Chamachos& Quiros - Markov-switching dynamic factor models in real time.pdf

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Markov-switching dynamic factor models in real time

Maximo Camacho

Universidad de Murcia

[email protected]

Gabriel Perez-Quiros

Banco de España and CEPR

[email protected]

Pilar Poncela

Universidad Autónoma de Madrid

[email protected]

(Preliminary draft)

Abstract

We extend the Markov-switching dynamic factor model to account for the speci-

…cities of the day to day monitoring of economic developments such as ragged edges

and mixed frequencies. We also analyze the consequences of increasing the number of

series N in the model. All these issues have been contemplated in the linear frame-

work, but they have not been considered yet in the Markov switching context. We

also show that extracting a linear univariate common factor and applying afterwards a

Markov switching mechanism leads to wrong inference about the probability of being

in a certain state.

We will use the dynamic common factor Markov switching model to compute

inferences of the percentage chance that the Euro area economy will face a recession

in the short term. Applied to a real time dataset, we provide examples which show the

nonlinear nature of the relations between data revisions, point forecasts and forecast

uncertainty.

Keywords: Business Cycles, Output Growth, Time Series.

JEL Classi…cation: E32, C32, E27

1



1 Introduction

Diebold and Rudebusch (1996) were the …rst to suggest a uni…ed explanation of the two

business cycle features, comovements of economic aggregates and business cycle asymme-

tries, which were embedded in the seminal description developed by Burns and Mitchell

(1946). They argued that comovements among individual economic indicators can be

modelled by using the linear coincident indicator approach described in Stock and Watson

(1991), while the existence of two separate business cycle regimes can be modelled by us-

ing the Markov-switching speci…cation advocated by Hamilton (1989). Integrating these

suggestions, Kim and Yoo (1995), Chauvet (1998) and Kim and Nelson (1998) combined

the dynamic-factor and Markov-switching frameworks to propose di¤erent versions of sta-

tistical models which capture both comovements and regime shifts. Recently, Chauvet

and Hamilton (2006) and Chauvet and Piger (2008) examine the empirical reliability of

these models in computing real time inferences of the US business cycle states.

Markov switching dynamic factor models were originally designed to deal with bal-

anced panels of business cycle indicators. We consider that they do not show the degree

of re…nement that are needed nowadays when applied to the (timely) day to day moni-

toring of the economic activity. In the linear framework, the models used for real time

forecasting have been updated and are able to incorporate the new information as soon

as it is available, in spite of its frequency and the possibility of ragged ends. See, for

instance, Mariano and Murasawa (2003) for mixing frequencies, Giannone, Reichlin and

Small (2008) for ragged ends and Arouba, Diebold and Scotti (2009) and Camacho and

Perez-Quiros (2009) for an application of all these issues with data from the US and the

Euro Area in the linear setup. Another important issue that is largely analyzed in the

recent literature is the role of increasing the number of variables in the model. On the

one hand, Stock and Watson (2002) and Forni et al (2000, 2005) show the consistency of

the principal components estimator as both N (the number of time series) and the time

dimension T tend to in…nity. On the other hand, Boivin and Ng (2006) and Bai and Ng

(2008) show through simulation studies that not always more information means better

forecasting performance. Keeping in mind that we are in a nonlinear setup (and there-

2



fore, a central limit theorem type of argument keep us in …nite N ), we show the role of

increasing the number of variables for forecasting purposes.

The primary goal of this paper is to update the Markov switching dynamic factormodels for real time forecasting of the state of the business cycle analyzing the role of

each of the previous features as it has been done in the linear framework. In order to so,

we will analyze how to incorporate each one of the characteristics of the data in real time

forecasting.

Form a practical point of view, the …rst question that an analyst faces is how many

series to introduce in the model. What is the gain of using more than one series as in

Chauvet and Piger (2008) and Chauvet and Hamilton (2006) versus the original univariate

model as in Hamilton (1989) ? We answer this question in terms of the change in the

relative entropy.

Additionally, the analyst faces data with mixing frequencies. Some of the typical eco-

nomic indicators that are observed to infer business cycle states are available monthly while

others are available quarterly. For example, the National Bureau of Economic Research

(NBER) dating committee acknowledges that recessions are de…ned as signi…cant declines

in economic activity normally visible in real GDP, real income, employment, industrial

production, and wholesale-retail sales, which are clearly available at di¤erent frequencies.

Finally, the analyst confronts data with ragged ends due to the typical lack of synchronic-

ity that characterizes the daily ‡ow of macroeconomic information. Not accounting for

this publication pattern would imply that making a forecast using traditional Markov-

switching dynamic factor models to develop early assessments of the economic evolution

can involve substantial costs since the forecasts will be constrained just to the information

of the balanced panel.

Since all these speci…cities of real time forecasting have been already included in thelinear framework, it is very tempting to …t a linear dynamic factor model to the data

and then to apply a univariate Markov switching model to the linear common factor as

suggested by Diebold and Rudebusch (1996). There are two reasons why the analysts

might prefer this option: First, the linear indicator is already built in several agencies and

it is straightforward to use the indicator to analyze the state of the business cycle with

3



a univariate …lter. Second, the estimation of multivariate Markov switching models does

su¤er the curse of dimensionality when evaluating the likelihood function. Nevertheless,

we show that this shortcut results in a worst forecasting performance due to the inertiapresent in the extracted linear common factor.

In the empirical section, we apply the model to a real time Euro area data set. We …nd

that the model provides a reliable description of the Euro area turning points. Additionally,

we show that the model gives an early signal of the current recession, before negative

growth rates of GDP were published. Overall, these results suggest that the Markov-

switching dynamic factor model proposed in this paper is a potentially very useful tool to

be used in the day to day monitoring of the economy.1

The structure of this paper is organized as follows. Section 2 shows that a Markov

switching model …tted to a univariate common factor extracted linearly leads to wrong

inference about the probability of being in a certain state. Section 3 analyzes each of the

di¤erent features of the data and its role in forecasting turning points in the context of

Markov switching models. Section 4 presents the model. Section 5 presents the empirical

application. And section 6 concludes.

2 Multivariate vs univariate analysis of the Markov switch-

ing model

In this section we want to address the point of the performance (in terms of detecting

the probability of being on a given state) of a multivariate Markov switching factor model

versus a univariate Markov switching model …tted to a linear common factor extracted

from the set of time series available. Of course, if the data generating process is the

multivariate Markov switching, this option should be the correct one, but this alternative

option is easier to implement. Besides, there are several agencies that have already built

their own indicator and they may want to use it to analyze the state of the business cycle.

1 In the empirical application, the model also accounts for data revisions for GDP growth, by assuming

that preliminary estimates are equal to the true data plus an uncorrelated noise as in Evans (2005).

4



In order to address the previous issue, we will assume a one factor model where the

common factor follows a Markov switching process, but we …t a linear AR(1) model to

capture the autocorrelation induced by the common switch of the N variables. In particu-lar we assume that the vector of N measured or observed time series yt = (y1;t;:::;yN;t)0

is generated by r = 1 non-observed common factor f t and N speci…c or idiosyncratic

components

yt = f t + ut

N 1 N 1 1 1 N 1(1)

where ut is multivariate white noise (0; u) and = (1; 2;:::; N )0 is the factor loading

matrix. As in classical factor analysis, we will assume that u is a diagonal matrix. For

the moment, we will assume that the common factor loads contemporaneously on the

observed variables, but we will also relax this assumption later on.

To highlight the point we want to make in this section, we will assume that all the

dynamics in the observed series are due to a common factor, given by a Markov switching

mean plus noise

f t = st + at; (2)

where at is univariate white noise (0; 2a) and st is assumed to evolve according to an

irreducible 2-state Markov chain whose transition probabilities are de…ned by

p(st = j jst1 = i; st2 = h; :::; I t1) = p(st = j jst1 = i) = pij (3)

where i; j = 0; 1 and I t is the information set up to period t:

Nevertheless, instead of …tting equation (2) we estimate the factor as an AR(1) process.

We will denote the misspeci…ed factor with an ’*’. Then,

f

t = f

t1 + a

t; (4)

with var(at ) = 1, in order to identify the model. After some straightforward algebra, it

can be shown that the AR parameter is given by

= E (f tf t1)

E (f 2t1) =

211 p11 + 2

00 p00 + 10(1 p10 + 0 p01)

211 + 2

00 + 2a

(5)

5



where i is the steady state probability of state i, such that 1 + 0 = 1 . For this two

state Markov chain the steady state probabilities are given by (see, for instance, Hamilton,

1994)

1 = 1 p00

2 p11 p00

0 = 1 p11

2 p11 p00:

The AR parameter re‡ects the serial correlation induced by the common switching

factor in all the series. Furthermore, we will assume that we are able to extract the common

factor with the true remaining parameters of the model, that is, the only misspeci…cation

comes from the fact that we try to model the dynamics of the observed series througha linear common factor, rather through the Markov switching mechanism. As it is well-

known, the …ltered linear common factor f tjt can be expressed as a weighted sum of past

and present observations

f tjt =tX

=1

w0t; y (6)

where

wt;t = 1

ct01

u

wt; = 1

c

1

V j 1wt; +1 = B wt; +1 for = t 1; :::; 1 (7)

where ct = 1V tjt1

+ 01u and V tjt1 is the MSE of the misspeci…ed state estimated at

t with information up to time t and B = 1c

1V j 1

.

In the case of diagonal noise variance-covariance matrix u = diag(21;:::;2

N ), the

previous weighting scheme gets the intuitive form

f tjt = 1

ct 1

V tjt1f tjt1 +

N Xi=1

2

i2i

(yi;t

i)!

with ct = 1V tjt1

+PN

i=12i2i

: From the formula, it is clearly seen that f tjt is the estimation

of the factor with information up to time t as a weighted mean. The …rst term is the

estimation of the factor at time t with the information up to time t 1, and it has a

weight proportional to the precision of this estimation. The second term is the estimation

6



of the factor provided by the information contained at time t on the vector of observed

series yt: The ratio 2i2i

; i = 1;:::;N; is the precision of the estimation of the common factor

from the i-th series yi;t at each time t: They are known as signal to noise ratio (SNR, fromnow on). These ratios determine how new information is incorporated into the forecasts.

The greater the SNR is, the stronger is the signal of the common information on the series

and vice-versa, the smaller it is, the weaker is the common information signal in relation

to the speci…c noise of each series.

From (7), we can see that the memory of the misspeci…ed process is bigger (and this is

re‡ected in larger weights for past observations), the larger are B and/or : Both things

will di¢cult the estimation of the …ltered and smoothed probabilities of a state, since if a

switch has happened at t, it will be obscured in f tjt due to the fact that f tjt is a weighting

sum of past and present information. The higher the weights on the past information,

the more di¢cult will be to detect a change of regime. The AR parameter depends of

the parameters of the Markov switching (the di¤erence in means among regimes, and the

transition and state probabilities). As regards the B coe¢cients, we will study them in

more detail in what follows.

Since we are dealing with stationary processes, the …lter reaches its steady state and

V tjt1 = V . In this case, the solution of the algebraic Ricatti equation for the misspeci…ed

linear …lter is given by

V =01

u (1 2) +

q01

u (1 2)2

+ 401u

201u

=

PN i=1

2i2i (1 2) +

r PN i=1

2i2i (1 2)

2+ 4

PN i=1

2i2i

2PN i=1

2i2i

and, therefore the weighting scheme becomes

7



wt;t =01

u (1 2) +

q01

u (1 2)

2

+ 401u

01u + (1 + 2) + q01u (1 2)2 + 401u

01u

0

1u PN

i=12i2i (1 2) +

r PN i=1

2i2i (1 2)

2+ 4

PN i=1

2i2iPN

i=12i2i

+ (1 + 2) +

r PN i=1

2i2i (1 2)

2+ 4

PN i=1

2i2i

01uPN

i=12i2i

wt; = 2PN

i=12i2i

+ (1 + 2) +

r PN i=1

2i2i (1 2)

2+ 4

PN i=1

2i2i

)

wt; +1

for = t 1; :::; 1

and does not depend on t since the initial value for the backward recursion wt;t does not

depend on t. As regards B , it is given by

B = 2

01u + (1 + 2) +

q01

u (1 2)2

+ 401u

= 2PN

i=12i2i

+ (1 + 2) +

r PN i=1

2i2i (1 2)

2+ 4

PN i=1

2i2i

)

:

Notice that the greater is the sum of signal to noise ratios 01

u =PN

i=1

2i

2

i

, the smaller

is B ; so we can conclude that we give more weight to present instead of past information

as the common signal weights more relative to the noises in the variance matrix.

If the noise has dynamic structure, we just have to substitute in the above formulae

u by the corresponding expression. For instance, if the noise follows a diagonal VAR(1)

process (that is, each of the components of the idiosyncratic noise is a univariate AR(1)),

such that

ut = ut + t

where var(t) = diag(21;:::;2

N ) and = diag(1;:::;N ); then u = diag( 2

1

12

1

; :::; 2N

12

N

)

in the above formulae.

Once we have the linear common factor f tjt, we will …t a univariate Markov switching

model to it. Let I t the information set available up to time t; that is I t =n

(f j ) =t =1

o:

Since maximum likelihood is computationally infeasible as the number of states grows

8



with t, such that at t we have 2t possible paths (and therefore a mixture of 2t Gaussian

distributions), we will estimate the model by approximate maximum likelihood. Assume,

for instance, that at each t we will approximate the mixture of 2

t

Gaussian distributionsby 22 Gaussian distributions. For instance, we will evaluate the Gaussian approximation

of coming from the four paths {ij}being i; j = 0; 1 (expansion/recession) each of the two

possible states where i is referred to the state at t 1 and j to the state at t: In this

case the …ltered prob(st = j jI t ) and smoothed prob(st = j jI T ) probabilities of being in a

particular state are estimated as (see Kim, 1994)

prob(st = jjI t ) =1

Xi=0

prob(st = j; st1 = ijI t ) =1

Xi=0

f (f tjt; st = j; st1 = ijI t1)

f (f tjtjI t1)

=1X

i=0

f (f tjtjst = j; st1 = i; I t1)

f (f tjtjI t1) pij

and

prob(st = j jI T ) =M Xi=1

prob(st = j; st+1 = ijI T ) M Xi=1

prob(st+1 = ijI T ) prob(st = j jI t ) pij prob(st+1 = ijI t )

:

It is easily seen that all the probability density functions f (f tjtjst = j; st1 = i; I t1) are not

correctly evaluated since f tjt is a linear combination of present and past yt, and therefore

depends not only on st and st1; but on all the remaining lags of the state variable: The

higher the persistence in the model, the worse the approximation is. Intuitively, this

will result in a slower detection of turning points, precisely when we want to update the

Markov switching dynamic factor model to obtain more timely estimates of the state of

the business cycle.

3 Features of the data

We analyze the role of each of the features of the data has on improving the forecasting

performance of the Markov switching model. The section is conceived as a guide for an

analyst who is searching for indications on how to build a Markov switching model to

characterize and forecast the state of the business cycle.

9



3.1 The role of N

The …rst issue that an analyst faces is how many series to introduce in the model. Although

the economic theory might provide an initial set of series to study the business cycle,

these series do not necessarily provide accurate and early forecasts of the turning points.

The recent literature about the linear framework provides mixed evidence about how

many series to consider for forecasting. While consistency results in the estimation of the

common factors are available as both N and T tend to in…nity, the forecasting performance

of the common factors does not necessarily improve with N.

As mentioned in the introduction, to maintain the assumption of non-linearity we need

to keep N …nite. Additionally, in non linear models, the curse of dimensionality appliesand, even though we can handle N relatively large, we can not handle an N of the size of

the estimated in large scale linear models. However, the question of the optimal number

of variables still holds. If our analyst starts with a single variable that gives a reasonable

signal of turning points, under what circumstances should we add more variables?

To illustrate the answer to this question, suppose that we have a simple model where

all the variables share a common switch. In particular, we assume that the vector of

N

measured or observed time series yt = (y1;t;:::;yN;t)0 is generated by a non-observed

common factor f t plus some idiosyncratic component ut as in (1), where ut is assumed to

be white noise. To further simplify things, we will assume …rst that the equation for the

factor is

f t = st ; (8)

where st evolves as in (3).

In practice, the analyst does not know the value of N as well as the speci…c series

included in yt: To simplify things, assume that N = 2 and the analyst …ts a univariate

model for the …rst series y1;t: We want to compare the …ltered probabilities prob(stjy1;t; y2;t)

and prob(stjy1;t), that we will refer as multivariate and univariate probabilities, respec-

tively. The second variable y2;t will be useful in detecting the turning points if prob(st =

1jy1;t; y2;t) > prob(st = 1jy1;t) when st = 1 (for instance, recessions) and prob(st =

10



1jy1;t; y2;t) < prob(st = 1jy1;t) when st = 0 (i.e., expansions). Notice that

prob(st = 1jy1;t; y2;t) = prob(st; y2;tjy1;t)

f (y2;tjy1;t)

= f (y2;tjst = 1; y1;t)

f (y2;tjy1;t) prob(st = 1jy1;t)= wt prob(st = 1jy1;t);

where wt = f (y2;tjst=1;y1;t)

f (y2;tjy1;t) : Therefore, y2;t will be useful if the weight

f (y2;tjst=1;y1;t)f (y2;tjy1;t)

> 1

when st = 1 and f (y2;tjst=1;y1;t)

f (y2;tjy1;t) < 1 when st = 0: It is straightforward to show that if

prob(st = ijy1;t) = 1, then prob(st = ijy1;t) = prob(st = ijy1;t; y2;t) = 1. So, assume that

0 < prob(st = 1jy1;t) < 1: Since

f (y2;tjst = 1; y1;t)

f (y2;tjy1;t) = f (y2;tjst = 1; y1;t)

f (y2;tjst = 1; y1;t) prob(st = 1jy1;t) + f (y2;tjst = 0; y1;t)(1 prob(st = 1jy1;t));

then f (y2;tjst=1;y1;t)

f (y2;tjy1;t) > 1 when st = 1 if

f (y2;tjst = 1; y1;t) > f (y2;tjst = 1; y1;t) prob(st = 1jy1;t)+f (y2;tjst = 0; y1;t)(1 prob(st = 1jy1;t))

that leads tof (y

2;tjst = 1; y

1;t)

f (y2;tjst = 0; y1;t) > 1 when st = 1: (9)

Since (9) does not need to be true for all possible values of y2;t (for instance, in the case

of overlapping Gaussian density functions) we will evaluate when the second variable is

useful on average. Taking natural logarithms, the previous condition implies that

ln f (y2;tjst = 1; y1;t) ln f (y2;tjst = 0; y1;t) > 0 when st = 1: (10)

Denote by 22j1 the conditional variance of y2;t given y1;t and the state, which we assume

to be the same in both states. Taking into account all possible outcomes when st = 1, theexpected value of the di¤erence between the two conditional densities when st = 1 under

Gaussianity is given byZ ln f (y2;tjst = 1; y1;t)f (y2;tjst = 1; y1;t)dy2;t

Z ln f (y2;tjst = 0; y1;t)f (y2;tjst = 1; y1;t)dy2;t(11)

= 2

2(1 0)2

222j1

11



which is greater than zero if 1 6= 0 (the two states are distinguishable) and 2 6= 0 (the

second variable is also generated by the Markov switching mechanism). Notice that minus

the …rst integral represents the entropy of the density f (y2;tjst = 1; y1;t) and thatZ f (y2;tjst = 1; y1;t) ln

f (y2;tjst = 1; y1;t)

f (y2;tjst = 0; y1;t)dy2;t (12)

is a Kullback-Leibler divergence measure which is always 0. Although (11) could depend

on y1;t, in this simple model it does not. In more general settings, to obtain a similar result

that does not depend on y1;t, we could …nd the expected values of the conditional densities

with respect to the joint distribution of y1;t and y2;t. Instead of (11), in general we will

have Z Z f (y1;t; y2;tjst = 1) ln

f (y2;tjst = 1; y1;t)

f (y2;tjst = 0; y1;t)dy1;tdy2;t =Z Z

f (y2;tjst = 1; y1;t) ln f (y2;tjst = 1; y1;t)

f (y2;tjst = 0; y1;t)dy2;t

f (y1;tjst = 1)dy1;t (13)

although in this particular simple model (13) is again equal to 2

2(10)

2

222j1

since f (y1;t; y2;tjst =

1) = f (y2;tjst = 1; y1;t)f (y1;tjst = 1), the integral with respect to y2;t is a constant and the

integral of a density function over all its support is 1. Notice that (13) is 0 since (11) is

0 and f (y1;tjst = 1) is a density function.In a similar way, averaging over all possible values of y2;t when st = 0, the expected

value of the di¤erence between the two conditional densities when st = 0 under Gaussianity

is given byZ ln f (y2;tjst = 1; y1;t)f (y2;tjst = 0; y1;t)dy2;t

Z ln f (y2;tjst = 0; y1;t)f (y2;tjst = 0; y1;t)dy2;t

= 2

2(1 0)2

222j1

which is strictly negative if 1 6= 0 (the two states are distinguishable) and 2 6= 0 (the

second variable is also generated by the Markov switching mechanism). Notice also that

Z

f (y2;tjst = 0; y1;t) ln f (y2;tjst = 0; y1;t)

f (y2;tjst = 1; y1;t)dy2;t

is minus the Kullback-Leibler divergence measure which therefore is always 0: In both

cases the greater is the conditional variance 22j1, the less informative is y2;t given y1;t:

12



As an overall conclusion, for this simple model, given the …rst variable y1;t, the second

variable y2;t will be, on average, more informative about the …ltered state probabilities,

the greater is the di¤erence between the means in the two states 10, the larger is thefactor loading 2 and the smaller is the conditional variance 2

2j1: Notice also that in the

case of u diagonal, given the state, the two variables become conditionally independent

and the conditional variance 22j1 is the marginal variance of the second series.

In the previous discussion we have only considered the contemporaneous dependence

among variables. In order to generalize the previous conclusion taking into account the

past history of the variables, notice that the posterior probability of st = 1 is calculated

as

prob(st = 1jy1;t; y2;t; I t1) = f (y2;tjst = 1; y1;t; I t1)

f (y2;tjy1;t; I t1) prob(st = 1jy1;t; I t1);

while conditions (9), (10) and (11) are maintained since for this simple example f (y2;tjst =1; y1;t; I t1) = f (y2;tjst = 1; y1;t), j = 0; 1: Nevertheless, this simple result generalizes to

more complex models since it is a consequence of Gibbs inequality. This enables us to relax

the assumption of Gaussianity as well as consider more complex models for the common

factor and idiosyncratic components. Consider again equation (1) where the common

factor is generated now by

f t = st + at

(B); (14)

with (B) = 11B ::: pB p; that is, the common factor is a Markov switching autore-

gressive process with changing mean. We also consider that the idiosyncratic component

may have dynamic structure

F(B) ut = t

N N N 1 N 1(15)

where F(B) = diag(F i(B)) is a diagonal matrix that collects the dynamics speci…c to

each idiosyncratic shock, with F i(B) = 1 F i;1B ::: F i;piB pi ; i = 1;:::;N and t is

multivariate white noise (0; ) with diagonal. The model allows for shocks to the

economy (at) as well as speci…c shocks for each of the variables (t). Furthermore, the

result also applies to factors that load dynamically onto the observed series in (1) as long

13



as we correctly specify the conditional densities of the observed series. For illustrative

purposes, let (B) = 1 1B in (14) and the speci…c components be white noise. Then,

the posterior probability of being in recession is given by

prob(st = 1jI t) =1X

i=0

prob(st = 1; st1 = ijI t)

=1X

i=0

f (y2;tjst = 1; st1 = i; y1;t; I t1)

f (y2;tjy1;t; I t1) prob(st = 1; st1 = ijy1;t; I t1):

A su¢cient condition for prob(st = 1jI t) > prob(st = 1jI t1; y1;t) when st = 1 is that

f (y2;t

jst = 1; st1 = i; y1;t; I t1)

f (y2;tjst = 0; st1 = i; y1;t; I t1) > 1 when st = 1; (16)

that on average is the change in relative entropy from one state to the other with respect

to the conditional density function of the true state, that is,

Z f (y2;tjst = 1; st1 = i; y1;t; I t1) ln

f (y2;tjst = 1; st1 = i; y1;t; I t1)

f (y2;tjst = 0; st1 = i; y1;t; I t1)dy2;t (17)

As an overall conclusion, the change in relative entropy quanti…es the usefulness of an

additional variable to detect the true state at t: If the change in relative entropy between

the conditional densities is large, the additional variable is able to better separate the two

states.

3.2 Mixing frequencies

The fact that some economic indicators are available monthly while others are available

quarterly raises the question of how to combine them into a uni…ed forecasting model.2

To deal with this data problem, this section describes a method to weight monthly obser-

vations to form quarterly predictions and analyzes the peculiarities of its implementation

in this non-linear setup.

Quarterly series which refer to stocks can be converted easily in monthly observations

since they simply refer to quantities which are measured at a particular time and do not

2 Aruoba, Diebold and Scotti (2009) describe a linear model to combine time series which are available

at higher-than-monthly frequencies.

14



require any time restriction. Accordingly, these series can be treated as observed the

month that they are issued and as unobserved otherwise. However, ‡ow variables are

measured during some time periods and must be temporally aggregated. In this paper,we follow Mariano and Murasawa (2003) to describe a time aggregation which is based on

the notion that quarterly time series can be viewed as sums of underlying monthly series

in the corresponding quarter. Assuming that arithmetic means can be approximated by

geometric means, quarter-on-quarter growth rates (gt) of quarterly series are weighted

averages of the monthly-on-monthly past growth rates (xt) of the (assumed to be known)

monthly underlying series

gt = 1

3 xt + 2

3 xt1 + xt2 + 2

3 xt3 + 1

3 xt4: (18)

In empirical applications, the underlying monthly series are not usually available but can

be treated as missing and estimated by using an appropriate speci…cation of the Kalman

…lter.

It is worth mentioning that the performance of the …lter relies on the accuracy of

geometric means to approximate arithmetic means. In practice, it is hard to believe that

monthly changes of quarterly series could be high enough to invalidate the approximation,

however. For example a constant growth of 1% each month in a particular quarter (annual

growth of more than 12%), would imply a di¤erence between arithmetic and geometric

means of less than 0:4 percentage points. In addition, other approaches in the literature

which try to skip the approximation are not exempt of problems. The exact nonlinear

…lter of Proietti and Moauro (2006) involves approximations in its own and the exact

linear …lter of Aruoba, Diebold and Scotti (2009) assumes all indicators to be polynomial

trends.

Now we address the point of mixing monthly and quarterly indicators of data in thecontext of Markov switching models. We have to take into account two characteristics

of the mixed process: (i) it is a linear combination of present and past monthly random

variables that depend of present and lagged hidden discrete state variables st;:::;stk; and

(ii) we do not observe the complete monthly series of quarterly growth rates, but instead

we observe one every three outcomes.

15



First, we are going to analyze the maximum lag order k in stk that we need to correctly

specify the probability distribution function of a given variable. We need it in order to

properly apply the Bayes theorem to compute the probability of being in a certain state,expansion or recession. To …x ideas, we want the monthly GDP growth to be as close as

possible to the common factor, so let xt = f t and to simplify things assume …rst that it is

given by (14) with (B) = 1: Later on, we will see what happens when (B) is of order

p > 0: Assume that xt = st + at, with at N (0; 2) and st =

8<: 0 if st = 0

1 if st = 1

9=;. If st

takes two possible values, then xtjst follows the Gaussian distribution3

f (xt

jst = i) =

1p 2

2exp

(xt i)2

22 ; i = 0; 1

and therefore xt is a mixture of two Gaussian distributions

f (xt) =2X

i=1

if (xtjst = i)

where i is the unconditional probability of being in state i.

By equation (18), we know that the common factor will a¤ect the monthly series of

quarterly GDP growth rates contemporaneously and lagged. Since the monthly GDP

growth xt can be in either of the two states i = 0; 1 (expansion or recession), and the

monthly series of quarterly growth gt is a linear combination of xt and four lags, de…ne

the 25 = 32 Markov process st as

st = 1 if st = 0; st1 = 0;:::;st4 = 0

st = 2 if st = 1; st1 = 0;:::;st4 = 0

st = 3 if st = 0; st1 = 1;:::;st4 = 0

...

st = 32 if st = 1; st1 = 1;:::;st4 = 1:

The probability distribution of gt given st and the parameters of the Gaussian distributions

for the two possible states of st; (0; 1; 2), is given by

f (gtjst = j) = 1p

22

exp

(gt (st = j))2

2

; j = 1; 2; :::; 32

3 We are assuming the same variances for the two regimes, although it is straightforward to generalize

to di¤erent variances in the two regimes as well.

16



where

(st = 1) = 1

30 +

2

30 + 0 +

2

30 +

1

30 = 30

(st = 2) = 13

1 + 23

0 + 0 + 23

0 + 13

0 = 13

1 + 83

0

...

(st = 32) = 1

31 +

2

31 + 1 +

2

31 +

1

31 = 31

and

2 =

19

9 2:

So the probability distribution of gt is

f (gt) =32X

j=1

jf (gtjst = j) (19)

where j is the unconditional probability of being in state j: If we assume that not all

the dynamics in the common factor are collected by the changing mean st , then we can

allow (B) in (14) to be an AR process of order p: Then, xt = f t will depend directly

on st; st1;:::;st p and by (18), gt will depend on st;:::;st4; :::st p4: This might result

in a very complex algorithm. To simplify the estimation procedures in the linear setup,

Chauvet (1998) uses a factor model with changing intercept, rather than changing mean

where

f t = st + 1f t1 + ::: + pf t p + at (20)

and argues that it obtains similar, although rather superior results that with the model

with changing mean implied by (14).

The second point we would like to stress in this section is how to extend the procedure

of Mariano and Murasawa (2003) for mixing frequencies in the linear case to the Markov

switching setup. Anticipating the main conclusion, this procedure can be applied to the

nonlinear setup as well. Let be the parameter vector and without loss of generality let

gt be the …rst component of the vector of time series fy1;t = gtgT t=1 : Finally, let

y+1;t :=

8<: y1;t if y1;t is observable

zt otherwise

9=;17



where zt is a random variable whose distribution is independent of ; for instance,

zt N (0; R): Let y+t = (y+1;t;:::;yN;t)0: Consider now the joint distribution function of

(y+

1 ;:::;y+

T ): By construction, since f (zt) does not depend on

f (y+1 ; :::; y+

T ; ) = f (y1; :::; yT ; )Yt2A

f (zt)

where A f1;:::;T g is the subset of missing observations for y1;t: Now we will show that

the inference based on fy+t gT t=1 is the same as the inference based on fytgT t=1 and the

former one is preferred since it does not contain missing observations.

Given a state, the model is linear and we can apply the equations for the Kalman

…lter and smoother as in Mariano and Murasawa (2003), that is, we will zero out all the

parameters in the measurement equation for y+1;t whenever gt is unobserved and equate the

error term to zt: We will use an approximation of 25 Gaussian distributions to evaluate

the probability distribution at each t. This corresponds, for instance, to a common factor

generated by (14) with (B) = 1 and to approximate gt by (18). The generalization for a

process with AR polynomial (B) of order p > 0 is straightforward. So, let us approximate

the true likelihood by a mixture of 25 Gaussian distributions as indicated in (19). Now we

are going to show how to carry out the inference for the …ltered probabilities. Let I +t be

the information set generated by (y+1 ; :::; y

+t ). Then, given the initial state probabilities

prob(s0 = l; s1 = m; s2 = n; s3 = o)

we are going to show how to compute the probability of being in a particular state with

information up to time t prob(st = ijI +t ) and that this probability is independent of zt; since

st is independent of zt: Accepting as input p(st1 = l; st2 = m; st3 = n; st4 = ojI +t1);

we will show how to compute prob(st = ijI +t ) (the quantity of interest) being i=expansion,

recession and prob(st = h; st1 = l; st2 = m; st3 = njI +t ) (the next input needed to run

the algorithm to compute the required probabilities). The initial probabilities can be

equate at 1/16 if we do not have any prior belief about the initial state. At each t

prob(st = ijI +t ) =32X

j=1

prob(st = j \ st = ijI +t )

that is, the probability of being in a certain state i (expansion or recession) in a given t

is the sum of the probabilities of all paths from t 4 to t such that the last state is i:

18



Since i can take two possible values, we sum 16 joint probabilities for each value of i. Let

C = f j 2 f1; :::; 32gjst = ig. Then

prob(st = ijI +t ) =

32X j=1

prob(st = j \ st = ijI +t ) = X j2C

prob(st = j jI +t ): (21)

Each of the 16 terms in the last summation is given by

prob(st = jjI +t ) = prob(st = j jy+t ; I +t1) (22)

=f (y+

t ; st = j jI +t1)

f (y+t jI +t1)

:

Now if yt is completely observed y+t = yt and we can compute the previous posterior

probability as

prob(st = jjI +t ) =f (yt; st = j jI +t1)

f (ytjI +t1)

=f (ytjst = j; I +t1) prob(st = j jI +t1)

32Xk=1

f (ytjst = k; I +t1) prob(st = kjI +t1)

; (23)

where

f (ytjst = j; I +t1) = 1

r (2)N j

tjt1expf1

2(yt y

jtjt1

)0

jtjt1

1

(yt y jtjt1

)g (24)

with y jtjt1

the one step ahead forecast of yt given information up to time t 1 if st = j

and jtjt1

its variance-covariance matrix. Now for each value of j 2 C of st

prob(st = jjI +t1) = prob(st = h; st1 = l; st2 = m; st3 = n; st4 = ojI +t1) (25)

= plh prob(st1 = l; st2 = m; st3 = n; st4 = ojI +t1);

l ;h;m;n;o 2 f0; 1g: So by (25), (24) and (23), we can compute the desired …ltered prob-

abilities of being in expansion/recession given by (21). We also obtain the next input

needed by the algorithm prob(st = h; st1 = l; st2 = m; st3 = njI +t ) as

prob(st = h; st1 = l; st2 = m; st3 = njI +t ) =

=2X

o=1

p(st = h; st1 = l; st2 = m; st3 = n; st4 = ojI +t )

=X j2O

p(st = j jI +t )

19



where j takes two possible values given in O = fst = j jst = h; st1 = l; st2 = m; st3 =

ng for given h; l;m; n.

Assume now that we do not observe y1;t and therefore y+

t = (zt; y2;t;:::;yN;t)

0

=(zt; y0

t )0 where y0t are the observed variables at t: Then, since zt is independent from

the remaining variables in the model with density f (zt), (23) is computed as

prob(st = jjI +t ) =f (y

t ; st = j jI +t1)f (zt)

f (yt jI +t1)

(26)

=f (y

t jst = j; I +t1) prob(st = j jI +t1)f (zt)32Xk=1

f (yt jst = k; I +t1) prob(st = kjI +t1)f (zt)

(27)

= f (y

t js

t = j; I

+

t1) prob(s

t = j jI

+

t1)32Xk=1

f (yt jst = k; I +t1) prob(st = kjI +t1)

(28)

where f (yt jst = j; I +t1) can be automatically obtain with a time dependent state space

model for y+t where all the system matrices are zero out when y1;t = gt is not observed

and the error term of the measurement equation is equated to zt with variance R as

proposed in Mariano and Murasawa (2003). So the scale factor that appears both in the

numerator and denominator of (27) f (zt) cancels out, equations (28) and (23) resemble

each other and therefore no modi…cations of the computing algorithm are needed (further

than considering the time varying Kalman …lter to zero out the missing observations).

The intuition behind the result is that if we …ll the data holes with outcomes of a random

variable that is independent of the remaining variables in the model and, in particular,

independent of the state, the estimated probabilities are not a¤ected by them.

3.3 Ragged edges

In this section we analyze how we handle the di¤erent timing of new information and

how this a¤ects our estimation of the state probabilities. With a simple example we will

illustrate the importance of processing the new information as soon as it arrives, instead

of waiting to have all the information for all the variables at each period of time. For

instance, assume that with the information set I t, the probability of being in recession is

1, that is prob(st = 1jI t) = 1 and prob(st = 0jI t) = 0: The arrival of the …rst GDP data

20



(‡ash) is about 45 days after the end of the quarter. On a monthly basis, this means that

we have to make 2 steps ahead forecasts (before this arrival) or 1 step ahead forecasts

after the arrival of the ‡ash data. What can we say about the actual GDP at t + 2 bythen? For illustrative purposes assume that p00 = 0:9 and p11 = 0:7 (of course, p01 = 0:1

and p10 = 0:3). These values are about the ones estimated in many empirical applications.

Then, can we say that we are still in a recession now in t + 2? Taking into account that

prob(st+2 = 1jI t) =1X

i=0

prob(st+1 = ijI t) pi1

=1

Xi=0

1

X j=0

p ji pi1 prob(st = j jI t)

then, prob(st+2 = 1jI t) = 0:52 for the values of this example. That is, at time t + 2, we

can hardly say anything about being in either of the two states, even though at t we were

certainly in one of them. How can we improve our believes of being in a certain state

in real time? If instead of waiting to have all the values for all the variables at each t

we process the information as it arrives, we can modulate our believes according to the

evaluation of the likelihood of the variables already available. Actually, monthly indicators

are usually published much more timely than quarterly series. In addition, indicators based

on surveys (soft indicators) are more promptly issued than economic activity indicators

(hard indicators) and their samples are usually longer. This implies that forecasters need

a model to compute forecasts from unbalanced sets if they do not want either to loose

valuable information at the time of the forecast or to wait until balanced panels become

available.

4 Speci…cation of the model

As we have seen, the Markov-switching dynamic factor model consists of a factor model

which decomposes the joint dynamics of the business cycle indicators into two components:

common and idiosyncratic.

In the related literature, several speci…cations of the nonlinear dynamics of the common

factor dynamics have been suggested. Kim and Yoo (1995) and Chauvet (1998) allowed

21



intercept term to be regime dependent. In the speci…cation of Kim and Nelson (1998)

it is the mean instead of the intercept what is allowed to exhibit regime shifts. In this

paper, we follow Camacho and Perez Quiros (2007) to assume that the factor dynamicscan be captured by shifts between the business cycle states and we set the autoregressive

coe¢cients equal to zero in (14). Within this framework, we can label st = 0 and st = 1

as the expansion and recession states at time t if 0 > 0 and 1 < 0. Hence, the common

factor is expected to exhibit positive rates of growth in expansions and negative rates of

growth in recessions.

To specify the dynamic factor model of ‡ash, …rst, second, employment, hard and

soft indicators, let us …rst assume that missing data do not appear in the data set so

that quarterly series are observed monthly and vintage panels are balanced. We assume

that the factor captures the common dynamics in the growth rates of real activity data.

However, since survey indicators in Europe are designed to capture annual growth rates

of the reference series (see European Commission, 2006), we impose that the levels of soft

indicators depend on the sum of current values of the common factor and its last eleven

lagged values.

Let us collect the rh hard indicators in the vector Zht and the rs soft indicators in

the vector Zst . Let lt be the quarterly employment growth rate, and let u1t, u2t, Uht ,

and Ust be the scalars and rh-dimensional and rs-dimensional vectors which determine

the idiosyncratic dynamics of GDP. The dynamic of the business cycle indicators can be

22



stated as

0BBBBBBBBBBBB@

y2ndt

Zht

Zst

lt

y1stt

yf t

1CCCCCCCCCCCCA=

0BBBBBBBBBBBBBBB@

1

13f t +

23f t1 + f t2 + 2

3f t3 + 13f t4

2f t

3

11X j=0

f t j

413f t +

23f t1 + f t2 + 2

3f t3 + 13f t4

113f t +

23f t1 + f t2 + 2

3f t3 + 13f t4

113f t +

23f t1 + f t2 + 2

3f t3 + 13f t4

1CCCCCCCCCCCCCCCA

+

0BBBBBBBBBBBB@

13u1t +

23u1t1 + u1t2 + 2

3u1t3 + 13u1t4

Uht

Ust

13u2t +

23u2t1 + u2t2 + 2

3u2t3 + 13u2t4

13u1t +

23u1t1 + u1t2 + 2

3u1t3 + 13u1t4

13u1t +

23u1t1 + u1t2 + 2

3u1t3 + 13u1t4

1CCCCCCCCCCCCA

+

0BBBBBBBBBBBB@

0

0

0

0

e2t

e1t + e2t

1CCCCCCCCCCCCA

;(29)

where Uht = (v1t;:::;vrht)

0, Ust = (vrh+1t;:::;vrt)0, and r = rh + rs. The factor loadings,

=

1 02 0

3 4

0, measure the sensitivity of each series to movements in the

latent factor and have dimensions that lead them to be conformable with each equation.

The dynamics of the model is achieved by assuming that

u1t = b1u1t1 + ::: + bm2u1tm2

+ u1t ; (30)

v jt = c j1v jt1 + ::: + c jm3v jtm3

+ vjt ; (31)

u2t = d1u2t1 + ::: + dm4u2tm4

+ u2t ; (32)

where f t i:i:d:N

0; 2

f

, u1t i:i:d:N

0; 2

u1

,

vjt i:i:d:N

0; 2

vj

, with j = 1; :::; r;

and u2t i:i:d:N

0; 2

u2

. All the covariances are assumed to be zero and we set the

variance of the common factor, 2f , equal to one.

4

Consider the following state space representation of the Markov-switching dynamic

factor model

yt = Hht + wt; (33)

ht = Dst + Fht1 + t; (34)

4 This identifying assumption is standard in dynamic factor models.

23



where Dst =

st 01;n1

, st = i; j, and

0@ wt

t1A ~iidN 0@0;0@ R 0

0 Q1A1A : (35)

The Appendix provides more details on the model structure and the speci…c forms of these

matrices.

Let us now describe how to handle missing data. For this purpose, we follow Mariano

and Murasawa (2003) and substitute missing observations with random draws t from

N (0; 2). This implies replacing the i-th row of yt; Ht; wt and the i-th element of the

main diagonal of Rt, by yit,H it, w

it, and Riit. The starred expressions are yit, H it, 0, and

0 if variable yit is observable at time t, and t, 01, t, and 2 in case of missing data.

Accordingly, this transformation converts the model in a time-varying state space model

with no missing observations and the nonlinear version of the Kalman …lter can be directly

applied to yt , H

t , wt , and R

t .

Maximizing the exact log likelihood function is computational burdensome since at

each time t, the …lter produces a 2-fold increase in the number of cases to consider (since

at each t, the variable st can take two new values and therefore, at each t we have 2t possible

paths to consider when evaluating the likelihood). Two solutions have been proposed in

the literature. The …rst solution based on collapsing, that is to approximate the mixture of

2t Gaussian distributions with a mixture of m Gaussians. It was proposed by Kim (1994)

and used by Kim and Yoo (1995) and Chauvet (1998). The second solution, which is

based on and Bayesian estimation methods of Gibbs sampling, was proposed by Kim and

Nelson (1998) and gets approximation-free inference at the cost of being computationally

harder. Based on the results of Chauvet and Piger (2005), who show that the approximated

method works very well in practice, we use the Kim’s algorithm to compute inference in

the nonlinear Kalman …lter. In this paper we will use two di¤erent mixtures of Gaussians

in the approximation, a simple one with m = 22 densities and a more complex one with

m = 25 Gaussians motivated by equation (18).

To describe how the model can be estimated in the case of m = 22 Gaussians, let h(i;j)tj

be the forecast of ht based on information up to period and the realized states st1 = i

24



and st = j , and let P(i;j)tj

be its covariance matrix. The prediction equations become

h(i;j)tjt1

= j + Ht hi

t1jt1; (36)

P(i;j)tjt1

= Ht Pi

t1jt1H0

t + Q: (37)

The conditional forecast errors are (i;j)tjt1

= yt H

t h(i;j)tjt1

and (i;j)tjt1

= Ht P

(i;j)tjt1H0

t + Rt

is its conditional variance. Hence, the log likelihood can be computed in each iteration as

l(i;j)t = 1

2 ln

2 (i;j)tjt1

1

2(i;j)0

tjt1

(i;j)tjt1

1(i;j)tjt1: (38)

The updating equations become

h(i;j)tjt

= h(i;j)tjt1

+ K(i;j)t (i;j)

tjt1; (39)

P(i;j)tjt

= P(i;j)tjt1

K(i;j)t H

tP(i;j)tjt1; (40)

where the Kalman gain, K(i;j)t , is de…ned as K

(i;j)t = P

(i;j)tjt1

H0t

(i;j)tjt1

1.

To collapse the means and variances in order to apply (36) and (37) next period, Kim

(1994) approximates h jtjt

and P jtjt

by the weighted averages of the updating equations

where the weights are given by the probabilities of the Markov state:

h jtjt

=

1Xst1=0

p (st = j; st1 = ijI t) h(i;j)tjt

p (st = j jI t) (41)

P jtjt

=

1Xst1=0

p (st = j; st1 = ijI t)

P(i;j)tjt

+

h jtjt h

(i;j)tjt

h j

tjt h

(i;j)tjt

0 p (st = j jI t)

: (42)

To conclude this section, let us point out one additional advantage of this proposal

against standard Markov-switching dynamic speci…cations applied to balanced data sets:

our model can easily compute GDP growth forecasts. Recall that our method mixes

frequencies and …lls in outliers following the rule of replacing missing by random numbers

which allows us to include GDP growth as an additional business cycle indicator. In

this context, if we call T the last month for which we have observed GDP growth and

h(i;j)T +1jT

( j) the j -th element of h(i;j)T +1jT

, forecasts for month T + 1 can be computed by the

25



model as

y2ndT +1=T = 11

3h(i;j)T +1jT

(1) + 2

3h(i;j)T +1jT

(2) + h(i;j)T +1jT

(3) + 2

3h(i;j)T +1jT

(4) + 1

3h(i;j)T +1jT

(5)

+

1

3h(i;j)T +1jT

(13) + 2

3h(i;j)T +1jT

(14) + h(i;j)T +1jT

(15) + 2

3h(i;j)T +1jT

(16) + 1

3h(i;j)T +1jT

(17)

; (43)

It is worth noting that including a missing observation y2ndT +1 in the data set, the model

will automatically replace the missing by a dynamic forecast. Following the same reason-

ing, forecasts for longer horizons and forecasts for other indicators can be automatically

computed.

5 Empirical results

5.1 Data description

The empirical analysis focuses on thirteen business cycle indicators. The forecasting sam-

ple goes from April 2004 to January 2009. According to Camacho and Perez Quiros (2009),

the set of business cycle indicators include: (1) three quarterly series, second GDP growth

releases, its two preliminary announcements ‡ash and …rst, and employment, all of them

in quarterly growth rates, (2) monthly hard indicators, Euro area Industrial Production

Index (IPI, excluding construction), the Industrial New Orders index (INO, total manu-

facturing working on orders), the Euro area total retail sales volume, and the extra-Euro

area exports, all of them in monthly growth rates, and (3) …ve soft indicators, the Euro-

zone Economic Sentiment Indicator (ESI), the German business climate index (IFO), the

Belgian overall business indicator (BNB), and the Euro area Purchasing Managers con…-

dence Indexes (PMI) in the services and manufactures sectors, which are loaded in levels.

In the analysis, data are standardized by substracting the sample mean from each variable

and dividing by its standard deviation.

Table 1, which reports the last …gures of the time series, illustrates the main character-

istic of how the ‡ow of macroeconomic information may a¤ect real time forecasting. Since

GDP and Employment releases appear quarterly, the two …rst months of each quarter are

treated as missing data. Surveys have very short publishing delays of one (or even less)

months while hard data are released with a relatively longer delay of about two months.

26



Finally, forecasts for particular quarters of GDP spread over a period of nine months.

Accordingly, the nine months of missing data after the last GDP growth observation (Oc-

tober 2008 to January 2009) will be replaced by short-term forecasts by the model. Assoon as the GDP …gure for the last quarter is available, the nine-moth forecasting horizon

will be moved forward conveniently.

5.2 In-sample analysis

The in-sample analysis was carried out with the vintage data set that was available on

January, 21th 2009. The unsynchronized way on which data are published is illustrated

in Table 1. In this table we can observe the particularities of real-time forecasting. Data

for quarterly series appear just in the third month of each quarter and, although the

vintage refer to 2008, their …gures for the fourth quarter of 2007 are not available yet.

Soft indicators contain data until January 2008 while hard indicators exhibit their typical

publication delays of one and two months. In the next forecasting dates but not in this

vintage, preliminary advances of GDP growth (‡ash and …rst) were already available for

the last quarter of 2008.

The model speci…cation has proceeded under several assumptions regarding regime

switching. We need to perform several exercises to provide suggestive evidence as to

whether the model accords to the model assumptions. First, we assumed that the positive

autocorrelation of the common underlying economic activity can be captured by regime

switching rather than by autoregressive parameter. To provide evidence that this assump-

tion is realistic, we estimate the linear version of the common factor model of Camacho

and Perez Quiros (2009) using the same data set and we obtain that the sample correlation

between both factors is 0.97.5

The second exercise to assess the robustness of our assumptions has to do with the viewthat the factor exhibits a business cycle dynamics. To evaluate the extent to which data

reinforce this thesis we examine the factor dynamics. The maximum likelihood estimates

of parameters show that factor is expected to be signi…cantly positive (value of 0:37 with

standard deviation of 0:10) in the state st = 0 while it is signi…cantly negative (2:04

5 Normalizing linear and nonlinear factors leads to very similar graphs.

27



with standard deviation of 0:31) in the state st = 1. Accordingly, we can associate these

states as expansions and recessions. In addition, each regime is highly persistent, with

estimated probabilities of one regime to be followed by the same regime of 0:97 and 0:93(standard deviations of 0:02 and 0:06), respectively. Finally, another interesting business

cycle implication of the Markov framework is that one can derive the expected number

of quarters that the business cycle phases prevail. Conditional on being in state 1, the

expected duration of a typical US expansion is (1 b p)1 or 32:33 months, and the expected

duration of recession is likewise (1 bq )1 or 14:28 months. These estimates accord with

the well-known fact that expansions are longer than contractions on average.

The last exercise is related to the statistical signi…cance of the nonlinear model. Ap-

plied to the same data set, the log likelihood rises from 178:10 to 128:27 when non-

linearities are accounted for in the dynamic factor model so that the likelihood ratio test

statistics would be about 100. Although standard econometrics cannot be employed in this

context, we consider that the increases in likelihood are signi…cant enough to be con…dent

that nonlinearities appear to be part of data generating process.

Although the scope of this paper is more ambitious than constructing a coincident

index, we must check if its dynamics is consistent with the Euro area business cycles since

the model was constructed under the assumption that the indicators share the underlying

aggregate economic activity dynamics whose pattern is captured by the common factor.

For this purpose, the switching factor coincident index estimated in this paper is compared

with the Eurocoin which is published each month by CEPR and is considered the leading

coincident indicator of the euro area business cycle. A visual inspection of Figure 1 suggests

that the common factor and the Eurocoin move together synchronously. Although the

Eurocoin is designed to track the medium term trend by removing short-run ‡uctuations

from a large data set (so that the index moves smoothly), the sample correlation betweenthese two series is about 0:7. Remarkably, there seems to be commonality among switch

times. While the indicators ‡uctuate around their respective means, the broad changes

of direction in the series seem to mark quite well the cycles. In particular, they exhibit

periods of pronounced drops in dates for which GDP growth rates deteriorate signi…cantly:

1992-1993, 2001 and 2008. Of special interest for nowcasting is the most recent period for

28



which both indicators reached a peak in the beginning of 2008 and have declined since

then.

To examine the correlation of these indicators and the factor, Table 2 reports the maxi-mum likelihood estimates of the factor loadings (standard errors within parentheses). The

estimates are always positive and statistically signi…cant, which agrees with the standard

view that the indicators are procyclical. With respect to the size of the correlations, the

economic indicators with larger loading factors are those corresponding to IPI ( 0:36), INO

(0:33) and GDP (0:29). Soft indicators exhibit much lower loading factors, with a maxi-

mum of 0:11 in the case of PMI in manufactures. This results is tempted to be interpreted

in contrast to using surveys as coincident indicators. However, Camacho and Perez Quiros

(2009) show that the in-sample estimates of the loading factors do not re‡ect the timely

advantages observed in real time exercises.

Table 3 shows some of the key outputs of the model: forecasts of GDP growth and

the corresponding inferences about the business cycle provided by the Markov-switching

speci…cation for quarters 2008.4, 2009.1 and 2009.2. We call them backcasting, nowcasting

and forecasting …gures, respectively. In line with to the current pessimism about the short

term evolution of the underlying economic activity, this table suggests that output is

expected to fall in the next quarter although the intensity of the falls are expected to

decline from 1:23 in backasting to 0:07 in forecasting. According to these estimates,

the expected probability of staying in recession in the next future is very high. Finally,

this table shows the forecasts for each of the business cycle indicators used in the model.

One additional application of the Markov-switching dynamic factor speci…cation de-

veloped in this paper is that the model provides an ideal framework to date the historical

Euro area business cycle phases. For this purpose, we show in Figure 3 the monthly full

sample smoothed inferences that the economy is in recession. To confront them with thedata, this …gure adds the quarterly GDP growth estimates which are equal to the actual

…gures in the third month of each quarter. For international comparisons, the US reces-

sions dated by the NBER are included in the graph as shaded areas. From this …gure, we

observe that the inferred probabilities create clear signals about the business cycle states.

High probabilities of recessions appear in 1992-1993, 2001 and 2008 which correspond to

29



low (or even negative) growth. They also reveal one episode of uncertainty in the econ-

omy in 2002. This …gure reveals that the Euro area economy su¤ers in 2002 from great

uncertainty in the economy since recession probabilities rise up to 0.7 in this year. Finally,the …gure shows that the business cycle concordance between the Euro area and the US

has increased signi…cantly during the last two decades. While US clearly leads the 1991

recession, the 2001 and 2008 recessions are highly synchronized.

Since we are interested in obtaining speci…c turning point dates, we will require a rule

to convert the recession probabilities into a dichotomous variable which signals whether

the economy is in an expansion or a recession. Following Chauvet and Piger (2008), we

take a conservative approach.6 To establish a peak (trough) at time t, we require that

the probability of recession moves from above 0:8 (below 0:2) and remains above (below)

this threshold for three consecutive months. Table 3 reports the turning points derived

from this criterion. For comparison purposes, the NBER o¢cial dates are also shown

in the table. In relation with the US, the recession in the early nineties clearly …nishes

later, but the historical turning points dates in 2001 and 2008 recessions roughly coincide.

According to this exercise, we conclude that discrepancies in business cycle synchronicity

between the US and the Euro area have been recently diminished.

To illustrate the usefulness of Markov-switching models to transform the information

about the economic evolution that is contained in business cycle indicators, we develop the

following exercise. When a business cycle indicator is published, the statistical agency in

charge of its release tries to provide the economic agents with an outlook of the economy

which is supposed to be contained in the indicator release. However, the intuition behind

the indicator releases are not so easy since they are no more than numbers. The Markov-

switching dynamic factor model becomes a …ltering rule which extracts the indicator’s

information about the state of the economy, by transforming the indicator release intoprobabilities of recession which are much easy to interpret.

Suppose we are in the January 2006 which was a year that can be considered as an

economic expansion, and we simulate the possible outcomes of the following BNB release

6 Applying the second step in the procedure described by these authors does not change our turning

point dates.

30



from about -32 to 2. Using these potential outcomes, we infer which is the probability

of recession for that month. Figure 4 (bottom line) displays the predicted probability

of recession, associated to each BNB potential issue growth. In addition, we present intins …gure the results of a similar analysis, but applied to the probability of recessions for

January 2009, which can be considered as an economic recession. It is worth mentioning

that we are using exclusively the information available at the dates of the forecasts. As

we can see from the pictures, the curve associated to 2006 is clearly shifted down. This

implies that the same BNB value contains very di¤erent information about the probability

of an imminent recession depending on the period that we consider. Speci…cally, in 2009,

a BNB value of -20 would be associated with a probability of recession of almost 0.8.

However, in 2006, the same value of BNB would have implied a recession probability next

period close to 0.3. The intuition is clear. In order to predict that a recession is coming,

we need stronger evidence in the BNB behavior in expansions that in recessions to believe

that a recession is imminent.

The Markov-switching behavior assumed in this speci…cation also implies richer rela-

tionships between the business cycle indicators and GDP previsions than those suggested

by linear dynamic factor models. The intuition behind the nonlinear responses is clear:

new releases are be converted into inferences about the state of the business cycle which

are used in computing output predictions by the model. To illustrate this nonlinear e¤ect,

we plot in Figure 5 the expected GDP growth rates that would be forecasted from dif-

ferent potential realizations of BNB. For this purpose, we call the Kalman …lter with the

historical time series of BNB which is enlarged with each of these simulated values and we

plot in the graph the forecasts of the di¤erent expected values of output growth. For ex-

treme negative values of the indicator, the model would inferred probabilities of recession

close to one and GDP which are used to forecast growth rates values which are close to1:5. As the values of BNB increase, the model predicts relatively better values of GDP

growth which increase almost linearly with BNB since then until values the indicator of

about 20. Around this value, which correspond to the values for which Figure 4 showed

a substantial decline in the inferred probability of recession, the line suddenly increases

its shape and the responses of expected GDP to BNB values dramatically increase. As

31



documented in Figure 4, for values of BNB about 9, the inferred probabilities of recession

become very low indicating that the economy would be in the expansionary phase. Since

then, the expected growth to BNB become quasi linearly trended again.

5.3 Real-time analysis

We examine the real time performance of the model in predicting turning points in the

last 2008 recession.

Figure 3 shows the probabilities of recessions that would be inferred daily by a fore-

caster who used the information available at the day of the forecast. Although we dated

the last peak in February 2008 by using the full sample estimated probabilities, in the

beginning of the year the probability of recession is almost zero and remains negligible

until Summer. The …ve-month lag in identify the peak re‡ects the typical uncertainty

regarding real time analyses. However, this lag is reasonably short if we recall that the

NBER waited one year in dating the last peak. Since bad news had been accumulated

in that period, on the 9th of July the probability of recession has a pronounced increase

to 0:34 due to the negative …gure of Industrial Production. One of the worse historical

records of Exports let the recession probability to reach the phycological threshold of 0:5

on the 18th of July. At the end of this month, the data that were published were among

the worse in the history of almost all the business indicators which implied that the prob-

ability of recession became greater than 0.9. Consecutive good news were not observed

since then so that the probability of recession remained around this value until the last

vintage. Hence, the Markov switching dynamic factor models had unequivocally signaled

in July 2008 that a peak in the euro area had occurred.

6 Conclusion

Markov-switching dynamic factor models are becoming very popular in empirical analyses.

However, they do not account for some speci…cities which are typical of real time fore-

casting exercises. They do not mixes frequencies, do not model data revisions and do not

account for ragged edges. Not accounting for these publication patterns would imply that

32



forecasters using traditional Markov-switchcing dynamic factor models to develop early

assessments of the economic evolution can involve substantial costs since forecasters are

restricted either to loose valuable information at the time of the forecast or to wait untilbalanced panels become available. We propose a model in this paper which is able to deal

with all of them and we show that it is a potentially very useful tool to be used in the day

to day monitoring of the Euro area economy

33



Appendix ATo illustrate how the matrices stated in the measurement and transition equations look

like, let 0i;j be a matrix of (i j) zeroes, I r be the r-dimensional identity matrix, and be the Kronecker product. According to the empirical application, let us assume that

m1 = m2 = m4 = 6, m3 = 2, rh = 4, and rs = 5. For simplicity, let us assume that all

variables are always observed at a monthly frequency.

In this example, the measurement equation, yt = Hht + wt; with wt i:i:d:N (0; R),

can be expressed as

yt =

y2ndt Z h

0

t Z s0

t lt y1stt yf t

0

; (44)

wt = 01;r+4; (45)

R = 0r+4;r+4; (46)

ht = (f t;:::;f t11; u1t;:::;u1t5; v1t; v1t1;:::;vrt; vrt1; u2t;:::;u2t5; e1t; e2t)0 : (47)

The matrix H is in this case

H =

0BBBBBBBBBBBB@

H 11 01;6 H 12 01;8 01;10 01;6 0 0

H 21 0rh;6 0rh;6 H 22 0rh;10 0rh;6 0rh;1 0rh;1

H 31 H 31 0rs;6 0rs;8 H 32 0rs;6 0rs;1 0rs;1

H 4 01;6 01;6 01;8 01;10 H 12 0 0

H 11 01;6 H 12 01;8 01;10 01;6 0 1

H 11 01;6 H 12 01;8 01;10 01;6 1 1

1CCCCCCCCCCCCA

; (48)

where

H 11 =

1

32

1

3 11

32

1

3 0

; (49)

H 12 = 13

23 1 1

323 0 ; (50)

H 22 = I rh 1 0 ; (51)

H 32 = I rs

1 0

; (52)

H 4 =

43

243 4

43

243 0

; (53)

H 21 is a (rh 6) matrix of zeroes whose …rst column is 2, and H 31 is a (rs 6) matrix

whose columns are 3.

34



Using the assumptions of the underlying example, the transition equation, ht = st +

Fht1 + t, can be stated as follows. Let Q be a diagonal matrix in which the entries

inside the main diagonal are determined by the vector

diag(Q) =

2f 01;11 2

u1 01;5 2v1 0 ::: 2

vr 0 2u2 01;5 2

e1 2e2

0; (54)

The matrix F becomes

Fst =

0BBBBBBBBBBBBBBB@

a 012;6 012;8 012;10 012;6 0 0

06;12 b 06;8 06;10 06;6 0 0

08;12 08;6 ch 08;10 08;6 0 0

010;12 010;6 010;8 cs 010;6 0 006;12 06;6 06;8 06;10 d 0 0

01;12 01;6 01;8 01;10 01;6 0 0

01;12 01;6 01;8 01;10 01;6 0 0

1CCCCCCCCCCCCCCCA

; (55)

where

a =

0BBBBBB@

0 ::: 0 ::: 0 0

1 ::: 0 ::: 0 0...

... . . .

... ...

...

0 ::: 0 ::: 1 0

1CCCCCCA

; (56)

b =

0BBBBBB@b1 ::: b5 b6

1 ::: 0 0...

. . . ...

...

0 ::: 1 0

1CCCCCCA ; (57)

ci =

0BBBBBBBBB@

c11 c12 ::: 0 0

1 0 ::: 0 0...

... . . .

... ...

0 0 ::: cr1 cr2

0 0 ::: 1 0

1CCCCCCCCCA; (58)

35



d =

0BBBBBB@

d1 ::: d5 d6

1 ::: 0 0.

.. .

. .

.

..

.

..

0 ::: 1 0

1CCCCCCA : (59)

36



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