corr forecasting dfm Lux - gouvernement · 2018-08-09 · nowcasting, backcasting and forecasting alike. To examine the real-time forecasting accuracy, a pseudo real-time analysis

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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:


xst

)=

(γh · ftγs · ft

)+

(uh

t

ust

)(20)(

y1stt

yft

)=

(ω

ω

)· [γqft + ut,q] +

(0

εt

)(21)

with ω = 1


(1− ϕε(L))εt = με + et, et ∼ NID(0, σ2e) (22)


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

φi,2 -0.82 0.07 -12.22 0.00pmi US manufacturing φi,1 0.92 0.06 15.09 0.00

φi,2 -0.01 0.06 -0.11 0.91steel production φi,1 0.41 0.05 7.56 0.00

φi,2 0.07 0.11 0.61 0.54steel production EU φi,1 1.16 0.08 15.13 0.00

φi,2 -0.30 0.07 -4.57 0.00industrial production φi,1 0.52 0.07 7.54 0.00

φi,2 0.17 0.07 2.51 0.01prices of iron and steel products φi,1 1.03 0.11 8.99 0.00

φi,2 -0.16 0.12 -1.36 0.18PPI EA φi,1 1.46 0.06 25.38 0.00

φi,2 -0.51 0.05 -9.20 0.00GDP EA φi,1 1.13 0.33 3.47 0.00

φi,2 -0.20 0.32 -0.61 0.54factor φf,1 1.82 0.05 34.44 0.00

φf,2 -0.87 0.05 -16.92 0.00revision termmean με 0.30 0.19 1.56 0.12

The stability of the estimated parameters and the overall model fit are assessed

using the evidence presented in Figures 5 and 6. The red lines in Figure 6 refer to

the band (min vs. max) of the variation of the point estimate of each factor loading

from an estimation where the sample size is extended recursively; and the boxplots

show the point estimates and a one-standard and two-standard deviation confidence

interval based on an estimation using the whole sample. As can be seen the variation

of the factor loadings from an estimation based on recursively extending the sample

size is rather small. In general it lies within the one-standard deviation interval from

the estimation using the whole sample. Against this background, the evidence points

towards rather stable parameter estimates over time. This argument is supported

additionally by the evidence presented in Figure 5; as can be seen, the temporal

variation of the overall model fit is rather small. The correlation coefficient of the

extracted factor with the final measure for real goods exports varies in a close band

ranging from 0.92 - 0.94.

16

Figure 4: Model fit

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Std. Residuals

Actual

Predicted

17

XB_R_Q XB_FR PMIMANUS_M

PRODSTEELLU_M PRODSTEELEU_M PRODINDLU_M

P_MET_M P_PPIEA_M PIBEA_R_Q

Figure 5: Parameter Stability - Dynamic Correlation Coefficient

2004 2006 2008 2010 2012 2014 2016 20180.85

0.9

0.95

1

Figure 6: Parameter Stability - Factor loadings

expo pmi_us steel prod steel prod EU IP price PPI EA GDP EA global IP-0.05

0

0.05

0.1

0.15Factor loadings: γ

3.1 Forecast evaluation

The forecast evaluation is done by considering standard statistical measures, as for

instance, mean-squared errors, and mean-absolute error statistics. However, before

18

Figure 7: Forecast trajectory for annual growth rates

Oct 2007 Apr 2008 Oct 2008 Apr 2009 Oct 2009 Apr 2010 Oct 2010-20

-15

-10

-5

0

5

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15

Gro

wth

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e [in

%]

200820092010

turning to these measures in more detail, it is convenient to analyze the model’s

methodology in generating forecasts. Figure 7 is intended to provide some insights

for this. The figure shows predictions for the annual growth rate of real goods exports

for the years surrounding the global financial crisis (2008-2010). Each square refers

to the point in time when a forecast was made - that is, always in the middle (second

month) of a quarter. In particular, consider for instance the first black square -

this square is a forecast for the annual growth rate of real goods exports made in

November 2007. The forecast for the 2008 annual growth rate made in November

the year before is already close to the realized growth rate (black horizontal line).

This is especially due to the fact that this year’s annual growth rate turned out to

be close to the long run mean growth rate which is around 4%. For the following two

years, the realized growth rate is, however, far off the mean. The forecast trajectory

in the figure highlights the extent to which new information in form of updates of

the monthly series, triggers a change in the model’s forecast. This turns out to be

particularly striking for the forecasts of the year 2009 made in late 2008. At that time,

the indicators, as for instance the US PMI, industrial production in Luxembourg and

several other indicators declined drastically, inducing the dynamic factor model to

change the forecast. Within two quarters only, the growth forecast was reduced from

19

+4% (September 2008) down to -14% (March 2009), which turned out to be already

rather close to the observed rate of around -17%. It has to be stressed here that

the realized growth rate of the year 2009 is more than two standard deviations off

the unconditional mean. Against this background, the model produces a reliable

prediction for the annual growth rate already as early as in March of the same year.

The model’s core structure is autoregressive - this can be seen in the model’s

forecasts. The forecast trajectory for the annual growth rate for the year 2010 fol-

lows the forecast trajectory of the year 2009 to some extent; however, once valuable

information for the year 2010 is available, the model quickly adjusts its forecasts.

This explains the tremendous forecast revision from around -2% to up to +14% for

the growth forecast for the year 2010.

Table 2: Forecast evaluation for exports - annual growth rates

full sample sub-sample

MSE DFM 47.201 57.080ARMA(2,2) 58.818 79.991RW 192.297 292.702Expert 63.268

Relative performance of DFM toARMA(2,2) 0.803 0.714RW 0.245 0.195Expert — 0.902

MAE DFM 4.751 4.878ARMA(2,2) 6.004 7.797RW 11.457 13.805Expert 6.183


Figure 7 only shows the forecast trajectory of the dynamic factor model. Table 2

in turn confronts the dynamic factor model’s forecasting performance with alternative

models: an ARMA(2,2) model and a Random-Walk (RW). In line with Figure 7, the

focus in the table is on annual growth rates. As can be seen, the mean-squared error

and the mean-absolute error of the dynamic factor model are noticeably smaller than

those of the two alternative models. The comparison of the dynamic factor model with

the ARMA(2,2) highlights the extent to which high frequency information improves

the forecasts. The dynamic factor model’s stochastic structure is basically of the

form of an ARMA(2,2) model: the latent factor is governed by an auto-regressive

process of order two, and the error terms of equation (20) and (21) are comparable to

20

a moving-average process of order two. Hence the key difference arises from the high-

frequency information that enters the dynamic factor model but not the ARMA(2,2)

model. The smaller MSE and MAE statistics in the case of the dynamic factor model

relative to the ARMA(2,2) model hence show in how far this additional information

increases the precision of the forecast.

Table 2 also offers a comparison of the dynamic factor model with the forecasts

of an expert-panel. As can be seen the MSE and MAE statistics of the factor model

are slightly smaller than the ones of the expert panel; however care has to be taken

here as the number of observations in this case is fairly small.

Table 3: Forecast evaluation for exports - quarterly growth rates

1 quarter 2 quarters 4 quarters

MSE DFM 0.0061 0.0069 0.0096ARMA(2,2) 0.0066 0.0087 0.0126RW 0.0094 0.0167 0.0314

Relative performance of DFM toARMA(2,2) 0.9187 0.7862 0.7627RW 0.6484 0.4100 0.3064

MAE DFM 0.0643 0.0705 0.0763ARMA(2,2) 0.0632 0.0754 0.0885RW 0.0777 0.1067 0.1430


Equal predictive accuracy test (p-values)ARMA(2,2) 0.5562 0.0851 0.0424RW 0.0814 0.0000 0.0000

Table 7 assessed the model’s performance only by means of annual growth rates.

This leaves in general too few observations to carry out statistical tests. Table 3

carries out the same exercise using the y-o-y growth rate for each quarter starting in

the year 2004 until 2016. As can be seen, the dynamic factor model again performs

better than the two alternative models. Since now there is a sufficient number of

observations, the Diebold and Mariano (1995) test can be applied. The test results

provide evidence for a significant difference of the forecasts of the ARMA(2,2) and

RW model in relation to the dynamic factor model. The dynamic factor model’s

forecasts outperform those of the other two models especially for the two, three (not

shown in the table) and four quarter ahead predictions. As concerns the one-quarter

ahead forecast, the MSE and MAE statistics of the dynamic factor model are smaller

than those of the other two models, however, the Mariano-Diebold test does not find

21

evidence for a statistically significant difference for that.

4 Specifying a DFM for real private household

consumption

The dynamic factor model specified for real private household consumption is con-

structed in the same fashion as the one for real goods exports discussed in the previous

Section. Again, the model is based on equations (20)-(26), where equation (22) is

specified as follows:

(1− ϕε · L)εt = et, et ∼ NID(0, σ2e) (30)

and L is the lag operator. The variables used for the estimation are given by:

xht =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

employment Dyoy,Mt

social contributionsyoy,Mt−3

unemploymentyoy,Mt−5

car registrationyoy,Mt−1

GDPyoy,Qt

house pricesyoy,Qt−2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)

xst =

⎡⎢⎣ CSI: car purchaseMt

CSI: economic situation in 12 monthMt−7

yield curveMt

⎤⎥⎦ (32)

where employment D refers to employment excluding commuters and CSI refers

to the indicators in the Consumer Sentiment Indicator set; yft and y1stt are given be

the y-o-y growth rate of the final real private household consumption series as well as

its first estimate respectively. The data are depicted in Figure 8 and further details

are provided in Section B in the Appendix; the data for the two measures of real

private household consumption are shown in Figure 9. As can be seen in the figure,

the two measures for the growth rate 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 (30). Figure 9 already gives a first visual hint

towards the possibility of a positive autocorrelation in the revision term.

22

Figure 8: Data used - DFM for real private household consumption

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23

CFIN_R_Q CFIN_FR CONJCONF12_M

CONJCONF20_M

COSOCEFF_Q EMMPNAR_M IMAVOITPART

P_IMMOLU_Q

PIB_R_Q U_M YIELD_CURVE

Figure 9: Data used for real private household consumption

-.02

.00

.02

.04

.06

.08

.10

94 96 98 00 02 04 06 08 10 12 14 16 18

The selection of the variables is based on the same methodology as already used

in the previous section. The variables capture consumption related elements along

various dimensions. There are contemporaneous wage and income related elements

taken into account by GDP and social contributions; there are in turn also ele-

ments capturing the expectation of wage and income paths which is accounted for

by employment, unemployment and the assessment of the economic situation in 12

months. The selected variables also capture elements related to wealth effects by

means of house prices. The two variables for car registration - one soft indicator and

one hard indicator - capture consumption relevant elements related to durable goods

consumption. Finally the yield curve captures uncertainty and hence allows for the

possibility of consumption being adjusted due to precautionary savings motives. In

general, the variables are, however, closely interrelated, implying that a particular

variable could in principle capture consumption related elements across several of the

aforementioned dimensions.

Table 4 lists all the parameter estimates of the dynamic factor model for real

private household consumption. As can be seen there is evidence for a high first-order

autocorrelation of the revision term εt. The coefficient is 0.85 and highly significant

- again one has to take into account here that the estimation of the equation of the

24

CFIN_R_Q CFIN_FR

Figure 10: Factor and Revision Term

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S1 – 2 RMSE

Smoothed Factor

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RVER_0 – 2 RMSE

Smoothed Revision Error

revision term is based on a few observations only, since the first estimate for the

growth rate of real private household consumption starts only in the year 2009.

The autoregressive coefficients for the factor equation imply that both parameters

are significantly different from zero. The two estimated parameters imply that the

stochastic process is governed by complex roots. As before, including a third lag

would result in a parameter estimate for the coefficient of the third lag which is

rather small and statistically not different from zero.

Table 4 also provides the weights of the variable, measured by means of each

variables’ factor loading relative to the one of consumption. As can be seen, the

measure for house prices has the highest weight suggesting that wealth effects might

be an important element in forecasting household consumption in Luxembourg. The

weights of the other variables are still sizable, though, noticeably lower than the one

for house prices.

Figure 10 shows the temporal trajectory of the extracted latent factor. As can

be seen, the factor looks rather different from the one of the factor model for goods

exports. The global financial crisis shows up in the factor only by a modest swing;

much larger swings can be observed in the run up to the financial crisis. This how-

ever, should not be interpreted as a misspecification of the model. In fact Mody et al.

25


Coeff. St. Dev. t-Value Prob. rel. Weight

factor loadingsconsumption γi 0.18 0.04 4.07 0.00 1.00employment γi 0.07 0.03 2.32 0.02 0.38car purchase (CSI) γi 0.08 0.04 2.06 0.04 0.43house prices γi 0.13 0.08 1.68 0.09 0.72yield curve γi 0.05 0.03 1.79 0.07 0.26economic situation (CSI) γi 0.07 0.05 1.47 0.14 0.38social contributions γi 0.05 0.03 1.65 0.10 0.30GDP γi 0.07 0.04 1.95 0.05 0.40unemployment γi -0.04 0.02 -2.34 0.02 0.23car registration γi 0.05 0.02 2.75 0.01 0.27autoregressive coefficientsconsumption ϕi,1 -1.86 0.03 -70.19 0.00

ϕi,2 -0.94 0.02 -38.66 0.00employment ϕi,1 0.91 0.03 31.53 0.00

ϕi,2 0.01 0.03 0.37 0.71car purchase (CSI) ϕi,1 0.94 0.18 5.24 0.00

ϕi,2 -0.05 0.18 -0.25 0.80house prices ϕi,1 0.37 0.49 0.75 0.45

ϕi,2 0.44 0.50 0.87 0.38yield curve ϕi,1 1.40 0.05 27.12 0.00

ϕi,2 -0.43 0.05 -8.50 0.00economic situation (CSI) ϕi,1 0.81 0.09 8.93 0.00

ϕi,2 0.11 0.09 1.32 0.19social contributions ϕi,1 1.06 0.39 2.70 0.01

ϕi,2 -0.14 0.37 -0.37 0.71GDP ϕi,1 0.99 0.48 2.08 0.04

ϕi,2 -0.08 0.46 -0.18 0.86unemployment ϕi,1 0.94 0.05 18.55 0.00

ϕi,2 0.02 0.05 0.32 0.75car registration ϕi,1 0.22 0.06 3.86 0.00

ϕi,2 0.11 0.06 1.99 0.05factor ϕi,f 1.57 0.13 11.73 0.00

ϕi,f -0.63 0.13 -5.03 0.00revision termAR-coefficient ϕε 0.85 0.07 11.69 0.00

(2012) document for many countries that the global financial crisis had only a com-

paratively small effect on real private household consumption. The second subplot

in the same figure shows the estimated revision term. As can be seen, the revisions

show a sizable degree of autocorrelation.

Finally, Figure 11 gives some insights to 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.

The stability of the estimated parameters and the overall model fit are assessed

using the evidence presented in Figures 12 and 13. The red lines in Figure 13 refer to

the band (min vs. max) of the variation of the point estimate of each factor loading

from an estimation where the sample size is extended recursively; the boxplots in turn

26

Figure 11: Model fit

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Std. ResidualsActualPredicted

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27

CFIN_R_Q CFIN_FR EMMPNAR_M CONJCONF20_M

P_IMMOLU_Q YIELD_CURVE CONJCONF12_M COSOCEFF_Q

PIB_R_Q U_M IMAVOITPART


2004 2006 2008 2010 2012 2014 2016 20180.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1


cons empl car purch (CSI) hou pri y curve econ sit (CSI) soc contr GDP unempl car regist

-0.1

0

0.1

0.2

0.3


show the point estimates and a one-standard and two-standard deviation confidence

interval based on an estimation using the whole sample. As can be seen, the variation

28

of the factor loadings from an estimation based on various sub-samples is rather small.

In general it lies within the one-standard deviation interval from the estimation using

the whole sample. Against this background, the evidence points towards rather

stable parameter estimates over time. This argument is supported additionally by

the evidence presented in Figure 12; as can be seen, the temporal variation of the

overall model fit is rather small. The correlation coefficient of the extracted factor

with the final measure for real private household consumption varies in a close band

ranging from 0.86 - 0.88.


Oct 2007 Apr 2008 Oct 2008 Apr 2009 Oct 2009 Apr 2010 Oct 20100.5

1

1.5

2

2.5

3

Gro

wth

rat

e [in

%]

200820092010


The forecast evaluation of the dynamic factor model for real private household con-

sumption is carried out in the same way as for the export model. Before discussing

some particular forecast related statistical measures, it is convenient to look on how

the dynamic factor model adjusts its forecasts for consumption as a reaction to new

information. For this, Figure 14 gives some insights. The figure replicates what was

done for exports in Figure 7, however, now for consumption growth rates. The figure

plots the projections for the annual consumption growth rate as well as the final pub-

29

Table 5: Forecast evaluation for consumption - annual growth rates

full sample sub-sample

MSE DFM 0.837 0.355ARMA(2,2) 2.661 1.054RW 1.493 0.579Expert — 0.669


MAE DFM 0.649 0.453ARMA(2,2) 1.277 0.853RW 0.939 0.663Expert 0.599


lished values (horizontal lines). The forecast trajectory for the 2008 growth rate does

not show any strong adjustment - this is in line with the indicators in the model. As

concerns the 2009 growth rate, the picture of the indicators used in the model turns

fairly negative in September/October 2008; the model takes this into account by ad-

justing the annual growth forecast for the year 2009 - the growth forecast of 2008 is

left rather unaffected by the steady deterioration of the indicators. From November

2008 onwards, the growth forecast for the annual growth rate of consumption in the

year 2009 was already fairly close to the realized rate. The same applies more or less

also for the forecast for the year 2010. All in all, Figure 14 offers an optimistic picture

of the performance of the dynamic factor model in producing forecasts. However, yet

this comprises only a rough assessment; in what follows several statistical concepts

will be used again for a more thorough evaluation, following broadly Section 3.1.

Table 5 confronts the dynamic factor model’s forecasting performance with alter-

native models: an ARMA(2,2) model and a Random-Walk (RW). As can be seen in

the table, the mean-squared error and the mean-absolute error of the dynamic factor

model are noticeably smaller than those of the two alternative models. The same

applies also once the forecast of an expert-panel is taken into account - however care

has to be taken here as the number of observations in this case is fairly small. Again,

the Mariano-Diebold test has not been applied here since the number of observations

available to implement this test is fairly small, rendering questionable the test results.

Table 6 carries out the same exercise using the y-o-y growth rate for each quarter

30

Table 6: Forecast evaluation for consumption - quarterly growth rates

1 quarter 2 quarters 4 quarters

MSE DFM 0.4827 0.6836 0.9262ARMA(2,2) 0.8817 1.0051 1.0338RW 1.3715 1.7970 2.2774


MAE DFM 0.5379 0.6528 0.7185ARMA(2,2) 0.7054 0.7334 0.7526RW 0.8932 1.0013 1.1391


Equal predictive accuracy test (p-values)ARMA(2,2) 0.0069 0.0990 0.1642RW 0.0000 0.0001 0.0000

starting in the year 2004 until 2016. As can be seen, the dynamic factor model

again performs better than the two alternative models. Since now there is a sufficient

number of observations, the Diebold and Mariano (1995) test can be applied. The test

results provide evidence for a significant difference of the forecasts of the ARMA(2,2)

and RW model in relation to the dynamic factor model. The dynamic factor model’s

forecasts outperform those of the other two models especially for the one and two

quarter ahead predictions.

5 Specifying a DFM for the labour market

This section develops a dynamic factor model (DFM) for the Luxembourgish labour

market. Key variables in this respect are (i) employment and (ii) unemployment.

Yet there is one additional variable that plays an important role in Luxembourg. A

large part of the employees commute every day. As depicted in Figure 15, the share

of commuters has risen steadily over the last decades. From the year 2009 onwards,

this share stabilized and is currently at around 40 % of total employment. The

flattening out, however, will pose a particular challenge for the temporal stability

of the estimated parameters; more on that below. The high share of commuters

extends the domestic workforce far beyond domestic borders and hence demonstrates

the attractiveness of the Luxembourgish labour market among foreigners.

The dynamic factor model specified here for the labour market focuses in partic-

31

Figure 15: Share of commuters in total employment

.08

.12

.16

.20

.24

.28

.32

.36

.40

.44

1980 1985 1990 1995 2000 2005 2010 2015

ular on the following three variables: employment, cross border workers and unem-

ployment. Since these three variables are highly interrelated, only one model will be

developed, involving all of the previously mentioned three key variables. The employ-

ment variable used here includes commuters; hence it is distinct to the one used in

Section 4 where commuters were excluded; Other indicators used will be selected in

line with the procedure applied in the previous two sections. The final factor model

will be based on the equations system (20)-(26); though there are a few noteworthy

differences to the models specified in the previous sections. In contrast to consump-

tion and exports, the key three variables in this section are characterized by two

distinct characteristics: (i) they are available at a monthly frequency and (ii) they

are not revised. The former feature is tacitly solved by the Kalman filter, and hence

needs no further consideration; the latter one renders obsolete equation (21) and in

turn also equation (22). Hence the model is characterized only by the variables in xht

32

Figure 16: Data used

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-1

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1

2

1985 1990 1995 2000 2005 2010 2015

-4

-2

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1985 1990 1995 2000 2005 2010 2015

-4

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1985 1990 1995 2000 2005 2010 2015

33

EMP_M U_M FRIN_M

CONJIND_SA_M

CONJEMP_M EAMPRODIND PIB_R_Q

PIBEA_R_QNC2LUX OENSFLUX_M

Figure 17: Model Fit

-8

-4

0

4

8

-4

-2

0

2

4

1985 1990 1995 2000 2005 2010 2015

-5.0

-2.5

0.0

2.5

5.0

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4

1985 1990 1995 2000 2005 2010 2015

-4

0

4

-2

-1

0

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2

3

1985 1990 1995 2000 2005 2010 2015

-4

-2

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1985 1990 1995 2000 2005 2010 2015

-4

-2

0

2

4

-4

-2

0

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1985 1990 1995 2000 2005 2010 2015

-4

-2

0

2

4

-4

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0

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1985 1990 1995 2000 2005 2010 2015

-4

-2

0

2

4

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4

1985 1990 1995 2000 2005 2010 2015

Std. ResidualsActualPredicted

-4

-2

0

2

4

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-2

0

2

1985 1990 1995 2000 2005 2010 2015

-4

-2

0

2

4

-4

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1985 1990 1995 2000 2005 2010 2015

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1985 1990 1995 2000 2005 2010 2015

34

EMP_M U_M FRIN_M CONJIND_SA_M

CONJEMP_M EAMPRODIND_M PIB_R_Q PIBEA_R_Q

NC2LUX_G UFLUX_M_G

Figure 18: Factors

-12

-8

-4

0

4

8

1985 1990 1995 2000 2005 2010 2015

Smoothed Factor_1st

-60

-40

-20

0

20

40

60

1985 1990 1995 2000 2005 2010 2015

Smoothed Factor_2nd

and xst , which are specified as follows:

xht =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

employmentyoy,Mt

unemploymentyoy,Mt

commutersyoy,Mt

vacanciesyoy,Mt

industrial productionyoy,Mt

GDPyoy,Qt

EA GDPyoy,Qt

unemployment gapyoy,Mt

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

xst =

[ESI (industry)MtESI (employment)Mt

](34)

where as before yoy indicates that the particular variable enters in terms of year-

on-year growth rates. The two soft indicators are taken from the set of European

Sentiment Indicators (ESI) and used in levels. M and Q indicate if the corresponding

variable is observed at a monthly (M) or quarterly (Q) frequency. The unemploy-

ment gap-variable is constructed as a weighted average of unemployment in Belgium,

35


Coeff Std t-Val Prob Coeff Std t-Val Prob rel. Weight

factor loadingsemployment γi,1 0.13 0.02 5.36 0.00 γi,2 0.05 0.01 4.11 0.00 1.00unempl. γi,1 0.02 0.02 0.59 0.63 γi,2 -0.05 0.01 -4.17 0.00 0.86commuters γi,1 0.05 0.01 5.44 0.00 γi,2 0.04 0.00 4.21 0.00 0.64esi (industry) γi,1 0.03 0.03 0.30 0.76 γi,2 0.03 0.01 2.61 0.01 0.22esi (empl.) γi,1 -0.04 0.05 -0.04 0.86 γi,2 0.05 0.02 3.18 0.00 0.75indust. prod. γi,1 0.32 0.05 6.94 0.00 γi,2 0.01 0.01 0.80 0.43 0.29GDP γi,1 0.25 0.06 4.02 0.00 γi,2 0.03 0.01 2.25 0.02 0.41EA GDP γi,1 0.29 0.05 5.78 0.00 γi,2 0.03 0.01 3.58 0.00 0.54unempl. gap γi,1 0.26 0.06 5.06 0.00 γi,2 -0.04 0.02 -3.61 0.00 0.69vacancies γi,1 0.29 0.06 5.21 0.00 γi,2 -0.01 0.01 -0.50 0.62 0.23

autoregressive coefficients

employment ϕi,1 0.89 0.05 19.34 0.00 ϕi,2 0.03 0.05 0.72 0.47unempl. ϕi,1 0.85 0.06 14.64 0.00 ϕi,2 0.09 0.06 1.55 0.12commuters ϕi,1 0.94 0.08 11.54 0.00 ϕi,2 0.06 0.08 0.68 0.50esi (industry) ϕi,1 0.83 0.06 14.54 0.00 ϕi,2 0.10 0.06 1.73 0.08esi (empl.) ϕi,1 0.82 0.08 9.99 0.00 ϕi,2 0.05 0.08 0.66 0.51indust. prod. ϕi,1 0.37 0.08 4.69 0.00 ϕi,2 0.17 0.09 1.95 0.05GDP ϕi,1 -0.58 0.12 -4.98 0.00 ϕi,2 -0.75 0.07 -10.99 0.00EA GDP ϕi,1 0.18 0.22 0.82 0.41 ϕi,2 0.53 0.29 1.81 0.07unempl. gap ϕi,1 -0.41 0.21 -1.95 0.05 ϕi,2 -0.20 0.16 -1.25 0.21vacancies ϕi,1 0.31 0.07 4.43 0.00 ϕi,2 0.17 0.08 2.19 0.031st factor ϕf1,1 0.55 0.11 5.09 0.00 ϕf1,2 0.36 0.11 3.40 0.00

2nd factor ϕf2,1 1.87 0.05 40.02 0.00 ϕf2,2 -0.89 0.05 -19.15 0.00

France and Germany relative to unemployment in Luxembourg. This variable acts

as an attractor variable: relatively low unemployment in Luxembourg increases the

country’s attractiveness among foreigners as a work destination. Hence an increase

in the gap should ultimately trigger and increase in commuters and employment;

concerning unemployment there is no specific effect to be expected.

The remaining variables are rather standard - the two GDP series capture domes-

tic and foreign aggregate demand conditions; they act as contemporaneous indicators.

The remaining variables - vacancies, industrial production and the two soft indicators

from the ESI - are in turn purely leading indicators.

As before, all the data in xst and xh

t are standardized before being used in the

model. The standardized data are shown in Figure 16. The Maximum likelihood

estimates of the factor loadings are depicted in Table 7 and Figure 17 provides a

graphical overview of the model fit. As concerns the particular specification of the

dynamic factor model, each lag polynomial is again specified with two lags; however,

in contrast to the previous model for goods exports and consumption, the model

here considers two factors. Hence ft is two-dimensional rendering equation (24) to a

36

multivariate auto-regressive equation. This in turn implies that νft and σ2

f are now a

2× 1 vector and a 2× 2 matrix respectively. Identification of the two factors is based

on the following two assumptions: (i) φf (L) is a diagonal matrix polynomial, and (ii)

σ2f = I2 (I2 refers to the identity matrix of order two). These two assumptions imply

that equation (24) features two independent auto-regressive processes of order two.


2008 2010 2012 2014 2016 20180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Correlation with 1st Factor

2008 2010 2012 2014 2016 20180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Correlation with 2nd Factor

EmploymentCommutersUnemployment

As concerns the estimated coefficients outlined in Table 7 the auto-regressive

coefficients for the factor equation imply that the parameters are both significantly

different from zero. The two estimated parameters for each factor imply that the

stochastic process 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 two factors are shown in Figure 18 and Figure 19 displays the correlation of

the two factors with the three variables of interest (employment, unemployment and

commuters). As can be seen, the second factor has a high correlation with employ-

ment and unemployment, however, a comparatively low correlation with commuters.

37

The first factor has a negligibly low correlation with employment and unemployment

while a decent correlation with commuters. Considering the two factors in a linear

regression of each of the three key variables of interest implies that, the two fac-

tors explain 73%, 79% and 62% of the variation in employment, unemployment and

commuters respectively.

The Maximum likelihood estimates of the factor loadings reflect the degree to

which the variation in each observed variable can be explained by the the two latent

series. Each series has at least one significant factor loading and the corresponding

sign indicates that, with the exception of unemployment, all series are procyclical.

The unemployment-gap variable depicts a slightly different pattern as concerns its

relation with the factors. It has a positive factor loading with the first factor and a

negative one with the second one. This accounts for the circumstance that employ-

ment in Luxembourg and the amount of commuters to Luxembourg can both increase

or decrease once unemployment is high in the neighbour countries. For instance, high

unemployment in neighbour countries might induce workers in these countries to start

working in Luxembourg. This might in turn increase Luxembourgish employment,

and also raises the volume of commuters. On the other hand, high unemployment in

neighbour countries might in turn also be a signal of an economic downswing in these

countries. Since the Luxembourgish economy is highly interrelated with neighbour-

ing economies, a slack abroad is quickly imported worsening domestic labour market

conditions. In this respect the estimates of the dynamic factor model allow for both

of these situations to occur.

The selection of the variables has been applied to the whole sample. However,

structural changes in the underlying economic dynamics could influence the choice

of indicators over time or lead to time-variation in the estimated parameters. For

instance, it might be possible that the variable selection could motivate a different

set of indicators once applied to a time span ending before the global financial crisis.

Against this background, the model is analyzed concerning its stability over time.

This is assessed based on the factor loadings γh and γs; using a measure for the

model fit as done by means of the correlation coefficients within a time-span starting

in January 2008 and ending in December 2016. In order to introduce time-variation

in the parameters, a rolling regression approach is applied: the model is estimated

based on a sample starting in 01:1985 until 01:2008 and then the sample-window is

recursively extend adding an additional month until 12:2016. For each step the model

38


empl unempl commu esi ind esi empl IP GDP EA GDP unempl gap vacan

-0.2

-0.1

0

0.1

0.2

0.3

0.4


1

empl unempl commu esi ind esi empl IP GDP EA GDP unempl gap vacan-0.15

-0.1

-0.05

0

0.05

0.1

Factor loadings: γ2

is fully estimated. Figure 19 displays the time variation by means of the dynamic

correlation coefficient and Figure 20 shows boxplots for the factor loadings to assess

the overall change in the model fit over time.

The figures highlight that time variation is present. On the other, however, it

is fairly small for most parameters. The variation of the estimated parameters for

the factor loadings (red lines) is smaller than the estimate of the in-sample standard

deviation (black boxplots) depicted in Table 7. All the time-variation of the param-

eters is still within the one-standard-deviation confidence interval of the full-sample

estimates. Hence, the time-variation is of negligible statistical importance. A similar

reasoning applies when considering the temporal trajectory of the model fit - the

correlation coefficient between the model and the final value for the three variables

of interest shows a limited degree of variation over time and there appears to be no

systematic pattern of time variation in the model fit.

39



Oct 2007 Apr 2008 Oct 2008 Apr 2009 Oct 2009 Apr 2010 Oct 20100

2

4

6

Gro

wth

rate

[in

%]

Employment

Oct 2007 Apr 2008 Oct 2008 Apr 2009 Oct 2009 Apr 2010 Oct 20100

2

4

6

8

10

Gro

wth

rate

[in

%]

Commuters

Oct 2007 Apr 2008 Oct 2008 Apr 2009 Oct 2009 Apr 2010 Oct 2010-10

0

10

20

30

40

50

Gro

wth

rate

[in

%]

Unemployment

200820092010

The forecast evaluation of the dynamic factor model for the three labour market

variables is carried out in the same way as for the previous models. Before discussing

some particular forecast related statistical measures, it is convenient to look on how

40

the dynamic factor model adjusts its forecasts as a reaction to new information. For

this, Figure 21 shows the forecast trajectory for the annual growth rates for the

years surrounding the global financial crisis (2008-2010) within the pseudo real-time

analysis. The figure plots the annual forecast for each year made at different points in

time for all of the three variables of interest. Considering for instance the employment

projections first, the first black square is the forecast of the growth rate of employment

in the year 2008 made in November 2007. The next square corresponds to a forecast

for employment growth rate for the year 2008 made in March 2008. In each subplot,

the solid lines refer to the observed annual growth rate for the three years. As can

be seen, the model projections converge to the observed values rather quickly. As of

the third quarter, the model’s projections are in most cases already indistinguishable

from the observed value of the same year. This also applies to the forecast trajectory

for commuters and unemployment. All in all, Figure 21 offers an optimistic picture

of the performance of the dynamic factor model in producing forecasts. However, yet

this comprises only a rough assessment; in what follows several statistical concepts

will be used again for a more thorough evaluation, following broadly the previous

Sections.

Table 8: Forecast evaluation - annual growth rates

Employment Commuters UnemploymentMSE DFM 0.0000 0.0001 0.0056

ARMA(2,2) 0.0001 0.0003 0.0079RW 0.0156 0.0014 0.0140Expert 0.0001 0.0002 0.0042

Relative performance of DFM toARMA(2,2) 0.5854 0.5689 0.7137RW 0.0031 0.1052 0.4021Expert 0.6145 0.7332 1.3510

MAE DFM 0.0054 0.0081 0.0559ARMA(2,2) 0.0068 0.0116 0.0716RW 0.0992 0.0310 0.1004Expert 0.0057 0.0106 0.0582

Relative performance of DFM toARMA(2,2) 0.8034 0.6983 0.7807RW 0.0548 0.2619 0.5570Expert 0.9598 0.7648 0.9604

In order to better judge the precision of the dynamic factor model’s forecasts,

several alternative model forecasts are utilized. The forecasts of each model are

evaluated by means of mean-squared error and mean-absolute error statistics. The

41

alternative models comprise (i) a random walk model, (ii) and ARMA(2,2) model and

(iii) the forecast of an expert -panel (again only as concerns the annual forecasts).

Tables 9 and 8 provide the results.

Table 9: Forecast evaluation - monthly frequency

Forecast horizon: 1 month 3 months 6 months 12 months

Employment MSE DFM 0.0438 0.0929 0.2670 0.8106ARMA(2,2) 0.0492 0.1151 0.3256 0.8303RW 0.0574 0.1673 0.4970 1.3654

Relative performance of DFM toARMA(2,2) 0.8912 0.8073 0.8201 0.9762RW 0.7643 0.5553 0.5372 0.5937

MAE DFM 0.1435 0.2170 0.3580 0.6854ARMA(2,2) 0.1592 0.2493 0.4136 0.6848RW 0.1768 0.3033 0.5043 0.8647


Equal predictive accuracy test (p-values)ARMA(2,2) 0.7379 0.0160 0.0120 0.2340RW 0.0110 0.0000 0.0000 0.0000

Commuters MSE DFM 0.0074 0.0177 0.0529 0.1710ARMA(2.2) 0.0079 0.0202 0.0634 0.2082RW 0.0093 0.0297 0.0912 0.2654


MAE DFM 0.0606 0.0938 0.1564 0.2984ARMA(2,2) 0.0668 0.1071 0.1834 0.3340RW 0.0731 0.1291 0.2199 0.3796



Unemployment MSE DFM 0.0297 0.1007 0.2661 0.6635ARMA(2,2) 0.0342 0.1063 0.2865 0.6701RW 0.0398 0.1547 0.4552 1.2071


MAE DFM 0.1248 0.2257 0.3660 0.6369ARMA(2,2) 0.1350 0.2449 0.4159 0.6412RW 0.1414 0.2871 0.4940 0.8271



Table 8 compares the dynamic factor model’s forecasting performance with al-

42

ternative models: an ARMA(2,2) model and a Random-Walk (RW). As can be seen

in the table, the mean-squared error and the mean-absolute error of the dynamic

factor model are noticeably smaller than those of the alternative models. The same

applies also to the forecast of the expert-panel; however care has to be taken here

as the number of observations in this case is fairly small. Note that in the case

of unemployment the expert-panel’s forecast has a noticeably smaller mean-squared

error than the dynamic factor model; the contrary applies to the mean absolute er-

ror, implying that outliers from exceptional periods play a role (the years 2008 and

2009 can be identified as the two most important ones for these outliers). Again,

the Diebold and Mariano (1995) test has not been applied here since the number of

observations available to implement this test is fairly small, rendering questionable

the test results.

In order to assess the statistical evidence concerning the difference of the forecasts

of the different models - dynamic factors model, random-walk, and ARMA(2,2) - the

forecast evaluation has been run additionally on a quarterly basis. This generates

enough observation rendering feasible the Diebold and Mariano (1995) test. Table

9 shows the results for the three key variables - employment, commuters and un-

employment - for various horizons. As can be seen there is statistically significant

evidence that the dynamic factor model performs better in forecasting employment,

commuters and unemployment at the horizon three months and six months than the

alternative models. The evidence for the one month and twelve months horizon is

mixed.

5.2 An extension: MF-FAVAR model

The dynamic factor model specified for the labour market variables considers three

variables modeled jointly within the same model. An extension that is capable of

generating slightly more accurate forecasts is considered here by means of a Mixed

Frequency-Factor Augmented Vector Autoregressive (MF-FAVAR) model. In order

to motivate the model and show its performance, it is only applied to employment.

In principle it could also be extended involving yet further variables, however, care

has to be taken here as one might be quickly confronted with problems related to the

over- dimensionality of the model. The model reads as follows:

43

MF-FAVAR model:


xst

)=

(γh · ftγs · ft

)+

(uh

t

ust

)(35)



t ∼ NID(0, σ2

q

)(36)

φf (L) ·[

ft

empt

]= νf

t (37)

Φu(L)

(uh

t

ust

)= νt (38)(

νft

νt

)∼ NID

(0,

[σ2f 0

0 Σν

])(39)

where empt refers to the employment series used before. The vectors of observed

variables are defined as follows:

xht =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

unemploymentyoy,Mt

commutersyoy,Mt

vacanciesyoy,Mt

industrial productionyoy,Mt

GDPyoy,Qt

EA GDPyoy,Qt

unemployment gapyoy,Mt

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(40)

xst =

[ESI (industry)MtESI (employment)Mt

](41)

The model is different to the extent that the latent factor ft is extracted excluding

employment. The relation between the factor and employment is than re-established

by means of equation (37) - this is the FAVAR equation as it relates the latent factor

ft directly to the observed variable, in this case, employment. Again, a lag order of

44

two has been used for all lag-polynomials and φ(L), σ2f and νf

t are now 2×2 matrices

and a vector respectively. The model is identified assuming that the (1,1) element

in the variance-covariance matrix σ2f is unity. The remaining elements of σ2

f are

estimated. The (2,1) - and due to symmetry also the (1,2) element - of σ2f captures

the contemporaneous relation between the latent factor ft and employment where

the lag-polynomial φ(L) establishes the dynamic interaction of these two variables.

Table 10: MF-FAVAR model: Forecast evaluation - monthly frequency

Forecast horizon: 1 month 3 months 6 months 12 months

Employment MSE MF-FAVAR 0.0401 0.0850 0.2371 0.8020DFM 0.0438 0.0929 0.2670 0.8106

Relative performance of DFM toMF-FAVAR 1.092 1.093 1.127 1.010

Equal predictive accuracy test (p-values)MF-FAVAR 0.7571 0.1201 0.1101 0.8502

The model is estimated making use of the same estimation procedure as before.

For brevity, only the statistics for the forecast evaluation are shown and are depicted

in Table 10. The mean-squared error statistics of the MF-FAVAR are smaller than

those of the dynamic factor model for any horizon considered. The statistical evidence

for a better forecast precision is limited - the p-values for the three months and six

months horizon are slightly above the 10% significance level.

All in all this is to show in how far small variations of the specification of the

dynamic factor model motivated in Section 2 can quickly alter the model to another

popular group of models commonly used. Even though the forecast precision of the

MF-FAVAR is slightly better, clear preference is given to the dynamic factor model

of Section 5 here for at least two reasons: (i) the dynamic factor model is more

parsimonious than the MF-FAVAR which allows for a significantly faster estimation

of the model; and (ii) the variable selection process might turn out more challenging

in case of the MF-FAVAR model since one has to decide not only which variables to

include in xst and xh

t , but also as to which variables to include in the FAVAR equation.

The latter adds an additional search dimension within the variable selection process

rendering the specification of the model more tedious.

45

6 Conclusion and outlook

The dynamic factor models developed and presented in the previous sections are based

on a small set of indicators. The selection of them takes into account specificities of

the Luxembourgish economy. The models showed a decent forecasting performance

on a pseudo real-time basis.

The three dynamic factor models are all independent to each other; merging

them into one large-scale model might quickly render the estimation intractable due

to problems related to the dimensionality of the model. However, there is another

possibility to merge them. The idea is not to merge the models, yet, only their indi-

vidual forecasts. Starting for instance with the model for goods exports, it produces

forecasts not only for goods exports, but also for the other variables, among oth-

ers, industrial production and Euro-zone GDP growth. As these two variables show

up also in the dynamic factor model for the labour market, the forecasts from the

dynamic factor model for exports could then be used in the dynamic factor model

for the labour market to produce conditional forecasts of the remaining variables.

In other words, the forecasts from the dynamic factor model for exports enter the

dynamic factor model for the labour market as if they were observed. The forecasts

of certain variables of the dynamic factor model for the labour market can then be

used in the dynamic factor model for consumption. This applies in particular to GDP

growth, and partly to employment. By this one can construct a consistent forecast

involving all of the three models, where in turn each dynamic factor model is used

individually, however, they are merged by making use of the concept of conditional

forecasts. Continuing in this fashion and constructing similar models for yet other

important macroeconomic variables would allow for a consistent projection of yet

other variables despite using independent models in each case.

46

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48

A State-space representation of the dynamic fac-

tor model

This section provides details on how the dynamic factor model spelled out in equa-

tions (20)-(26) can be cast into a state-space representation; the following considers

in particular a version of the dynamic factor model with one factor and the revision

term εt is specified as a white noise process. To illustrate how the corresponding

matrices of the transition and measurement equation look like, the following defini-

tions are used: 0(i,j) is a matrix of zeros of dimension i × j, Ir is an identity matrix

of dimension r; xst is a ns dimensional vector of soft-indicators, xh

t is a nh dimen-

sional vector of hard-indicators. For simplicity, it is assumed here that the variables

are all observed at a monthly frequency and that there are no missing observations.

Relaxing this assumption would require to extend yt, wt, R and H by means of an

indicator function as discussed in Section 2.

Transition equation: This equation relates the observed variables to the factor

and can be expressed as:

yt = Hst +wt, wt ∼ NID (0, R) (42)

We use the following defintion of the vectors yt, st wt and variance co-variance matrix

R:

yt =[yft , y

1stt , (xs

t )′, (xh

t )′]′

(43)

wt = 0(nX ,1) (44)

R = 0(nX ,nX) (45)

with n = nh + ns, nX = n+ 2 and

st = [ft, ft−1, ut,q, ut−1,q, εt, ...

ust,1, u

st−1,1, ..., u

st,ns

, ust−1,ns

, ...

uht,1, u

ht−1,1, ..., u

ht,nh

, uht−1,nh

]′(46)

Given these definitions, the matrix H will be the following:

49

H =

⎛⎜⎜⎜⎜⎝

ft, ft−1︷︸︸︷η11 0(1,2)

ut, ut−1︷︸︸︷η12

εt︷︸︸︷0

ust︷︸︸︷

0(1,2·ns)

uht︷︸︸︷

0(1,2·nh)

η11 0 η12 1 0(1,2·ns) 0(1,2·nh)

η31 0 0(ns,1) 0(ns,1) η32 0(ns,2·nh)

η41 0(nh,1) 0(nh,1) 0(nh,1) 0(nh,2·ns) η42

⎞⎟⎟⎟⎟⎠ (47)

with ust =(ust,1, u

st−1,1, us

t,2, ust−1,2, ..., u

st,ns

, ust−1,ns

); uh

t =(uht,1, u

ht−1,1, uh

t,2, uht−1,2, ..., u

ht,ns

, ust−1,nh

),

and

η12 =(

1 0)

(48)

η32 = Ins ⊗(

1 0)

(49)

η42 = Inh⊗(

1 0)

(50)

and η11 = γq, η31 = γs, and η41 = γh.

State equation: Using the previous definitions of the vectors, the state equation

can be expressed as:

st = Fst−1 + vt, vt ∼ NID (0, Q) (51)

where Q is a matrix whose off-diagonal elements are all zero and its diagonal is given

by:

diag(Q) =[σ2f , 0, σ

2q , 0, σ

2ε , diag (Σν)

′ ⊗(

1 0)]′

(52)

where diag (Σν) =(σ2νs,1, ..., σ

2νs,ns

, σ2νh,1

, ..., σ2νh,nh

)′; and the error term vt is given by:

vt =[νft , 0, ν

qt , 0, εt, [ν

st,1, 0], ..., [ν

st,ns

, 0], [νht,1, 0], ..., [ν

ht,nh

, 0]]′

(53)

The matrix F becomes:

50

F =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ft−1, , ft−2︷︸︸︷φf,1 φf,2

ut−1, ut−2︷︸︸︷0(1,2)

εt−1︷︸︸︷0

ust−1,1, u

st−2,1︷︸︸︷

0(1,2)

[· · · ]︷︸︸︷· · ·

ust−1,ns

, ust−2,ns︷︸︸︷

0(1,2)

uht−1,1, u

ht−2,1︷︸︸︷

0(1,2)

[· · · ]︷︸︸︷· · ·

uht−1,nh

, uht−2,nh︷︸︸︷

0(1,2)

1 0 0(1,2) 0 0(1,2) · · · 0(1,2) 0(1,2) · · · 0(1,2)

0(2,1) 0(2,1) fγq 0 0(2,2) · · · 0(2,2) 0(2,2) · · · 0(2,2)

0 0 0(1,5) 0 0(1,2) · · · 0(1,2) 0(1,2) · · · 0(1,2)

0(2,1) 0(2,1) 0(2,2) 0 f s1 0(2,·) 0(2,2) 0(2,2) · · · 0(2,2)

......

......

. . . . . . . . ....

. . ....

0(2,1) 0(2,1) 0(2,2) 0 0(2,2) · · · f sns

0(2,2) · · · 0(2,2)

0(2,1) 0(2,1) 0(2,2) 0 0(2,2) · · · 0(2,2) fh1 0(2,·) 0(2,2)

......

......

.... . .

.... . . . . .

...

0(2,1) 0(2,1) 0(2,2) 0 0(2,2) · · · 0(2,2) 0(2,2) · · · fhnh

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

fγq =

(φq,1 φq,2

1 0

)(55)

f s1 =

(φs1,1 φs

1,2

1 0

)(56)

f sns

=

(φsns,1 φs

ns,2

1 0

)(57)

fh1 =

(φh1,1 φh

1,2

1 0

)(58)

fhnh

=

(φhnh,1

φhnh,2

1 0

)(59)

51

B The data

Table 11: Data used

Variable Description Ctry Unit DFM for Frequency

CFIN R Q House hold final consumption (chainedlinked volumes 2010)

LU Mio Euro Consumption Quarterly

CONJCONF12 M Consumer Confidence, General Eco-nomic Situation over Next 12 Months,Balance

LU Index Consumption Monthly

CONJCONF20 M Consumer Confidence, Intention toBuy a Car within the Next 12 Months

LU Index Consumption Monthly

CONJEMP M Average employment expectations LU Index Labour market MonthlyCONJIND SA M Business Surveys, Industry Confidence

Indicator, Balance, SALU Percent Labour market Monthly

COSOCEFF Q Social contributions receivable - ofwhich employers actual social contri-butions

LU Mio Euro Consumption Quarterly

EAMPRODIND M Industrial production 2010 =100 LU Index Labour market MonthlyEMMPNAR M National employment LU 1000 Persons Consumption Monthly

EMP M Total employment; interior concept LU Persons Labour market MonthlyFRIN M Cross border workers in Luxembourg LU Persons Labour market Monthly

IMAVOITPART Luxembourg, Domestic Trade, VehicleRegistrations (ACEA), New, Passen-ger Cars

LU Number of cars Consumption Monthly

OENSFLUX M New job offers (flows) LU Persons Labour market MonthlyPIBEA R Q Gross domestic Product Luxembourg

(chained linked volumes 2010)EA Mio Euro Labour mar-

ket, ExportQuarterly

PIB R Q Gross domestic Product Luxembourg(chained linked volumes 2010)

LU Mio Euro Consumption,Labour market

Quarterly

P IMMOLU Q House Price Index LU Index Consumption QuarterlyP MET M Price index of iron and steel products,

tubes and othersLU 2010=100 Exports Monthly

PMIMANUS M United States, Business Surveys, ISM,Manufacturing, Purchasing Managers’Index

US Rolling index Exports Monthly

P PPIEA M Industrial Produces Price Index EA 2010=100 Exports MonthlyPRODINDLU M Working day adjusted production in-

dex total economyLU 2010=100 Exports Monthly

PRODSTEELEU M EU, Metal Production, Crude Steel EU metric tonnes Exports MonthlyPRODSTEELLU M Luxembourg, Metal Production,

Crude SteelLU metric tonnes Exports Monthly

TICTEUR M Interbank Rates, EURIBOR, 3 Month,Fixing

EA Percent Consumption Monthly

TILTEUR M Government Benchmarks, Eurostat,Government Bond, 10 Year, Yield

EA Percent Consumption Monthly

U BE M Unemployment Belgium ILO defini-tion

BE Persons Labour market Monthly

U DE M Unemployment Germany ILO defini-tion

DE Persons Labour market Monthly

U FR M Unemployment France ILO definition FR Persons Labour market MonthlyU LU M Unemployment Luxembourg ILO defi-

nitionLU Persons Consumption Monthly

U M Number of registered unemployed. LU Persons Consumption,Labour Market

Monthly

XB R Q Exports of goods (chained linked vol-umes 2010)

LU Mio Euro Exports Quarterly

52

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