Workshop DGECFIN, Bruxelles, 16 November 010 Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Mahieu CORNEC INSEE Business Surveys Uni Draf, version 11/4/010 Absrac Among economic forecasers, i has become a more common pracice o provide poin projecion wih a densiy forecas. This realisic view acknowledges ha nobody can predic fuure evoluion of he economic oulook wih absolue cerainy. Inerval confidence and densiy forecass have hus become useful ools o describe in probabiliy erms he uncerainy inheren o any poin forecas (for a review see Tay and Wallis 000). Since 1996, he Cenral Bank of England (CBE) has published a densiy forecas of inflaion in is quarerly Inflaion Repor, so called fan char. More recenly, INSEE has also published a fan char of is Gross Domesic Producion (GDP) predicion in he Noe de Conjoncure. Boh mehodologies esimae parameers of exponenial families on he sample of pas errors. They hus suffer from some drawbacks. Firs, INSEE fan char is uncondiional which means ha whaever he economic oulook is, he magniude of he displayed uncerainy is he same. On he conrary, i is common belief among praciioners ha he forecasing exercise highly depends on he sae of he economy, especially during crisis. A second limiaion is ha CBE fan char is no reproducible as i inroduces subjeciviy. Evenually, anoher inadequacy is he parameric shape of he diribuion. In his paper, we ackle hose issues o provide a reproducible condiional and semi-parameric fan char. For his, following Taylor 1999, we combine quanile regression approach ogeher wih regularizaion echniques o display a densiy forecas condiional on he available informaion. In he same ime, we build a Forecasing Risk Index associaed o his fan char o measure he inrinsic difficuly of he forecasing exercise. The proposed mehodology is applied o he French economy. Using balances of differen business surveys, he GDP fan char capures efficienly he growh sall during he crisis on an real-ime basis. Moreover, our Forecasing Risk Index increased subsanially in his period of urbulence, showing signs of growing uncerainy. Key Words: densiy forecas, quanile regression, business endency surveys, fan char. JEL Classificaion: E3, E37, E66, C
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa 1. Inroducion Usually, insiuions display economic poin forecass. However, he forecas is no free of uncerainy: assuming ha forecass are no biased, hey may be considered as he expeced figure given he available informaion. Indeed, many shocks can affec he forecas in regard o oil prices, exchange raes, ineres raes, and oher variables. Anoher reason is ha he behaviors of economic agens can only be esimaed imprecisely over he pas (when hey do no change). The poenial scenarios are herefore numerous: forecasers hus condense heir forecas ino a single, baseline scenario. In he end, readers may lose sigh of he uncerainy inheren in his ype of exercise. Among economic forecasers, i has herefore become a more common pracice o provide poin projecion wih a densiy forecas. The longes running series of macroeconomic densiy forecass daes back o 1968, when he ASA and he NBER iniiaed a survey of forecasers. For a deailed hisorical review, we refer o Tay and Wallis (000). In paricular, o dispel heir risk, he Cenral Bank of England (cf. Brion e al 98) and INSEE display a fan char o provide a concise illusraion of he uncerainy affecing poin forecass. This realisic view recognizes ha fuure evoluion of he economic oulook canno be prediced wih absolue cerainy. Confidence inervals and densiy forecass have appeared as useful ools o describe in probabiliy erms he uncerainy inheren o any poin forecas (for a review, see Tay and Wallis 000). Boh mehodologies suffer from some drawbacks. Firs, he INSEE fan char is uncondiional which means ha he magniude of he displayed uncerainy is he same, whaever he economic oulook is. On he conrary, i is a common belief among praciioners ha he forecasing exercise highly depends on he sae of he economy, especially during crisis. Secondly, CBE s fan char is condiional bu his condiionaliy comes from he subjecive assessmens by he members of he Moneary policy comiee, and is herefore no reproducible. Evenually, anoher limiaion is he parameric shape of he disribuion, which is usually assumed o be exponenial, ha is wihou fa ails. In his paper, we ackle hese issues o provide a reproducible mehodology o build a non parameric condiional densiy forecass. This paper is organized as follows: in secion 1, we review he main mehods o describe uncerainy used by boh INSEE and CBE, namely confidence inervals and densiy. We inroduce our main noaions and we give a brief descripion of French business surveys in secion. In secion 3, we inroduce our mehodology o derive condiional densiy forecass and o consruc our Forecasing Risk Index. Evenually, he proposed mehodology is applied o he French economy in secion 4. Uncerainy descripion A common way o describe uncerainy is o consider he fuure evoluion of he economy (i.e. GDP growh rae) as he realizaion of a coninuous random variable. The uncerainy is hen fully characerized by he random variable densiy. A peaky densiy means a small uncerainy. On he conrary, a large uncerainy is ranslaed ino a loose disribuion.
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Confidence inervals Confidence inervals may be he firs simple mehod o describe he poin forecas uncerainy (for a review see Tay and Wallis 000). The principle is he following : he economic forecaser provides an inerval ogeher wih his poin forecas. The fuure observed value is hen supposed o lie wihin his inerval wih a specified probabiliy. For example, if he forecaser provides 95%-confidence inervals, he observed value is supposed o lie around 95% of he ime ino hese inervals. I is also common o provide confidence inervals a differen probabiliy levels, i.e. 0%, 50%, 99% confidence inervals. In he case of confidence inervals, he lengh of he inerval measures he level of uncerainy. For example, we usually expec uncerainy o grow as he forecasing horizon increases. This is displayed by an increasing lengh of confidence inervals for a specified level of probabiliy (cf. figures 1 and ). However, if heir simpliciy makes confidence inervals very aracive, hey do no compleely describe uncerainy. Densiy forecass Densiy forecass have hus become more and more appealing since hey fully describe uncerainy. Noice ha hey can also be used o easily derive confidence inervals based on he appropriae disribuion quaniles. In he case of densiy, he level of uncerainy is measured by he inverse of he sharpness of he disribuion. Many indicaors have been proposed o characerize he level of uncerainy, among hem variance and enropy (see. Lugosi). Since 1996, he Cenral Bank of England has a published a densiy forecas of inflaion in is quarerly Inflaion Repor, so called fan char (cf. figure 1). More recenly, INSEE has also published a fan char of is GDP predicion in he Noe de Conjoncure (cf. figure ). 3
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 1 CBE inflaion fan char Figure INSEE GDP fan char 4
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa INSEE and CBE densiy esimaion Boh mehodologies assume ha he error densiy belongs o exponenial families. Esimaions of he parameers are based on he sample of pas forecas errors. INSEE fan char displays a normal disribuion: 1 ( y ) f ( y) exp( ). Thus INSEE perceives he possible GDP oucomes symmerically dispersed around he cenral mos probable value. This is he famous bell-shaped curve (cf. figure 3 for he densiy of a sandard normal). Figure 3 densiy of a sandard normal disribuion Bell curve Densiy 0.0 0.1 0. 0.3 0.4-3 - -1 0 1 3 x Cenral Bank of England percepion is ha in some circonsances he forecas error is more likely o be in one direcion han he oher. In saisical erms, heir fan char disribuion is skewed. Tha is why hey chose a paricular form of saisical disribuion called wo-piece normal (cf. figure 4): 5
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa 1 1 x f ( y) exp( ( ) ( )( ) ) x x. x Evenually, CBE s mehodology allows he injecion of subjeciviy by deforming he densiy shape. Figure 4: example of a skewed disribuion by CBE Even if boh insiues display densiy forecass, a fundamenal philosophical difference remains beween boh mehodologies. INSEE mehodology aemps o capure an inrinsic uncerainy. assuming no change in he volailiy of growh figures and he mehodology used by INSEE forecasers during he period, he disribuion of forecasing errors calculaed from pas exercises is a reliable indicaor of he disribuion of fuure errors (cf. Noe de conjoncure) we have chosen no o depar from he hisoric variance of forecasing errors, as he injecion of a dose of subjeciviy seems hard o jusify ( cf. Noe de conjoncure) 6
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa On he conrary, CBE s fan char somehow measures he CBE subjecive uncerainy abou: The aim of he fan char has been o convey o he reader a more accurae represenaion of he Bank s subjecive assessmen of medium-erm inflaionary pressures. I is herefore a forward-looking view of he risks o he forecas, no an exrapolaion of pas uncerainy. For more insighs abou his debae, we refer o B. de Finei (1975) and references herein. Drawbacks Condiional versus uncondiional The firs pracical and imporan consequence is ha he INSEE fan char is uncondiional. Thus, whaever he economic oulook is, he magniude of he displayed uncerainy is he same. I means ha on a long-erm basis i is supposed o be correc on average. In oher words, during a recession, he usual uncondiional cenral confiden inerval can be significanly wrong. Le s ake a simple example by comparing forecasing wih flying a plane: when crossing a urbulence area, flying ges more difficul. In he same way, during a crisis period, he inrinsic uncerainy of he forecasing exercise increases. Insead of uncondiional forecasing, condiional forecasing which can change from one period o anoher should be favoured o describe uncerainy. We illusrae he ineres of condiional forecasing versus uncondiional forecasing by a simple oy model. Toy Model Le Z be a random variable ha describes he sae of he economy a ime and ha can only ake wo values: A for acceleraion wih probabiliy p, D for deceleraion 1 p. E Y Le Y be a random variable describing he GDP growh rae. To make i simple, we suppose he condiional disribuions of ( y Z ) Z, Var( y Z Z ) wih D A A D (bu he volailiy is higher). Y o be such ha: (he growh rae is smaller during recession) equal o Then he descripion of uncerainy by an uncondiional forecaser is E Var( y I )) p D ( 1 p) A. ( On he conrary, for a condiional forecaser, uncerainy as measured by Var y I ) equal o D during recession and o A during acceleraion. ( which is would be We can deduce ha on a long-erm average he uncondiional descripion is correc. However, a each dae, knowing he informaion Z, i is eiher oo small or oo big. 7
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Indeed, since A p D ( 1 p) A D, he uncondiional descripion of uncerainy is eiher oo loose (during acceleraion period) or oo sharp (during crisis). The uncondiional forecas error neglecs he informaion embedded in i. On he conrary, since CBE s mehodology allows he injecion of subjeciviy from one period o anoher, heir fan char is hus condiional. However, heir mehod is no reproducible as i is a measure of CBE s subjeciviy. Exponenial disribuion versus nonparameric To check a densiy forecas, we can use Talagrand s diagram. The principle is he following: Fˆ ( Y ) is supposed o be a sequence of independen idenically disribued uniform variables on 0,1 wih Fˆ he forecased cumulaive disribuion funcion. The hisogram of Fˆ ( Y ) is called Talagrand s diagram and is supposed o be a sraigh line if our forecas is correc (cf. Dowd, K., 004). Figure 5 displays Talagrand s diagram for he INSEE fan char. Insead of a sraigh line, i exhibis a concave profile indicaing ha he probabiliies of righ exreme risks are overesimaed. 8
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 5: Talagrand s diagram of he INSEE fan char INSEE Talagrand diagram Frequency 0 5 10 15 0 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 The aim of his paper is o sugges a reproducible mehodology o build condiional densiy forecass. In he same ime, we will release he consrain of he parameric disribuion shape. Noaions and business surveys descripion 9
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Recall ha Z describes he sae of he economy in our oy model. The role of Z will be played by business surveys. Indeed, business surveys are a useful source of informaion when forecasing, for hey presen hree ypes of advanages: (1) hey provide reliable informaion coming direcly from he economic decision makers, () hey are rapidly available (abou a monh afer he quesionnaires are sen), on a monhly, bimonhly or quarerly basis, and (3) hey are subjec o small revisions (each publicaion presens a generally negligible revision, only on he preceding poin). The disseminaed saisics compiled from hese surveys are usually balances of opinions. Since here are a large number of available survey variables (cf. able 1), composie indexes (CI) have been developed over he years o provide suiable summaries by exracing he common rend, and suppressing he undesirable "noise" of numerous daa. In paricular, he French composie indicaor (cf. figure 6) gives an assessmen of he global climae of he whole French economy. Figure 6: French composie indicaor and year-on-year GDP growh rae French Composie Indicaor and GDP growh rae 130 6,5 10 5,0 110 3,5 100,0 90 0,5 80-1,0 70 -,5 60 199 1994 1996 1998 000 00 004 006 008 010-4,0 CI GDP growh rae 10
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Table 1: balances of opinions used in he French CI (able from Bardaji 09) Denoe he French CI by (i ) 1. Recall ha our goal is o forecas he firs release of he quarerly French GDP growh rae, denoed by y. The firs release will be published only 45 days afer he end of he curren quarer. Usually, economiss also forecas he nex quarers. For he sake of simpliciy, we will resric ourselves o he forecas of he curren quarer. Finally, we define y ( y,, y ). Our quarerly hisorical daa of French GDP firs release sars from 1988 Q1. : 1 1 To be more precise, we define i as he mean of he hree las known monhly releases when forecasing akes place. 11
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Mehodology Adaping he sequenial framework in Biau and Para (009): a each quarer 1 T, we observe business surveys daa z bu he firs release of he q-o-q GDP growh rae y is unknown. To model uncerainy, we consider ( z ) and ( y ) as he oucome of random variables ( Z ) and (Y ) such ha he process ( Z, ) is joinly coninuous saionary ergodic. Y Z In a sequenial version of he densiy predicion problem, he forecaser is asked o guess he nex condiional densiy of Y of a sequence of random variables Y,..., 1 Y 1 wih knowledge of he pas observaions y 1 y1,, y 1 and z z, 1, z. In oher words, he observaions y1, y and z1, z are revealed one a a ime, beginning wih ( y0, z1),( y1, z ), Quanile regression echniques provide a suiable semi-parameric ool o achieve a proper densiy forecas. We inroduce here he main oulines of he quanile regression mehodology. Brief inroducion o quanile regression (Koenker and Basse 78) developed a heory for he esimaion of he quaniles of a variable Y based on pas observaions. The saring poin is o noice ha many probabiliy quaniies can be characerized by a minimizaion problem. For example, he expecaion of a L random variable saisfies E( Y ) m arg min m R E( Y ). In he same way, he median corresponds o med( Y ) arg min E Y m. m R The naural quesion is hen: can he quanile funcion Q ( ) : inf R : F ( ) Y Y be described by a minimizaion problem? (Koenker and Basse 78) generalizes he observaion ha 1 minimizing he L1 loss yields o he median. To ha end, hey ransform he L loss ino a suiable loss funcion called pinball loss funcion or also check loss (cf. figure 7). ( y ) : y (1 ) y.wih y : max(0, y), y : max(0, y) 1
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 7: pinball loss Then hey obain (for a proof see Biau and Para (009)): Q ( ) arg min E ( Y m) Y m R These ideas exend readily o condiional quaniles Q ( z) : inf R : F ( z) We can hen verify ha Q ( z) arg min (.) E ( Y m( z)) X z) Y m Y Y. (Koenker and Basse 78) considered m(z) of a linear form z ' and he coefficiens are esimaed by minimizing ' ( y i z i ) on pas sample observaions. Their esimaed quanile i 1 curve is hen defined by ˆ ' Q ( z) : z i ˆ. Y To jusify quanile regression, assume he daa o be generaed by he following model wih heeroscedasiciy: Y ( Z ) ( Z ) wih Z ) : E( Y Z ) and Z ) : V ( Y Z ) ( (, and an error erm independen of Z. 13
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Based on he aforemenioned model, he condiional quanile of Y given Q ( Z ) ( Z ) ( Z ) Q ( ). Y Z has he form (Koenker and Basse 8) showed ha if ( Z ) and ( Z ) are linear funcions of Z hen quanile regression esimaes are asympoically consisen. The appeal of quanile regression is ha pas observaions of he quaniles are no required. Reference Model The firs sep is o find a proper se of explicaive variables o boh: -reduce he uncerainy since Var( Y ) E( Var( Y Z )) Var( E( Y Z )) E( Var( Y Z )) In oher words, he square error of an uninformed forecaser is equal o he error of an informed forecaser plus a residual erm (knowledge erm). -and explain correcly he remaining uncerainy (i.e. Var ) / Var( ( Z )) small). ( As menioned above, here is a huge number of balances of opinions (hundreds). Thus, we face he curse of dimension. The French composie indicaor (FCI) is a way o reduce he dimension by assessing he global climae of he whole French economy. In a simplified framework, FCI would be roughly equal o GDP growh rae Y I. However, in hese surveys, enrepreneurs are asked o give a qualiaive appreciaion on he occurred or expeced changes of some variables of ineres (oupu, order book, foreign order book, invenories,...) hrough he hree following caegories: increase, no change, decrease. There is hus a difference beween qualiaive answers from microeconomic surveys and quaniaive macroeconomic measures. To fill he gap, we sugges a model of he following form: Y Z ( Z ) u ' wih Z (1, Y, I, I I ), : (,,, and u available informaion. : 1 0 1 3) a whie noise independen of he 14
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Poin forecas Once he model se, he forecaser needs firs a poin forecas. I is usual (hough no opimal for predicion as we do no need he esimaor o be unbiased) o esimae by Ordinary Leas Squares (OLS) and we ge: Yˆ ' ˆ Z ' 1 ' wih ˆ : ( Z Z ) Z Y. The uncondiional variance E Var( Y Z )) of he error erm can hen be esimaed by ( 1 E ˆ ( Var( Y )) : ˆ I Y Y. T Condiional confidence inervals A simple way o derive condiional confidence inervals for Y is o use quanile regressions. In order o consruc a confidence inerval wih probabiliy a leas 90%, i is sufficien o esimae he quanile curve a levels 5% and 95% by quanile regression on he pas available observaion ( y0, z1),...,( y 1, z ) : Qˆ Y (0.05 z) and Qˆ Y (0.95 z). Our confidence inerval a quarer can be wrien as CI 90% : [ Qˆ Y (0.05 z ), Qˆ Y Condiional Densiy forecass (0.95 z )] Recall ha he quanile curve z ) compleely describes he disribuion. Thus, i would QY ( be sufficien o give he esimaed quanile curve Qˆ Y ( z ). Insead, i is a common pracice among forecasers o display a densiy forecas y fˆ ( y ). z We give here a simple heurisic o derive a densiy forecas from he esimaed quanile curve ˆ ( ). QY z 15
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Recall ha QY ( U z ) wih U a uniform disribuion is disribued as Y z. Moreover, y Y f 1 Y ( y z ) : K( ) Th h Y z f ( y z ) Y under suiable condiions. Box 1 Heurisic pah o esimae ( y z) f Y Se N a large number (a pracical choice is 100). Compue a each u i Evenually, compue bandwidh i : ; 1 i N, y : Q ˆ ) N ( i ui z y yi f ˆ 1 Y ( y z) : K( ) wih K a Epanechnikov kernel and h he Th h Forecasing Risk Index: FRI In his secion, we aim a building a Forecasing Risk Index for each quarer, associaed o his fan char o measure he inrinsic difficuly of he forecasing exercise. A quarer, he difficuly of forecasing is linked o he sharpness of he disribuion: he sharper he disribuion, he easier forecasing is. We mus hus se a quaniaive indicaor o measure how much he disribuion is spread. In he lieraure, classical measures are variance or enropy. As far as he roo mean square error is concerned, he L norm (roo of variance) suis our needs. The definiion of condiional variance is given by: Var( Y z) We esimae Var( Y z) by Vˆ ar( Y z) : Var ˆ ( Y ) f z We are now in posiion o define our Forecasing Risk Index for each quarer as FRI ˆ : Var( Y z ) 16
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa When he poin forecas is based on our reference model, he expeced forecasing error is approximaely equal o he value of he Forecasing Risk Index. Ou-of-sample Validaion Recall he forecaser is asked o guess he nex condiional densiy of Y of a sequence of random variables Y,..., 1 Y 1 wih knowledge of he pas observaions y 1 y1,, y 1 and z z,,. 1 z Thus, all previous quaniies mus be compued on a real ime basis: ' 1 ' 90% ˆ : ( Z Z ) Z Y, CI, ˆ ( y ) Inerval forecas fy z Yˆ ˆ Z ' wih If he confidence inervals for Y are correc, y should lie in CI 90% around 90% of he ime. Densiy forecas To check our assumpions on real ime basis, we can noice ha Y is supposed o follow fˆ ( y ). Thus he cumulaive funcion Fˆ ( Y x ) : fˆ ( y z ) dy should be independen random Y z variables disribued as uniform variable on [ 0,1]. The hisogram of F ˆ ( Y z )) Y ( can be ploed and compared wih a uniform densiy: his is he classical Talagrand s diagram. Oher more sophisicaed ess migh be considered (see Berkowiz 003, Clemen 004, Wallis 003 and references herein). 17
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Numerical resuls Poin forecass Z : 1 (1, Y, I, I I ) () Parameers esimaion by OLS (from 1988 Q1 o 010Q1) leads o he following char: Table 1 Parameers esimaion Esimae Sd. Error T value p-value Inercep 0.61 0.05 11.84 *** Y -0.41 0.09-4.4 *** 1 I 0.09 0.01 6.89 *** I I 0.10 0.01 8.8 *** Figure 8 displays he ou of sample forecass ogeher wih he firs GDP release. The residual sandard error is 0.31. Var( Y ) E( Var( Y Z )) Var( E( Y Z )) Recall ha we have. In oher words, he square error of an uninformed forecaser is equal o he error of an informed forecaser plus a residual erm Var( Y ) 0.4 (knowledge erm). We esimae E( Var( Y )) 0.1 and Z. Thus, our reference model gives us a 50% gain of accuracy for he L loss. 18
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 8: ou of sample poin forecass Ou-of-sample forecass GDP growh rae (%) -1.0-0.5 0.0 0.5 1.0 cpib_pr OoS Forecass 1995 1997 1999 001 003 005 007 009 19
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Condiional confidence inervals In able, we obain he esimaes (from 1988 Q1 o 010Q1) for he quanile regression coefficiens. Ineresingly, he coefficien of he acceleraion erm is almos zero for he 95% quanile whereas i is significan for he 5% quanile. In oher words, roughs are much worse han peaks are grea. Table Parameers esimaion hea = 0.05 Thea = 0.95 Inercep 0.17 1.3 Y -0.35-0.46 1 I 0.09 0.06 I I 0.10-0.00 Figure 9 displays GDP firs releases ogeher wih 90% confidence inervals of is las forecas. Noice ha he lengh of he inerval depends on he forecasing dae. The percenage of GDP firs releases ouside our 90%-confidence inervals is 13% esimaed on our hisorical daa. 0
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 9: Ou of sample 90% confidence inervals Ou-of-sample forecass GDP growh rae (%) -1.5-1.0-0.5 0.0 0.5 1.0 1.5 cpib_pr OoS Quan forecass 5 % OoS Quan forecass 95 % 1995 1997 1999 001 003 005 007 009 1
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Condiional Fan char Figures 10 show densiy forecass (mehodology from box1) ogeher wih GDP firs release (verical line) a differen quarers (from 008 Q o 010Q). Figure 10: densiy forecass and GDP firs release (verical line) As menioned above, we draw Talagrand s diagram for our new fan char (cf. figure 11). In comparison wih INSEE diagram (cf. figure 11), we can see ha our new mehod leads o a beer esimae of probabiliy ails. The bumpy fan chars come from he semi-parameric mehodology as no unimodal disribuion has been imposed. This raised he philosophical quesion if he error disribuion should be unimodal, parameric or nonparameric. If necessary, i is always possible o display a unimodal densiy bu wih a condiional variance. For his, i is sufficien o plug he forecasing risk index, as he condiional sandard error of a gaussian densiy forecas.
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 11: Talagrand s diagram for he new densiy forecas Talagrand diagram for he new fan char Frequency 0 5 10 15 0.0 0. 0.4 0.6 0.8 1.0 Forecasing Risk Index Figure 1 displays he Forecasing Risk Index. On an ou-of-sample basis, he FRI exhibis clear signs of urbulence during he previous crisis. This gives an early signal of growing uncerainy. For example, when FRI is equal o 0.6 in 008Q4, i can be inerpreed as an expeced error of 0.6 for a poin forecas based on our reference model. I is worh recalling ha our reference model is only based on surveys daa. Thus, any uncerainy ha may be no refleced by business surveys such as uncerainy on oil prices or exchange raes will no be displayed by our forecasing risk index. However, Talagrand s diagram shows ha our fan chars enjoy nice empirical properies. Moreover, in a bayesian manner, an economic forecaser could always sar from he uncerainy as measured by he surveys daa and add his own subjecive assessmen of uncerainy. 3
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 1: Forecasing Risk Index Forecasing Risk Index sqr(variance) 0. 0.3 0.4 0.5 0.6 0.7 1995 000 005 010 Time Focus on he crisis period Our new GDP densiy forecas capures efficienly he growh sall during he crisis on a real ime basis. Indeed, he firs release of 008 Q4 (-1.3% in volume) lies inside he confidence inerval (cf. figure 13). On he conrary, i was almos considered as an oulier by he INSEE fan char (cf. figure 15). I is also ineresing o compare resuls during he rebound of 009 Q. The firs release (+0.3%) was in our confidence inerval bu i was far ouside he INSEE fan char (cf. figure 16). I could be surprising ha our confidence inerval capures efficienly he growh sall during he crisis on an realime basis. Indeed, before 008 Q4, he minimum q-o-q GDP growh rae of our hisorical daa was - 0.6%, far above he -1.3% of 008 Q4. This feaure is made possible boh by he linear form of every quanile regression and by new exreme business surveys values during he las crisis. 4
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 13: Ou-of sample confidence inervals during he las crisis Ou-of-sample forecass GDP growh rae (%) -1.5-1.0-0.5 0.0 0.5 1.0 1.5 cpib_pr OoS Quan forecass 5 % OoS Quan forecass 95 % 006 007 008 009 010 5
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 14: densiy forecass during crisis Char on 008 Q Char on 009 Q1 Densiy 0.0 0.5 1.0 1.5 GDP Firs release Densiy 0.0 0.4 0.8 1. GDP Firs release 0.0 0.5 1.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 N = 99 Bandw idh = 0.10 N = 99 Bandw idh = 0.1098 Char on 008 Q3 Char on 009 Q Densiy 0.0 0.4 0.8 GDP Firs release Densiy 0.0 0.4 0.8 1. GDP Firs release -0.5 0.0 0.5 1.0 1.5-0.5 0.0 0.5 1.0 1.5.0 N = 99 Bandw idh = 0.168 N = 99 Bandw idh = 0.1531 Char on 008 Q4 Char on 009 Q3 Densiy 0.0 0.4 0.8 GDP Firs release Densiy 0.0 0.5 1.0 1.5 GDP Firs release - -1 0 1-1.0-0.5 0.0 0.5 1.0 N = 99 Bandw idh = 0.179 N = 99 Bandw idh = 0.1088 6
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Figure 15: INSEE fan char on 008 Q3 GDP firs release 008Q4 Figure 16: INSEE fan char on 009 Q1 GDP firs release 009Q 7
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Conclusion In his paper, we developed a reproducible mehodology o build condiional densiy forecass based on business surveys daa. Even hough resuls are no fully comparable since he forecasing device is no exacly he same, his mehodology seems able o beer cach he uncerainy paern associaed wih forecass of French GDP. In paricular, we derive a Forecasing Risk Index from he condiional densiy forecass, which aims a quanifying he inrinsic uncerainy of a poin forecas. Ineresingly, our index mehodology leads o uncerainy ha increases on recessions, a characerisic ha Insee forecass have wienessed in he recen crisis. References Bardaji, J., Clavel L. and Talle, F. (010). Consrucing a Markov-Swiching Turning Poin Index Using Mixed Frequencies wih an Applicaion o French Business Survey Daa, Journal of Business Cycle Measuremen and Analysis, forhcoming. Berkowiz, J. (001), Tesing Densiy Forecass, Wih Applicaions o Risk Managemen, Journal of Business and Economic Saisics, 19: 465-474. Biau, G. and Para, B. (009). Sequenial quanile predicion of ime series. Brion, Erik, Paul Fisher, and John Whiley (1998),.The Inflaion Repor Projecions: Undersanding he Fan Char, Bank of England Quarerly Bullein, 38(1), 30-37 B. de Finei. Theory of probabiliy. Vol. 1-. John Wiley & Sons Ld., Chicheser, 1990. Reprin of he 1975 ranslaion. Clemens, M. P. (004), Evaluaing he Bank of England Densiy Forecass of Inflaion», Economic Journal, 114, 844 866. Diebold, F.X., Tay, A.S. and Wallis, K.F. (1997). Evaluaing densiy forecass of inflaion: he Survey of Professional Forecasers. Discussion Paper No.48, ESRC Macroeconomic Modelling Bureau, Universiy of Warwick and Working Paper No.68, Naional Bureau of Economic Research, Cambridge, Mass. Dowd, K., (004). The inflaion fan chars: An evaluaion. Greek Economic Review 3, 99 111. INSEE Noe de Conjoncure for June 008, pages 15 o 18 Koenker, Roger W. and Gilber W. Basse, (1978). Regression quaniles. Economerica 46, 33-50 Koenker RW and Basse GW (198). Robus ess for heeroscedasiciy based on regression quaniles. Economerica 49, 3-51 G. Lugosi, Concenraion-of-measure Inequaliies, Lecure Noes. 8
Consrucing a condiional GDP fan char wih an applicaion o French business survey daa Taylor, J.W. (1999b). A quanile regression approach o esimaing he disribuion of muliperiod reurns, Journal of Derivaives, 7 Fall, 64-78. Tay, A.S. and Wallis, K.F. (000). Densiy forecasing: a survey.journal of Forecasing, 19, 35-54. Wallis, K. F. (003), Chi-squared Tess of Inerval and Densiy Forecass, and he Bank of England s Fan Chars, Inernaional Journal of Forecasing, 19: 165-175. 9