AN APPLICATION OF THE CHOLESKIZED MULTIVARIATE DISTRIBUTIONS FOR CONSTRUCTING INFLATION FAN CHARTS

Transcription

1 AN APPLICATION OF THE CHOLESKIZED MULTIVARIATE DISTRIBUTIONS FOR CONSTRUCTING INFLATION FAN CHARTS Wojciech Charemza* ), Piotr Jelonek* ) an Svetlana Makarova** ) *) University of Leicester, UK **) University College Lonon, UK June 2010 Preliminary an incomplete. In this raft most, but not all, computation results have been so far replicate, which means that in subsequent versions of the paper some numerical results might slightly change. ACKNOWLEDGEMENT We are inebte to Mikhail Lifshits for his helpful remarks an iscussions which precee the paper an to Imran Shah for his help with ata collection. Neeless to say, all the mistakes here are ours.

2 ABSTRACT The paper postulates applying the multivariate non-normal an skewe istributions with epenent components to construct probabilistic forecasts of inflation, usually presente in a form of fan charts. The rationale of this came from the empirical analysis of naïve an econometric forecasts annual inflation (measure monthly) for 28 OECD countries from January 1957 (or later, for some countries) to September Naïve forecasts are obtaine by simple extrapolation of rates of growth, while econometric forecasts are given by the stationary bilinear ARMA (BARMA) moels. The forecasts horizons are from one to 8 months. It has been foun that, for the naïve forecasts, the empirical istributions of forecasts errors (which are the empirical base for constructing the fan charts) are in most cases stably istribute. For the econometric forecasts the picture is less clear, but in most cases the tempere stable istribution exhibits the best fit. It has also been foun that istributions of multi-step forecasts errors are mutually epenent. Hence, a logical step forwar seems to be constructing fan-charts by simulation of forecasts outcomes from istributions obtaine by Choleskization of the tempere or tempere-stable istributions, which parameters have been evaluate using past forecast errors. This approach seems to be particularly attractive for evaluation of probabilities of the appearance of turning points an probabilities of runs of inflation of certain length which might appear in future. Expecte lengths of continuation of inflation tenencies (to rise or to fall) have been compute for all 28 OECD countries. Empirical illustration containing the fanchart an etaile evaluation of probabilities of turning points occurrences is given for Japan. The paper inclues two appenices; one on ranom number generation from the Choleskize tempere stable istributions an another on the gri-search comparison of fitte stable an tempere stable istributions.

3 1. INTRODUCTION In the literature evote to the theoretical an practical aspects of probabilistic forecasting of inflation, as well as some other macroeconomic inicators one might ientify two istinctive tenencies. The first of these tenencies emphasises the ynamic nature of inflation an evelops the forecast as some sort of extrapolation of the probabilistic an ynamic features of the moel, as observe in the past, imposing in forecasting the parameters estimate from a ynamic moel (see e.g. Kemp, 1991, 1999, Chauhuri et al., 2008, Cogley et al., 2004). A more pragmatic an popular tenency is relate more loosely to the ynamic nature of the process an apply, somewhat arbitrarily, probability istributions which emulate uncertainty relate to further evolution of the forecaste phenomena. Parameters of such istributions are often either calibrate or set with the help of poole forecasts, experts opinions, past forecasts errors etc. This methoology is usually preferre by professional forecasters an central banks practitioners in situation when forecasting is to be repeate frequently an spee of preparation of forecasts is at a premium (for a escription of traitional methoology see Wallis, 1999, an for some further evelopment Giorani an Söerlin, 2003). There are also papers in which these two approaches are combine (e.g. Diebol et al., 1999; this paper, however, is not ealing with inflation forecasting). Applications of first of these methoologies methoology is, so far, limite to rather simple linear autoregressive moels. The secon, pragmatic, approach requires accurate external information as this affects the uncertainly impose to the forecasts. Common rawbacks of these two streams are that they rely, so far, on rather unrealistic assumptions regaring the types of the istributions use. The unerlying istributions are either normal, of skewenormal, or resulting from conitional-autoregressive heterosceastic processes. It is often assume, especially by the pragmatists, that the istributions of forecasts mae for ifferent horizons are mutually inepenent. In this paper, which is evelope primarily from finings of the pragmatic school, we are trying to relax the constraints impose by unrealistic assumptions of normality an inepenence. We start with the wiely use heuristic ientification of inflation istribution with the istribution of its past point forecast errors. However, in relation to earlier papers, the similarity stops here. It is known (an confirme further in Section 3 of this paper) that both inflation itself an inflation forecasts errors are not normally istribute. Episoes of high inflation (usually followe by rapi isinflation) often in practice create the heavy tails effect an skewness. Moreover, as errors of forecasts mae for ifferent horizons are rarely inepenent, the istributions of inflation forecasts shoul be epenent as well. Nevertheless, attempts to moel inflation uner non-normality an mutual epenence have been limite (for theoretical justification of inflationary asymmetry see e.g. Ball an Mankiw, 1994, 1995, Corrao an Holly, 2003; some limite empirical results are given in Charemza et. al, 2005a an Pou an Dabus, 2008). Uner linear preiction an homogeneity, forecast errors shoul have a similar istribution to that of inflation, subject to a shift. Uner nonlinear or ynamic preiction schemes, istributions of forecasts errors might more complex. This shoul be reflecte in the way the fan charts are constructe. With this in min, we have evelope the concept of a simple pragmatic inflation forecasting with the use of the so-calle Choleskize istributions, that is the istributions for which the epenence has been impose by a simple positively efine matrix. We have Choleskize the stable an tempere stable, possibly skewe, istributions, as they reasonably escribe the empirical istributions of the inflation forecast errors. In Section 2 of the paper these

4 2 istributions are briefly escribe. In Section 3 they are applie in the analysis of forecast errors obtaine for monthly time series of annual inflation for 28 OECD countries. The general conclusion here is that stable an tempere stable istributions fit much better to the empirical frequencies of forecast errors than the normal an skewe normal ones. In general, the stable istributions escribe well the naïve errors, while the tempere stable fits better to the forecast errors obtaine from more sophisticate moels (here we have use bilinear autoregressive-moving average processes). In Section 4 we analyse the OECD ata more etaily, by computing expecte lengths of continuations of inflationary runs (that is, the consecutive perios of increasing an ecreasing inflation). The results show that, allowing for epenence in the multivariate istribution from which the fan charts are generate, leas to more realistic results regaring the evaluation of tenencies an estimation of turning points. Further in this section this approach is illustrate by the results for Japan. 2. SOME RELEVANT SKEWED AND HEAVY-TAILED DISTRIBUTIONS We have introuce here some istributions that will be further use in constructing fan charts. STABLE DISTRIBUTIONS Stable istributions (calle also the α-stable istributions) constitute a popular family of istributions, renowne for their capacity to reprouce fat tails an skewness. Since their introuction to finance in the 1960 s they have been frequently use in applie finance an financial engineering. Below in this section the main basic efinitions an properties of stable istribution are summarize. More etails can be foun in, e.g. Samoronitsky an Taqqu (2000, chapters 1 an 2). In the general (multivariate) case the stable istribution may be efine as following: Definition 1: Stable ranom vector. A ranom vector X, X R, has a imensional stable istribution if, for any, n 2 there is a positive number B n an a real number D n such that: 1 2 where X1, X 2,, X n are inepenent copies of X an X X X B X D, (1) n n n " " enotes equality in istribution. Such vectors are often referre to as infinitely ivisible. It is possible to show that a ranom 1/ vector X is stable if an only of the constant Bn in (1) is Bn n, where (0,2], which allows to attribute inex to X an to refer to X as to be -stable an to refer to as to inex of stability. If Dn 0 in (1) the ranom variable X is calle strictly stable. Stable istribution becomes Gaussian when 2. For most values of the probability ensity function (pf) an cumulative istribution function (cf) are not known in close form even in the univariate case (see e.g. Zolotarev, 1986). The corresponing technique of using stable ranom variables relies, to great extent, on the characteristic functions an spectral measures. In the univariate case the characteristic function of -stable variable X has the form: exp (1 i (sgn )tan ) i, if 1, 2 E exp( i X ) (2) 2 exp (1 i (sgn )ln ) i, if 1.

5 3 where: inex of stability (0, 2], scale parameter 0, skewness [ 1,1] an the 1 location parameter. The usual notation for univariate -stable istribution, also applie here, is S (,, ). If 1 then the shift parameter equals to the mean of the istribution (see Samoronitsky an Taqqu, 2000, chapter 1, for iscussion). In the multiimensional case, when > 1, the parameterisation of stable istribution is not as straightforwar as in the one-imensional case an is base on the Lévy-Khintchine (see e.g. Sato, 1999, chapter 2) or the spherical (see (3) below) representations. However in case of univariate istribution, the parameterization S (,, ) introuce above is convenient an the notation X ~ S (,, ) is use. In the -imensional case we use notation S (,, ), where i, i 1 i i 1 i i 1 i i 1 for component-wise stable vector (with the possibly ifferent inex of stability) with the i th component istribute as S (,, ) i i i i. Consiering the problems tackle in this paper, the relevant properties of a primary interest of stable ranom variables may be summarize as following: Statement 1 (properties of stable istributions, see Samoronitsky an Taqqu, 2000, chapters 1 an 2, for proofs an iscussion): 1. Shift. Let X is -stable vector in with the same inex of stability. an b. Then X + b is also -stable 2. Linearity: linear combination of -stable vectors with the same inex of stability is -stable. 3. Let X S (,0, ) with 2. Then there exist two i.i.. ranom variables Y S (1,0, ), i = 1, 2, an constants A, B an C which epen on α an β i that X A Y B Y C., 1, 2, 4. Absolute moments. Let the univariate ranom variable X is stable with 2. Then: p X for any 0 p, X p for any p. Note, that 1) for 2, when stable istribution becomes Gaussian, the property 4 oes not hol any more as all moments exist; 2) in case of 1 even the first moment oes not exist. In the multiimensional case the linear combination of marginal components of a - imensional stable ranom vectors is also stable (this follows from Statement 1). However the reverse statement in more complicate an the corresponing results are summarize in the following: Statement 2 (see Samoronitsky an Taqqu, 2000, chapter 2, for proofs an iscussion): 1. Let X X,..., 1 X is (strictly) -stable in, then all linear combinations bx k k are also (strictly) stable with the same inex of stability. k 1

6 4 2. If all linear combinations bx k k are stable with inex of stability 1 k 1 X X,..., 1 X is stable with the same inex of stability. 3. If all linear combinations then X X X, then bx k k are strictly stable with inex of stability k 1,..., is also strictly stable with the same inex of stability. 1 When 2, the ranom -stable vector becomes Gaussian an its epenence structure is ientifie by the covariance matrix. However, for 2 secon moments no longer exist an thus this concept is not applicable. Two popular ways to circumvent this obstacle is to introuce the notions of either covariation, for 1 2 or coifference, efine for 0 2 (see Samoronitsky an Taqqu, 2000, chapter 2). For 2 both are equivalent to covariance. TEMPERED STABLE DISTRIBUTIONS Tempere stable istribution was introuce, in its most general form, in physics by Koponen (1995) an in finance by Boyarchenko an Levenorskiy (2000) as a class of infinitely ivisible istributions that combines, to some extent, properties of stable an Gaussian istributions. Since then this approach have been moifie in various ways an use extensively in finance. The general approach for tempering, which is common to all approaches, is base on tilting the heavy tails of probability ensity function of stable istribution in such a way that allows to obtain the property of having all moments finite an, at the same time, to keep some esirable properties of stable istributions (such as infinite ivisibility). Those properties mae the tempere stable family an increasingly popular tool for moelling pricing erivatives in finance. A practical approach which is use in this paper has been suggeste by Rosiński (2007); for further evelopment see Imai an Kawai (2010) 1. In orer to introuce tempere stable istribution on Rosiński s sense the spherical representation of stable istribution is use. It is known (see e.g. Sato, 1999, chapter 3) that -stable istribution, (0,2) allows the spherical parameterization of the form: M r u r r u, (3) (, ) 1 ( ) 0 where M 0 is a measure in that correspons to the stable istribution, is inex of 1 stability, an is a measure on a unit sphere S in, calle spectral measure (see Samoronitsky an Taqqu, 2000, p. 66). The tempere stable istribution in Rosiński s sense allows the parameterization that is similar to spherical parameterization of -stable istribution given above. The formal efinition is the following: Definition 2 (see Rosiński, 2007): A ranom variable X in is sai to be a tempere stable if the corresponing measure M in R in polar coorinates has a form M r u r q r u r u, (4) (, ) 1 (, ) ( ) q 1 Rosiński s concept of tempere stable istributions, albeit very wie, oes not cover all cases regare as tempere stable in the literature. However, it covers the case consiere in this paper.

7 5 where q for each 1 : (0, ) S (0, ) u is a Borel function such that q(., u ) is completely monotone 1 S, q(0, u) 1 an q(, u) 0. Rosiński (2007) prove that the tempere stable istributions are infinitely ivisible, the corresponing spectral measure is unique an all moments (incluing some exponential moments) are finite. In a one-imensional case the simple tempering might be ientifie with the parameter 0 by having q( r, u) exp( r) in (4) above. For simulation purposes it is convenient to refer to this parameter instea of function q. The technical etails of this might be foun in Rosiński (2007), Cohen an Rosiński (2007) an Riout (2009). Further on, in the univariate case, we use notation T or T (,,, ) which is similar to the traitional notation of univariate stable istribution an convenient in ientifying parameters of simulate istributions. Similarly to the case of the component-wise stable istribution (see above) the notation T (,,, ) for component-wise tempere stable case is use. The following linear properties of tempere istribution, which will be neee further, is originally erive by Rosiński (2007). Statement 3 (Rosiński, 2007). Let X, X1, X 2be ranom vectors in a linear map. Then: 1) if X i ~ T, i = 1, 2, are inepenent, then X ~ 1 X T 2, 2) if X ~ T, then V ( X ) ~ T. k an let V : be CHOLESKIZED MULTIVARIATE STABLE AND TEMPERED STABLE DISTRIBUTIONS The practical problem with the multivariate istributions efine above is that the epenence between components of the multivariate ranom vector is not easy to figure out from its spectral representation. However, some knowlege of this epenence is crucial for moeling the istributions of forecast errors. A practical solution coul be obtaine by assuming that the interepenence within the multivariate stable an tempere stable istributions can be approximate by a positively efine matrix Σ. Conitions an constraints of such approximation are iscusse at length by Cohen an Rosiński (2007). Let X have a multivariate istribution. Uner Choleskize istribution of X we unerstan a istribution of multivariate ranom variable Z CX where C is a Cholesky upper triangular matrix, such that CC, ' where is a positively efine matrix. Further in the paper we will refer to X as the initial vector from which the Choleskization has been obtaine. Moreover, as a fan chart of inflation forecasts is constructe with the use of a multivariate istribution with each marginal istribution representing forecast for ifferent horizon, it might be unrealistic to assume that forecasts for ifferent forecasts horizons are constraine by the same inex of stability α. In fact the empirical estimates shown in Section 3 further on suggest that inex of stability might be ifferent there. With this in min we apply a somewhat more vaguely efine concept of multivariate Cholskize istributions, where the initial istributions might have ifferent inices of stability. Eviently this class of multivariate istributions is not α-stable (tempere stable) any more. From the theoretical point of view little is known about properties of such kin of ranom vectors so we applie an empirical approach of moeling such kin of istributions as a first step to this irection.

8 6 The propose application of tempere stable istributions for the analysis of inflation an inflationary forecast errors is markely ifferent than that in finance. In finance, when the interest is in moeling the tails of the istributions of pricing erivatives, usually multivariate tempere stable istributions are consiere, often for the positively efine values of the ranom variables. In our case, for the analysis of istributions forecasts errors, which can be either positive or negative, a larger class has to be use an the eviations from normality, expresse by α, vary with the change of forecast horizons. If one allows for the parameter α to vary, the aitivity an linearity properties of the istributions are lost, which poses aitional theoretical an practical problems. Although the theoretical properties of the Choleskize stable an tempere stable istributions are not yet explicitly known, an important avantage of the Choleskize istributions to moeling inflationary forecast errors is an ease of application. In orer to simulate ranom variables with the Choleskize istributions it is enough to know the parameters of the initial univariate istributions an the Gaussian correlation matrix. After that, straightforwar simulation algorithms for generating ranom numbers form univariate istributions an a Cholesky ecomposition can be applie (for etails see Appenix A). A possible isavantage is in the lack of precise knowlege about the joint istribution an possible violation of the Cohen an Rosińki (2007) conitions, which are ifficult to evaluate. Also, the ensity an characteristic functions are not known in close form. Hence, in practice, stochastic simulation seems to be the best way to apply these istributions. Figures 1-9 below shows the simulate istributions of the T 1.5 (2;,0,1) an S (,0,1) 1.5 ranom variables. It is compare with the simulate skewe normal istributions A(,0,1) (see e.g. Azzalini, 1985). 2 Figures 1-3 show the symmetric istributions with the parameter β = 0 (in this case the Azzalini s istribution becomes stanar normal) an Figures 4-6 show asymmetric istributions with β = Figures 7-9 present the two-imensional histograms erive from Choleskize two-imensional stable S (,0, ) an tempere stable T (,,0, ) istributions with (2, 2), (1.5,1.0), (0.75, 0.75), an the covariance matrix { ij }, where an Analogous parameters have been use for simulation of Azzalini an Valle (1996) bivariate istribution, enote as MA(,0, ). For construction of each figure, 250,000 replications have been mae. At the first sight, the simulate istributions of the stable an tempere stable istributions o not look interesting for a macroeconomist. They are heavily concentrate in a relatively small interval, an the concentration of the stable istribution is greater than that of the tempere stable. On the other han the Azzalini s univariate an bivariate istributions look more natural an appears to be more appropriate for the approximation of econometric forecast errors use for construction fan charts. However, the empirical results shown in the next section prove otherwise. 2 In practice fan charts are often constructe using ifferent forms of skewe normal istributions, e.g. the Bank of Englan uses the two-peace normal istribution (see Wallis, 1999). 3 As in skewe normal istribution β is not limite to the [-1,1] interval, there is no exact comparison with β s of the stable an tempere stable istributions.

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10 8 3. FORECAST ERRORS AND THEIR DISTRIBUTIONS OF THE OECD INFLATION In this section we present the results of the istribution analysis of forecast errors mae while forecasting annual inflation (efine as the inflation in a given month in relation to the corresponing month of the previous year) using monthly ata for the OECD countries. From the IMF Financial Statistics, through the Datastream, we have obtaine ata for 28 out of 30 OECD countries: Austria, Belgium, Canaa, Czech Republic, Denmark, Finlan, France, Germany, Greece, Hungary, Icelan, Italy, Japan, Korea, Luxembourg, Mexico, Netherlans, Norway, Polan, Portugal, Slovak Republic, Spain, Sween, Switzerlan, Turkey, Unite Kingom an Unite States. Data for Australia an New Zealan have been iscare as incomplete. All the series en with September 2009 except for Italy, Japan an Mexico, which ens a month earlier. The starting perios vary; the longest series is that for Norway, from January 1957 an the shortest is for Irelan, which starts at January Initially the analysis of naïve one- an two-step ahea forecast errors have been performe, where the forecasts have been obtaine by using the observe ata on inflation as a preictor (the martingale preiction). We starte with the evaluation of parameters of the istributions iscusse in Section 2 above an the analysis of their gooness of fit to the ata. Estimation of parameters of stable an tempere stable istributions is cumbersome. For stable istribution some algorithms are known (McCulloch, 1986, Nolan, 2001), but they are limite to a rather narrow range of parameters. Moreover, even if the estimation of these parameters by a maximum likelihoo metho is possible, a lack of possibility to use ensity functions for irect computations of gooness of fit tests makes them unattractive. For the tempere stable istributions only very general an not irectly applicable results are known (Sztonyk, 2010). Hence, we have ecie to restore to the minimum χ 2 metho on simulate ensities. In current context, it has two avantages. It allows irectly for a comparison of the gooness of fit across the istributions, as the one for which the minimum χ 2 value is the smallest, exhibits better fit. Seconly, the χ 2 gooness of fit measure is often use for the comparison inflation fan charts an is regare as being superior to alternative measures in practical fan chart type applications (see Wallis 2003). Similar approach has been use for estimation of kinetic rates in biology, except that the optimisation criterion statistic was the maximum likelihoo rather than χ 2 (Tian et. al, 2007). However, it is known that the χ 2 statistic is, in most cases, equivalent to the maximum likelihoo an, in a number of practical applications, it performs better (see Berkson, 1980). A istinct isavantage of the χ 2 base best fit analysis is its epenence on the way the intervals for the ensities are constructe, which is always arbitrary. We have trie to overcome this problem by aopting an algorithm which sets the cells with a minimum esirable number of observations (see Appenix B for etails). Moreover, a large number of observations we use (over 10,000) makes the problem of inaequate cells selection less important. Figures show the univariate istributions of inflation an the naïve forecast errors an also the bivariate istribution of the one an two steps errors. Tables 1-3 show the numerical results. In the parentheses below the parameters the Monte-Carlo stanar errors are given, that is the errors which characterise ispersion in the parameters between results obtaine in particular simulation runs (see Appenix B).

11 9 Table 1:Gooness of fit results, level of inflation, No. of obs. 14,258 α-stable (29.21) (0.025) (0.050) Tempere stable (217.7) (0.071) (0.050) (0.764) Skewe normal (115.4) (0.018) Stanar normal 2957 ()

12 10 Table 2: Gooness of fit results, one-step ahea naïve forecasts errors No. of obs. 14,230 α-stable (7.032) (0.033) (0.061) Tempere stable (104.50) (0.050) (0.0532) (1.041) Skewe normal (82.857) (0.0056) Stanar normal () Table 3: Gooness of fit results, two-steps ahea naïve forecasts errors No. of obs. 14,202 α-stable (6.058) (0.0257) (0.045) Tempere stable (126.43) (0.038) (0.0503) (0.289) Skewe normal (93.38) (0.0051) Stanar normal () Figure 10 inicates that the istribution of OECD inflation is strongly asymmetric with a very long right tail. This is confirme by the estimation results (Table 1), where the coefficients expressing skewness for all compare methos are positive. However, the univariate istributions of one an two-steps ahea naïve forecast errors are practically symmetric, albeit exhibiting heavy tails, with the right tail being heavier that the left one (note the nonlinear scale on the horizontal axis for all graphs). The non-normality of the istributions have been confirme by the minimum χ 2 statistics, which inicates that the best fit has been achieve for the α-stable istribution, followe by that of the tempere stable istribution. The fit of both

13 11 skewe normal an symmetric normal istributions to the inflation ata an to the naïve forecasts errors is much worse. Next we have ecie to apply a more sophisticate forecasting technique. For each country we have estimate the bilinear autoregressive-moving average moel BARMA(p,r,m,h) in the form: p r m k y a y c b y, (5) h, t h, j h, t j h, j h, t j h, l1l 2 h, t l1 h, t l2 j 1 j 0 l1 1 l2 1 where y h,t is inflation in each of the OECD countries, h = 1,2, 28, a h, j, c h, j, b h, l12 l are the parameters an h,t is white noise. Stationary moels have been introuce by Sabba Rao an Gabr (1974) an evelope further by Granger an Anersen (1978). Further results can be foun in Sabba Rao (1981), Kim an Basava (1990), Liu (1990), Tong (1990), Grahn (1995), Brunner an Hess (1995) an Terik, (1999). Economic applications can be foun in Peel an Davison (1998) an Charemza et al. (2005b). Following the results of Romero-Ávilia an Usabiaga (2009), we have assume stationarity or near stationarity of the series. There is no structural breaks ummies in the moel, as the eventual structural breaks shoul be capture by the bilinear part of (5). The estimation metho is the maximum likelihoo Kalman filter metho applie for the steay-state transformation of (5). The selection of lags has been performe through the general to specific approach, so that a congruent moel is erive. Distributions of forecasts errors for up to 8 steps ahea have been obtaine through cutting the sample for each country from top by 20, estimating the moel, forecasting, saving the ex-post forecast errors, separately for each forecast horizon, aing the next observation, reestimating the moel, forecasting, saving the ex-post forecast errors, etc., until the en of the sample is reache. As the result, for each country we have 12 forecasts errors for the horizon of 8, 13 forecasts errors for the horizon of 7, etc. Figures show the univariate istributions of the forecast errors for the forecasts mae from one to eight steps ahea for all analyse OECD countries.

14 12 The figures above inicate that the istributions of forecasts errors are usually concentrate aroun the highest moe an the bimoality effects are presumably cause by the unfortunate ivision into subinterval (the ivision into subintervals applie here has been appropriate for the estimation, but on the graphs it sometimes gives an effect of bimoality). Most of the istributions are clearly asymmetric, where the asymmetry can be either positive (for the forecast errors for 7 an 8 steps ahea), or negative (for the forecasts errors for 1 an 6 steps ahea). Figures show the bivariate istributions of the one-step forecasts an forecasts mae respectively for 2,3,,7 steps ahea.

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16 14 The figures above clearly inicate substantial positive epenence of the istributions. This is confirme by simple correlation coefficients of the one-step forecast errors with the remaining ones, given in Table 4. Table 4: Correlation coefficients between one-step ahea an more istant forecasts errors for BARMA forecasts 2-step 3-step 4-step 5-step 6-step 7-step 8-step Corr.coefs Although the correlation seems to iminish slightly with the increase in istance between the forecasts, it remains high for all forecasts. In practical applications, however, this correlation might be smaller, as we have use purely mechanistic forecasts, without any ajustments an experts corrections. If such ajustments are implemente, this woul likely reuce the bias of the forecasts an the epenence of forecasts errors. Estimation results are given below in Tables Table 5: Gooness of fit results, 1-step errors, BARMA forecasts, No. of obs. 476 α-stable (0.131) (0.076) (0.047) Tempere stable (0.0478) (0.145) (0.020) (0.289) Skewe normal (0.117) (0.0164) Stanar normal () Table 6: Gooness of fit results, 2-steps errors, BARMA forecasts, No. of obs. 476 α-stable (0.0483) (0.165) (0.196) Tempere stable (0.0837) (0.0643) (0.477) (0.500)

17 15 Skewe normal (0.0430) (0.0009) Stanar normal () Table 7: Gooness of fit results, 3-steps errors, BARMA forecasts, No. of obs. 420 α-stable (0.0733) (0.0292) (0.266) Tempere stable (0.0855) (0.3439) (0.0993) (0.289) Skewe normal (0.0665) (0.101) Stanar normal () Table 8: Gooness of fit results, 4-steps errors, BARMA forecasts, No. of obs. 392 α-stable (0.0560) (0.0578) (0.196) Tempere stable (0.0612) (0.115) (0.152) (0.289) Skewe normal (0.154) (0.325) Stanar normal ()

18 16 Table 9: Gooness of fit results, 5-steps errors, BARMA forecasts, No. of obs. 364 α-stable (0.0956) (0.0255) (0.118) Tempere stable (0.0743) (0.288) (0.213) (1.323) Skewe normal (0.0345) (0.0985) Stanar normal () Table 10: Gooness of fit results, 6-steps errors, BARMA forecasts, No. of obs. 336 α-stable (0.106) (0.106) (0.0751) Tempere stable (0.0391) (0.0436) (0.202) 3.00 (0.000) Skewe normal (0.0336) (0.0969) Stanar normal () Table 11: Gooness of fit results, 7-steps errors, BARMA forecasts, No. of obs. 308 α-stable (0.245) (0.115) (0.0256)

19 17 Tempere stable (0.0222) (0.231) (0.0137) (0.500) Skewe normal (0.0165) (0.0577) Stanar normal () Table 12: Gooness of fit results, 8-steps errors, BARMA forecasts, No. of obs. 280 α-stable (0.109) (0.215) (0.0603) Tempere stable (0.0222) (0.0289) (0.0289) (0.144) Skewe normal (0.146) (0.120) Stanar normal () Results shown in Tables 5-12 o not confirm the superiority of the α-stable istribution any more. With the minimum χ 2 as the criterion, the α-stable istribution exhibits better fit than the other istributions compare only for the 4-steps ahea forecasts errors. For the two-steps ahea forecasts errors the fit of α-stable an tempere stable istributions is practically ientical. For the remaining cases, tempere stable prove to be the best for the forecasts errors of 1, 3, 5 an 8 steps ahea. In two cases (6 an 7 steps ahea), the skewe normal istribution exhibits the best fit. In neither case the symmetric normal istribution isplays the best fitting. It is worth noting that, with the increase in forecast horizon, the number of observations use for estimation iminishes, which affects the reliability of the results. Nevertheless, the overall picture is such that α-stable istributions approximate forecasts errors well when these errors are obtaine by naïve an ruimentary methos, while the tempere stable istributions are more appropriate for more sophisticate forecasts.

20 18 4. SERIES OF RUNS AND TURNING POINTS IN OECD FAN CHARTS Results presente in Section 3 inicate that fan charts shoul be constructe with the use of stable or tempere stable istributions, as they represent inflationary forecasts errors in much better way than symmetric an skewe normal istributions. Clearly, if one is intereste only in the construction of fan charts for performing probability forecasting of the realisation of future inflation, it might be straightforwar to use just a series of inepenent α-stable or tempere stable istributions. If, however, the fan charts are to be use for more in-epth analysis of future inflation, an application of the multivariate Choleskize istribution might be of an avantage. As the istributions of forecast errors are markely epenent, such epenency can be use for the evaluation of the probabilities that future rise (or fall) of inflation will continue or, alternatively, that there will be a turning point in the series. If one simulates the multivariate istribution representing the probability forecast, it is straightforwar to count the frequencies of runs (that is, series with an ientical sign of first ifferences), their means, which correspon to the expecte length of runs, frequencies of turning points, etc.. This is illustrate below by the analysis of the expecte length of continuation of runs. As the forecast is usually mae upon information available up to a certain time point, a vali empirical question seems to be whether the current run of inflation will continue an, if so, for how long. Hence, for all 28 OECD countries analyse here, fan charts with the forecasting horizons of 1,2,...,12 months have been compute using the simulate 12-imensional α- stable, stable, skewe normal an normal istributions. These istributions have been initially scale by stanar eviations obtaine for the empirical istributions of the forecast errors calculate separately for each forecast horizon. The empirical istributions have been obtaine by recursive ex-post forecasting, as escribe in the previous section. As the result, the ispersions of the fan charts obtaine by ifferent istributions for the same country o not iffer much. After scaling, appropriate multivariate istributions have been simulate an use for constructing the fan charts. Two alternatives have been use here. Within the first one, the multivariate istributions are regare as mutually inepenent, with a unitary Cholesky matrix. Within the secon one, they are becoming epenent, with non-zero covariations impose. From each of these joint 12-imensional istributions 10,000 runs (multivariate realisations of length 12 each) have been rawn an their average run length compare with that of an average continuation of a run compute for the historical ata for each country. It is important not to confuse the run length with the length of run continuation. In a comparison of simulate ex-ante runs with the historical runs the latter concept is more appropriate, as all simulate runs start at the same time. For instance, if x 1 < x 2 < x 3 < x 4 < x 5 but x 5 > x 6, the run length is 4, but the run continuation lengths are 4 for time 1, 3 for time 2, 2 for time 3 an 1 for time 4. In Table 13 the average continuation of a run is abbreviate as ACR. Table 13: Expecte length of runs continuation Country ACR.; No. obs Austria 1.567; 599 Belgium 1.76; 599 α-stable temp. stable skew. norm normal inepenent epenent inepenent epenent

21 19 Country ACR.; No. obs Canaa 1.55; 599 Czech Rep 2.08; 167 Denmark 1.62; 479 Finlan 1.73; 599 France 2.30; 599 Germany 1.68; 191 Greece 1.92; 599 Hungary 2.00; 3.71 Icelan 1.98; 267 Irelan 2.01; 119 Italy 2.23; 599 Japan 1.41; 598 Korea 1.49; 313 Luxembourg 1.62; 599 Mexico 4.05; 598 Netherlans 1.63; 599 α-stable temp. stable skew. norm normal inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent

22 20 Country ACR.; No. obs Norway 1.69; 611 Polan 2.27; 127 Portugal 1.64; 599 Slovak Rep. 1.82; 167 Spain 1.70; 599 Sween 1.51; 599 Switzerlan 1.80; 599 Turkey 1.93; 455 U.K. 2.03; 599 U.S.A. 2.02; 599 α-stable temp. stable skew. norm normal inepenent epenent inepenent epenent inepenent epenent inepenent 1,31 1, epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent inepenent epenent The criterion of comparing the forecaste continuation runs with the average historical ones is somewhat fuzzy, as the historical runs are presumably autocorrelate. It is likely that runs in the current perio of stabilize inflation are getting shorter. Nevertheless, Table 13 shows a clear ifference between the lengths of runs obtaine from the inepenent an epenent istributions. For the inepenent istributions the continuations of runs are clearly shorter, markely below the average length of the historical continuation of runs. The average lengths of the ex-ante continuations of runs obtaine from the epenent istributions are, in all cases, much closer to the historical averages. In most cases the lengths for the skewe normal an symmetric normal istributions are longer than these for the α-stable an tempere stable ones. This, however, oes not necessarily prove their superiority. It is not known to what extent they correspon to runs obtaine uring the current perio of inflation stabilisation. It woul not be approptiate to compare them with a small number of recent ex-post runs, as their aim is for the ex-ante rather than ex-post analysis. The α-stable an tempere stable

23 21 istributions might be more appropriate, ue to more accurate estimation of the tails, for the evaluation of probabilities of high inflation an rapi isinflation. In orer to illustrate the practical aspects of forecasting of runs an turning points in a greater etail, the results for Japan are presente below. Out of all OECD countries Japan has been chosen as the example, as it is expecte that, after a perio of eflation, in 2010 it graually returns to a positive, albeit not very strong, inflation. Hence, a clear turning point in inflation is expecte here. The forecast is obtaine from the BARMA(2,0,2,2) moel, which is in turn combine with the experts forecasts. As before, two types of forecasts have been compare here, which iffer by the assume epenence between the istributions. The epenent forecast has the covariance matrix built in such a way that the iagonal elements correspon to the variances of the istributions of the ex-post forecasts errors an the non-iagonal elements are obtaine from the correlation coefficients 0.7 i j ij for i > j an ij ji for i < j. Hence, the correlation between particular forecasts iminishes with the increase in the forecast horizon. The inepenent forecast is compute with the use of iagonal covariance matrix, so that no epenence between the istributions is allowe. The forecast perio is from July 2009 to June 2010, that is, the maximum forecast horizon is 12 months. Figures 29 an 30 represent the epenent an inepenent forecasts obtaine with the use of Choleskize tempere stable istribution with the parameter α = 1.80, θ = 2 an the skewness varying accoring to the bias of historical forecast errors. In the case of Japan skewness is minimal an the istributions are nearly symmetric. The istributions are constructe aroun the point forecasts. It is preicte that the perio of eflation, which ha begun in January 2009, shoul en in November In the first half of 2010, inflation, explaine by the mean of the forecasts istribution, is likely to reach a level of 2.2% in February-April an then fall to aroun 1-1.2%. The fan charts for the epenent an inepenent forecasts are virtually ientical, as the both forecasts has the same ispersion, representing the ex-post forecasts errors. However, the ifferences between the both forecasts become evient in the analysis of the expecte turning points an continuation of runs. The probabilities of turning points an continuation of runs of the epenent an inepenent forecasts are shown by Figure 31. The soli blue an re lines represent the epenent forecast an the otte lines represent the inepenent one. The upper part of the iagram (re lines) reflects the probabilities of turning points to appear an a continuation afterwars. Let r t be the event that, in time t, a continuation of the current tenency appears, that is

24 22 sgn( ) sgn( ) t t 1, an s t be the event of the tenency reversion, t t 1 sgn( ) sgn( ) k Hence, points on the re line represent Pr rt i st for k = 1,2,...5. These probabilities are i 1 measure on the right axis, ownwars (this is inicate by the right arrow pointing ownwars). For the epenent forecast the first point of the first re line shows the probability that a turning point appeare June 2009 (the last observation in the sample). It is equal to (measure from top). The same point, measure from the bottom, represents the probability that the tenency of inflation to ecrease will be continue until April 2009 ( =0.595). Subsequent points on the first re line relate to the probabilities that, if a turning point appears in June 2009, the new tenency (of inflation to rise) will be continue for 1, 2, 3 months. Hence, they express joint conitional probabilities of the current tenency to continue. Further re lines, which start for July, August, September, October, November an December 2009 show analogous conitional an joint probabilities assuming that the current tenency of inflation to fall is continuing until the month specifie. 4. The blue line represents the probabilities that the actual tenency of inflation to fall will continue for 1, 2,, etc. months. Using the notation implemente above, points on the blue k line represent Pr rt i rt for k = 1,2,...,5. Initial points on each blue line are simple i 1 complements of the initial points of the re line as in the initial month only two alternative events are possible: that a tenency will continue an that there will be a turning point. Subsequent points represent conitional probabilities that, if the tenency is maintaine until a given month, it will be continue further for the number of perios as inicate by the point on the line. The otte lines mark the analogous probabilities for the inepenent forecasts. Figure 31 epicts the events which might happen until December Obviously the tenencies evelope prior to January 2010 might continue in later months. Hence, the time axis is longer here an time points for January, February, March an April 2010 are marke in re. For June an July 2009 the probability of continuation of a run is greater than that of a turning point, as it is represente by the joint points of the blue an re lines being above the 0.5 mark. After that, the reverse appears an the probabilities of turning points are greater than the probabilities of a continuation. For the forecast for July-November 2009 the re lines for the epenent forecast are visibly flatter than for the inepenent one. It reflects the situation 4 Note that the efinition of the turning point applie here is more restrictive than a efinition use often in the financial literature (see e.g. Malevergne an Sornette, 2006).

25 23 when the estimate conitional probabilities that a new tenency (of inflation to rise) will continue after a reverse is greater for the epenent rather than for inepenent forecast. For the perio from July to August the en points of the re lines are relatively low, which inicates, that the probabilities of the tenencies to continue further for more than five months is not negligible. At the same time, the blue lines for the epenent forecast are steeper than for the inepenent one (except for June 2009). It inicates that the joint conitional probabilities of a continuation of the current tenency of inflation to fall woul go to zero more rapily for the latter forecast rather than for the former. If one regars the epenent forecast as being more reliable than the inepenent one, the results escribes the forthcoming change in tenency of the inflation ynamics in Japan in a much greater etail, than a point forecast woul o. The ifferences between the epenent an inepenent forecasts are further illustrate by the comparison of the expecte uration of runs presente in Table 14. By the expecte uration of current run we unerstan the expecte length of the tenency observe in the previous month. For June 2009 the tenency of inflation was to fall, so that the expecte uration of the continuation of this run is 1.58 months accoring to the epenent forecast an 1.33 months accoring to the inepenent forecast. If, in June 2009, the tenency to fall is to be reverse, the expecte uration of the new tenency is 1.72 months accoring to the epenent forecast an 1.37 months accoring to the inepenent forecast. Further ata in Table 14 are interprete analogously. Table 14: Expecte urations of current run continuation an after the turning point Month Current run After turning point Dep. forecast Inep. forecast Dep. forecast Inep. forecast Jun Jul Aug Sep Oct Nov Dec CONCLUSIONS The finings presente in this paper inicate that fan charts built uner the assumption of the normalityan skewe normality might not lea to precise results, as normal istributions o not approximate the empirical istributions of forecast errors well. Much better fit is obtaine by the α-stable an tempere stable istributions. It is also important to incorporate epenence into the multivariate istributions which are use for constructing the fan charts. The grouns for this are twofol. Firstly, the real life forecast errors are usually epenent, which makes the fan charts built from epenent multivariate istributions more realistic. Seconly, if the epenent istributions are use, there are new possibilities, going beyon the traitional fan charts of computing the probabilities of turning points to appear at a specific time in future, expecte lengths of the increasing an ecreasing runs of inflation, etc.. These computations are relatively uncomplicate, as they can be obtaine by running simple simulations on multivariate istributions. The concept of the Choleskize multivariate istributions seems to be of a particular use here, as it is computationally straightforwar an intuitively simple.

26 24 However, the approach propose here is not without limitations. Firstly, the parameters of the Choleskize istributions cannot be easily estimate, especially if the sample size is limite. Seconly, little is known of the theoretical properties of the Choleskize stable an tempere stable istributions. In particular, their characteristic functions an parameters are not known, which means that the only practical way of using the Choleskize istributions is by simulation. As this is clearly a shortcoming of the metho, more analytical inference is neee here.

27 25 APPENDIX A: SIMULATION OF CHOLESKIZED STABLE AND TEMPERED STABLE DISTRIBUTIONS The rawing proceure consists of the following steps: 1. Drawing from a -stable istribution (possibly skewe) for Applying tilting (if rawing from tempere istribution is require). Step 1 Without loss of generality we can assume, by choosing an appropriate normalisation of the Cholesky matrix C, that the scale parameter amounts to unity. From Statement 1 it follows that when a ecomposition of stable ranom variable is applie, it is enough to generate raws for 1. Thus in orer to procee we just nee to generate ranom raws from S (1, 0,1). For generating raws from S (1, 0,1), the metho of Chambers, Mallows an Stuck (1976), originally implemente by McCulloch (1986) an evelope further by Weron (1996), is applie. Let u an u be inepenent raws from U(0,1) (the uniform istribution on [0,1]). 1 2 Define: v ( u 0.5), w log u. 1 2 Let z be such that: z tan( ). 2 1/(1 an: y ( w ) cos v) ( sin v z cos v) ( cos(1 ) v z sin(1 ) v ). Then y constitutes a realization of a ranom variable Y ~ S (1, 0,1). Step 2 In orer to raw from the tempere T istribution the Brix (1999) rejection proceure is use. The algorithm is initialize by setting value of 0 which is a parameter of tilting. Next it raws u ~ U (0,1) an y ~ S (1, 0,1). We accept y as a vali raw an set y y if the conition y u e is fulfille. Then y constitutes a realization of a ranom variable Y ~ T (,1,0,1). We can now raw two ii ranom variables such that: Y ~ T (,1,0,1) i = 1 1 1, 2. Then the linear combination of the form X Y 2 2 realization of X of the tempere stable analogy for T (,,0,1). 1 1 ( ) ( ) i 1 2 Y gives a