Les articles publiés dans la série "Économie et statistiques" n'engagent que leurs auteurs. Ils ne reflètent pas forcément les vues du STATEC et n'engagent en rien sa responsabilité. N N ° 1 1 0 2 2 A Août 2 2018 A Auteur: Christian Glocker, Austrian Institute of Economic Research – Department for Macroeconomics (WIFO) A state space approach to forecasting short-term dynamics in Luxembourg Abstract This paper presents small-scale dynamic factor models in order to compute short-term forecasts for the Luxembourgish economy. Particular models are designed for each of the following variables (i) real goods exports, (ii) real private household consumption, and (iii) employment, commuters and unemployment, where the latter three variables are considered jointly within one model. The models are estimated using the Kalman filter, which in turn allows for a straightforward application of the dynamic factor models for nowcasting, backcasting and forecasting alike. To examine the real- time forecasting accuracy, a pseudo real-time analysis has been applied; the results highlight the superior forecasting performance of the small scale factor models to various alternatives, including experts' forecasts.
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Les articles publiés dans la série "Économie et statistiques" n'engagent que leurs auteurs. Ils ne reflètent pas forcément lesvues du STATEC et n'engagent en rien sa responsabilité.
NN° 11022 AAoût 22018
AAuteur: Christian Glocker, Austrian Institute of Economic Research –Department for Macroeconomics (WIFO)
A state space approach to forecasting short-term dynamics in Luxembourg
Abstract
This paper presents small-scale dynamic factor models in order to compute short-term forecasts for the Luxembourgish economy. Particular models are designed for each of the following variables (i) real goods exports, (ii) real private household consumption, and (iii) employment, commuters and unemployment, where the latter three variables are considered jointly within one model. The models are estimated using the Kalman filter, which in turn allows for a straightforward application of the dynamic factor models for nowcasting, backcasting and forecasting alike. To examine the real-time forecasting accuracy, a pseudo real-time analysis has been applied; the results highlight the superior forecasting performance of the small scale factor models to various alternatives, including experts' forecasts.
dynamics in Luxembourg
Abstract
This paper presents small-scale dynamic factor models in order to com-
pute short-term forecasts for the Luxembourgish economy. Particular models
are designed for each of the following variables (i) real goods exports, (ii) real
private household consumption, and (iii) employment, commuters and unem-
ployment, where the latter three variables are considered jointly within one
model. The models are estimated using the Kalman filter, which in turn allows
for a straightforward application of the dynamic factor models for nowcasting,
backcasting and forecasting alike. To examine the real-time forecasting accu-
racy, a pseudo real-time analysis has been applied; the results highlight the
superior forecasting performance of the small scale factor models to various
alternatives, including experts’ forecasts.
1 Introduction
Early assessment of the current economic conditions is of importance for economic
agents’ decision making and economic policy alike. The lack of timely information
associated with the publication of macroeconomic variables, the presence of missing
values in historical time series, and the problem surrounding mixed frequencies im-
pede the day to day monitoring of the economic activity. This applies also when
econometric models are used, which are estimated relying on traditional methods.
*Project Report written by Christian Glocker, Spring 2018. Thanks to Ferdy Adam and histeam, Lionel Fontagne, and Massimiliano Marcellino for support. Les opinions exprimees dans lapresente publication sont celles des auteurs et ne refletent pas forceement les opinions du STATECet de l’ANEC.
1
A state space approach to forecasting short-term
In this context the main challenges within model-based projections consists of
finding an appropriate statistical framework which allows to (1) analyze jointly data
of different sampling frequencies (i.e. monthly business cycle indicators and quarterly
National Account figures), (2) to deal with missing observations at various instances
of the time-series used, and (3) to take account of data revisions. The problem of
mixed frequencies could be solved in a straightforward fashion by aggregating all
monthly series to quarterly ones. This would allow standard estimation techniques,
though this approach is associated with a considerable loss of information as the
dynamics within a quarter are left un-modeled. Furthermore, this approach does
not solve the problem of how the latest available information from monthly business
cycle indicators could be used if observations are available only until the first or
second month within a quarter. Hence the method should allow for a quick update
of any forecast to incorporate new information on the highest possible frequency.
New information can also be in the form of data revisions. As this can at times be
useful, it could also lead to significant changes in the forecast trajectory rendering
the projections too sensitive to data revisions; this constitutes a key drawback of
auto-regressive models in forecasting time series as their forecasts are highly sensitive
to data revisions. Against this background, an adequate model should be able to
produce forecasts which are as far as possible invariant to data revisions.
Finally, as concerns the time span that the data series cover - sophisticated tech-
niques require a decent number of observations; series too short render the estimation
unfeasible and the results unreliable. This poses a particular challenge for variables
whose statistical data definitions is altered regularly. This problem applies to many
macroeconomic time series involving national accounts data, labour market data as
well as financial data, of which credit stock data in particular. Against this back-
ground the statistical method applied should be able to allow for the possibility of
missing observations in general.
Once having specified an appropriate statistical model which is capable of dealing
with all these aforementioned problems, the next challenge arises - forecasting. Per-
forming multi-period forecasts requires to decide whether to use a recursive model
based upon one-step-ahead forecasts, or a multi-period model that is estimated with
a loss function tailored to the forecast horizon. Although the recursive method pro-
vides more efficient parameter estimates than the direct method and does not require
different models for different forecasting horizons, it is prone to distortion if the
2
one-step-ahead model is incorrectly specified and usually requires separate forecast-
ing models for the explanatory variables. Which approach is better depends on the
properties of the forecast model and will ultimately be an empirical matter.
In this setup, one convincing approach is the VAR framework with mixed frequen-
cies. However, the calculation of the predictions from this approach usually suffers
from serious dimensionality related problems, especially when the number of time
series or their frequency increases. One possibility to account for this is by relying
on Bayesian methods. The parameter proliferation problem can also be addressed by
means of the well-known dimensionality reduction allowed for by factor models. Since
macroeconomic data are usually collinear, it is reasonable to assume that these are
multiple, indirect measurements of some low-dimensional sources that can be used
to reproduce most of the variability of a data set, although they cannot normally be
measured directly. Factor models in turn allow to calculate indices of macroeconomic
activity, which can be useful in tracking economic developments.
Against this background the dynamic factor model as proposed by Stock and Wat-
son (1992) and extended by Camacho and Perez-Quiros (2010) comprises a promis-
ing econometric approach. The basic idea is to separate a possibly large number of
observable variables into two independent and unobservable, yet estimable, compo-
nents: a common component that captures the main bulk of co-movement between
the observable variables, and an idiosyncratic component that captures any remain-
ing individual movement. The common component is assumed to be driven by a
few common factors, thereby reducing the dimension of the system. In this context
the models are converted into state space representations and estimated using the
Kalman filter. Since the estimation of a model by means of the Kalman filter is re-
cursive, the approach is able to take missing observations in the data set into account
in a relatively straightforward manner. The strategy is to skip some calculations
while others do not need to be changed so that the basic Kalman filter remains valid
and the parameters of the model can be estimated with maximum likelihood meth-
ods. This characteristic is of practical relevance when calculating forecasts, since
the future values of the time series can be considered as (yet) missing observations.
Consequently, the Kalman filter also provides the necessary calculations for forecasts.
Moreover, the estimation of a model by means of the Kalman filter occurs via two
steps which are repeated consecutively; one of these steps involves the computation
of forecasts. Hence the application of the Kalman filter to estimate models which are
3
in turn used for forecasting purposes is a natural extension of the Kalman filtering
technique. Considering the population parameters, the Kalman filter also provides
the mean square prediction error (MSFE).
In what follows this econometric methodology will be applied to key variables of
different sectors of the Luxembourgish economy. Section 2 introduces basic elements
of the dynamic factor model and discusses aspects relevant for the current application.
Section 3 establishes a dynamic factor model for real goods exports and Section 4
develops a model for private household consumption. Section 5 establishes a dynamic
factor model for key labour market variables; the analysis draws particular attention
to (i) employment, (ii) unemployment and (iii) commuters. Each section involves a
discussion on the precision of the forecasts of the dynamic factor model relative to
various alternative models as well as the forecasts of an expert panel. Moreover, the
analysis also features a discussion on the extent to which new information changes
the forecast trajectory. Finally Section 6 concludes.
2 Dynamic factor models
The basic idea of dynamic factor models (DFM) is that the information of an ob-
servable vector of time series under investigation can be explained by a vector of
unobserved components with the requirement that the vector of unobserved com-
ponents has a lower dimension than the vector of time series under investigation.
By this, dynamic factor models capture the most important co-movements of the
variables in the vector of observed time series.
Dynamic factor models are motivated by theory, which predicts that macroeco-
nomic shocks should be pervasive and affect most variables within an economic sys-
tem. They have therefore become popular among macroeconometricians (see, e.g.,
Breitung and Eickmeier 2006, for an overview). In particular, it has been demon-
strated that dynamic factor models are valuable in business cycle analysis (e.g. Forni
and Reichlin 1998; Eickmeier 2007; Ritschl, Sarferaz, and Uebele 2016), forecasting
(e.g. Stock and Watson 2002a,b) and nowcasting the state of the economy, that is,
forecasting of the very recent past, the present, or the very near future of indicators
for economic activity (see, e.g., Banbura, Giannone, Modugno, and Reichlin 2012,
and references therein).
Let yt =((xh
1,t)′, (xs
1,t)′, y1stt , yft
)′be a vector of N distinct time series available
4
at time t and It = {y1, ...,yt} be the information set including all information up
to and including time t. Then a dynamic factor model is usually specified such
that all observable variables in yt can be represented as the sum of two independent
components: a common component ft which is common to all variables in yt and
the remaining idiosyncratic component et (see for instance Bai and Ng, 2008). These
idiosyncratic disturbances arise from measurement error and from special features
that are specific to an individual series. The latent factors follow a time series process,
which is commonly taken to be a vector autoregression (VAR). In equations, the
dynamic factor model is,
• System of static equations(xht
xst
)=
(γh · ft
γs ·∑11
j=0 ft−j
)+
(uh
t
ust
)(1)(
y1stt
yft
)=
(ω
ω
)· [γqft + ut,q] +
(0
εt
)(2)
with ω := 13+ 2
3· L+ L2 + 2
3· L3 + 1
3· L4
• System of dynamic equations
(1− ϕε(L))εt = με + et, et ∼ NID(0, σ2e) (3)
φu(L) · ut,q = νqt with νq
t ∼ NID(0, σ2
q
)(4)
φf (L) · ft = νft (5)
Φu(L)
(uh
t
ust
)= νt (6)(
νft
νt
)∼ NID
(0,
[σ2f 0
0 Σν
])(7)
where xht is a vector of nh hard indicators on a monthly frequency, xs
t is a vector of
ns soft indicators on a monthly frequency, y1stt and yft typically a national account
variable involving its first release (y1stt ) and final value (yft ) with ε being the revision
term. uht and us
t are some residuals, allowed in turn to follow an autoregressive
process, and finally - the key feature of the dynamic factor model - is to assume a
5
dynamic equation for the factor ft which is done by equation (5). The idiosyncratic
disturbances are assumed to be uncorrelated with the factor innovations at all leads
and lags, that is, Eνtuit+k = 0 ∀k ∈ Z and i ∈ {s, h}.
In this set-up, y1stt and yft are usually specified as quarter-on-quarter (q-o-q)
growth rate. The term ω hence decomposes the q-o-q to a month-to-month growth
rate and relates it to the variables in xht which are specified in terms of month-
to-month growth rates. Since soft indicators typically have a high correlation with
the year-on-year growth rates of the variables in xht , y
1stt and yft , the second line in
equation (1) hence relates the soft indicators to the approximate year-on-year growth
rate of the factor ft. This model set-up has been frequently used in the literature
(REF) and has become a standard for short-term forecasting exercises.
Estimation of dynamic factor models concern foremost the common component.
The idiosyncratic component is generally considered as a residual. The common com-
ponent of the dynamic factor model may be consistently estimated in the frequency
domain by spectral analysis; see Forni, Hallin, Lippi, and Reichlin (2000, 2004). The
main benefit of the factor model, is that the common component may be consistently
estimated in the time domain, which reduces the computational complexity. The
workhorse for estimating the factor is the method of principal components (PC). In
a second step, the dynamic equation (5) can be estimated by standard ordinary-
least-squares method treating the estimated factors in ft as observed variables. This
method is easy to compute, and is consistent under quite general assumptions as long
as both the cross-section and time dimension grow large. It suffers, however, from
a large drawback: the data set must be balanced, where the start and end points of
the sample are the same across all observable variables. In practice, data are often
released at different dates. A popular approach is therefore to cast the dynamic factor
model in a state space representation and then estimate it using the Kalman filter,
which allows unbalanced data sets and offers the possibility to smooth missing val-
ues. The state space representation contains a signal equation, which links observed
variables to latent states, and a state equation, which describes how the states evolve
over time. The Kalman filter and smoother provide mean-square optimal projections
for both the signal and state variables. In what follows the dynamic factor model will
be cast into the structure of a state space model where the measurement equation
6
and the transition equation read as follows:
yt = Hst +wt, wt ∼ NID (0, R) (8)
st = Fst−1 + vt, vt ∼ NID (0, Q) (9)
with a corresponding definition of the matrizes H, F , R, Q, the vectors yt, st, wt
and vt and their relation to the equation system (1)-(7).
If all series in the model were observable at a monthly frequency and the data panel
was balanced, then the estimation of the dynamic factor model could be implemented
using standard maximum likelihood methods in conjunction with the Kalman filter.
This assumption is, however, rather unrealistic, since in our empirical application
we have to deal with mixed frequencies and with time series which are published at
different time lags; moreover the series start at different points in time.
According to Mariano and Murasawa (2003), with the subtle transformation of
replacing missing observations by random draws rt ∼ iidN(0, σ2r), the system of
equations remains valid. Importantly, if the distribution of rt does not depend on
the parameter space that characterizes the Kalman filter, then the matrices in the
state-space representation are conformable and do not have an impact on the model
estimation since the missing observations just add a constant term in the likelihood
function to be estimated.
Let yi,t be the ith element of vector yt and let Rii be its variance. Let Hi,t be the
ith row of matrix Ht, which has z columns. The measurement equation can then be
replaced by the following expressions
y∗i,t =
⎧⎨⎩yi,t if yi,t is observable
rt otherwise,(10)
H∗i,t =
⎧⎨⎩Hi if yi,t is observable
01×z otherwise,(11)
w∗i,t =
⎧⎨⎩0 if yi,t is observable
rt otherwise,(12)
7
R∗ii,t =
⎧⎨⎩0 if yi,t is observable
σ2r otherwise.
(13)
With this transformation the time-varying state-space model can be treated as having
no missing observations and the Kalman filter can be directly applied to y∗t , H
∗t , w
∗t
and R∗t . The implementation of this algorithm corresponds to expanding yt, H, wt
and R in equation (8) by means of an indicator function which takes into account if
yi,t ∈ yt is observed or not.
The estimation of the model’s parameters can be developed by maximizing the
log-likelihood of {y∗t }t=T
t=1 numerically with respect to the unknown parameters in
matrices. Let st|t−1 be the estimate of st based on information up to period t − 1.
Let Pt|t−1 be its covariance matrix. The prediction equations are:
st|t−1 = F st−1|t−1, (14)
Pt|t−1 = FPt−1|t−1F′ +Q. (15)
The predicted value of yt with information up to t − 1, denoted yt|t−1 is yt|t−1 =
H∗t st|t−1, such that the prediction error is ηt|t−1 = y∗
t − yt|t−1 = y∗t − H∗
t st|t−1 with
covariance matrix ξt|t−1 = H∗t Pt|t−1H
∗t +R∗
t . In each iteration, the log-likelihood can
therefore be computed as
logLt|t−1 = −1
2ln(2π∣∣ξt|t−1
∣∣)− 1
2η′t|t−1
(ξt|t−1
)−1ηt|t−1. (16)
The updating equations are:
st|t = st|t−1 +K∗t ηt|t−1 (17)
Pt|t = Pt|t−1 −K∗t H
∗t Pt|t−1 (18)
in which K∗t is the Kalman gain defined as K∗
t = Pt|t−1H∗′t (ξt|t−1)
−1. The initial
values: s0|0 = 0 and P0|0 = I used to start the filter are a vector of zeros and the
identity matrix, respectively.
Computing short-term forecasts in real-time from this model is straightforward.
The future values of the time series can be regarded as missing observations at the
end of the sample periods. The Kalman filter accounts for the missing data which
8
are replaced by forecasts. Particularly, the k-period ahead forecasts are
yt+k|t = H∗t st+k|t, (19)
with st+k|t = F kst|t.
To conclude, the Maximum-Likelihood estimation of the dynamic factor using
the Kalman filter can smooth over missing values, allowing an unbalanced panel with
missing values at the end or start of the panel; the so called ragged-edge-problem.
This feature is very valuable for economic forecasting, because key economic indica-
tors tend to be released at different dates. For any estimation approach, the number
of factors q is generally unknown, and needs to be either estimated or assumed. Pop-
ular estimators for the number of factors in approximate factor models can be found
in Bai and Ng (2002), Onatski (2010) and Ahn and Horenstein (2013). Throughout
the paper, the number of factors will be treated as known.
An important step in developing an appropriate model addresses the selection of
the variables. This applies not only to the specific variables used, but also on the
time dimension; that is, assume the interest lies in setting up a model in order to
better forecast a core-variable xt and to use an appropriate auxiliary variable yt−k.
The selection along the time-dimension would result in judging the forecast of xt in
relation to k ∈ Z.
The process surrounding the selection of appropriate variables and their time-
structure can hence be endless. For this it is convenient to rely on an efficient strategy
in order to judge if a variable is accepted or not. In what follows, this will involve two
steps. The first step is based on the experience of professional forecasters in order
to gather a pre-selected list of variables. The second step now involves the dynamic
factor model. Once a pre-selected list has been specified, each particular variable is
analyzed and included as an additional variable only if it increases the percentage
of the variance of the core-variable explained by the common factor(s). Within this
step, the time-dimension is considered as an additional search-direction.
2.1 Some practical issues
The model outlined in equations (1)-(7), has been frequently used when the variable
yft has some (significant) positive degree of (first-order) autocorrelation. The perfor-
mance of this model deteriorates significantly if quarterly growth rates of the data in
9
yft are used. This is due to the zero autocorrelation or even negative autocorrelation.
An approach that seems more promising in this case, is to utilize year-on-year (y-o-y)
growth rates instead. Extending the use of y-o-y growth rates also for the variables
in xht yields a factor ft that can be interpreted in terms of y-o-y growth rates. Fi-
nally, since the variables in xst have a high correlation with the y-o-y growth rates of
the variables in xht and yft , the model hence simplifies significantly to the following
structure:
• System of static equations(xht
xst
)=
(γh · ftγs · ft
)+
(uh
t
ust
)(20)(
y1stt
yft
)=
(ω
ω
)· [γqft + ut,q] +
(0
εt
)(21)
with ω = 1
• System of dynamic equations
(1− ϕε(L))εt = με + et, et ∼ NID(0, σ2e) (22)
φu(L) · ut,q = νqt with νq
t ∼ NID(0, σ2
q
)(23)
φf (L) · ft = νft (24)
Φu(L)
(uh
t
ust
)= νt (25)(
νft
νt
)∼ NID
(0,
[σ2f 0
0 Σν
])(26)
The simplification is due to the fact that the state space of the model is reduced.
In what follows, this set-up will be used, modified with minor extensions in order
to adjust the dynamic factor model to specific features of the data. The state space
representation of this model structure is presented in Section A in the Appendix.
The selection of the indicators is based on three steps. First of all, a set of
core variables is chosen utilizing the advice of professional forecasters at Statec; this
comprises in general around thirty variables for each model. This amount of variables
is too large of being used effectively in the model as computational difficulties arise.
10
Hence, as a second step the long list of variables is reduced by searching along two
dimensions: cross-sectional and temporal. Each variable has then been analyzed if
it should be added to the model; if added the analysis also evaluates an appropriate
lag. It has been decided to include an additional variable when it increases the
percentage of variance explained by the common factor. This procedure is applied
for each dynamic factor model as discussed in Sections 3 to 5. In addition, to be
included in the dynamic factor model, all of these series have been normalized to
have zero mean and unit variance.
The key element in judging the adequacy of the proposed models addresses their
forecast performance. In this context Stark and Croushore (2002) among others sug-
gest that the analysis of in-sample forecasting performance of models is questionable
since the result can be deceptively less conclusive when using a real-time data set.
This is due to the fact that the in-sample analysis misses to adequately capture the
following three elements in the analysis: (i) the recursive estimation of the parame-
ters of the model; (ii) the flow of the data in real time (this addresses the problem
that data are released at different points in time); and (iii) revision of the data.
Producing real-time data vintages is, however, not feasible for several variables
within the current application. For this reason the out-of-sample analysis carried out
here follows instead the example of Stock and Watson (2002) in which (i) holds. The
method consists of computing forecasts from successive enlargements of a partition
of the latest available data set. Using this recursive sample structure, the model is
fully estimated and h-step ahead forecasts are computed. This procedure continues
iteratively until the final iteration, which is computed from a model that uses the
complete latest available observation.
Within the iterative enlargement of the data set, the analysis takes the publication
lag of certain variables into account. This procedure is based on trying to mimic as
closely as possible the real time analysis that would have been performed in case a
truly real-time data set were available. The current analysis can hence be considered
as a pseudo real-time analysis.
In what follows this procedure is applied to all factor models developed here in
order to (i) assess the forecast trajectory of the model within periods characterized
by exceptional economic fluctuations, and (ii) to assess the precision of the forecasts
of the dynamic factor model relative to alternative models.
Finally it has to be stressed that the specification of the DFMs should not be
11
understood as a structural model; instead the models focus purely on a Granger-
causality objective.
3 Specifying a DFM for real goods exports
The following describes a dynamic factor model for real goods exports. The particular
model used is given by equations (20)-(26), where (22) is specified as follows:
εt = με + et, et ∼ NID(0, σ2e) (27)
The variables used for the estimation are given by:
xht =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
steel productionyoy,Mt−7
steel prodroduction EUyoy,Mt−3
industrial productionyoy,Mt−2
prices of iron and steel productsyoy,Mt
PPI EAyoy,Mt
GDP EAyoy,Qt
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(28)
xst =
[PMI US manufacturingMt−7
](29)
where yoy indicates that a variable enters in year-on-year growth rates and the su-
perscripts M or Q indicate if the time series is available on a monthly or quarterly
frequency. yft and y1stt represent the growth rate of the final real goods export series
as well as its first estimate respectively. The data are depicted in Figure 1 and fur-
ther information can be found in Table 11; data for the two measures of real goods
export are shown in Figure 2; as can be seen in the figure, the two measures for the
growth rate of real goods exports follow on average a rather similar path, however,
there are indeed several years where the difference is sizable. The model takes this
into account by means of equation (27). A detailed overview of the data used can be
found in Section B in the Appendix.
The variables were selected following the procedure described in Section 2. As can
be seen, the final list of variables contains three indicators which are directly related
to the iron&steel market (steel production in Luxembourg; steel production in the EU
and the (domestic) price of iron and steel products). In the case of Luxembourg this
12
Figure 1: Data used - DFM for real goods exports
-4
-2
0
2
4
1985 1990 1995 2000 2005 2010 2015 2020
-3
-2
-1
0
1
2
3
1985 1990 1995 2000 2005 2010 2015 2020
-6
-4
-2
0
2
1985 1990 1995 2000 2005 2010 2015 2020
-4
0
4
8
12
1985 1990 1995 2000 2005 2010 2015 2020
-4
-2
0
2
4
6
1985 1990 1995 2000 2005 2010 2015 2020
-4
-2
0
2
4
1985 1990 1995 2000 2005 2010 2015 2020
-3
-2
-1
0
1
2
3
1985 1990 1995 2000 2005 2010 2015 2020
-4
-2
0
2
4
1985 1990 1995 2000 2005 2010 2015 2020
-4
-3
-2
-1
0
1
2
1985 1990 1995 2000 2005 2010 2015 2020
-6
-4
-2
0
2
4
1985 1990 1995 2000 2005 2010 2015 2020
13
XB_R_Q XB_FR PMIMANUS_M
PRODSTEELLU_M
PRODSTEELEU_M PRODINDLU_M P_MET_M
P_PPIEA_M
PIBEA_R_Q PRODINDHW_M
Figure 2: Data used for real goods exports
-.3
-.2
-.1
.0
.1
.2
.3
96 98 00 02 04 06 08 10 12 14 16 18 20
seems appropriate, since steel products comprise a major part of real goods exports.
Table 1 shows the estimated coefficients of the complete model jointly with com-
mon further statistical measure to evaluate the parameters’ statistical importance,
and also a measure for the weight of each variable in the model. As can be seen
there is some evidence for a non-zero mean με of the revision term εt. One has to
take into account here that the estimation of the equation of the revision term is
based on a few observations only, since the first estimate for the growth rate of real
goods exports starts only in the year 2009. Any extension of equation (27) involving
auto-regressive or moving-average elements did not contribute to improve the model
fit.
The autoregressive coefficients for the factor equation are significantly different
from zero. Their particular parameter values imply that the stochastic process for
ft is governed by complex roots. Including a third lag would results in a parameter
estimate for the third lag which is rather small and statistically not different from
zero. The extent to which the dynamic factor is governed by complex roots can also
be seen in Figure 3 where the left subplot shows the path of the latent factor over time.
Moreover, the subplot highlights in how far the factor captures the recent peaks and
troughs surrounding the global financial crisis and its aftermath. The second subplot
14
XB_FR XB_R_Q
Figure 3: Factor and Revision Term
-80
-60
-40
-20
0
20
40
1985 1990 1995 2000 2005 2010 2015 2020
S1 – 2 RMSE
Smoothed Factor
-1
0
1
2
3
1985 1990 1995 2000 2005 2010 2015 2020
RVER_0 – 2 RMSE
Smoothed Revision Error
in Figure 3 shows the estimated revision term. Revisions were sizable especially in
the year 2016; on average there seems to be no systematic autocorrelation structure,
however, considering the parameter estimate for με, the first release of the growth
rate of real goods exports seems to be systematically lower than the final value.
Turning to the factor loadings, the estimates imply a rather high value for exports,
steel production in the EU and the price measure for iron and steel products. The
factor loadings of the remaining variables are slightly lower. To the extent that factor
loadings show the contribution of a particular variable in shaping the path of the
extracted latent factor, one can interpret these loadings also as weights. Considering
the factor loading of each variable relative to the factor loading of the export variable
implies that the contribution of the price measure of iron and steel products has a
weight of around 75%, and of the variable capturing steel production in the EU of
around 70%, and so on. This information can be used to judge the contribution of
an update of a particular variable in shaping the model’s prediction for the growth
rate of real goods exports.
Finally, Figure 4 shows the overall model fit. The residuals have been checked for
normality and autocorrelation; the corresponding tests do not provide evidence for
non-normality or autocorrelation.
15
Table 1: Estimated coefficients
Coeff. St. Dev. t-Value Prob. rel. Weight
factor loadingsexports γi 0.09 0.02 5.58 0.00 1.00pmi US manufacturing γi 0.05 0.02 3.31 0.00 0.55steel production γi 0.05 0.02 2.37 0.02 0.37steel production EU γi 0.08 0.02 3.58 0.00 0.70industrial production γi 0.06 0.02 3.91 0.00 0.65prices of iron and steel products γi 0.08 0.02 3.41 0.00 0.75PPI EA γi 0.06 0.02 3.33 0.00 0.54GDP EA γi 0.07 0.02 3.46 0.00 0.57autoregressive coefficientsexports φi,1 -1.71 0.07 -22.84 0.00