Folie 1. Folie 2. Folie 3. Evaluating Econometric Forecasts of. Economic and Financial Variables. Chapter 2 (

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1 Folie 1 Evaluating Econometric Forecasts of Economic and Financial Variables Michael P. CLEMENS Palgrave macmilian, Basingstoke, 005 Chapter (.1.3.) Point Forecasts Zenaty Patrik Folie CONEN Introduction esting the Forecasts Forecast precision Rival forecasts, forecast combination and encompassing Conclusion Folie 3 Introduction constructed ex post Forecast: any statement about the future point forecast: quantitative forecast of the level or rate of change of a continuous variable

2 Folie 4 esting the Forecasts forecast is unbiased E t(y t+h t y t+h) = 0 sample mean of the forecast errors e t+h t y t+h y t+h t over t = 1,,, is significant different from zero Problem: forecasts unbiased and efficient but highly inaccurate take into account the variance Folie 5 he test of rationality y t+1 = α + β y t+h t + e t+1 null hypothesis α = 0 and β = 1 entails unbiasedness also as a test for efficiency unbiasedness & efficiency: minimum requirements for optimal or rational forecasts Folie 6 Forecast precision(1) use the available information efficiently? able to avoid making systematic errors? difficult to judge Assume: y t = φ y t-1 + υ t φ < 1

3 Folie 7 Forecast precision () minimum attainable forecast error variance for an h step ahead forecast : V(et+h t) E[(et+h t - E(et+h t)) ]= E h h 1 i (1 φ ) φ υ = t + h i σ i= 0 1 φ as a benchmark approximate forecast error variance: 1 V(êt+h t) h φ ( h 1) 1 φ φ h 1 + y σˆ υ t 1 φ Folie 8 Forecast precision (3) forecast bias and forecast-error variance expected squared forecast error E(e t+h t) = V(e t+h t) + [E(e t+h t)] minimum mean squared error predictor (MMSEP) yt = φy t 1 + yt + h t υ t h = φ y t y t + ht = Et(yt+h) Folie 9 Forecast precision (4) mean squared forecast error E(e t+h t) = V(e t+h t) + [E(e t+h t)] for the sample of h stepahead forecast is: 1 MSFEh = e t= 1 t+ h t

4 Folie 10 Rival forecasts assume loss function: squared error loss corresponding sample measure of forecast accuracy, (R)MSFE calculated for each set of forecasts most accurate: smallest MSFE Folie 11 est of equal variances MORGAN-GRANGER-NEWBOLD test u1,t+1 t = ê t+1 t - e ~ t+1 t u,t+1 t = ê t+1 t + e ~ t+1 t E(u 1,t+1 t u,t+1 t) = E(ê t+1 t) E(e ~ t+1 t) = 0 est statistic is (for h=1): r ~ t u 1u 1 r = ( 1) (1 r ) u uu u uí = (ui, 1,.., ui, +t ), i = 1, 1 1 Folie 1 test DIEBOLD and MARIANO: test statistic for h > 1 app d ~ 1 N (0,1) πfˆ d (0) d app (0,1) ˆ ~ ( ˆ) N V d

5 Folie 13 Forecast combination (or pooling) and encompassing(1) basic idea: although one forecast may be superior to another (test of equal forecast accuracy) a combined forecast may be still better conditionally efficient variance of error of forecast from combination not significantly less than that of original forecast alone If rival s forecasts: no additional explanatory power (contributing to lower MSFE or forecast error variance) then a model encompass rival forecast. Folie 14 Forecast combination () ~ y ˆ t + ht and y t + ht one step forecast: Assumption: y ˆ t +1 and ~ y t + 1 f yˆ t f ~ y t t and Combination: fct = ( 1 λ ) f1 t + λft e = ( 1 λ ) ct e + λe t e y ct t fct and e y it t f it, i = 1, Folie 15 Forecast combination (3) combined forecast error variance: V e ) = (1 λ) V ( e ) + λv ( e ) + λ(1 λ) C( e, e ) ( ct t t Minimize: * V ( e1 t ) C( e1 t, et ) λ = V ( e1 t ) + V ( et ) C( e1 t, et ) using the optimal weight λ*: MSFE ( f ct ) min{msfe ( f ), MSFE ( f t )}

6 Folie 16 Forecast combination (4) calculate the weights: (1/ ) e t = t e t = te 1 1 (1/ ) 1 1 t ) e + t (1/ ) e t t (1/ ) t t = t ˆλ = = ( e1 t et ) e1 1 t (1/ e = = = te t e = t e ( 1 1 t ) optimal weight by OLS Folie 17 Forecast combination (5) t-test of the null that λ = 0 in: e e e = λ( ) + t e ct that f() forecast encompass f(t) one sidedtest against alternative λ>0 Folie 18 Forecast combination (6) Monte Carlo study for = (8, 16, 3, 64, 18) two data generating processes samples of size are generated from: e1 t = ε1 t = ε + 0, ε et 5 t satisfies that forecast 1 encompasses forecast E( ε ε ) 0 1 t t =

7 Folie 19 Forecast combination (7) following tests were calculated Standard: he standard t-statistic for λ = 0 was compared to the standard normal R1: he t-statistic was calculated using a White HCSE and compared to a Studend t (-1) reference distribution (DIEBOLD MARIANO est): he test for equal forecast accuracy was compared to the standard normal distribution Folie 0 Forecast combination (8) M (Modification of the ): modification to improve the small sample performance. Here it is M = SR: Spearman s rank correlation test. distribution free test that determines whether there is a monotonic relation between two variables Folie 1 est statistic =8 Normal errors Student t errors Standard R M SR SR =16 Standard R M SR SR =3 Standard R M SR SR est statistic =64 Standard R M SR SR =18 Standard R Normal errors M SR SR Student t errors

8 Folie Results (1) for = 8 and =16: R1 and are over sized M improves the performance nd column show size estimates for heavy tailed forecast errors if absolute errors occasionally observed forecast error distribution likely to be heavy tailed standard t-test will be oversized Folie 3 Results () Forecast errors are generated from: e = it u it ( vt x / v u1 t = ε1 t ut = ε ε t test becomes oversized as increases R1 correctly sized for large samples other tests quite reasonable Folie 4 hank you for your attention!