The Geometric Combination of Forecasting. Models

Transcription

1 The Gemetric Cmbinatin f Frecasting Mdels A. E. Faria and E. Mubwandarikwa Department f Statistics, The Open University MK7 6AA, U.K. Abstract The mst cmmnly used methd fr cmbining prbability mdels is the linear cmbinatin (LCmb) als knwn as mixture. In this reprt we prpse an alternative apprach, the gemetric cmbinatin (GCmb), which vercmes sme f the main disadvantages f the linear methds. In fact, the prpsed GCmbs nt nly preserve the distributinal frm f the cmbinatin fr many distributin types (including thse f the expnential family and t-distributins), but are als externally Bayesian. While clsedness t distributin frms can preserve unimdality, external Bayesianity will preserve immunity f influence n decisin making. The classificatin therem f catastrphe thery is used t establish the unimdality cnditins fr the GCmb f densities frm the Student t family. We apply bth the LCmb and the GCmb t a case f beverage sales frecasting in Zimbabwe. Several plausible (nn-similar) regressin dynamic linear mdels which include bth ecnmic and weather variables are entertained. The perfrmances f the tw cmbinatin 1

2 1 INTRODUCTION. 2 methds are cmpared in terms f symmetric lss functins such as the quadratic and the expnential lsses as well as the nn-symmetric lgarithmic lss. Cnsequences f decisins fr such lsses under multimdal predictive densities are cmpared when the mean and the largest mde are chsen as pint frecasts. Keywrds: Bayesian frecasting, nn-linear mdel cmbinatin, regressin dynamic linear mdels, multimdal predictive densities, lss functins 1 Intrductin. A statistical mdel like any mathematical mdel cnsists f a simple representatin f a usually cmplex real prcess r phenmena. As such it is nt always apprpriate t assume that a mdel is the true representatin f the underlying prcess. We assume here that a plausible (r apprpriate) mdel irrespective f its truth is useful fr the purpses f frecasting a time series. In ur cntext, we assume that a plausible mdel is mainly characterised by its set f regressrs. In particular, we will cnsider a class f Bayesian time series frecasting mdels, the regressin dynamic linear mdels (RDLMs) prpsed by West and Harrisn (1997). One f the main features f the Bayesian frecasting mdelling apprach is t allw fr mdel interventin t accmmdate subjective infrmatin. This is particularly imprtant at times f majr changes in the time series prcess. Interventin allws fr bth parametric and structural changes t be made t a mdel. Changes t existing parameters f a mdel allw it t adapt faster t changes in the series. Fr example, suppse a mdel is structured such that it has a specific parameter t accunt

3 1 INTRODUCTION. 3 fr the series mean level at each perid f time. Thus, by prperly mdifying the mdel s prir prbability distributin fr the mean level, a significant change in the level f the series can be better fitted by the mdel. Such a mdel shuld als prduce imprved predictins in the shrt term. Similarly, the mdel can be made t adapt t stchastic changes in the variance f the series by having the prir distributin fr the bservatinal variance changed accrdingly. Certain (structural) changes in the prcess generating the data, where prir expert interventin nly is nt apprpriate, may require changes in the structure f the Bayesian mdel itself. Thse changes can be implemented by mdificatins t either (i) the mdel s parameter set (by augmentatin, fr example, t allw an explicit mdelling f a newly bserved characteristic r change in the prcess), r (ii) the mdel s design matrix (ne example being t include a new causal variable in the regressr set). Further t mdel interventin, mdel mnitring fr cntinual assessment and detectin f mdel inadequacies, is an implicit recgnitin that a single mdel may nt always be apprpriate t represent the glbal behaviur f the series. In effect, despite a particular RDLM with a specific frmulatin being mre apprpriate fr a time series during a perid f time, it will nt necessarily be apprpriate at all times. Als, in many situatins, there will be a number f mdels with distinct frmulatins which can be cnsidered t be apprpriate (r plausible) fr the time series prcess even during a cmmn perid f time. In this paper, we assume that a Bayesian decisin maker (BDM) entertains a number f RDLMs, say M 1,...,M k (k 2) she believes are gd (plausible) mdels fr a time series prcess, represented by the randm variable Y t (t = 1,2,...), but is interested in determining a single mdel she can use fr frecasting. The BDM pssesses a set f histrical time series

4 1 INTRODUCTION. 4 data f dimensin T, y T = (y 1,...,y T ), abut the underlying prcess Y t as well as n a set f auxiliary variables X it (we call influential variables) which realisatins we represent by the vectr x T i = (x i,1,...,x i,t ), (i = 1,...,r) she believes relate in a causal sense t Y t. The ntatin adpted thrughut, underlined characters represent vectrs, bldfaced characters represent matrices and prime dentes transpsitin. Each RDLM that the BDM entertains at a given perid f time can be seen as cming frm her belief abut hw the influential variables X t relate t Y t. Als, this relatinship is nt fixed but may change ver time, giving rigin t further alternative mdels that reflect the BDM s believed causal assciatins at different perids. The BDM s task is t btain, at time t, a single predictive distributin fr future values f Y t t suprt her decisin making. There are basically tw curses f actin that the BDM culd take t btain a single predictive density: (i) she can chse a single mdel frm the set M = {M 1,... M k } based n sme mdel selectin criteria and use that mdel fr frecasting r (ii) she culd btain a mdel frm the cmbinatin f M 1,... M k and use the cmbined mdel fr frecasting. Within the actin curse (i), mdel selectin, there are a number f different appraches the BDM culd adpt. Sme f these are based n cmparing the mdels predictive perfrmances and chsing the best accrding t a defined measure. Others, like the Bayes factr methds, are based n calculating the mdels predictive likelihd ratis and selecting the mdel with the largest likelihd. Als, the BDM culd chse a mdel assciated with the belief net she thinks is the mst apprpriate fr the frecasting perid. In any case, a single mdel frm the set f plausible mdels is selected frm her set f mdels and used fr

5 1 INTRODUCTION. 5 predictin. Under ptin (ii) abve, mdel cmbinatin, there is a main class f available methds all based n the linear cmbinatin f mdels usually referred t as mixture mdels. In the Bayesian cntext, Raftery, Madigan and Heting (1997) refer t the cmbinatin f the predictive distributins weighted by the mdels prbabilities as Bayesian mdel averaging (BMA). West and Harrisn (1997) cnsidered classes f dynamic linear mdels (DLMs) with their linear cmbinatins (r mixtures) referred t as multi-prcess mdels. Hwever, they nly cnsider mdels whse parameter sets are identical and differ nly n their variances. Perhaps the main advantage f adpting a mdel cmbinatin ver a mdel selectin apprach is that the uncertainty abut the mdels can be explicitly accunted fr in the analysis. Cnditining n a single selected mdel ignres mdel uncertainty, and thus leads t the underestimatin f uncertainty when making inferences abut quantities f interest. Please see fr example Raftery et. al. (1997) fr mre details. Als, the cmbinatin f mdels can be interpreted as a frm f aggregating infrmatin frm different surces (mdels) with bvius advantages. Hwever, the linear cmbinatins f prbability distributins such as the BMA als has its disadvantages. The exact resulting cmbinatin, in certain cases, can be difficult t btain in practice. This is remedied by adpting apprximate slutins like Markv Chain Mnte Carl (MCMC) simulatin. Furthermre, in many situatins the resulting distributin presents multi-mdality which is an undesirable characteristic frm a decisin making standpint as a decisin usually requires a single estimate fr the lcatin parameter f the predictive distributin. In this article we prpse an alternative apprach t the linear cmbinatin, the gemetric

6 1 INTRODUCTION. 6 cmbinatin, which reslves sme f the main disadvantages f the linear methds. The main reasns we prpse the gemetric cmbinatin f prbability distributin mdels are twfld. First, gemetric cmbinatins are externally Bayesian thus pssessing the advantages f such types f cmbinatins (e.g. immunity f influence n decisin making). External Bayesianity, Madansky (1964), ensures that the cmbinatin rule will give the same result a psteriri, independently f being btained befre r after the individual distributins are updated by new data. In the case where the individual distributins are subjective prbability distributins prvided by experts, Raiffa (1968) illustrated hw the relevance ver the rder in which the cmbinatin and updating are dne can lead t subjects trying t increase their influence n the cnsensus r resulting cmbinatin f distributins by insisting that their pinins be cmputed befre the utcme f an experiment is knwn. External Bayesianity makes such argument pintless. The secnd reasn, is that gemetric cmbinatins preserve the distributin frm f the cmbined mdels fr many distributins. In particular we shall shw that the gemetric cmbinatin f expnential family distributins is als a member f the expnential family. This result means that the gemetric cmbinatin f expnential family densities is unimdal. This is nt the case with linear cmbinatins where, in general, unimdality ccurs nly under certain cnditins, see e.g. Titteringtn, Smith and Makv (1985). Further, whilst linear cmbinatins f (Student) t-distributins are necessarily multimdal under linear cmbinatin, it is nt always s under the gemetric rule as we als shall see. Student t-distributins play an imprtant rle in Bayesian frecasting. The remainder f this paper is structured as fllws. In Sectin 2 we intrduce sme further ntatin and define the structure f plausible RDLM s and their assciated prper-

7 2 REGRESSION DYNAMIC LINEAR MODELS 7 ties. In Sectin 3 we intrduce bth the linear the gemetric cmbinatins f prbability distributins. We shw their main characteristics and prperties in terms f cnditins fr unimdality when applied t Gaussian and Student-t mdels in particular, and mre generally t the expnential family. In Sectin 4, we prduce a cmparative analysis f the perfrmances f the tw cmbinatin methds t a time series f beverage sales in Zimbabwe. Sectin 5 cncludes the paper and pints ut sme future research. 2 Regressin Dynamic Linear Mdels At a fixed perid f time t, a regressin dynamic linear mdel (RDLM), M j, (j = 1,2,...,k), fr a time series, Y t, is characterised by the quadruple {F,G,V,W} j, where F j = (X 1,...,X r ) j is the (r j 1) regressin vectr, X ij being the i th regressin variable (i = 1,2,...,r j ), G j is the (r j r j ) system evlutin matrix, V j is the bservatinal variance (i.e. the variance f Y t ), and W j is the (r j r j ) state evlutin cvariance matrix. Fr simplicity and withut lss we have mitted the subscript t in defining M j. Nte that the regressin vectr F j, the evlutin matrix G j and the evlutin cvariance matrix W j are defined by the BDM during the mdel specificatin stage f the mdelling prcess. The evlutin matrix W j can, fr instance, be chsen with the use f discunt factrs, δ j (0,1), interpreted as measures f hw quickly the value f the current infrmatin set D t = {y t,d t 1 } is expected t decay in time. The bservatinal variance V j is usually unknwn (and large relative t the evlutin variance W j ). Als, it is usually the majr surce f uncertainty but apprpriate Bayesian learning prcedures can be used in its specificatin and estimatin. Please see West and Harrisn (1997) fr further details. The fllwing subsectin describes hw in general terms (fr any assumed errr distri-

8 2 REGRESSION DYNAMIC LINEAR MODELS 8 butin) the sequential Bayesian (prir-t-psterir) updating f parameters f a RDLM is carried ut. 2.1 Sequential Bayesian Mdel Updating A RDLM M jt = {F,G,V,W} jt is sequentially updated in time in the fllwing manner. Let θ jt = (θ 1,...,θ rj ) jt be the (r j 1) state vectr assciated with M jt. At each time t, Y t cnditinal n θ jt, has a knwn bservatin prbability density (r mass) functin p j (Y t θ jt ). Cnditinal n θ jt, Y t is assumed t be independent f Y s and θ js fr all past and future values f s (s t). The parameter vectr, θ jt, evlves sequentially in time accrding t a knwn Markvian prcess described by an evlutin density p j (θ jt θ j,t 1 ). Ntice that θ jt is assumed independent f θ js fr s t 1 given θ j,t 1. Like a dynamic linear mdel (DLM), a RDLM M jt can be represented by an bservatin and an evlutin equatin: Observatin: Y t = F jtθ jt + ν jt ; ν jt p(ν jt ) Evlutin: θ jt = G jt θ j,t 1 + ω jt ; ω jt p(ω jt ), where p(ν jt ) is a density (r mass) functin with zer mean and variance V jt fr the bservatinal errr ν jt, and, p(ω jt ) is a jint density with zer mean and cvariance matrix W jt fr the evlutin errr vectr ω jt. Within the Bayesian dynamic frecasting framewrk we adpt here, the predictive densities fr Y t are btained sequentially fr each mdel M j,t (j = 1,...,k) frm the marginalisatin f the mdel s parameter vectr θ j,t = (θ 1,t,...,θ nj,t) f dimensin n j frm the jint density

9 2 REGRESSION DYNAMIC LINEAR MODELS 9 p j (Y t+1,θ j,t+1 D t ) = p j (Y t+1 θ j,t+1,d t )p j (θ j,t+1 D t ) where the prir density p j (θ j,t+1 D t ) = p j (θ j,t+1 θ j,t,d t )p j (θ j,t D t )dθ j,t. (1) The psterir density at time t, p j (θ j,t D t ) p j (θ j,t D t 1 )p j (Y t θ j,t,d t 1 ) (2) is determined by Bayes therem. The ne step ahead predictive density is then p j (Y t+1 D t ) = p j (Y t+1 θ j,t+1 )p j (θ j,t+1 D t )dθ j,t+1 where the integral dentes multiple integratin ver the parameter space generated by θ j,t+1. An imprtant prperty that effectively enables dynamic mdeling is that f cnditinal independence, in which, given θ j,t at time t, Y t+h (h = 0,1,...) is independent f Y t i (i = 1,2,...). That is, given the present θ j,t, the past, present, and future bservatins are mutually independent. Als, given D t, all infrmatin cncerning the future is cntained in the psterir density p j (θ j,t D t ) s that its hyperparameters are sufficient fr {Y t+h,θ j,t+h } (h = 1, 2,...). The predictive (r frecasting) densities can be cmputed sequentially fr larger frecasting hrizns. In general, the h steps ahead predictive density fr Y t+h calculated at time t (sequentially) fr h = 1, 2,... will be: p j (Y t+h D t ) = p j (Y t+h θ j,t+h )p j (θ j,t+h D t )dθ j,t+h. (3) In its simple frm, a DLM M j (at a fixed time t), characterised by {F,G,V,W} j, will assume specific knwn prbability density functins fr the bservatinal and the evlutin

10 2 REGRESSION DYNAMIC LINEAR MODELS 10 errrs such that the general sequential updating abve can be perfrmed in analytical clsed frm. Perhaps the best knwn case is the Gaussian DLM where the bservatinal errr ν jt is assumed t fllw a nrmal density, i.e. ν jt N[0,V jt ]. Unknwn mdel parameters such as in sme cases, the bservatinal variance V jt, can be dealt with in a Bayesian framewrk by assuming they fllw a density functin which is als sequentially updated within the Bayesian paradigm. In this case, the evlutin errr ω jt is assumed t fllw a multivariate (Student) t-density with n j,t 1 degrees f freedm, zer mean vectr and cvariance matrix W jt, i.e. ω jt St nj,t 1 [0,W jt ]. Recall that, in general, if θ St n [m,c] then { } ( r+n p(θ n,m,c) = c n + (θ m) C 1 2 ) (θ m). (4) where c is a cnstant such that p(θ n,m,c)dθ = 1, n is the number f degrees f freedm, m is the r 1 mean vectr f θ, C is the r r cvariance matrix and r is the dimensin f the vectr θ. In cases where it is nt pssible t adpt a cnjugate analysis in (2), numerical integratin methds can be emplyed t determine the psterir parametric density in the sequential updating described abve. One f the mst ppular methds is the Markv chain Mnte Carl (MCMC), which we shall adpt in ur applicatin sectin. Nw, that we have defined a RDLM and shwed hw it can be updated in time, we shall next define hw we cnsider tw RDLMs t differ frm ne anther.

11 3 CHARACTERISING COMBINATIONS OF MODELS Similar RDLMs Frm a pragmatic pint f view, it makes sense in cmbining nly RDLMs that represent distinct characteristics f the underlying prcess. Therefre, we will be interested in RDLMs which are assciated with distinct causal structures f assciatin between the underlying variables. Fr that reasn it is imprtant that we characterise when tw RDLMs are cnsidered t differ frm ne anther. Within the thery f DLMs, any tw mdels prducing the same frecasts are said t be equivalent mdels. Similarly, any tw mdels with the same qualitative (i.e. algebraic) frm f frecast functins are said t be similar mdels. In this paper, tw RDLMs M i and M j (i j) are cnsidered t differ frm ne anther if they are nt similar, i.e. if they d nt have frecast functins f exactly the same algebraic frm. Frmally, M i and M j are similar mdels, M i M j, if and nly if their evlutin matrices G i and G j have identical eigenvalues. Equivalently, G i and G j are similar matrices such that there exists a nn-singular similarity matrix H such that G i = HG j H 1. Please refer t West and Harrisn (1997) fr further details. Nw that we have revised hw a RDLM is sequentially updated in time, as well as intrduced nn-similar RDLMs as the types f mdels the BDM will be entertaining, we can mve t the prblem f hw t cmbine them tgether t btain a single frecasting mdel. 3 Characterising cmbinatins f mdels Let (Ω,µ) be a measure space. Als, let be the class f all µ-measurable functins p : Ω [0, ) such that pdµ = 1 with µ almst everywhere (a.e.). A (generic) cmbinatin functin P : k, is ne which maps a vectr f prbability density functins (p 1,...,p k ),

12 3 CHARACTERISING COMBINATIONS OF MODELS 12 where p j = p( M j ) (fr j = 1,...,k), int a single density p( ) als in. In the fllwing subsectins we define bth the linear and the gemetric cmbinatins fr univarite distributins. We als shw unimdality cnditins fr mdels within the expnential family as well as fr mdels with t-distributins fr bth types f cmbinatin. 3.1 Linear Cmbinatin f Predictive Mdels The linear cmbinatin P L : k f k (predictive) densities fr a time series Y t Ω, given the actual infrmatin D t = (y t,d t 1 ), has the fllwing general frm: k P L (p 1,...,p k )(Y t+h D t ) = w jt p j (Y t+h D t ) (5) where p j (Y t+h D t ), the h-steps-ahead (h = 1,2,...) predictive density f mdel M jt, is sequentially btained frm (3) and w jt, j = 1,...,k, are arbitrary weights (nt necessarily nnnegative) adding up t ne. As P L (Y t+h D t ) is a prbability density functin (pdf) it means P L 0 fr all Y t Ω, care must be taken when chsing negative weights. The weights culd be elicited by the BDM based n her knwledge abut the relative predictive capabilities f the individual mdels fr the perid f interest. There are a number f methds -including Bayesian, such as Bunn s (1975) utperfrmance- available fr determining the weights based n the mdels past predictive perfrmances. In the Bayesian mdel averaging framewrk, the weight w jt is treated as the psterir prbability fr mdel M jt, that is w jt = p(m jt D t ) btained by Bayes therem via MCMC. Please see Raftery et. al. (1997) fr further details. In the applicatin sectin f this paper, we have adpted such interpretatin and btain their values sequentially in time. j=1

13 3 CHARACTERISING COMBINATIONS OF MODELS 13 Nw, in the cntext where the cmpnent mdels are nn-similar RDLMs, the cmbinatin (5) abve is applied t the predictive densities btained frm the psterir distributins f the parameter vectr θ jt, i.e. p j (θ jt D t ). The predictive density will therefre be btained frm the cmbined psterir densities by the integratin in (3). Cnsider the functin P L (Y t+h D t ) as the BDM s linear cmbinatin f all h-steps-ahead (h = 1,2,...) predictive prbability density functins p j (Y t+h D t ) btained frm the RDLM M jt. Nte that the future values f regressrs are required when frecasting ahead. There are varius pssible ways in which thse frecasts can be btained. A mre cmplex way is t use multivariate time series mdelling and frecasting t btain a jint mdel fr frecasting the regressr variables as time series alng with the variable f interest Y. We adpt the simpler apprach here f estimating the future values f regressrs by individual univariate mdels fr thse. In the Gaussian case, when all the elements f M jt are knwn, the frecast ahead t time t + h frm time t, p j (Y t+h D t ), (h 1) is als Gaussian with mean f jt (h) = F j,t+h a jt (h) and variance Q jt (h) = F j,t+h R jt(h)f j,t+h + V j,t+h can be calculated recursively fr h 1 using a jt (h) = G j,t+h a jt (h 1) and R jt (h) = G j,t+h R jt (h 1)G j,t+h + W j,t+h,

14 3 CHARACTERISING COMBINATIONS OF MODELS 14 with the initial values a jt (0) = m t and R jt (0) = C jt. The vectr m jt and the matrix C jt are the psterir lcatin and spread parameters respectively f the state vectr θ jt. Please see West and Harrisn (1996) fr further details. A case f particular interest ccurs in situatins where sme f the elements f M jt are unknwn (e.g. V jt ) r they are knwn but the sample sizes are small. In such cases, the h-steps-ahead predictive density p j (Y t+h D t ) will typically be the density f a t-distributin. This density will have n jt = n j,t degrees f freedm, mean f j,t (h) and variance Q j,t (h), that is (Y t+h D t ) St nj,t (f jt (h),q jt (h)). The mean and variance are btained as fr the Gaussian case abve but with the sample variance S jt used as estimatr f the unknwn V jt. The psterir fr θ jt is (θ jt D t ) St njt (m jt,c jt ) with n jt, m jt and C jt determined recursively by the Kalman filter. 3.2 Unimdality in Linear Cmbinatins This sectin presents the unimdality cnditins fr the linear cmbinatin f Gaussian and t-distributins. It is well knwn that the linear cmbinatins f Gaussian densities are unimdal nly under rather restrictive cnditins, see e.g. Titteringtn, Smith and Makv (1985). Fr example, fr the linear cmbinatin f tw Gaussian densities p j (y µ j,σ 2 j ), with means µ j and variances σj 2, (j = 1,2), Eisenberger (1964) shwed that independently f the cmbining weights, a sufficient cnditin fr unimdality f the cmbined density is that (µ 2 µ 1 ) 2 < 27 4 σ 2 1 σ2 2 (σ1 2 + (6) σ2 2 ). This cnditin means that t btain unimdality, the distance between the lcatin parameters f the cmpnents densities must be small enugh relative t a rati f their spread

15 3 CHARACTERISING COMBINATIONS OF MODELS 15 parameters relative t the ttal spread. Otherwise, the cmbined density will present bimdality. This result when extended fr mre than tw densities yields a similar interpretatin, that is, unimdality f the cmbined density will result when the distances between the cmpnents means are small enugh relative t a certain rati f their standard deviatins. Otherwise, the cmbined density will have between tw and k mdes where k > 2 is the number f cmpnents Linear Cmbinatin f t-distributins In the case where the cmbining densities in (5) are t-distributins, the resulting cmbinatin can als be unimdal under cnditins which are even mre restrictive than that fr the Gaussian case. Withut lss (fr simplicity), we mit the time index t in this subsectin. Fr k = 2 and fr sme fixed weight w, the linear cmbinatin (5) can be written as P L (p 1,p 2 )(Y ) = wp 1 (Y n 1,µ 1,σ 2 1) + (1 w)p 2 (Y n 2,µ 2,σ 2 2) where p j (n j,µ j,σ 2 j ) (j = 1,2)is the density f a t-distributin with n j degrees f freedm, mean µ j and variance n j (n j 2) σ2 j (n j > 2 fr finite variance). Nte that despite being very similar in shape t a nrmal distributin, the t-distributin has heavier tails. The smaller the number f degrees f freedm the heavier the tails. In practice, this means that samples frm a t-distributin will have less bservatins away frm the mean than samples frm a nrmal distributin. Als, recall that fr a nrmally distributed randm variable X N(µ,σ 2 ), and a chi-square distributed randm variable nu χ 2 (n) with n degrees f freedm, we have that Y = µ + X will fllw a t-distributin with n U/n

16 3 CHARACTERISING COMBINATIONS OF MODELS 16 degrees f freedm, mean µ and variance n n 2 σ2, i.e. Y St n (µ,σ 2 ). Nw, if we intrduce chi-square latent variables n j u j χ 2 (n j ) (j = 1,2), we can use them t re-scale the variances f nrmal distributins (s as t make them resemble t-distributins with heavier tails) such that the linear cmbinatin f t-densities abve can be apprximated by the linear cmbinatin f nrmal densities: P L (p 1,p 2 )(Y ) wp 1(Y µ 1, σ2 1 u 1 ) + (1 w)p 2(Y µ 2, σ2 2 u 2 ) where p j (Y µ j, σ2 j u j ) is the density f a nrmal distributin with mean µ j and variance σ2 j u j (j = 1,2). Therefre, the unimdality cnditin (6) fr the linear cmbinatin f nrmal distributins can be rewritten as (µ 2 µ 1 ) 2 < 27 4 σ 2 1 σ2 2 (u 2 σ u 1σ 2 2 ). Nte that, fr heavier tails, u 1,u 2 > 1 imply that σ1 2 + σ2 2 < u 2σ1 2 + u 1σ2 2, and thus unimdality f the linear cmbinatin f the tw t-densities wuld be achieved nly when the means f the tw t-distributed cmpnents are clser t each ther cmparatively with the required distance between the means f the Gaussian cunterparts. That is, it is harder (in the sense that shrter distances between the means are required) t btain unimdality in the linear cmbinatin f t-densities than it is in the linear cmbinatin f Gaussian densities. 3.3 The Gemetric Cmbinatin The gemetric cmbinatin P G : k f k predictive densities fr a time series Y t Ω, given the infrmatin set D t = {y t,d t 1 }, has the fllwing general frm (fr h = 1,2,...): P G (p 1,...,p k )(Y t+h D t ) = c k j=1 p w jt j (Y t+h D t ), (7)

17 3 CHARACTERISING COMBINATIONS OF MODELS 17 where c 1 = k j=1 pw jt j (Y t+h D t )dy t+h, w jt R, j = 1,...,k, is the weight assciated with the predictive density p j (Y t+h D t ) (btained frm mdel M jt ) such that k j=1 w jt = 1. As mentined in the intrductin, ne f the advantages f adpting the gemetric ver the linear cmbinatin methd is that gemetric cmbinatins are externally Bayesian. The advantage prved by Raiffa (1968), being that f immunity f influence n the decisin making in cases where predictive distributins are subjective pinins frm experts (r interested parties). External Bayesianity is defined in the fllwing sectin External Bayesianity In general terms, an externally Bayesian (EB) cmbinatin functin P is characterised as ne satisfying the fllwing cnditin: P ( ) p1 Lp1 dµ,..., p k = LP(p 1,...,p k ) Lpk dµ LP(p1,...,p k )dµ, µ a.e., (8) where L : Ω (0, ) is a likelihd functin fr the actually bserved data, such that 0 < Lp j dµ < (j = 1,...,k). Briefly, an EB cmbinatin plicy ensures that the cmbinatin rule will give the same result a psteriri, independently f being btained befre r after each individual cmbining density is updated when new data is bserved. It can be easily seen that the gemetric cmbinatin P G P in (7) satifies (8) and therefre is EB. The reader can refer t Faria (1996) r Genest et. al. (1986) fr mre details.

18 3 CHARACTERISING COMBINATIONS OF MODELS The Gemetric Cmbinatin f Expnential Family Densities In this sectin we shw that the gemetric cmbinatin f densities frm the regular expnential family f distributins has a density which is strngly unimdal. A prbability measure is said t be strngly unimdal if it is lg-cncave (i.e. its lgarithm is a cncave functin) ver its parameter space. In fact, the strng unimdality f the gemetric cmbinatin cmes frm the fact, shwn by the fllwing therem, that the gemetric cmbinatin f strngly unimdal densities frm the regular expnential family als belngs t that family. A density p(y η) where η = (η 1,...,η n ) Ω belngs t the n-parameter expnential family has the natural representatin p(y η) = h(y)c(η)exp[η d(y)] (9) where h(y) 0 des nt depend n η and d(y) = (d 1 (y),...,d n (y)) with d i (y) : Ω R nt depending n η. The natural parameter space Ω is the set where the kernel functin has a finite integral (r sum): Ω = {(η 1,...,η n ) : 1 c(η) = + h(y)exp[η d(y)]dy < }. The expnential family is said t be regular if (i) the elements f η and thse f d are linearly independent, and (ii) Ω is a n-dimensinal pen set. Recall that the elements f d(y) are linearly independent if n i=1 a id i (y) = b fr all y if and nly if a 1 =... = a n = 0 where a i and b are cnstants. We can state the fllwing result: Therem 3.1 Let p 1 (Y t+h D t ),...,p k (Y t+h D t ) be strngly unimdal densities frm the regu-

19 3 CHARACTERISING COMBINATIONS OF MODELS 19 lar expnential family. Then, the density f the gemetric cmbinatin P G (p 1,...,p k )(Y t+h D t ) in (7) is als a strngly unimdal density frm the regular expnential family. Prf. First, assume that a randm variable y whse density under a mdel M, p(y η) belngs t the n-parameter strngly unimdal regular expnential family. Then, p(y η) raised t any cnstant pwer w (0,1) (nt a functin f y r η) is a density which als belngs t the n-parameter strngly unimdal regular expnential family. In fact, fr a fixed w R we can write: p w (y η) = [h(y)] w [c(η)] w exp[wη d(y)] = h (y)c (y)exp[η d (y)] (10) where h (y) = h w (y), c (η) = c w (η) and d (y) = (wd 1 (y),...,wd n (y)). Therefre p w is frm the expnential family. Nte that because ln p(y η) is cncave, ln p w (y η) = w[ln h(y) + ln c(η)] + ηd(y) is als cncave, and thus p w is strngly unimdal. Nw, the prduct f densities frm the strngly unimdal regular expnential family als belngs t the same family. In fact, fr j = 1,...,k, let p j (y η j ) belng t the n j -parameter strngly unimdal regular expnential family as abve. Thus, given w = {w 1,...,w k } with w j (0,1) : k j=1 w j = 1, we can write that P G (y η) = a( η) k [h j (y)] w j [c j (η)] w j exp[w j η j d j(y)] j=1 = h(y) c(η)exp[ k j=1 η j d j (y)], (11) where η is the parameter set f P G (y η), a 1 ( η) = k j=1 [h j(y)] w j [c j (η)] w j exp[w j η j d j (y)]dy,

20 3 CHARACTERISING COMBINATIONS OF MODELS 20 h(y) = k j=1 [h j(y)] w j and c( η) = a( η) k j=1 [c j(η j )] w j. Therefre, P G (y η) is a nk-parameter density frm the regular expnential family, where n = k j=1 n j. Nw, ln P G (y η) = ln a( η) + k w j ln h j (y) + j=1 is cncave and thus P G is unimdal. k w j ln c j (η j )) + j=1 k η j d j(y) j=1 Ntice that this result is rather interesting frm a decisin analysis viewpint. It may in many cases give a decisin maker a very gd reasn t adpt the gemetric rather than the linear cmbinatin f mdels as we shall see. In the particular case where p j (y µ j,τ j ) is a nrmal density with mean µ j and precisin τ j = σ 2 j (j = 1,...,K), we have that P G (y λ) is als Gaussian with mean µ =ÈK j=1 τ iµ i ÈK j=1 τ i precisin τ = K j=1 w iτ i. As mentined befre, there are situatins in which the predictive densities belng t the family f Student t-distributins. The fllwing sectin shws that (similarly t the linear cmbinatin f Gaussian distributins) the density f the gemetric cmbinatin f t-distributins may be multimdal. and Unimdality in the gemetric cmbinatin f t-distributins Let p j (Y ) (j = 1,...,k) be the density f a t-distributin with ν j degrees f freedm, mean µ j and variance σj 2, i.e. Y M j St νj (µ j,σj 2 ). In this case, the gemetric cmbinatin f the

21 3 CHARACTERISING COMBINATIONS OF MODELS 21 frm (7) will have a density f the frm: P G (p 1,...,p k )(y) = c k [ j=1 1 + τ 1 j (y µ j ) 2] w j (ν j +1) 2, (12) where c is a nrmalizing cnstant, and τ j = ν j σ 2 j. The (natural) lgarithmic kernel f P G (y) has the frm L G (y) = k j=1 w j 2 (ν j + 1)ln[1 + τ 1 j (y µ j ) 2 ]}. (13) Nw, if we represent by D i y the i-th derivative w.r.t y, and equal the first derivative f L G (y) w.r.t. y t zer, i.e. D y L G (y) = d dy L G(y) = 0, we have statinary pints defined by the equatin: k j=1 w j (ν j + 1)τj 1 (y µ j ) [τj 1 = 0, (14) (y µ j ) 2 + ν j ] and thus, k j=1 w j (ν j + 1)τ 1 j (y µ j ) k i=1,i j Nte that this equatin is a cubic functin f y. [τ 1 i (y µ i ) 2 = ν i ] = 0. (15) Nw, assuming fr simplicity that k = 2 and ν j = ν (j = 1,2), equatin (15) can be rewritten in the frm x 3 bx a = 0 (16) where x = (y µ) with µ = 1 3 [(2 w)µ 1 + (1 + w)µ 2 ] ; a = 1 27 (M3 + νt) with M 3 = w 1 µ3 1 3w 12 µ2 1 µ 2 + 3w 21 µ 1µ 2 2 w 2 µ3 2 where

22 3 CHARACTERISING COMBINATIONS OF MODELS 22 w 1 = (w 2)(4w 2 7w + 7), w 12 = 4w 3 9w w 8, w 21 = 4w3 3w 2 + 9w 2, w 2 = (w + 1)(4w 2 w + 4) and T = 9{(w 1)[(w 2)µ 1 (w + 4)µ 2 ]τ 1 + w[(w 5)µ 1 (w + 1)µ 2 ]τ 2 } ; b = S 2 ν T with T = (1 w)τ 1 + wτ 2 and S 2 = 1 3 (w2 w + 1)(µ 1 µ 2 ) 2. Smith (1978) btain a simpler result in determining the critical pints fr the likelihd functin f t-distributins. Using Thm s Classificatin Therem f catastrphe thery (see e.g. Pstn and Stewart, 1978), this crrespnds t the manifld f a cusp catastrphe fr the ptential functin P G (Y w,ν,µ 1,µ 2,τ 1,τ 2 ) fr tw t-distributins with differing means and precisins but identical degrees f freedm. In this manifld, the axis b in (16) lying alng the cusp is the splitting factr (and the rthgnal axis a is the nrmal factr). Nte that, the splitting factr b = S 2 T ν abve can be seen as representing the symmetrical split in P G, while the nrmal factr a = 1 27 (M3 + νt) as incrprating all assymetrical cmpnents f P G. The prjectin f the cusp pint (a pint in the manifld frmed by the set Q = {y,w,ν,µ 1,µ 2,τ 1,τ 2 } f all pints at which the derivative f P G with respect t y vanish) nt the cntrl space C frmed by (w,ν,µ 1,µ 2,τ 1,τ 2 ) divides it int regins which depict the statinary pints f P G. The cusp manifld and its prjectin n the plane (a, b) can be seen in the fllwing Figure (1).

23 3 CHARACTERISING COMBINATIONS OF MODELS 23 Figure 1: A cusp manifld with its prjectin ver the (a, b) plane frmed by the nrmal and the splitting factrs. There will be be three rts as the slutin f the cubic equatin (16). Thse rts fr values f the cntrl variables will determine whether the gemetric cmbinatin is unimdal r nt. A practical way f determining the regins f unimdality in the (a,b) plane is by using Cardan s discriminant functin δ(a,b) defined as: δ(a,b) = 27a 2 4b 3. (17) When δ is psitive (δ > 0) there are three real rts fr (16) with tw pssible values fr the mde f P G, that is we have bimdality. When δ < 0 there is a single real rt (and a cnjugate pair f cmplex rts) and P G will be unimdal. In the case that δ = 0 there are three real rts, f which tw have the same value. This indicates that the a mde f P G is lcated n a catastrphe pint. The cusp pint ccurs in the cntrl space at a = b = 0, that

24 3 CHARACTERISING COMBINATIONS OF MODELS 24 is δ(0, 0) = 0, and there are three real rts with the same value (all equal t 0). See Sectin 5.2 f Pstn and Stewart (1978) fr mre details. Here, we are interested n thse pints (a,b) fr which δ(a,b) < 0 s that P G is unimdal. Thse pints satisfy 27a 2 < 4b 3 r (S 2 ν T) 3 < (M3 + νt) 2. (18) Ntice that while the gemetric cmbinatin f natural expnential family densities is always unimdal, fr τ 1 = τ 2 and ν > 1 the weighted average mean µ in the gemetric cmbinatin f t-distributins ges frm being a unique psterir mde in the regin δ > 0 t the unique anti-mde in the regin δ < 0 as the distance between the means µ 1 and µ 2 increases in the cntrl space. Figure (2) shws a trajectry acrss the cntrl space as µ 2 µ 1 increases. Ntice that fr τ 2 = τ 1 and ν > 1, the mean µ changes frm the unique mde t the unique anti-mde f P G as the distance between the individual means increases. µ 1 =µ 2 * * a b µ 1 <<µ 2 µ 1 >> µ 2 Figure 2: A trajectry acrss the cntrl space as the abslute distance between the

25 3 CHARACTERISING COMBINATIONS OF MODELS 25 means µ 2 µ 1 increases. As in ur case the predictive densities are univariate, the first tw mments f the gemetric cmbinatin f t-distributins can easily be btained cmputatinally by numerical integratin. Hwever, while the mean and variance can characterise P G when its density is unimdal they are nt apprpriate parameters t characterise multimdal densities. In such cases, mdes and spread arund thse wuld give a better descriptin f the density s shape. In general, accrding t Cbb (1978), the first fur mments wuld be apprpriate fr a prper characterisatin. In this manuscript we lk at the cnsequences n decisins f adpting the mean and the largest mde as pint estimates in predictin as we shall see in the applicatin sectin. S far, we have nt yet mentined hw the weights w 1,...,w k ( j w j = 1) can be determined. In the fllwing sectin we intrduce sme f the main methds fr btaining weights. In particular, fr ur applicatin in this paper, we will adpt a Bayesian methd called utperfrmance which allws an interpretatin f a weight in terms f the prbability f a scenari (which the mdel assciated with that weight represents) ccurring during the predictive hrizn. 3.4 Weights fr mdel cmbinatins One f the majr difficulties assciated with btaining weights fr the mdel cmbinatin appraches we have described in this paper, is that there is, up t date, n nrmative thery behind them t supprt their chice. Hwever, there is a bdy f literature with several peratinal methds each giving a particular interpretatin fr the weights. Many f them were develped fr linear cmbinatins mre due t the lack f alternative cmbining appraches

26 3 CHARACTERISING COMBINATIONS OF MODELS 26 than t a methdlgical requirement. The reader can refer t Winkler (1968), Bates and Granger (1969), DeGrt (1974), Bunn (1975), Smith and Makv (1978) and Cke (1991) amngst thers fr different interpretatins and methds fr btaining the weights. In this paper, we adpt the utperfrmance apprach prpsed by Bunn (1975, 1978) which interprets a cmpnent w jt f the weights vectr w t = (w 1,...,w k ) as the prbability that mdel M jt will prduce the mst apprpriate frecast f Y t. The sufficient statistics with which t learn abut w t is assumed t be bth w t 1 and the identity f the mdel that prduced the clsest frecast f Y t+h. This identity, viewed as a randm variable, is assumed t fllw a Multinmial (1,θ t ) distributin, θ t = (θ 1t,...,θ k 1,t ) where θ jt is the parameter assciated with the weight w jt, while the w t is interpreted as the prir mean f θ t. The parameter vectr w t is then successively updated in the usual Bayesian framewrk in the light f frecasts. Using this methd with the assumptin that the j-th mdel relative perfrmance abut frecasting Y t is independent f every ther mdel, it is easily checked that fr t 1 w j,t = (1 ρ t 1 )w j,t 1 + ρ t 1 (t 1) 1 r j,t 1 fr j = 1,...,k, where ρ t 1 = (t 1)/[ᾱ t 1 + (t 1)], with ᾱ t 1 = k j=1 α j,t 1, where α j,t 1 are the parameters f the cnjugate Dirichlet prir distributin f θ t, and r j,t 1 is the number f successes f frecasting mdel j up t time t 1. A mre attractive frmulatin f this apprach (Bunn, 1978) allws the prbability θ jt that M jt will be a mre apprpriate mdel than mdel M it at time t, t be revised in the light f all the mdels relative perfrmances. Pairwise cmparisns between mdels are set up and a relative perfrmance ranking is btained. The weight w jit is assumed t be

27 4 DECISION UNDER MULTIMODAL PREDICTIVE DENSITY 27 the psterir mean f the θ ji,t 1 whse density functin is nw assumed t be a beta with parameters (α ji,α ij ) t 1. These parameters are updated in the usual Bayesian way. Als assuming utperfrmance independence amng estimatrs, the estimate f the prbability f mdel j utperfrm all ther mdels, w jt, can be btained fr i j as k w jt w ujt. u=1 Such a methd f updating weights transfers directly nt bth the linear and the gemetric cmbinatins. Ntice that the rule is fair in that it is symmetric in a mdel s success if a priri we set w j,0 = w i,0 fr j,i = 1,...,k. Nw that we have intrduced the gemetric cmbinatin and sme f its main characteristics as well as described a Bayesian methd fr btaining the cmbining weights, we will shw in the next sectin an applicatin t the sales f beer in Zimbabwe. 4 Decisin under multimdal predictive density A Bayesian ptimal decisin Ŷt+h is the ne that minimises the expected value f a specified lss functin L(Ŷt+h,Y t+h ) w.r.t. a density functin P(Y t+h D t ) fr Y t+h, i.e. the Ŷt+h fr which the infimum value f EL P (Ŷ ) = E P[L(Ŷt+h,Y t+h )] = L(Ŷt+h,Y t+h )P(Y t+h D t )dy t+h, (19) is btained. E P dentes expectatin w.r.t. the prbability density functin P. In cases in which the expected lss has a single lcal minimum the ptimal decisin is that based n that chice. Hwever, in cases in which the expected lss has mre than ne pint f minimum, the selectin f a particular pint fr decisin must be made. Hwever,

28 4 DECISION UNDER MULTIMODAL PREDICTIVE DENSITY 28 an infinitesimal change in the parameters f L and/r P can result in a majr change in the decisin. It may be intuitive t think that when a decisin maker s infrmatin is characterised by a cntinuus unimdal prbability density functin and the lss functin is symmetric with just ne minimum, the expected lss will have a single pint f minimum. Hwever, as shwn by Smith et. al. (1981), this is nt the case in general except fr bunded mntnic increasing lss functins f Ŷt+h Y t+h and strictly psitive twice differentiable prbability density functins. Nte that the later cnditin include all symmetric unimdal densities as well as the Gamma and the Beta distributins. Nnetheless, there will exist a large set f lss functins which expected lss will have at least tw lcal pints f minima if the prbability density is nt strictly psitive and the first derivative f its lgarithm transfrmatin tends t zer as Y t+h tends t infinity. This is the case fr the lg-nrmal, the inverted Gamma, Paret and mst distributins in the F family. Again, refer t Smith et. al. (1981) fr a prf. We have seen in Sectin 3 that the linear and the gemetric cmbinatins f predictive densities may nt be unimdal. This can be the case fr linear cmbinatins f bth nrmal and t-distributed cmpnent densities as well as fr gemetric cmbinatin f t-distributed cmpnents. In the case f the linear cmbinatin f nrmal densities Smith et al. (1981) btained the cusp catastrphe crdinates (i.e the nrmal and the splitting factrs a and b in an equatin similar t (16) f sectin 3.3.3) fr the expected lss when Lindley s (1976) cnjugate (t the Gaussian distributin) lss is adpted. In that case, the expected lss E PL [Ŷt+h] =

29 5 FORECASTING BEVERAGE SALES IN ZIMBABWE 29 k j=1 w je pj [Ŷt+h µ j ] where E pj [Ŷt+h] = Ω L(Ŷt+h,Y t+h )p j (y t+h D t )dy t+h and L(Ŷt+h,Y t+h ) = L(Ŷt+h Y t+h ) = 1 exp{ 1 2 λ 1 (Ŷt+h Y t+h ) 2 } (20) with λ > 0. Fr k = 2, a cmmn variance σ 2 1 = σ2 2 = σ2 and a large λ, Smith et. al. (1981) shwed that the expected lss abve will have tw pints f minimum if (µ 1 µ 2 ) 2 > 4(σ 2 + λ). In this case, using the apprximate symmetry f the expected lss, the lwest minimum will be the ne nearest t µ 2 if w > 1 2, and the ne nearest t µ 1 if w < 1 2. A similar apprach can be used fr bth the linear and the gemetric cmbinatins f t-densities. Certainly that if L(Ŷt+h,Y t+h ) = (ŶT+h Y T+h ) 2, then the expected lss functin is always quadratic in Ŷt+h and thus a minimum pint can be btained. The use f cnvex lss functins with unbunded decisin parameter range, such as the quadratic lss abve, has been criticised by Kadane and Chuang (1978). The use f bunded (cnjugate) lss functins has been prpsed by Lindley (1976). 5 Frecasting Beverage Sales in Zimbabwe In this sectin, the predictive perfrmances f the gemetric and the linear cmbinatin methds are cmpared when applied t three plausible nn-similar RDLMs frmulated fr a series f beer sales frm a brewery in Zimbabwe. We als shw the cnsequences f decisins under the quadratic, the expnential and the lgarithmic lss functins. Fr that we frmulate three plausible RDLMs (each representing a past ecnmic and weather scenari) t be cmbined

30 5 FORECASTING BEVERAGE SALES IN ZIMBABWE 30 and tested in the last 12 mnth f data in the time series. The weight f each mdel in the cmbinatin methds culd be chsen by the BDM as her subjective prbability that the assciated mdel represents the ecnmic and weather scenari that she believes will prevail during the frecasting hrizn. In this applicatin the utperfrmance methd described in Sectin 3.4 was adpted instead as a surrgate fr the BDM assessment. The underlying series cmprises f 120 mnthly deseasnalised lg-transfrmed bservatins f ttal beer sales (Y ) frm April 1991 t December The series was deseasnalised as the riginal data presented a very strng and predictable seasnal behaviur with peaks at the end-f-the-year seasns (which cincide with the spring-summer seasns - frm Nvember t April) and trughs during autumn-winter seasns (frm May t Octber). The deseasnalised sales data was then further transfrmed with the use f a lgarithmic functin in rder t btain a series with an apprximately cnstant variability. The resulting series displayed in Figure 3(a) shws distinct trend patterns during three distinct perids f time. During the first perid, which we call perid A (frm April 1991 t December 1994), there was a linear but steep decline in sales. This was fllwed by a perid B (frm January 1995 t December 1997) f a psitive linear sales trend, and a perid C (frm January 1998 t December 2000) f a negative linear trend. Perid D in Figure 3 cmprises f the last year f data and was used t test the mdels in this sectin.

31 5 FORECASTING BEVERAGE SALES IN ZIMBABWE 31 A B C D A B C D Y X Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (a) Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (b) A B C D A B C D X X Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (c) Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (d) Figure 3: Deseasnalised series frm April 1991 t March 2001 f (a) ttal beer sales (Y ), (b) average deflated beer price per unit (X 1 ), (c) average maximum temperature (X 2 ) and (d) average rainfall (X 3 ). Figure 3 (b), (c) and (d) shws the deseasnalised time series f beer price, maximum temperature and rainfall respectively. 5.1 The frmulated RDLMs A plausible RDLM were determined frm the believed causal structure fr each ne f the perids A, B and C abve. In fact, there were a number f ecnmic and weather variables which had marked influence n sales at thse perids. Obviusly the mnthly (deflated)

32 5 FORECASTING BEVERAGE SALES IN ZIMBABWE 32 average beer price (X 1 ) as well as the mnthly average maximum temperature (X 2 ) were explanatry variables which were believed t influence the mnthly average beer sales (Y ) at any time perid. Hwever, each perid A, B and C had ther distinct explanatry factrs believed influential nly at thse perids. Each frmulated mdel is believed t represent the ecnmic and envirnmental scenari at the crrespnding perid f time it was btained. During perid A, the decline in beer sales was heavily influenced nt nly by gvernmental plicy f general price de-regulatin (fllwing the intrductin f a Wrld Bank and Internatinal Mnetary Fund (IMF) frmulated ecnmic structural adjustment prgramme (ESAP) implemented in Zimbabwe frm 1990 t see e.g. Wrld Bank publicatins, 1996) which had a strng effect n increasing prices, but als by a drught which ccurred between 1991 and 1992 and was characterised by a cmbinatin f lw rainfall and high temperatures. The plausible mdel M A frmulated fr perid A s scenari included X 1,X 2 as well as mnthly average rainfall (X 3 ) as explanatry variables. The regressin vectr was set as F A = (1,t,X 1,X 2,X 3 ) with the tw initial terms (1 and t) used t mdel level and trend respectively. The initial mean and variance values were set as (2,0.2) fr X 1, (27,2) fr X 2 and (50,10) fr X 3. The evlutin matrix G A was set as an (5 5) identity matrix. Discunt factrs were used t determine the bservatinal (V a ) and the evlutin (W A ) variances. The bservatinal variance was assumed unknwn but cnstant was btained by a discunt factr f The evlutin variance was determined with the use f blck cmpnent discunt with the trend blck f the variance having a factr f 0.99 and the regressin cmpnent blck having a 0.9 factr. In perid B, beer sales was thught t have been influenced by beer prices, which cntinued t rise steadily in nminal (but nt in deflated) terms, cmpunded by the lng lasting

33 5 FORECASTING BEVERAGE SALES IN ZIMBABWE 33 drught (started in perid A) with a severe impact n agricultural seasns leading t high temperatures, lw rainfall and hence lw crp sales fllwing pr harvests. The mdel M B fr perid B be included X 1, X 2, X 3 and ttal crp sales (X 4 ). The regressin vectr is F B = (1,t,X 1,X 2,X 3,X 5 ) with initial mean and variance values f (2,0.2), (27,2), (50,10) and (6,2) fr X 1,X 2,X 3,X 5 respectively. The evlutin matrix G B was set as a (6 6) identity matrix. The bservatinal variance, V B, was set with the use f a discunt factr f 0.99, while W B had a trend blck discunt factr f 0.99 and a regressin cmpnent discunt factr f 0.9. A B C D A B C D X X Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (a) Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (b) X A B C D Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (c) X A B C D Apr 1991 Mar 1993 Feb 1995 Jan 1997 Dec 1998 Nv 2000 (d) Figure 4: Transfrmed series frm April 1991 t March 2001 f (a) CPI (X 4 ), (b) ttal crp sales (X 5 ), (c) emplyment (X 6 ) and (d) ttal maize prductin (X 7 ). Perid C had beer sales influenced by the well dcumented scial, plitical and ecnmical

34 5 FORECASTING BEVERAGE SALES IN ZIMBABWE 34 events that ccurred in Zimbabwe in the late nineties. In fact, tw plitical events in 1997 and 1998 were thught t be mst influential fr the sudden turn in the ecnmy. First, the awarding f large grants and pensins t liberatin war veterans which were paid fr by an increase in sales taxes. Secnd, the gvernment annunced and began implementing the 1993 Land Designatin Act which saw the redistributin f agricultural land with cmpensatin cvering buildings and infrastructure rather than land value. Thse had an effect f significant utput decrease n the cmmercial agricultural sectr and related industries, Bnd (1999) and Brett (2004). Thse changes in the ecnmy were believed t have caused the decrease in sales f beer seen at that perid. Thse effects were represented in mdel M C by: (i) the large increase in the (deseasnalised and lg-transfrmed) cnsumer price index (CPI) (X 4 ); (ii) the decrease in (deseasnalised) ttal crp sales (X 5 ); (iii) the decrease in emplyment levels (X 6 ); as well as (iv) the decrease in (deseasnalised) ttal maize sales (X 7 ). Thse explanatry variables can be seen in Figure 4(a), (b), (c) and (d) respectively. The regressin vectr fr mdel M C was then set as F C = (1,t,X 1,X 2,X 4,X 5,X 6,X 7 ). The initial mean and variance values f (2,0.2), (27,2), (100,10), (6,2), (7.1,0.2) and (8,6) were chsen fr X 1,X 2,X 4,X 5,X 6 and X 7 respectively. The evlutin matrix G C was set as a (8 8) identity matrix. A discunt factr f 0.99 was chsen fr the bservatinal variance, V C. The system variance, W C, had a trend blck discunt factr f 0.99 while the regressin cmpnent factr was Predictive perfrmances and lsses As mentined abve, each f the plausible RDLMs M A, M B and M C were frmulated t represent a causal relatinship (r scenari) prevalent in each perid f time (A, B and