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Economic literature concerning leading economic indicators
WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL PAULO PICCHETTI Fundação Getúlio Vargas – EESP/IBRE Abstract. The search for business cycles leading indicators in the econometrics literature has considered a large set of statistical methods. The main di¤erence among these methodologies can be considered the treatment of the evident non-stationarity of the data. While a very large class of models is based on some sort unit-root type stochastic trend, others try to incorporate a more formal probabilistic structure for regime shifts and yet another large class of models considers di¤erent …lters and decompositions aimed to reveal and anticipate major turns in economic activity. Data on the Brazilian economy is particularly characterized by relatively short time-series, while containing signi…cant structural changes, seriously weakening the performance of these classes of models. An alternative set of decompositions, based on Wavelets and Multi-Resolution Analysis [Priestley (1996), Percival and Walden (2000)] have been providing a very promising alternative to the analysis of time-series based on time-scales that expose their changing behavior across time, both individually and in terms of correlations with each other [Ramsey and Lampart (1998)]. Wavelet covariances between series are estimated on di¤erent time scales, thus allowing the whole sample to be retained in the analysis, even in the presence of structural changes. A leading indicator for Brazilian industrial activity is constructed upon these results. Another contribution of this paper will be to re-construct a set of survey data available on di¤erent frequencies to make it compatible with the series it will try to anticipate. Keywords: Business cycles, Brazilian industrial production, Leading indicators, Wavelets, Structural time-series models JEL: C32, C42, C53, E32, E47 1. Introduction Surveys on expectations of industrial activity are mainly motivated by the need to obtain leading indicators of activity cycles. How the di¤erent surveyed aspects relate to actual industrial production in the future is a very active topic of research, both at the theoretical and applied levels. Traditional methods for dynamical correlation between historical data on variables representing expectations and variables representing actual realizations that can be associated with these expectations provide results which are both highly relevant and interesting. Some shortcomings of this type of analysis are fundamentally linked to the nature of the non-stationarity of the available data. Speci…cally, with historical data highly characterized by structural breaks and outliers, …rst-di¤erencing methods on the time domain are usually incapable of inducing stationarity. Likewise, traditional methods for frequency domain decomposition and coherence analysis between this kinds of variables also produces results whose usefulness is considerably limited. 1 2 PAULO PICCHETTI In this context, Lucas (1981) stresses that the importance of looking at comovements between variables: "Technically, movements about trends in GDP in any country can be well described by a stochastically disturbed di¤ erence equation of very low order. These movements do not exhibit uniformity of either period or amplitude... Those regularities which are observed are in the co-movements among di¤ erent series. ... The central …nding, of course, is the similarity of all cycles with one another, once variation in duration was controlled for, in the sense that each cycle exhibits about the same pattern of co-movements as do the others." Harding and Pagan (2000), however, consider a fundamental issue concerning the implementation of this idea: "How exactly could Lucas conclude that there is no uniformity in temporal movements in output and yet be con…dent that there are uniform co-movements ? ... The academic literature has mostly identi…ed comovements with covariances, and then estimated the latter with a sample period that includes many cycles. Hence, it is assumed that the co-movements are the same across cycles." Wavelet decompositions can be employed to capture these co-movements between economic series at the …ner detail of di¤erent time-scales. Insofar as covariances are regularly estimated averaging several di¤erent frequencies, it seems highly desirable to isolate these associations. The objective of this paper is to decompose series on expectations based on di¤erent time-scales, and assess how each of these components relate dynamically the equivalent components in actual industrial production measures. This decomposition is based on wavelet coe¢ cients which have been widely used in a series of applications across di¤erent …elds, including economics. Section 3 below very quickly summarizes some of the main concepts needed to understand and interpret the results here, and provides additional references for details on aspects of both theory and implementation. Section 2 describes the data, Section 4 shows the results. In section 5 we build a leading indicator for industrial activity based on expectations from survey data. Section 6 discusses the implications for growth-cycle analysis, and Section 7 concludes. 2. Data Description The main variable for measuring industrial activity is the monthly industrial production index calculated by Instituto Brasileiro de Geogra…a e Estatística – PIM/IBGE. This series has been calculated since the mid-seventies, and has urdegone periodic methodological revisions. The last of these revisions established new weights at the …rm and sectoral levels, and was made compatible with the previous methodology going back to January 1991. Data is available on di¤erent regional and sectoral aggregations, but here we concentrate at the whole-industry level for all regions, i.e., the most aggregated version. Future research may contemplate …ner resolutions at both dimensions. Our analysis will decompose the time-scale relationships between this main variable and di¤erent indicators from survey data provided by Sondagem Conjuntural da Indústria de transformação –Sondagem/FGV, calculated by Fundação Getúlio Vargas in Brazil since the mid-sixties. Sondagem/FGV surveys the general performance related to the most relevant products at the …rm level. While some of the items on the questionnaire relate to …rm-speci…c measures (such as employment level and capacity utilization), others such as demand, output, stocks and prices are WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 3 taken at the speci…c product level. In order to make the sample periods compatible with PIM/IBGE, we will consider data from January 1991 until March 2008. More speci…cally, the variables in our analysis are: Global Demand Level Inventory Levels Capacity Utilization Expected Production Expected Employment Inventory Levels and Capacity Utilization are surveyed in relation to the time of the response, whereas all other variables are qualitative measures of three-month ahead expectations. Capacity Utilization is the only quantitavely measured variable, on a 0– 100 scale. All other variables are indexes representing the net result of qualitative questions, such as Inventory Levels above, equal, or below a desired number, Expected Production less, equal or above the current value, etc. Raw data at the …rm level are weighted according to each respondent …rm’ gross receipts, s the only exception being the Expected Employment variable, which is weighted according to each …rm’ number of employees. Data at the sectoral and regional s levels are accordingly weighted with respect to industry-wide and country-wide signi…cances to compose the aggregate measures which will be used in this study. Data for these variables from Sondagem/FGV is also available with …ner resolutions in terms of industrial sectors and regions, mostly in consistence with data for PIM/IBGE, warranting the kind of further research outlined above. 2.1. Frequency Compatibility. Data on Sondagem Industrial is gathered by FGV since 1966 , …rst on a quarterly basis, and more recently (since mid-2005) on a monthly basis. We reconstruct the series on a monthly basis for the whole sample period (beginning on january 1991) in order to make it compatible with monthly data from PIM/IBGE, the variable for which we seek a leading indicator. The methodology is to estimate the components of a structural model (Harvey(1989)), and treat the months between quarterly observations as missing values. The formulation of the structural model in state-space and the estimation of its components by the Kalman Filter easily allows the estimation of these missing values. The state-space representation of the structural model is yt t+1 t+1 = = = = = = t t t + + + t t t; + t; + t; t t t N ID(0; N ID(0; 2 2 ) ) 2 N ID(0; ) [s=2] t X j=1 jt jt j;t+1 j;t+1 cos sin j + j jt sin j + ! jt ; j ! jt ; ! jt N ID(0; 2 !) jt + jt cos + ! jt ; j = 1; : : : ; [s=2] where [s=2] is the largest integer of the seasonal frequency divided by two, and 2 j j = 1; : : : ; [s=2]. This choice of speci…cation for the seasonal component j = s ; is based on considerations by Durbin and Koopman (2001). t For one of the variables (NUCI) these model was estimated both in the univariate context and the bi-variate, where an alternative measurement of the same variable 4 PAULO PICCHETTI (NUCI-CNI) is available on a monthly basis for the whole sample period. The latter approach has the advantage of more e¢ cient estimation of the missing data, given that the state components of the structural model are usually correlated. Insofar as the results from both approaches were very similar, the univariate model was used to provide the complete monthly observations for all the series in Sondagem. 3. Building Blocks for Wavelet Decompositions Wavelet theory has been developing for a long time, and has relatively recently been consolidated in a single …eld of theoretical and applied research, with a fast growing literature (Percival and Walden(2000)). Economic applications of the method are still far behind traditional time-series and …ltering methods, but are growing very rapidly (Ramsey(2000,2002), Crowley(2007)). A good introduction to the current literature on wavelets and its applications to economics is Crowley(2007). Here we only attempt a rapid introduction, to put the methodology in context and to help the interpretation of the results. Wavelets transforms are analogous to Fourier transforms in the sense that the original series is projected on a set of basis functions, which are related to di¤erent frequencies. However, the main limitation to Fourier analysis is the requirement of a stationary time-series, which are unable to represent interesting economic timeseries exhibiting stochastic trends, structural breaks, outliers, and changing variance. Wavelets provide a new set of basis functions which are ‡ exible enough to represent these features commonly found in economic data, decomposing the original series across di¤erent time-scales. These time-scales are related to the inverse of di¤erent frequencies intervals. With data sampled at discrete points in time, we concentrate on the Discrete Wavelet Transform (DWT) methods. Initially, for a series fyt g of T = 2J observations (J being the number of time-scales represented), we can obtain a vector of wavelet coe¢ cients represented as W = [w1 ; w2 ; : : : ; wJ ; vJ ]0 where each wj is T =2j vector of coe¢ cients associated with variations within a scale of length j = 2j 1 , and vj is a T =2J vector of coe¢ cients associated with averages os a scale of length 2J = 2 J . These coe¢ cients are obtained by the expression w = Wy where W is a T T orthonormal matrix de…ning the DWT. Details of di¤erent algorithms for implementing this matrix can be found in Percival and Walden (2000). With these estimated coe¢ cients, the series yt has a multiresolution analysis (MRA) representation given by yt = SJ + 0 J X j=1 Dj;t ; t = 1; : : : ; T 0 where Dj = Wj wj ; j = 1; : : : ; J and SJ = VJ VJ de…ne the j-th level "wavelet detail" associated with variations in yt at scale j . While the individual wavelet coe¢ cients are normally referred to as "atoms", their sums over each time-scale are named "crystals", and summarize the behavior of the series at the corresponding level or time-scale. In the context of our monthly frequency dataset ranging from January 1991 through March 2008, we are able to estimate crystals for six di¤erente WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 5 time-scales. The monthly resolutions for the time-scales corresponding to crystals D1– are, respectively {[2-4], [4-8], [8-16], [16-32], [32-64], [64-128]}. Each of D6 these will be interpreted according to the usual analysis of time-series components in terms of seasonality and cycles. The smooth component SJ represents the longterm trend of the series which does not necessarily conforms to periodic behavior. Here, we utilize a variation of the DWT known as the Maximal Overlap Discrete Wavelet Transform (MODWT). Details for the algorithms employed can be found in Percival and Walden (2000). The motivation is summarized by: whereas sample size is restricted to T = 2J observations in DWT, it can have any number of observations in MODWT, in multiresolution analysis, the wavelet details can be perfectly aligned in time with the original series, which is not the case in DWT, and the MODWT wavelet variance estimator is asymptotically more e¢ cient compared to the estimator based on DWT. This last item is particularly important for the analysis conducted here since we use the estimated wavelet variances for calculating the wavelet cross-correlation between a pair of series. This cross-correlation in wavelet analysis is the analogue of the coherence in Fourier analsyis. Formally, the wavelet cross-correlation between a pair of series Yt = [y1;t ; y2;t ] for di¤erent leads/lags is de…ned as Y; ( j) = Y; ( j) 1( j ) 2( j ) where Y ( j ) is the wavelet covariance of between the series in Yt on time-scale j and the denominator represents the product of the wavelet standard-deviations of both series on the same time-scale. Again, details of the estimation for these statistics can be found in Percival and Walden (2000). In what follows, we use these results to base our analysis on the lag/lead relationships between variables of sondagem and the industrial production index on a scale-by-scale basis. 4. Results Decomposing the PIM/IBGE by the MODWT produces a set of wavelet coef…cients that can be better understood visually. One of the ways they are usually depicted can be seen on Figure 1. Figure 1 The absolute magnitude of the estimated coe¢ cients at each point in time, and at each of the six time-scales considered, is represented by a darker colour. therefore, 6 PAULO PICCHETTI the relative darkness of each decomposition relates to the relative importance of that level of time-scale at that point in time for explaining the observed variation in the series. The generalization of this type of analysis compared to a standard Fourier frequency-domain decomposition is clearly seen in this real-world example, since the contribution of di¤erent frequencies for the total variation of the series is not constant across time. Another yet more powerful visual tool is the evolution of the MRA elements compared to the actual observed series in the sample period, as seen on Figure 2. PIM 140 120 100 80 60 150 SJ0 1992 1994 1996 1998 2000 2002 2004 2006 2008 100 50 10 D1 D2 D3 D4 D5 D6 0 -10 10 0 -10 10 0 -10 5 0 -5 5 0 -5 5 0 -5 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure 2 Crystal D1 captures the random behavior on the short time-scale, analogous to the estimated residuals in standard time-series models. The seasonal pattern of the series is characterized by variations which are neither monotonic inside each year, nor constant across di¤erent years. Crystals D2 and D3 are associated with this important behavior of the analyzed series. Cristals D4-D6 are associated with longer time-scales variability, usually related to cycles. The relative estimated wavelet variances of these di¤erent time-scales can be seen in Figure 3. WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL P IM 7 10 2 W avelet V ariance 10 1 10 0 0 1 2 3 Level 4 5 6 Figure 3 For the industrial production index, the greatest wavelet variance is estimated at level 3 –in relation to the seasonal time-scale, while short-time scales variances are relatively bigger than the ones associated to the longer time-scales. Similar estimations are conducted to the survey data series, the graphical results of which can be seen on the appendix. In what follows, we focus on the wavelet crosscorrelations between the series in the survey data and the industrial production index. 4.1. Global Demand Level. Estimated crystals for global demand level are dynamically correlated to industrial production at all time-scales. Whereas it positively leads and lags at the J4 throuth J6 time-scales, at the main seasonal time-scale (J3) it positively lags and negatively leads. In other words, at the approximately half-year resolution, a larger output increases the percepetion of the global demand level, but this increased demand perception is associated with a smaller output at the seasonal frequency. Global D emand Lev el J1 1 0 -1 -6 1 0 -1 -6 0.5 0 -0.5 -6 1 0 -1 -6 1 0.5 0 -6 1 0.5 0 -6 -5 -4 -3 -2 -1 0 J2 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J3 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J4 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J5 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J6 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Figure 4 8 PAULO PICCHETTI This suggests that in this time-scale, a current increase in the perception of demand tends to anticipate production, leading to a smaller output …ve to six months ahead. 4.2. Inventory Levels. Inventories are only correlated to output at the longer time-scales, positively leading and lagging it. Inv entory Lev els J1 0.5 0 -0.5 -6 0.5 0 -0.5 -6 0.5 0 -0.5 -6 1 0 -1 -6 1 0.5 0 -6 1 0.5 0 -6 -5 -4 -3 -2 -1 0 J2 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J3 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J4 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J5 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J6 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Figure 5 Inventories below the optimal level (associated with increases in the values of these variables) lead to greater future output, but not as strongly as greater realized output are associated with inventories below the optimal level. This suggests an interesting asymmetry concerning the planning and management of inventories. 4.3. Capacity Utilization. At the longer time-scales, capacity utilization significantly and positively leads and lags industrial output, as expected. WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 9 C apac ity L ev el U tiliz ation J1 1 0 -1 -6 1 0 -1 -6 1 0 -1 -6 1 0 -1 -6 1 0.5 0 -6 1 0.5 0 -6 -5 -4 -3 -2 -1 0 J2 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J3 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J4 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J5 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J6 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Figure 6 At the J3 seasonal frequency, bigger levels of capacity utilization are associated with smaller output at future periods beginning around 4 months. This suggests an interesting pattern of changes in the intra-year allocation of production, related to decisions justifying anticipations in planned output. 4.4. Expected Employment. For the seasonal component J3, correlations are near perfect (close to 1) at zero lags, and close to -1 at the six-months lags and leads, indicating a perfect forecast for the peaks and troughs of the seasonal frequency between production and expected employment. For the cycles components J4 and J5, correlations are also close to one at the zero lead/lag, and present a symmetrical declining pattern towards the leads and lags directions. 10 PAULO PICCHETTI Ex pe c ted Employ ment J1 1 0 -1 -6 1 0 -1 -6 1 0 -1 -6 1 0 -1 -6 1 0.5 0 -6 0.5 0 -0.5 -6 -5 -4 -3 -2 -1 0 J2 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J3 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J4 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J5 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J6 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Figure 7 Expected employment can then be taken as leading and lagging industrial production at the J4 and J5 time-scales, and displaying zero correlation at the longer J6 time-scale at all leads and lags. The combined results suggest that employment being one of the relatively least volatile variables in the economy, adjustments in expected values are clearly related to predictable seasonal e¤ects, and to longer time-scales movements. 4.5. Expected Production. At the J4 time-scale of the cyclical components, PIM negatively leads expected production between 6 and 4 months, whereas PIM positively lags expected production from zero to 3 months. For the J5 and J6 timescales, PIM positively lags and leads expected production from zero to six months, more strongly to the J6 larger cycle. WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 11 Ex pec ted Produc tion J1 0.5 0 -0.5 -6 1 0 -1 -6 1 0 -1 -6 1 0 -1 -6 1 0 -1 -6 1 0.5 0 -6 -5 -4 -3 -2 -1 0 J2 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J3 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J4 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J5 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 J6 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Figure 8 As in the case of capacity utilization, these results suggest interesting patterns of intra-year allocation of output, isolating this e¤ect from the longer time-scales strong positive correlations. Of all the survey-based variables considered here, expected production is the natural candidate for anticipating actual output. Next section provides a methodology for implementing this idea in the context of the …ner time-scale resolution. 5. Constructing a Leading Indicator using Expected Production The survey measure of expected production is traditionally used as a leading indicator for actual future industrial production. Quantitatively, we can estimate future production using currently available information on both actual and survey measures through a dynamic linear model of the form: p X i=1 p X i=1 P IMt = i P IMt i + i EPt i + t The estimated parameters for a dynamic speci…cation chosen by the Schwarz Information Criteria, along with their associated standard errors (in parentheses) are shown in Table 1 below. 12 PAULO PICCHETTI PIMt PIMt PIMt PIMt 2 3 4 5 ExpProdt ExpProdt ExpProdt ExpProdt R Q-stat 2 1 2 5 6 Table 1 Estimated Coe¢ cients 0:285 (0:071) 0:188 (0:072) 0:213 (0:073) 0:253 (0:072) 0:203 (0:041) 0:096 (0:043) 0:076 (0:036) 0:072 (0:032) 0:93 0:795 The coe¢ cient of multiple linear determination shows a high value, in accordance with the range usually obtained in this class of dynamic linear models. The Portmanteau-test statistic for residual auto-correlation up to six lags shows no sign of dynamic speci…cation problems. Whereas this result represent dynamic relationships across all time-scales simultaneously, we attempt to assess potential gains in estimating particular dynamic relations for each of the estimated time-scale crystals, both for actual industrial production and the survey expected variable. Figure 9 compares the estimated crystals for industrial production and expected production for each of the timescales considered. 2. 5 0. 0 D6t_PIM D6t_EP 2 0 -2 2000 2005 20 0 D4t_PIM D4t_EP D5t_PIM D5t_EP -2.5 1995 5 0 -5 1995 2000 2005 -20 2000 2005 10 0 D3t_PIM D3t_EP 1995 1995 2000 2005 10 -10 D2t_PIM D2t_EP -10 2000 2005 1995 2000 D1t_PIM D1t_EP 1995 2005 WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 13 Figure 9 There is a clear dynamical association at each time-scale, which can be estimated by: p X i=1 p X i=1 P IM [T Sj ]t T Sj = i P IM [T Sj ]t i+ i EP [T Sj ]t i + t = SJ0; D1 D6: Particular dynamical speci…cations are chosen again on basis of optimal information criteria. The corresponding statistics are now presented for each particular timescale in table 2: Table 2 Estimated Coe¢ cients SJ0 PIMt PIMt PIMt PIMt PIMt PIMt 1 D1 D2 D3 D4 D5 D6 2 3 4 5 6 ExpProdt ExpProdt ExpProdt ExpProdt ExpProdt ExpProdt R2 Q-stat 1 4:876 (0:054) 9:893 (0:259) 10:806 (0:514) 6:819 (0:529) 2:411 (0:282) 0:381 (0:061) 0:242 (0:119) 2 3 4 5 6 0:772 (0:344) 1:141 (0:266) 0:706 (0:116) 0:168 (0:022) 0:99 0:814 2:442 (0:058) 3:564 (0:136) 3:659 (0:190) 2:769 (0:192) 1:372 (0:138) 0:369 (0:059) 0:521 (0:074) 0:946 (0:105) 1:155 (0:114) 1:075 (0:100) 0:662 (0:067) 0:258 (0:031) 0:97 1:013 1:531 (0:066) 2:550 (0:113) 2:179 (0:165) 1:818 (0:160) 0:745 (0:107) 0:256 (0:056) 2:869 (0:072) 3:698 (0:223) 2:446 (0:348) 0:860 (0:343) 4:192 (0:068) 7:609 (0:280) 7:729 (0:501) 4:757 (0:491) 1:743 (0:263) 0:309 (0:061) 4:471 (0:068) 8:228 (0:307) 7:846 (0:583) 3:928 (0:589) 0:867 (0:316) 0:236 (0:090) 0:169 (0:028) 0:389 (0:097) 0:397 (0:168) 0:507 (0:141) 0:254 (0:090) 0:104 (0:053) 0:137 (0:026) 0:98 0:981 0:061 (0:030) 0:99 0:813 0:99 1:957 0:627 (0:151 0:149 (0:030) 0:99 1:835 5:292 (0:056) 11:860 (0:268) 14:475 (0:524) 10:195 (0:527) 3:943 (0:271) 0:655 (0:057) 1:471 (0:136) 2:833 (0:318) 2:895 (0:424) 1:646 (0:337) 0:485 (0:150) 0:99 0:875 Once again, the auto-correlation tests for the residuals of each equation show no signs of dynamic misspeci…cations, while the linear association measure shows a near-perfect …t for each of the equations. The leading indicator is now constructed summing all the individual estimated components of the MRA, and then compared to the …tted values of the previous 14 PAULO PICCHETTI model (averaging the time-scales). Figure 10 compares the actual industrial production index to both series of …tted values throughout the sample period, as well as the estimated residuals for both models. 12 5 fitted_average fitted_MRA PIM 10 0 75 1995 10 5 0 -5 -10 1995 2000 2005 residuals_average residuals_MRA 2000 2005 Figure 10 Clearly, the magnitude of the residuals from the model averaging the time-scales is signi…cantly and consistently bigger than the residuals from the MRA approach. As a simple exercise, the RMSE for one-step ahead forecasts during the last 12 months of the sample period is 3.71 for the previous model, compared with 0.58 for the prediction based on individual time-scales. In conclusion, there are substantial gains in precision by building a leading indicator based on the MRA approach. 6. Implications for Cycles Analysis The Brazilian economy underwent a series of signi…cant shocks and structural modi…cations during the past two decades, which mainly coincides with the sample period in our analysis. The characterization of standard business-cycles in this context with the available data is troublesome (Chauvet (2002)). However, shifting the attention to the annual growth rate of industrial production, a growth-cycle regularity seems to apply, as can be seen in Figure 11. WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 15 10 .0 PIM 1 2mon ths(year_ t)/1 2mon ths(year_t-1) D5 D6 2 7.5 1 5.0 0 2.5 0.0 -1 -2 .5 -2 19 95 20 00 20 05 Figure 11 Measured on the left scale is the percent variation of the twelve-month accumulated PIM index over the previous twelve-month period. On the right scale, we measure the estimated D5 and D6 wavelet crystals from the MODWT decomposition. The combined e¤ect of the D5 and D6 crystals appears to match the major turning points of the accumulated industrial production over the sample period. While the D5 crystal overall dynamics is very similar to the industrial production growth cycle, the D6 crystal – related to a longer time-scale – seems to reinforce it. Given the high dynamic correlation of the estimated crystals for the survey-based expected production and actual industrial output pointed out in the previous section, it appears that the wavelet decomposition o¤ers good insights for predicting growthcycle turning points. 7. Conclusions and Further Research Correlations at di¤erent time-scales between survey-based expectations and industrial production show a set of interesting dynamic relationships between these variables. The expected production variable was used to construct a leading indicator for actual output, and the information at the …ner resolution of di¤erent time-scales provides encouraging improvements over the results obtained through the usual methodologies using information corresponding only to time variation. These …ndings provide a considerable incentive to extend this kind of analysis along di¤erent dimensions. First, the same kind of leading indicator methodology can be tried to relate survey-based expectations to actual future realizations of other important variables such as employment. Second, the survey data from Sondagem/FGV employed here was aggregated accross di¤erent sectors providing industry-wide measures, but these variables are also available for di¤erent sectors, strati…ed consistently with the industrial output measured by PIM/IBGE. Therefore, the usual heterogeneity considerations motivating the analysis disagreggated by sectors apply to the methodologies considered here, with potential bene…ts for 16 PAULO PICCHETTI interpretation of the relationships, as well as construction of leading indicators. Finally, on the methodology side, the wavelet transform comprises nowadays a large amount of alternative speci…cations and estimation algorithms, and so additional research is by all means necessary to gain an understanding on the relative merits of these alternatives. 8. References Abberger, K. (2007): IFO Survey on Employment Plans: Sectoral Evaluation, in Handbook of Survey-Based Business Cycle Analysis. Chauvet, M. (2002): The Brazilian Business Cycle and Growth Cycle, Revista Brasileira de Economia. Crowley, P. (2007): A Guide to Wavelets for Economists, Journal of Economic Surveys, Vol. 21, No.2. Durbin, J. and Koopman, S. J. (2001): Time Series Analysis by State Space Methods, Oxford University Press. 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Appendix – MODWT decompositions of the Survey Data Gl bal D o emand Level 150 100 50 0 120 S J0 1992 1994 1996 1998 2000 2002 2004 2006 2008 100 80 10 D1 D2 D3 D4 D5 D6 0 - 10 10 0 - 10 20 0 - 20 20 0 - 20 20 0 - 20 20 0 - 20 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure A1 I nvent ory Level s 120 100 80 60 100 S J0 1992 1994 1996 1998 2000 2002 2004 2006 2008 90 80 5 D1 D2 D3 D4 D5 D6 0 -5 10 0 - 10 20 0 - 20 20 0 - 20 10 0 - 10 10 0 - 10 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure A2 18 PAULO PICCHETTI Level of C t y U i at i n apaci tl o z 90 80 70 60 90 S J0 1992 1994 1996 1998 2000 2002 2004 2006 2008 80 70 2 D1 D2 D3 D4 D5 D6 0 -2 2 0 -2 5 0 -5 5 0 -5 2 0 -2 5 0 -5 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure A3 C U . Level - C r uct on Mat er as ap. t l i onst i il 90 80 70 60 50 100 S J0 1992 1994 1996 1998 2000 2002 2004 2006 2008 80 60 5 D1 D2 D3 D4 D5 D6 0 -5 5 0 -5 5 0 -5 5 0 -5 5 0 -5 5 0 -5 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure A4 E xpect ed E oyment mpl 150 100 50 120 S J0 1992 1994 1996 1998 2000 2002 2004 2006 2008 100 80 10 D1 D2 D3 D4 D5 D6 0 - 10 10 0 - 10 20 0 - 20 10 0 - 10 10 0 - 10 5 0 -5 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure A4 WAVELET-BASED LEADING INDICATORS OF INDUSTRIAL ACTIVITY IN BRAZIL 19 E xpect ed P oduct i n r o 200 150 100 50 125 S J0 1992 1994 1996 1998 2000 2002 2004 2006 2008 120 115 20 D1 D2 D3 D4 D5 D6 0 - 20 50 0 - 50 50 0 - 50 10 0 - 10 5 0 -5 5 0 -5 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure A5