Keywords: Forecasting; Business cycles; Leading indicators; Coincident indicators; Turning points
1. Introduction
In the recent past, there has been a renewed interestin coincident and leading indicators as a tool formonitoring the economy and predicting its futurebehaviour. Such an interest was also in part stimulatedby a set of theoretical developments, which havepresumably eliminated or at least reduced the draw-backs of traditional composite coincident and leadingindexes (CCI and CLI, respectively).
Stock and Watson (1989, SW) improved, in fivemain respects, what was at that time the currentpractice in indicator analysis. First, they formalizedBurns and Mitchell's (1946) notion that businesscycles represent co-movements in a set of series byestimating a coincident index of economic activity asthe unobservable factor in a dynamic factor model.Second, they stressed the importance of the choice ofcandidate leading indicators and introduced a system-atic regression based selection. Third, they jointlymodelled the coincident and leading indicators.Fourth, they introduced a state space framework thatallows the joint resolution of a set of data problems,such the identification and removal of outliers, thetreatment of data revisions, and the use of indicators
rs. Published by Elsevier B.V. All rights reserved.
1 The single coincident indicators present either a trendingbehaviour, or at least persistent deviations from the mean. Thesefeatures are confirmed by ADF unit root tests, which do not rejecthe null hypothesis of a unit root for any of the indicators. Insteadthe results of cointegration tests conducted in a VAR framework arenot conclusive; they substantially depend on the lag length, thedeterministic component included in the VAR, and the type ocointegration test applied (Johansen's, 1988, trace or eigenvaluestatistic). When BIC is used to select the lag length of the VAR andthe deterministic component, it also selects models withoucointegration. Since all factor based methods require the inpuvariables to be weakly stationary, we model the log differences othe single coincident indicators.
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whose most recent unavailable data can be substitutedwith forecasts. Finally, they developed an index ofleading indicators that produces early warnings ofrecession, in the form of a probability that a recessionwill take place in the next six months. The topics werefurther developed in Stock and Watson (1991, 1992).
Another very important article in this contextis Hamilton (1989), whose main contributions to thebusiness cycle literature are threefold. First, heshowed that, using the available data on one or morecoincident series, it is possible to infer the probabilityof being in an expansion or a recession. Second, it ispossible to jointly model coincident and leadingindicators, since the latter should be driven by thesame Markov process but with a lead (Hamilton &Perez-Quiros, 1996). Third, and related to theprevious point, the method can easily be used toproduce point or probability forecasts of the coinci-dent variable, but it can also be useful for analyticalforecasts of the probability of being at a recession in orwithin a certain future date.
Stock and Watson (1989) and Hamilton (1989) havegenerated an impressive amount of subsequent research,and in this paper we wish to provide an overview ofthose contributions more closely related to the construc-tion of CCIs and CLIs. Additional theoretical details canbe found in Marcellino (2006). As an illustration of theimplementation of the newer techniques, we willconstruct and compare a variety of CCIs and CLIs forthe UK.
The paper is organized as follows. In Section 2we describe and implement alternative techniques forthe construction of composite coincident indexes. InSection 3 we present and apply a variety of methods forbuilding composite leading indexes, and for using themas forecasting devices. In Section 4 we summarize themain findings and conclude.
2. Alternative methods for the construction of CCIs
The variables we combine into a composite coin-cident index (CCI) for the UK are very similar to thosetraditionally considered for the US, see e.g. Marcellino(2006), and coincide with those selected by theConference Board (CB) for their CCI, which repre-sents our benchmark due to its long establishedtradition. Specifically, we consider Industrial Produc-tion, Retail Sales, Employment, and Real Household
Disposable Income over the sample 1978–2004, at amonthly frequency.
There exist different methods of summarizing theinformation in the four series into a single CCI, see e.g.Marcellino (2006) for details. The simplest procedurerequires the single components of the CCI to betransformed so that they have similar ranges, and thenaggregated using a set of weights. We will refer to theresulting index as the non-model based (NMB) CCI.The NMB CCI is very similar to the one produced bythe Conference Board, and we will use it as abenchmark for the more sophisticated CCIs in theensuing analysis.
A second procedure for the construction of a CCIwas introduced by Stock and Watson (1989, SW) forthe US, and it is based on a parametric factor model forthe components of the composite index. FollowingStock and Watson, we have considered the followingspecification:
yit ¼ giDCt þ uit; i ¼ 1; N ; 4uit ¼ w1iuit−1 þ w2iuit−2 þ eit;eitfiidNð0; r2i Þ; i ¼ 1; 2; 3; 4;
ð2Þwhere yit indicates the demeaned log difference ofeach indicator, ΔCt the common factor driving all theindicators (the log difference of the CCI), and uit theidiosyncratic component of each indicator.1 Thecumulative values of the estimated common factorwill be referred to as the parametric factor based (SW)CCI.
t,
f
ttf
3 The filtered CCIs are obtained by applying the bandpass HPfilter proposed in Artis, Marcellino, and Proietti (2004) toemphasize the business cycle frequencies between 1.5 and 8 years4 Detailed results are available upon request.
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One problem with the SW CCI is that standarddiagnostic tests on the residuals εbit of the model inEq. (1) in the UK indicate a lack of normality and serialcorrelation. The former seems to be due to a few outlyingobservations, but can hardly be eliminated by insertingdummy variables into the model. Increasing the numberof lags is not effective for eliminating the detected serialcorrelation of the residuals either. However, thesechanges in model specification do not substantiallyalter the estimated factor, which provides evidence infavour of the robustness of the computed SW CCI.
An alternative solution which addresses the mis-specifications of the parametric factor model in Eq. (1),is to resort to non-parametric techniques to estimate thecommon factor and obtain alternative factor-basedCCIs. The two most common techniques weresuggested by Stock and Watson (2002a,b, SW2) in thetime domain, and Forni, Hallin, Lippi, and Reichlin(2000, FHLR) in the frequency domain. Basically, SW2suggested estimating the common factors as the staticprincipal components of the variables, namely, of thesingle coincident indexes used in our case. FHLRproposed the use of dynamic principal components tobetter capture the dynamics of the model. The latterapproach has the disadvantage that the estimated factorsare combinations of lags, contemporaneous values andleads of the single series, and the use of leads prevents areal time implementation. A modified procedure whichallows a real time implementation was suggested byForni, Hallin, Lippi, and Reichlin (2005, FHLR2).Details of these methods can be found in the originalpapers and in Altissimo et al. (2001) and Marcellino(2006) in a CCI context. Both methods are particularlysuitable when the number of variables under analysis islarge, but the evidence in Marcellino (2006), for the US,and Carriero and Marcellino (2006), for euro areacountries, suggests that they can also produce reliableCCIs when applied to a limited number of coincidentindicators. We will refer to the non-parametric factorbased indicators as SW2 and FHLR2 CCIs.2
In the three panels of Fig. 1, we graph the levels, six-month percentage changes and filtered versions of the
2 More specifically, the SW2 CCI is the first static principalcomponent of the four coincident series, while to construct theFHLR2 CCI we apply their two-step procedure, set the bandwidthparameter at M=12, and use one factor both in the first step (i.e. tocompute the variance covariance matrix of the common componentsobtained using FHLR) and in the second step.
(standardized) NMB CCI and of the three versions ofthe factor based CCIs, namely, SW, SW2 and FHLR2.3
The different CCIs seem to move closely together, forany transformation. The use of growth rates or filtereddata also emphasizes the close similarity of the indexesat turning points. The visual impression is confirmedby their correlations, which are often higher than 0.80.The smallest similarity is between the SW and SW2CCIs, and the latter seems more reliable due to theproblems of the estimated parametric factor modelmentioned previously. We then aggregate the monthlyvalues to produce quarterly data, and compare theindexes both across themselves and with real GDPgrowth. The similarity across the indexes is confirmedat the quarterly frequency. In terms of correlation withGDP growth, the lowest value is 0.62 for the SWCCI.4
However, the comparison with GDP growth should beinterpreted with care. Even though such a comparisonis standard in the literature, it is not clear that GDP is agood overall measure of the status of the economy,since its growth can be uncorrelated with higheremployment or disposable income, as the prolongedjobless recovery in the US at the beginning of the newmillennium, or the persistently high unemploymentrates in Europe, testify.5
Both theNMBCCI and the factor-basedCCIs assumethat the economic conditions can be summarized by acontinuous variable. An alternative approach treats thesingle unobservable force underlying the evolution of thecoincident indicators as discrete rather than continuous.Basically, the CCI in this context represents the statusof the business cycle (expansion/recessionmapped into a0/1 variable), which determines the behaviour of all thecoincident indicators that can change substantiallyover different phases of the cycle. Hamilton's (1989)Markov switching model provides a convenient statis-tical framework for estimating such a discrete CCI.More
5 If GDP is accepted as an overall measure of the status of theeconomy, a monthly estimate of GDP evolution would represent anatural alternative CCI. For the UK, Mitchell, Smith, Weale, Wrightand Salazar (2005) suggest a formal procedure for combininginformation about a range of monthly series into indications of shortterm movements in output. Their assessment of the efficacy of theapproach to evaluate the state of economic activity is rather satisfactory
.
,
-
.
Fig. 1. Coincident indexes. The upper panels graph the levels and the six-month percentage change in the CCIs, and the lower panel graphs thefiltered CCIs.
222 A. Carriero, M. Marcellino / International Journal of Forecasting 23 (2007) 219–236
specifically, the Markov switching (MS) CCI coincideswith an estimate of the unobservable current status of theeconomy, sbt|t. Hamilton (1994) or Krolzig (1997) providedetails on the computation of sbt|t, and also formulae forcalculating sbt|T, i.e., the smoothed estimator of theprobability of being in a given status in period t.6
In the upper panels of Fig. 2 we compare the six-month growth rate in the NMB CCI with the smoothedand filtered probability of recession (sbt|T and sbt|t)resulting from a MS-VAR(1) for the four componentsof the NMB CCI. We would expect higher probabilityof recessions to be associated with marked slowdownsin the growth of the NMB CCI, which indeed appearsto be the case. The recessions at the end of the '70s and
6 The basic Markov switching model can be extended in severaldimensions, for example, to allow for more states and cointegrationamong the variables, see e.g. Krolzig, Marcellino, and Mizon(2002), or time-varying probabilities, as e.g. in Diebold, Lee, andWeinbach (1994) or Filardo (1994).
in the early '90s are correctly identified, and minorepisodes of declining growth in the NMB CCI are alsogenerally associated with an increase in the estimatedprobability of recession.
Factor-based and Markov switching models aresuitable for capturing two complementary characteris-tics of business cycles, namely, the diffusion of cyclicalfluctuations across many series, and the differentbehaviours of the variables during expansions andrecessions. Diebold and Rudebusch (1996) suggestedthat the two approaches be combined by allowing theunderlying factor in the SW model to evolve accordingto a Markov switching model. Estimation of the factor-basedMSmodel using the Gibbs sampler technique wasproposed by Kim and Nelson (1998) and Filardo andGordon (1999). Following Kim and Nelson (1998), it istherefore possible to obtain both a continuous anda discreet CCI, labeled KN1 and KN2 respectively.The former is the accumulated value of the estimatedcommon factor, as for the SWCCI, while the latter is the
Fig. 2. Probabilities. The upper panels are smoothed and filtered Markov Switching-based probabilities of recession, and six-month percentagechanges in the NMB CCI. The lower panel is discrete and continuous CCI (KN1 and KN2) from the Markov Switching factor model.
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estimated probability of recession, sbt|t, as in theMSCCI,but estimated from the MS factor model.
In the third panel of Fig. 2, we plot the six-monthpercentage change in the KN1 CCI, and the discreteKN2 CCI. The former is similar to the SW2 and FHLR2factor-based indexes (also in terms of correlations), andthe latter tends to increase during recessions. However,KN2misses the recession in the early '90s, and providesweaker signals in the late '70s. This is clear from Fig. 3,where we report the dating of the UK recessions (basedon the Bry–Boschan algorithm applied to the SW CCI),and the probability of recessions from the Markovswitching VAR and factor models.
A final interesting comparison concerns the datingof the UK business cycle. For example, a comparisonof the Economic Cycle Research Institute (ECRI)peak and trough dates with those in Artis (2002)
reveals that the UK recessions lasted longer accordingto the former. Similarly, the recession of the early '90slasted longer according to Birchenhall, Osborn, andSensier (2001) than according to Artis et al. (2004).Moreover, Artis et al. (2004) also indicate a highprobability of recession in 2001. These differencescan be due to the use, either of alternative datingtechniques, or of a different CCI.
To understand whether the construction method ofthe CCI matters, we have applied the Bry–Boschandating algorithm to the IP series, to the NMB CCI andto the SW CCI. The results indicate that the choice ofthe CCI can indeed play a role, even when the samedating technique is used. In particular, the recessions ofthe late '70s and early '90s seem longer whenmeasuredon the NMB or SW CCI than with IP. Moreover, IPsuggests the presence of a recession at the beginning of
Fig. 3. Bry–Boschan dating of UK recessions and probability of recession from the Markov Switching VAR and the factor model.
224 A. Carriero, M. Marcellino / International Journal of Forecasting 23 (2007) 219–236
the new millennium, which is absent according to theSW CCI and very short with the NMB CCI.7
In summary, all the methods for the construction ofcontinuous composite coincident indexes yield similarresults when measured in terms of correlation. How-ever, there are some differences in the exact dating ofrecessions, in particular when IP is used as a singlecoincident indicator; and the discrete indexes appear toprovide imprecise indications of the arrival of arecessionary period.
3. Alternative methods for the construction of CLIs
As in the case of the CCIs, the selection of theindicators to be included in a Composite Leading
7 Detailed results are available upon request.
Index (CLI) is a fundamental first step. FollowingMoore and Shiskin (1967), a leading indicator shouldpossess the following properties: (i) consistent timing(i.e., systematically anticipate peaks and troughs in thetarget variable, possibly with a rather constant leadtime); (ii) conformity to the general business cycle(i.e., have good forecasting properties not only atpeaks and troughs); (iii) economic significance (i.e.,being supported by economic theory either as possiblecauses of business cycles or, perhaps more impor-tantly, as quickly reacting to negative or positiveshocks); (iv) statistical reliability of data collection(i.e., provide an accurate measure of the quantity ofinterest); (v) prompt availability without later majorrevisions (i.e., being timely and regularly available foran early evaluation of the expected economic condi-tions, without requiring subsequent modifications ofthe initial statements); (vi) smooth month to month
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changes (i.e., being free of major high frequencymovements).
Most of these properties can be formally evaluated,but selecting the indicators and testing their propertiescan be a very time-demanding task. Therefore, we relyon the selection of the leading indicators made by theConference Board.8
The use of a single leading indicator is dangerousbecause economic theory and experience teach thatrecessions can have different sources and characteristics.Combining of leading indicators into composite indexescan therefore be more useful for capturing the signalscoming from different sectors of the economy. Theconstruction of a composite index can be undertakeneither in a non-model-based framework, or withreference to a specific econometric model of the evo-lution of the leading indicators, possibly jointly with thetarget variable.
A non-model-based (NMB) CLI is constructedfollowing the procedure outlined in the previoussection for the NMB-CCI; namely, the single leadingindicators are averaged, possibly after a set of suitabletransformations such as seasonal adjustment,differencing and standardization. Our NMB CLI forthe UK is very similar to the one constructed by theConference Board, and it will be our benchmark forthe comparison with the more sophisticated CLIs.
The main advantage of NMB CLIs is simplicity.Non-model-based indexes are easy to build, easy toexplain, and easy to interpret — all of which are veryvaluable assets, in particular for the general public andfor policymakers.Moreover, simplicity often improvesforecasting as well. With an NMB CLI there is noestimation uncertainty, no major problems of over-fitting, and the literature on forecast pooling suggeststhat equal weights work pretty well in practice, see e.g.Stock and Watson (2003), even though their variablesare pooled rather than forecasts. However, from aneconometric point of view, NMB CLIs are also subjectto several criticisms, see e.g. Emerson and Hendry(1996) and Marcellino (2006). First, there is no explicitreference to the target variable in the construction of the
8 Specifically, the single leading indicators we consider are OrderBook Volume, Volume of Expected Output, House Building Starts,Fixed Interest Price Index, All Share Price Index, New Orders ofEngineering Industries, Productivity, and Operating Surplus ofCorporations.
composite index, e.g. in the choice of the weightingscheme. Second, the weighting scheme is fixed overtime, with periodic revisions which are mostly dueeither to data issues, such as changes in the productionprocess of an indicator, or to the past unsatisfactoryperformance of the index. Third, lagged values of thetarget variable are typically not included in the leadingindex, though there can be economic and statisticalreasons underlying the persistence of the target variablethat would favor such an inclusion. Fourth, laggedvalues of the single indicators are typically not used inthe index, although they could provide relevantinformation, e.g. because not only the point value ofan indicator, but also its evolution over a period of time,matter for anticipating the future behavior of the targetvariable. Finally, if some indicators and the targetvariable are cointegrated, the presence of short rundeviations from the long run equilibrium could provideuseful information on future movements of the targetvariable.
Most of the issues raised regarding the NMBCLIs are addressed by the model based procedures,where the single leading indicators are combined tomake a CLI in a formal econometric context. Thistopic is analyzed in detail in Marcellino (2006); herewe summarize the main results, focusing on thosewhich are useful for interpreting the empiricalfindings for the UK.
3.1. Linear methods
A linear VAR provides the simplest model basedframework to understand the relationship betweencoincident and leading indicators, the construction ofregression based composite indexes, the role of thelatter in forecasting, and the consequences of invalidrestrictions or unaccounted cointegration. FollowingMarcellino (2006), we group the m coincidentindicators in the vector yt, and the n leading indicatorsin xt. For the moment, (yt, xt) is assumed to be weaklystationary and its evolution is described by the VAR(1):
ytxt
� �¼ cy
cx
� �þ A B
C D
� �yt−1xt−1
� �þ eyt
ext
� �;
eytext
� �fi:i:d:
0
0
� �;
Ryy Ryx
Rxy Rxx
� �� �:
ð3Þ
226 A. Carriero, M. Marcellino / International Journal of Forecasting 23 (2007) 219–236
It immediately follows that the expected value ofyt+1 conditional on the past is cy+Ayt+Bxt, so that forxt to have a useful set of leading indicators it must beB≠0. When A≠0, lagged values of the coincidentvariables also contain useful information for forecast-ing. Both hypotheses are easily testable and, in thecase where both A=0 and B=0 are rejected, acomposite regression based leading indicator for yt+1(considered as a vector) can be constructed as
CLI1t ¼ bcy þbAyt þbBxt; ð4Þ
where the b indicates the OLS estimator. Standard errorsaround this CLI can be constructed using standardmethods for VAR forecasts, see e.g. Lütkepohl (2006).Moreover, recursive estimation of the model provides aconvenient tool for continuous updating of the weights.
A similar procedure can be followed when the targetvariable is dated t+h rather than t+1. For example,when h=2,
As an alternative, the model in Eq. (3) can be re-written as
ytxt
� �¼ ecyecx� �
þeA eBeC eD
!yt−hxt−h
� �þ eeyteext� �
ð6Þ
where a e indicates that the new parameters are acombination of those in (3), and eext and eeyt are correlatedof order h−1. The specification in Eq. (6) can beestimated by OLS, and the resulting CLI is written as
gCLI1ht ¼ becy þbeAyt þbeBxt: ð7Þ
The main disadvantage of this latter method, oftencalled dynamic estimation, is that a different model hasto be specified for each forecast horizon h. On the otherhand, no model is required for the leading indicators,and the estimators of the parameters in Eq. (6) can bemore robust than those in Eq. (3) in the presence of mis-specification, see e.g. Clements and Hendry (1996) fora theoretical discussion, and Marcellino, Stock, andWatson (2006) for an extensive empirical analysis of
the two competing methods (showing that dynamicestimation is on average slightly worse than the iteratedmethod for forecasting US macroeconomic timeseries). For the sake of simplicity, in the rest of thepaper we will focus on h=1 whenever possible.
Consider now the case where the target variable is acomposite coincident indicator,
CCIt ¼ wyt; ð8Þwhere w is a 1×m vector of weights. To construct amodel based CLI for the CCI in Eq. (8), two methodsare available. First, and more commonly, we couldmodel CCIt and xt with a finite order VAR, say
CCItxt
� �¼ dCCI
dx
� �þ eðLÞ FðLÞ
gðLÞ HðLÞ
� �CCIt−1xt−1
� �þ uCCIt
uxt
� �;
ð9Þwhere L is the lag operator and the error process iswhite noise. Repeating the previous procedure, thecomposite leading index for h=1 is
CLI2t ¼ bdCCIþbeðLÞCCIt þbFðLÞxt: ð10Þ
However, in this case the VAR is only anapproximation for the generating mechanism of (wyt,xt), since in general the latter should have either aninfinite number of lags or an MA component.
The alternative method is to stick to the model inEq. (3), and construct the CLI as
CLI3t ¼ wCLIt; ð11Þthat is, aggregate the composite leading indicators foreach of the components of the CCI, using the sameweights as in the CCI. Lütkepohl (1987) showed in arelated context that, in general, aggregating theforecasts (CLI3) is preferable to forecasting theaggregate (CLI2) when the variables are generatedby the model in Eq. (3); but that is not necessarily thecase if the model in Eq. (3) is also an approximationand/or the y variables are subject to measurementerror. Stock and Watson (1992) found little differenceoverall in the performance of CLI2 and CLI3 for theUS.
9 A linear VAR model also underlies the construction of the welknown Stock and Watson's (1989, SW) CLI for the US. Theintuition is that if the single leading indicators are also driven bythe (leads of the) common cyclical force, then a linear combinationof their present and past values can contain useful information fopredicting the CCI; see Marcellino (2006) for details. Acomparison of the NMB and SW CLIs for the UK indicates thathe former provides earlier and more reliable signals of recessionsdetails are available upon request. Therefore, in the followingempirical analysis we will focus on the NMB CLI and evaluatewhether it is possible to improve upon it.
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Both CLI2 and CLI3 are directly linked to the targetvariable, incorporate distributed lags of both thecoincident and the leading variables (depending on thelag length of the VAR), have weights that can easily beperiodically updated using recursive estimation of themodel, and have standard errors around the pointforecasts (or thewhole distribution under a distributionalassumption for the error process in the VAR) that arereadily available. Therefore, this simple linear modelbased procedure already addresses several of the maincriticisms of the non-model-based composite indexconstruction.
One assumption that we have maintained so far isthat both the coincident and the leading variables areweakly stationary, while in practice it is likely that thebehaviour of most of these variables is closer to that ofintegrated processes. Following Sims, Stock, andWatson (1990), this is not problematic for theconsistent estimation of the parameters of VARs inlevels such as Eq. (3), and therefore for theconstruction of the related CLIs, even though inferenceis complicated and, for example, hypotheses on theparameters could not be tested using standardasymptotic distributions. An additional complicationis that in this body of literature, when the indicators areI(1), the VAR models are typically specified in firstdifferences rather than in levels, without prior testingfor cointegration. Continuing the VAR(1) example, theadopted model would be
DytDxt
� �¼ cy
cx
� �þ eyt
ext
� �; ð12Þ
rather than possibly
DytDxt
� �¼ cy
cx
� �−
Im 0
0 In
� �−
A B
C D
� �� �yt−1xt−1
� �þ eyt
ext
� �¼ cy
cx
� �−ab V yt−1
xt−1
� �þ eyt
ext
� �;
ð13Þwhere β is the matrix of cointegrating coefficients andα contains the loadings of the error correction terms.As usual, the omission of relevant variables yieldsbiased estimators of the parameters of the includedregressors, which can translate into biased and
inefficient composite leading indicators. See Emersonand Hendry (1996) for additional details and general-izations, and, e.g., Clements and Hendry (1999) for theconsequences of omitting cointegrating relationshipswhen forecasting. As long as m+n is small enoughwith respect to the sample size, the number andcomposition of the cointegrating vectors can readily betested (see e.g. Johansen, 1988, for tests within theVAR framework), and the specification in Eq. (13)used as a basis for constructing model based CLIs thatalso take cointegration into account properly.9
To illustrate the empirical implementation of thetechniques described so far, we now consider forecastingthe (one-month symmetric percentage change in the)NMB CCI, using six alternative linear specifications:
1. A bivariate VAR for the NMB CCI and the NMBCLI;
2. A univariate AR for the NMB CCI;3. A bivariate ECM for the NMB CCI and CLI, as in
Eq. (13), where one cointegrating vector is imposedand its coefficient recursively estimated;
4. A VAR for the four components of the NMB CCIand the NMB CLI, as in Eq. (9).
5. AVAR for the NMB CCI and the eight componentsof the NMB CLI; and
6. A VAR for the four components of the NMB CCIand the eight components of the NMB CLI, as inEq. (3).Notice that most of these models are non-nested,
except for the AR which is nested in some of theVARs, and for the bivariate VAR which is nested in theECM.
The models are compared on the basis of theirforecasting performance one and six months ahead overthe period 1985:1–2004:12. The forecasts are computedrecursively, with the first estimation sample being
l
r
t;
Table 1A forecast comparison of alternative VAR models for CCI NMB and CLI NMB
1 step-ahead 6 step-ahead 6 step-ahead
DYNAMIC ITERATED
Relative Relative Relative Relative Relative RelativeMSE MAE MSE MAE MSE MAE
MSE MAE MSE MAE MSE MAEVAR(2) 0.000 0.001 0.000 0.001 0.000 0.002
Note: The forecast sample is 1985:1–2004:12. The first estimation sample is 1977:4–1984:12 (for 1 step-ahead) or 1977:4–1984:6 (for 6 steps-ahead), recursively updated. The lag length selection is done using the BIC. MSE and MAE are mean square and absolute forecast error,respectively. The benchmark is the VAR for CCI CB and CLI CB. ⁎, ⁎⁎, and ⁎⁎⁎ indicate significance at 10%, 5%, and 1% of the Diebold–Mariano test for the null hypothesis of no significant difference in MSE or MAE with respect to the benchmark.
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1977:4–1984:12 for one step ahead forecasts and1977:4–1984:6 for six step ahead forecasts. The laglength of the models is chosen by the BIC over the fullsample. The recursive BIC selects smaller models for theinitial samples, but their forecasting performance isslightly worse. The forecasts are computed using boththe standard iteratedmethod, and dynamic estimation (asdescribed in Eq. (6)).We use final vintage data, since realtime vintages were not available to us. This can bias theevaluation towards the usefulness of a CLI when thecomposition of the latter is modified during the eval-uation period. However, we use fixed weights and com-ponents over the whole period under analysis.Moreover,we focus on the comparison of alternative models for thesame vintage of data, rather than on showing that aspecific method performs well.
The comparison is based on the MSE and MAErelative to the bivariate VAR for the NMB CCI andCLI. The Diebold and Mariano (1995) test for thestatistical significance of the loss differentials is alsocomputed. The results are reported in Table 1, and afew comments can be made. First, as for the US, thesimple AR model performs very well; it generates thelowest MSE and MAE at both forecast horizons, withstatistically significant gains of about 20% in terms ofthe MSE at six-steps ahead. This finding indicates thatthe lagged behaviour of the CCI contains usefulinformation that should be included in a leading index.Second, taking cointegration into account does notimprove the forecasting performance with respect to a
VAR in differences. Third, forecasting the four compo-nents of the CCICB and then aggregating the forecastsdecreases the MSE at the longer horizon. Finally, theranking of iterated forecasts and dynamic estimation isquite clear cut: the former is systematically better thanthe latter in our application.
3.2. Markov switching models
Up until now we have implicitly assumed that thegoal of the CLI is forecasting a continuous variable,the CCI. However, leading indicators were originallydeveloped for forecasting business cycle turningpoints. Simulation based methods can be used toderive forecasts of a binary recession/expansionindicator within a linear framework, and these in turncan be exploited to forecast the probability that arecession will take place within, or at, a certainhorizon.
For example, let us consider the model in Eq. (9)and assume that the parameters are known and theerrors are normally distributed. Then, drawing randomnumbers from the joint distribution of the errors forperiod t+1,…, t+n and solving the model forward, it ispossible to get a set of simulated values for (CCIt+1,Δxt+1),…,(CCIt+n,Δxt+n). Repeating the exercise manytimes, a histogram of the realizations provides anapproximation for the conditional distribution of(CCIt+1, Δxt+1),…,(CCIt+n, Δxt+n), given the past.Given this distribution and a rule to transform the
Table 2Turning point predictions
Target Model RelativeMSE
RelativeMAE
NBER(1 step-ahead)
Univariate 1.097 1.123
Univariate MS 1.269 0.874Bivariate 1.092 1.077Bivariate MS 1.291 0.939Probit CLI_CB 1 1
MSE MAEProbit 0.1547 0.2882
Note: These are one-step ahead turning point forecasts for the BBexpansion/recession indicator. Linear and MS models (as inHamilton & Perez-Quiros, 1996) are used for CCI CB and CLCB. Six lags of CLI CB are used in the probit model. ⁎, ⁎⁎, and ⁎⁎⁎
indicate significance at 10%, 5%, and 1% of the Diebold–Marianotest for the null hypothesis of no significant difference in MSE oMAE with respect to the benchmark.
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continuous variable CCI into a binary recessionindicator, e.g. the three months negative growth rule,the probability that a given future observation can beclassified as a recession is computed as the fraction ofthe relevant simulated future values of the CCI thatsatisfy the rule. The procedure can easily be extended toallow for parameter uncertainty by drawing parametervalues from the distribution of the estimators ratherthan treating them as fixed. Normality of the errors isnot strictly required either, since re-sampling can beused, see e.g. Wecker (1979), Kling (1987) and Fair(1993) for additional details and examples.
As an alternative procedure, the MS model intro-duced in Section 2 to define the MS CCI can also beexploited to evaluate the forecasting properties of asingle or composite leading indicator. In particular, asimplified version of the model proposed by Hamiltonand Perez-Quiros (1996) can be written as
ð14Þwhere y and x are univariate, st evolves according to aMarkov chain, and the leading characteristics of x arerepresented not only by its influence on future valuesof y, but also by its being driven by future values of thestate variable, st+r.
Hamilton and Perez-Quiros (1996) found that theirmodel provides only a weak signal in the case of theUS recessions of 1960, 1970 and 1990. Moreover, theevidence in favor of the non-linear cyclical factor isweak, and the forecasting gains for predicting GNPgrowth or its turning point are minor with respect to alinear VAR specification. Even weaker evidence infavor of the MS specification was found when acointegrating relationship between GNP and laggedCLI was included in the model.10
To evaluate the usefulness of the MS feature forforecasting UK recessions, we compare univariate andbivariate models, with and without Markov switching,for predicting the turning points of the IP index one step
10 Lahiri and Wang (1994) for the first time successfully utilizedthe Hamilton model to generate recession probabilities from theindex of leading indicators.
I
r
ahead, using the NMB CLI as the leading indicator(jointly with the NMB CCI in the VAR), and the sameestimation and forecast sample as in the linear VARexample. The turning point probabilities for the linearmodels are computed using simulations, as described atthe beginning of this section, using a two consecutivenegative growth rule to identify recessions. For theMS,we use the filtered recession probabilities; see Marcel-lino (2006) for details on their computation. We alsoadd a probit model to the comparison, where theexpansion/recession indicator (Bry–Boschan based onIP) is regressed on six lags of the NMB CLI. Thismodel will be analyzed in detail in the next subsection.
The results of the turning point forecast comparisonare summarized in Table 2, where we report the MSEand MAE for each model relative to the probit. TheDiebold and Mariano (1995) test does not signalstatistical significance of the loss differentials at the 1,5 and 10 percent confidence level. Notice that the MSEis just a linear transformation of the QuadraticProbability Score (QPS) criterion of Diebold andRudebusch (1989). The figures indicate that the probitmodel produces the lowest MSE, while the univariateMS model is best when it is based on the MAEcriterion. The turning point probabilities for the fivemodels are graphed in Fig. 4, together with the Bry–Boschan dated recessions (shaded areas). The figurehighlights that the MS models correctly assign a high
Fig. 4. Turning point probabilities from alternative models. The shaded areas are recessionary periods, according to the Bry–Boschan algorithmapplied to the IP index.
230 A. Carriero, M. Marcellino / International Journal of Forecasting 23 (2007) 219–236
probability of recession in the early 90s, but for toolong a period of time. In addition, they only give a lightsignal of recession at the beginning of the new millen-
nium. The performance of the probit model is alsounimpressive, with an estimated probability of reces-sion not higher than .80 even during recessions, and a
Table 3Forecasting performance of alternative CLIs using probit models fothe BB recession/expansion classification
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few false alarms both in the first part of the forecastsample and in the late '90s. Finally, there are no majorchanges in the results when the target variablebecomes the turning points in the NMB CCI.11
3.3. Binary models
In the models we have analyzed so far to relatecoincident and leading indicators, the dependentvariable is continuous, even though forecasts of businesscycle turning points are feasible either directly (usingMS models) or by means of simulation methods (linearmodels). A simpler and more direct approach treats thebusiness cycle phases as a binary variable, and models itusing a logit or probit specification.
In particular, let us assume that the economy is inrecession in period t, Rt=1, if the unobservablevariable st is larger than zero, where the evolution ofst is governed by
st ¼ b Vyt−1 þ et: ð15Þ
Therefore,
PrðRt ¼ 1Þ ¼ PrðstN0Þ ¼ Fðb Vyt−1Þ; ð16Þwhere F(·) is either the cumulative normal distributionfunction (probit model), or the logistic function (logitmodel). The model can be estimated by maximumlikelihood, and the estimated parameters combined withcurrent values of the leading indicators to provide anestimate of the recession probability in period t+1, i.e.,
bRtþ1 ¼ PrðRtþ1 ¼ 1Þ ¼ Fðbb VytÞ: ð17Þ
The logit model was adopted for the US by Stock andWatson (1991), among others, and the probit model byEstrella and Mishkin (1998), while Birchenhall, Jessen,Osborn, and Simpson (1999) provided a statisticaljustification for the former in a Bayesian context. Binarymodels for European countries were investigated by
11 Artis et al. (1995) consider the possible contribution of the‘longer’ and ‘shorter’ leading indicators published by the CentralStatistical Office (CSO) to predict the turning-points in theeconomic cycle. They find that the longer index leads the coincidentseries by about ten months at the peak and thirteen at the trough, onaverage, and the shorter index by five months at the peak and nine atthe trough; but there is substantial variation.
Estrella and Mishkin (1997), Bernard and Gerlach(1998), Birchenhall et al. (2001), and Osborn, Sensier,and Simpson (2001). Marcellino (2006) summarizes thepros and cons of this class of models.
Notice that, as in the case of MS or linear models,the estimated probability of recession, Rbt+1, should betransformed into a 0/1 variable using a proper rule. Thecommon choices are of the type Rbt≥c, where c istypically 0.5.
We now consider the turning point forecastingperformance of the probit model for the UK in moredetail, which from the previous subsection was good incomparison to MS and linear models, but not so goodin absolute terms.
In particular, we consider whether any of the CLISW,or CLISW and CLINMB jointly, or the three-month ten-year interest rate spread, or the latter and the two CLIsjointly, have a better predictive performance than theCLINMB only. The estimation and forecasting sample isas in the first empirical example, and the specificationof the probit models is as in the second example,namely, six lags of each CLI are used as regressors(more specifically, the symmetric one-month percent-age changes for CLINMB and the one-month growthrates for the other CLIs).
Note: The forecast sample is 1985:1–2004:12. The first estimationsample is 1977:4–1984:12 (for 1 step-ahead) or 1977:4–1984:6 (fo6 steps-ahead), recursively updated. The lag length selection is usingthe BIC. MSE and MAE are mean square and absolute forecaserrors, respectively. The benchmark is the VAR for CCI CB and CLCB. ⁎, ⁎⁎, and ⁎⁎⁎ indicate significance at 10%, 5%, and 1% of theDiebold–Mariano test for the null hypothesis of no significandifference in MSE or MAE with respect to the benchmark.
r
r
tI
t
Fig. 5. Turning point probabilities from alternative probit models. The shaded areas are recessionary periods, according to the Bry–Boschanalgorithm applied to the IP index.
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From Table 3, the model with the two CLIs isfavoured for one-step ahead turning point forecasts.Repeating the analysis for six-month-ahead forecasts,
the gap across models shrinks, the term spread modelremains worst, and the model with the CLINMB yieldsthe lowest MSE and MAE. However, the recession
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probabilities derived from these models, graphed inFig. 5, never reach one and are sometimes rather high,even if there is no subsequent recession.
3.4. Pooling
Since the pioneering work of Bates and Granger(1969), it is well known that pooling several forecastscan yield a mean square forecast error (MSFE) lowerthan that of each of the individual forecasts; seeTimmermann (2005) for a comprehensive overviewand Clements and Hendry (2004) for possible explana-tions. Hence, rather than selecting a preferred forecast-ing model, it can be convenient to combine all theavailable forecasts, or at least some subsets.
Several pooling procedures are available. The threemost common methods in practice are linear combina-tion, with weights related to the MSFE of each forecast(see e.g. Granger & Ramanathan, 1984), median fore-cast selection, and predictive least squares, where asingle model is chosen, but the selection is recursivelyupdated at each forecasting round on the basis of pastforecasting performance. Stock andWatson (1999) andMarcellino (2004) presented a detailed study of therelative performance of these pooling methods, using alarge dataset of, respectively, US and Euro areamacroeconomic variables, and taking as basic fore-casts those produced by a range of linear and non-
Table 4Evaluation of forecast pooling
Combine Relative MSE
MSE-weighted
Predicting CCI_CB growth6 linear models (1 month) 0.7437⁎⁎⁎
6 linear models (6-month dynamic) 0.82446 linear models (6-month iterated) 0.6457⁎⁎⁎
Predicting NBER turning point4 linear and MS models (1 month) 1.24864 linear and MS models+probit (1 month) 1.09943 single index PROBIT (1 month) 1.15063 single index PROBIT+all (1 month) 1.11893 single index PROBIT (6 months) 1.28803 single index PROBIT+all (6 months) 1.2711
Note: The forecast sample is 1985:1–2004:12. The first estimation sampleahead), recursively updated. The lag length selection is using the BIC. MSEThe benchmark is the VAR for CCI CB and CLI CB. ⁎, ⁎⁎, and ⁎⁎⁎ indicatenull hypothesis of no significant difference in MSE or MAE with respect
linear models. In general, simple averaging with equalweights produces good results, more so for the US thanfor the Euro area.
Camacho and Perez-Quiros (2002) focused onpooling leading indicator models using regressionbased weights, as suggested by Granger and Rama-nathan (1984). Hence, the pooled forecast is obtainedas
and the weights, wi, are obtained as the estimatedcoefficients from the linear regression
yt ¼x1bytjt−1;1þx2bytjt−1;2þ: : :þxpbytjt−1;pþut ð19Þwhich is estimated over a training sample using theforecasts from the single models to be pooled,ybt|t−1,i,and the actual values of the target variable.
Camacho and Perez-Quiros (2002) evaluated therole of pooling, not only for GDP growth forecasts, butalso for turning point prediction. The pooled recessionprobability is obtained as
is 1977:4–1984:12 (for 1 step-ahead) or 1977:4–1984:6 (for 6 steps-and MAE are mean square and absolute forecast errors, respectively.significance at 10%, 5%, and 1% of the Diebold–Mariano test for theto the benchmark.
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obtained as the estimated parameters in the probitregression
ð21Þwhich is again estimated over a training sampleusing the recession probabilities from the singlemodels to be pooled, rbt|t−1,i, and the actualvalues of the recession indicator, rt.
12
To assess the role of pooling for forecasting thegrowth rate of the UK NMB CCI, we combine theforecasts from the six linear models in Section 3.1(i.e., AR, VAR, ECM and the three VARs withdisaggregated components of the CCI, or of the CLI,or of both), using either equal weights or the inverseof the MSEs obtained over the training sample1985:1–1988:12. The results are reported in theupper panel of Table 4. They indicate very clearly thatpooling works: the gains are large, over 30%, and arestatistically significant. Moreover, the simple aver-age works at least as well as the more sophisticatedweighting scheme.
For IP turning point prediction, see the middlepanel of Table 4, pooling linear and MS models cannotbeat the benchmark probit model, even when using thebetter performing equal weights for pooling, or addingthe probit model with the CLINMB index into theforecast combination as a regressor.
Finally, also in the case of probit forecasts for theUK IP turning points (lower panel of Table 4), a singlemodel performs better than the pooled forecast for bothone and six-month horizons, and equal weightsslightly outperforms MSE based weights for pooling.Marcellino (2006) reached similar conclusions for theUS.
4. Conclusions
In this paper, we have provided an overview of recentdevelopments in the methodology for the constructionof composite coincident and leading indicators.We havethen applied several methods for the construction andevaluation of CCIs and CLIs for the UK.
12 The pooling method described above was studied from atheoretical point of view by Li and Dorfman (1996) in a Bayesiancontext.
Regarding coincident indexes, factor based tech-niques are promising for building continuous CCIs.They can handle very large information sets, and, inthe more sophisticated versions, automatically lead/lag the component series. Moreover, they can takecointegration into account, and provide a unifiedframework for handling data problems, such asmissing observations or data revision (see e.g.Angelini, Henry, & Marcellino, 2006). However, inpractice, the results are not very different from thoseobtained from a simple average of the standardizedindex components.
Discrete CCIs, in the form of probabilities ofrecessions, can be obtained within the framework ofMarkov switching models. While the results for theUK are interesting, an accurate fine tuning of themodels is important to obtain reliable results.
Regarding leading indexes, the target can be acontinuous variable, such as a CCI, or a discretevariable, such as the turning points of a CCI. Differentmodels can be adopted to relate a set of leadingvariables to the target, e.g. linear or Markov switchingspecifications, and the leading variables could besummarized in a first step by means of a factor model.The results for the UK suggest that lagged values ofthe CCI contain useful information in addition to thatprovided by standard leading variables, and that probitmodels are better than linear or Markov switchingspecifications for predicting turning points, using anMSE criterion.
Another interesting empirical result is that forecastpooling seems to be quite useful in predicting futurevalues of a UK CCI, but much less so for its turningpoints, as is in line with previous results for the US.
The main implication of the findings in this paperfor the future of economic forecasting is that attentionshould be focussed more on the construction ofcomposite leading indexes than of coincident indexes.Moreover, the selection of the components of the indexis very important, since the best leading indicatorschange over time. Finally, the procedures for turningpoint forecasts should be refined, since most of theexisting methods do not yet produce systematicallysatisfying results. The many important improvementsin the construction and evaluation of compositeleading indexes in the recent past that are documentedand applied in this paper suggest that these additionalissues can also be addressed successfully.
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Acknowledgements
We are grateful to Ullrich Heileman, HermanStekler, two anonymous referees, and participants atthe Conference on “The Future of Forecasting”, held atthe University of Leipzig, for useful comments on thetopics discussed in this paper. We are also grateful tothe Conference Board for providing the data. The usualdisclaimers apply.
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Andrea Carriero is a Lecturer of Economics at Queen Mary,University of London. His research interests include time seriesanalysis and applied macroeconometrics. He has published in theJournal of Econometrics and in the Oxford Bulletin of Economicsand Statistics.
Massimiliano Marcellino is a Professor of Econometrics atBocconi University, the Deputy Director of IGIER, and a researchaffiliate of CEPR. He earned his PhD in 1996 from the EuropeanUniversity Institute. His research interests include time seriesanalysis, econometrics and applied macroeconomics. He is aDepartmental Editor for the Journal of Forecasting and he haspublished in several journals including the Journal of Econometrics,Journal of Applied Econometrics, Econometrics Journal, OxfordBulletin of Economics and Statistics, Journal of EconomicDynamics and Control, International Journal of Forecasting,Journal of Forecasting, Journal of Business and EconomicStatistics, Journal of Time Series Analysis, Empirical Economics,Economic Modeling, European Economic Review, MacroeconomicDynamics, Journal of Macroeconomics, and the Journal of PolicyModeling.
