On the Use of Density Forecasts to Identify Asymmetry in Forecasters Loss Functions

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1 Business and Economic aisics ecion JM 9 On e Use o Densiy Forecass o Ideniy Asymmery in Forecasers Loss Funcions Kajal Lahiri 1, Fushang Liu 1 Deparmen o Economics, Universiy a Albany-UNY, 14 Washingon Avenue, Albany, NY 16 Massachuses Deparmen o Revenue, 1 Cambridge ree, Boson, MA 114 Absrac We consider how o use inormaion rom repored densiy orecass rom surveys o ideniy asymmery in orecasers loss uncions. We show a, or e ree common loss uncions - Lin-Lin, Line, and Quad-Quad - we can iner e direcion o loss asymmery by jus comparing poin orecass and e cenral endency (mean or median o e underlying densiy orecass. I we know e enire disribuion o e densiy orecas, we can calculae e loss uncion parameers based on e irs order condiion o orecas opimaliy. This meod is applied o orecass or annual real oupu grow and inlaion obained rom e urvey o Proessional Forecasers (PF. We ind a orecasers rea underpredicion o real oupu grow more dearly an overpredicion, reverse is rue or inlaion. Key Words: Asymmeric loss uncion; Densiy orecass; e urvey o Proessional Forecasers; 1. Inroducion Alough e imporance o asymmeric loss uncion or model esimaion, selecion, predicion, orecas evaluaion, and raionaliy es is widely recognized, ew sudies ry o esimae e loss uncion rom daa direcly. One ecepion is Ellio, Komunjer, and Timmermann (5, 8, hereaer reerred o as EKT. They propose o esimae e parameers o loss uncion by GMM meod based on momen condiions implied by orecas raionaliy. This meod relies on wo assumpions. Firs, e loss uncion parameers are consan over ime; and second, orecass are raional. As saed in EKT, ey back ou e loss uncion parameers consisen wi e orecas being raional. In is paper, we consider how o use inormaion rom densiy orecass o learn abou e loss uncions. ince orecasers orm eir poin orecass based on wha ey believe o be e daa generaing processes (densiy orecass and eir loss uncions, we can reverse is process and learn abou orecasers loss uncions by comparing orecasers poin orecass and densiy orecass or e same arge. The advanage o is meod is a we can rela e wo assumpions needed in EKT s GMM meod: e poin orecass and densiy orecass need no o be raional and e loss uncion parameers need no o be consan over ime. Moreover, we do no need o know e acual values o e arge variable. We jus compare e wo ypes o orecass. We show how is meod can be applied or e common loss uncions in empirical work Lin-Lin, Line, and Quad-Quad loss uncions. 396

2 Business and Economic aisics ecion JM 9 The res o e paper is organized as ollows. In secion, we discuss e daa used in e paper - e urvey o Proessional Forecasers (PF and e real ime macro daa. From ese wo daa ses we consruc poin and densiy orecass or e same arge: annual inlaion rae and real oupu grow rae. In secion 3, we se up e general ramework o opimal orecass under asymmeric loss uncion. In secion 4, we show a i we know only e cenral endency (mean or median o densiy orecass, we can iner e direcion o loss asymmery by jus comparing poin orecass and e cenral endency o densiy orecass or e ree common loss uncions menioned above. In secion 5, we show a i we know e enire densiy orecass, we can learn no only e direcion bu also e degree o loss asymmery. ecion 6 concludes e paper.. Daa Two daa ses are used in is paper. One is e urvey o Proessional Forecasers (PF, which provides densiy orecass or annual inlaion and real oupu grow, as well as poin orecass or quarerly oupu-price inde and real oupu. The oer is e real ime macro daa, which provides e acual values o quarerly price inde and real oupu in real ime. We combine ese acual values wi e PF poin orecass or quarerly price inde and real oupu o calculae e implied poin orecass or annual inlaion and real oupu grow. Bo e PF daa and e real ime macro daa are available rom e Federal Reserve Bank o Philadelphia web sie. PF was sared in e our quarer o 1968 by e American aisical Associaion and e Naional Bureau o Economic Research, and was aken over by e Federal Reserve Bank o Philadelphia in June 199. The respondens are proessional orecasers rom e academia, governmen, and business. The survey is mailed our imes a year, e day aer e irs release o e NIPA (Naional Income and Produc Accouns daa or e preceding quarer. Mos o e quesions ask or poin orecass on a large number o variables or e curren and e ne our quarers, including e level o quarerly price inde and real oupu. A unique eaure o e PF daa se is a respondens are also asked o provide densiy orecass or year-over-year grow raes in aggregae oupu and oupu price inde in e curren and ollowing year. Or more speciically, ey are asked o provide probabiliies a e grow raes will all in dieren inervals 1. In is paper we use inlaion orecass during 1968Q4-3Q4 and real oupu grow orecass in 1981Q3-3Q4. Forecass on real oupu grow are no available beore 1981Q3. To calculae e implied poin orecass or annual inlaion and real oupu grow, we need o know e acual values o price inde and real oupu in quarers beore e orecasing period. Theoreically, we should no use e mos recen daa because a was no available o orecasers when ey made e orecass. The real ime daa se provided by e Federal Reserve Bank o Philadelphia is a good choice. This daa se repors values o variables as ey eised in e middle o each quarer rom November 1965 o e presen. Thus, or each vinage dae, e observaions are idenical o ose one would have observed a a ime. Forunaely, is is also approimaely e dae when orecasers o PF are asked o submi eir orecass. In addiion, e deiniion o e price inde and real oupu in is daa se is consisen wi a in e PF daa se. 1 More descripions o is daa se can be ound in Lahiri and Liu (6. ee Croushore and ark (1 or descripions o is daa se. 397

3 Business and Economic aisics ecion JM 9 o we can convenienly combine e PF daa and e real ime macro daa o calculae e implied poin orecass or annual inlaion and real oupu grow. To see how is is done, consider a densiy orecas π i,, which is made h quarers beore e end o year abou e arge variable (annual inlaion or real oupu grow in year by orecaser i. A poin orecas or e same arge and wi e same orecas horizon can be consruced as ollows. I e densiy orecas is made in e irs quarer o year, e corresponding poin orecas or e value o e arge variable in year is where P i, j P i,,1+ Pi,,+ Pi,,3+ Pi,,4 i, = 1 1 (1 A 1,1+ A 1,+ A 1,3+ A 1,4, is responden i s prediced value o e price inde or real oupu in e j quarer o year and A, is e real ime acual value o e price inde or real oupu j in e j quarer o year. imilarly i e densiy orecas is made in e second quarer o year, e corresponding poin orecas is A,1+ Pi,,+ Pi,,3+ Pi,,4 i, = 1 1 ( A 1,1+ A 1,+ A 1,3+ A 1,4 Poin orecass in e ird and our quarer can be consruced similarly. 3. Opimal Poin Forecas under Asymmeric Loss Funcion uppose we wan o orecas e value o y in year, h quarers ahead. When viewed h quarers beore e end o year, y is a random variable and is bes described by a densiy uncion condiional on e inormaion available a ime -h, I. This densiy uncion is oen reerred o as orecas densiy and can be denoed as, where e superscrip o means a e orecas densiy describes e rue daa generaing process. Forecasers may no know e rue daa generaing process. Wha ey believe o be e rue daa generaing process, or eir densiy orecass or e same arge, are denoed as, which we call subjecive densiy. Typically, people will repor a poin orecas o represen e uure occurring random variable. I is chosen, e resuling orecas error is e = y. The loss uncion associaed wi is error could be epressed as L( e ; η 3, characerized by some shape parameer η. When a orecaser decides on his poin orecas, he minimizes E [ L( e ] ; I L O ;η, i.e. min η (3 3 We assume a e loss is a uncion o jus e size and sign o e orecas error. 398

4 Business and Economic aisics ecion JM 9 I everying is well behaved, we ge e irs order condiion as E ( L ( e ; η =, or e ; = L η e where L e denoes e derivaive o L wi respec o e error e. This irs order condiion is equivalen o where ( e I ( e ; ( e de = L η (4 e is derived immediaely rom given a e = y and is jus a consan given I. (4 is e momen condiion implied by orecas opimaliy. For any variables in e inormaion se I e, say V, (4 implies E ( L ( e ; η V =. (5 Using consan as e insrumen variable, (5 means a e opimal poin orecas under e asymmeric loss uncion mus saisy e E ( L ( e ; η =. (6 Conversely, given e poin orecas and densiy orecas or e same arge we can use equaion (6 o learn abou e loss uncion parameers. 4. Comparison o Poin Forecass and e Cenral Tendency o Densiy Forecass 4.1 Inerence abou e Direcion o Loss Asymmery In mos cases, we don know e enire disribuion o y. Thereore we canno ( calculae e loss uncion parameers using equaion (6. Bu we may have some inormaion abou e cenral endency o y, such as is mean, or median, or ( mode. The ollowing ree eorems prove a, or e common loss uncions (Lin-Lin, Quad-Quad and Line and any belie abou e daa generaing process, a non-zero dierence beween e opimal poin orecas and e mean (or Quad-Quad and Line or e median (or Lin-Lin o e densiy orecas is bo a suicien and a necessary condiion o loss uncion asymmery. This means a we can deermine i e loss uncion is asymmeric or no by jus comparing e poin orecas wi e mean or e median o e underlying densiy orecas. Theorem 1: For Lin-Lin loss uncion, symmery o loss uncion is a suicien and necessary condiion or e equaliy o opimal poin orecas and e median o e underlying densiy orecas. Consider e ollowing Lin-Lin loss uncion 399

5 Business and Economic aisics ecion JM 9 (,1 L( y α y, = y ( 1α i, i y y α. This loss uncion is symmeric when α =. 5. The opimal poin orecas solves > min E [ L( y α ] = (1α The irs-order condiion is ( 1α F ( α ( 1 F ( + = where F ( is e cumulaive disribuion uncion. The above irs-order condiion is equivalen o F ( = α (7 I e Lin-Lin loss uncion is symmeric, i.e. i α =. 5, (7 implies a F ( = α =.5, or is e median o e densiy orecas. This esablishes e suicien par o eorem 1. Conversely, i is e median o e densiy orecas, α F ( =.5. This esablishes e necessary par o eorem 1. = We can also iner e direcion o asymmery o Lin-Lin loss uncion by comparing e opimal poin orecas and e median o e underlying densiy orecas. For eample, i is less an e median, en F ( = α <. 5. imilarly, i is more an e median, en F ( α >. 5. = Theorem : For e Quad-Quad loss uncion, symmery o loss uncion is a suicien and necessary condiion or e equaliy o opimal poin orecas and e mean o e underlying densiy orecas. Consider e ollowing Quad-Quad loss uncion (,1 α ( y, L = (1α ( y, α. This loss uncion is symmeric when α =. 5. The opimal poin orecas solves i i y y > 4

6 Business and Economic aisics ecion JM 9 min E [ L( y α ] = + (1α The irs-order condiion is α + (1α = (8 I e loss uncion is symmeric, i.e. i α =. 5, e irs order condiion (8 implies = + = = E i.e., is e mean o e densiy orecas. This esablishes e suicien par o eorem. Now, consider e necessary par. Use o denoe e mean o e densiy orecas E. I = E =, equaion (8 can be rewrien as α α α α + (1α + α + (1α + (1 α = = = Noe a e irs erm is equal o zero. o we have ( 1 α = (9 ince <, (9 implies a ( 1 α = α =. 5 This esablishes e necessary par o Theorem. As beore, we can also iner e direcion o asymmery o Quad-Quad loss uncion by comparing e opimal poin orecas and e mean o e densiy orecas. uppose e opimal poin orecas = E + m = + m. Le = y = y m, en E( = E( y m = m. The irs-order condiion (8 can be rewrien as 41

7 Business and Economic aisics ecion JM 9 α α (α + (1α (α (α ( ( ( d ( ( + (1α d d d d + (1α + (1α = + (1α (1α m ( = ( d ( = d d = = since E ( ( d = m. o we have = (α 1 ( d (1α m = (1 I e opimal poin orecas is less an e mean o e densiy orecas, m < by deiniion. (1 implies a (α 1 (α 1 ( ( d d (1α m = = (1α m < (11 ince ( d >, (11 implies a α 1< α <. 5. imilarly, i e opimal poin orecas is more an e mean o e densiy orecas, we can show a α >. 5. Theorem 3: For e Line loss uncion, symmery o loss uncion is a suicien and necessary condiion or e equaliy o opimal poin orecas and e mean o e underlying densiy orecas. Consider e ollowing Line loss uncion [ α ] α 1, α L = ep Assuming a we may inerchange e epecaion and diereniaion operaors, e irsorder condiion or e opimal poin orecas,, under e Line loss uncion akes e orm 4

8 Business and Economic aisics ecion JM 9 L E E = α α E [ ep{ α }] = 1 [ ep{ α }] = (1 ince ep(. is a conve uncion, by Jensen s inequaliy and assume a e densiy orecas o e arge variable, y, is no degenerae, we have [ ] [ ep{ α }] > ep E { } 1 = E α (13 Taking naural logarim o bo sides o (13, we obain E [ ] < α [ E ] < α (14 From (14, we can easily prove a loss symmery is bo a suicien and a necessary condiion or e equaliy o poin orecas and e mean o e densiy orecas. By (14 α > E ( y < and α < E ( y > Theorem 1 shows a, or Lin-Lin loss uncion, we can iner e direcion o loss asymmery by comparing e opimal poin orecas and e median o e densiy orecas; Theorem and 3 show a, or Quad-Quad and Line loss uncion, we can iner e direcion o asymmery o e loss uncion by comparing e opimal poin orecas and e mean o e densiy orecas. 4. Empirical Resuls To compare e poin orecass and e mean or median o e underlying densiy orecass or annual inlaion and real oupu grow, we ollow e meod proposed by Engelberg, Manski and Williams (9. They compare e poin orecass wi e cenral endency (mean, median and mode o e underlying densiy orecass in e PF daa. Their sample period is 199Q1-4Q4, ecluding 1996Q1. They calculae e poin orecass or annual inlaion rae and real oupu grow by using e annual level o oupu-price inde and real oupu in e previous year (provided by e PF surveys, and e poin orecass or e same wo variables in e curren year and e ne year. These daa are available rom e PF only since 199Q1. As discussed in secion, we calculae e poin orecass or annual inlaion and real oupu grow by combining e PF daa and e real ime macro daa. This allows us o calculae e poin orecass or a longer sample period. To ind e relaionship beween orecasers poin orecass and e cenral endency o eir densiy orecass, Engelberg, Manski and Williams (9 employ bo nonparameric analysis and parameric analysis. The nonparameric analysis does no assume densiy orecass o have any speciic shape bu e parameric analysis assume a each densiy orecas has a Bea or isosceles-riangle shape. We will ocus on e nonparameric analysis here. Engelberg, Manski and Williams (9 noice a e PF densiy orecass repor e subjecive probabiliies a real oupu grow or inlaion will lie in given inervals. Thus, ese orecass do no ully reveal e subjecive disribuions a respondens hold and, hence, e cenral endency canno be calculaed precisely. However, ey do imply bounds on e subjecive means and medians. By 43

9 Business and Economic aisics ecion JM 9 assuming a e mode is conained in e inerval wi e greaes probabiliy mass, ey also sugges a way o ind e bounds on e mode. Having compued e bounds on e cenral endency o densiy orecas, Engelberg, Manski and Williams (9 check i e poin orecas lies wiin e bounds on e cenral endency. I no, ey rejec e hypoesis a e poin orecas is e cenral endency eamined. They also couned how many imes e poin orecass are below e lower bounds and how many imes e poin orecass are above e upper bounds. We apply eir meod o our longer sample period and e resuls are presened in Table 1. Our inding is similar o eirs. Firs, mos poin orecass are consisen wi e cenral endency o densiy orecass (alling wiin e bounds on e cenral endency. Bu sill or a signiican racion o observaions, ey are no. This racion is usually beween 5% and 5% and varies over orecas horizons and across dieren measures o e cenral endency. econd, orecasers who skew eir poin orecass end o presen rosy scenarios. For real oupu grow, orecasers are more likely o repor a poin orecas a is above e upper bound on e cenral endency; or inlaion, however, orecasers are more likely o repor a poin orecas a is below e lower bound on e cenral endency. Engelberg, Manski and Williams (9 do no provide an eplanaion or is phenomenon. A possible one may be a e associaed loss uncions are asymmeric. Based on secion 4.1 and indings in able 1, we may iner a, or real oupu grow, e cos o underpredicion may be higher an overpredicion. As a resul, orecasers end o repor an opimisic orecas. The opposie may be rue or inlaion. Third, Table 1 also shows a as e orecas horizon shorens, e poin orecass are more consisen Table 1: Percenage o Poin Forecass Falling below Lower Bounds, inside Bounds and above Upper Bounds o Various Momens o Densiy Forecass Real Oupu Grow (1981Q3-3Q4 Mean Median Mode 4Q Ahead Forecas 41/.4/.8/ /.1/.73/ /.6/.85/.9 3Q Ahead Forecas 495/.4/.84/.13 66/.7/.78/ /.4/.87/.8 Q Ahead Forecas 518/.4/.87/.8 616/.9/.8/.9 614/.5/.89/.6 1Q Ahead Forecas 57/./.94/.3 646/.8/.87/.5 645/.4/.9/.4 Inlaion (1968Q4-3Q4 Mean Median Mode 4Q Ahead Forecas 93/./.71/.7 111/.5/.66/.9 118/.16/.77/.7 3Q Ahead Forecas 99/./.76/.5 1/./.73/ /.13/.8/.5 Q Ahead Forecas 861/.14/.77/.1 18/.16/.7/.13 15/.1/.79/.1 1Q Ahead Forecas 67/.1/.88/.3 717/.11/.83/.7 714/.8/.88/.4 Noe: For each enry, e irs number is e oal number o observaions. The second number is e percenage o poin orecass alling below e lower bounds o mean/median/mode. The ird number is e percenage o poin orecass alling beween e lower bounds and upper bounds o mean/median/mode. The our number is e percenage o poin orecass alling above e upper bounds o mean/median/mode. wi e cenral endency o e densiy orecass. For eample, or e real oupu grow orecass, when e orecas horizon is 4 quarers, 8% o e poin orecass are consisen wi e means o e densiy orecass. As e orecas horizon alls o 1 quarer, is acion increases o 94%, implying a a longer orecas horizon, loss uncion is more likely o be asymmeric. Above preliminary analysis provides some inormaion abou asymmery in loss uncions (when e loss uncion is Lin-Lin, Line 44

10 Business and Economic aisics ecion JM 9 or Quad-Quad associaed wi e orecass or real oupu grow and inlaion, and can be used o check e validiy o various meods o esimaing loss uncion parameers. 5. Combining e Poin Forecass and e Densiy Forecass o Calculae e Loss Funcion Parameers 5.1 Calculaion o Common Loss Funcion Parameers I we know e enire disribuion o y, we can derive e enire disribuion o ( e I (. We can en calculae e loss uncion parameers using equaion (6. Aer some manipulaion, we can show a e loss uncion parameer can be calculaed 1 as e ˆ α = F ( = 1 E or Lin-Lin loss uncion, e 1 E = ( e ˆ α 1 or Quad-Quad loss uncion, and soluion o E[ ep ( α e ] = 1 E e or Line loss uncion. ince e PF densiy orecass are no coninuous and orecasers repor jus e probabiliies wi which e arge variables all in dieren inervals, e calculaed disribuions o orecas errors are also hisograms elling us e probabiliies wi which e orecas errors all in dieren inervals. To calculae e loss uncion parameers, we could ollow wo meods: nonparameric analysis and parameric analysis as in Engelberg, Manski and Williams (9. The nonparameric analysis does no assume e probabiliy disribuions o e arge variables and orecas errors o have any speciic shape. As a resul, we canno calculae e loss uncion parameers eacly. However, or Lin-Lin loss uncion we can calculae bounds on e loss uncion parameer 4. The calculaion o bounds on e Quad-Quad and Line loss uncion parameer is more complicaed and no considered in is paper. We also do parameric analysis by making some assumpions abou e disribuions underlying e hisograms o e arge variables, or equivalenly e orecas errors. We considered wo ypes o disribuions: (1 Assume wiin each inerval, e probabiliy alls on e midpoin o a inerval; ( The underlying disribuions o e arge variables, or equivalenly e orecas errors are normal. For e irs ype o disribuions, e calculaion o e momens o e orecas errors (and eponenial o orecas errors is sraighorward. For e normal disribuion, i can be shown a e esimaed αˆ depends only on e mean and variance o e subjecive densiy o e orecas error or e Lin-Lin, Quad-Quad and Line loss uncions 5. For eample, suppose a = N(, where and are e mean and variance o e densiy orecas o y when orecas horizon is h. Then e subjecive densiy o e orecas error is ( e N( b, =, where Then e esimaed αˆ or Lin-Lin loss uncion is b = is e epeced orecas bias. 4 ince ˆ α = F (, i e poin orecas alls wiin e inerval [a, b], e lower bound o αˆ would en be F ( a and e upper bound o αˆ would be F ( b. 5 For more general disribuion and loss uncion, higher order momens are also needed. 45

11 Business and Economic aisics ecion JM 9 ˆ α = F ( =Φ = 1Φ( b (15 where, Φ ( is e cumulaive disribuion uncion o e sandard normal disribuion. For Quad-Quad loss uncion, e esimaed αˆ is 1 E( e 1 b ˆ α = 1 = E e 1 φ( b + b φ( b + bφ( b b D b = = φ( b + b [ Φ( b 1] D b [ Φ( b 1] (16 where D = φ b + b Φ( b. ( For Line loss uncion, since e is normally disribued, e disribued as ( α e = N( α b, α. ( disribuion wi mean [ ep ( e ] = 1 E α, we have ep αb α is also normally α ep e en ollows a lognormal 1 ep α b+ α. Based on e momen condiion (1 1 + α (17 1 b = 1 αb + α = α = 5. Empirical Resuls Figure 1 and Figure show e disribuion o calculaed Lin-Lin loss uncion based on (15 across orecasers by orecas horizon 6. Figure 1 is or real oupu grow orecass. Figure is or inlaion orecass. As shown in e igures, or inlaion, ere are more orecasers wi e loss uncion parameer o be less an.5, which means a overpredicion is more cosly an underpredicion. For real oupu grow, ere are more orecasers wi e loss uncion parameer o be more an.5, which means a overpredicion is less cosly an underpredicion. Figure 1 and Figure do no show clearly a e loss uncions are more likely o be symmeric as e orecas horizon shorens. Bu noe a e esimaion is based on e assumpion a e densiy orecass are normal. This assumpion may be no valid, see Lahiri and Teigland ( The resuls or Quad-Quad and Line loss uncion, and e assumpion a probabiliy mass alls on e midpoin o each inerval are similar and no repored here o save space. 46

12 Business and Economic aisics ecion JM 9 Alernaively we conduc a nonparameric analysis or Lin-Lin loss uncion as discussed in secion 5.1. The resul o is analysis is repored in able. We couned how many imes e calculaed bounds on loss uncion parameers cover.5 -- e value or symmeric loss uncion, and how many imes.5 is less an e lower bounds and how many imes.5 is larger an e upper bounds. Our indings are summarized as ollows. Firs, in mos cases, e bounds cover.5. Bu sill or a signiican racion o observaions, ey do no, implying asymmeric loss uncions. This racion is usually beween 1% and 5% and varies over orecas horizons. econd, or real oupu grow, ere are more cases a.5 is less an e lower bounds an e cases a.5 is larger an e upper bounds. This means a orecasers are more likely o have a loss uncion parameer larger an.5. For inlaion, e opposie is rue. This inding is consisen wi wha we ound in secion 4. Third, able also shows a as orecas horizon shorens, bounds are more likely o cover.5, or in oer words, loss uncions are more likely o be symmeric. For eample, or real oupu grow orecass, when e orecas horizon is 4 quarers, 8% o bounds cover.5. As e orecas horizon alls o 1 quarer, is racion increases o 9%. This is consisen wi our inding in secion

13 Business and Economic aisics ecion JM 9 Table : Percenage o.5 Falling below Lower Bounds, inside Bounds and above Upper Bounds o Lin-Lin Loss Funcion Parameer Real Oupu Grow (1981Q3-3Q4 4Q Ahead Forecas 3Q Ahead Forecas Q Ahead Forecas 1Q Ahead Forecas 47/.1/.8/.6 434/.1/.84/.5 46/.7/.89/.4 358/.4/.9/.6 Inlaion (1968Q4-3Q4 4Q Ahead Forecas 3Q Ahead Forecas Q Ahead Forecas 1Q Ahead Forecas 834/.6/.77/ /.5/.8/ /.9/.79/.1 444/.5/.87/.8 Noe: For each enry, e irs number is e oal number o observaions. The second number is e percenage o cases a.5 alling below e lower bounds o loss uncion parameers. The ird number is e percenage o cases a.5 alling beween e lower bounds and upper bounds o e loss uncion parameers. The our number is e percenage o cases a.5 alling above e upper bounds o e loss uncion parameers. 6. Conclusion In is paper, we consider how o use inormaion rom densiy orecass o recover e loss uncion parameers. We prove a or Lin-Lin, Line and Quad-Quad loss uncions we could iner abou e eisence and direcion o asymmery by comparing e poin orecass wi dieren measures o cenral endency o e underlying densiy orecass. When we know e enire disribuion o e densiy orecas, we can calculae e loss uncion parameers based on e irs order condiion o orecas opimaliy. This meod is applied o orecass or annual real oupu grow and inlaion obained rom e urvey o Proessional Forecasers (PF. We ind a orecasers rea underpredicion o real oupu grow o be more cosly an overpredicion; e reverse is rue or inlaion. Thus, or bo variables, orecass end o be opimisic. In addiion, as orecas horizon shorens, loss uncions are more likely o be symmeric. Reerences Croushore, D., ar, T., 1. A real-ime daa se or macroeconomiss. Journal o Economeric 15, Ellio, G., Komunjer, I., Timmermann, A., 8. Biases in macroeconomic orecass: Irraionaliy or asymmeric loss? Journal o European Economic Associaion 6(1, Ellio, G., Komunjer, I., Timmermann, A., 5. Esimaion and esing o orecas raionaliy under leible loss. Review o Economic udies 7, Engelberg, J., Manski, C. F., Williams, J., 9. Comparing e Poin Predicions and ubjecive Probabiliy Disribuions o Proessional Forecasers. Journal o Business and Economic aisics 165, Lahiri, K., Liu, F., 6. Modelling muli-period inlaion uncerainy using a panel o densiy orecass. Journal o Applied Economerics 1, Lahiri, K., Teigland, C., On e normaliy o probabiliy disribuion and GNP orecass. Inernaional Journal o Forecasing 3,

Robust estimation based on the first- and third-moment restrictions of the power transformation model

Robust estimation based on the first- and third-moment restrictions of the power transformation model h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,