Adaptive state space models with
applications to the business cycle and
financial stress∗
Davide Delle Monache†
Ivan Petrella‡
Fabrizio Venditti§
Banca d’Italia
Bank of England
Banca d’Italia
February 5, 2016
Abstract
The estimation of state-space models with time-varying parameters typically implies
the use of computationally intensive methods. Moreover, when volatility evolves stochastically the model ceases to be conditionally Gaussian and requires nonlinear filtering techniques. In this paper, we model parameters’ variation in a Gaussian state-space model by
letting their dynamics to be driven by the score of the predictive likelihood. In this setup,
conditionally on past data, the model remains Gaussian and the likelihood function can
be evaluated using the Kalman filter. We derive a new set of recursions running in parallel with the standard Kalman filter recursions that allows us estimate simultaneously
the unobserved state vector and the time-varying parameters by maximum likelihood.
Given that a variety of time series models have a state space representation, the proposed
methodology is of wide interest in econometrics and applied macroeconomics. Specifically, we use it for improving GDP measurement based on alternative noisy measures and
to construct an index of financial conditions that can be used to nowcast GDP in real
time.
JEL codes: C22, C32, C51, C53, E31.
Keywords: time-varying parameters, score driven models, state space models, dynamics factor models, financial indicator.
∗
The views expressed in this paper are those of the authors and do not necessarily reflect those of the
Bank of England or the Banca d’Italia. While assuming the scientific responsibility for any error in the paper,
the authors would like to thank. We thank the participants at the workshop on “Dynamic Models driven by
the Score of Predictive Likelihoods-Tenerife 2014”, the “9th CFE Conference-London 2015”, and the seminar
participants at the University of Glasgow.
†
Monetay Policy and Economic Outlook Directorate. Email: [email protected]
‡
Bank of England, Birkbeck University London, and CEPR. Email: [email protected]
§
Financial Stability Directorate. Email: [email protected]
1
1
Introduction
After two decades of Great Moderation, a period of low macroeconomic volatility and stable
correlations across macroeconomic time series, the Great Recession has brought the issue of
structural breaks back in the spotlight, spurring a wealth of new research on modeling and
forecasting economic time series in the presence of parameter instability. Recent work in the
field builds on two well established research agendas. The former maintains that parameter
instability manifests itself through large, infrequent breaks, and it is typically associated with
institutional changes (shifts in the monetary policy strategy or changes in the wage bargaining
process). A second view is based on the idea that the collection of information is costly and
sticky, and that the learning process through which economic agents adapt to shocks results
in a slow and continuous adjustment of their behaviour, which is reflected in a gradual change
of the parameters of empirical models.1 Born and bred in different contexts, the former being
more oriented to solving forecast failure, the latter more often used for structural modeling, the
two modeling approaches have recently crossed their paths. Alessandri and Mumtaz (2014),
for example, use threshold VARs for analyzing the response of the macroeconomy to a financial
shock, while D’Agostino, Gambetti and Giannone (2013) document the forecasting properties
of models with gradual time variation in coefficients.
However, the quest for flexibility that underscores these econometric tools, does not come
without costs, since the non-linearities introduced in the models often require the use of computationally intensive estimation methods and sophisticated filtering techniques. These complications can be particularly burdensome for unobserved component models, such as the dynamic
factor models, where the hurdle implied by time varying coefficients is compounded by the
presence of latent states, which need to be inferred on the basis of the observed data.
Motivated by these challenges, in this paper we develop a theoretical framework for the
estimation of state space models with time-varying parameters and since a variety of univariate
and multivariate time-series models can be cast in state space, the proposed approach is of
interest for a wide spectrum of empirical applications. The estimator, which retains most of
the desirable properties of its linear counterpart, generalizes existing algorithms in the adaptive
models literature, like the one proposed by Koop and Korobilis (2013), justifying the definition
of Adaptive State Space models adopted in the title of the paper. The building block of the
method consists in positing a law of motion for the parameters that is a (linear) function of
the score of the predictive likelihood, in line with Creal at al. (2013) and Harvey (2013). In
this setup, conditionally on past data, we show that the model remains Gaussian and that
the likelihood function can be evaluated using the Kalman filter (KF). Still, the nonlinear
setup poses an important challenge. In the linear case the accrual of new information (i.e.
a data release) allows to compute a prediction error that, through the KF recursions, is the
basis for updating and forecasting the unobserved components. In practice, conditional on the
1
Unfortunately the question of which view of the world is more appropriate can not really be settled with an
econometric test, as tests designed for detecting large breaks typically perform poorly when the data generating
process features slow, continuous changes in the model parameters, see Benati (2007).
2
model parameters, new information allows us to refine our view on the most likely location
of the unobserved states and to form the best possible guess of where the data will be in the
future. Time variation in the parameters introduces an additional margin of adjustment to this
process. Since past forecast errors could be due to both a poor estimate of the unobserved states
and to an estimation error of the model parameters, new information calls for a simultaneous
update of both these ingredients. A crucial contribution of our paper consists in deriving the
analytical expressions for a new set of recursions that, running in parallel with the standard
KF recursions, allow to update at each point in time both the unobserved states as well as the
time varying parameters. Therefore, by means of the proposed augmented KF, the unobserved
state vector and the time-varying parameters can be estimated simultaneously by maximum
likelihood.
After presenting the main theorems we discuss two extensions of the theoretical framework
that could be particularly valuable in applied work. First, since in some applications the
researcher would like to impose parameters restrictions, such as stationarity and non-negative
variances, we show how to incorporate general restrictions on the model parameters. Second,
we illustrate how the model can be extended to deal with data at mixed frequencies and
missing observations. We conclude the theoretical part of the paper by briefly discussing how
our estimation strategy is sufficiently general to nest existing adaptive algorithms used in the
literature as a special case.
Given that a wide range of time series models have a state space representation, the estimation method we propose is of interest for a large number of applications in econometrics
and applied macroeconmics. Specifically, we illustrate the potential of the method through two
empirical applications.
In the first exercise we focus on business cycle measurement. In particular, we take as a
starting point the model that Aruoba et al. (2016) have developed to estimate GDP on the
basis of underlying (noisy) measures and extend it to account for time variation in the model
parameters. The measure of GDP developed by Aruoba et al. (2016), labelled GDPplus, is
produced by combining expenditure- and income-side measures of GDP (GDPE and GDPI
respectively) through optimal filtering methods in a constant parameter framework. We show
that most of the parameters of the state space model used by Aruoba et al. (2016) are subject
to frequent breaks, which are well captured by our score driven model. Our approach can
then retrieve some well known stylized facts of the U.S. business cycle that are overlooked
in a constant parameter framework. From our estimates, for instance, four distinct phases
of U.S. long-run growth (measured by the long-run forecast of the estimated model) emerge:
the first (in the 1950s and 1960s) characterized by a steady high level of growth, the second
(from the mid-1970s) marked by a rapid deceleration, a third period of resurgence starting
from mid-1990s and, finally, a sharp decline since the burst of the dotcom bubble in the early
2000s, a development which has been further reinforced by the Great Recession. As a result, we
currently estimate long-run growth in the U.S. to stand at levels between 1.5 and 2 percent, in
3
line with current Congressional Budget Office estimates2 Our model also succesfully captures
the decline of GDP growth volatility during the Great Moderation, as first documented by
Perez-Quiros and McConnell (2000), and the subsequent leap in volatility due to the Great
Recession.
In a second application we derive from a panel of business cycle and financial variables
an indicator of financial stress for the euro area. In this respect, we follow a literature that
has grown tremendously in the recent past, as the financial crisis has shown that the financial
sector can be the origin of powerful shock that have disruptive repercussions on the business
cycle. As a result a large number of Financial Condition Indexes (FCIs) have been developed,
aimed at summarising in a single indicator the (sometimes conflicting) signals coming from
different segments of the financial system. A non exhaustive list of contributions on the topic
includes Illing and Liu (2006), Hakkio and Keeton (2009), Hatzius et al. (2010), Matheson
(2011), Brave and Butters (2012), Hollo’ et al (2012) and Koop and Korobilis (2014).3 Our
modeling approach allows us to address three related challenges in the construction of measures
of financial stress. The first relates to model instability. Since the financial/real economy nexus
is not necessarily stable over time, being shaped by the evolution of technological progress
and by financial innovation, our time varying parameters model seems well suited to capture
shifts in the correlations between the involved variables. The second issue that we address is
endogeneity. Indeed, the relevant information that one would like to extract from financial
variables regards primitive shocks that are coincident or leading with respect to the business
cycle, rather than an endogenous response of the financial sector to past shocks originated in the
real economy. We show how to place appropriate restrictions in the State-Space representation
of our model that allow us to extract a financial component that Granger-causes the business
cycle. Third, since our estimation method handles both missing data and mixed frequencies
we can (i) deal with the fact that some financial indicators are very short and have irregular
patterns (ii) model montly data together with quarterly GDP. We assess the performance
of our model in a nowcasting exercise in which financial variables are used to complement
standard business cycle indicators in monitoring GDP developments in real time. The model
detects significant shifts in the parameters and delivers GDP forecasts that are on average more
accurate than those obtained on the basis of simple univariate benchmarks.
The rest of the paper is structured as follows. Section 2 constitutes the theoretical body of
the paper, where we present the main theorems as well as the extensions discussed above. In
Section 3 we apply the model to the problem of measuring GDP in the U.S. In Section 4 we
use it to measure financial stress in a hte euroa area. In Section 5 we conclude. A technical
Appendix collects the proofs.
2
Up-to-date estimates of potential output for the U.S. are available here https://research.stlouisfed.
org/fred2/series/GDPPOT
3
Most of the available FCI papers borrow their methodological setup from the factor model literature that
was developed in the 2000s to forecast GDP and inflation in a data-rich environment, see Stock and Watson
(2002), Forni, Hallin, Lippi and Reichlin (2000), Doz, Giannone and Reichlin (2012) and the review article by
Stock and Watson (2010).
4
2
The score driven state space model
Let’s assume that a given time series model has the following state space representation
yt = Zt αt + εt , εt ∼ N (0, Ht ),
αt+1 = Tt αt + ηt , ηt ∼ N (0, Qt ),
t = 1, ..., n,
(1)
where yt is the N × 1 vectors of observed variables, εt is the N × 1 vectors of measurement
errors that are Gaussian distributed with zero mean and covariance matrix Ht , αt is the m × 1
vector of state variables and ηt is the corresponding m × 1 vector of Gaussian disturbances with
zero mean and covariance matrix Qt . Thus, the system matrices are Zt that is N × m matrix,
Ht that is N × N , and Tt and Qt that are m × m.
The observation and the state vector are conditionally Gaussian given the information
set Yt−1 = {yt−1 , ..., y1 } and the vector of parameters θ; namely yt |Yt−1 θ ∼ N (Zt at , Ft ) and
αt |Yt−1 θ ∼ N (at , Pt ), and the conditional log-likelihood function is equal to
`t (yt |Yt−1 ; θ) = −
N
log 2π + log |Ft | + vt0 Ft−1 vt .
2
(2)
The prediction error, vt , its covariance matrix Ft , the conditional mean of the state vector
at and its covariance matrix Pt , are estimated optimally (in the sense of the minimum mean
square error) using the KF recursions
vt = yt − Zt at , Ft = Zt Pt Zt0 + Ht , Kt = Tt Pt Zt0 Ft−1 ,
at|t = at + Pt Zt0 Ft−1 vt , Pt|t = Pt − Pt Zt0 Ft−1 Zt Pt ,
at+1 = at + Kt vt , Pt+1 = Tt Pt Tt0 − Kt Ft Kt0 + Qt , t = 1, ..., n,
(3)
where at = E(αt |Yt−1 ) and at|t = E(αt |Yt ) are the predictive filter and real-time filter respectively together with their MSEs Pt = E[(at −αt )(at −αt )0 |Yt−1 ] and Pt|t = E[(at −αt )(at −αt )0 |Yt ].
For the Gaussian model, the matrices Zt , Ht , Tt and Qt can be time-varying but are assumed
to be given (or predetermined); see details in Harvey (1989). In case those matrices contain
time-varying parameters driven by a additional stochastic processes the KF looses its optimality
and Bayesian simulation techniques are usually needed; see Durbin and Koopman (2001).
In this paper we propose a new approach in which the parameters variation is driven by past
observations only. In particular, we exploit the score driven models recently proposed by Creal
et al (2013) and Harvey (2013). Specifically, first we assume a given conditional distribution
of the observations yt ∼ p(yt |ft , Yt−1 , θ), where we condition to past informations Yt−1 , to the
vector of static parameters θ and to the vector ft containing the last estimated time-varying
parameters. Secondly, we posit a law of motion for the the vector of time-varying parameters
ft+1 = ω + Φft + Ωst ,
5
st = St ∇t ,
(4)
where the driving mechanism st is equal to the scaled-score of the conditional likelihood:
∂`t
∇t =
,
∂ft
St = −Et
∂`2t
∂ft ∂ft0
−i
,
with `t = log p(yt |ft , Yt−1 , θ).
Therefore, for i = 0 we have a gradient based algorithm in which st = ∇t is zero mean and
variance equal to It which is the Fisher information matrix. Alternatively, for i = 1 we have
that St = It−1 and st has variance equal to It−1 , and for i = −1/2 we have that st has variance
equal to identity matrix I. If it is not differently stated we consider the case of i = 1. Note
that in order to avoid numerical instability it is often desirable to replace St with a smoothed
estimator S̃t = (1 − λ)St + λS̃t−1 .
In this setup the system matrices are function of the past estimated parameters, ft , and
the static parameters θ; namely, Zt = Z(ft , θ), Tt = T (ft , θ), Ht = H(ft , θ) and Qt = Q(ft , θ).
The the conditional Gaussianity is preserved and the predictive likelihood (2) is amended to
account for the parameters’ variation.
Let’s focus briefly on the updating rule (4); at each point in time the vector of time-varying
parameters is updated so as to maximise the local fit of the model. Specifically, the size
of the update depends on the slope and curvature of the likelihood function. As such, the
updating law of motion (4) can be rationalised as a stochastic analog of the Gauss–Newton
search direction for the time-varying parameters. Blasques et al. (2014) show that updating
the parameters using the score is optimal, as it locally reduces the Kullback-Leibler divergence
between the true conditional density and the one implied by the model.
Creal et al (2008) propose the use the score-driven approach to model the parameters’
variation for univariate unobserved component model and we generalised their result for any
Gaussian state space model with time-varying parameters.4 Specifically, the two theorems
below describe how the standard KF is augmented by a new set of recursions that compute
the scaled-score st driving the parameters’ variation.
Let first introduce some notation. Given a N × m matrix X, vec(X) is the vector stacking
the columns of X one underneath the other, thus vec(X) is a N m×1 vector. Moreover, vech(X)
eliminates all supradiagonal elements of X from vec(X), thus vech(X) is a N (N + 1)/2 × 1
vector. Following Abadir and Magnus (2005, ch 11), we define the following matrices: the N 2 ×p
duplication matrix D and the p × N 2 cancellation matrix D+ , such that Dvech(X) = vec(X)
and D+ vec(X) = vech(X). It is worth stressing that the cancellation matrix is equal to
the left inverse of duplication matrix, namely D+ = (D0 D)−1 D0 . Moreover, the N m × N m
commutation matrix CN,m is such that CN,m vec(X) = vec(X 0 ), and for N = m the m2 × m2
commutation matrix is denoted by Cm . The identity matrix of order N is denoted by IN . We
also use the symemtrizer matrix Nn = 21 (In2 + Cn ), such that Nn vec(A) = vec 12 (A + A0 ), and
for A symmetric Nn vec(A) = vec(A). Finally, “⊗” is the Kronecker product.
4
Creal et al (2008, sec. 4.4) focus on the local level model with stochastic volatility proposed by Stock and
Watson (2007).
6
Theorem 1 Given the Gaussian model (1) with the conditional log-likelihood (2), the gradient
and information matrix are obtained as follows
•0
1 • 0 −1
−1
−1
F t (Ft ⊗ Ft )[vt ⊗ vt − vec(Ft )] − 2V t Ft vt ,
=
2
•
•0
•
1 • 0 −1
−1
−1
=
F t (Ft ⊗ Ft )F t + 2V t Ft V t , t = 1, ..., n,
2
∇t
It
•
where V t =
∂vt
∂ft0
•
and F t =
∂vec(Ft )
.
∂ft0
(5)
(6)
Proofs in the Appendix A.1.
The previous theorem show how to compute analytically the score and the information
matrix, which depend on the prediction error vt and its covariance matrix Ft that are recursively
•
•
computed using the KF (3). Furtheremore, we have two new elements V t and F t and the
following theorem shows the analytical expressions to recursively compute them.
Theorem 2 In order to compute the gradient and the information matrix we need the following
set of additional recursions:
•
•
•
V t = −[(a0t ⊗ IN )Z t + Zt At ],
•
t = 1, ..., n,
•
•
(7)
•
F t = 2NN (Zt Pt ⊗ IN )Z t + (Zt ⊗ Zt )P t + H t ,
•
•
(8)
•
K t = (Ft−1 Zt Pt ⊗ Im )T t + (Ft−1 Zt ⊗ Tt )P t
•
•
+(Ft−1 ⊗ Tt Pt )CN m Z t − (Ft−1 ⊗ Kt )F t ,
•
•
•
•
•
•
(9)
•
At+1 = (a0t ⊗ Im )T t + Tt At + (vt0 ⊗ Im )K t + Kt V t ,
•
(10)
•
P t+1 = (Tt ⊗ Tt )P t − (Kt ⊗ Kt )F t + Qt
•
•
+2Nm [(Tt Pt ⊗ Im )T t − (Kt Ft ⊗ Im )K t ],
•
where Z t =
∂vec(Zt )
,
∂ft0
•
Ht =
∂vec(Ht )
,
∂ft0
•
Tt =
∂vec(Tt )
∂ft0
•
and Qt =
∂vec(Qt )
.
∂ft0
(11)
Proofs in the Appendix A.2.
Remark 3 The expressions (5)-(11) generalize the results in Harvey (1989, pp. 140-3) for the
case of time-varying parameters.
•
•
•
•
It is worth to stress that the Jacobians matrices Z t , H t , T t and Qt are model specific,
i.e. they depend on the specific time-variation of the model under consideration. In the
simplest case they result in selection matrices, but in general they incorporate Jacobians due
to parameters restrictions. This point will be clarified in later on.
Putting together Theorem 1 and 2, we obtain the new set of recursions (5)-(11) running in
parallel with the KF (3) to compute the scaled score st and thus the time-varying parameters
in (4). To summarise, the augment algorithm runs as described below.
•
•
Algorithm 4 Given the initial values a1 , P1 , A1 , P 1 and f1 , for t = 1, ..., n:
• evaluate Zt , Ht , Tt , Qt ;
7
•
•
•
•
• evaluate Z t , H t , T t , Qt ;
• compute vt , Ft , `t , Kt ;
•
•
•
• compute V t , F t , K t , ∇t , It , st ;
•
•
• update at+1 , Pt+1 , At+1 , P t+1 , ft+1 .
All the static parameters in the law of motion (4), and possibly any static element in Zt ,
Tt , Ht and Qt , are collected in the vector θ, which is then estimated by maximum likelihood
P
(ML); θb = arg max nt=1 `t (θ). The evaluation of the log-likelihood is straightforward and the
maximisation can be obtained numerically. Following Harvey (1989, p. 128) we have that
√ b
n(θ − θ) → N (0, Ξ), where is the asymptotic variance Ξ is evaluated by numerical derivative
at the optimum as discussed in Creal et al (2013, sec. 2.3).
2.1
State space models with mean adjustment
It is sometimes convenient to include mean adjustment terms in (1) resulting in
yt = Zt αt + dt + εt , εt ∼ N (0, Ht ),
αt+1 = Tt αt + ct + ηt , ηt ∼ N (0, Qt ),
(12)
where dt and ct are known vector possibly time-varying. While the specification (1) is adequate
for most purposes, the specification (12) may be suitable for occasional use and the filtering
(3) is amended only in the following elements
vt = yt − Zt at − dt ,
at+1 = Tt at + ct + Kt vt ,
(13)
thus expressions (7) and (10) are modified as follows
•
•
•
•
V t = −[(a0t ⊗ IN )Z t +dt + Zt At ],
•
•
•
•
•
•
At+1 = (a0t ⊗ Im )T t +ct + Tt At + (vt0 ⊗ Im )K t + Kt V t ,
•
where dt =
2.2
∂dt
∂ft0
•
and ct =
∂ct
∂ft0
(14)
will be model specific.
Missing observations
Assume to have a data set containing missing observations, namely the observed vector
is Wt yt , where Wt is Nt × N selection matrix with 1 ≤ Nt ≤ N, meaning that at least one
observation is available, Wt is obtained by eliminating the i − th row from IN if the i − th
variable is missing at time t and if there is no missing observations Wt = IN . If no data is
available Nt = 0, the KF computes the prediction step only and the likelihood is equal to a
constant so that st = 0, and the time-varying parameters are not updated.
8
The measurement equation (12) is therefore modified taking into account the missing observations as following
Wt yt = Wt Zt αt + Wt dt + Wt εt , Wt εt ∼ N (0, Wt Ht Wt0 ).
(15)
Thus, at each point in time the likelihood `t is computed using Nt observations, the KF
recursions (3) are modified as follows
vt = Wt (yt − Zt at − dt ),
Ft = Wt (Zt Pt Zt0 + Ht )Wt0 ,
Kt = Tt Pt Zt0 Wt0 Ft−1 ,
(16)
while the expression for at+1 and Pt+1 are still the same. Therefore, the formulae (7), (8) and
(9) need to be accommodated as following
•
•
•
•
V t = −[(a0t ⊗ Wt )Z t + Wt dt + Wt Zt At ],
•
•
•
•
F t = 2NNt (Wt Zt Pt ⊗ Wt )Z t + (Wt Zt ⊗ Wt Zt )P t + (Wt ⊗ Wt )H t ,
•
•
•
K t = (Ft−1 Wt Zt Pt ⊗ Im )T t + (Ft−1 Wt Zt ⊗ Tt )P t
•
(17)
•
+(Ft−1 Wt ⊗ Tt Pt )CN m Z t − (Ft−1 ⊗ Kt )F t
•
•
while recursion (10) and (11) for At+1 and P t+1 are still the same. While the formulae in (16)
can be find in standard textbook such as Durbin and Koopman (2001), the formulae (17) can
be obtained following similar steps as the Appendix by taking into account the selection matrix
Wt .
2.3
Mixed frequencies and temporal aggregation
This section deals with the relation between the high frequency variable xt , which for key
economic concepts are unobserved, and the corresponding observed low frequency series, xkt
with k > 1. The relation between the observed low frequency variable and the corresponding
indicator depends on whether the variable is a flow or a stock variable and on how the variable is
transformed before entering the model. In all cases the variable can be rewritten as a weighted
average of the unobserved high frequency indicator, specifically
xt =
2k−2
X
ωjk xkt−j
(18)
j=0
Here a summary of the implied weights (see e.g. Banbura and Modugno 2014):
• If the variable enters in level and is a stock variable then ωjk = 1 for j = 0 and ωjk = 0
for j > 0.
• If the variable enters in level and is a flow then ωjk = 1 for j = 0, ..., k − 1 and ωjk = 0 for
9
j ≥ k.
• If the variable enters in first difference and is a stock ωjk = 1 for j = 0, ..., k − 1 and
ωjk = 0 for j ≥ k.
• If the variable enters in first difference and is a flow ωjk = j + 1 for j = 0, ..., k − 1 and
ωjk = 2k − j − 1 for j ≥ k.
Assume that the unobserved high frequency variable follows the state space model
2
yi,t = Zi,t αt + εi,t , εi,t ∼ N 0; σi,t
,
(19)
we consider a single indicator (e.g., the GDP) to be aggregated from quarterly to monthly (i.e.
k = 3). Therefore, at quarterly frequency we observe
q
yi,t
=
4
X
ωj3 Zi,t−j αt−j +
j=0
4
X
ωj4 ε4i,t−j ,
(20)
j=0
and the state space needs to be accommodated taking into account the aggregation (20) and
the implied missing observations.5 In empirical section we will describe in detail the state space
specification.
2.4
Relationship with the existing literature
Koopman et al (2010) introduce time-varying parameters in the dynamic Nelson–Siegel
yield curve model of interest rates of different maturities. This results in a state space model
in which the factor loading are time-varying and the volatility follows a GARCH process. The
state vector is augmented in order to include the time-varying loading and as such they obtain
a non linearity masurement equation which is solved by means of the extended KF. Moreover,
they allow for time-varying volatility. In the spirit of Harvey, Ruiz, and Sentana (1992) they use
the common GARCH specification where the time-varying volatility is driven by the squared
error estimated within the KF as additional element of the state vector.
SI POTREBBE CERCARE DI VEDERE SE APPROCCIO DI Harvey, Ruiz,
and Sentana (1992) SI PUO RAZIONALIZZARE NEL NOSTRO FRAMEWORK.
Koop and Korobilis (2013) propose the use of the adaptive algorithm, specifically forgetting
factor algorithm, to deal with time-varying parameters and stochastic volatility in large VARs
models. In a different paper, Koop and Korobilis (2014) exploit the adaptive algorithm to model
the parameters variation in a state space framework using a two-step procedure, in which the
unobserved state vector is extract by means of KF. Here, we show that this approach is a
special case of our general framework. Specifically, if the state vector αt is observable (and
5
Note that although the aggregation implies that the measurement error now follows a moving average
process of order related to the dimension of the high frequency, it nevertheless remains white noise when
observed at the low frequency frequency (see Auroba et al., 2011). Hence we treat the measurement error at
the low frequency as white noise in what follows.
10
denoted with at ) we have that both the state space model reduces to two regression models
with rime-varying parameters:
yt = XZt βZt + εt , εt ∼ N (0, Ht ),
at+1 = XT t βT t + ηt , ηt ∼ N (0, Qt ),
(21)
where βZt = vec(Zt ), XZt = (a0t ⊗ IN ), βT t = vec(Tt ) and XT t = (a0t ⊗ Im ). For simplicity, we
consider only the first equation of (21), while the derivation for the second equation follows
accordingly. The conditional log-likelihood for yt is
`t = −
N
log (2π) + log |Ht | + ε0t Ht−1 εt ,
2
0
and the vector of time-varying parameters is ft = (βZt
, vec(Ht )0 )0 . Given the formulae (5)-(6),
the scaled score is qeual to st = (s0βt , s0Ht )0 with:
0
−1 0
•
•
•
−1
sβt = − B t Ht B t
B t Ht−1 εt ,
0
−1 0
•
•
•
−1
−1
−1
−1
sHt = H t (Ht ⊗ Ht )H t
H t (Ht ⊗ Ht )[εt ⊗ εt − vec(Ht )] .
•
•
•
∂εt
= −XZt , H=D if we model the vech(Ht ), or H t = IN 2 if we model vec(Ht ).
with B t = ∂β
0
Zt
Finally, the two scaled score driving the drifing coefficients and volatilities are:
sβt = Xt0 Ht−1 Xt
−1
Xt0 Ht−1 εt ,
sHt = [εt ⊗ εt − vec(Ht )].
Assuming a random walk law of motion for ft we obtain the following filter to6
βZt+1 = βZt +κβ Xt0 Ht−1 Xt
−1
Xt0 Ht−1 εt ,
vec(Ht+1 ) = vec(Ht )+κH (εt ⊗εt −vec(Ht )). (22)
If we replace the scaling matrix Xt0 Ht−1 Xt with its smoothed estimator
Sβt = (1 − κβ )Sβt−1 + κβ Xt0 Ht−1 Xt
we have that the filter used in Koop and Korobilis (2013, 2014) boils down to the filter in (22).
For details on the exact equivalence see Delle Monache and Petrella (2014).
2.5
Parameters restrictions
Applications of time-varying parameters models often require imposing restrictions on the
parameters space. For instance, stationarity restrictions may be imposed or positivity of the
volatilities. When restrictions are implemented within a score-driven setup, the resulting model
can still be estimated by MLE without the need of computational demanding simulation meth6
Starting from the law of motion (4), we assume that ω = 0, Φ = I and Ω is block diagonal that depends
on two scalar only; κβ and κH regulating the time-variation of the coefficients and volatility, respectively.
11
ods. This requires the re-parameterization of the vector of time-varying parameters as follows
f˜t = ψ(ft ),
(23)
where ft is the unrestricted vector of parameters we model, and f˜t is the vector of interest
restricted through the function ψ(·). The latter is a time invariant, continuous and twice
differentiable function, often called link function (Creal et al., 2013, and Harvey, 2013). The
vector ft continues to follow the updating rule (4), but the score needs to be amended taking
into account the Jacobian of ψ(.). In practice, we model ft = h(f˜t ), where h(·) is the inverse
function of ψ(·). Given a continuous and differentiable function ψ(·), Ψt is a deterministic
function given past information, whose role is to re-weight the original score such that the
restrictions are satisfied at each point in time.
•
•
•
In our setup this transformation will have an effect on the Jacobian matrices Z t , H t , T t and
•
Qt . Specifically, we propose the general form for a generic time-varying matrix of the system:
vec (Mt ) = cM + SM 1 ψ (SM 2 ft ) ,
(24)
where Mt denotes a nr × nc matrix containing both constant and time-varying parameters, cM
is a nr nc × 1 vector with constant elements, and SM 1 and SM 2 are selection matrices. SM 1
is obtained as follows: take Inr nc , retain the columns associated to the time varying elements
of vec (Mt ), while SM 2 selects the sub-vector of ft belonging to Mt . Finally, ψ (.) denotes a
(time-invariant) link function and its Jacobian is Ψt = ∂ψ(ft )/∂ft0 . The representation (24)
helps to compute
•
M t = SM 1 Ψt SM 2 .
In the application, we will impose restrictions on the time-varying autoregressive coefficients
such that they have stable roots; this is implemented by re-parameterizing the autoregressive
model with respect to the partial autocorrelations as in Delle Monache and Petrella (2014). A
second set of restrictions concern a generic time-varying covariance matrix Σt to be positive
defined and this achived by means of log-Cholesky transformation. Specifically, we have Σt =
Lt L0t , with Lt being a lower triangular matrix, and we model vech(L̃t ), where L̃ij,t = Lij,t for
i 6= j, and L̃ii,t = log Lii,t . Thus, the Jacobian is equal to
Ψt =
∂vec(Σt )
= (IN 2 + KN )(Lt ⊗ IN )SL Slog t ,
∂vech(L̃t )0
where KN commutation matrix, SL is a selection matrix such that SL vech(Lt ) = vec(Lt ) and
it is obtained form diag(vec(Lt )) by dropping the columns containing only 0s and replacing
∂vech(Lt )
the no-zeros elements with 1s. Finally, Slog t = ∂vech(
= diag(exp(L̃ii,t )). Note that in case
L̃t )0
2
Σt = σ is the time-varying variance, ψ(˜lt ) = exp(2˜lt ), where ˜lt = log σt , and Ψt = 2 exp(2˜lt ).
t
12
3
Empirical application 1: GDP plus revisited
Our first empirical application consists of extending the model proposed by Aruoba et al.
(2016) in the context of GDP measurement. Because the source of all incomes lies in the
flow of value added generated by production and all production must be either consumed at
home or abroad or else invested, Gross Domestic Product (GDP) can be measured in different
ways. In the U.S. two GDP estimates exist, a widely-used expenditure-side version, GDPE ,
and a much less widely-used income-side version, GDPI . These two related concepts can,
at times, diverge quite significantly. In the top panel of Figure (1) we show the annualized
percentage quarterly rate of growth of both GDPE (green line) and GDPI (blue line) as well
as their difference (bottom panel). The discrepancy between the two concepts is generally non
negligible and it can be as large as five percentage points on an annualized basis. The question
of which GDP measure is more reliable is of particular interest to policy makers who need to
monitor business cycle developments in real time, since some researchers argue that GDPI ,
which has traditionally received much less attention than GDPE , has actually done a better
job at recognizing the start of recessions, see Nalewaik (2012).
Aruoba et al. (2016) suggest to use the two existing measure of GDP computed from the
expenditure side and from the income side to extract, through optimal filtering techniques, a
measure of underlying GDP that is relatively cleaner of measurement error. To this end they
work with a dynamic factor model where both statistical measures of output are assumed to
be driven by a common latent component (which they label GDP plus). An optimal estimate
of GDP plus is then delivered by the Kalman filter conditional on the parameters of the model,
which are estimated via maximum likelihood. In our empirical exercise we extend their framework by allowing for the parameters of the dynamic factor model to change over time, driven
by the score of the conditional likelihood.
Formally, the baseline constant parameter specification from which we start is described in
the following State-Space system:7
"
GDPE,t
GDPI,t
#
"
=
1
1
#
"
αt +
εE,t
εI,t
#
"
,
αt+1 = ρ0 + ρ1 αt + ρ2 αt−1 + et ,
εE,t
εI,t
#
∼ N (0, H)
(25)
2
et ∼ N (0, σ )
where H is a full 2 × 2 matrix.8 It can be readily recognized that the model is a traditional
small-scale dynamic factor model in the spirit of Sargent and Sims (1977). We extend the
7
Aruoba et al. (2016) consider up to five alternative model specifications, which differ from each other with
respect to the correlation structure of the measurement errors. Given that the focus of our paper is mainly
methodological, in our application we consider only one of their specifications.
8
In the original specification, Aruoba et al. (2016) work with an AR(1) transition equation. We prefer an
AR(2) specification that is more in line with traditional business cycle models. Notice, in particular, that the
spectrum of an AR(1) model has its peak at zero and most of its mass on the left hand side of the frequency
band, while in an AR(2) model the peak of the spectrum can occur at cyclical frequencies.
13
model to allow for all the seven parameters involved to change over time:
"
GDPE,t
GDPI,t
#
"
=
1
1
#
"
αt +
εE,t
εI,t
#
"
,
αt+1 = ρ0,t + ρ1,t αt + ρ2,t αt−1 + et ,
εE,t
εI,t
#
∼ N (0, Ht )
et ∼ N
(26)
(0, σt2 )
In the vector ft we collect the three group of parameters, namely the intercept and autoregressive elements of the transition equation,
the variance of the erorr
the elements of Ht and
0 0
term in the transition equation, i.e. ft = ρ0,t , ρ1,t , ρ2,t , σt2 , vech L̃t
, where L̃t is a lower
tringular matrix containing the elements of the log-Cholesky transformation of Ht as described
earlier. We assume that
ft+1 = ft + Ωst
where Ω is a diagonal matrix function of only 3 static coefficients: κρ , κσ , κl , one for each group
of parameters. These parameters are then estimated by maximum likelihood.
The results of the analysis are shown in Figures 2 to 9.
• In Figure 2 we report the estimated time varying intercept ρ0,t (blue line), together
with the estimate obtained on the basis of the constant parameters model. Two points
are worth highlighting. First, there is a secular downward trend that drives this term,
indicating that the mean growth rate of the U.S. economy has progressively fallen over
time. Second, there are marked falls corresponding to some recessions, namely those of
the early 70s and early 2000s. As we will see below, this has important implications
for the estimated long-run growth of the economy. Third the constant parameter model
severely overestimates this term at the end of the sample.
• The evolution of the autoregressive parameters is shown in Figure 3. Notice that the second autoregressive term (red line) fluctuates markedly over time, although it is reasonably
centered around zero, the value to which is constrained in the original specification by
Aruoba et al. (2016). The first AR term (blue line), also displays non-negligible time
variation, especially at the end of the sample. Aruoba et al. (2016) place a lot of emphasis on this parameter, which they estimate at around 0.5 on the whole sample. This
value is higher than the one estimated for either GDPE or GDPI , which leads them to
conclude that the filtered measure is more predictable than its components. Our findings
cast some doubt over the possibility of exploiting this auto-correlation in real time, since
it seems to be the result of the prolonged fall in macroeconomic activity during the Great
Recession rather than a robust, full-sample, feature of GDP plus. Notice that both the
autoregressive terms in Figure 2 and the time-varying interecept in Figure 1 contribute
ρ0,t
. This implies
to defining the long-run forecast of GDP, which is given by µt = (1−ρ1,t
−ρ1,t )
that underestimating or overestimating any of this terms can lead to a poor estimate
of the long-run GDP growth, so that letting both vary over time can have important
implications for the real-time assessment of long-run growth.
14
• Next, in Figure 4 we report the estimated long-run growth path implied by our model,
ρ0,t
. It is worth stressing that this is a key variable for both
computed as µt = (1−ρ1,t
−ρ1,t )
monetary and fiscal policy. Orphanides (2003), for instance, stresses how poor real-time
estimates of potential growth can generate sizeable monetary policy mistakes. According
to his analysis, for instance, part of the monetary policy mistake that led to the Great
Inflation of the 70s can be attributed to a failure in identifying a permanent slowdonw in
productivity growht in the first part of the 70s. Long-run projections are also at the center
of the fiscal policy debate as they determine the solvency of pension systems and, more
generally, the sustainability of public debt. According to our estimates, four distinct
phases of U.S. long-run growth (measured by the long-run forecast of the estimated
model) emerge: the first (in the 1950s and 1960s) characterized by a steady high level of
growth, the second (from the mid-1970s) marked by a rapid deceleration, a third period
of resurgence starting from mid-1990s and, finally, a sharp decline since the burst of the
dotcom bubble in the early 2000s, a development which has been further reinforced by
the Great Recession. As a result, we currently estimate long-run growth in the U.S. to
stand at levels between 1.5 and 2 percent, in line with current Congressional Budget
Office estimates. These results are strikingly in line with both the analysis in Benati
(2007) and with the conclusions reached by Antolin-Diaz et al. (2015).
• Figure 5 shows the variance of the error of the common component, a measure of macroeconomic volatility. It is interesting to notice that, after a prolonged period of decrease
in the 50s and 60s, volatility reaches a plateau where it stays up to the mid-80s. From
1984 onwards there is a second marked fall of volatility, corresponding to the Great
Moderation, which is interrupted, and partially reversed by the Great Recession in 2008.
• Next, in Figure 6 we report the Kalman filter gain of GDPE relative to that of GDPI over
the whole sample, which can be interpreted as the relative weights that the two observed
measures receive in the construction of GDP plus. Given how this is computed, we have
that values below 1 imply that a relatively larger weight is assigned to GDPI . From
the picture it emerges that, indeed, as claimed by part of the literature, GDPI better
captured the behaviour of GDP plus in the middle of the sample. However, in recent
years, an equal weighting scheme seems warranted. This result squares well with the fact
that from July 2015 the BEA started publishing the mean of GDPE and GDPI as an
indicator of the business cycle. Our results corroborate the choice of an equal weighting
scheme at the current juncture.
• In Figure 7 we show the resulting GDP plus estimate (blue line) together with the two
underlying components. From this picture it is clear that the estimated latent factor is
considerably smoother than the two observed series, so that in many periods GDPI and
GDPE fall outside of the confidence intervals. In Figures 8 and 9 we zoom these comparisons in two recession periods, namely the 1970s recessions and the Great Recession
of 2008. It is evident that during these turbulent periods GDP plus gives less volatile
15
indications on output dynamics than the other two measures.
Concluding, when applied to the problem of extracting a cleaner measure of output from
observed series of Gross Domestic Product, our model reveals a number of interesting aspects
that are overlooked in the constant parameters framework proposed by Aruoba et al. (2016). In
particular, our model delivers a real time/time-varying assessment (i) of the long-run economic
growth (ii) of macroeconomic volatility and (iii) of the relative importance of GDPI and GDPE .
All these three ingredients are of great interest to policy makers and business cycle analysts.
4
Empirical application 2: measuring financial stress in
the euro area
We focus our second empirical application on a measure of financial stress for the euro
area. Since the focus is on the methodological aspects of our model, we do not engage in a
search of new financial indicators but rely on the 15 sub-indexes that form the ECB-CISS as
the representative indicators of financial stress for the euro area.9 and model these 15 variables
together with 4 business cycle indicators, namely GDP, industrial production and two survey
indicators, the composite Purchasing Manager Index (PMI) and its orders subcomponent. The
financial stress variables are:
• Realised volatility of the 3-month Euribor rate, Interest rate spread between 3-month
Euribor and 3-month French T-bills, Monetary Financial Institution’s (MFI) emergency
lending at Eurosystem central banks
• Realised volatility of the German 10-year benchmark government bond index, Yield
spread between A-rated non-financial corporations and government bonds (7-year maturity bracket), 10-year interest rate swap spread: weekly average of daily data
• Realised volatility of the Datastram non-financial sector stock market index, CMAX
for the Datastream non-financial sector stock market index (Maximum cumulated index
losses over a moving 2-year window), Stock-bond correlation
• Realised volatility of the idiosyncratic equity return of the Datastream bank sector stock
market index over the total market index, Yield spread between A-rated financial and
non-financial corporations (7-year maturity), CMAX as defined above interacted with the
inverse price-book ratio (book-price ratio) for the financial sector equity market index
• Realised volatility of the euro exchange rate vis-à-vis the US, dollar, Realised volatility of
the euro exchange rate vis-à-vis the Japanese Yen, Realised volatility of the euro exchange
rate vis-à-vis the British Pound
9
We thank the authors of the CISS for sharing their data with us.
16
See Hollo’ et al. (2012) for an accurate description of their construction. In the next two
subsections we discuss more in details the dynamic factor model specification and how this can
be nested in the more general State Space framework discussed in Section 2. Next we present
the results of full sample estimation and of a real time forecast exercise.
4.1
Financial stress and economic conditions
The vector of observable is yt = (ytr0 , ytx0 )0 , in which ytr0 in the vector macroeconomic variable
including also the quarter on quarter growth rate of GDP and other cyclical indicators, and
ytx0 , the chosen financial indicators. Although financial variables are potentially available at
frequencies higher than monthly, we restrict ourselves to the use of monthly averages in order to
limit the dimension of the state vector and also because a monthly update of such an indicator
seems sufficiently informative for policy makers.
Hatzius et al. (2012) define the financial condition as ”the current state of financial variables
that influence economic behaviour and (thereby) the future state of the economy”. Furthermore
they add: ”ideally, a FCI should measure financial shocks – exogenous shifts in financial conditions that influence or otherwise predict future economic activity. True financial shocks should
be distinguished from the endogenous reflection or embodiment in financial variables of past
economic activity that itself predicts future activity”. The definition above is implicitly a statement about Granger-causality; i.e. we look for an indicator of financial activity that predicts
output beyond the information in the past levels of economic activity. However, in Hatzius et
al. (2010) the financial assets beside responding to financial shocks also react to real shocks as
well as the predictable component of real activity. The latter though contradicts the idea that
all available information is already priced, which implies that asset prices should only respond
to the unpredictable component, i.e. the news, in real activity. Therefore, our proposed FCI is
defined as the summary of all the relevant information in the asset prices that helps to predict
the future behaviour of the coincident indicator of economic activity; it is a leading indicator
of future activity.
Formally, we can cast the model in the following state space form. The measurement
equation is block diagonal and can be written as
"
ytr
ytx
#
"
=
λrt 0
0 λxt
#"
αtr
αtx
#
"
+
εrt
εxt
#
"
,
εrt
εxt
#
∼ N (0, Ht ).
(27)
Time variation in the loading vector allows for the relevance of the variables included in the
system for the factors to vary over time. For instance the impact of financial variables that
are related to the real estate market is likely to have increased during the early part of the
latest recession. More generally, the information content of different asset classes could change
in different stages of the business cycle.
17
The transition equation imposes the Granger-causality restrictions, specifically
"
r
αt+1
x
αt+1
#
"
= Φ1t
r
αt−1
x
αt−1
#
"
+ Φ2t
r
αt−2
x
αt−2
#
+ ut ,
ut ∼ N (0, Σt )
(28)
where Φjt are upper triangular matrices. The time variation in the feedback matrices in the
transition equations is meant to capture possible time variation in the leading properties of
the financial indicator. Furthermore, time varying volatilities are allowed for both in the
measurement error and to the innovations of the transition equations. The former are included
so as to pick up changes in the volatility that are specific to the variables included in the
system, whereas the latter capture common changes in volatilities (and the associated time
varying correlation), i.e. the great moderation (see e.g. Perez-Quiros and McConnell, 2000) or
the business cycle related fluctuations in uncertainty documented by Jurado et al. (2015) as
well as the broader increase of the volatility of financial markets.
Note that the contemporaneous relation between the factors is fully captured by the covariance matrix Σt on which we have made no restriction. Therefore, current development in
real activity, i.e. in the coincident indicator αxt , affect contemporaneously asset prices in this
context. However, it is important to stress that it is only the unpredictable component, i.e. the
shock to real activity, that affects the asset prices (therefore implicitly incorporating the idea
that financial market are efficient or that informational frictions are not relevant at monthly
frequency). To illustrate this point let us consider the identification that financial shocks affect
real activity with a (1 month) delay, financial market react instantaneously to any development
in the real economy. This ordering restriction allows to identify real shocks, urt , from financial
shocks, uxt ,
#
#
"
"
#"
ert
h11t 0
ert
∼ N (0, I).
(29)
,
ut =
ext
ext
h21t h22t
Substituting this into the measurement equation we have that the unpredictable component of
the real activity, urt , has a direct impact on asset prices. Therefore assuming a block diagonal
measurement equation ensures that is only the shocks, i.e. the news, in real economic activity
that move the financial markets (and not their predictable component). Note that even though
the measurement equation is block diagonal the fact that the covariance matrix is unrestricted
implies that financial variables help to nowcast current economic activity. In fact, it is easy
to show that the Kalman gain is not diagonal in this setting. Intuitively, if financial markets
react systematically to macroeconomic developments, their reaction helps to pin down the real
shocks and therefore to improve the nowcast of the coincident indicator and its components.
4.2
The state space specification
Here we discuss the details of the state space representation of the factor model (27)-(28)
taking into account the mixed frequency and aggregation. For the macro variables ytr , we have
three types of indicators: one quarterly (i.e. GDP, ytq ), three monthly business cycle indicators
18
ytm (industrial production and two surveys) and 15 monthly financial indicators, ytx . Therefore,
the vector of observables yt = (ytq , ytm0 , ytx0 )0 have dimension 19. The state vector is
0
y
y
y
y
x
αt = αty , αtx , αt−1
, αt−1
, αt−2
, αt−3
, αt−4
, εqt , εqt−1 , εqt−2 , εqt−3 , εqt−4
where αty is the real factor αtx is the financial factor and εqt is the measurement error of the
quarterly macro variable (i.e., GDP) and due to the time aggregation we have the moving
average terms. The corresponding time-varying matrix of factor loading, consistent with the
time aggregation, is equal to



Zt = 


1 q
λ
3 t
1 s
λ
3 t
λm
t
0
0
0
0
λxt
2 q
λ
3 t−1
1 s
λ
3 t−1
0
0
0
0
0
0
λqt−2
1 s
λ
3 t−2
0
0
2 q
λ
3 t−3
1 q
λ
3 t−4
0
0
0
0
0
0
1
3
2
3
1 23 31
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0






x
where λqt is scalar, λst is ns × 1 vector, λm
t is nm × 1 vector and λt is nx × 1 vector. Note that we
impose restrictions on the factor loadings such that the one relative to the first macro variables
(i.e. the GDP) and the one relative to the first financial variables are both normalize to one.
m0 x0 0
The measurement errors is εt = (0, εs0
t , εt , εt ) ∼ N (0, Ht ), where Ht is diagonal and thus
we impose positive volatility on the elements of Ht by log transformation element by element.
Consistently with the previous discussion on the identification of real and financial factors,
the transition equation is specified as follows

Tt
Qt
φ11,t φ12,t φ13,t φ14,t
"
#

 0 φ22,t
T11,t T12
0 φ24,t
=
, T11,t = 

T21 T22
0
0
0
 1
0
1
0
0
h
i
= blockdiag Σt 05×5 σε2q t 04×4






while T12 , T21 and T22 are sub-matrices of 0s and 1s. In order to impose stable roots in the
transition matrix Tt , we use the transformation described early on for the two pairs
φr,t = (φ11,t , φ13,t )0 ∈ S 2 and φx,t = (φ22,t , φ24,t )0 ∈ S 2
where S 2 is the space with stable roots, while φ12,t and φ14,t are left unrestricted. Finally, we
restrict the matrix Σt to be positive definite using the log-Cholesky transformation. Collecting
all the time varying parameters in the vector ft we specialize their score driven law of motion
as
ft+1 = ft + Ωst
We restrict the matrix Ω to be diagonal and to depend on two constants, κs and κv , where
the former drives the amount of time variation in the factor loadings and in the AR coefficients,
19
while κv is a smoothing parameter that governs the time variation of the model volatilities.
Given that the model dynamics depend only on these constants and that irregular behaviour of
the financial variables over the considered sample makes likelihood estimation sensitive to the
starting parameter values, we estimate the model via dynamic model selection and dynamic
model averaging, see the Appendix for a discussion.
We start from a description of the full sample parameter estimates and of the estimated
business cycle and financial factors.
In Figures 10 and 11 we report a time series plot of the weights assigned by the Dynamic
Model Averaging algorithm and the indicator of the best model as selected by Dynamic Model
Selection. It can be noticed that the model achieves model sparsity, in the sense that, after a
necessary period of learning, it assigns nontrivial weights only to a limited number of models.
Figure 11, on the other hand, reveals that the highest weights are typically assigned to models
in the upper part of grid, where time variation is relatively more pronounced.
Figure 12 shows the estimated factors. For comparison, we also report the ECB-CISS,
rescaled to have zero mean and unit variance. Two observations are in order. First, the
business cycle factor captures the main features of the euro-area business cycle, namely the
two expansion phases before the Great Recession, as well as the double dip downturn generated
by the Great Recession first and by the Sovereign Debt Crisis. The estimated financial factor,
our measure of financial stress, also captures correctly the two period of financial stress after
the Lehman and the Greek crises. Unlike the CISS, however, it also signals the build up of
financial vulnerabilities in the early part of the sample.
Next, in Figures 13 to 14 we show the time varying elements of the covariance matrix of
the unobserved states, i.e. the volatilities of the business cycle and of the financial factor, as
well as their time varying covariance. The model detects significant shifts in the volatilities in
correspondence with the two recent recessions, although, financial stress volatility is quite high
already at the beginning of the 2000s. The covariance between the disturbances of these two
factors, rather stable for most of the sample, turns significantly negative after 2007, a period
of both severe financial stress and real economy recession. The volatilities of the measurement
equations, shown in Figure 16, similarly display significant spikes towards the end of the sample.
Finally, in Figures 17 and 18 we show the time varying factor loadings. Although some variation
can be noticed in these coefficients, especially when DMS is used, it seems the most of the time
variation in the whole model originates from the variances rather than from the factor loadings.
Finally, in Figure 19 we report the results of a forecast exercise in which we predict GDP
1 to 18 months ahead and contrast the results of our model with that of two simple univariate
benchmarks, namely an AR and a random walk model. The figure shows the evolution of the
RMSE relative to that of the AR model for the various forecast horizons of the model selected
with DMA, DMS or the simple median model. Notice that the RMSE of our factor models
decline as the forecast horizon shortens, meaning that they use new information efficiently.
Also the factor model always outperforms the AR model at short-medium horizons (between
1 and 10 months ahead).
20
5
Conclusions
In this paper we develop a score-driven approach to estimate state-space models with timevarying parameters. By letting the dynamics of the parameters be driven by the score of
the predictive likelihood we show that the model retains many desirable properties, namely it
can be analyzed using an extended KF procedure. We derive the new set of recursions that,
running in parallel with the standard KF , allows us estimate simultaneously the unobserved
state vector and the time-varying parameters. Given that a variety of time series models
have a state space representation, the proposed methodology has to be considered
of wide interest in econometrics and applied macroeconomics.
After presenting the main theorems we discuss two extensions of the theoretical framework
that could be particularly valuable in applied work. First, since in some applications the
researcher would like to impose parameters restrictions, such as stationarity and non-negative
variances, we show how to incorporate general restrictions on the model parameters. Second,
we illustrate how the model can be extended to deal with data at mixed frequencies and
missing observations. We conclude the theoretical part of the paper by briefly discussing how
our estimation strategy is sufficiently general to nest existing adaptive algorithms used in the
literature as a special case. Given that a wide range of time series models have a state
space representation, the estimation method we propose is of interest for a large
number of applications in econometrics and applied macroeconmics.
Specifically, we illustrate the potential of the method through two empirical applications.
In the first exercise we focus on business cycle measurement. In particular, we take as a
starting point the model that Aruoba et al. (2016) have developed to estimate GDP on the
basis of underlying (noisy) measures and extend it to account for time variation in the model
parameters. Our model delivers a real time and time-varying assessment (i) of the long-run
economic growth (ii) of macroeconomic volatility and (iii) of the relative importance of GDPI
and GDPE . In a second application we extract from a panel of business cycle and financial
variables an indicator of financial stress for the euro area. Our modeling approach allows us
to address three related challenges in the construction of measures of financial stress. The
first relates to model instability. The second issue that we address is endogeneity and we
place appropriate restrictions in the State-Space representation of our model that allow us
to extract a financial component that Granger-causes the business cycle. Third, since our
estimation method handles both missing data and mixed frequencies we can (i) deal with the
fact that some financial indicators are very short and have irregular patterns (ii) model montly
data together with quarterly GDP. We assess the performance of our model in a nowcasting
exercise in which financial variables are used to complement standard business cycle indicators
in monitoring GDP developments in real time. The model detects significant shifts in the
parameters and delivers GDP forecasts that are on average more accurate than those obtained
on the basis of simple univariate benchmarks.
21
Figure 1: GDP(E) vs. GDP(I)
22
Figure 2: Time varying intercept
Figure 3: Time varying AR coefficients
23
Figure 4: Long-run growth
Figure 5: Variance of the error of the common component
24
Figure 6: Relative Kalman Filter Gain
Figure 7: Estimated Common Factor, GDPE and GDPI
25
Figure 8: Zooming in the 1970s recession
Figure 9: Zooming in the Great Recession
26
Figure 10: Model weights (DMA)
Model weights
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2010
2013
Figure 11: Best Model (DMS)
Best model (Dynamic Model Selection)
70
60
50
40
30
20
10
0
1986
1989
1992
1995
1998
2001
27
2004
2007
Figure 12: Estimated Factors
Estimated Business Cycle and Financial factors
4
3
2
1
0
−1
−2
−3
−4
−5
0
DMS−FinCycle
DMA−FinCycle
median model−FinCycle
DMS−BusCycle
DMA−BuCycle
median model−BusCycle
CISS
GDP
20
40
60
80
100
120
140
160
180
200
Figure 13: Business Cycle factor volatility
Bus−cycle volatility
0.4
DMS
median model
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1986
1989
1992
1995
1998
2001
28
2004
2007
2010
2013
Figure 14: Financial factor volatility
Fin−factor volatility
0.7
DMS
median model
0.6
0.5
0.4
0.3
0.2
0.1
0
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
Figure 15: Covariance between financial and business cycle factors
Covariance between financial and business cycle factors
0.05
DMS
median model
0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
1986
1989
1992
1995
1998
2001
29
2004
2007
2010
2013
Figure 16: Measurement equation volatilities
0.8
0.08
0.6
0.4
0.06
2.5
3
0.3
2
2
1.5
0.4
0.04
0
0.02
0
1996 2006
2.5
0.1
1996 2006
3
2
0
1996 2006
1.5
1
1
0
1.5
1996 2006
1.4
1.2
1
0
1996 2006
2
0.8
1.5
0.6
1
0.4
0.5
0.2
1996 2006
0
1996 2006
0
2
4
1
1.5
3
0.8
1
2
0.6
0.5
1
0.4
1996 2006
1996 2006
0.5
0.6
0.4
0
1
0.8
1996 2006
1996 2006
1.5
1
0.5
0
0.5
0.5
1996 2006
0
1
1.5
0.5
0
1996 2006
0.5
2
1
0
0
1
1
0.5
2.5
2
1.5
1.5
0.2
1
0.2
2.5
2
1996 2006
0
1996 2006
0
1996 2006
0.2
1996 2006
0
1996 2006
Figure 17: Financial Cycle factor loadings
Real−loadings
4
2
0
1986
DMS
median model
1989
1992
1995
1998
2001
2004
2007
2010
2013
2007
2010
2013
2007
2010
2013
Real−loadings
4
2
0
1986
DMS
median model
1989
1992
1995
1998
2001
2004
Real−loadings
4
2
0
1986
DMS
median model
1989
1992
1995
1998
2001
30
2004
Figure 18: Financial Cycle factor loadings
2
2
2
4
1
2
0
0
0
−2
−1
1994 2004 2014
0
1994 2004 2014
−2
5
2
2
0
0
0
1994 2004 2014
−2
1994 2004 2014
4
2
0
−5
−2
1994 2004 2014
2
1994 2004 2014
2
−2
1994 2004 2014
−2
2
2
0
0
1994 2004 2014
0
0
−2
−2
−4
1994 2004 2014
5
2
0
0
−5
−2
1994 2004 2014
1994 2004 2014
−2
1994 2004 2014
−2
1994 2004 2014
1994 2004 2014
Figure 19: Root Mean Square Forecast Errors
RMSE relative to AR model
1
0.9
0.8
0.7
0.6
0.5
0.4
19
AR
RW
DMS model
DMA model
Median model
17
15
13
11
9
Months to GDP release
31
7
5
3
1
A
Appendix
We follow the notation and the main results on the matrix differential calculus in Abadir
and Magnus (2005, ch 13)
A.1
Proof of theorem 1
The gradient vector is
0
0
∂`t
1 ∂ log |Ft | ∂vt0 Ft−1 vt
=−
+
∂ft0
2
∂ft0
∂ft0
0
1 1
∂|Ft | ∂vec(Ft ) ∂vt0 Ft−1 vt ∂vt
∂vt0 Ft−1 vt ∂vec(Ft−1 ) ∂vec(Ft )
−
+
+
2 |Ft | ∂vec(Ft )0 ∂ft0
∂vt0
∂ft0 ∂vec(Ft−1 )0 ∂vec(Ft )0
∂ft0
•
•
•
1
− [vec(Ft−1 )0 F t + 2vt0 Ft−1 V t − (vt0 ⊗ vt0 )(Ft−1 ⊗ Ft−1 )F t ]0
2
•0
•0
1 •0
−1
−1
−1
−1
− F t vec(Ft ) − F t (Ft ⊗ Ft )(vt ⊗ vt ) + 2V t Ft vt
2
0
•0
•0
1 •
−1
−1
−1
−1
−1
F t (Ft ⊗ Ft )(vt ⊗ vt ) − F t vec(Ft Ft Ft ) − 2V t Ft vt
2
0
•0
•0
1 •
−1
−1
−1
−1
−1
F t (Ft ⊗ Ft )(vt ⊗ vt ) − F t (Ft ⊗ Ft )vec(Ft ) − 2V t Ft vt
2
•0
1 • 0 −1
−1
−1
F t (Ft ⊗ Ft )[vt ⊗ vt − vec(Ft )] − 2V t Ft vt .
(30)
2
∇t =
=
=
=
=
=
=
We compute the information matrix as the expected value of the Hessian
It = −Et
∂ 2 `t
.
∂ft ∂ft0
(31)
Let re-write the Gradient (30) as follows
∇t =
=
=
=
=
=
•0
1 • 0 −1
−1
0
−1
− F t (Ft ⊗ Ft )[vec(Ft ) − vec(vt vt )] + 2V t Ft vt
2
•0
1 •0
−1
−1
−1
−1
0
−1
− F t [(Ft ⊗ Ft )vec(Ft ) − (Ft ⊗ Ft )vec(vt vt )] − 2V t Ft vt
2
•0
1 •0
0 −1
−1
−1
−1
− F t [vec(Ft ) − vec(Ft vt vt Ft )] + 2V t Ft vt
2
•0
1 •0
−1
−1
−1
0 −1
− F t vec(Ft − Ft vt vt Ft ) + 2V t Ft vt
2
•0
1 •0
−1
0 −1
−1
− F t vec{Ft (I − vt vt Ft )} + 2V t Ft vt
2
•0
1 •0
−1
−1
0 −1
− F t (I ⊗ Ft )vec(I − vt vt Ft ) + 2V t Ft vt .
2
32
(32)
The negative Hessian is equal to
∂∇t
1
∂ 2 `t
=− 0 =
−
0
∂ft ∂ft
∂ft
2
|
•0
∂ F t (I ⊗
Ft−1 )vec(I
−
vt vt0 Ft−1 )
0
•
−1
∂ V t F t vt
+
∂f 0
{z t
}
Φ1t
.
∂ft0
{z
|
(33)
}
Φ2t
We start by computing the first term of (33)
0
•
∂ F t (I⊗Ft−1 )vec(I−vt vt0 Ft−1 )
•0
∂vec(F t )
•0
∂ft0
0
∂vec(F t )
0
•
∂ F t (I⊗Ft−1 )vec(I−vt vt0 Ft−1 )
∂vec(I⊗Ft−1 ) ∂vec(Ft−1 ) ∂vec(Ft )
1
+2
−1 0
−1 0
0
∂ft0
∂vec(I⊗F
)
t
∂vec(Ft ) ∂vec(Ft )
1
2
Φ1t =
•0
∂ F t (I⊗Ft−1 )vec(I−vt vt0 Ft−1 )
∂vec(I−v v 0 F −1 )0
t t t
+ 21
∂ft0
∂vec(I−vt vt0 Ft−1 )0
#
−1
0 −1 0
−
v
= 12 vec(I
t vt Ft ) (I ⊗ Ft ) ⊗ I F t
•0
•
∂vec(I⊗F −1 )
−1
− 12 vec(I − vt vt0 Ft )0 ⊗ F t ∂vec(F −1t )0 (Ft−1 ⊗ Ft−1 )F t
(34)
t
∂vec(vt vt0 Ft−1 )
− 21 F t (I ⊗ Ft−1 )
∂ft0
|
{z
}
•0
Γt
•0
#
where F t =
∂vec(F t )
∂ft0
Γt =
and
∂vec(vt vt0 Ft−1 ) ∂vec(vt vt0 ) ∂vt
∂vec(vt vt0 )0
∂vt0
∂ft0
Ft−1 ⊗ I (vt ⊗ I + I ⊗
=
∂vec(vt vt0 Ft−1 ) ∂vec(Ft−1 ) ∂vec(Ft )
∂ft0
∂vec(Ft−1 )0 ∂vec(Ft )0
•
•
vt ) V t − (I ⊗ vt vt0 ) (Ft−1 ⊗ Ft−1 )F t .
+
(35)
Putting together (34) and (35)we obtain
#
−1
0 −1 0
vec(I
−
v
t vt Ft ) (I ⊗ Ft ) ⊗ I F t
•0
•
∂vec(I⊗F −1 )
1
0 −1 0
= − 2 vec(I − vt vt Ft ) ⊗ F t ∂vec(F −1t )0 (Ft−1 ⊗ Ft−1 )F t
Φ1t =
1
2
(36)
t
•0
− 21 F t (Ft−1 ⊗ Ft−1 ) (vt ⊗ I + I ⊗
•0
•
+ 12 F t Ft−1 ⊗ Ft−1 vt vt0 Ft−1 F t .
•
vt ) V t
The second term of (33) is equal to
Φ2t =
0
•
∂ V t Ft−1 vt
•0
∂vec(V t )0
•0
∂vec(V t )
∂ft0
#
+
0
•
∂ V t Ft−1 vt
∂vec(Ft−1 )
•
∂vec(Ft−1 ) ∂vec(Ft )
∂vec(Ft )0
∂ft0
•
+
•0
0
•
∂ V t Ft−1 vt
∂vt0
•
= (vt0 Ft−1 ⊗ I)V t − (vt ⊗ V t )0 (Ft−1 ⊗ Ft−1 )F t + V t Ft−1 V t ,
33
∂vt
∂ft0
(37)
•0
#
where V t =
∂vec(V t )
.
∂ft0
Putting together (36) and (37) we obtain the following expression for (33)
2
∂ `t
=
− ∂f
0
tf
t
#
vec(I − vt vt0 Ft−1 )0 (I ⊗ Ft−1 ) ⊗ I F t
•0
•
∂vec(I⊗F −1 )
1
0 −1 0
− 2 vec(I − vt vt Ft ) ⊗ F t ∂vec(F −1t )0 (Ft−1 ⊗ Ft−1 )F t
1
2
t
•0
•
− 21 F t (Ft−1 ⊗ Ft−1 ) (vt ⊗ I + I ⊗ vt ) V t
•0
•
+ 12 F t Ft−1 ⊗ Ft−1 vt vt0 Ft−1 F t
(38)
#
+(vt0 Ft−1 ⊗ I)V t
•
•
−(vt ⊗ V t )0 (Ft−1 ⊗ Ft−1 )F t
•0
•
+V t Ft−1 V t .
Following Harvey (1989, p.141), taking the conditional expectation of (??)the only random
#
•
is the prediction error vt and its first and second derivatives are V t and V t . However, such
derivatives are fixed given the conditional expectation at time t − 1. Moreover, we have that
Et (vt ) = 0, Et (vt vt0 ) = Ft and Et [vec(I − vt vt0 Ft−1 )] = 0. Therefore, applying the expectations
Et (.), the fourth and the seventh term in (38) are the only nonzero elements and the information
matrix is equal to
•0
•
•
1 •0
It = F t Ft−1 ⊗ Ft−1 F t + V t Ft−1 V t .
(39)
2
A.2
Proof of theorem 2
Given the model-specific Jacobian matrices
•
Zt =
N m×k
∂vec(Zt )
,
∂ft0
•
Ht =
N 2 ×k
•
∂vec(Ht ) •
∂vec(Tt )
∂vec(Qt )
, Tt =
and Qt =
.
0
0
∂ft
∂ft
∂ft0
m2 ×k
m2 ×k
We show how compute the following Jacobian matrices
•
Vt
=
∂vt
∂ft0
=
∂Zt at
∂fht0
N ×k
∂Zt at ∂vec(Zt )
t at ∂at
+ ∂Z
∂vec(Zt )0
∂ft0
∂a0t ∂ft0
•
•
−[(a0t ⊗ IN )Z t + Zt At ],
= −
=
•
Ft
=
∂vec(Ft )
∂ft0
=
∂vec(Ft ) ∂vec(Zt Pt Zt0 ) ∂vec(Zt )
∂ft0
∂vec(Zt )0
∂ft0
N 2 ×k
+
•
∂vec(Zt Pt Zt0 ) ∂vec(Pt )
+
∂vec(Pt )0
∂ft0
•
•
= (IN 2 + CN 2 )(Zt Pt ⊗ IN )Z t + (Zt ⊗ Zt )P t + H t
•
•
•
= 2NN (Zt Pt ⊗ IN )Z t + (Zt ⊗ Zt )P t + H t
34
(40)
i
•
Ht
(41)
•
Kt
=
mN ×k
=
=
=
∂vec(Kt )
∂ft0
∂vec(Tt Pt Zt0 Ft−1 ) ∂vec(Tt )
∂vec(Tt Pt Zt0 Ft−1 ) ∂vec(Pt )
+
∂vec(Tt )0
∂ft0
∂vec(Pt )0
∂ft0
∂vec(Tt Pt Zt0 Ft−1 ) ∂vec(Zt0 ) ∂vec(Zt )
∂vec(Tt Pt Zt0 Ft−1 ) ∂vec(Ft−1 ) ∂vec(Ft )
+ ∂vec(Z 0 )0
+ ∂vec(F −1 )0 ∂vec(Ft )0 ∂f 0
∂vec(Zt )0
∂ft0
t
t
t
•
•
•
−1
−1
−1
(Ft Zt Pt ⊗ Im )T t + (Ft Zt ⊗ Tt )P t + (Ft ⊗ Tt Pt )CN m Z t −
•
•
•
(Ft−1 Zt Pt ⊗ Im )T t + (Ft−1 Zt ⊗ Tt )P t + (Ft−1 ⊗ Tt Pt )CN m Z t −
•
(IN ⊗ Tt Pt Zt0 )(Ft−1 ⊗ Ft−1 )F t
•
(Ft−1 ⊗ Kt )F t .
(42)
•
At
m×k
=
∂at
∂ft0
=
∂Tt at ∂vec(Tt )
∂vec(Tt )0
∂ft0
•
(a0t ⊗ Im )T t +
=
∂Tt at ∂at
∂a0t ∂ft0
•
Tt At + (vt0
+
+
∂Kt vt ∂vec(Kt )
∂vec(Kt )0
∂ft0
•
•
+
(43)
∂Kt vt ∂vt
∂vt0 ∂ft0
⊗ Im )K t + Kt V t
,
•
Pt
=
∂vec(Pt )
∂ft0
=
∂vec(Tt Pt Tt0 ) ∂vec(Tt )
∂vec(Tt )0
∂ft0
m2 ×k
+
∂vec(Tt Pt Tt0 ) ∂vec(Pt )
∂vec(Kt Ft Kt0 ) ∂vec(Kt )
− ∂vec(K
0
∂vec(Pt )0
∂ft0
∂ft0
t)
•
•
−
∂vec(Kt Ft Kt0 ) ∂vec(Ft )
∂vec(Ft )0
∂ft0
•
•
+ Qt
•
•
= (Im2 + Cm )(Tt Pt ⊗ Im )T t + (Tt ⊗ Tt )P t − (Im2 + Cm )(Kt Ft ⊗ Im )K t − (Kt ⊗ Kt )F t + Qt
•
•
•
•
•
= 2Nm [(Tt Pt ⊗ Im )T t − (Kt Ft ⊗ Im )K t ] + (Tt ⊗ Tt )P t − (Kt ⊗ Kt )F t + Qt
(44)
A.3
Dynamic model averaging and selection
Estimation via model averaging and selection proceeds as follows. We specify a grid of
values for the parameters λs and λv . Each point in this grid defines a new model. Weights for
each model j (defined πt|t−1,j ) are obtained as a function of the predictive density at time t − 1
through the following recursions:
α
πt−1|t−1,j
πt|t−1,j = PJ
α
l=1 πt−1|t−1,l
πt|t−1,j pj (yt |y t−1 )
πt|t,j = PJ
l=1
πt|t−1,l pl (yt |y t−1 )
(45)
(46)
where pj (yt |y t−1 ) is the predictive likelihood. Since this is a function of the prediction errors
and of the prediction errors variance, which are part of the output of the KF, the model weights
can be computed at no cost along with the model parameters estimation. Note that here a new
forgetting factor appears, α, which discounts past predictive likelihoods. We set this parameter
to 0.9. At each point in time, forecast are obtained on the basis of the model with the highest
weight πt|t−1,j or by averaging on the basis of the model weights, see also Koop and Korobilis
(2013).
35
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