthly GDP directly. In
terms of modeling methodology, some authors construct monthly GDP based on univariate
models for real GDP, e.g. Bernanke et al. (1997) and Liu and Hall (2001), and others apply
multivariate approach, such as Mariano and Murasawa (2003, 2010). These studies are in
line with the ‘common factor’ approach proposed by Stock and Watson (1989, 1991). Their
basic statistical method is to build state-space models with mixed-frequency series. Being
abstract from structural modeling, the common factor approach like the VAR approach is
a reduced-form method. The coefficients estimated in such a model are not subject to any
structural restrictions.
Different from the aforementioned studies that build upon reduced-form time series frameworks, some recent studies consider to merge structural macroeconomic model with the mixed
frequency strategy. Two important contributions are Giannone et al. (2009) and Foroni and
Marcellino (2014). Giannone et al. (2009) develop a framework to incorporate monthly information in quarterly dynamic stochastic general equilibrium (DSGE) models. They take the
parameter estimates from the quarterly DSGE as given and obtain increasing accurate early
forecasts of the quarterly variables. Foroni and Marcellino (2014) demonstrate that temporal aggregation bias, as pointed out in Christiano and Eichenbaum (1987), may arise when
economists estimate a quarterly DSGE while agents’ true decision interval is on a monthly
basis. They propose a mixed-frequency strategy to estimate the DSGE model and find that
the temporal aggregation bias can be alleviated.1 However, there are no general rules on to
1
Some other recent studies with the mixed frequency strategy include the fixed-frequency VAR model in
Schorfheide and Song (2011) and Rondeau (2012). The latter combines quarterly series with annual series
in an effort to estimate a DSGE model for emerging economies.
1
what extent such a complicated framework helps in forecasting real GDP, since it depends
on the structure of the DSGE model and on the content of higher frequency available.
In this paper we develop a mixed-frequency structural model for a small open economy.
The main purpose is to assess the advantage of estimating real GDP from a mixed frequency
model with a structural context. We assume that economic agents make decision monthly.
Because real GDP is a quarterly series, mixed-frequency technique is adopted to provide
early estimates and forecasts. Based on a monthly small open DSGE, we derive its mixfrequency state-space representation. This estimation framework is applied to Taiwan. For
comparison, we also estimate a reduced-form mixed frequency model by applying Kalman
filter estimation technique of Acruoba et al. (2009).
A few other studies are related this paper. Boivin and Giannoni (2006) incorporate
a large data set that contains additional variables (i.e. non-core variables) that are not
considered in a DSGE model. Their approach is appealing conceptually, because it exploits
information contained in the other indicators when making inferences about the latent state
of the economy. The DSGE model parameters as well as the factor loadings for the noncore variables are jointly estimated using Bayesian methods. Nevertheless, their study solely
employs data at the quarterly frequency. In reality, higher frequency information may arrive
and central banks or institutional forecasters would like to include the additional information
in their forecasting framework. Rubaszek and Skrzypczynski (2008) and Edge et al. (2008)
have surveyed the literature on evaluating the forecasting properties of the DSGE model in
a real-time environment. Schorgheide, Sill, and Kryshko (2010) examine whether a DSGE
model could be used to forecast variables that are not included in structural model. Instead
of jointly estimating all the parameters in the system, they suggest a two-step Bayesian
method to reduce the computational burden.
The next section lays out the models we use, including a small open DSGE, its implied
structural mixed frequency state space model, and a reduced-form mixed frequency state
space model. representation. Section 3 presents the empirical findings and the final section
concludes.
2
The Models
The goal of this paper is to study the extent to which the incorporation of monthly observations via DSGE framework improves estimates of the model parameters and produce forecast
2
gains. We assume that the agents in the economy make decisions on a monthly basis. Based
on a small open economy DSGE model, we derive its mixed-frequency state-space representation and estimate the model with maximimum likelihood method. To assess the forecast
performance, we also estimate a monthly reduced-form mix-frequency model.
2.1
A Log-Linearized Small Open DSGE Model
The structural small open-economy model used in this paper is taken from Lubik and
Schorfheide (2007), which is a simplified version of Gali and Monacelli (2005). The model is
derived from a structural New Keynesian DSGE model that consists of households, firms and
a central bank. Given that the decision rules of agents form a system of nonlinear difference
equations with rational expectations, a log-linear approximation to this system around its
steady state is derived. The log-linearized model consists of the following equations:
γ
∗
,
yt = Et yt+1 − [τ + γ)](Rt − Et πt+1 + αEt ∆qt+1 ) − ρa at + Et ∆yt+1
τ
πt = βEt πt+1 + αβEt ∆qt+1 − α∆qt +
κ
(yt − ȳt ),
τ +γ
∆et = πt − πt∗ − (1 − α)∆qt ,
∆qt =
(1)
(2)
(3)
1
(∆yt∗ − ∆yt ) + εq,t ,
τ +γ
(4)
Rt = ρR Rt−1 + (1 − ρR )[ψπ πt + ψy (yt − ȳt ) + ψe ∆et ] + εR,t ,
(5)
where
at = ρa at−1 + εa,t
(6)
∗
yt∗ = ρy∗ yt−1
+ εy∗ ,t
(7)
∗
+ επ∗ ,t
πt∗ = ρπ∗ πt−1
(8)
and γ = α(2 − α)(1 − τ ).
Eq (1) is essentially an open-economy IS curve. It is derived from the consumption Euler
equation of a representative household who makes decisions on consumption of domestic
goods and imported goods, labor supply, and investment. It describes how aggregate output
(yt ) is related to its future expected value, the expected real interest rate (Rt − Et πt+1 ),
the expected change in terms of trade (Et ∆qt+1 ), the growth rate of world technology (at ),
∗
and the expected world output growth (Et ∆yt+1
). In this equation, τ is the intertemporal
3
substitution elasticity, ρa is the first-order autocorrelation coefficient in the dynamic process
of at , and α is the import share that measures the degree of openness with α = 0 in a
closed economy. The terms of trade, qt , is defined as the relative price of exports in terms
of imports. The world output, yt∗ , is exogenous to the small open economy.
Eq (2) is the Phillips curve derived from the optimal pricing setting behavior of domestic
firms. Under the New Keynesian pricing scheme considered in Calvo (1983), there are
nominal rigidities in the domestic goods sector. In this equation, β is the discount factor,
κ measures the degree of price stickiness and ȳt = [−α(2 − α)(1 − τ )/τ ]yt∗ is the potential
output in the absence of nominal rigidities.
Eq (3) illustrates the dynamics in the nominal exchange rate (et ), with rising value in et
indicating a depreciation in the domestic currency. This equation states that dynamics in
the nominal exchange rate is explained by the deviation from the purchasing power parity,
πt −πt∗ , and is adjusted by a fraction of changes in the terms of trade, ∆qt , when the economy
is not completely open to the world economy.
Eq (4) describes the behavior of the terms of trade as a function of the relative demand
changes in the world and the domestic market. When the growth of world output is higher
than the growth of domestic output, the demand for the domestically produced goods rises
so that the terms of trade improve. In the equation, εq,t is an exogenous terms of trade shock
or measurement error.
Eq (5) describes the central bank’s monetary policy rule, that is to raise its interest rate
in response to an increase in CPI inflation, positive output gap, and currency depreciation,
where ρR is an indicator of interest rate smoothing, ψi ’s are the policy reaction coefficients,
and εR,t is the exogenous shock to monetary policy and is assumed to be a white noise
process.
Following Lubik and Schorfheide (2007), in equations (6)- (8) we assume the world technology growth (at ), the world output (yt∗ ), and the world inflation shock (πt∗ ) all evolve
according to univariate AR(1) processes with autoregressive coefficients ρj ’s. Together with
the shocks to monetary policy (εR,t ) and the shocks to terms of trade (εq,t ), there are five
structural shocks. These shocks are assumed to be mutually independent and distributed as
εj,t ∼ iidN (0, σj2 ), for j = a, R, q, π ∗ and y ∗ .
4
2.2
State-Space Representation of the DSGE Model
The log-linearized rational expectations model can be solved with a numerical method and
the solution is transformed into a state-space representation, where the measurement equation relates the model’s variables to the observable ones and the state equation describes the
law of motion of the endogenous and driving forces in the model.2
0
Let St = [Rt , yt , πt , ∆et , ∆qt , yt∗ , πt∗ , at ] denote the vector of non-predetermined endogenous variables that contains endogenous state variables and exogenous driving force variables
0
and εt = [εa,t , εR,t , εq,t , επ∗ ,t , εy∗ ,t ] denote the vector of exogenous structural shocks. The solution to the log-linearized rational expectations model has the following form of transition
for state variables:
St = Φ1 (θ)St−1 + Φ2 (θ)εt ,
(9)
where Φ1 and Φ2 have elements as functions of the deep parameters (θ) from the model and
we denote Σε as the diagonal variance matrix of underlying structural shocks εt . To estimate
the DSGE model, a measurement equation based on a set of observables Yt is added as
Yt = Λ(θ)St + ut ,
(10)
where Λ(θ) defines the relationship between the observed variables and states and ut is the
measurement error term. Jointly, equations (9) and (10) form a state-space representation.3
In a conventional DSGE model estimation, eq (9) is usually timed at the quarterly frequency. This assumption is imposed mainly because real output measure such as GDP does
not have monthly observations. However, as argued in Aadland and Huang (2004), Kim
(2010), and Foroni and Marcellino (2014), if the true decision period is a month, then assuming instead a quarter may lead to misspecification error or temporal aggregation bias.
In the following, we describe how to estimate the monthly DSGE model of (9)-(10) with a
mixed-frequency framework.
2
The most popular solution methods are Sims (2002), Blanchard and Kahn (1980), Klein (2000), and
Uhlig (1999).
3
In the empirical analysis of Lubik and Schorfheide (2007), the choice for the vector of of observable
Yt is composed of annualized interest rates, annualized inflation rates, GDP growth, currency depreciation
0
rate, and changes of the terms of trade. That is, Yt = [4Rt , 4πt , ∆yt + at , ∆et , ∆qt ] . We will estimate
the model with another set of observables, including variables that are available at high frequency, in the
mixed-frequency model.
5
2.3
A Structural Mixed-Frequency Model
Given the assumption that agents made decisions at monthly frequency, our next task is
to estimate the monthly model of (9)-(10). Because there are many missing values in this
monthly econometric model, our first task is to handle unobserved flow and stock variables
differently.
The time index t is at monthly frequency. Let yit be the ith element of Yt , and ỹit be the
same variable in month t that is observed at the quarterly frequency. If it is a variable with
monthly observations, then
ỹit = yit ∀t.
If it is a stock variable with quarterly observations, ỹit is equal to yit in the last month of
each quarter and is a missing value otherwise. Specifically,
yit if t = MAR, JUN, SEP, DEC,
ỹit =
NA otherwise.
When it is a flow variable with quarterly observations, the temporal aggregation problem
arises. In the last month of each quarter, ỹit is the sum of yit over the past three months,
and is a missing value otherwise.
yit + yit−1 + yit−2 if t = MAR, JUN, SEP, DEC,
ỹit =
NA
otherwise.
For example, GDP is a quarterly flow variable, and the relationship between yit and ỹit is
˜ t = GDPt + GDPt−1 + GDPt−2 .
GDP
We can write compactly the aggregation rule of monthly observations into quarterly ones as
Ỹt = H(L)Yt = H0 Yt + H1 Yt−1 + H2 Yt−2 .
(11)
Then, the measurement equation (10) can be modified as

Ỹt = [ H0 Λ(θ) H1 Λ(θ) H2 Λ(θ) H0 H1
6



H2 ] 


St
St−1
St−2
ut
ut−1
ut−2 .




.


(12)
Next, the state equation (9) is

 
St
Φ1 (θ) 0
 St−1   I
0

 
I
 St−2   0
 u = 0
0
 t  
 ut−1   0
0
ut−2
0
0
extended as

0 0 0 0
0 0 0 0 

0 0 0 0 

0 0 0 0 

0 I 0 0 
0 0 I 0
St−1
St−2
St−3
ut−1
ut−2
ut−3


 
 
 
+
 
 
Φ2 (θ)
0
0
0
0
0
0
0
0
I
0
0



 εt
 u .

t

(13)
0
Define a new vector of state variables as ft = [St , St−1 , St−2 , ut , ut−1 , ut−2 ] and a new vector
0
of measurement errors as ηt = [εt , ut ] . Then, eq. (12)-(13) form a mixed-frequency statespace representation based on the structural model described in the previous section and
can be expressed compactly as
Ỹt = Gθ ft ,
(14)
ft = Mθ ft−1 + Pθ ηt ,
(15)
with the covariance matrix of the new error term as
0
Σε 0
.
Eηt ηt =
0 Σu
To sum up, our DSGE empirical model with mixed frequency consists of the state equation (15) and the measurement equation (14). They form a mixed-frequency state-space
representation because there are missing values in Ỹt when the variable is only available at
quarterly frequency. The goal is to jointly estimate the structural parameters of the theoretical model {θ, Σε , Σu } and the measurement equation parameters in (14), if any. Based
on the monthly frequency state space representation, the model is estimated with maximum likelihood method with the likelihood function being evaluated by the Kalman filter
algorithm proposed in Acruoba, Diebold and Scotti (2009).
2.4
A Reduced-Form Mixed-Frequency Model
To assess whether the structural model helps in short-term forecast, we estimate a reducedform mixed-freqency model with the same set of observable variables as stated in the previous
section with structural model. We assume in the reduced-form model that the endogenous
variables in vector Yt links to a single business condition indicator, or a single dynamic factor
along the line of Stock and Watson (1989, 1991), in the following form:
Yt = Axt + wt ,
(16)
7
where xt is the single indicator that summarizes the current economic and business condition
and wt is a vector of measurement errors with covariance matrix Σw . In this reduced-form
equation, the coefficient vector A contains free parameters to be estimated. We assume that
the single business condition indicator follows an AR(1) process:
xt = ρxt−1 + ζt ,
(17)
where ζt is the shock to xt with zero mean and variance σζ2 .
Given the measurement equation (16) and state equation (17), a state-space for this
reduced-form model is derived after incorporating the aggregate rule (11):
Ỹt = G∗ ft∗ ,
(18)
∗
ft∗ = M ∗ ft−1
+ P ∗ ηt∗ ,
(19)
0
where the corresponding state vector is ft∗ = [xt , xt−1 , xt−2 , wt , wt−1 , wt−2 ] , the vector of
0
measurement errors is ηt∗ = [ζt , wt ] .
3
Empirical Results
In this section, the estimation precision and forecast assessment is conducted for the Taiwanese data. The sample period covers from 1983:Q1 to 2007:Q4. For the variables in the
DSGE model, R is the annualized call rate, y is detrended real GDP in logarithm, π is
demeaned GDP deflator inflation rate in annual percentage, ∆e is the depreciation rate of
Taiwan dollars against U.S. dollars, ∆q is the logarithm of the terms of trade index, y ∗ is
detrended U.S. real GDP in logarithm.
(To be continued)
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11
Table 1: Parameter Estimation
Parameter Estimate S.D.
ψπ
ψy
ψe
κ
τ
ρR
ρa
ρy ∗
ρπ∗
ρa
σq
σa
σy ∗
σπ∗
σR
1.31
0.37
0.44
0.40
0.22
0.87
0.79
0.92
0.74
0.88
1.4
1.1
1.62
2.3
0.35
12
(0.25)
(0.16)
(0.32)
(0.14)
(0.09)
(0.22)
(0.20)
(0.32)
(0.15)
(0.29)
(0.44)
(0.21)
(0.20)
(0.75)
(0.08)