Empirical Simultaneous Confidence Regions for Path-Forecasts

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(1)DEPARTMENT OF ECONOMICS. EUI Working Papers ECO 2010/18 DEPARTMENT OF ECONOMICS. EMPIRICAL SIMULTANEOUS CONFIDENCE REGIONS FOR PATH-FORECASTS. Òscar Jordá, Malte Knüppel and Massimiliano Marcellino.

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(3) EUROPEAN UNIVERSITY INSTITUTE, FLORENCE. DEPARTMENT OF ECONOMICS. Empirical Simultaneous Confidence Regions for Path-Forecasts ÒSCAR JORDÁ, MALTE KNÜPPEL and MASSIMILIANO MARCELLINO. EUI Working Paper ECO 2010/18.

(4) This text may be downloaded for personal research purposes only. Any additional reproduction for other purposes, whether in hard copy or electronically, requires the consent of the author(s), editor(s). If cited or quoted, reference should be made to the full name of the author(s), editor(s), the title, the working paper or other series, the year, and the publisher. ISSN 1725-6704. © 2010 Òscar Jordá, Malte Knüppel and Massimiliano Marcellino Printed in Italy European University Institute Badia Fiesolana I – 50014 San Domenico di Fiesole (FI) Italy www.eui.eu cadmus.eui.eu.

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(164) .

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(169) .

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(174) .

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(183) .

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(219).

(220) Table 1. MC results, well specified model, nominal coverage 68%, N=40 VAR. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. 1 P UN FF P UN FF 4 P UN FF P UN FF 8 P UN FF P UN FF 12 P UN FF P UN FF. FWE control Bonf. Bonf. emp.. Marg.. Marg. emp.. Schef. Schef. emp.. 62.2 63.4 62.3 66.8 66.9 67.1. 66.7 67.5 66.4 67.1 67.0 67.0. 62.2 63.4 62.3 66.8 66.9 67.1. 66.7 67.5 66.4 67.1 67.0 67.0. 62.2 63.4 62.3 66.8 66.9 67.1. 24.2 37.0 30.5 27.7 42.1 35.7. 30.2 43.1 36.8 28.6 42.6 35.7. 67.4 74.9 71.6 74.6 81.2 78.2. 74.4 80.3 78.0 73.7 80.2 77.2. 11.1 21.0 18.1 13.6 25.1 20.6. 17.0 26.1 24.3 14.2 25.4 21.3. 69.5 76.1 73.1 78.3 83.6 82.1. 6.7 12.8 11.0 6.8 15.8 13.8. 11.9 18.1 16.7 8.3 16.6 14.3. 70.9 75.8 73.2 79.8 84.3 83.2. FDR control Bonf. Bonf. emp.. Marg.. Marg. emp.. Schef. Schef. emp.. 66.7 67.5 66.4 67.1 67.0 67.0. 62.2 63.4 62.3 66.8 66.9 67.1. 66.7 67.5 66.4 67.1 67.0 67.0. 62.2 63.4 62.3 66.8 66.9 67.1. 66.7 67.5 66.4 67.1 67.0 67.0. 62.2 63.4 62.3 66.8 66.9 67.1. 66.7 67.5 66.4 67.1 67.0 67.0. 48.3 57.8 54.8 54.5 63.5 60.9. 55.0 63.0 60.9 53.7 63.4 60.3. 16.9 12.2 12.6 20.2 14.2 14.8. 18.3 13.7 13.8 20.7 14.0 15.0. 63.9 52.9 56.3 71.9 59.7 62.6. 65.7 55.4 58.6 69.0 57.2 60.0. 58.5 59.4 58.6 65.7 66.6 65.2. 62.5 63.3 62.5 62.4 63.6 62.3. 76.5 80.7 78.6 75.8 81.1 78.8. 46.3 55.4 53.0 53.7 61.3 59.0. 52.8 60.0 58.7 51.5 59.5 57.5. 0.9 1.4 1.0 1.5 2.0 1.1. 1.2 1.7 1.2 1.6 2.3 1.4. 42.5 48.7 40.6 52.6 58.2 49.0. 41.1 46.4 38.8 47.1 50.6 42.1. 54.9 56.3 56.2 65.4 65.7 65.1. 53.2 52.2 52.6 53.9 54.3 53.2. 75.8 78.7 78.3 75.2 79.7 78.5. 46.5 52.8 52.4 52.9 60.6 59.8. 51.6 53.6 56.7 50.1 54.3 56.8. 0.0 0.3 0.0 0.0 0.5 0.1. 0.1 1.5 0.1 0.1 1.3 0.2. 28.7 56.0 30.1 35.6 67.8 37.5. 25.1 51.6 27.0 29.7 54.7 31.2. 54.4 54.9 54.3 64.4 65.6 64.7. 43.1 42.3 41.9 43.1 43.1 42.6. Marg.. Marg. emp.. Local Projections. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. 1 P UN FF P UN FF 4 P UN FF P UN FF 8 P UN FF P UN FF 12 P UN FF P UN FF. FWE control Bonf. Bonf. emp.. Schef. Schef. emp.. FDR control Bonf. Bonf. emp.. Marg.. Marg. emp.. Schef. Schef. emp.. 62.4 63.7 62.6 66.9 66.9 67.2. 66.7 67.5 66.4 67.1 67.0 67.0. 62.4 63.7 62.6 66.9 66.9 67.2. 66.7 67.5 66.4 67.1 67.0 67.0. 62.4 63.7 62.6 66.9 66.9 67.2. 66.7 67.5 66.4 67.1 67.0 67.0. 62.4 63.7 62.6 66.9 66.9 67.2. 66.7 67.5 66.4 67.1 67.0 67.0. 62.4 63.7 62.6 66.9 66.9 67.2. 66.7 67.5 66.4 67.1 67.0 67.0. 62.4 63.7 62.6 66.9 66.9 67.2. 66.7 67.5 66.4 67.1 67.0 67.0. 21.1 33.6 26.8 27.1 41.0 35.0. 30.8 43.2 37.0 28.5 42.4 35.9. 62.2 70.6 66.9 73.5 80.7 77.4. 74.0 79.9 78.1 73.6 80.2 77.3. 44.0 56.0 52.2 53.6 63.2 60.5. 55.1 63.2 60.9 53.6 63.4 60.2. 17.7 11.3 12.2 20.5 13.9 14.8. 18.3 13.3 14.1 20.6 14.0 15.3. 65.0 51.1 55.2 72.2 59.5 62.5. 65.6 55.4 58.7 69.0 57.1 59.8. 55.0 56.3 55.3 65.1 65.7 64.7. 62.4 63.0 62.5 62.6 63.3 61.9. 6.7 14.9 12.3 12.1 23.9 19.3. 17.1 26.8 24.7 14.3 25.5 21.4. 56.6 64.4 61.2 76.3 82.2 80.4. 76.6 80.6 78.9 75.6 81.4 78.8. 37.3 48.8 46.9 52.7 60.8 58.5. 52.8 60.0 58.8 51.6 59.6 57.4. 1.2 1.6 0.8 1.7 2.1 1.0. 1.2 1.9 1.3 1.8 2.3 1.3. 45.1 48.8 38.9 53.4 58.2 48.6. 42.1 46.3 39.5 47.3 50.6 42.4. 45.4 45.6 44.8 63.8 64.2 63.8. 52.7 52.0 52.2 53.8 54.2 53.4. 2.4 7.0 5.3 5.9 14.4 12.0. 12.3 18.3 17.1 8.3 16.7 14.0. 51.0 59.1 54.1 76.9 81.8 80.6. 76.0 79.4 78.5 75.1 79.7 78.8. 33.8 34.4 40.7 51.8 59.0 59.3. 51.7 51.9 56.4 50.0 54.1 56.4. 0.1 1.2 0.2 0.0 0.5 0.1. 0.1 2.6 0.4 0.1 1.4 0.1. 32.3 62.3 32.1 36.2 69.1 38.0. 26.8 52.0 28.3 29.8 54.6 31.2. 35.9 36.8 35.7 61.2 61.8 61.7. 42.2 41.1 40.7 42.9 43.1 42.5. Notes: 10,000 samples generated from VAR(4) for three variables (P, UN, FF) with stable parameters. Model coincides with DGP. Each estimated VAR or local projection (LP) on these 10,000 samples generates a forecast error variance (which includes estimation uncertainty) for the forecast path of length h, and hence the sets of bands (marginal, Bonferroni, and Scheffé) used in the analysis. Similarly, each estimated model generates N forecast paths whose associated error paths are used to generate a forecast error variance for the forecast path and hence the set of bands (marginal emp, Bonferroni emp, and Scheffé emp). Hence 10,000 actual paths are then compared with each set of 10,000 bands to determine the appropriate coverage rates. FWE control stands for “family-wise error control” and simply computes the proportion of paths strictly inside the bands. FDR control instead is the proportion of forecast paths whose Mahalanobis distance attains a value that is lower than the chi-square statistic for probability equal to nominal coverage and degrees of freedom equal to H. See text for more details..

(221) Table 2. MC results, well specified model, nominal coverage 95%, N=40 VAR. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. 1 P UN FF P UN FF 4 P UN FF P UN FF 8 P UN FF P UN FF 12 P UN FF P UN FF. FWE control Bonf. Bonf. emp.. Marg.. Marg. emp.. Schef. Schef. emp.. 92.0 92.4 92.2 94.7 94.8 94.3. 94.4 94.4 94.4 94.4 94.6 94.4. 92.0 92.4 92.2 94.7 94.8 94.3. 94.4 94.4 94.4 94.4 94.6 94.4. 92.0 92.4 92.2 94.7 94.8 94.3. 75.9 81.9 79.2 82.8 87.6 86.5. 81.7 86.3 84.5 81.5 86.7 85.2. 90.5 92.8 91.7 95.0 96.3 96.2. 93.8 95.4 94.5 93.9 95.5 94.9. 65.2 73.1 69.9 74.3 81.0 79.4. 72.0 77.8 76.1 71.2 78.1 76.1. 89.5 92.7 91.1 95.3 97.1 96.3. 58.6 65.1 62.1 68.7 74.8 73.1. 66.4 69.8 69.0 64.3 71.2 68.7. 89.3 91.9 90.1 95.4 96.6 96.5. FDR control Bonf. Bonf. emp.. Marg.. Marg. emp.. Schef. Schef. emp.. 94.4 94.4 94.4 94.4 94.6 94.4. 92.0 92.4 92.2 94.7 94.8 94.3. 94.4 94.4 94.4 94.4 94.6 94.4. 92.0 92.4 92.2 94.7 94.8 94.3. 94.4 94.4 94.4 94.4 94.6 94.4. 92.0 92.4 92.2 94.7 94.8 94.3. 94.4 94.4 94.4 94.4 94.6 94.4. 86.7 90.3 89.4 91.8 93.7 93.7. 90.3 93.1 92.5 90.0 93.1 92.7. 74.5 64.0 67.3 82.1 70.8 75.3. 76.0 66.5 69.4 79.2 67.8 71.8. 92.1 85.8 87.8 96.3 90.8 93.4. 92.0 86.7 88.9 93.9 87.0 90.3. 92.1 90.5 90.8 96.2 94.4 95.3. 92.7 90.8 91.4 93.6 90.7 92.4. 92.5 93.7 93.1 92.7 94.5 93.7. 85.5 89.5 88.3 91.5 93.7 93.2. 88.7 91.4 90.9 88.7 92.0 91.7. 37.1 42.6 34.9 45.3 51.3 41.8. 36.1 41.4 33.7 40.0 45.1 37.2. 81.0 85.9 79.9 90.2 92.5 87.2. 75.9 79.9 74.4 80.7 83.7 77.3. 89.5 91.9 90.4 95.9 96.7 95.8. 84.3 86.5 85.2 86.8 88.4 86.9. 91.2 92.2 92.2 91.4 93.4 92.4. 85.8 87.1 87.6 91.5 93.3 92.7. 87.7 83.9 89.3 88.0 85.3 89.9. 14.9 36.3 16.6 18.8 46.3 20.1. 14.1 37.3 15.7 17.7 40.5 18.5. 70.5 89.6 71.5 80.9 96.4 82.5. 57.6 79.0 60.5 65.0 83.6 66.8. 89.3 92.6 90.2 96.0 97.4 96.6. 74.4 80.4 77.4 78.0 83.2 79.7. Local Projections. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. H: T=100. T=400. 1 P UN FF P UN FF 4 P UN FF P UN FF 8 P UN FF P UN FF 12 P UN FF P UN FF. FWE control Bonf. Bonf. emp.. Marg.. Marg. emp.. 92.2 92.5 92.3 94.7 94.8 94.4. 94.4 94.4 94.4 94.4 94.6 94.4. 92.2 92.5 92.3 94.7 94.8 94.4. 94.4 94.4 94.4 94.4 94.6 94.4. 92.2 92.5 92.3 94.7 94.8 94.4. 71.0 77.7 74.5 81.9 87.3 85.8. 81.5 86.2 84.7 81.3 86.7 85.0. 86.9 89.7 88.5 94.6 96.1 95.9. 93.7 95.1 94.3 93.9 95.4 94.9. 52.7 61.2 58.4 72.2 79.2 77.8. 71.9 77.6 76.4 71.3 78.3 76.2. 80.0 84.5 81.9 94.4 96.5 95.6. 39.7 47.2 43.0 65.5 71.6 70.2. 66.2 69.7 69.2 64.8 70.8 68.7. 73.1 78.9 74.7 93.9 95.7 95.3. Notes: See notes to Table 1.. Schef. Schef. emp.. FDR control Bonf. Bonf. emp.. Marg.. Marg. emp.. Schef. Schef. emp.. 94.4 94.4 94.4 94.4 94.6 94.4. 92.2 92.5 92.3 94.7 94.8 94.4. 94.4 94.4 94.4 94.4 94.6 94.4. 92.2 92.5 92.3 94.7 94.8 94.4. 94.4 94.4 94.4 94.4 94.6 94.4. 92.2 92.5 92.3 94.7 94.8 94.4. 94.4 94.4 94.4 94.4 94.6 94.4. 82.9 89.0 87.4 91.6 93.7 93.6. 90.2 93.1 92.4 90.0 93.1 92.6. 75.5 63.1 66.1 82.3 70.7 75.1. 76.1 66.7 69.6 79.0 67.8 72.1. 92.3 83.9 86.4 96.4 90.8 93.3. 92.1 86.7 88.7 94.0 87.0 90.3. 90.4 88.4 88.8 96.1 94.3 95.1. 92.5 90.6 91.4 93.6 90.6 92.6. 92.6 93.6 93.3 92.9 94.4 93.8. 77.2 83.9 82.7 91.1 93.6 93.1. 88.5 91.2 90.8 88.6 91.9 91.7. 39.5 43.7 34.8 45.8 51.5 41.7. 36.9 41.5 34.0 40.1 45.1 36.9. 80.8 84.6 76.4 90.6 92.7 87.1. 75.9 79.6 74.4 81.1 83.8 77.3. 83.0 85.9 83.6 95.4 96.2 95.4. 84.0 86.3 85.0 86.7 88.4 86.8. 91.6 92.0 92.1 91.7 93.4 92.6. 71.8 64.7 74.5 91.2 92.3 92.4. 87.6 81.9 88.7 88.0 85.0 89.8. 18.5 44.7 19.5 19.5 48.7 20.0. 15.0 38.5 17.5 17.9 40.9 18.6. 68.4 89.8 69.2 80.8 96.5 82.4. 58.3 79.1 61.4 65.4 83.8 66.6. 74.6 80.7 76.6 95.0 96.6 95.6. 73.7 79.6 75.8 77.8 83.1 79.4.

No results found