A Forecast Rationality Test that Allows for Loss Function Asymmetries Andrea A. Naghi Department of Economics, University of Warwick I I I I I Forecast rationality is tested under different assumptions regarding the forecaster’s underlying loss function Instrument A symmetric loss function may not be flexible enough to capture the loss structures that forecasters face. An asymmetric loss function, could be more representative for the forecaster’s intentions; see Zellner (1986), Christoffersen and Diebold (1997), Elliott, Komunjer and Timmermann (2005, 2008; EKT hereafter), Patton and Timmermann (2007a), Komunjer and Owyang (2007) R, P R=250, P=150 R=250, P=200 R=250, P=250 R=300, P=200 EKT provide a forecast rationality testing framework based on a general class of loss functions that allows for a parametrization of the asymmetry in the loss function a0=-5 0.0482 0.0494 0.0509 0.0482 a0=-3 0.1326 0.1331 0.1346 0.1321 a0=-1 0.3461 0.3470 0.3488 0.3484 a0 = 3 0.8621 0.8606 0.8608 0.8647 a0=1 0.6532 0.6540 0.6516 0.6522 a0 = 5 0.7769 0.7769 0.7751 0.7745 However, as I show in this paper, their methodology is loss function sensitive. Also, the EKT test is based on the assumption that the forecasts were generated using a linear model. Thus, their test may not detect nonlinear dependencies This paper proposes an new test for forecast rationality that allows for an asymmetric loss function, relaxes the assumption that the forecaster’s loss belongs to the parametrization of EKT and allows for nonlinear dependencies between the forecast error and the information set a0=-5 0.0870 0.0930 0.1010 0.0870 a0=-3 0.0610 0.0580 0.0710 0.0670 a0=-1 0.0360 0.0450 0.0320 0.0380 a0 = 3 0.2130 0.2950 0.3260 0.3070 a0=1 0.0510 0.0400 0.0310 0.0420 a0 = 5 0.3950 0.5130 0.5890 0.4880 Empirical Size and Power Comparison Linex and EKT-Loss Graphs 3 J-Stat g(x) = x The EKT Framework -4 -3 -2 -1 General Class of Loss Functions: MT Stat 2 0 1 2 3 4 δ = 0.2 δ = 0.5 δ=1 δ=2 5 -1 p L1 (t+h; p, α) = [α + (1 − 2α) · 1(t+h < 0)] · |t+h| -2 0.186 0.244 0.242 0.310 0.436 0.662 0.592 0.800 The shape parameter α describes the degree of asymmetry in the forecaster’s loss function 0 TX +τ −1 1 p0−1 −1 vt [1(êt+1 < 0) − α̂T ]|êt+1| J= Ŝ T t=τ TX +τ −1 p0−1 2 vt [1(êt+1 < 0) − α̂T ]|êt+1| ∼ χd−1 Figure 1: Red L2(t+1; a) = exp(a · t+1) − a · t+1 − 1 with a = 1 Blue: L1(t+1; p, α) = [α + (1 − 2α) · 1(t+1 ≤ 0)] · |t+1|p with p = 2, α = 0.65 3 g(x) = arctan(x) 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 New Forecast Rationality Test 0.4630 0.4759 0.4651 0.4715 0.4683 0.0488 0.0484 0.0483 0.0483 0.0483 -0.7586 -0.4966 -0.7216 -0.5890 -0.6570 1.2149 0.5967 0.2062 0.0640 1.6501 2.71 2.71 2.71 2.71 4.60 Case 1 Case 2 Case 3 Case 4 Case 5 Consumption Case 1 Case 2 Case 3 Case 4 Case 5 0.5541 0.5676 0.5626 0.5610 0.5621 0.0471 0.0463 0.0460 0.0459 0.0458 1.1480 1.4601 1.3622 1.3312 1.3549 0.4478 2.6596 0.9246 0.6171 0.9456 2.71 2.71 2.71 2.71 4.60 0.2760 0.3075 0.2815 0.3057 0.2732 0.0502 0.0522 0.0503 0.0519 0.0498 -4.4603 -3.6884 -4.3453 -3.7425 -4.5532 5.9011 0.7563 4.1666 0.3205 6.0723 2.71 2.71 2.71 2.71 4.60 MT Test for Forecast Rationality Based on Median Forecasts Info Set δ = 0.2 δ = 0.5 δ=1 δ=2 0.296 0.302 0.298 0.320 0.610 0.724 0.762 0.812 δ = 0.2 δ = 0.5 δ=1 δ=2 0.290 0.324 0.362 0.430 0.616 0.736 0.856 0.816 δ=0 0.104 0.116 -2 Idea: asymmetric preferences imply an unconditional bias of the forecast error but not a conditional bias H0 : E(t+1|Wt ) = E(t+1) H1 : E(t+1|Wt ) 6= E(t+1) H0 : E[(t+1 − E(t+1))|Wt ] = 0 H1 : Pr [E[(t+1 − E(t+1))|Wt ] = 0] < 1 Conditional moment type test in the spirit of Bierens (1982,1990), de Jong (1996): MT = supγ∈Γ|mT (γ)| mT (γ) = √1 T t=0 (êt+1 − e) w P t−1 0 j=0 γj Φ(Wt−j ) 2014 NBER-NSF Time Series Conference September 26-27, 2014 Figure 2: Red: L2(t+1; a) = exp(a · t+1) − a · t+1 − 1 with a = 3 Blue: L1(t+1; p, α) = [α + (1 − 2α) · 1(t+1 ≤ 0)] · |t+1|p with p = 2, α = 0.86 Test Stat BootCV at 5% BootCV at 10% Case 1 Case 2 Case 3 Case 4 Case 5 0.0229 0.0224 0.0200 0.0164 0.0235 0.0581 0.0439 0.0457 0.0554 0.0470 0.0466 0.0439 0.0391 0.0493 0.0370 Case 1 Case 2 Case 3 Case 4 Case 5 0.0316 0.0321 0.0322 0.0336 0.0308 0.0340 0.0257 0.0419 0.0319 0.0315 0.0303 0.0219 0.0302 0.0281 0.0305 Case 1 Case 2 Case 3 Case 4 Case 5 0.0152 0.0114 0.0171 0.0123 0.0220 0.0378 0.0397 0.0512 0.0410 0.0540 0.0338 0.0347 0.0400 0.0377 0.0471 Price Index -3 PT −1 CV at 10% Case 1 Case 2 Case 3 Case 4 Case 5 g(x) = exp(x) 2 t=τ I J-Stat GNP/GDP -3 I t-Stat 1 -5 I StdErr Price Index Table 2: Rejection frequencies for the J-test, forecast evaluation under a misspecified loss R/ P R=250, P=150 R=250, P=200 R=250, P=250 R=300, P=200 α GNP/GDP Table 1: Average GMM estimates for α across 1000 MC, obtained under the misspecified loss 2 I J Test for Forecast Rationality Based on Median Forecasts The Effect of a Misspecified Loss Function in Forecast Evaluation Introduction and Motivation Consumption 0 0 DGP : Yt = θ Wt + δg(φ Wt ) + Ut T = 500, M = 1000 Concluding Remarks New forecast rationality test that allows for asymmetric preferences, without the necessity to assume any particular functional form for the forecaster’s loss function. It is a conditional moment type test in the spirit of Bierens (1982,1990) I Monte Carlo evidence shows that the forecast rationality test of EKT (2005, 2008) is loss function sensitive. The proposed test has higher power than the J-test in the presence of nonlinear dependencies between the forecast error and the info. set used in forecasting I Empirical application using data from the Survey of Professional Forecasters (SPF) I Email: [email protected] Acknowledgments Part of this paper was written while the author was visiting the UCSD Economics Department, whose warm hospitality is gratefully acknowledged. I would like to thank Michael Clements, Valentina Corradi, Ivana Komunjer and Jeremy Smith for their helpful comments and suggestions. Financial support from the ESRC is thankfully acknowledged. Web: http://www.wawick.ac.uk/anaghi

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# A Forecast Rationality Test that Allows for Loss Function Asymmetries