Markov-switching mixed-frequency VAR models

International Journal of Forecasting (

)

–

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International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast

Markov-switching mixed-frequency VAR models Claudia Foroni a , Pierre Guérin b , Massimiliano Marcellino c,d,∗ a

Norges Bank, Norway

b

Bank of Canada, Canada

c

Bocconi University, IGIER, Italy

d

CEPR, United Kingdom

article

info

Keywords: Markov-switching MIDAS Mixed-frequency VAR Nowcasting Forecasting

abstract This paper introduces regime switching parameters to the Mixed-Frequency VAR model. We begin by discussing estimation and inference for Markov-switching Mixed-Frequency VAR (MSMF-VAR) models. Next, we assess the finite sample performance of the technique in Monte-Carlo experiments. Finally, the MSMF-VAR model is used to predict GDP growth and business cycle turning points in the euro area. Its performance is then compared with those of a number of competing models, including linear and regime switching mixed data sampling (MIDAS) models. The results suggest that MSMF-VAR models are particularly useful for estimating the status of economic activity. © 2014 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction Macroeconomic variables are sampled at a range of different frequencies. Thus, in empirical studies, the econometrician has to choose the sampling frequency of the model. The first solution consists of aggregating all of the high-frequency variables to the frequency of the lowest frequency variable. This is a common choice in VAR studies. Another solution is to interpolate the low-frequency variable to the frequency of the high-frequency variable, which is usually done using the Kalman filter (see e.g. Aruoba, Diebold, & Scotti, 2009; Foroni & Marcellino, 2014; Frale, Marcellino, Mazzi, & Proietti, 2011; Giannone, Reichlin, & Small, 2008; Mariano & Murasawa, 2003). Bai, Ghysels, and Wright (2013) and Kuzin, Marcellino, and Schumacher (2011) use the Kalman filter for VAR forecasting with mixed-frequency data. See also Foroni, Ghysels, and Marcellino (2013) for an overview. Ghysels (2012) recently introduced a mixed-frequency VAR model where the vector of dependent variables includes both low-frequency and high-frequency variables, with the

∗

Corresponding author at: Bocconi University, IGIER, Italy. E-mail address: [email protected] (M. Marcellino).

latter being stacked depending on the timing of the release. The advantage of this approach is that it avoids the estimation of a mixed-frequency VAR model via the Kalman filter, which has often proven computationally cumbersome. Alternatively, in a Bayesian context, Chiu, Eraker, Foerster, Kim, and Seoane (2011) developed a Gibbs sampler that allows the estimation of VAR models with irregular and mixed-frequency data, see also Schorfheide and Song (2012). Finally, one can also work with mixedfrequency data directly. This is the approach that is used by MIDAS specifications, where the aggregation of the high-frequency variable is carried out in a parsimonious and data-driven manner (see e.g. Andreou, Ghysels, & Kourtellos, 2013; Francis, Ghysels, & Owyang, 2011; Ghysels, Santa-Clara, & Valkanov, 2004). Banbura, Giannone, Modugno, and Reichlin (2012) and Foroni and Marcellino (2013) provide overviews on the use of mixed-frequency data in econometric models. Increasing numbers of works are evaluating the effects of the choice of the sampling frequency in a macroeconomic forecasting context (starting with Clements & Galvao, 2008, but see also Banbura et al., 2012; Foroni & Marcellino, 2014; Foroni, Marcellino, & Schumacher, in press,

http://dx.doi.org/10.1016/j.ijforecast.2014.05.003 0169-2070/© 2014 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

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C. Foroni et al. / International Journal of Forecasting (

for example) but others consider structural studies (Foroni & Marcellino, in press; Kim, 2011). However, relatively few studies have considered the issue of time variation in models dealing with mixed-frequency data (see e.g. Camacho, Perez-Quiros, & Poncela, 2012; Carriero, Clark, & Marcellino, 2013; Galvão, 2013; Guérin & Marcellino, 2013; Marcellino, Porqueddu, & Venditti, 2013), and most of these few studies have only considered the univariate case. In this paper, we allow for regime switching parameters in the mixed-frequency VAR model, by introducing the Markov-Switching Mixed-Frequency VAR model (MSMF-VAR). The relevance of Markov-Switching models for econometrics is now well established, and large numbers of studies have been published since the seminal paper by Hamilton (1989), see in particular Krolzig (1997) in a VAR context; however, all of these studies are based on variables of the same frequency. The MSMF-VAR model permits us to model time variation explicitly in the relationship between the high- and low-frequency variables. As a by-product of our estimation, we also obtain highfrequency estimates of the low-frequency variable(s) in the case of the mixed-frequency VAR model estimated via the Kalman filter. For example, in a system with quarterly GDP and monthly indicators, we can obtain monthly estimates of GDP, which are of direct interest. Modeling the time variation through Markov-Switching models is also attractive, since it allows us to estimate and forecast the probabilities of being in a given regime endogenously. We introduce regime switching parameters to the mixed-frequency VAR model estimated via the Kalman filter, as well as to the mixed-frequency VAR model of Ghysels (2012), which does not require the estimation of a state space model, thus permitting a straightforward estimation. We compare these new models with the Markov-switching MIDAS model of Guérin and Marcellino (2013), and discuss the distinctive features of these models. The paper is structured as follows. Section 2 introduces the Markov-switching mixed-frequency VAR models and discusses their use for forecasting the variables of interest and the underlying unobservable states. In a Monte-Carlo experiment, Section 3 evaluates the forecasting performances of the MSMF-VAR models relative to those of a number of alternative forecasting models. Section 4 presents an empirical application to the prediction of GDP growth and business cycle turning points in the euro area. Section 5 concludes.

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2.1.1. Mixed-frequency data modeled via a state- space representation For the sake of clarity, we illustrate the model using a single low-frequency variable, sampled at the quarterly frequency, and a single high-frequency variable, sampled at the monthly frequency. This corresponds to the standard macroeconomic forecasting context, where one wants to forecast a quarterly variable, say GDP growth, using a monthly indicator, such as industrial production, an interest rate spread, or a survey of economic conditions. Models with more low- or high-frequency variables can be specified along the same lines. A key feature of the mixed-frequency VAR (MF-VAR) model is that it works at the frequency of the highfrequency variable. Thus, the quarterly GDP is disaggregated at the monthly frequency using the geometric mean: Yt = 3(Yt∗ Y ∗

t − 13

Y∗

t − 23

1

)3 ,

(1)

where Yt is the observed quarterly GDP in quarter t and Yt∗−h is the unobserved monthly GDP of the months h =

{0, 13 , 23 } belonging to quarter t. After taking logarithms

and performing some transformations, this yields:

2 2 1 1 ∗ yt + y∗ 1 + y∗ 2 + y∗t −1 + y∗ 4 , (2) t− 3 3 3 t− 3 3 3 t− 3 where yt is the observed quarterly growth rate in quarter t and y∗t −h is the monthly growth rate of GDP in months t − h. Note that we use the geometric mean rather than the arithmetic mean. This is because using the arithmetic mean rather than the geometric mean would involve the estimation of a non-linear state space model, which would complicate the estimation. Besides, the approximation involved in the use of the geometric mean is typically small when dealing with GDP growth rates in developed economies. Finally, it is the approach most commonly used in the literature, see for example Mariano and Murasawa (2003).1 The state vector of unobserved variables st includes the monthly changes in GDP growth y∗t −h , as well as the high frequency indicator xt −h . It is defined as follows:

yt =





zt

 . 

st =  ..  , zt − 4 3

2. Modeling and forecasting with Markov-switching mixed-frequency models

where:

2.1. Markov-switching mixed-frequency VAR

The state space representation of the MF-VAR(p) is described by the following transition and measurement equations:

In this subsection, we introduce two alternative Markov-switching mixed-frequency VAR (MSMF-VAR) models. First, we present the MSMF-VAR model in state space form so as to accommodate the different frequency mixes. This model is labeled MSMF-VAR (KF), where KF stands for the Kalman filter. Second, we introduce regime switching parameters to the mixed-frequency VAR model proposed recently by Ghysels (2012), where mixed frequency data are analyzed through a stacked-vector system. This model is labeled MSMF-VAR (SV), where SV stands for stacked-version.

 zt −h =

y∗t −h . x t −h



st +h+ 1 = Ast +h + Bvt +h+ 1 , 3



yt +h x t +h

(3)

3



 =

µy + Cst +h , µx 

(4)

1 Note that one could also handle missing observations by varying the dimension of the vector of observables as a function of time t (see for example Schorfheide & Song, 2012).

C. Foroni et al. / International Journal of Forecasting (

where vt +h ∼ N (0, I2 ) and the system matrices A, B and C are given by:

 

A1 , A2

A=

A1 = [81 . . . 8p 02∗2(5−p) ], A2 = [I8 08∗2 ], B=

 1/2  Σ 08∗2

,

and C = [H0 . . . H4 ], where the matrix C contains the coefficient matrices in 4 i the lag polynomial H (Lm ) = i=0 Hi Lm , which is defined according to the aggregation constraint defined in Eq. (2):

 H(Lm ) =

1

0

3 0

 +



 +

1 2 3 0

0

2 3 0



0

+

 Lm +

0

 L3m

0



1 3 0

1 0



0 2 L 0 m



0 0

L4m .

Note that this specification of the MF-VAR differs from that reported by Bai et al. (2013) and Camacho and PerezQuiros (2010), where the low-frequency variable is related to the high-frequency indicators through common factors (i.e., they use a mixed-frequency factor model). In contrast, the specification we consider is similar to that of Kuzin et al. (2011) and Zadrozny (1990). We now include regime switches in potentially all of the parameters of the MF-VAR(p) model: the intercepts, the autoregressive parameters and the variance of the innovations.2 The state space representation of the Markovswitching MF-VAR (MSMF-VAR) is then given by:



st +h+ 1 = A St +h+ 1 3 3



yt +h x t +h







st +h + B St +h+ 1 3

  µy (St +h ) = + Cst +h , µx (St +h )



vt +h+ 1 ,

(5)

3

(6)

where St is an ergodic and irreducible Markov-chain with a finite number of states St = {1, . . . , M } defined by the following constant transition probabilities:





pij = Pr St +h = j|St +h− 1 = i ,

(7)

3

M 

pij = 1 ∀i, jϵ{(1, . . . , M )}.

(8)

j =1

Following Mariano and Murasawa (2003), we replace the missing observations with zeros, and rewrite the measurement equation in such a way that the Kalman filter skips the missing observations. The model is estimated using the Kalman filter combined with the Hamilton filter for Markov-switching models (see Appendix A for details). We also need to make 2 Note that Camacho (in press) also considers regime switches in a mixed-frequency VAR similar to ours. However, he concentrates on the in-sample performance of the MSMF-VAR model for estimating business cycle turning points in the US, and does not perform a forecasting exercise.

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an approximation at the end of the Kalman and Hamilton filters to avoid the proliferation of cases to be considered, as per Kim and Nelson (1999). Appendix A reports the equations of the Kalman filter as well as the Kim and Nelson (1999) approximation that is necessary for state space models with regime switches. The model parameters are estimated by maximum likelihood using the Expectation Maximization (EM) algorithm. The expectation step (‘‘E’’ step) uses the Kalman and Hamilton filters to obtain an estimate of the log-likelihood, and the maximization step (‘‘M’’ step) maximizes the log-likelihood obtained from the expectation step using the BFGS algorithm from the OPTMUM library of the Gauss software. We iterate over the ‘‘E’’ and ‘‘M’’ steps until the algorithm has converged. 2.1.2. Mixed frequency data modeled via a stacked-vector system Here, we present the regime switching version of the mixed-frequency VAR model of Ghysels (2012). In this case, the high-frequency variable is stacked in the vector of dependent variables, depending on the timing of the release of the high frequency data. This makes it possible to avoid the estimation via the Kalman filter that is often so computationally demanding. As in the previous subsection, we describe the model using a single low-frequency variable and a single high-frequency variable to facilitate the notation. In particular, we illustrate the model in the case where one wants to forecast one quarterly variable using one monthly variable that is released three times a quarter. However, extensions to larger systems involving more low- or high-frequency variables naturally follow. Alternatively, a model considering more than two different sampling frequencies could be specified along the same lines. The crucial point with this type of mixedfrequency VAR model is to stack observations from the high-frequency variable xt in the vector of dependent variables depending on the timing of the release of the highfrequency indicator. Denote the quarterly variable by yt , and the quarterly variables representing the release of the monthly indicator in the first, second and third months of quarter t by xt − 2 , xt − 1 , xt , respectively. This class of mixed-frequency 3

3

VAR models can therefore be written as follows: zt = A0 +

p 

Aj zt −j + ut ,

(9)

j =1

where zt = (xt − 2 , xt − 1 , xt , yt )′ is the vector of observ3 3 able time series variables, A0 is the vector of intercepts, Aj (j = 1, . . . , p) are the coefficient matrices, and ut is the white noise error term with mean zero and positive definite variance covariance matrix Σu ; that is, ut ∼ (0, Σu ). Potentially, regime switching can be included in all parameters of the model: intercepts, autoregressive coefficient matrices and the variance–covariance matrix of the error term, so that the MSMF-VAR (SV) model can be specified as follows: zt = A0 (St ) +

p 

Aj (St )zt −j + ut (St ),

(10)

j =1

where St is an ergodic and irreducible Markov-chain with a finite number of states, as defined in Eqs. (7) and (8). The

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conditional distribution of ut given St is assumed to be nori

mal ut |St ∼ N (0, Σu (St )). In the Monte-Carlo experiments and the empirical application, we consider two different versions of the MSMFVAR (SV) model. The first version assumes no restrictions on the coefficient matrices Aj , and is therefore denoted MSMF-VAR(SV-U). The second version, on the other hand, assumes that the coefficient matrices have particular structures. In detail, we assume that the high-frequency indicator follows an ARX(1) process, following an example by Ghysels (2012). More details about the restrictions we impose are reported in Appendix B. The restricted version of the MSMF-VAR (SV) model is therefore denoted MSMFVAR(SV-R). The estimation of this type of MSMF-VAR is similar to the estimation of a single frequency MS-VAR, and is done in a classical maximum likelihood framework, as was discussed by Ghysels (2012) for the linear case.3 We therefore use the EM algorithm without requiring the Kalman filter, as per Krolzig (1997).

Markov-switching parameters were first introduced in MIDAS models by Guérin and Marcellino (2013). The Markov-switching MIDAS model can be written as follows: (m)

yt = β0 (St ) + β1 (St )B(L1/m ; θ (St ))xt −h + ϵt (St ), where B(L

1/m

; θ (St )) =

(m) xt −1−s/m

i

j =1

b(j; θ (St ))L

(j−1)/m

(11)

,

(m) Ls/m xt −1

and ϵt |St ∼ N (0, σ (St )). The MIDAS weight function b(j; θ (St )) aggregates the high-frequency variable to the frequency of the dependent variable yt through a parametric weight function. The MIDAS parameters θ govern the aggregation process so that one can include a large number of lags for the high frequency variable with only a limited number of parameters. Thus, a key feature of the MIDAS aggregation scheme is the parsimonious aggregation of the high frequency variable. A common choice for the weight function b(j; θ (St )) (see e.g. Ghysels, Sinko, & Valkanov, 2007) is the exponential Almon lag specification. A standard choice in the literature is to use two parameters for the MIDAS weight function (i.e., θ = {θ1 , θ2 }). The exponential Almon lag specification can then be written as follows:

=

b(j; θ(St )) =

2

exp(θ1 (St )j + θ2 (St )j2 ) K



–

As in the VAR case, the regime generating process is an ergodic Markov-chain with a finite number of states St = {1, . . . , M } defined by the following constant transition probabilities: pij = Pr (St +1 = j|St = i), M 

pij = 1

(13)

∀i, jϵ{1, . . . , M }.

(14)

j=1

The ADL-MIDAS type of specifications introduces autoregressive dynamics in MIDAS models in a straightforward manner (see e.g. Andreou et al., 2013), so that the MSADL-MIDAS model can be written as follows: (m)

yt = β0 (St ) + β1 (St )B(L1/m ; θ (St ))xt −h

+ λ(St )yt −d + ϵt (St ).

(15)

In the remainder of the paper, we will use the MIDAS and ADL-MIDAS models with the exponential Almon lag specification. 2.3. Comparison between the MS-MIDAS and MSMF-VAR models

2.2. Markov-switching MIDAS model

K

)

.

(12)

exp(θ1 (St )j + θ2 (St ) ) j2

j =1

3 Chauvet, Goetz, and Hecq (2013) study the link between economic fluctuations and financial markets using the MF-VAR(SV-R) model estimated via seemingly unrelated regressions (see e.g. Greene, 2011). However, their specification of the MF-VAR(SV-R) differs from ours, in that they assume that the indicator follows an ARX(1) process, but they also make use of MIDAS restrictions for the equation describing the relationship between the low-frequency and high-frequency variables. In addition, they also consider a mix of monthly and daily data, so that the dimension of their models is very large. As a result, they implement a two-step estimation procedure to facilitate the estimation. In our case, however, we do not impose MIDAS restrictions, but consider a relatively small system (i.e., a mix of quarterly and monthly data). Maximum likelihood estimation therefore remains fairly straightforward.

The main differences between the MS-MIDAS and MSMF-VAR models are as follows. First, MIDAS models produce direct forecasts, whereas the forecasts from MF-VAR models (both the state space and stacked vector system versions) are usually calculated as iterated forecasts. For a comparison of direct and iterated forecasts, see for example Chevillon and Hendry (2005) and Marcellino, Stock, and Watson (2006). Second, MIDAS models do not model the behavior of the high-frequency variable, whereas MF-VAR models model the dynamics of the high frequency variable explicitly. As a result, provided that the MF-VAR models are specified well, one can expect to achieve more precise forecasts than with MIDAS models. However, MIDAS models are more parsimonious, meaning that they are less prone to misspecifications than MF-VAR models. In particular, the MF-VAR model version of Ghysels (2012) is typically subject to parameter proliferation, especially in systems with different frequency mixes and large numbers of lags. Third, the estimates and forecasts of the regime probabilities from MS-MIDAS models are made at a quarterly frequency, albeit they can be updated on a monthly basis. The MSMF-VAR (SV) models also estimate the regime probabilities at a quarterly frequency. In contrast, MSMFVAR models estimate and forecast the regime probabilities at the monthly frequency directly. Finally, the MSMF-VAR (KF) model permits a monthly estimate of the quarterly variable to be obtained, which is often of interest in itself. 3. Monte Carlo experiments 3.1. In-sample estimates Our first Monte Carlo experiment is designed to check how well our estimation algorithm performs in finite samples. First, we assume that the data are generated by a regime-switching VAR(1) process. The population model is

C. Foroni et al. / International Journal of Forecasting (

defined as follows:

  yt xt

 =

  µ1 (St ) 0.5 + µ2 (St ) 0.1

+ ϵt (St ),

t =

α



yt − 1

–

5

Following Ghysels (2012), the variance–covariance matrix of the residuals takes the following form:



3

0.5

xt − 1

Σ (St ) 

3

1 2

, , 1, . . .

(16)

3 3

where

µ1 (St ) = {−1, 1}, µ2 (St ) = {−1, 1},    0 1 ϵt |(St = 1) ∼ N , 0 0.3    0 2 ϵt |(St = 2) ∼ N , 0 0.4

)

σxx (St )  ρH σxx (St ) = ρ 2 σ (S )  H xx t



ρH σxx (St )

ρH2 σxx (St )

σxy (St )

(1 + ρH )σxx (St )

ρH σxx (St )

ρH σxx (St )

(1 + ρH + ρ )σxx (St )

σxy (St )

σxy (St )

 σxy (St ) . σxy (St )  σyy (St )

σxy (St )

2 H

(18)

We assume the following parameter values: 0.3 1

,

0.4 2

,

 

µi (St ) = {−1, 1}, ∀i = {1, 2, 3, 4} ρH = 0.5, ρL = 0.5 ai = {0.1}, bi = {0.1}, ∀i = {1, 2, 3} σxx (St ) = {1, 2}, σyy (St ) = {1, 2}, σxy (St ) = {0.3, 0.4}.

α = {0.1, 0.5}. The parameter α therefore governs the extent to which the first dependent variable is affected by the second dependent variable. Note that, unlike the model generally discussed in Section 2, we only consider switches in the intercepts and in the variance–covariance matrix of the shocks, not in the autoregressive matrix. There are no conceptual issues in terms of estimation which prevent us from having switches in the autoregressive term. We consider changes in the intercepts because they are the most common sources of forecast failures (see e.g. Clements & Hendry, 2001). Moreover, considering regime changes in the autoregressive coefficient matrix for all of the regime switching models we analyze would complicate the estimation substantially by increasing the computational time and the convergence problems of the algorithm. We also consider differences in the duration of the regimes. The first set of transition probabilities is defined as follows:

(p11 , p22 ) = (0.95, 0.95). On the other hand, the second set of transition probabilities implies that the second regime has a shorter duration than the first regime:

(p11 , p22 ) = (0.95, 0.85). The data are generated at a monthly frequency. We therefore refer to this DGP as the single-frequency DGP. However, in the estimation, we assume that xt is observed every month, whereas yt is only observed every third month. Second, we assume that the data are generated by a Markov-switching mixed-frequency VAR, or, more specifically, by the stacked vector system version with restrictions on some of the parameters (i.e., the data are generated from the MSMF-VAR(SV-R)). The model is defined as follows:       xt − 2 0 0 ρ H a1 µ1 (St )

x 31  µ2 (St )  0 0 ρH2  t− 3  =  +   x  µ3 (St )  0 0 ρH3 t µ ( S ) b1 b2 b3 4 t yt     x1 xt − 5 ut (St )  2 x 43   uxt (St ) t−  ∗ . 3 x 3  +  ux (St ) t −1 t y yt −1 ut (St )

a2   a3

ρL

(17)

Also, for this DGP, we use two sets of transition probabilities, (p11 , p22 ) = (0.95, 0.95) and (p11 , p22 ) = (0.95, 0.85). The sample size, expressed at the frequency of the low frequency variable, is T = 250. Note that, in this second case, the DGP is set at a quarterly frequency, and, using the notation introduced in the previous section, xt − 2 , xt − 1 , and xt represent the realiza3

3

tions of the variable x in the first, second and third months of the quarter, respectively. We will refer to this DGP as the mixed-frequency DGP. We compare the in-sample fits of the models under scrutiny by calculating the in-sample mean squared error, defined as follows: IS − MSE =

T  (yˆt − yt )2 /T .

(19)

t =1

The in-sample MSE is computed for each replication n = {1, . . . , N }, with N = 1000 being the number of MonteCarlo replications. The median across replications is then reported. The Quadratic Probability Score (QPS) and Log Probability Score (LPS) can also be calculated for assessing the abilities of non-linear models to detect turning points. QPS and LPS are defined as follows: QPS =

T 2

T t =1

(P (St = 1) − St )2 ,

T 1

(1 − St ) log(1 − P (St = 1)) T t =1 + St log(P (St = 1)),

LPS = −

(20)

(21)

where St is a dummy variable that takes the value one if the true regime is the first regime in period t, and P (St = 1) is the smoothed probability of being in the first regime in period t. The median across replications is then reported. In this exercise, we consider fifteen different models, a list similar to that used in the empirical application in Section 4. Specifically, we have AR(1), ADL(1,1), linear MIDAS, linear ADL-MIDAS, MS-MIDAS, MSADL-MIDAS, univariate MS, linear VAR, regime-switching VAR, linear mixedfrequency VAR (both the state space and stacked-vector system versions) and regime switching mixed-frequency

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C. Foroni et al. / International Journal of Forecasting (

VAR (both the state space and stacked-vector system versions) models. For the SV models, we consider both the restricted (R) and unrestricted (U) versions. Appendix B provides more details of each of the models under scrutiny. Table 1 presents the results from the Monte Carlo experiments. Panels A and C report the results for the DGP with equal transition probabilities for the single-frequency and mixed-frequency DGPs, respectively. Panels B and D show the results for the DGP that implies a lower duration for regime 2 than for regime 1 for the two types of DGP. For panels A and B, we present results with the two different parameter values for α (i.e., α = 0.1 and α = 0.5). Several main findings emerge from Table 1. First, for the single-frequency DGP, across the four different cases, the MSMF-VAR model estimated via the Kalman filter (i.e., the MSMF-VAR(KF) model) obtains the best results in terms of QPS and LPS. In other words, our simulation results indicate that, based on these DGPs, the MSMF-VAR (KF) model yields the best estimates of the in-sample regime probabilities. Second, the MSMF-VAR(SV-R) model yields the best in-sample estimates across all four DGPs, with the MSMIDAS being a very close second best for α = 0.1 and the MSADL-MIDAS model being a very close second best for α = 0.5. Third, in the case of a mixed-frequency DGP, the MSMF-VAR(SV-R) performs best in terms of continuous forecasts, which is not surprising given that the DGP is exactly a MSMF-VAR. Fourth, the best regime probabilities are obtained by either the MSMF-VAR(SV-U) when looking at the QPS, or the KF version (when the second regime has a lower duration) and a standard MS-VAR (when the duration is the same for both regimes) when looking at the LPS. Finally, it is also worth mentioning that linear models’ fit to the data is much worse than that of the MS specifications. As we will see, this finding no longer holds generally in an out-of-sample context. 3.2. Out-of-sample estimates We now assess the finite sample forecasting performances of the Markov-switching mixed-frequency VARs for the low frequency variable, comparing them with a number of competing forecasting models in a controlled setup. The design of this Monte Carlo experiment is as follows. For each draw, we set the evaluation sample to Tev al = 50, and calculate one-step-ahead forecasts recursively until we reach the end of the full estimation sample. When calculating one-step-ahead forecasts, we make sure that all forecasting models use the same information set. Thus, when we forecast the next quarter, we do not use any monthly information on that quarter. As a result, the design of our Monte-Carlo experiment is relatively unfavorable to mixed-frequency data models, in that this class of models could accommodate more timely information explicitly. Also, to reduce the computational time, we fix the parameter estimates at the values obtained using the first estimation sample, rather than updating the values recursively. A limited evaluation of fully recursive estimation has produced similar results. The one-step-ahead forecasts are generated from the fifteen forecasting models described in the previous subsection. Forecasts for regime switching models are calculated in the standard way by weighting the conditional

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forecasts on each regime by the predicted probabilities of being in a given regime. In the case of two regimes, onestep-ahead forecasts are calculated as: yˆ t +1|t = yˆ t +1|t ,1 P (St +1|t = 1) + yˆ t +1|t ,2 P (St +1|t = 2), (22) where:

• yˆ t +1|t ,i is the one-step-ahead forecast for the dependent variable yt conditional on the process being in state i at time t + 1 given the past observable information. • P (St +1|t = i) is the conditional probability of the process being in state i at time t + 1 given the past observable information. Generalizing to h-step-ahead forecasts and M regimes, Eq. (22) becomes: yˆ t +h|t =

M 

yˆ t +h|t ,i P (St +h|t = i).

(23)

i=1

We compare the forecasting performances for the levels of the low-frequency variable using the Mean Squared Forecast Error (MSFE). When using a regime switching model, we also calculate the in-sample Quadratic Probability Score (QPS) and Log Probability Score (LPS) in order to assess the ability of non-linear models to predict turning points in a real-time context. QPS and LPS are then defined as follows: QPS =

Tev al 1 



Tev al i=1

T i t =1



LPS =

Tev al 1 

Tev al i=1



Ti 2 

−

(P (St = 1) − St )

2

,

(24)

Ti 1 

T i t =1

(1 − St ) 

× log(1 − P (St = 1)) + St log(P (St = 1)) , (25) where St is a dummy variable that takes the value one if the true regime is the first regime in period t, and P (St = 1) is the smoothed regime probability of being in the first regime in period t. Ti is the size of the sample for evaluation period i of the estimation sample, and Tev al is the size of the evaluation sample. Table 2 reports the results with N = 1000 Monte Carlo replications. Conditional on the single-frequency DGP with equal transition probabilities and α = 0.1, the best forecasting model is the MSMF-VAR(SV-U) model (Panel A). The MSMF-VAR(SV-U) also obtains the best forecasting results with α = 0.5. In addition, conditional on the singlefrequency DGP with transition probabilities p11 = 0.95 and p22 = 0.85 with α = 0.1, the MS-VAR model yields the best continuous forecasts, while for α = 0.5, the best model in terms of continuous forecasts is again the MSMFVAR(SV-U) model. For the mixed-frequency DGPs, the MSMF-VAR(SV-U) model is again the best performer, closely followed by the MSMF-VAR(SV-R). Table 2 also shows the QPS and LPS for each model across the four DGPs. The results indicate that the MSMF-VAR (KF) model systematically yields the best discrete estimates of the regimes with the single-frequency DGP (Panels A and B),

ADL

MS-AR

MIDAS

MS-MIDAS

IS-MSE QPS LPS

IS-MSE QPS LPS

5.966

– –

5.648 2.743 0.236 0.624

10.959 10.801 4.747 – – 0.303 – – 0.906

– –

6.094

10.652 – –

– – 4.130 0.247 0.654

2.553 0.237 0.588

IS-MSE QPS LPS

IS-MSE QPS LPS

6.081

– –

5.881 2.642 0.233 0.575

10.876 10.508 4.672 – – 0.309 – – 0.765

– –

5.929

10.650 – –

– – 4.255 0.247 0.672

2.408 0.244 0.567

11.659 11.355 9.105 – – 0.549 – – 1.062

12.600 – –

8.752 0.505 1.135

11.652 11.273 9.337 – – 0.602 – – 1.304

12.582 – –

8.755 0.513 1.119

11.379 – –

9.411 0.410 0.744

9.436 0.399 0.745

4.134 0.228 0.597

2.563 0.228 0.516

3.962 0.232 0.588

2.623 0.224 0.526

MSADL-MIDAS

MS-VAR

11.314 9.857 – 0.415 – 0.716

11.343 9.889 – 0.504 – 0.700

10.688 4.555 – 0.237 – 0.792

5.828 3.045 – 0.228 – 0.707

10.801 4.501 – 0.254 – 0.757

5.648 3.283 – 0.240 – 0.755

VAR

– – –

– – –

– – –

– – –

– – –

– – –

MFVAR (KF)

– 0.296 0.493

– 0.302 0.704

– 0.191 0.319

– 0.183 0.302

– 0.221 0.353

– 0.205 0.331

MSMFVAR (KF)

11.167 – –

11.156 – –

9.662 – –

5.746 – –

9.612 – –

5.792 – –

MFVAR (SV-U)

9.996 0.263 1.092

10.132 0.252 1.054

4.830 0.234 0.871

3.218 0.224 0.835

4.786 0.234 0.854

3.297 0.224 0.811

MSMFVAR (SV-U)

11.449 – –

11.487 – –

9.822 – –

5.766 – –

9.774 – –

5.832 – –

MFVAR (SV-R)

7.214 0.591 1.735

7.118 0.589 1.710

3.215 0.348 0.980

2.249 0.326 0.823

3.325 0.341 0.972

2.357 0.315 0.879

MSMFVAR (SV-R)

Note: This table reports the in-sample MSEs (IS-MSE) as defined in Eq. (19). For regime switching models, we also report the QPS and LPS, as defined in Eqs. (20) and (21). Bold entries are the smallest IS-MSE, QPS and LPS for the indicated DGP. Note that the table does not include the in-sample MSEs for the MF-VARs estimated with the Kalman filter (with and without regime switches). This is because, with the Kalman filter, the in-sample fitted monthly values are constrained to add up to the observed quarterly values. Therefore, at the quarterly frequency, there is a perfect fit by construction.

IS-MSE QPS LPS

9.943

5.615

9.832

5.773

11.390 – –

– –

– –

– –

– –

ADL-MIDAS

)

Panel D: Mixed-frequency MS DGP with p11 = 0.95 and p22 = 0.85

IS-MSE QPS LPS

Panel C: Mixed-frequency MS DGP with p11 = 0.95 and p22 = 0.95

α = 0.5

α = 0.1

Panel B: Single-frequency MS DGP with p11 = 0.95 and p22 = 0.85

α = 0.5

α = 0.1

Panel A: Single-frequency MS DGP with p11 = 0.95 and p22 = 0.95

AR(1)

Table 1 In-sample results: Monte Carlo experiments.

C. Foroni et al. / International Journal of Forecasting ( – 7

ADL

MSAR

MIDAS MSMIDAS

ADLMIDAS

MSFE QPS LPS

MSFE QPS LPS

6.047

– –

5.829

6.300 6.842 0.235 – 0.583 –

5.951 0.231 0.610

10.862 10.699 10.300 11.555 10.316 – – 0.318 – 0.251 – – 0.980 – 0.705

– – 10.053 – –

6.049 – –

MSFE QPS LPS

MSFE QPS LPS

6.050

5.783 – –

5.688 6.608 0.241 – 0.585 –

5.796 0.244 0.593

10.760 10.598 10.206 11.788 10.591 – – 0.332 – 0.259 – – 1.091 – 0.725

– – 10.180 – –

5.958 – –

11.711 – –

11.649 0.409 0.742

12.233 0.416 0.711

5.858

5.741

11.217 – –

11.420 – –

10.597 – –

– –

10.603 – –

– –

11.212 0.301 0.690

11.291 0.301 0.691

10.026 0.234 0.713

5.547 0.228 0.680

10.081 0.235 0.691

5.590 0.228 0.699

11.153 0.301 0.690

11.278 0.301 0.691

10.161 0.234 0.713

5.554 0.228 0.680

10.146 0.235 0.691

5.646 0.228 0.699

Av. VAR

19.695 – –

17.732 – –

20.081 – –

7.902 – –

19.778 – –

22.585 0.293 0.488

23.144 0.294 0.485

20.414 0.203 0.341

6.425 0.203 0.339

20.306 0.205 0.342

6.332 0.200 0.336

(KF)

(KF)

7.738 – –

MSMFVAR

MFVAR

20.880 0.293 0.488

21.194 0.294 0.485

19.873 0.203 0.341

6.754 0.203 0.339

19.560 0.205 0.342

6.636 0.200 0.336

Av. MFVAR (KF)

6.245

9.595

5.882

11.947 – –

12.060 – –

10.383 – –

– –

– –

– –

(SVU)

MFVAR

11.005 0.259 1.079

11.228 0.253 1.059

9.617 0.233 0.851

5.576 0.232 0.853

8.722 0.234 0.843

5.256 0.220 0.849

(SVU)

MSMFVAR

5.956

9.360

5.607

11.444 – –

11.833 – –

10.426 – –

– –

– –

– –

(SVR)

MFVAR

11.405 0.592 1.736

11.765 0.587 1.710

10.627 0.345 0.950

6.019 0.326 0.919

9.632 0.356 1.090

5.600 0.314 0.952

(SV-R)

MSMFVAR

11.177 0.313 0.575

11.296 0.306 0.560

10.010 0.240 0.555

5.863 0.237 0.543

8.892 0.247 0.556

5.428 0.235 0.526

Av. MFVAR (SV)

Note: This table reports the out-of-sample MSEs. For regime switching models, we report QPS and LPS as defined in Eqs. (24) and (25). Bold entries are the smallest MSE, QPS and LPS for the indicated DGP. Appendix B describes each of the models under scrutiny. Av. MIDAS, Av. MF-VAR(KF) and Av. MF-VAR(SV) report the MSE, QPS and LPS obtained by taking averages (equal weights) of the continuous forecasts and smoothed probabilities over the MIDAS, MF-VAR(KF) and MF-VAR(SV) models, respectively.

11.720 11.474 11.691 14.812 13.127 – – 0.607 – 0.508 – – 1.326 – 1.125

12.033 0.505 0.700

10.409 0.245 0.585

5.829 0.234 0.507

10.215 0.240 0.558

5.997 0.233 0.503

MSVAR

)

MSFE QPS LPS

11.576 0.397 0.745

9.849 0.238 0.638

5.798 0.229 0.543

9.759 0.234 0.630

5.889 0.229 0.545

MSADL- AvMIDAS.VAR MIDAS

C. Foroni et al. / International Journal of Forecasting (

MSFE 11.590 11.429 11.538 14.848 13.042 11.518 QPS – – 0.533 – 0.504 – LPS – – 1.080 – 1.121 – Panel D: Mixed-frequency MS DGP with p11 = 0.95 and p22 = 0.85

Panel C: Mixed-frequency MS DGP with p11 = 0.95 and p22 = 0.95

α = 0.5

α = 0.1

Panel B: Single-frequency MS DGP with p11 = 0.95 and p22 = 0.85

α = 0.5

α = 0.1

Panel A: Single-frequency MS DGP with p11 = 0.95 and p22 = 0.95

AR(1)

Table 2 Out-of-sample results: Monte Carlo experiments.

8 –

C. Foroni et al. / International Journal of Forecasting (

and also in the mixed-frequency regimes when we consider the LPS. On the other hand, the MSMF-VAR (SV-U) model tends to obtain the best estimates of the regimes in terms of QPS with the mixed-frequency DGP (Panels C and D). Overall, the results of the Monte Carlo experiments suggest that our estimation procedure for MSMF-VARs also performs well in finite samples. Moreover, not accounting for MS features leads to a deterioration in the in- and outof-sample performances of all models, the former more than the latter. 4. Empirical application 4.1. Data The empirical application focuses on euro area macroeconomic variables. In particular, we use the euro area GDP growth as our quarterly variable, together with a set of four monthly indicators which are often considered to be good predictors of growth: the Economic Sentiment Indicator (ESI), the M1 monetary aggregate, headline industrial production and the slope of the yield curve, see Table 3. We use real-time data obtained from the ECB real time database (see Giannone, Henry, Lalik, & Modugno, 2012, for a description of the database), except for the slope of the yield curve, where the data were downloaded from Haver Analytics.4 Our full estimation sample ranges from 1986:Q1 to 2012:Q2. The euro area GDP is taken to be 400 times the quarterly change in the logarithm of GDP, in order to obtain the quarterly GDP growth at an annual rate. Note that euro area GDP values before 1995Q1 are taken from the euro area business cycle network backcasted GDP series. This allows us to have a long enough sample, which is helpful for the inference on regimes. 4.2. In-sample estimates First, we focus on the in-sample analysis. Table 4 reports the in-sample parameter estimates for the four indicators obtained using a MSMF-VAR(KF) model. The second regime is associated with higher volatility and lower GDP growth across all four indicators, meaning that this regime can be interpreted as the recessionary regime. Fig. 1 shows the monthly estimates of GDP growth that we obtain from the MSMF-VAR (KF) model with the ESI. These estimates are a byproduct of our estimation. In Fig. 1, we also report the Eurocoin indicator, which is a monthly estimate of the medium- to long-run component of quarterly euro area GDP growth, see Altissimo, Cristadoro, Forni, Lippi, and Veronese (2010). Our monthly estimates of GDP growth are relatively close to the Eurocoin indicator, albeit they tend to be more volatile than Eurocoin, as expected. Fig. 2 shows the in-sample monthly probability of recession from the MSMF-VAR (KF) model with the ESI. The 4 The slope of the yield curve is defined as the difference between the yields on a 10-year government bond (national data are aggregated at the euro area level by Eurostat) and the yields on a 3-month money market interest rate (i.e., the EURIBOR).

)

–

9

recession and expansion phases of the euro area economic activity are all captured very well by the MSMF-VAR (KF) model, including the latest euro area recession. However, it should be noted that this model identifies the 2001–03 period as recession, while the CEPR qualifies this episode as being a ‘‘prolonged pause in the growth of economic activity’’.5 In Table 5, we report the in-sample QPS and LPS for each model with MS parameters, using the four indicators sequentially. We also report the QPS and LPS values for an average (equal weights) of the smoothed probability over all models for a given indicator (see the last rows of Panels A and B)6 and the LPS and QPS for an average (equal weights) of the smoothed probability over all indicators for a given model (see the last columns of Panels A and B). First, the ESI yields the best discrete estimates of the business cycle conditions pooling across all models (i.e., the lowest QPS and LPS values, see the last rows of Panels A and B). Second, when pooling across all indicators, the MSMF-VAR (KF) obtains the lowest average QPS, while the MSMF-VAR (SV-R) gets the lowest LPS (see the last columns of Panels A and B). Third, for a given indicator, taking a simple average of the smoothed probability over all models generally leads to substantial improvements in the classification of economic activity. Finally, and in line with the results of Guérin and Marcellino (2013) for the US, the MS-MIDAS with the slope of the yield curve performs very well in terms of both LPS and QPS. 4.3. Forecasting results The available sample is split between an estimation sample and an evaluation sample. The first estimation sample runs from 1986Q1 to 2006Q1, then this is recursively expanded until 2011Q2. Within the evaluation sample, we calculate three nowcasts and six forecasts, reflecting the information set available at the end of each month m of quarter t , m = {1, 2, 3}. For example, for the initial evaluation quarter 2006Q2, three nowcasts for quarter 2006Q2 are calculated, corresponding to the information sets available at the end of the months of April, May and June. Next, three forecasts for quarter 2006Q3 and three forecasts for quarter 2006Q4 are calculated using the information sets available at the end of the months of April, May and June. As a result, the forecasting evaluation sample is [T1 , T2 + h], where T1 is 2006Q2, T2 is 2011Q3, and h denotes the maximum forecast horizon in quarters (i.e., 2). Our forecasting exercise uses real-time data and takes into account the different publication lags for the monthly indicators we consider.7 The actual GDP growth for evaluating our forecasts is taken from the September 2012 data 5 One reason for differences in the estimation of turning points is that the CEPR business cycle dating committee concentrates on dating the classical business cycle (i.e., based on the level of economic activity), whereas regime-switching models estimate a growth cycle where the turning points are defined by deviations from trend. 6 In this case, we exclude the probability from the standard univariate Markov-switching model (MS-AR) when taking the average. 7 Specifically, the economic sentiment indicator and the slope of the yield curve are available readily at the end of each month. We consider that industrial production and the M1 monetary aggregate have a twomonth publication lag (since we assume that forecasts are generated on the last day of each month).

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)

–

Table 3 List of monthly indicators. Indicator

Transformation

Economic sentiment indicator M1 monetary aggregate Industrial production Slope of the yield curve

Quarterly change in the logarithm of monthly ESI Quarterly change in the logarithm of monthly M1 Quarterly change in the logarithm of monthly IP Quarterly change in the level of the slope of the yield curve

This table reports the list of monthly indicators we use in the empirical application. The second column of the table shows the transformation we applied to each indicator. Data were downloaded from the ECB real-time database and Haver Analytics. Table 4 In-sample results: MSMF-VAR (KF) model.

p11 p22

µ1y µ1x µ2y µ2x σy1 σx1 σy2 σx2

ESI

M1 monetary aggregate

Industrial production

Slope of the yield curve

0.975 [0.012] 0.922 [0.039] 2.597 [0.239] −0.458 [0.800] −0.999 [0.706] −1.868 [1.227] 0.891 [0.168] 1.131 [0.057] 1.994 [0.435] 2.993 [0.273]

0.961 [0.016] 0.724 [0.108] 2.096 [0.280] 1.628 [0.141] 0.545 [1.383] 2.880 [0.373] 0.962 [0.274] 0.610 [0.033] 2.583 [0.820] 1.854 [0.299]

0.992 [0.006] 0.904 [0.078] 2.053 [0.242] 0.465 [0.171] −0.486 [2.463] −1.078 [1.296] 1.819 [0.167] 0.693 [0.046] 4.603 [1.024] 2.244 [0.597]

0.970 [0.015] 0.952 [0.029] 2.234 [0.227] −0.067 [0.069] 1.094 [0.732] 0.131 [0.145] 0.540 [0.085] 0.211 [0.013] 1.598 [0.275] 0.401 [0.033]

This table reports the in-sample parameter estimates for the MSMF-VAR (KF) model with the quarterly GDP and monthly indicators. p11 is the transition probability of staying in the first regime. p22 is the transition probability of staying in the second regime. The µs are the intercepts for each equation of the VAR in each regime, while the σ s are the estimates of the variance of the innovation in each regime. Standard deviations are reported in brackets and are calculated from the outer product estimate of the Hessian. Table 5 In-sample QPS and LPS. ESI

M1 monetary aggregate

Industrial production

Slope of the yield curve

Pooling across indicators

Panel A: Quadratic probability score (QPS) MS-AR 0.239 MSADL-MIDAS 0.289 MS-MIDAS 0.688 MS-VAR 0.489 MSMF-VAR (KF) 0.201 MSMF-VAR (SV-U) 0.273 MSMF-VAR (SV-R) 0.262 Pooling across models 0.172

0.239 0.260 0.457 0.558 0.364 0.415 0.494 0.210

0.239 0.223 0.446 0.354 0.458 0.637 0.447 0.341

0.239 0.233 0.159 0.268 0.390 0.663 0.305 0.233

0.239 0.245 0.342 0.227 0.199 0.255 0.203 0.191

Panel B: Log probability score (LPS) MS-AR 0.601 MSADL-MIDAS 0.676 MS-MIDAS 1.673 MS-VAR 1.428 MSMF-VAR(KF) 0.439 MSMF-VAR (SV-U) 0.699 MSMF-VAR (SV-R) 0.596 Pooling across models 0.304

0.601 0.686 1.101 1.114 0.784 1.449 1.800 0.347

0.601 0.533 1.033 0.867 0.852 1.845 1.265 0.516

0.601 0.554 0.297 0.612 0.632 2.452 0.684 0.404

0.601 0.567 0.581 0.363 0.321 0.397 0.316 0.324

This table reports the in-sample quadratic probability score (QPS) and log probability score (LPS). We use the chronology of the euro area business cycle from the CEPR business cycle dating committee to identify the euro area expansions and recessions. This chronology is only available at the quarterly frequency. To calculate QPS and LPS for MSMF-VAR (KF) models that estimate the model at the monthly frequency directly, we disaggregated the CEPR business cycle chronology at the monthly frequency by assuming that the euro area was in recession in month m of quarter t if the CEPR business cycle dating committee considered that the euro area was in recession in quarter t. The probabilities of a euro area recession are from the September 2012 data vintage, with the last observation being in June 2012 for the MSMF-VAR (KF) model and 2012Q2 for the other models. Bold entries are the smallest QPS and LPS for the indicated indicator.

C. Foroni et al. / International Journal of Forecasting (

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Fig. 1. Monthly GDP from the Markov-switching mixed-frequency VAR. Note: This figure shows the monthly estimate of the euro area quarterly GDP growth at an annual rate from the MSMF-VAR (KF) model with the quarterly GDP and monthly ESI, along with the Eurocoin indicator, also at an annual rate.

Fig. 2. Monthly probability of recession: in-sample estimates. Note: The above probabilities of a euro area recession are from a MSMF-VAR(KF) model with the quarterly GDP and monthly ESI. The shaded areas represent the CEPR recessions. Note that the model identifies the period 2001–2003 as a recession, while the CEPR qualifies this episode as being a ‘‘prolonged pause in the growth of economic activity’’.

vintage, with the last observation being T = 2012:Q2. The forecasts are evaluated based on the RMSE, using an AR(1) model as the benchmark model. Our forecasting exercise considers linear MIDAS models, Markov-switching MIDAS models, linear MF-VAR models (also in a stacked version), and Markov-switching

MF-VAR models (in both the state space and stacked versions). For the stacked version, we have both restricted and unrestricted specifications. We also include as competitors a bivariate Markov-switching model, an ADL(1,1) model and a VAR(1) model. Overall, we estimate 14 models for each indicator, plus an AR(1) model. We use the Clark

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)

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Table 6 Forecasting euro area GDP growth: RMSE. Forecasting horizon (months)

0

1

2

3

4

5

6

7

8

Economic sentiment indicator MIDAS ADL-MIDAS MS-MIDAS MSADL-MIDAS VAR MS-VAR MF-VAR (KF) MSMF-VAR (KF) MF-VAR (SV-U) MSMF-VAR (SV-U) MF-VAR (SV-R) MSMF-VAR (SV-R) ADL(1,1)

0.890*** 0.724*** 0.755** 0.655** 0.796** 0.773** 0.651** 0.673** 0.556** 0.643** 0.429** 0.413** 0.781**

0.804*** 0.664*** 0.720** 0.623** 0.796** 0.773** 0.694** 0.709** 0.545** 0.542** 0.513** 0.433** 0.781**

0.805*** 0.641** 0.669** 0.545* 0.796** 0.773** 0.836** 0.894* 0.627** 0.533*** 0.626*** 0.539** 0.781**

0.650** 0.557** 0.772** 0.595** 0.996 1.056 0.853** 0.666* 0.630** 0.562** 0.528** 0.679** 0.923**

0.774** 0.725** 0.824** 0.698** 0.996 1.056 0.955** 0.942 0.720** 0.592** 0.739** 0.737** 0.923**

0.820*** 0.811*** 0.810** 0.817** 0.996 1.056 1.014 1.018 0.846** 0.704** 0.866** 0.981** 0.923**

0.873*** 0.836** 0.902** 0.872** 1.034 1.178 0.952 1.084 0.868*** 0.844* 0.898*** 0.908** 0.973*

0.893*** 0.887*** 0.979** 0.912** 1.034 1.178 0.963** 1.085 0.887*** 0.842** 1.044** 0.930** 0.973*

0.902*** 0.893** 0.953** 0.954** 1.034 1.178 0.958* 1.114 0.974** 0.908** 1.081** 0.998** 0.973*

Industrial production MIDAS ADL-MIDAS MS-MIDAS MSADL-MIDAS VAR MS-VAR MF-VAR (KF) MSMF-VAR (KF) MF-VAR (SV-U) MSMF-VAR (SV-U) MF-VAR (SV-R) MSMF-VAR (SV-R) ADL(1,1)

0.552* 0.506* 0.586* 0.590* 1.121 0.846 0.662 1.093 0.729* 0.853* 0.703* 0.678* 1.029

0.733 0.733 0.977 1.052 1.121 0.846 0.741 1.251 1.116 1.357 1.097 0.924 1.029

1.094 1.061 1.200 1.147 1.121 0.846 0.733 1.362 1.253 1.474 1.181 1.006 1.029

1.117 1.125 0.829 0.911 1.183 1.010 0.791** 0.706* 1.205 1.233 1.242 1.132 1.088

1.254 1.231 0.929 0.917 1.183 1.010 0.831** 0.867** 1.345 1.383 1.347 1.266 1.088

1.215 1.203 0.991 0.952 1.183 1.010 0.879** 0.890** 1.387 1.416 1.348 1.264 1.088

1.082 1.106 1.044 1.023 1.205 1.277 0.826** 0.900** 1.296 1.538 1.305 1.224 0.976*

1.072 1.082 1.134 1.157 1.205 1.277 0.832** 0.878** 1.343 1.628 1.316 1.384 0.976*

1.035 1.025 1.184 1.148 1.205 1.277 0.831** 0.885** 1.377 1.701 1.325 1.391 0.976*

Slope of the yield curve MIDAS ADL-MIDAS MS-MIDAS MSADL-MIDAS VAR MS-VAR MF-VAR (KF) MSMF-VAR (KF) MF-VAR (SV-U) MSMF-VAR (SV-U) MF-VAR (SV-R) MSMF-VAR (SV-R) ADL(1,1)

1.156 0.922** 0.994 0.749 1.011** 0.922** 1.080 1.223 1.258 1.173 1.271 1.162 1.011

1.144 1.020** 0.800 0.552 1.011** 0.922** 1.106 1.384 1.233 1.136 1.256 1.156 1.011

1.317 1.026 1.493 1.077 1.011** 0.922** 1.078 1.435 1.282 1.268 1.250 1.130 1.011

1.046 1.051 1.004 1.029 1.047* 0.988 0.944* 1.045 1.061 1.048 1.071 1.028 0.978

1.042 1.007* 1.011 1.007 1.047* 0.988 0.946 1.032 1.070 1.046 1.080 1.036 0.978

1.033 1.000* 0.874 1.059* 1.047* 0.988 0.955 1.025 1.142 1.060 1.189 1.005 0.978

1.050 0.992 1.163 0.980 1.025 1.018 0.931** 0.994 1.029 1.073 1.031 1.030 0.972*

1.048* 0.994 1.185 1.094 1.025 1.018 0.932** 0.990 1.028 1.074 1.043 1.044 0.972*

1.050 0.986 1.179 1.066 1.025 1.018 0.932** 0.992 1.065 1.112 1.108 1.062 0.972*

Univariate MS model

0.851*

0.851*

0.851*

0.947

0.947

0.947

1.042

1.042

1.042

Monetary aggregate M1 MIDAS ADL-MIDAS MS-MIDAS MSADL-MIDAS VAR MS-VAR MF-VAR (KF) MSMF-VAR (KF)

0.960* 0.772*** 0.839** 0.633* 0.920** 1.111 0.787** 1.273

0.980* 0.782*** 0.859*** 0.871 0.920** 1.111 0.722** 1.266

0.954* 0.875** 0.895*** 0.621** 0.920** 1.111 0.866*** 1.226

0.793** 0.771** 0.948** 1.007 0.946 0.953* 0.704** 1.037

0.766** 0.733*** 0.946** 0.734*** 0.946 0.953* 0.714*** 1.039

0.736** 0.713*** 0.822*** 0.756*** 0.946 0.953* 0.831*** 0.968

0.702** 0.692*** 0.831*** 0.752*** 1.342 1.007 0.854** 0.951*

0.698** 0.675*** 0.859*** 0.767*** 1.342 1.007 0.938** 0.959*

0.717** 0.702*** 0.758*** 0.740*** 1.342 1.007 1.030* 0.953*

(continued on next page)

and West (2007) test to compare the predictive accuracy of a given model against that of the more parsimonious (nested) AR(1) model.8 Table 6 reports the out-of-sample forecasting results, in terms of RMSEs. For forecast horizons h = {0, . . . , 5}, the ESI is the best indicator when combined with MSMF-VAR (SV-R) (h = 8 Admittedly, this test overlooks the real-time nature of the data. However, the Clark and McCracken (2009) test of equal predictive ability with real-time data is not straightforward to implement in the context of MIDAS models.

{0, 1}), MSMF-VAR (SV-U) (h = {2, 4, 5}), or MF-VAR (SVR) (h = {3}). The good performance of the MSMF-VAR (SV) model is also in line with the Monte Carlo results. Interestingly, for two-quarter-ahead forecasts (h = 6, . . . , 8), M1 becomes the best indicator, particularly when combined with an ADL-MIDAS specification. Overall, the results for nowcasting and short term forecasting of GDP growth provide good support for the MSMFVAR, with the MSMF-VAR (SV) being best or second best for h = {0, 1, 2, 3, 4, 5}, in combination with the ESI. This is even more noticeable in regime 2 (recessions), since it has

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Table 6 (continued) Forecasting horizon (months)

0

1

2

3

4

5

6

7

8

0.997 0.933* 0.957 0.856*** 1.011

0.989 0.963 0.940 0.879** 1.011

0.990 0.937* 0.875** 0.783** 0.819***

1.129 0.948* 0.891** 0.848** 0.819***

1.106 0.940* 0.889** 0.927** 0.819***

1.012 1.007 0.878*** 0.726** 0.847***

0.931 0.962* 0.765** 0.706** 0.847***

0.933** 0.969* 0.789** 0.922** 0.847***

Model average across all indicators (equal weights) MIDAS 0.732*** 0.783*** ADL-MIDAS 0.627** 0.698** MS-MIDAS 0.530** 0.557** MSADL-MIDAS 0.471** 0.550* VAR 0.862** 0.862** MS-VAR 0.838** 0.838** MF-VAR (KF) 0.537** 0.644** MSMF-VAR (KF) 0.953 0.955 MF-VAR (SV-U) 0.660** 0.683** MSMF-VAR (SV-U) 0.616** 0.658** MF-VAR (SV-R) 0.623** 0.659** MSMF-VAR (SV-R) 0.547** 0.586** ADL(1,1) 0.900* 0.900*

0.828*** 0.782** 0.725** 0.636* 0.862** 0.838** 0.734** 1.063 0.706** 0.704** 0.683** 0.613** 0.900*

0.721** 0.748** 0.722** 0.896 0.955 0.956 0.698** 0.966 0.781** 0.798** 0.788** 0.769** 0.914**

0.801*** 0.801** 0.803** 0.793** 0.955 0.956 0.800* 0.969** 0.871** 0.846** 0.858** 0.852** 0.914**

0.828*** 0.839** 0.799** 0.841** 0.955 0.956 0.865** 0.979 0.920** 0.850** 0.889* 0.858* 0.914**

0.831*** 0.842** 0.863*** 0.835** 1.059 1.076 0.838** 0.984 0.959** 1.023 0.926** 0.912** 0.927***

0.849*** 0.848** 0.909*** 0.867** 1.059 1.076 0.844** 0.999* 0.946** 1.015 0.907** 0.936* 0.927***

0.870*** 0.855*** 0.893*** 0.873*** 1.059 1.076 0.855** 1.007 0.954* 1.013 0.931* 0.965* 0.927***

0.851*

0.947

0.947

0.947

1.042

1.042

1.042

MF-VAR (SV-U) MSMF-VAR (SV-U) MF-VAR (SV-R) MSMF-VAR (SV-R) ADL(1,1)

1.129 0.963* 1.030 0.895** 1.011

0.851*

Univariate MS model

0.851*

**

This table reports the relative mean squared forecast error (RMSE) for forecasting euro area GDP growth. The benchmark model is an AR(1) model. Bold entries are the smallest RMSE for the relevant horizon and indicator. * Indicates cases in which the Clark and West (2007) test rejected the null hypothesis of equal forecast accuracy at the 10% significance level (one-sided tests, where the parsimonious model is always the AR(1) model). ** Indicates cases in which the Clark and West (2007) test rejected the null hypothesis of equal forecast accuracy at the 5% significance level (one-sided tests, where the parsimonious model is always the AR(1) model). *** Indicates cases in which the Clark and West (2007) test rejected the null hypothesis of equal forecast accuracy at the 1% significance level (one-sided tests, where the parsimonious model is always the AR(1) model). Table 7 QPS and LPS: Recursive forecasting exercise. ESI

M1 monetary aggregate

Industrial production

Slope of the yield curve

Pooling across indicators

Panel A: Quadratic probability score (QPS) MS-AR 0.201 MSADL-MIDAS 0.394 MS-MIDAS 0.615 MS-VAR 0.480 MSMF-VAR (KF) 0.236 MSMF-VAR (SV-U) 0.364 MSMF-VAR (SV-R) 0.276 Pooling across models 0.194

0.201 0.254 0.444 0.624 0.330 0.444 0.511 0.211

0.201 0.251 0.486 0.371 0.379 0.670 0.454 0.285

0.201 0.205 0.194 0.422 0.597 0.895 0.386 0.294

0.201 0.237 0.353 0.252 0.187 0.299 0.217 0.193

Panel B: Log probability score (LPS) MS-AR 0.597 MSADL-MIDAS 1.018 MS-MIDAS 1.725 MS-VAR 1.480 MSMF-VAR(KF) 0.526 MSMF-VAR (SV-U) 1.128 MSMF-VAR (SV-R) 0.812 Pooling across models 0.322

0.597 0.774 1.296 1.272 0.769 1.558 1.929 0.348

0.597 0.668 1.269 0.964 0.879 2.330 1.276 0.429

0.597 0.597 0.451 1.045 1.282 3.287 1.000 0.461

0.597 0.636 0.671 0.401 0.313 0.488 0.336 0.332

This table reports the quadratic probability score (QPS) and log probability score (LPS) calculated for each quarter of the recursive forecasting exercise, then averaged over the size of the evaluation sample. We use the chronology of the euro area business cycle from the CEPR business cycle dating committee to identify the euro area expansions and recessions. This chronology is only available at the quarterly frequency. To calculate QPS and LPS for MSMF-VAR (KF) models that estimate the model at the monthly frequency directly, we disaggregated the CEPR business cycle chronology at the monthly frequency by assuming that the euro area was in recession in month m of quarter t if the CEPR business cycle dating committee considered that the euro area was in recession in quarter t. Bold entries are the smallest QPS and LPS for the indicated indicator.

a limited number of observations in the sample, and, as we have seen from the Monte Carlo experiments in this case, simpler models, possibly even without MS, perform well. To assess whether the MSMF-VAR also performs well at predicting turning points, Figs. 3–8 report the recursively estimated vintages of the real-time probability of a euro area recession, obtained from the various MS specifications. The different vintages for the probabilities are

obtained from the recursive forecasting exercise. Table 7 reports the average of the QPS and LPS values calculated for each quarter of the evaluation sample. First, a simple univariate MS-AR model performs best in terms of QPS relative to models that use the M1 monetary aggregate and industrial production as a monthly indicator. However, a visual inspection of Fig. 4 shows that the model performs relatively poorly for detecting the beginnings of

14

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Fig. 3. Euro area probability of recession obtained from the recursive forecasting exercise: MSMF-VAR (KF) model. Note: This figure reports the monthly real-time probability of a euro area recession obtained from the MSMF-VAR (KF) model with the quarterly GDP and monthly ESI. The probabilities are from the recursive forecasting exercise using real-time data vintages. For example, the June 2006 data vintage uses information up to 2006:Q1 for GDP and March 2006 for ESI.

Fig. 4. Euro area probability of recession obtained from the recursive forecasting exercise: univariate MS model. Note: This figure reports the quarterly real-time probability of a euro area recession obtained from a univariate MS model. The probabilities are from the recursive forecasting exercise using real-time data vintages. For example, the June 2006 data vintage uses information up to 2006:Q1 for GDP.

recessions. Second, when pooling the results across indicators, the MSMF-VAR (KF) model yields the best results in terms of QPS and LPS (see the last column of Table 7).

In particular, Fig. 3 shows that the MSMF-VAR (KF) model can indeed capture euro area recessionary episodes very well in a pseudo real-time forecasting context, including

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Fig. 5. Euro area probability of recession obtained from the recursive forecasting exercise: MS-MIDAS model. Note: This figure reports the quarterly realtime probability of a euro area recession obtained from a MS-MIDAS model with the quarterly GDP and monthly ESI. The probabilities are from the recursive forecasting exercise using real-time data vintages. For example, the June 2006 data vintage uses information up to 2006:Q1 for GDP and March 2006 for ESI.

Fig. 6. Euro area probability of recession obtained from the recursive forecasting exercise: bivariate MS model. Note: This figure reports the quarterly real-time probability of a euro area recession obtained from a bivariate MS model with the quarterly GDP and quarterly ESI. The probabilities are from the recursive forecasting exercise using real-time data vintages. For example, the June 2006 data vintage uses information up to 2006:Q1 for GDP and March 2006 for ESI.

the last euro area recession. Third, the stacked-system version of the MSMF-VAR is nearly always outperformed by its Kalman filter version counterpart (except when compared with the MSMF-VAR(SV-R) with the slope of the yield curve

as a monthly indicator), which is in line with the simulation results. Fourth, as Guérin and Marcellino (2013) found for the US, the MS-MIDAS model with the slope of the yield curve also works very well for the euro area.

16

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Fig. 7. Euro area probability of recession obtained from the recursive forecasting exercise: MSMF-VAR (SV-U) model. Note: This figure reports the quarterly real-time probability of a euro area recession obtained from a MSMF-VAR (SV-U) model with the quarterly GDP and monthly ESI. The probabilities are from the recursive forecasting exercise using real-time data vintages. For example, the June 2006 data vintage uses information up to 2006:Q1 for GDP and March 2006 for ESI.

Fig. 8. Euro area probability of recession obtained from the recursive forecasting exercise: MSMF-VAR (SV-R) model. Note: This figure reports the quarterly real-time probability of a euro area recession obtained from a MSMF-VAR (SV-R) model with the quarterly GDP and monthly ESI. The probabilities are from the recursive forecasting exercise using real-time data vintages. For example, the June 2006 data vintage uses information up to 2006:Q1 for GDP and March 2006 for ESI.

On balance, our results suggest that there are advantages in using both simple models based on GDP alone and more elaborate models involving different frequency

mixes, such as the MSMF-VAR(KF) model, for detecting business cycle turning points. This is in line with the Monte Carlo results and the conclusions of Hamilton (2011).

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5. Conclusions

)

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17

for state space models with regime switching are:

In this paper, we introduce regime switching parameters to the mixed-frequency VAR model. We first discuss estimation and inference for the resulting Markovswitching mixed frequency VAR (MSMF-VAR) model, based on either a combination of the Kalman and Hamilton filters, as per Kim and Nelson (1999), or the Hamilton filter applied on a stacked version of the MF-VAR. Next, we assess the finite sample performance of the technique in Monte- Carlo experiments, relative to a large set of alternative models for mixed frequency data, with or without Markov switching. We find that the estimation of the MSMF-VAR is fairly complex, such that its forecasting performances for the variables in levels are not so good unless there are frequent changes in regimes and a fairly long estimation sample. The MSMF-VAR (SV) seems to work better than the MSMF-VAR (KF), while the opposite is true when estimating regime changes. For the latter target, the MSMF-VAR (KF) becomes very competitive. Finally, the MSMF-VAR model is used to predict GDP growth and business cycle turning points in the euro area. Again, its performance is compared with those of a number of competing models, including linear and regime switching mixed data sampling (MIDAS) models. In line with the Monte Carlo experiments, the MSMF-VAR (KF) is particularly useful for estimating the status of economic activity, while the MSMF-VAR (SV) seems to work well for nowcasting and short term forecasting of the euro area GDP growth (when using the ESI as a timely monthly indicator). Acknowledgments We would like to thank the handling Editor, George Athanasopoulos, two anonymous referees, and seminar participants at the Multivariate Time Series Modelling Workshop in Melbourne, Norges Bank, Bank of Canada, the 21st SNDE conference in Milan, the 47th Annual conference of the CEA in Montréal, the 2013 CEF conference in Vancouver, the 2013 JSM in Montréal and the 2013 EC(2) conference in Nicosia for useful comments on a previous draft. The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada or the Norges Bank.

j) βt(|it,− ˜ j + Fj βti−1|t −1 1 = µ

(28)

i,j P t | t −1

(29)

= Fj Pti−1|t −1 Fj′ + GQj G′j

j) ηt(|i,tj−) 1 = yt − Hj βt(|it,− 1 − Aj zt (i,j) (i,j) ′ ft |t −1 = Hj Pt |t −1 Hj + Rj j) (i,j) ′ (i,j) −1 (i,j) βt(|it,j) = βt(|it,− ηt |t −1 1 + Pt |t −1 Hj [ft |t −1 ] (i,j) (i,j) (i,j) ′ (i,j) −1 Pt |t = (I − Pt |t −1 Hj [ft |t −1 ] Hj )Pt |t −1 .

(30) (31) (32) (33)

When there is regime switching, it is also necessary to introduce approximations at the end of the Kalman and Hamilton filters in order to avoid the proliferation of cases to be considered: M 

β

j t |t

=

=

(34)

Pr [St = j|Ψt ] M 

j Pt |t

(i,j)

Pr [St −1 = i, St = j|Ψt ]βt |t

i=1

(i,j)

j

(i,j)

j

(i,j)

Pr [St −1 = i, St = j|Ψt ]{Pt |t + (βt |t − βt |t )(βt |t − βt |t )′ }

i=1

.

Pr [St = j|Ψt ]

(35)

Appendix B. Additional details of the Monte Carlo experiments In the Monte Carlo experiments, we estimate 15 models in order to evaluate the forecasting performances of the Markov-switching mixed frequency VAR models. Denote the low frequency variable sampled at the low frequency unit by yt , the low frequency variable at the high frequency unit by y∗t , the high frequency variable aggregated at the low frequency unit via a simple arithmetic mean by xt , and the high frequency variable sampled at the high frequency unit by x∗t . Also, assuming a mix of quarterly and monthly variables, denote the releases of the high frequency variable in the first, second and third months of quarter t by x∗ 2 , x∗ 1 and x∗t , respectively. When considering regime t− 3

t− 3

switches, we assume that St is an ergodic and irreducible Markov-chain with two regimes. The models under comparison are listed below.

• AR(1) model: yt = α0 + α1 yt −1 + ϵt ,

ϵt ∼ N (0, σ )

(36)

• ADL(1,1) model: Appendix A. Kalman and Hamilton filters for estimating state space models with regime switching The equations of the basic Kalman filter can be found in standard time series textbooks such as that of Lütkepohl (2005). The general representation of a state space model with regime switching in both the measurement and transition equations is given by: yt = H (St )βt + A(St )zt + et

(26)

βt = µ( ˜ St ) + F (St )βt −1 + G(St )vt ,

(27)

where et ∼ N (0, R(St )), vt ∼ N (0, Q (St )), and et and vt are not correlated. Kim and Nelson (1999) show how the Kalman and Hamilton filters can be combined in a tractable way. The equations of the Kim and Nelson (1999) filtering procedure

yt = α0 + α1 yt −1 + β1 xt + ϵt ,

ϵt ∼ N (0, σ )

(37)

• linear MIDAS model: yt = β0 + β1 B(L(1/m) ; θ )x∗t −h + ϵt ,

ϵt ∼ N (0, σ ) (38)

• linear ADL-MIDAS model: yt = β0 + β1 B(L(1/m) ; θ )x∗t −h + λyt −1 + ϵt ,

ϵt ∼ N (0, σ ) • univariate Markov-switching model:

(39)

yt = α0 (St ) + α1 yt −1 + ϵt (St ),

ϵt |St ∼ N (0, σ (St )) • Markov-switching MIDAS model

(40)

yt = β0 (St ) + β1 B(L(1/m) ; θ )x∗t −h + ϵt (St ),

ϵt |St ∼ N (0, σ (St ))

(41)

18

C. Foroni et al. / International Journal of Forecasting (

• Markov-switching ADL-MIDAS model yt = β0 (St ) + β1 B(L

(1/m)

; θ )x∗t −h

+ λyt −1 + ϵt (St ),

ϵt |St ∼ N (0, σ (St ))

(42)

• Kalman filter version of the linear mixed frequency VAR (MF-VAR (KF)): t − 31

+ ut ,

ut ∼ N (0, Σ )

(43)

where Yt∗ = (y∗t , x∗t )′ • Kalman filter version of the Markov-switching mixed

+ ut (St ),

t − 31

ut |St ∼ N (0, Σ (St ))

(44)

∗ ′

Yt = A0 + A1 Yt −1 + ut ,

ut ∼ N (0, Σ )

(45)

where Yt = (yt , xt )′ • Markov-switching VAR(1) model:

(46)

ut ∼ N (0, Σ )

(47)

where Zt = (xt − 2 , xt − 1 , xt , yt ) 3 3 • stacked vector system version of the Markov-switching ′

mixed frequency VAR with no restrictions on the parameters (MSMF-VAR (SV-U)): Zt = A0 (St ) + A1 Zt −1 + ut (St ), ut |St ∼ N (0, Σ (St ))

(48)

where Zt = (xt − 2 , xt − 1 , xt , yt )′ 3 3 • stacked vector system version of the linear mixed frequency VAR with restrictions on some of the parameters (MF-VAR (SV-R)): Zt = A0 + A1 Zt −1 + ut ,

ut ∼ N (0, Σ ),

(49)

where Zt = (xt − 2 , xt − 1 , xt , yt ) and the A1 matrix has a 3 3 ′

particular structure. We assume that the high frequency process is ARX(1), with the impact of the low frequency series being constant throughout the period. Then, following Ghysels (2012), Eq. (49) can be rewritten as follows:



xt − 2

3





0

yt

b1



ρH ρH2 ρH3

0 0 0 b2

x 1  0  t− 3  =    x  0 t uxt

∗,1

b3

t

y

ut

 x 5  a1 t− 3 a2   xt − 4   ∗ 3 a3 x t −1  ρL yt −1

σyy

(52)

where Zt = (xt − 2 , xt − 1 , xt , yt )′ and the A1 matrix has a 3 3 structure similar to that in Eq. (50) above. The variance–covariance matrix of the residuals takes the following form: 

 σxx (St ) ρH σxx (St ) ρH2 σxx (St ) σxy (St )   ρH σxx (St ) (1 + ρH )σxx (St ) ρH σxx (St ) σxy (St ) . (53) Σ = ρ 2 σ (S ) 2 ρH σxx (St ) (1 + ρH + ρH )σxx (St ) σxy (St )  H xx t  σxy (St )

σxy (St )

σyy (St )

Appendix C. Nowcasting exercise with the stacked vector system version of the mixed-frequency VAR model In the case of the stacked vector system version of the mixed-frequency VAR model, nowcasts are obtained as follows. First, denote the reduced-form version of the VAR(1) model as follows: Zt = AZt −1 + et .

(54)

Assume that E (et e′t ) = V and V = P −1 D(P −1 )′ , where P has a unit main diagonal and D is a diagonal matrix. Premultiplying Eq. (54) by P, we obtain the structural VAR: PZt = PAZt −1 + ϵt . (55) The structural shocks, ϵt , are given by ϵt = Pet For simplicity of notation, we assume in what follows that there is only one low-frequency variable and one highfrequency variable, and consider the standard macroeconomic forecasting context of a mix between quarterly and monthly variables, but note that this framework can be extended easily to larger systems or different frequency mixes. We then assume that Zt = (xt − 2 , xt − 1 , xt , yt )′ , 3

3

where: • xt − 2 is the monthly indicator corresponding to the first 3

month of the quarter. • xt − 1 is the monthly indicator corresponding to the 3

second month of the quarter. • xt is the monthly indicator corresponding to the third month of the quarter. • yt is the quarterly indicator. The first possibility is that we want to forecast yt +k without knowing any observations of xt +k− 2 , xt +k− 1 and 3



 x∗,2    + ut ∗,3  .  ux 

H

mixed frequency VAR with restrictions on some of the parameters (MSMF-VAR (SV-R)):

σxy (St )

where Yt = (yt , xt )′ • stacked vector system version of the linear mixed frequency VAR with no restrictions on the parameters (MF-VAR (SV-U)): Zt = A0 + A1 Zt −1 + ut ,

σxy

• stacked vector system version of the Markov-switching

Yt = A0 (St ) + A1 Yt −1 + ut (St ), ut |St ∼ N (0, Σ (St ))

σxy

ut |St ∼ N (0, Σ (St ))

where Yt = (yt , xt ) • VAR(1) model: ∗

σxy

Zt = A0 (St ) + A1 Zt −1 + ut (St ),

frequency VAR (MSMF-VAR (KF)): Yt∗ = A0 (St ) + A1 Y ∗

–

Likewise, in line with Ghysels (2012), we also impose the following restrictions on the variance–covariance matrix of the residuals:   σxx ρH σxx ρH2 σxx σxy ρH σxx (1 + ρH )σxx ρH σxx σxy  . (51) Σ = ρ 2 σxx ρH σxx (1 + ρH + ρ 2 )σxx σxy  H

Yt∗ = A0 + A1 Y ∗

∗

)

(50)

3

xt +k , where k = {1, 2, 3, . . . , H }. This is the standard case where one uses the reduced-form VAR model described in Eq. (54). In contrast, if one wants to forecast yt +k provided that the first release of the high frequency indicator has been

C. Foroni et al. / International Journal of Forecasting (

released (i.e., xt +k− 2 ), one can use certain elements of the

)

–

19

References

3

structural VAR matrix P to capture the contemporaneous correlation between the variables for nowcasting (and performing within-quarter updates of the nowcasts). Let us define the P matrix of contemporaneous correlations that is obtained via a Choleski decomposition as follows:



1 A P = B C

0 1 D E

0 0 1 F



0 0 . 0 1

(56)

Then, Eq. (55) above can be rewritten as follows:



1 A B C

0 1 D E





 xt − 2 0 3  0  x t − 31  ∗   x  0 t 1 yt

0 0 1 F

 x 5  α11 α12 α13 α14 t− 3 xt − 4  α21 α22 α23 α24    = ∗ 3 α31 α32 α33 α34   xt −1  α41 α42 α43 α44 yt −1  x(1)  ϵt ϵ x(2)   , (57) +  tx(3)  ϵt  y ϵt where PA = [α]ij is the matrix of structural-form VAR co

efficients. From Eq. (57), the optimal forecast for yt +1 given xt + 1 is 3

yt +1 = α41 xt − 2 + α42 xt − 1 + α43 xt + α44 yt − Cxt + 1 3  3    − E E xt + 2 |xt + 1 − FE xt +1 |xt + 1 , 3

3

3

3

where



E xt + 2 |xt + 1 3 3



= α21 xt − 2 + α22 xt − 1 3

3

+ α23 xt + α24 yt − Axt + 1 3

and



E xt +1 |xt + 1



3

= α31 xt − 2 + α32 xt − 1 + α33 xt 3 3   + α34 yt − Bxt + 1 − DE xt + 2 |xt + 1 . 3

3

3

Along the same lines, the optimal nowcast for yt +1 given xt + 1 and xt + 2 is 3

3

yt +1 = α41 xt − 2 + α42 xt − 1 + α43 xt + α44 yt − Cxt + 1 3

3

3





− Ext + 2 − FE xt +1 |xt + 2 , 3

(58)

3

where



E xt +1 |xt + 2 3



= α31 xt − 2 + α32 xt − 1 + α33 xt 3

3

+ α34 yt − Bxt + 1 − Dxt + 2 . 3

(59)

3

Finally, if we know xt + 1 , xt + 2 and xt +1 , the nowcast for 3

yt +1 is given by:

3

yt +1 = α41 xt − 2 + α42 xt − 1 + α43 xt + α44 yt 3

3

− Cxt + 1 − Ext + 2 − Fxt +1 . 3

3

(60)

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Schorfheide, F., & Song, D. (2012). Real-time forecasting with a mixedfrequency VAR. FRB Minneapolis Working Paper 701. Zadrozny, P. A. (1990). Forecasting U.S. GNP at monthly intervals with an estimated bivariate time series model. Federal Reserve Bank of Atlanta Economic Review, Nov., 2–15. Claudia Foroni is an economist at the Norges Bank. She obtained her Ph.D. in Economics from the European University Institute in 2012, with a dissertation on mixed frequency models in econometrics. She has written several papers in this area, one of which has already been published in the International Journal of Forecasting. Pierre Guérin is a senior analyst at the Bank of Canada. He obtained his Ph.D. in Economics from the European University Institute in 2011, with a dissertation on applied time series econometrics. He has written several papers in this area, one of which has been published in the Journal of Business and Economic Statistics. Massimiliano Marcellino is a professor of Econometrics at Bocconi University. He is also a fellow of the Center for Economic Policy Research and Scientific Chairman of the Euro Area Business Cycle Network. He has published a large number of academic articles in leading international journals on applied macroeconomics, econometrics and forecasting, his main areas of research and teaching. He is currently an editor of the Journal of Forecasting and the coordinator of the European Forecasting Network.