Forecasting with Judgment

Transcription

1 Forecasing wih Judgmen Simone Manganelli DG-Research European Cenral Bank Frankfur am Main, German) Disclaimer: he views expressed in his paper are our own and do no necessaril reflec he views of he ECB or he Eurossem

2 Classical heor of Forecasing Forecass serve o make decisions abou he fuure. Forecas errors impose coss on he decision-maker. Agens wan o minimize expeced loss associaed o forecas errors. Classical forecas is he minimizer of he sample equivalen of he expeced loss. See, e.g., Haavelmo 944), Granger and Newbold 986), Granger and Machina 005).

3 Opimal Poin Forecas Simples case: i.i.d. normall disribued observaions wih variance quadraic loss funcion } { = Wan o forecas + minq0 ) # E[ + " ) ] FOC: E[ + " ] = 0 * = + E[ ] #ˆ = " =

4 he Problem ˆ is he minimiser of: " = min Qˆ $ ) # [ " $ ) $ ] min{ E[ " + # = " ) ] + [ " ) ] " E[ + " ) # " ) ]} min{ E [ + # ) ] + " )}

5 Ouline Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

6 Sar from a Subjecive Guess Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian minq0 ) # E[ + " ) # Q0 $ ) " E[ + $ ] = 0 ] he sample equivalen of he FOC is: ˆ " $ ) # % Q = [ " ] = ˆ " where is a subjecive guess of he decision maker. NOE: his is a random variable, which ma be differen from zero jus because of saisical error.

7 hen Do Hpohesis esing H 0 : is he rue mean ˆ " N0,/ ) Choose a confidence level α η α/ is he criical value Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian If null is rejeced: Firs derivaives saisicall 0 Can be confiden o decrease he rue objecive funcion Onl up o he poin * ˆ where new H 0 canno be rejeced

8 Graphical Illusraion Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian Disribuion of FOC under H 0 / "Qˆ ) " ˆ ) ˆ ˆ * "Q ) Q 0

9 E[ " ) ] Inuiion Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian " ) " ) 00*-α)% confidence inerval ˆ # $ " / # +" / ˆ = ˆ Firs derivaives no saisicall differen from zero

10 New esimaor: " # $ < % % + < % > % % = / / / / / * ˆ if ˆ ˆ if ˆ if ˆ & & & & & ' ' ' ' ' Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

11 Inerpreaion Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian α is he probabili of commiing pe I errors, i.e. of rejecing he null when i is rue. Choose low α when confiden in subjecive guess or if he cos of pe I errors is high. Classical paradigm ses α =: Alwas FOC equal o zero; No room for subjecive guess; I commis pe I errors wih probabili.

12 Problems wih Prees Esimaors es he following null hpohesis, for given confidence level α: H hen: : ˆ 0 = If do no rejec, keep he subjecive guess If rejec, ake he maximum likelihood esimaor Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian his is wrong

13 E[ " ) ] Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian " ) " ) -α)% confidence inerval ˆ # $ " / # +" / ˆ = ˆ Firs derivaives no saisicall differen from zero

14 Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

15 Esimae he mean, given a single observaion Magnus 0),)... 0 N d i i Le " # $ < % % + < % > % % = / / / / / * if if if & & & & & ' ' ' ' ' Compare he risk properies of wo esimaors: ) Subjecive classical esimaor coincides wih Magnus 0) " # $ > % < % = / / if if ˆ & & ' ' P ) Prees esimaor Risk Analsis of New Esimaor Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

16 E 0 0 [ f ) " ) ], " = " 4 Risk Associaed o f) 0 Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian OLS _=) Prees _=0.0) Subjecive Guess _=0) Subjecive Classical _=0.0)

17 Mone Carlo Simulaion Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian Random draws from sandard normal Sample sizes = 5, 0, 60, 0, 40, 000 Quadraic loss funcion wo esimaors: classical, new α=0.0) Evaluaed expeced loss wih MC simulaion Repea 5000 imes and average

18 0.7 Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian Mean

19 Implicaions Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian Good guess gu feelings) are as imporan as good economeric models. Organizaion of forecasing process: Subjecive guess based on maximum likelihood esimaes can never be rejeced b consrucion; Clear separaion b/w: Who provides guess, based on judgmen; Who ess he guess, based on economeric models. Shared responsibili for he quali of he forecass: High confidence in bad judgmen resuls in bad forecas.

20 Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

21 Generalizaion Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian ˆ 0 $ $ # U )" N0, ) d $ ) # $% ˆ + " $ )% $ [0,] * % Under he null H = * 0 : ") * ˆ ' * ˆ )) )) ˆ ' z & # U " $% $ # 0 Uˆ * " " " " )) if if zˆ zˆ $ $ * * 0)) 0)) &# > #, k, k % % " ˆ = " ˆ = 0 arg maxuˆ "'[0,] $ * ")) s.. zˆ $ * ")) = #, k

22 Mean-Variance Asse Allocaion Mean-variance opimizers end o produce porfolio allocaions wih lile or negaive invesmen value. Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian [he] overuse saisicall esimaed informaion and magnif he impac of esimaion errors. I is no simpl a maer of garbage in, garbage ou, bu, raher, a molehill of garbage in, a mounain of garbage ou Michaud 998)

23 Se up Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian N+ asses,i reurn of asse i θ i weigh of asse i he sum o ) N + i= i, i " ) = " porfolio reurn

24 Opimizaion Opimizaion } )] [ )] [ { )] [ } ] [ ' { ] [ ' )] [ )] [ ] ; [ " " " E E E V E V E U # # = # = # = } )] [ ) { ) ] ; [ ˆ = " = " = " " " = U # # $ # # Risk Analsis of New Esimaor Example : Asse Allocaion Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

25 Implemenaion Deails Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian Monhl log reurns from DJIA From Januar 987 o Jul 005 λ = 4 coefficien of risk aversion) Rolling windows M=60 and 0 as in DeMiguel e al. 007) Benchmark porfolio: equall weighed Repor ou of sample differences in average uiliies beween opimal and benchmark porfolios

26 Ou of Sample Evaluaion Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian $ U ) = ) $ "{ ) $ [ $ M ) # )] = M + )} Z ˆ)) U $ )] " * " M # " M ) = + [ U " " M $ % es for saisical significance using Giacomini and Whie 006)

27 Resuls Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian M=60 M=0 N=4 N=6 N=30 N=4 N=6 N=30 α= *** *.6 α= * * 4.9 ***.48 * α= ** 0

28 Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

29 Implemenaion Issues Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian Difficul o formulae guess on absrac model parameers. Assume he decision-maker can formulae a guess on he variable o be forecas U.S. GDP, in his example). his guess can be mapped ino a guess for model parameers as follows: = s.. ˆ arg maxuˆ + ) = ) + Model s forecas Subjecive guess

30 Implemenaion Deails Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian he model, AR4): = + # 4 0 " i = i $ i " + he objecive funcion quadraic): U ˆ # ) = " [ " ˆ # )] " = he daa: Quarerl U.S. real GDP growh raes FRED daa base, seasonall adjused From Q 983 o Q3 005

31 Resuls Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian ˆ ) % % + = 3% ˆ * ˆ) " % % + = 5% " * ˆ) %

32 Risk Analsis of New Esimaor Example : Asse Allocaion Example : Forecasing US GDP Relaionship wih Baesian

33 Relaionship wih Baesian Economerics Non sample informaion is summarised b: priors, in Baesian economerics subjecive guess and confidence associaed o i, in our case Special cases:. No uncerain abou opimal value of θ 0 : Baesian: prior is degenerae disribuion wih oal mass on θ 0 ; Classical: subjecive guess=θ 0, α=0 i.e. never rejec he null). No informaion, besides he sample: Baesian: diffuse prior Classical : α= i.e. se FOC equal o zero) In oher inermediae cases, no clear mapping b/w he wo. When non sample info if formulaed: via prior disribuions, be Baesian via subjecive guess and confidence, use subjecive classical esimaor In general, he choice is dicaed b he decision maker, hrough he forma in which s/he provides he non sample info.

34 Ignoring non sample informaion and esimaion error are conneced problems in he classical heor of forecasing. Forecass should maximize he objecive funcion in a sochasic sense, no deerminisic. Sar from subjecive guess and consruc esimaor insead of he oher wa round) es if FOC are saisicall differen from zero: If no, subjecive guess is reained as forecas If es, objecive funcion is increased as long as FOC are = 0