Abstract
Prediction bands for time series are usually generated point-wise by bootstrap methods. Such bands only convey the prediction uncertainty for each horizon separately. The joint distribution is not taken into account. To represent the forecast uncertainty over the entire horizon, methods for constructing joint prediction bands for path forecasts from SETAR models are proposed. This class of nonlinear models is increasingly used in time series analysis and forecasting as it is useful for capturing nonlinear dynamics. Approaches based on statistical theory and explicit sequential and global optimization methods are both considered. Monte Carlo simulation is used to assess the performance of the proposed methods. The comparison is done with regard to the actual coverage of the constructed prediction bands for full path forecasts as well as with regard to the width of the bands. An empirical application demonstrates the relevance of the choice of bands for indicating the uncertainty of path forecasts in nonlinear models.
Acknowledgements
Support from the MNiSW/DAAD PPP Grant (56268818) and the National Science Center, Poland (NCN) through HARMONIA 6:UMO-2014/14/M/HS4/00901 are gratefully acknowledged.
A Bootstrap procedure for residual rescaling
Algorithm 3: Bootstrap procedure for residual rescaling
1: Estimate SETAR parameters
2: Compute the standard deviation σ(r1) of the residuals
3: Re-center
4: forc = 1 to Cdo
5: Construct bootstrapped series
6: Estimate the SETAR parameters for
7: Compute the standard deviation
8: end for
9: The estimated rescaling factor is given by
B Simulation results for heteroscedastic error terms
Estimated coverage probabilities and band width and respective standard deviations for the joint coverage of the prediction bands for DGPs with regime dependent standard deviation. The effect of different standard deviations is shown.
| β11 = 0.8, β21 = 0.8, T = 60 | ||||||
| σε1 = 0.9, σε2 = 1.1 | σε1 = 0.8, σε2 = 1.2 | σε1 = 0.5, σε2 = 1.5 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 82.96 | 64.79 | 82.46 | 65.14 | 80.04 | 64.68 |
| (11.22) | (20.50) | (12.19) | (21.32) | (16.22) | (26.27) | |
| Bt | 96.39 | 122.66 | 95.82 | 123.70 | 93.06 | 144.06 |
| (4.29) | (141.22) | (5.23) | (101.13) | (9.63) | (205.52) | |
| Badj | 94.31 | 89.17 | 93.50 | 90.01 | 90.16 | 92.02 |
| (5.83) | (29.63) | (7.12) | (31.19) | (11.43) | (40.80) | |
| TA | 94.99 | 84.19 | 94.21 | 84.79 | 90.76 | 85.57 |
| (5.88) | (22.29) | (7.12) | (23.48) | (11.41) | (29.94) | |
BPI, Bt, Badj and TA stand for the bootstrap interval method, the traditional Bonferroni method, the adjusted Bonferroni method and the method using Threshold Accepting, respectively. The results were obtained using 2000 Monte Carlo replications and 2000 bootstrap replications.
Estimated coverage probabilities and band width and respective standard deviations for the joint coverage of the prediction bands for DGPs with regime dependent standard deviation. The effect of different sample sizes is shown.
| β11= 0.8, β21 = 0.8, σε1 = 0.8, σε2 = 1.2 | ||||||
| T = 60 | T = 120 | T = 240 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 82.46 | 65.14 | 80.53 | 52.98 | 78.62 | 48.78 |
| (12.19) | (21.32) | (10.58) | (8.68) | (9.63) | (4.71) | |
| Bt | 95.82 | 123.70 | 95.83 | 78.20 | 95.72 | 70.37 |
| (5.23) | (101.13) | (4.38) | (13.80) | (3.58) | (6.95) | |
| Badj | 93.50 | 90.01 | 93.88 | 70.82 | 93.77 | 65.42 |
| (7.12) | (31.19) | (5.37) | (10.91) | (4.36) | (5.94) | |
| TA | 94.21 | 84.79 | 93.97 | 70.69 | 93.78 | 65.42 |
| (7.12) | (23.48) | (5.35) | (10.68) | (4.36) | (5.94) | |
BPI, Bt, Badj and TA stand for the bootstrap interval method, the traditional Bonferroni method, the adjusted Bonferroni method and the method using Threshold Accepting, respectively. The results were obtained using 2000 Monte Carlo replications and 2000 bootstrap replications.
Estimated coverage probabilities and band width and respective standard deviations for the joint coverage of the prediction bands for DGPs with regime dependent standard deviation. The effect of different autoregressive coefficients is shown.
| β21 = 0.8, σε1 = 0.8, σε2 = 1.2, T = 60 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 78.59 | 55.11 | 82.46 | 65.14 | 88.45 | 355.69 |
| (13.31) | (39.93) | (12.19) | (21.32) | (18.75) | (5131.30) | |
| Bt | 94.45 | 89.76 | 95.82 | 123.70 | 97.27 | 1062.70 |
| (7.11) | (82.15) | (5.23) | (101.13) | (9.91) | (15064.0) | |
| Badj | 92.62 | 74.77 | 93.50 | 90.01 | 94.57 | 472.79 |
| (7.92) | (53.25) | (7.12) | (31.19) | (14.49) | (6260.80) | |
| TA | 92.84 | 71.79 | 94.21 | 84.79 | 95.33 | 338.84 |
| (7.99) | (36.91) | (7.12) | (23.48) | (13.99) | (4453.30) | |
BPI, Bt, Badj and TA stand for the bootstrap interval method, the traditional Bonferroni method, the adjusted Bonferroni method and the method using Threshold Accepting, respectively. The results were obtained using 2000 Monte Carlo replications and 2000 bootstrap replications.
C Simulation results for alternative threshold values
Estimated coverage probabilities and band width and respective standard deviations for the joint coverage of the prediction bands for DGPs with threshold value γ = 0.5.
| sample size: T=60 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 78.99 | 49.46 | 83.2 | 64.91 | 85.69 | 428.98 |
| (11.83) | (17.41) | (11.13) | (19.85) | (20.58) | (3286.50) | |
| Bt | 95.37 | 78.85 | 96.51 | 122.53 | 97.03 | 1391.4 |
| (5.14) | (39.35) | (4.17) | (105.18) | (10.85) | (11734.0) | |
| Badj | 93.75 | 66.682 | 94.54 | 89.70 | 93.48 | 638.86 |
| (6.09) | (23.59) | (5.58) | (29.71) | (16.33) | (5169.60) | |
| TA | 93.99 | 64.877 | 95.23 | 84.13 | 94.05 | 394.41 |
| (6.11) | (18.88) | (5.66) | (21.59) | (16.32) | (2541.60) | |
| sample size: T=120 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 76.04 | 43.06 | 80.99 | 52.43 | 86.17 | 128.03 |
| (8.47) | (6.27) | (8.12) | (7.89) | (17.11 ) | (159.95) | |
| Bt | 95.49 | 60.94 | 96.49 | 76.17 | 97.82 | 217.35 |
| (3.44) | (9.51) | (3.08) | (12.03) | (7.36) | (317.82) | |
| Badj | 94.23 | 57.56 | 94.79 | 69.25 | 94.52 | 162.93 |
| (4.00) | (7.89) | (3.78) | (9.46) | (12.75) | (209.89) | |
| TA | 94.25 | 57.51 | 94.85 | 69.14 | 94.62 | 147.67 |
| (4.00) | (7.66) | (3.81) | (9.59) | (13.25) | (148.2) | |
| sample size: T=240 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 75.01 | 41.62 | 78.39 | 48.15 | 84.46 | 100.42 |
| (6.34) | (3.78) | (6.14) | (4.02) | (16.52 ) | (62.80) | |
| Bt | 95.82 | 58.59 | 96.27 | 68.24 | 97.40 | 159.74 |
| (2.43) | (5.49) | (2.26) | (5.87) | (8.73) | (104.24) | |
| Badj | 94.63 | 56.00 | 94.46 | 63.70 | 93.94 | 125.50 |
| (2.81) | (4.87) | (2.73) | (5.05) | (12.50) | (71.99) | |
| TA | 94.63 | 55.99 | 94.46 | 63.69 | 93.75 | 119.65 |
| (2.81) | (4.81) | (2.74) | (5.05) | (12.28) | (63.47) | |
BPI, Bt, Badj and TA stand for the bootstrap interval method, the traditional Bonferroni method, the adjusted Bonferroni method and the method using Threshold Accepting, respectively. The results were obtained using 2000 Monte Carlo replications and 2000 bootstrap replications.
Estimated coverage probabilities and band width and respective standard deviations for the joint coverage of the prediction bands for DGPs with threshold value γ = 1.
| sample size: T=60 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 79.85 | 51.08 | 83.92 | 68.67 | 87.39 | 253.05 |
| (12.16) | (18.70) | (11.54) | (25.17) | (16.90) | (1477.50) | |
| Bt | 95.58 | 83.74 | 96.66 | 135.82 | 97.57 | 752.55 |
| (5.13) | (50.97) | (4.31) | (180.87) | (7.74) | (5380.20) | |
| Badj | 94.10 | 69.25 | 94.84 | 95.07 | 94.99 | 363.50 |
| (6.04) | (26.86) | (5.67) | (37.88) | (11.43) | (2346.40) | |
| TA | 94.37 | 66.99 | 95.60 | 88.44 | 95.48 | 254.88 |
| (6.06) | (19.86) | (5.52) | (28.04) | (11.8) | (1195.00) | |
| sample size: T=120 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 76.48 | 44.21 | 80.86 | 52.72 | 85.78 | 120.74 |
| (8.64) | (7.03) | (8.33) | (8.21) | (17.116 ) | (124.88) | |
| Bt | 95.65 | 62.37 | 96.48 | 76.52 | 97.77 | 200.56 |
| (3.45) | (10.16) | (3.08) | (12.62) | (7.13) | (245.08) | |
| Badj | 94.50 | 59.01 | 94.74 | 69.63 | 94.43 | 152.38 |
| (3.92) | (8.67) | (3.91) | (10.16) | (12.14) | (152.09) | |
| TA | 94.52 | 58.95 | 94.82 | 69.52 | 94.52 | 140.90 |
| (3.92) | (8.34) | (3.93) | (10.01) | (12.87) | (119.55) | |
| sample size: T=240 | ||||||
| β11 = 0.2 | β11 = 0.8 | β11 = 1 | ||||
| Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | Coverage (SD) | Width (SD) | |
| BPI | 74.89 | 42.31 | 78.80 | 48.36 | 84.80 | 97.56 |
| (6.14) | (3.58) | (6.14) | (4.00) | (15.88 ) | (59.68) | |
| Bt | 95.90 | 59.46 | 96.38 | 68.50 | 97.75 | 153.98 |
| (2.31) | (5.18) | (2.17) | (5.77) | (8.31) | (101.97) | |
| Badj | 94.80 | 56.89 | 94.64 | 63.95 | 94.08 | 121.83 |
| (2.61) | (4.65) | (2.63) | (4.98) | (12.18) | (68.20) | |
| TA | 94.80 | 56.89 | 94.64 | 63.94 | 94.01 | 116.77 |
| (2.61) | (4.65) | (2.63) | (4.98) | (12.76) | (60.15) | |
BPI, Bt, Badj and TA stand for the bootstrap interval method, the traditional Bonferroni method, the adjusted Bonferroni method and the method using Threshold Accepting, respectively. The results were obtained using 2000 Monte Carlo replications and 2000 bootstrap replications.
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Supplemental Material
The online version of this article offers supplementary material (DOI:https://doi.org/10.1515/snde-2016-0066).
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