Approximating and forecasting macroeconomic signals in real-time

International Journal of Forecasting 29 (2013) 479–492

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

Approximating and forecasting macroeconomic signals in real-time João Valle e Azevedo a,∗ , Ana Pereira b a

Banco de Portugal, Nova School of Business and Economics, Portugal

b

Banco de Portugal, Portugal

article

info

Keywords: Dynamic factor models Band-pass filter Business cycle fluctuations Coincident indicator Leading indicator Small economy

abstract We incorporate factors extracted from a large panel of macroeconomic time series in the predictions of two signals related to real economic activity: business cycle fluctuations and the medium- to long-run component of output growth. The latter is simply output growth short of fluctuations with a period below one year. For forecasting purposes, we show that targeting this object rather than the original (noisy) time series can result in gains in forecast accuracy. With conventional projections, high-frequency fluctuations are always fitted, despite being (mostly) unpredictable or idiosyncratic. We illustrate the methodology and provide forecast comparisons for the U.S. and Portugal. © 2013 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction We present a method designed for predicting two measures of economic activity in real-time: business cycle fluctuations of aggregate output and the smooth component of output growth. Following Baxter and King (1999), business cycle fluctuations are usually defined as ‘‘fluctuations with a specified range of periodicities’’ in the spectrum of the time series of interest. We will pick the standard [6, 32] quarters band (see for example Stock & Watson, 1999). The smooth component of output growth (hereafter, smooth growth) is defined as output growth short of fluctuations with a period below one year. The signals just defined can be extracted through the application of well-known twosided filters to the series of interest. Extraction in real-time is therefore restricted by the availability of data, and is thus a difficult task. Christiano and Fitzgerald (2003) and Wildi (1998) provided a univariate minimum mean squared solution to the endpoints problem, while Valle e Azevedo

∗ Correspondence to: Av. Almirante Reis 71 6th floor 1150-012 Lisboa, Portugal. Tel.: +351 213130163. E-mail address: [email protected] (J. Valle e Azevedo).

(2011) showed the usefulness of considering a multivariate solution. Our main contribution is to show that it can be useful to wed multivariate filtering with factor analysis in order to improve signal extraction. We extend the analysis of Valle e Azevedo (2011), who focused solely on business cycle fluctuations, by investigating the usefulness of multivariate predictions of smooth growth as coincident/leading indicators of economic activity, and also as forecasts of the quarterly Gross Domestic Product (GDP) itself. Furthermore, we evaluate our predictions within a simulated realtime environment, in contrast to Valle e Azevedo (2011), who estimated second moments using the full sample, as opposed to using the data that were actually available at each prediction moment, for example. The resulting real activity indicators have several desirable properties: (i) they are timely, since we take into account the release delays of all of the series used in the exercise; (ii) they display few short-run oscillations, and therefore provide a clear picture of growth prospects and cyclical developments; (iii) they are based on a comprehensive panel of predictors, whose idiosyncratic components are eliminated through factor analysis; and (iv) predictions of smooth growth for period t + h constructed using data available at time t forecast GDP growth itself at t + h remarkably well (with h less

0169-2070/$ – see front matter © 2013 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijforecast.2012.12.005

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than one year). We conclude that, for forecasting purposes, targeting a smooth version of a time series may be more efficient than targeting the original series. With conventional projections, short-run fluctuations are always fitted, despite being (mostly) unpredictable or idiosyncratic. At least in the context of forecasting output growth, it seems clear that a superior forecast method is not characterized by its ability to nail the high frequency fluctuations, especially at long horizons. Our approach can be summarized as follows: we assume that the panel of predictors is described appropriately by a factor model, as was originally proposed by Geweke (1977) and Sargent and Sims (1977). We estimate the common factors using principal components, as did Stock and Watson (2002a,b).1 Next, we use the estimated factors and the series of interest in a feasible (minimum mean squared error) projection targeting business cycle fluctuations or smooth growth. The resulting multivariate filter is an extension of the univariate filter developed by Christiano and Fitzgerald (2003) and Wildi (1998), which is itself an extension of the two-sided symmetric filter of Baxter and King (1999).2 Following Stock and Watson (1989), model-based (or parametric) methods assuming a factor structure have also been used to construct business cycle (or growth) indicators. Harvey and Trimbur (2003) propose unobserved components models for which the extraction of a cycle component is equivalent to applying a band-pass filter. Additionally incorporating an extension proposed by Rünstler (2004) that allows for phase shifts in the cyclical components of multiple time series, Valle e Azevedo, Koopman, and Rua (2006) construct a business cycle indicator which can be seen as a multivariate band-pass filter. In their paper, although a common factor structure is assumed to describe a small set of time series, the representation is far from general and the method does not aim to predict a pre-defined range of frequencies. That is therefore the aim of this paper. The approach closest to ours is that of Altissimo, Cristadoro, Forni, Lippi, and Veronese (2010), which resulted in the New Eurocoin indicator. The indicator is obtained by projecting the smooth component of output growth onto estimated smooth factors. We will contrast the two approaches in more detail later. The outline of the paper is as follows. In Section 2 we define our targets precisely and describe how we predict them. In Section 3 we discuss the estimation of the factors which are used as covariates in the prediction of the signals. In Section 4 we assess the real-time performances of our predictions, analyzing results for the U.S., while in Section 5 we analyze the performances of predictions of smooth growth as forecasts of GDP growth for the U.S. and Portugal. Section 6 concludes.

1 We could have used generalized principal components instead; see Forni, Hallin, Lippi, and Reichlin (2005) and the working paper version of this paper, Valle e Azevedo and Pereira (2008). 2 Such solutions were provided in the case of stationary processes by Geweke (1978) and in a univariate context including unit-roots by Pierce (1980). Multivariate solutions were also analyzed by Wildi (2008) and Wildi and Sturm (2008).

2. Signals of interest and predictions 2.1. Business cycle fluctuations and smooth growth Throughout the paper, our variable of interest will be (the log of) real quarterly GDP, as the best available proxy for aggregate economic activity. Define xt as the log of real GDP and ∆xt = (1 − L)xt as its growth rate, where L is the lag operator. We define business cycle fluctuations as fluctuations in xt with periods in the range [6, 32] quarters. Smooth growth is defined as GDP growth short of fluctuations with a period below one year (or 4 quarters). Specifically, take the following decompositions of xt and ∆xt : xt = BC(L)xt + (1 − BC(L))xt

∆xt = SG(L)∆xt + (1 − SG(L))∆xt , ∞

(1) (2)

j j where BC(L) = j=−∞ BCj L and SG(L) = j=−∞ SGj L are the ‘‘ideal’’ filters isolating the [6, 32] and [4, ∞[ quarters bands, respectively. The filter weights, BCj and SGj , are well-known, see for example Altissimo et al. (2010) and Baxter and King (1999). Business cycle fluctuations are defined as BC(L)xt , and smooth growth is defined as SG(L)∆xt . Business cycle fluctuations can reasonably be interpreted as those fluctuations in real GDP which are not attributable to either long-run growth or high-frequency measurement error. Smooth growth is a measure of output growth which is free of the short-run oscillations that make the assessment of the current aggregate economic situation difficult. We believe that this is an extremely useful measure, especially for (small) economies in which GDP data are prone to exhibiting noisy behaviors, reflecting unrepeatable events or a mismatch between the concept of output and its measurement (as is quite evident in the case of Portugal).3 Obviously, it is not possible to extract these signals with an arbitrary precision in finite samples, let alone in a real-time context. However, accurate predictions can be obtained in the middle of the sample. For instance, the filter developed by Baxter and King (1999) (BK filter), which amounts to truncating the ideal filter at a specified lead/lag, provides very accurate predictions of the signals if the truncation lead/lag is sufficiently large. Fig. 1 presents two (approximate) decompositions of xt and ∆xt for the U.S. obtained with the BK filter. Given the fixity of the lag orders and of the filter weights, these predictions do not suffer from revisions once more GDP data become available. Furthermore, the consideration of additional leads and lags of GDP data would lead to negligible differences between these predictions and those obtained with the ‘‘ideal filters’’ BC(L) and SG(L), while implying the loss of even more filtered observations near the endpoints of the sample. Notice, however, that we are particularly interested in obtaining predictions of these signals for the current quarter (coincident indicator), the next quarter (leading indicator), or any other future quarter.

∞

3 Several examples can be given: for instance, a (large) purchase of aircraft by an airline can be registered (in national accounts) as investment in a given quarter and as imports next quarter. This creates (as it has done in the past) a peak in measured GDP growth, followed by a trough.

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481

i

ii

Fig. 1. (i) Business cycle fluctuations in U.S. real GDP (i.e., predictions of 100 × BC(L)xt , the percentage deviation of output from trend) obtained using the BK filter, which truncates the ideal filter at lead and lag j = 24, and guarantees the removal of one unit-root. End of sample predictions (which will suffer from revisions) are obtained using the best multivariate band-pass filter (MBPF) ‘‘quasi-final’’ prediction, which is described in detail in Section 4. Sample: 1966Q1–2009Q4. (ii) Smooth growth of U.S. real GDP (more precisely, the prediction of 100 × SG(L)∆xt ), obtained using the BK filter, which truncates the ideal filter at leads and lags of j = 6. Sample: 1960Q1–2009Q4. Grey areas represent the NBER’s recession datings.

2.2. How to predict the signals of interest Suppose that xt is integrated of order 1 and we are interested in predicting yt = B(L)xt (or alternatively yt = B(L)∆xt ), where B(L) is an arbitrary (absolutely summable and stationary) filter that defines a signal on xt (∆xt ). Without loss of generality, assume further that B(1) = 0 (respectively, B(1) = 1), which is verified by BC(L), defined in Eq. (1) (SG(L) in Eq. (2)). We want to predict the signal given observations of xt and of c series of covariates, z1,t , . . . , zc ,t , with 1 ≤ t ≤ T . We make the following assumption:

Assumption DGP 1. The vector (∆xt , z1,t , . . . , zc ,t ), where ∆ = 1 − L, is covariance-stationary with a finite moving average representation (of order M, say). ′

Now let x = (x1 , x2 , . . . , xT ) , and z1 = (z1,1 , z1,2 , . . . , ′ ′ z1,T ) , . . . , zc = (zc ,1 , zc ,2 , . . . , zc ,T ) , and let yt = B(L)xt (or, alternatively, yt = B(L)∆xt ) be the signal at time t. A prediction  yt of the signal yt is a weighted sum of the elements of x (or ∆x) and the elements of z1 , . . . , zc :

 yt =

p  j=−f

p,f

Bj xt −j +

p c   s=1 j=−f

p,f

Rs,j zs,t −j ,

(3)

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or, in the case where the target is yt = B(L)∆xt :

 yt =

p 

p,f

Bj ∆xt −j +

j=−f

p c  

p,f

Rs,j zs,t −j ,

(4)

s=1 j=−f

where p denotes the number of past observations that are considered and f the number of future observations that are considered (relative to time t). To obtain the minimum mean squared error prediction of yt , we choose p,f p,f p,f the weights { Bj ,  R1,j , . . . ,  Rc ,j }j=−f ,...,p associated with the series of interest and the available covariates that solve the following problem: p,f

p,f

Min

p,f

{Bj ,R1,j ,...,Rc ,j }j=−f ,...,p

E[(yt − yt )2 ],

(5)

where the information set is implicitly restricted by p and f . We use the solution to this problem under Assumption DGP 1 which is analyzed by Valle e Azevedo (2011). For the predictions of yt = B(L)xt , the restriction Bp,f (1) = 0 must p p,f j be imposed on Bp,f (L) = j=−f Bj L , since B(1) = 0. This ensures that the problem is well-defined (and ensures the stationarity of the extracted signal). We note that dropping the second term on the right hand side of Eqs. (3) and (4) delivers the univariate prediction of Christiano and Fitzgerald (2003) and Wildi (1998). Remark 1. It is straightforward to predict the signal yT +h = B(L)xT +h for h > 0, since f is allowed to be negative. More generally, if one is interested in yT +h and xT is released with a lag, then one should set f = −h-release delay. Finally, since f is the same for all of the variables in the projections in Eqs. (3) and (4), we should trivially relabel the time subscript t of all of the indicators zs,t , such that the last observation of each zs,t matches the last observation of xt . This avoids the potential to disregard observations of the zs,t indicators in the projections. Remark 2. We will seek predictions of the (quarterly) signal yt in each month. Hence, it is convenient to incorporate time series which are recorded at mixed frequencies, e.g., quarterly xt (say GDP) and quarterly zs,t s (say, preliminary estimates of GDP), as well as monthly zs,t ∗ s. In this paper, the zs,t ∗ s will be estimated factors which have been extracted from a large panel of time series. For dealing with mixed frequencies, we break the monthly factors into three quarterly series. Specifically, the monthly factors zs,t ∗ will be split into three ‘‘quarterly’’ series, zs1,t , zs2,t and zs3,t , such that zs1,t = zs,t ∗ , zs1,t −1 = zs,t ∗ −3 , zs1,t −2 = zs,t ∗ −6 , . . . , zs2,t = zs,t ∗ −1 , zs2,t −1 = zs,t ∗ −4 ,

zs2,t −2 = zs,t ∗ −7 , . . . and zs3,t = zs,t ∗ −2 , zs3,t −1 = zs,t ∗ −5 , zs3,t −2 = zs,t ∗ −8 . . . . The weights of the filter are obtained by simply solving a linear system with (p + f + 1) × (c + 1) equations and unknowns. The solution depends only on the second moments of (∆xt , z1,t , . . . , zc ,t )′ (which need to be estimated, see below), and on the weights of the ‘‘ideal’’ filter. For the real-time signal extraction exercises, we will always set p = 50 (larger values of p lead to negligible differences in the predictions). With regard to f , several situations deserve discussion:

(i) in order to predict yT +h for h ≥ 0, we set f = −h (in quarters) — release delay, where release delay is 2 in the first or second month of the current quarter (T ) and 1 in the third month of the current quarter. In this way we account for the release delays of the final estimate of GDP for the U.S. and Portugal. (ii) in order to obtain very accurate (but unfeasible in real-time) predictions of the signals, against which real-time predictions can be evaluated, we use the univariate filter with a non-parametric estimation of second moments (see below for details) and, for each t, set f = T − t (and also p = t − 1), guaranteeing that all of the available observations are used. These will be denoted ‘‘quasi-final’’ predictions. We note that, in the endpoints of the sample, say for  yT , yT −1 , . . . , yT −s , these predictions may suffer substantial revisions once additional (future) data become available. As s grows, the revisions become negligible and  yT −s will give an accurate picture of the signals we are targeting. In the case of predictions of business cycle fluctuations, dropping the last 6 observations is enough to ensure that negligible revisions occur once more data become available. For predictions of smooth growth, dropping the last two observations is enough (see the simulation results of Valle e Azevedo, 2011, for business cycle fluctuations, and Altissimo et al., 2010, for smooth growth).4 , 5 We are left with the estimation of the required second moments of (∆xt , z1,t , . . . , zc ,t ). We propose two alternatives for estimating the autocovariance function (or spectrum) of vector (∆xt , z1,t , . . . , zc ,t ). The first is based on a standard non-parametric estimator of the spectrum, given by:

(ω) = 1 ,...,zc

 S∆x,z



1

(0) + Γ

2π

M (T )



κ(k, T )

k=1

 (k)eiωk + Γ (k)′ e−iωk ) , × (Γ where κ(k, T ) =



1−

k M (T )+1



(6)

denotes the Bartlett lag

(k), k = 0, 1, . . . , M (T ) is the sample autowindow. Γ covariance of vector (∆xt , z1,t , . . . , zc ,t ) at lag k, and the truncation point M (T ) < T is typically required to grow more slowly than T in order to guarantee consistency of  S∆x,z1 ,...,zc (ω). In this estimator, for all empirical purposes, we set M (T ) = M = 50 for the U.S. (the results are very similar in the range 30 < M < 60) and M = 12 in the 4 The same analysis applies to the predictions at the beginning of the sample,  y1 , y2 , . . . , ys−1 . However, these predictions will not be a concern, since the choice of the evaluation periods (where we compare real-time predictions to ‘‘quasi-final’’ predictions) implies disregarding a greater number of initial observations. 5 We used simulations in order to verify that the conclusions of Altissimo et al. (2010) and Valle e Azevedo (2011) carry over to a data generating process close to that followed by the U.S. GDP. We have also verified that different methods of estimating second moments and/or multivariate filters deliver ‘‘quasi-final’’ predictions that are very close to these. These results are available upon request.

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case of Portugal (due to the shorter sample size available; the results are very similar in the range 8 < M < 16). Alternatively, following Den Haan and Levin (1996) and Priestley (1981), we first pre-whiten the vector (∆xt , z1,t , . . . , zc ,t ) by estimating a SUR-VAR, with the lag length in each equation being determined by the BIC.  (L), and the esGiven the estimated VAR polynomial, VAR timated autocovariance function of the resulting residuε (k), with the maximum order k = M chosen als (Γ as above), we derive the autocovariance function of the original process (∆xt , z1,t , . . . , zc ,t ) implicit in the estimated autocovariance generating function  G(z ) =   ε k −1 −1 ′  ( z ) −1 M  VAR Γ ( k ) z VAR ( z ) . In order to apply k=−M the filter, we then truncate the estimated autocovariance function at lead and lag k = 40. Increasing the number of covariates used in the predictions complicates the estimation of the VAR but makes no difference for the nonparametric approach, hence the consideration of the two methods. For the univariate filters, we also estimate second moments as described above, with the only difference being that (∆xt , z1,t , . . . , zc ,t ) should read as (∆xt ). 3. Multivariate information: factor model and estimation In quarter t, month t ∗ , the set of covariates used in the predictions of the signal yt (or yt +h ) consists of common factors extracted from a large panel of monthly time series, each split into three quarterly series, as described in Remark 2 of Section 2.2. Consider the following panel of ‘‘monthly’’ time series, W = {wit }, i = 1, . . . , n; t = 1, . . . , T ∗ . We use T ∗ instead of T to make it clear that wit are monthly time series. We assume that each variable can be decomposed into a common component, χit , driven by a small number of common factors Fst , s = 1, . . . , r, which are loaded differently across i, and an idiosyncratic component, ξit , which is orthogonal to the χjt , j = 1, 2, . . . , n, at all leads and lags. Specifically:

wit = χit + ξit , where χit = λi1 F1t + λi2 F2t + · · · + λir Frt .

(7)

We assume that the vector Ft = (F1t , F2t , . . . , Frt )′ is covariance-stationary with a finite moving average representation, which we require in order to be able to apply the multivariate filter and establish consistency of the predictions (see the online appendix for details). The factor space G(F, t ) generated by Ft = (F1t , F2t , . . . , Frt )′ can be approximated in a number of ways. We estimate the factor space using principal components, following Stock and Watson (2002b). Specifically, the space G(F, t ) spanned by the so-called static factors Fst is estimated by  FSW = ( F1t ,  F2t , . . . ,  Frt )′ =  Swt = ( S1 wt , t ′  S2 wt , . . . ,  Sr wt ) , where S is the matrix of row eigenvectors T∗ 1 t =1 T∗ FSW simply t

w,0 = (ordered according to the eigenvalues) of Γ



wt w′t , where wt = (w1t , w2t , . . . , wnt )′ . Thus,  contains the first r principal components of wt . The number of static factors, r, is usually estimated according to the criterion of Bai and Ng (2002). This is reasonable if the aim is to estimate the factor space G(F, t ), and

483

subsequently a common component χit . However, for signal extraction or forecasting purposes, one should check whether all r of the factors are useful. Stock and Watson (2002a,b) analyze this issue and conclude that the inclusion of r (or an estimate of r) static factors in factor augmented regressions does not necessarily result in a superior forecast performance. Hence, only the first few (estimated) static factors are usually considered, with the exact number (and lags) being determined by the BIC or AIC. Altissimo et al. (2010) also use the first generalized principal components (cleaned of short-run oscillations) as regressors, stopping the inclusion if the increase in the R2 value of their approximation becomes negligible. We therefore report results for some choices of the number of estimated static factors included in the projections (denoted by k). Specifically: (i) we use the first principal component (i.e., k = 1) split into three quarterly series or the first two principal components (i.e., k = 2) split into six quarterly series (see Remark 2 in Section 2.2). It turns out that most of the gains relative to the univariate filter arise from considering only these covariates. Further, the estimation of the SUR-VAR is only feasible with a small number of covariates. (ii) like Altissimo et al. (2010), we add static factors (each split into three quarterly series) up to the point where the increase in the estimated R2 value of the predictions becomes negligible (<1%). This R2 value is the squared empirical correlation between ‘‘quasifinal’’ predictions and the ‘‘quasi real-time’’ predictions over the sample 1966Q1–2008Q2 (U.S. case only), where these ‘‘quasi real-time’’ predictions are obtained using the multivariate filter in the third month of each quarter and use f = −1 and p = 50, but consider the full-sample for estimating the second moments and common factors non-parametrically.6 This leads to values of k that never exceed the estimates of r obtained by applying the criterion CP1 of Bai and Ng (2002) ( r = 6 for the U.S.). In order to analyze the value added by principal components, we will also consider only one monthly factor,  F11,t , which is simply the cross-sectional average of all of the (standardized) variables in the panel of indicators, n w , split into three quarterly series. i.e.  F11,t = 1n it i=1 Finally, our U.S. panel consists of n = 105 time series, covering the period February 1960–February 2010 (T ∗ = 601). Data constraints for Portugal lead us to restrict its sample to the period January 1990–March 2010. The monthly panel contains n = 84 time series (mostly survey data). For the details of our data, please refer to the online appendix.

6 Recall that ‘‘quasi-final’’ predictions are obtained by predicting the in-sample signals yt , 1 ≤ t ≤ T , using the univariate filter with f = T − t , p = t − 1, and non-parametric estimation of the second moments (exactly as described in Section 2.2 but without covariates), then dropping some of the initial and final observations (which give a poor picture of the true signal). Here, we focus on ‘‘quasi-final’’ predictions for the sample 1966Q1–2008Q2, but based on the full 1960Q1–2009Q4 sample.

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4. Performances of the indicators

4.2. Evaluation of the indicators in a pseudo real-time exercise

4.1. Implementation details We start with a brief description of the steps involved in the construction of our real-time predictions of U.S. business cycle fluctuations and smooth growth (updated monthly).7 Let the prediction moment be month t ∗ of quarter t. Real-time predictions of the signals for quarter t + h are constructed using only data that would have been available in month t ∗ of quarter t. In month t ∗ of quarter t, we start by re-aligning all of the series in the panel such that their last observation refers to t ∗ . For example, if an Industrial Production index is only and always released with a lag of one month (i.e., the latest observation in month t ∗ refers to month t ∗ − 1), we treat the latest available observation as referring to month t ∗ , thus effectively taking into account the release delay. Using this re-aligned panel, we standardize all of the variables and use principal components (or the cross-sectional average of the panel) to estimate monthly factors. Then, a multivariate prediction of business cycle fluctuations for quarter t + h (yt +h = BC(L)xt +h ), using information up to month t ∗ of quarter t, takes the form: t ,t    y t +h = yt +h =  ∗

p  j=−f

p,f   Bj xt +h−j +

p l  

p,f Q   Rs,j  F

s,t +h−j

(8)

s=1 j=−f

where, for ease of notation, we have dropped the superscript t ∗ , t. In the case of predictions of smooth growth (yt +h = SG(L)∆xt +h ), xt +h−j in Eq. (8) is substituted by Q ∆xt +h−j .  Fs,t denotes estimated quarterly factors, obtained

by splitting each of the estimated monthly factors into three (again, please refer to Remark 2 in Section 2.2). Now, the release delays of GDP data and the form of Eq. (8) restrict the values of f in the projections. To see this, suppose that one wants to predict yt (i.e., h = 0), and t is the first quarter of 2009. For predictions constructed at the end of January and February of 2009, the latest available ‘‘Final’’ U.S. GDP data refers to the third quarter of 2008, whereas the latest observation of x to be used in the filter refers to quarter t + f . This requires setting f = −2 for these projections. Similarly, ‘‘Final’’ data for the fourth quarter of 2008 are released in March 2009, and therefore predictions constructed in March 2009 require f = −1. Further, since f is the same for all variables, we relabel the time subscript t of all the covariates used in the filter such that their last observation matches the last observation of GDP; recall Remark 1 in Section 2.2. By the same token, in order to predict yt +h for h ̸= 0 using information up to quarter t, we set f = −h − 2 in the first or second month of quarter t and f = −h − 1 in the third month of quarter t. Needless to say, the necessary second moments of vecQ Q tor (∆xt ,  F1,t , . . . ,  Fl,t ) are estimated using only data which were available at the moment of prediction.

7 All of the exercises in this section have been repeated with Portugal and euro area data, but we omit the results here for the sake of brevity; see Valle e Azevedo and Pereira (2008) for the euro area. The results for Portugal are available upon request.

In line with the work of Orphanides and van Norden (2002), the performances of the proposed business cycle and growth indicators (the predictions of yt +h = BC(L)xt +h and yt +h = SG(L)∆xt +h ) will be assessed by looking at the prediction errors obtained in real-time.8 These prediction errors are the difference between the real-time predictions, i.e., predictions of yt +h using only information available at time t (as described above), and the very accurate predictions obtained by considering the full sample, the so-called ‘‘quasi-final’’ predictions, denoted here by yFt+h ; refer to Section 2.2 for details. Obviously, since yFt+h uses information up to T , it is not feasible in real-time. We focus on the following statistics in order to compare real-time and ‘‘quasi-final’’ predictions. (a) An estimate of Correlation[yt +h , yt +h ], where  yt +h is the optimal prediction of the signal yt +h at time t. Given fixed values of p and f , it is easy to show that if  yt +h minimizes E[(yt +h −  yt +h )2 ] (meaning that the second moments are known rather than estimated), then E[(yt +h −  yt +h )2 ] = (1 − Correlation[yt +h , yt +h ]2 )Var[yt +h ], or the noise to signal ratio, defined as E[(yt +h −  yt +h )2 ]/Var[yt +h ], is equal to (1 − Correlation[yt +h , yt +h ]2 ). Correlation [yt +h , yt +h ] is therefore a good measure of the variance of the prediction error. We compute the sample counterpart of this statistic, using the estimated signal (say   yt +h ) as  yt +h and approximating yt +h with the ‘‘quasifinal’’ predictions, denoted yFt+h .

  (b) The noise to signal ratio, computed as t (yt +h −  F F 2 F 2 yt +h ) / t (yt +h − y ) . The relationship in (a) says that this measure is redundant if Correlation[yt +h ,  yt +h ] is known, but in actual fact the correspondence is exact only if true second order moments are used, hence the consideration of this statistic. t   (c) The percentage of times that  yt +h −  yt +h−1 (where 

t   yt +h−1 is the prediction of the signal at time t + h − 1 us-

ing information up to time t) correctly signs the change yFt+h − yFt+h−1 (see Pesaran & Timmerman, 1992). (d) For the predictions of business cycle fluctuations only,  the percentage of times  yt +h and yFt+h share the same

 sign (which gives an indication of whether  yt +h indicates correctly whether GDP is below or above the long-term trend). Our benchmark will be the univariate filter of Christiano and Fitzgerald (2003), using estimated moments derived from a pre-whitening procedure using an AR model or from a non-parametric estimator, exactly as in

8 Obviously, predictions that explore additional information vary near the end of the sample. This variation is due to statistical revisions in the actual data (as a result of methodological changes in national accounting, changes in the calculation of deflators, corrections, the use of updated information, and so on), which we do not analyze here (see Croushore & Stark, 2001, 2003, for a thorough analysis of this issue), and revisions due to the nature of the one-sided filters used at the end of the sample (in our case, along with the re-estimation of factors and second moments).

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Table 1 Evaluation statistics for the predictions of U.S. business cycle fluctuations constructed in the third month of the quarter. Performances of predictions of business cycle fluctuations, 3rd month of current quarter Correlation

Noise to signal

% Correct ∆ sign

Sign concord.

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

Benchmark filters BPF AR BPF KERNEL

0.76 0.73

0.62 0.60

0.43 0.48

0.67 0.70

0.79 0.75

0.72 0.70

0.73 0.69

0.64 0.68

w/ factor, cross-section avg MBPF KERNEL MBPF VAR

0.77 0.77

0.56 0.69

0.42 0.44

0.75 0.60

0.79 0.79

0.64 0.81

0.76 0.80

0.62 0.67

w/ factors, k = 5 <  r MBPF PC KERNEL

0.80

0.54

0.37

0.76

0.80

0.66

0.73

0.60

w/ factors, k = 2 <  r MBPF PC KERNEL MBPF PC VAR

0.83 0.88

0.63 0.66

0.34 0.25

0.67 0.79

0.80 0.86

0.70 0.74

0.75 0.76

0.64 0.71

BPF AR - univariate filter with second moments estimated using an AR model (with the BIC for lag length); BPF KERNEL - univariate filter with second moments estimated non-parametrically; MBPF - multivariate band-pass filter; Cross-section avg - Cross-sectional average of the standardized panel; PC - factor space estimated by principal components; KERNEL - non-parametric estimation of second moments; and VAR - estimation of second moments through pre-whitening with a SUR-VAR (with the BIC for lag lengths in the various equations). Evaluation periods: (1) 1976Q1–1989Q4, and (2) 1990Q1–2008Q2, with the estimation sample always beginning in 1960Q1. In the first sub-sample, we reduce M to 30 for the estimation of second moments.

Section 2.2 (without covariates). In the multivariate predictions, we considered some variations in the filters’ settings. These were described in detail earlier: (i) either the estimation of second order moments is fully nonparametric (simpler) or it involves estimating a VAR for pre-whitening in a first step, see Section 2.2; (ii) we include the first (the first two) common factor(s) estimated by principal components, split into three (six) quarterly series, or split factors up to the point where the increase in the R2 value of the full-sample predictions is negligible, see Section 3; and (iii) we include an estimated factor which is simply the cross-sectional average of all of the (standardized) variables in the panel, split into three quarterly series. 4.3. Business cycle fluctuations We report here the evaluation of predictions of U.S. business cycle fluctuations, yt = BC(L)xt , using only data which are available at the end of the first, second or third  month of quarter t; i.e.,  yt +h with h = 0. The estimation sample always begins in 1960Q1, while we compute the aforementioned statistics for two sub-samples, 1976Q1–1989Q4 and 1990Q1–2008Q2, in order to analyze the effects on our predictions of the well-documented fall in macroeconomic volatility over the past 20–25 years (obviously excluding the events of 2008 and 2009), see for example Giannone, Lenza, and Reichlin (2008) and McConnell and Perez-Quiros (2000). Table 1 focuses on the evaluation statistics for predictions of yt constructed in the third month of quarter t, and the variations considered. Here, we analyze only k = 5 split monthly factors (obtained using the R2 criterion) and k = 2 (as it delivered the best results). Table 2 also contains the evaluation of predictions constructed in the first and second months of quarter t for a selection of (the best) methods from Table 1.

The main conclusions follow:

• using 5 split monthly factors instead of 2 (estimated by principal components) results in a clear deterioration of the multivariate predictions, surely resulting from overfitting. Also, using the pre-whitening VAR (AR) to estimate second moments results, generally, in a superior performance of the multivariate (univariate) predictions. In any case, it is worth noticing the deterioration of the noise to signal ratio of MBPF PC VAR predictions over the second sub-sample. • the predictions tend to be more accurate in the third month of the quarter, followed by the second month and then the first, but only clearly so over the first subsample. • more importantly, over the first sub-sample, the multivariate filters fuelled with principal components clearly outperform the other methods in the dimensions analyzed. Over the second sub-sample, virtually all of the accuracy measures deteriorate for all methods, whereas the multivariate predictions add little (if anything) to the univariate predictions. This is assumed to stem from the difficulty of identifying the (now relatively small) deviations of output from trend. Still, and over this subsample, it is worth noting the superior performance of the multivariate filter that uses the cross-sectional average of the panel in the third month of the quarter. Fig. 2 displays the best (in the full evaluation sample) multivariate prediction (MBPF PC VAR) constructed in  the third month of each ‘‘current’’ quarter (or  yt +h with h = 0), as well as the ‘‘quasi-final’’ predictions of U.S. business cycle fluctuations. We visually confirm the clear deterioration of the real-time predictions in the later part of the sample (qualitatively the same effect occurs with all of the other methods). In particular, these predictions struggle to identify the large fluctuations identified in the period 2006–2008.

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Table 2 Evaluation statistics for predictions of U.S. business cycle fluctuations, by month of the current quarter. Performances of predictions of business cycle fluctuations, current quarter Correlation

Noise to signal

% Correct ∆ sign

Sign concord.

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

BPF AR 1st/2nd months 3rd month

0.74 0.76

0.63 0.62

0.47 0.43

0.66 0.67

0.73 0.79

0.73 0.72

0.65 0.73

0.56 0.64

MBPF PC KERNEL (k = 2 <  r) 1st month 2nd month 3rd month

0.80 0.79 0.83

0.63 0.57 0.63

0.39 0.40 0.34

0.67 0.73 0.67

0.78 0.79 0.80

0.73 0.72 0.70

0.70 0.69 0.75

0.67 0.64 0.64

MBPF PC VAR (k = 2 <  r) 1st month 2nd month 3rd month

0.76 0.82 0.88

0.62 0.62 0.66

0.45 0.35 0.25

0.80 0.79 0.79

0.78 0.82 0.86

0.70 0.72 0.74

0.67 0.75 0.76

0.68 0.71 0.71

MBPF VAR Cross-section avg 1st month 2nd month 3rd month

0.75 0.75 0.77

0.61 0.62 0.69

0.46 0.46 0.44

0.68 0.66 0.60

0.76 0.80 0.79

0.76 0.76 0.81

0.74 0.82 0.80

0.60 0.64 0.67

For the designation of the methods, see the notes to Table 1. Evaluation period: (1) 1976Q1–1989Q4, and (2) 1990Q1–2008Q2. In the first sub-sample we reduce M to 30 for the estimation of second moments.

Fig. 2. U.S. GDP business cycle fluctuations: ‘‘quasi-final’’ predictions and real-time (MBPF PC VAR) predictions. Grey areas represent the NBER’s recession dating. In the sub-sample 1976Q1–1989Q4 we reduce M to 30 for the estimation of second moments (i.e., the autocovariance function of the pre-whitening residuals, see Section 2.2).

Still, predictions that include additional (or fewer) data points may reveal the usefulness of the multivariate predictions relative to the univariate predictions. In Fig. 3 we evaluate the performances of univariate and multivariate predictions of yt +h with h = −5, −4, −3, −2, −1, 0, 1, constructed in the third month of quarter t. As before, we compare multivariate filter (principal components based) predictions with ‘‘quasi-final’’ predictions. For example, h = −4 corresponds to predictions of the signal that use information from four additional quarters relative to quarter t (say, predictions of business cycle fluctuations for 2008Q2 using information available at 2009Q2), whereas h = 1 corresponds to predicting the ‘‘next’’ quarter’s business cycle fluctuations (say, predictions for 2008Q2 using

information available up to 2008Q1). The results for h = 0 are given in Table 1. All of the measures generally improve as more data become available, and in most cases one of the multivariate filters has the best performance. The differences across methods tend to disappear once four additional quarters of data are considered (h = −4). This suggests that multivariate predictions of business cycle fluctuations, while performing poorly for h = 0 (or at least without a clear advantage over the univariate predictions) over the past twenty years, can be useful for assessing the business cycle position at horizons that we believe are still relevant for the policy-maker. Suppose, for example, that Eurostat decides to calculate the cyclically adjusted budget balance (as a percentage of GDP) as the budget balance

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Fig. 3. Evaluation of predictions of U.S. business cycle fluctuations, yt +h , h = −5, −4, −3, . . . , 0, 1, constructed in the third month of quarter t. Correlations with ‘‘quasi-final’’ predictions, noise to signal ratios and sign concordance (the predictions are BPF AR, MBPF PC VAR and MBPF PC KERNEL with k = 2 estimated monthly factors). Evaluation period: 1990Q1–2008Q2.

divided by ‘‘trend’’ GDP (where ‘‘trend’’ GDP can be defined as GDP ×(1 − BC(L)) log(GDP)). Now, additional data are helpful in calculating BC(L) log(GDP) more accurately, and thus in determining (with some lag, obviously) when a country did not accomplish an agreed target for the cyclically adjusted budget balance. 4.4. Smooth growth We now turn to the evaluation of predictions of U.S. smooth growth, or yt = SG(L)xt , focusing on predictions for, say, the current quarter (i.e., a coincident indicator), as well as for the next quarter (i.e., a leading indicator). We keep the estimation sample beginning in 1960Q1 and analyze the results obtained with k = 4 split monthly factors (obtained with the R2 criterion) or k = 1 (as it delivered the best results). Unlike the results for business cycle fluctuations in Section 4.3, we have not found a significant deterioration of the predictions of smooth growth over the last twenty years of the sample. Hence, we will focus exclusively on the sub-sample 1990Q1–2009Q2. Table 3 contains the evaluation statistics for the coincident indicators constructed in the third month of quarter t,  i.e.,  yt , and the variations considered. Table 4 also contains an evaluation of the best coincident indicators constructed in the first and second months of quarter t, and also of  the leading indicators, i.e., predictions of yt +1 ( yt +1 ), using information up to the first, second or third month of quarter t. The main conclusions follow:

Table 3 Evaluation statistics for the predictions of U.S. smooth growth constructed in the third month of the current quarter. Performances of predictions of smooth growth, 3rd month of current quarter: coincident indicator Correlation

Noise to signal

% Correct ∆ sign

Benchmark filters BPF AR BPF KERNEL

0.73 0.75

0.53 0.52

0.71 0.70

w/ factor, cross-section avg MBPF KERNEL MBPF VAR

0.84 0.83

0.34 0.33

0.83 0.78

with factors, k = 4 <  r MBPF PC KERNEL

0.80

0.54

0.75

with factors, k = 1 <  r MBPF PC KERNEL MBPF PC VAR

0.85 0.85

0.31 0.30

0.74 0.82

For the designation of the methods, see the note to Table 1. Evaluation period: 1990Q1–2009Q2, with the estimation sample always beginning in 1960Q1.

• Table 3 (coincident indicators constructed in the third month of the quarter) reveals that MBPF PC KERNEL with k = 1 and MBPF PC VAR are slightly superior to the cross-sectional average based predictions. Also, the non-parametric or pre-whitening VAR estimation of second moments makes little difference in this case. However, for coincident indicators constructed in the first or second month of the quarter, as well as for

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Table 4 Evaluation statistics for predictions of U.S. smooth growth in the current quarter (coincident indicator) and the next quarter (leading indicator), by month of the quarter. Performances of predictions of smooth growth by month of the quarter: coincident and leading indicators Coincident indicator

Leading indicator

Correlation

Noise to signal

% Correct ∆ sign

Correlation

Noise to signal

% Correct∆ sign

BPF AR 1st/2nd months 3rd month

0.40 0.73

0.99 0.53

0.68 0.71

0.15 0.41

1.13 0.99

0.66 0.68

MBPF VAR Cross-section avg 1st month 2nd month 3rd month

0.67 0.72 0.83

0.61 0.56 0.33

0.73 0.75 0.78

0.46 0.50 0.68

0.89 0.85 0.59

0.65 0.61 0.69

MBPF PC VAR (k = 1 <  r) 1st month 2nd month 3rd month

0.74 0.76 0.85

0.49 0.48 0.30

0.70 0.75 0.82

0.52 0.57 0.69

0.78 0.74 0.56

0.64 0.58 0.66

For the designation of the methods, see the notes to Table 1. Evaluation period: 1990Q1–2009Q2.

leading indicators, we have verified that using the pre-whitening VAR to estimate second moments leads to a superior performance of the multivariate predictions, hence our focus on pre-whitening VAR predictions in Table 4. • Using four split monthly factors instead of one (estimated by principal components) results in a deterioration of the multivariate predictions. While the results are not reported in Table 4, the performances of the predictions deteriorated in the first and second months of the quarter if four split monthly factors were used and the moments were estimated non-parametrically. No relevant deterioration is found if only one split principal component is used and moments are estimated nonparametrically (MBPF PC KERNEL with k = 1).9 • The multivariate filters clearly outperform the univariate filters in the dimensions under consideration. Further, losing one observation of GDP, as in the first or second month of a quarter relative to the third month, leads to a clear deterioration in the univariate filter predictions. This is in contrast to what was observed with the multivariate predictions (see Table 4). • Multivariate predictions are more accurate in the third month of a quarter, followed (generally) by the second month and then the first. This occurs for predictions of smooth growth for either the current quarter (coincident indicators) or the next quarter (leading indicators). Furthermore, the leading indicators (predictions of yt +1 ) computed in the first or second month of quarter t still reveal signal (noise to signal ratios well below 1), unlike for the univariate predictions. We stress that in these two months the latest available observation of GDP refers to quarter t − 2.

9 These findings highlight the fact that using more factors does not necessarily lead to better out-of-sample results, justifying our claim that the approach of adding covariates until the in-sample fit gains are negligible can be misleading. We found that using k = 1 (no more, no fewer) split monthly factors always produced the best results. Perhaps a larger time dimension is needed to be able to incorporate a larger number of factors usefully.

• Table 4 reveals that the best principal components based prediction (MBPF PC VAR) is systematically superior (most often by a wide margin) to the best prediction based on the cross-sectional average of the panel (MBPF VAR). In order to inspect the quality of the predictions further, Fig. 4 displays the ‘‘quasi-final’’ predictions of U.S. smooth growth, as well as the best overall prediction of yt (MBPF PC VAR), constructed in the third month of quarter t. Clearly, this multivariate coincident indicator tracks the signal of interest very accurately. The extents of the downturns in 1981–1982, 1990–1991 and 2001 are predicted accurately in real-time, although the violence of the contraction in 2008Q4 is not fully anticipated (a prediction of only −0.6% for smooth growth compared to a ‘‘quasi-final’’ prediction of −1.6%). The predictions of smooth growth for 2009Q1 and 2009Q2 are accurate. 4.4.1. Comparison with the new Eurocoin The new Eurocoin indicator of Altissimo et al. (2010) targets a monthly measure of quarterly GDP growth free of fluctuations with a period below one year. This monthly GDP is obtained through a linear interpolation of quarterly figures. We instead target quarterly GDP growth short of fluctuations with a period below one year, and do so in the three months of each quarter. Thus, the two targets are only comparable in the third month of each quarter. New Eurocoin is obtained by projecting smooth growth onto smooth monthly factors estimated by generalized principal components. These smooth monthly factors span the subspace of the factor space G(F, t ) which does not contain fluctuations with a period below 12 months. The objective is to have an indicator which is free of shortrun oscillations, just as the target. Instead, we project smooth growth onto unsmoothed split monthly factors, while producing vintages of the indicators that are smooth by definition, while still being subject to revisions. The multivariate band-pass filter mitigates these revisions. More importantly, we incorporate in the projections not only covariates but also the available observations of

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Fig. 4. Smooth growth of U.S. GDP: ‘‘quasi-final’’ predictions and real-time (MBPF PC VAR) predictions. Grey areas represent the NBER’s recession dating.

GDP. These are the single most important observations in our multivariate predictions (as can easily be seen by comparing these to the performance of the univariate filter, see Table 4). 5. Forecast performance Here, we assess, for the U.S. and Portugal, whether our best multivariate predictions of smooth growth are useful for forecasting GDP growth itself. Specifically, we will predict yt +h = SG(L)∆xt +h , for h ≥ 0, given information available at quarter t, and compare these predictions to the actual observations of quarterly GDP growth, ∆xt +h . Although our objective so far has been to predict yt , we can justify the use of the method for forecasting if it is reasonable to assume from the outset the impossibility of forecasting (given a set of covariates) the high frequencies of GDP growth (those with a period below one year). If the noisy (or for that matter any other) fluctuations of a time series are unpredictable given a set of covariates, it may be more efficient (in finite samples) to focus on predictions of what is in fact predictable, or to use projections that impose the restriction of unpredictability at some frequencies. We start by arguing that this restriction is indeed highly reasonable, proceeding as follows: using a BK filter, for the U.S. and Portugal, we (approximately) isolate the high frequencies (those associated with periodicities below 4 quarters) of output growth and of the indicators (aggregated quarterly) in our panels.10 We then compute the correlations between these fluctuations of output growth and those of the indicators at various lags. Finally, we summarize the results by plotting whisker plots

10 Since the factors extracted from the panels will be linear combinations of the indicators, a zero correlation between each of the indicators and output growth at high frequencies would imply a zero correlation between the factors and output growth at high frequencies.

of (the absolute values of) these correlations, one for each lag. We repeat the exercise with the low frequencies. As can easily be concluded from Fig. 5 (for the U.S. and Portugal), the correlations between the indicators and output growth at high frequencies are tightly concentrated around zero, particularly when the lag is positive (where a high correlation would be necessary if the indicators were to lead output growth). This is in sharp contrast to the results for low frequencies in Fig. 6. Furthermore, we should point out that neither country has high frequencies of any indicator with three or more cross-correlations (between lags −4 and 4) above 0.3. Now, in the case of Portugal, the contribution of the high frequencies to the variance of output growth (estimated by the variance of the high frequencies divided by the total variance) is 51%, compared to 33% in the U.S. case. Thus, we should expect more relevant forecast improvements in the case of Portugal if we focus on forecasts of the (relatively more) predictable component of output growth. We will show that such is indeed the case. The argument above is weakened if the high frequency fluctuations in output growth are predictable given (past) output growth. In this case, it would be more reasonable to combine a projection at low frequencies (using covariates) with a projection at high frequencies (using only output growth). Attempts to combine these projections did not produce any relevant forecast gains. What seems more promising at very short horizons (basically nowcasting) is reducing (possibly totally) the degree of smoothing of output growth. In fact, in the case of Portugal, our method is clearly beaten by competing methods (that target unsmoothed growth) in the case of nowcasting. 5.1. Quarterly growth Here, we analyze forecasts of quarterly output growth obtained using the best multivariate predictions of smooth growth from Section 4.4. (U.S. case), extending the analysis to Portugal data. Nothing changes in the construction of the

J. Valle e Azevedo, A. Pereira / International Journal of Forecasting 29 (2013) 479–492 1.0 0.8 0.6 0.4 0.2 0.0

1.0 0.8 0.6 0.4 0.2 0.0

Correlation

Correlation

490

–4

–3

–2

–1

0

1

2

3

4

LAG

–4

–3

–2

–1

0

1

2

3

4

LAG

1.0 0.8 0.6 0.4 0.2 0.0

Correlation

Correlation

Fig. 5. U.S. (left) and Portugal (right). Whisker plots (with boxes covering 90% of the values) of (absolute value of) the correlations at various lags between the high frequencies (period below four quarters) of real GDP growth and the high frequencies of the indicators (each point is Correlation HF (GDPgrowthHF t , indicatort −Lag ), where HF denotes high frequencies). The filtered series are obtained using the BK filter with truncation lead and lag equal to 6. The sample is 1966Q1–2008Q2 for the U.S. and 1991Q2–2008Q3 for Portugal.

–4

–3

–2

–1

0

1

2

3

4

1.0 0.8 0.6 0.4 0.2 0.0

LAG

–4

–3

–2

–1

0

1

2

3

4

LAG

Fig. 6. U.S. (left) and Portugal (right). Whisker plots (with boxes covering 90% of the values) of (absolute value of) correlations between the low frequencies LF (period above four quarters) of GDP growth and the low frequencies of the indicators (each point is Correlation (GDPgrowthLF t , indicatort −Lag ), where LF denotes low frequencies). The filtered series are obtained as the original series minus the high frequencies above. The sample is 1966Q1–2008Q2 for the U.S. and 1991Q2–2008Q3 for Portugal.

predictions, with the only difference here being that they are compared with GDP growth itself. Our covariates are the first split principal component in the U.S. case (as in Section 4.4) and the first two split principal components in the case of Portugal (as this delivered the best results overall). Due to the short sample size, VAR estimation proved unfeasible for Portugal. The pseudo-out-of-sample exercise focuses on 0–3-quarter-ahead forecasts made in the third month of a quarter, referring to 1990Q1–2009Q4 for the U.S. and to 2000Q1–2009Q4 in the case of Portugal, i.e., including the recession of 2008–2009. In the U.S. case, the estimation sample begins in 1960Q1, whereas for Portugal it begins in 1991Q1. We compare forecasts obtained as multivariate predictions of smooth growth to forecasts obtained using the following competing methods:

The results for the U.S. and Portugal are presented in Table 5, which contains the ratio of the mean square forecast error (MSFE) obtained with each method to the MSFE obtained with a simple autoregression (AR) forecast (with the lag length determined by the BIC). The main conclusions are as follows.

• In the U.S. case, multivariate predictions of smooth

•

• Regression of h-quarter-ahead GDP growth on (a maximum of two) split principal components and past GDP growth, with the lag length determined by the BIC, exactly as was done by Stock and Watson (2002a). This is denoted as DI AR-SW. More factors did not lead to a better performance. • A multivariate linear projection, similar to the solution to the filtering problem in Eq. (5), with the only difference being that the target yt +h = SG(L)∆xt +h (smooth growth) is substituted by ∆xt +h (unsmoothed growth). These projections use either the first split principal component (U.S. case) or the first two split principal components (in the case of Portugal). Second moments are derived from the pre-whitening VAR procedure (again, only for the U.S., in which case we denote the projections by PC VAR) or estimated nonparametrically (projections denoted by PC KERNEL), using exactly the same settings of the multivariate filter predictions. The methods PC VAR and PC KERNEL allow us to assess the value added by targeting smooth growth instead of unsmoothed growth. Also, it is natural to consider the widely used factor augmented autoregression proposed by Stock and Watson (2002a).

•

•

•

growth (especially MBPF PC VAR) rank very well at all horizons, though there is no uniformly superior method. In any case, there does not seem to be a cost of targeting smooth growth instead of unsmoothed growth. In the case of Portugal, the multivariate prediction of smooth growth, MBPF PC KERNEL, is beaten by DI ARSW and PC KERNEL at one step ahead, while it is superior to all of the other methods for longer horizons. We recall that the comparable ‘‘no-smoothing’’ prediction, denoted PC KERNEL, uses exactly the same covariates and estimated second moments, but targets GDP growth instead of smooth growth. DI AR-SW performs poorly at horizons greater than 0 quarters in the case of Portugal and 1 quarter in the U.S. case. In the U.S. case, all methods perform rather poorly at four steps ahead (and longer horizons, results not reported), confirming the well-known difficulty of forecasting quarterly GDP growth at long horizons (for recent overviews, see Giannone, Reichlin, & Sala, 2005, or Rünstler et al., 2009). Clearly, the results of the forecast encompassing tests presented in Table 5 confirm the usefulness of considering methods other than the simple AR model.

Overall, the results above suggest that multivariate predictions of smooth growth, while having been designed for a different purpose, are useful for short-term forecasting of quarterly real GDP growth, especially in the case of Portugal (where the high frequencies account for a larger fraction of the GDP growth variation). We now argue that these predictions can also be useful for forecasting annual GDP

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Table 5 Ratio of the mean squared error of the forecasts with each method to the mean squared error of a univariate autoregression forecast (BIC for lag length; maximum lag length equal to 6). Simulated out-of-sample forecast results: GDP quarter-on-quarter growth rate Relative MSFE One step ahead (current quarter)

2 steps ahead (1 quarter ahead)

Method

U.S.

Portugal

U.S.

Portugal

U.S.

Portugal

U.S.

Portugal

PC KERNEL PC VAR

0.85∗ 0.66∗

0.55∗ –

0.95∗∗ 0.78∗

0.80∗∗ –

1.01∗∗ 0.88∗

0.80∗∗ –

1.13 0.96∗

0.77∗ –

MBPF PC KERNEL MBPF PC VAR

0.68∗ 0.65∗

0.67∗ –

0.81∗ 0.77∗

0.72∗∗ –

0.95∗∗ 0.88∗

0.75∗∗ –

1.10 0.96∗

0.73∗ –

DI AR-SW

0.63∗

0.49∗

0.88∗

0.93

1.18

1.03∗∗

1.24

1.03∗∗

0.0058

0.0089

0.0064

0.0089

0.0067

0.0091

0.0067

0.0090

√

MSFE, AR

3 steps ahead (2 quarters ahead)

4 steps ahead (3 quarters ahead)

Evaluation period: 1990Q1–2009Q4 for the U.S. and 2000Q1–2009Q4 for Portugal. For the U.S., the multivariate predictions use one factor (estimated by principal components), whereas for Portugal they use two factors. Cases in which the null hypothesis (of encompassing) of a forecast encompassing test relative to the AR model is rejected at the 5% and 10% significance levels, based on the two-sided test of forecast encompassing of Harvey, Leybourne, and Newbold (1998), are denoted by one and two asterisks, respectively.

Fig. 7. U.S. (left) and Portugal (right): Annual real GDP growth rates: actual rates and those derived from ‘‘quasi-final’’ predictions of smooth growth (denoted Implicit), using the method MBPF PC VAR of Section 4.4 for the U.S. and MBPF PC KERNEL for Portugal.

growth. To see this, we note that using quarter on quarter growth rates, or alternatively smooth growth rates, to compute annual GDP growth, delivers two series which are are almost indistinguishable (see Fig. 7, which contains the observed annual GDP growth and annual GDP growth constructed from ‘‘quasi-final’’ predictions of smooth growth, for the U.S. and Portugal).11 Hence, predictions of smooth growth can be used to forecast annual GDP growth, potentially accurately.12 6. Concluding remarks We have shown how to usefully integrate recent developments in the analysis of dynamic factor models into the approximation of band-pass filters (or predictions of distributed lags of a series of interest). The resulting multivariate band-pass filter, fuelled by factors extracted from a 11 The annual growth between year 0 and year 1 can easily be obtained from quarterly growth rates. The GDP for year 0 (year 1) is just the sum of the GDPs of the four quarters of year 0 (year 1), which can be obtained using the growth rate of quarterly GDP. Now, using quarter on quarter growth rates or an accurate prediction of quarterly smooth growth (say, the ‘‘quasi-final’’ predictions of smooth growth) in order to calculate annual GDP growth delivers very similar series, as Fig. 7 shows. 12 A detailed analysis of the forecasts of annual GDP growth constructed in this way is available upon request. Overall, these forecasts rank well compared to forecasts based on PC KERNEL, PC VAR or DI AR-SW.

large panel of time series, is reliable and outperforms the univariate predictions in various dimensions. We have focused on real-time predictions of two relevant macroeconomic signals related to real activity: business cycle fluctuations and the smooth growth of real GDP. In our analysis of the forecasting performance of smooth growth, we have highlighted an important insight: targeting a smooth version of a time series may be more useful than targeting the sometimes erratic (or unpredictable at high frequencies) original series, especially at long horizons. Conventional forecast models fit the variables of interest at every frequency, regardless of the predictive content of the available covariates at each frequency. Appendix. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijforecast.2012. 12.005. References Altissimo, F., Cristadoro, R., Forni, M., Lippi, M., & Veronese, G. (2010). New Eurocoin: tracking economic growth in real-time. Review of Economics and Statistics, 92(4), 1024–1034. Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70, 191–221.

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João Valle e Azevedo has been an Economist at the Research Department of Banco de Portugal since 2007, being also Invited Assistant Professor at Universidade NOVA de Lisboa. His main fields of interest are time series econometrics and macroeconomics. He has previously worked as a Researcher at the Free University of Amsterdam. João received a ‘‘Licenciatura’’ in mathematics applied to economics and business (Technical University of Lisbon, 2001), an M.Sc. in Statistics (London School of Economics and Political Science, 2002) and a Ph.D. in Economics (Stanford University, 2007). His work has been published at the Journal of Business and Economic Statistics, Oxford Bulletin of Economics and Statistics and Journal of the Royal Statistical Society.

Ana Pereira is an Economist/Statistician at the Research Department of Banco de Portugal since 2006. She holds a ‘‘Licenciatura’’ in mathematics applied to economics and business (Technical University of Lisbon, 2005) and a masters in applied econometrics and forecasting from the same University (2009).