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Journal of Business & Economic Statistics
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Markov-Switching MIDAS Models
Pierre Guérin & Massimiliano Marcellino
To cite this article: Pierre Guérin & Massimiliano Marcellino (2013)
Markov-Switching MIDAS Models, Journal of Business & Economic Statistics, 31:1, 45-56, DOI: 10.1080/07350015.2012.727721
To link to this article: http://dx.doi.org/10.1080/07350015.2012.727721
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Markov-Switching MIDAS Models
Pierre G
UERIN´
Bank of Canada, Ottawa, ON K1A 0G9 ([email protected])
Massimiliano M
ARCELLINOEuropean University Institute, Florence, Italy; Bocconi University, Milan, Italy; and CEPR, London, United Kingdom ([email protected])
This article introduces a new regression model—Markov-switching mixed data sampling (MS-MIDAS)— that incorporates regime changes in the parameters of the mixed data sampling (MIDAS) models and allows for the use of mixed-frequency data in Markov-switching models. After a discussion of estimation and inference for MS-MIDAS and a small sample simulation-based evaluation, the MS-MIDAS model is applied to the prediction of the U.S. economic activity, in terms of both quantitative forecasts of the aggregate economic activity and the prediction of the business cycle regimes. Both simulation and empirical results indicate that MS-MIDAS is a very useful specification.
KEY WORDS: Business cycle; Forecasting; Mixed-frequency data; Nonlinear models; Nowcasting.
1. INTRODUCTION
The econometrician often faces a dilemma when observations are sampled at different frequencies. One solution consists of estimating the model at the lowest frequency, temporally ag-gregating the high-frequency data. However, this solution is not fully satisfactory since important information can be discarded in the aggregation process. A second solution is to temporally disaggregate (interpolate) the low frequency variables. How-ever, there is no agreement on the proper interpolation method, and the resulting high-frequency variables would be affected by measurement error.
The third option is represented by regression models that combine variables sampled at different frequencies. They are particularly attractive since they can use the information of high-frequency variables to explain variables sampled at a lower frequency without any prior aggregation or interpolation. In this context, the MIDAS (Mixed Data Sampling) model by Ghysels, Santa-Clara, and Valkanov (2004) has recently gained consid-erable attention. A crucial feature of this class of models is the parsimonious way of including explanatory variables through a weighting function, which can take various shapes depending on the value of its parameters.
MIDAS models have been applied for predicting both macroeconomic and financial variables. Ghysels, Santa-Clara, and Valkanov (2006) used the MIDAS framework to predict the volatility of equity returns, while Clements and Galvao (2008) successfully applied MIDAS models to the prediction of quarterly U.S. GDP growth using monthly indicators as high-frequency variables. Andreou, Ghysels, and Kourtellos (2010) exploited the informational content of daily financial variables to predict quarterly GDP and inflation in the United States. In particular, they extend the standard MIDAS model to include factors in a dynamic framework, along the lines of Marcellino and Schumacher (2010).
MIDAS models are generally used as single-equation models where the dynamics of the indicator is not modeled. By contrast, system-based models such as the mixed-frequency VAR (MF-VAR) explicitly model the dynamics of the indicator. Kuzin,
Marcellino, and Schumacher (2011) compared the forecasting performance of MIDAS and MF-VAR models for the prediction of the quarterly GDP growth in the Euro area. They find that MIDAS models outperform MF-VAR for short horizons (up to 5 months), while MF-VAR tend to perform better for longer horizons. A similar comparison is provided by Bai, Ghysels, and Wright (in press).
An issue that has so far attracted limited attention in the MIDAS literature is the stability of the relationship between the high- and low-frequency variables. Time-variation in MIDAS models has been only introduced by Galvao (2009) via a smooth transition function governing the change in some parameters of the model. This Smooth Transition MIDAS is applied to the prediction of quarterly U.S. GDP using weekly and daily financial variables.
In this article, we propose an alternative way to allow for time-variation in the MIDAS model, introducing the Markov-switching MIDAS (MS-MIDAS) model. Regime changes may result from asymmetries in the process of the mean or variance. From an economic point of view, the predicting ability of the higher-frequency variables could change across regimes follow-ing, for example, changes in market conditions or business cycle phases. For example, the slope of the yield curve is often consid-ered as a strong predictor of U.S. recessions, an inverted yield curve signaling a forthcoming recession. However, Galbraith and Tkacz (2000) argued that the predictive power of the slope of the yield curve is limited in normal times. Therefore, it could be important to permit time-variation in the predictive ability of the high-frequency data. Indeed, our empirical applications show that in general the predictions from MS-MIDAS models are more accurate than those from simple MIDAS models.
An additional attractive feature of Markov-switching models is the possibility of estimating and predicting the probabilities of being in a given regime. The literature (see e.g., Estrella
© 2013American Statistical Association Journal of Business & Economic Statistics
January 2013, Vol. 31, No. 1
DOI:10.1080/07350015.2012.727721
45
and Mishkin1998; Birchenhall et al.1999) often uses binary response models to predict the state of the economy using the NBER dating of expansions and contractions as a dependent variable. However, this method can be problematic since the announcements of turning points may be published up to 20 months after the turning point has actually occurred. Our MS-MIDAS model instead allows for real-time evaluation and fore-casting of the probability of being in a given regime.
Finally, MS-MIDAS is also a convenient approach to allow for the use of mixed frequency information in standard Markov-switching models. Hamilton (2011) pointed out the importance of using models with mixed frequency data for predicting re-cessions in real time. In our applications, the forecasting perfor-mance of standard MS models is indeed improved by the use of higher frequency information. In addition, MS-MIDAS models clearly outperform the linear and nonlinear benchmark mod-els we consider at the nowcasting horizon in our two empirical applications.
The article is organized as follows. Section2reviews the MI-DAS approach, introduces the MS-MIMI-DAS, and discusses the estimation method. Section3presents Monte Carlo simulations to assess the accuracy of the proposed estimation method in fi-nite samples and its forecasting accuracy. Section4presents two empirical applications on modeling and forecasting the levels and turning points of quarterly GDP growth in the United States using monthly indicators, and monthly U.S. IP growth using weekly indicators. Section 5 concludes. An online Appendix with an in-sample Monte Carlo experiment and an additional empirical application with UK data completes the article.
2. MARKOV-SWITCHING MIDAS
2.1 MIDAS Approach
2.1.1 Basic MIDAS. The MIDAS approach by Ghysels, Santa-Clara, and Valkanov (2004) involved the regression of variables sampled at different frequencies. Following the no-tation by Clements and Galvao (2008), and assuming that the model is specified forh-step ahead forecasting, the basic uni-variate MIDAS model is given by
yt =β0+β1B(L1/m;θ)x variableyt andmto the time unit of the higher-frequency
vari-ablesxt(m−h).
The forecasts of the MIDAS regression are computed directly so that no forecasts for the explanatory variables are required. However, unlike iterated forecasts, direct forecasts require to re-estimate the model when the forecasting horizon changes (see Chevillon and Hendry 2005; Marcellino, Stock, and Watson 2006, for a comparison of the relative merits of iterated and direct forecasts).
The crucial difference between MIDAS and Autoregressive Distributed Lag models is that the content of the higher fre-quency variable is exploited in a parsimonious way through the polynomialb(j;θ), which allows to have a rich variety of shapes with a limited number of parameters. Ghysels, Sinko, and Valkanov (2007) detailed various specifications for the
polyno-mial of lagged coefficients b(j;θ). A popular choice for the weighting scheme is the exponential Almon lag:
b(j;θ)= exp(θ1j+ · · · +θQj
Note that the weighting function of the exponential Almon lag implies that the weights are always positive. Also, we impose the restrictionKj=1b(j;θ)=1 to ensure that the parametersβ1 andθ are identified. In the empirical applications, we employ the exponential Almon lag scheme with two parametersθ = {θ1, θ2}.
2.1.2 Autoregressive MIDAS. Introducing an autoregres-sive lag in the MIDAS specification is not straightforward, as pointed out by Clements and Galvao (2008) who showed that a seasonal response ofytoxcan appear. However, they find that this can be done without generating any seasonal patterns if au-toregressive dynamics is introduced through a common factor, so that Equation (1) becomes
yt =β0+λyt−d+β1B(L1/m;θ)(1−λLd)xt(−mh) +ǫt. (3)
Andreou, Ghysels, and Kourtellos (2011) instead reported that this specification has undesirable properties since the lag poly-nomial is characterized by geometrically declining spikes at the low frequency due to the interaction of the high-frequency polynomial with the low-frequency polynomial. Hence, we also consider the ADL-MIDAS specification described by Andreou, Ghysels, and Kourtellos (2010). This specification can be writ-ten as
yt=β0+λyt−d+β1B(L1/m;θ)x (m)
t−h+ǫt. (4)
2.2 Markov-Switching MIDAS
2.2.1 The Model. The basic idea behind Markov-switching models is that the parameters of the underlying data generating process (DGP) depend on an unobservable discrete variableSt, which represents the probability of being in a
differ-ent state of the world (see Hamilton1989). The basic version of the Markov-switching MIDAS (MS-MIDAS) regression model we propose is
yt =β0(St)+β1(St)B(L1/m;θ)x(t−m)h+ǫt(St), (5)
whereǫt|St ∼NID(0, σ2(St)).
The MS-MIDAS that includes autoregressive dynamics as in the literature by Clements and Galvao (2008) is instead defined as
yt=β0(St)+λyt−d+β1(St)B(L1/m;θ)(1−λLd)x
(m)
t−h+ǫt(St).
(6)
The MS-MIDAS with autoregressive dynamics without a common factor is instead written as
yt =β0(St)+λyt−d+β1(St)B(L1/m;θ)xt(−m)h+ǫt(St).
The regime-generating process is an ergodic Markov-chain with a finite number of statesSt = {1, . . . , M}defined by the
following transition probabilities:
Here, the transition probabilities are constant. This assump-tion was originally relaxed by Filardo (1994), who used time-varying transition probabilities modeled as a logistic function, while Kim, Piger, and Startz (2008) modeled them as a pro-bit function. However, we stick to the assumption of constant transition probabilities to keep the model tractable.
The parameters that can switch are the intercept of the equa-tion,β0, the parameter entering before the weighting scheme,
β1, and the variance of the disturbances, σ2. Changes in the interceptβ0are important since they are one of the most com-mon sources of forecast failure (see e.g., Clements and Hendry 2001). The switch in the parameter β1 allows the predictive ability of the higher-frequency variable to change across the different states of the world. Besides, we also allow the variance of the disturbancesσ2to change across regimes. This proves to be useful not only for modeling financial variables but also for applications with macroeconomic variables.
Note that we do not consider changes in all parameters of the model (i.e., we keep the autoregressive parameterλ and the MIDAS parametersθ constant). A switch in the MIDAS parameters would lead to a different shape of the weighting function across regimes. However, this increases the number of parameters to be estimated and makes the estimation more difficult since the weighting function becomes highly nonlinear. As a result, our algorithm encountered convergence problems and we often obtained outlier forecasts for the dependent vari-able. Besides, we already model a change in the forecasting ability of the high-frequency indicator through a switch in the parameterβ1.
Another attractive feature of the Markov-switching models is that they allow the estimation of the probabilities of being in a given regime. This is relevant, for example, when one wants to predict business cycle regimes. Indeed, studies about the identification and prediction of the state of the economy have gained attention over the last decade (see, e.g., Estrella and Mishkin1998; Stock and Watson2010; Berge and Jorda2011; and the literature review in Marcellino2006).
Table 1reports the classification of the models we use in the empirical experiments.
2.2.2 Estimation and Model Selection. In the literature, MIDAS models are usually estimated by nonlinear least squares (NLS). However, for implementing the filtering procedure de-scribed by Hamilton (1989), we estimate the MS-MIDAS via (pseudo) maximum likelihood. We thus need to make a normal-ity assumption about the distribution of the disturbances, which is not required with the NLS estimation. We aim at maximizing the log-likelihood function given by
Table 1. Classification of the models
Model Regime changes in AR component
MSIH(M)-MIDAS β0andσ2 No
MSH(M)-MIDAS β0,β1andσ2 No
MSIHAR(M)-MIDAS β0andσ2 Yes
MSHAR(M)-MIDAS β0,β1andσ2 Yes
MSIHADL(M)-MIDAS β0andσ2 Yes
MSHADL(M)-MIDAS β0,β1andσ2 Yes
NOTE: The suffix ”H” refers to models with a switch in the variance of the shocks. The suffix ”I” refers to models with a switch in the interceptβ0. The suffix ”AR” means that we include an AR component in the model through a common factor (see Equation (3)). The suffix ”ADL” means that we include an AR component in the model without a common factor (see Equation (4)).
can be rewritten as
The computations are carried out with the optimization package OPTMUM of GAUSS 7.0 using the BFGS algorithm. Appendix A (provided online) provides more details about the estimation method we use.
Choosing the number of regimes for Markov-switching mod-els is a tricky problem. Indeed, the econometrician has to deal with two problems: first, some parameters are not identified under the null hypothesis and, second, the scores are identi-cally equal to zero under the null. Hansen (1992) considered the likelihood function as a function of unknown parameters and uses empirical processes to bound the asymptotic distri-bution of a standardized likelihood ratio test statistic. Garcia (1998) pointed out that the test is computationally expensive if the number of parameters and regimes is high. Carrasco, Hu, and Ploberger (2009) recently proposed a new method for test-ing the constancy of parameters in Markov-switchtest-ing models. Their procedure is attractive since it only requires to estimate the model under the null hypothesis of constant parameters. However, this testing procedure does not allow one to discrimi-nate between Markov-switching models with different number of regimes since the parameters must be constant under the null hypothesis.
Psaradakis and Spagnolo (2006) studied the performance of information criteria based on the optimization of complexity-penalized likelihood for model selection. They find that the AIC, SIC, and HQ criteria perform well for selecting the correct number of regimes and lags as long as the sample size and the pa-rameter changes are large enough. Smith, Naik, and Tsai (2006) proposed a new information criterion for selecting simultane-ously the number of variables and lags of the Markov-switching models. Note that both studies run their analysis with models where all parameters switch across regimes, which might not always be desirable. For example, in Equations (5) and (6), we do not consider switches in the vector of parameters θ since we encountered serious convergence problems in the empirical applications due to the relatively small size of our sample (T=200). However, with larger sample sizes, the MS-MIDAS model could easily accommodate changes in theθ vector. In addition, Driffill et al. (2009) showed that a careful study of the
parameters that can switch is crucial for forecasting accurately bond prices with the CIR model for the term structure.
In the empirical part, we will follow Psaradakis and Spagnolo (2006) and use the Schwarz information criterion for selecting the number of regimes and deciding whether the variance of the disturbances should also change across regimes. We will also report results for different parameterizations of the Markov-switching models.
3. MONTE CARLO EXPERIMENTS
The DGP used in the Monte Carlo experiments is the MSHAR(2)-MIDAS model defined by Equations (6)–(8), that is, it is a model with two regimes and switches in the intercept
β0, in the parameter entering before the weighting functionβ1 and in the varianceσ2, since models with two regimes are often used in the literature. We consider two sample sizes for the sim-ulated seriesT =200 andT =500. The matrix of explanatory variables includes a constant and the process forxt(m)is an AR(1)
with a large autoregressive coefficient (0.95) and a small drift (0.025). We are primarily interested in the predictive content of monthly variables for forecasting quarterly variables, so we setK=3 and K=13. We use the following true parameter values:
(β0,1, β0,2)=(−1,1),(β1,1, β1,2)=(0.6,0.2),
(σ1, σ2)=(1,0.67), (11)
(θ1, θ2)=(2∗10−1,−3∗10−2). (12)
These parameter values are similar to those used by Kim, Piger, and Startz (2008) and closely match the in-sample param-eter estimates of our empirical applications when (i) predicting monthly industrial production with weekly variables as high-frequency predictors, and (ii) the UK quarterly GDP growth with monthly indicators (see Tables B and C in the online Ap-pendix). The transition probabilities are first set such that both regimes are equally persistent (p11=0.95, p22 =0.95). We also consider another set of transition probabilities:p11=0.85 andp22=0.95. Indeed, with these transition probabilities, if one thinks ofyt as quarterly observations, the duration of the
first regime (6.67 quarters) is lower than the duration of the sec-ond regime (20 quarters), which roughly correspsec-onds to the av-erage duration of recessions and expansions experienced by the United States.
We first simulate a Markov chain with two regimes using one of the two sets of transition probabilities. The dependent variableytis then constructed depending on the outcome of the
simulated Markov chain using the above parameter values and the simulated series forxt.
The sample sizeTis split between an estimation sample and an evaluation sample. We choose two different sizesHfor the evaluation sample,H= {50,100}. We then run the following out-of-sample forecasting experiment: we use the firstT −H
observations and compute one-step ahead forecasts . We recur-sively expand the estimation sample until we reach the end of the sampleTso that we computeHforecasts. The design of this forecasting experiment is very close to the empirical application we run later in the article.
We use 10 different models to compute the forecasts: the MSHAR(2)-MIDAS model (i.e., the true model), the MSH(2)-MIDAS model (i.e., the true model without an autoregressive lag), a standard MIDAS, ADL-MIDAS, and AR-MIDAS mod-els. We also include as competitors MS-MIDAS models with AR dynamics without a common factor: the MSHADL-MIDAS and MSIHADL-MIDAS models. We also consider an AR(1) model and a standard Markov-switching model with two regimes, a switch in the intercept and in the variance of the disturbances, as well as one autoregressive lag. Besides, we also show the results for an MSIHAR(2)-MIDAS model (i.e., a model with a constant
β1). Finally, we also consider as an additional benchmark the ADL(1,1) model, which is a usual benchmark in the forecast-ing literature (see e.g., Baffigi, Golinelli, and Parigi2004). We always use the true number of lags (K=13) for the high-frequency variablexm
t when the models have mixed-frequency
data. Also, all nonlinear models have innovations with a regime switching variance.
We repeat the forecasting experimentNtimes for each eval-uation sample and report inTable 2the average of the mean square forecast error over the number of replications for the 10 models under consideration. We also report the quadratic probability score (QPS) and log probability score (LPS) for the models with Markov-switching features to check how well these models can predict the true regimes. Here, the MSFE is a crite-rion that allows us to assess the quantitative forecasting abilities of the model under scrutiny, whereas QPS and LPS criteria eval-uate their qualitative forecasting abilities, that is, to what extent the true regimes are predicted.
Note that the QPS is bounded between 0 and 2 and the range of LPS is 0 to∞. LPS penalizes large forecast errors more than QPS. LPS and QPS are calculated as follows:
QPS= 2 probability of being in the first regime in periodt + 1.
Table 2presents the results for the two sample sizes we con-sider (i.e.,T =200 andT =500). The number of Monte Carlo experimentsNis set to 1000. The first panel reports the results when the full estimation sample size isT =200, while the sec-ond panel reports results when the full estimation sample size is T =500. The results for the true model (the MSHAR(2)-MIDAS) are reported in the top row of each panel. We then report in the following rows of each panel the results for each model, as well as the average loss functions across the linear MIDAS models and the nonlinear MIDAS models.
For the first estimation sample sizeT =200 (reported in the top panel ofTable 2), the true model obtains the best continuous forecasts when the evaluation sample is H=100. However, the MSHADL-MIDAS model—that is, the model equivalent to the true model except that the autoregressive component is
Table 2. Monte Carlo results: forecasting exercise
50 100
Number of out-sample
forecastsH: QPS LPS MSFE QPS LPS MSFE
First estimation sample size
True model 0.423 0.655 1.796 0.483 0.721 1.736
Standard MS 0.608 1.015 1.972 0.598 1.004 1.925
MSH(2)-MIDAS 0.413 0.623 1.819 0.416 0.619 1.784
MSIHAR(2)-MIDAS 0.484 0.603 1.812 0.432 0.673 1.785
MSIHADL-MIDAS 0.468 0.729 1.850 0.487 0.762 1.746
MSHADL-MIDAS 0.380 0.574 1.779 0.394 0.597 1.770
ADL-MIDAS – – 1.869 – – 1.739
AR-MIDAS – – 1.885 – – 1.789
MIDAS – – 2.525 – – 2.434
AR(1) – – 1.932 – – 1.749
ADL(1,1) – – 1.811 – – 1.784
Av. Linear MIDAS – – 2.093 – – 1.987
Av. Nonlinear MIDAS 0.433 0.637 1.811 0.443 0.674 1.764
Second estimation sample size
True model 0.345 0.528 1.721 0.378 0.578 1.624
Standard MS 0.684 1.181 1.868 0.653 1.141 1.830
MSH(2)-MIDAS 0.361 0.559 1.794 0.407 0.570 1.739
MSIHAR(2)-MIDAS 0.390 0.592 1.934 0.396 0.606 1.913
MSIHADL-MIDAS 0.416 0.638 1.766 0.434 0.676 1.717
MSHADL-MIDAS 0.318 0.500 1.732 0.319 0.507 1.611
ADL-MIDAS – – 1.805 – – 1.792
AR-MIDAS – – 1.848 – – 1.745
MIDAS – – 2.351 – – 2.167
AR(1) – – 1.889 – – 1.722
ADL(1,1) – – 1.793 – – 1.754
Av. Linear MIDAS – – 2.001 – – 1.901
Av. Nonlinear MIDAS 0.366 0.563 1.789 0.387 0.588 1.721
NOTE: This table reports the average QPS, LPS, and MSFE over 1000 Monte Carlo replications. In the first estimation sample, the initial estimation sample size isT−HwhereT=200. In the second estimation sample, the initial estimation sample size isT−HwhereT=500. Both estimation samples are recursively expanded until the end of the sample is reached. The true model is the MSHAR(2)-MIDAS model. A classification of the models is provided inTable 1.
not included through a common factor—yields the best results when the evaluation sample size isH =50 for both discrete and continuous forecasts. In addition, this model produces the best discrete forecasts with the evaluation sample H =100. The simple MIDAS model has a poor forecasting performance as compared to the AR-MIDAS model, and the nonlinear MIDAS models perform on average better than the linear MIDAS models for both evaluation samplesH = {50,100}.
Turning our attention to the second, longer, estimation sample size (i.e., the bottom panel ofTable 2), the true model obtains the best continuous forecasts in the first evaluation sample while the MSHADL-MIDAS model obtains the best discrete fore-casts. Furthermore, in the larger evaluation sampleH=100, the MSHADL-MIDAS models obtain the best forecasts for both discrete and continuous forecasts. In addition, on average, non-linear MIDAS models outperform the non-linear MIDAS models. Finally, the standard MS model and AR(1) model yield rather inaccurate continuous forecasts.
Overall, the true MS-MIDAS model is either ranked first or has a forecasting performance very close to the best model for discrete forecasts. In addition, our simulation experiments sug-gest that including an autoregressive component through a com-mon factor can be detrimental for both discrete and continuous
forecasts. Indeed, the true model is often outperformed by the MSHADL-MIDAS, which is equivalent to the true model ex-cept that the autoregressive component is not included through a common factor.
4. EMPIRICAL APPLICATIONS
4.1 U.S. Quarterly GDP Growth and Monthly Indicators
4.1.1 Design of the Real-Time Forecasting Exercise. We analyze quarterly data for the U.S. GDP, taken from the real-time dataset of the Philadelphia Federal Reserve, available athttp:// www.philadelphiafed.org/research-and-data/real-time-center/ real-time-data/data-files/ROUTPUT/, which originates from the work by Croushore and Stark (2001). The quarterly vintages we use reflect the information available in the middle month of each quarter. The dependent variable is taken as 100 times the quarterly change in the log of the United States’ real GDP fromt =1959:Q1 to 2009:Q4. The sample is then split into an estima-tion sample and an evaluaestima-tion sample. The evaluaestima-tion sample consists of quarterly GDP growth in the quarters 1998:Q1– 2009:Q4. For each of these quarters, we generate forecasts with horizons h= {0,1/3,2/3,1}. The initial estimation
sample goes from 1959:Q1 to 1997:Q4 and is recursively expanded over time until we reach the end of the sample.
The forecasting model is the MS-MIDAS model with three regimes (MSIHAR(3)-MIDAS), which an in-sample analysis reported in Appendix C (provided online) indicates as a proper specification for describing quarterly U.S. GDP growth, and consider alternative monthly indicators. First, we use the slope of the yield curve since its predictive power for GDP growth has been widely documented (Estrella and Hardouvelis 1991; Galvao2006; Rudebusch and Williams2009). We take the dif-ference between the monthly yields on 10-year Treasury bond and the monthly yields on the 3-month Treasury-bill as a proxy for the monthly slope of the yield curve. We also consider stock returns as a monthly indicator for forecasting quarterly aggre-gate economic activity. Stock returns are taken as 100 times the monthly change in the log of the S&P500 index. We then con-sider the monthly Federal Funds as a monthly indicator to take into account the stance of the monetary policy, which is often considered as an important determinant of economic activity. We take the first difference for both the slope of the yield curve and the Federal Funds since we achieve better forecasting results with this transformation. We also use industrial production and one survey of economic activity, the ISM manufacturing. The data for the 10-year Treasury bond yields, the 3-month Treasury bill, and the Federal Funds are taken from the Federal Reserve web site, while the data for the S&P500 index are downloaded from Yahoo Finance. The real-time data for monthly industrial production are downloaded from the Federal Reserve Bank of Saint Louis web site, while the data for the monthly ISM manu-facturing are downloaded from Haver Analytics. These last two variables are taken as 100 times the monthly change in the log of the variable in level.
The design of the exercise is similar to the one described in Section 3.1.1 of the article by Clements and Galvao (2008). We denoteyτ,νas output growth in periodτ released in the vintage νdataset. We aim at forecasting output growth as defined in the second vintage data releases of GDP. Note that for GDP, the vintage dataset released in quartert + 1 contains data up to quartert, and the quarterly vintages we use reflect information available in the middle month of each quarter.
A few additional comments are required. First, forecasts for the regime probabilitieskquarters ahead are computed recur-sively as
Note that the predicted probabilities only depend on the transi-tion probabilities and on the filtered probabilities.
Second, forecasts with a horizonh=0 (i.e., nowcasts) im-ply that we want to forecast output growth for the current quarter knowing the values of the monthly indicators for all months of the current quarter. The nowcasts are computed as follows: we first regressyt|t+1onB(L(1/3);θ)xt|t+1andyt−1|t+1,
whereyt|t+1=[y1|t+1, y2|t+1, . . . , yt−1|t+1, yt|t+1] andxt|t+1= [x1|t+1, . . . , xt−1|t+1, xt|t+1]. We then use these estimates, the
forecasts for the regime probabilitiesP(St+1 =j|xt), yt|t+1, and
xt+1|t+1to compute the forecasts ˆyt+1|t+1.
Forecasts with a horizon h=1/3 imply that we only know the values for the first 2 months of the monthly in-dicator. To obtain these forecasts, we first regress yt|t+1 on B(L(1/3);θ)x
t−1/3|t+1 and yt−1|t+1, where xt−1/3|t+1 = [x1−1/3|t+1, . . . , xt−4/3|t+1, xt−1/3|t+1]. We then use these
es-timates, the forecasts for the regime probabilities P(St+1=
j|xt), yt|t+1 and xt+2/3|t+1 to obtain forecasts for yt+1, which
is conditioned onxt+2/3|t+1andyt|t+1.
Similarly, forecasts with a horizon h=2/3 imply that we only know the values for the first month of the monthly indicator. To obtain these forecasts, we first regress
yt|t+1 onB(L(1/3);θ)xt−2/3|t+1andyt−1|t+1where xt−2/3|t+1 = [x1−2/3|t+1, . . . , xt−5/3|t+1, xt−2/3|t+1]. We then use these
esti-mates, the forecasts for the regime probabilitiesP(St+1=j|xt)
andxt+1/3|t+1to obtain forecasts foryt+1, which is conditioned
onxt+1/3|t+1.
Finally, forecasts with a horizonh=1 are computed from the regression ofyt|t+1onB(L(1/3);θ)xt−1|t+1andyt−1|t+1. Hence,
for example, forecasts for the first quarter Q1 of a given year are generated as described inTable 3.
4.1.2 Out-of-Sample Results. Table 4reports our out-of-sample forecasting results for forecasting quarterly GDP growth with monthly variables. We organize our results by report-ing in the first column of each panel, for each indicator, the MSFE of the average of the forecasts across the six nonlinear MIDAS models we consider. We then calculate three relative mean square forecast errors (RMSFE) in the next three columns of each panel, using three different benchmarks: (i) an AR(1) model, (ii) a standard ADL(1,1) model that regresses quarterly GDP on its first lag and a quarterly average of the monthly indi-cator, and (iii) the MSFE of the forecasts across the three linear MIDAS models we consider. Each of these three benchmarks has its own virtues: the AR(1) model is a standard benchmark in the forecasting literature; the ADL(1,1) model allows us to assess the ability of the nonlinear MIDAS models to outper-form a regression model that uses a simple rule of thumb for aggregating the high-frequency variable; and the last benchmark (the average of the forecasts across the linear MIDAS models) permits us to directly evaluate the benefits of using Markov-switching parameters in the context of MIDAS models. The complete results for each individual model we estimate at each
Table 3. Forecasting scheme forQ1
Forecast horizonh 0 1/3 2/3 1
Data up to month Readily available variables Marcht Febt Jant Dect−1
Data up to month Macroeconomic variables Febt Jant Dect Novt−1
NOTE: The variables available without a publishing lag are stock prices, the slope of the yield curve, Federal Funds, and ISM Manufacturing. The industrial production series are published with a 1-month lag.
Table 4. Forecasting U.S. quarterly GDP growth with monthly variables
Slope of the yield curve Federal Funds S&P500
Ratio to Ratio to Ratio to
Nonlinear Nonlinear Nonlinear
MIDAS Linear MIDAS Linear MIDAS Linear
Horizon (MSFE) MIDAS AR ADL (MSFE) MIDAS AR ADL (MSFE) MIDAS AR ADL
0 0.336 0.984 1.107 1.056 0.289a 1.037 0.952 0.991 0.234a 1.042 0.770 1.001 1/3 0.328 0.975 1.082 1.032 0.280b 0.955 0.923 0.960 0.239b 1.063 0.789 1.025 2/3 0.323 0.945 1.064 1.015 0.296 0.956 0.975 1.015 0.189b .894 0.623 0.810 1 0.308 0.901 1.013 0.967 0.311 0.980 1.023 1.065 0.241c 1.195 0.793 1.031 Total Av. 0.324 0.951 1.014 1.018 0.294 0.982 0.968 1.008 0.226 1.049 0.744 0.967
Industrial production ISM manufacturing
Ratio to Ratio to
Nonlinear Nonlinear
MIDAS Linear MIDAS Linear
(MSFE) MIDAS AR ADL (MSFE) MIDAS AR ADL Standard MS
0 0.209b 0.827 0.688 0.801 0.219b 0.758 0.720 0.780 1/3 0.193a 0.796 0.636 0.741 0.223a 0.778 0.734 0.795 2/3 0.243a 0.999 0.800 0.931 0.249b 0.926 0.820 0.888 1 0.275b 1.085 0.905 1.054 0.266a 0.999 0.876 0.949
Total Av. 0.230 0.927 0.757 0.882 0.239 0.865 0.788 0.853 1.120
NOTE: The first column of each panel reports the MSFE for the average of the forecasts across the nonlinear MIDAS models for each indicator at different forecasting horizons. The next three columns of each panel report the ratio of the MSFE for the average of the forecasts across the nonlinear MIDAS models using three benchmarks: the MSFE of the average across the three linear MIDAS models, the AR(1) model, and the ADL(1,1) model. Total Av. is the average over all forecasting horizonsh= {0,1/3,2/3,1}. The superscriptsa,b,cindicate that
a given model forecast at a given forecasting horizon outperforms the AR(1) model at 10%, 5%, and 1%, respectively, according to the Clark and West (2007) test of equal out-of-sample forecast accuracy. The larger model is the one that takes the average of the forecasts over each of the nonlinear MIDAS models, which nests the AR(1) model.
forecasting horizon are reported in Table F of the online Ap-pendix. Table F also reports the results of the Clark and West (2007) tests of equal out-of-sample accuracy for nested mod-els to assess the forecasting performance of the modmod-els under scrutiny with respects to the forecasts from the AR(1) model. Note that the asymptotic distribution of this test is not known in the case of MS-MIDAS models, although it is probably reason-ably approximated by the normal distribution.
Our results go as follows. On average, nonlinear MIDAS models yield sizeable gains over the linear MIDAS models for all monthly indicators, except the S&P500, and all forecast hori-zons. The best indicator is the S&P 500 index. The second best indicator is industrial production, and in this case the gains of MS-MIDAS over standard MIDAS are particularly large. In ad-dition, the forecasts from the average of the nonlinear MIDAS models using the ISM indicator yields better forecasts than its three benchmarks. Moreover, again on average, the nonlinear MIDAS models also produce better forecasts than the AR(1) and ADL(1,1) models. Overall, this confirms the results by Kuzin, Marcellino, and Schumacher (2012), who showed that pooling mixed-frequency forecasts produce in general good and robust results. The improvement in forecast accuracy is statistically significant in most cases.
In addition to predicting quarterly GDP growth, Markov-switching MIDAS models can endogenously generate proba-bilities of being in a given regime. As a result, we report in Table 5the QPS and LPS for the models with Markov-switching features to check how well these models can predict the true regimes. The first column of each panel reports the average of the QPS (or LPS) over each of the nonlinear MIDAS models, while the second column of each panel reports the results for
the nonlinear MIDAS model with the best forecasts (i.e., with the lowest QPS (or LPS)). We use the classification of the eco-nomic activity from the NBER so thatSt is a dummy variable
that takes on a value of 1 if the economy is in recession in quar-tertaccording to the NBER, whileP(St =1) is the probability
of being in the recession regime in period t. Forecasts with a horizonh= {0,1/3,2/3,1}predict the regime of the economy one quarter ahead.
In contrast to forecasting the level of GDP growth, the slope of the yield curve and the Federal Funds tend to better predict the state of the economy than stock prices and IP. This is in line with the results from binary recession models that emphasize the predictive power of the slope of the yield curve (see e.g., Estrella and Mishkin1998). ISM manufacturing, industrial production, and stock prices have a relatively poor forecasting performance in terms of discrete forecasts. In general, using information from the monthly indicators produces better regime forecasts than a standard MS model for quarterly GDP growth only when using the monthly slope of the yield curve as high-frequency indicator. Additional evidence on the predictive ability of the Markov-switching MIDAS specification is presented inFigure 1, where we report the nowcasted real-time probability of being in a recession with the slope of the yield curve as a monthly indicator using the MSH(3)-MIDAS model withh=0. These probabili-ties are generated from the recursive exercise and correspond to the filtered probabilities for the last observationT, whereT is recursively expanded over time fromt=1997:Q4 to 2009:Q4. This figure also shows three additional sets of real-time probabil-ities of recession: from a standard univariate regime-switching model that only uses GDP, the Econbrowser recession probabili-ties that also only uses real GDP, and the recession probabiliprobabili-ties
Table 5. Quadratic probability score and log probability score
Quadratic probability score
Slope of the yield Industrial
curve Federal funds S&P500 production ISM manufacturing
Best Best Best Best Best Standard
Horizon Av. model Av. model Av. model Av. model Av. model MS
0 0.328 0.270 0.409 0.391 0.405 0.384 0.445 0.439 0.470 0.382 1/3 0.335 0.287 0.393 0.375 0.384 0.282 0.419 0.406 0.451 0.419 2/3 0.351 0.306 0.397 0.388 0.422 0.385 0.436 0.444 0.509 0.333 1 0.384 0.370 0.379 0.391 0.412 0.402 0.415 0.364 0.420 0.366
Total av. 0.350 0.308 0.394 0.389 0.406 0.363 0.429 0.413 0.462 0.375 0.357 Log probability score
Slope of the yield Industrial
curve Federal funds S&P500 production ISM manufacturing
Best Best Best Best Best Standard
Horizon Av. model Av. model Av. model Av. model Av. model MS
0 0.579 0.437 0.841 0.734 0.750 0.741 0.809 0.782 0.720 0.571 1/3 0.575 0.473 0.707 0.621 0.715 0.442 0.787 0.792 0.685 0.624 2/3 0.614 0.465 0.745 0.722 0.825 0.714 0.784 0.734 0.880 0.531 1 0.728 0.690 0.666 0.722 0.787 0.558 0.781 0.575 0.720 0.647
Total av. 0.624 0.516 0.739 0.700 0.769 0.614 0.790 0.721 0.753 0.593 0.594
NOTE: QPS is calculated as
QPS= 2
F
F
t=1
(P(St+h=1)−NBERt+h)2.
LPS is instead calculated as
LPS= −1
F
F
t=1
(1−NBERt+h)log(1−P(St+h=1))+NBERt+hlog(P(St+h=1)),
whereFis the number of forecasts,P(St+h) are the predicted regime probabilities of being in the first regime, and NBERt+his a dummy variable that takes on a value of 1 if the U.S.
economy is in recession in quartert+haccording to the NBER. Forh={0,1/3,2/3,1}, we predict business cycle regimes one quarter ahead.
from a model using both GDP and GDI (see Nalewaik 2012).
Figure 1shows that the MS-MIDAS model with the slope of the yield curve gives a first signal of recession in the second quarter of 2001 using information up to August 2001. The prob-ability of recession then rises above 0.90 in the third and fourth quarters of 2001. Interestingly, the probability of recession stays above 0.35 until the second quarter of 2003 and only fall below 0.10 in the third quarter of 2003. This illustrates the slow eco-nomic recovery that followed the 2001 recession.Figure 1also shows that there is a first peak in the probability of recession in the fourth quarter of 2007 using information up to Febru-ary 2008. The probability of recession then jumps above 0.85 from the end of the first quarter of 2008 until the second quar-ter of 2009 and it starts declining in the third quarquar-ter of 2009. This confirms the NBER dating of the end of the last recession in June 2009. Note that our model gives the first signal of recession well before the announcement of the recession by the NBER that occurred in December 2008. In addition,Figure 1shows that this nonlinear MIDAS model outperforms the standard regime switching model as well as the Hamilton’s Econbrowser reces-sion indicator index for detecting the beginning of the last two recessions (available athttp://www.econbrowser.com/archives/
recind/description.html). Moreover, the real-time probability of recession from the MS-MIDAS declines at a slower pace at the end of the last two recessions than the standard MS model and the Econbrowser recession indicator index.
A crucial point of the MS-MIDAS specification is that unlike the probabilities of recession from the alternative models re-ported inFigure 1, the quarterly probabilities of recession from MS-MIDAS models can be updated on a monthly basis (i.e., at the frequency of thextm variable). This makes this class of models very attractive for real-time estimation of business cy-cle conditions. Indeed, Table D in the online Appendix reports the nowcasted probability of recession for the MSH(3)-MIDAS model with the slope of the yield curve as an indicator for three different forecast horizonsh={0,1/3,2/3}. The table confirms that using the slope of the yield curve as a monthly indicator provides strong calls of recession in the first quarter of 2008 since the probability of recession gradually increased in the first quarter of 2008 to reach 0.87 in March 2008 (using information available up to May 2008). Table D also shows that the prob-ability of recession is decreasing in the third quarter of 2009 in line with the NBER dating of the end of the last recession. However, the probability of recession remains fairly high in the third and fourth quarters of 2009.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
Markov-switching MIDAS model
Standard Markov-switching model
Econbrowser recession indicator index
ProbabiliƟes inferred from the joint behaviour of GDP and GDI
Figure 1. Quarterly real-time probabilities of a U.S. recession. Note: The dotted line plots the probabilities of recession from a standard regime-switching model. The solid line plots the probabilities of recession from the MS-MIDAS model that uses GDP and the monthly slope of the yield curve. The line with triangles plots the probabilities of recession from the Econbrowser web site that only uses real GDP. The line with squares plots the probabilities of recession inferred from the joint behavior of GDP and GDI. The shaded areas represent NBER recessions. The online version of this figure is in color.
We also conducted an empirical application for forecast-ing UK quarterly GDP growth with monthly indicators. Our results—reported in the online Appendix—show that MS-MIDAS models outperform linear MS-MIDAS models as well as the AR(1) and ADL(1,1) benchmarks for continuous forecasts of the economic activity. Besides, MS-MIDAS models also pro-vide relevant timely updates of the UK business cycle regimes. In summary, Markov-switching MIDAS models not only gen-erate good forecasting results for the level of GDP growth, but they also provide relevant information about the state of the economy. In particular, the combination of high-frequency in-formation and parameter switching performs better than using each of these two features separately, as in standard MIDAS and MS models, respectively.
4.2 Monthly U.S. Industrial Production Growth
and Weekly Indicators
In this subsection, we run another experiment with another mix of frequencies, that is, we forecast monthly industrial pro-duction (IP) using variables sampled at weekly frequency. IP is taken as 100 times the monthly change in the level of
in-dustrial production from 1970:M01 to 2009:M12. We consider the following three explanatory variables taken at the weekly frequency: the slope of the yield curve, the Federal Funds, and the S&P 500 index. We use the same transformation for these explanatory variables as in the previous analysis of GDP growth.
We use a model with two regimes and a switch in the variance of the innovations since this model captures well the U.S. business cycle regimes (see Appendix D provided online). The design of the real-time forecasting exercise is similar to the one described above although the frequency mix we consider here is different. The forecasting horizonhrefers to forecastsh -week-ahead. For example, forh=0, the forecasts are made using full information on the weekly indicator in the current month, while forh=4, forecasts have a 4-week horizon. The actual values for IP are taken from the vintage of dataT =2010 :M1. The first estimation sample goes from t =1970 : 01 to t =1997 : 12 and is recursively expanded until we reach the end of the sample. As a result, our evaluation sample covers the same time period as in the previous application when forecasting quarterly GDP growth with monthly indicators. This permits us to have a fair comparison of the forecasting performance of
Table 6. Forecasting U.S. monthly industrial production growth with weekly variables
Slope of the yield curve Federal Funds S&P500
Ratio to Ratio to Ratio to
Nonlinear Nonlinear Nonlinear
MIDAS Linear MIDAS Linear MIDAS Linear Standard
Horizon (MSFE) MIDAS AR ADL (MSFE) MIDAS AR ADL (MSFE) MIDAS AR ADL MS
0 0.502a 0.957 0.944 0.934 0.507a 0.982 0.953 0.939 0.518a 0.955 0.974 0.950 –
1 0.508a 0.978 0.956 0.945 0.518a 0.997 0.974 0.960 0.532 1.012 1.002 0.977 –
2 0.520a 0.978 0.978 0.967 0.517a 0.992 0.973 0.958 0.507c 0.942 0.954 0.931 –
3 0.516a 0.966 0.971 0.960 0.511b 0.984 0.960 0.946 0.498b 0.980 0.936 0.914 –
4 0.533 0.963 1.003 0.992 0.512 0.980 0.964 0.950 0.495b 0.993 0.931 0.909 –
Total Av. 0.516 0.968 0.970 0.960 0.513 0.987 0.965 0.951 0.510 0.977 0.959 0.936 1.071 NOTE: The first column of each panel reports the MSFE for the average of the forecasts across the nonlinear MIDAS models for each indicator at different forecasting horizons. The next three columns of each panel report the ratio of the MSFE for the average of the forecasts across the nonlinear MIDAS models using three benchmarks: the MSFE of the average across the three linear MIDAS models, the AR(1) model, and the ADL(1,1) model. Total Av. is the average over all forecasting horizonsh= {0,1/3,2/3,1}. The superscriptsa,b,cindicate that a given model forecast at a given forecasting horizon outperforms the AR(1) model
at 10%, 5%, and 1%, respectively, according to the Clark and West (2007) test of equal out-of-sample forecast accuracy. The larger model is the one that takes the average of the forecasts over each of the nonlinear MIDAS models, which nests the AR(1) model.
the high frequency indicators for predicting the U.S. business cycle regimes across the two frequency mixes we consider.
Table 6reports the continuous forecasting results. First, the average of the nonlinear MIDAS models always outperforms the average of the linear MIDAS models when using the slope of the yield curve and the Federal Funds. Second, the average of the nonlinear MIDAS models slightly outperforms the AR(1) and ADL(1,1) models when using the slope of the yield curve
and the Federal Funds as weekly indicators. Finally, Table G in the online Appendix reports the results for the individual models we estimate along with the tests of equal out-of-sample forecast accuracy from Clark and West (2007). The increase in forecast accuracy from the nonlinear MIDAS models with respect to the AR(1) model is statistically significant in many cases.
Table 7 presents the quadratic and logarithmic probabil-ity scores, which allows us to assess the performance of the
Table 7. Quadratic and logarithmic probability scores
Quadratic probability score
Slope of the yield curve Federal Funds S&P500
Horizon Av. Best model Av. Best model Av. Best model Standard MS
0 0.270 0.221 0.269 0.255 0.223 0.193
1 0.259 0.210 0.262 0.248 0.232 0.195
2 0.261 0.230 0.272 0.259 0.272 0.230
3 0.250 0.208 0.266 0.220 0.267 0.200
4 0.283 0.222 0.274 0.248 0.280 0.215
Total av. 0.264 0.218 0.269 0.246 0.255 0.207 0.588
Logarithmic probability score
Slope of the yield curve Federal Funds S&P500
Horizon Av. Best model Av. Best model Av. Best model Standard MS
0 0.434 0.370 0.429 0.419 0.359 0.311
1 0.419 0.347 0.418 0.397 0.377 0.317
2 0.417 0.379 0.421 0.408 0.434 0.374
3 0.399 0.346 0.429 0.368 0.427 0.332
4 0.449 0.369 0.437 0.401 0.447 0.355
Total av. 0.424 0.362 0.427 0.399 0.409 0.338 0.871
NOTE: QPS is calculated as follows:
QPS= 2
F
T
t=1
(P(St+h=1)−NBERt+h)2.
LPS is instead calculated as follows:
LPS= −1
F
F
t=1
(1−NBERt+h)log(1−P(St+h=1))+NBERt+hlog(P(St+h=1)),
whereFis the number of forecasts,P(St+h) are the predicted regime probabilities of being in the first regime, and NBERt+his a dummy variable that takes on a value of 1 if the U.S.
economy is in recession in montht+haccording to the NBER.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Dec-97 Ma
y-98
Oct-98 Mar-99 Aug-99 Jan-00 Jun-00 Nov-00 Apr-01 Sep-01 Feb
-0
2
Jul-02 Dec-02
Ma
y-03
Oct-03 Mar-04 Aug-04 Jan-05 Jun-05 Nov-05 Apr-06 Sep-06 Feb
-0
7
Jul-07 Dec-07
Ma
y-08
Oct-08 Mar-09 Aug-09
Figure 2. Monthly real-time probabilities of a U.S. recession. Note: The above real-time probabilities of recession are from the MSH(2)-MIDAS model with the weekly S&P500 index as high-frequency indicator. We use real-time data and the forecasting horizon ish=0. The shaded areas represent NBER recessions.
MS-MIDAS models in terms of discrete forecasts of the monthly business cycle. The first column of each panel reports the av-erage of the QPS (or LPS) over each of the nonlinear MIDAS models, while the second column of each panel reports the re-sults for the best nonlinear MIDAS model (i.e., with the lowest average QPS (or LPS)). First, the slope of the yield curve is the second best indicator for predicting business cycle turning points according to both QPS and LPS, while the S&P500 index produces the best regime forecasts in terms of QPS and LPS. This is in contrast with the results we obtained in the previ-ous sub-section with another mix of frequencies. Second, the Federal Funds is the second best indicator in terms of LPS. Fi-nally, the standard MS model is strongly outperformed by the nonlinear MIDAS models in terms of discrete forecasts.
Figure 2presents the monthly real-time probabilities of being in a recession according to the MSH(2)-MIDAS model using the S&P500 index. This model captures adequately the beginning and the end of the 2001 recession. However, it gives a false signal of recession at the end of 2005 and detects with a certain delay the last recession, but it provides a timely and precise call for its end since the probabilities of recession declined sharply in July 2009.
5. CONCLUSIONS
Mixed data sampling (MIDAS) models are attracting con-siderable attention in the literature for their ability to com-bine in a rather simple regression framework variables sampled at different frequencies. Time-varying parameter models, with changes both in the conditional mean and in the variance, are also more and more used in applied macroeconomics. In this article we combine these two strands of literature, and introduce the Markov-switching (MS-)MIDAS model, which allows for time-variation in the parameters of MIDAS models, and for the use of high-frequency information in standard MS models.
The MS-MIDAS model can be estimated by maximum like-lihood, and Monte Carlo experiments reported in the online Appendix indicate that the resulting estimates are rather accu-rate. Information criteria can then be used for the selection of the number of lags and regimes. Two empirical applications to nowcasting and forecasting economic activity in the United States confirm the good performance of the MS-MIDAS model. It can also rather accurately predict changes in regimes.
Due to its generality and ease of implementation, we be-lieve that the MS-MIDAS model can provide a convenient
specification for a large class of empirical applications in applied macroeconomics and finance.
ACKNOWLEDGMENTS
We thank the editor Jonathan Wright, two anonymous refer-ees, Karol Ciszek, Todd Clark, Matthieu Droumaguet, Laurent Ferrara, Eric Ghysels, Helmut Herwartz, Helmut Luetkepohl, Michael McCracken, Rossen Valkanov, and seminar partici-pants at the EUI, the MIDAS workshop at Goethe University Frankfurt, and the 6th Eurostat Colloquium on Modern Tools for Business Cycle Analysis for useful comments on a previous draft. The views expressed in this article are those of the authors. No responsibility for them should be attributed to the Bank of Canada.
[Received June 2011. Revised August 2012.]
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