Multiple Structural Breaks in Vector Error Correction Models
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Domenic Franjic
Abstract
The analysis of structural breaks in vector error correction models is often confined to possible level shifts and trend breaks. In contrast, only rudimentary tools are available to deal with breaks in the cointegration matrix and the adjustment towards equilibrium. Particularly, the possibility of multiple structural breaks during long sampling periods is often ignored which can lead to inconsistently estimated coefficients. In this paper, we study a two-step estimator based on a penalized regression to determine the number of structural breaks, their timing, and estimate the model’s coefficients for each regime. We focus on two important cases, namely, (i) constant dynamics but changing long-run equilibria, and (ii) convergence to new long-run equilibria with new adjustment dynamics. We use simulations to investigate the finite sample performance of the two-step estimator and provide an empirical illustration using data on the term structure of interest rates.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: SCHW 2062/1-1
Acknowledgments
The authors would like to thank Robert Jung for helpful comments and suggestions. Funding by the German Research Foundation (Grant SCHW 2062/1-1) is gratefully acknowledged.
Appendix A: Tables
Estimation of (multiple) structural breaks in the VECM using the group LASSO with BEA – Case 1 (including short-run dynamics).
| SB1: (τ = 0.5) | |||||
| T | pce | τ | |||
| 100 | 85.5 | 0.483 (0.068) | |||
| 200 | 95.8 | 0.497 (0.043) | |||
| 400 | 98.1 | 0.499 (0.024) | |||
| T | b 0,1 | b 1,1 | α 1 | α 2 | |
| 100 | −1.004 (0.071) | −1.980 (0.097) | −0.525 (0.094) | 0.494 (0.094) | |
| 200 | −1.006 (0.039) | −1.988 (0.060) | −0.511 (0.068) | 0.489 (0.066) | |
| 400 | −1.004 (0.028) | −1.992 (0.032) | −0.503 (0.054) | 0.488 (0.054) | |
| SB2: (τ1 = 0.33, τ2 = 0.67) | |||||
| T | pce | τ 1 | τ 2 | ||
| 150 | 75.5 | 0.323 (0.051) | 0.654 (0.043) | ||
| 300 | 84.6 | 0.333 (0.039) | 0.660 (0.029) | ||
| 600 | 86.3 | 0.335 (0.034) | 0.662 (0.025) | ||
| T | b 0,1 | b 1,1 | b 2,1 | α 1 | α 2 |
| 150 | −1.016 (0.097) | −1.964 (0.126) | −1.018 (0.079) | −0.526 (0.095) | 0.487 (0.089) |
| 300 | −1.011 (0.049) | −1.981 (0.061) | −1.010 (0.047) | −0.511 (0.072) | 0.486 (0.070) |
| 600 | −1.007 (0.036) | −1.989 (0.037) | −1.008 (0.042) | −0.504 (0.055) | 0.486 (0.054) |
| SB4: (τ1 = 0.2, τ2 = 0.4, τ3 = 0.6, τ4 = 0.8) | |||||
| T | pce | τ 1 | τ 2 | τ 3 | τ 4 |
| 250 | 47.3 | 0.197 (0.044) | 0.389 (0.051) | 0.595 (0.040) | 0.790 (0.032) |
| 500 | 58.0 | 0.202 (0.031) | 0.395 (0.038) | 0.599 (0.032) | 0.794 (0.020) |
| 1,000 | 64.8 | 0.203 (0.029) | 0.396 (0.035) | 0.602 (0.024) | 0.796 (0.016) |
| T | b 0,1 | b 1,1 | b 2,1 | b 3,1 | b 4,1 |
| 250 | −1.026 (0.130) | −1.925 (0.210) | −1.071 (0.190) | −1.936 (0.178) | −1.015 (0.078) |
| 500 | −1.018 (0.092) | −1.955 (0.140) | −1.042 (0.130) | −1.965 (0.097) | −1.016 (0.074) |
| 1,000 | −1.016 (0.087) | −1.962 (0.134) | −1.036 (0.126) | −1.980 (0.064) | −1.010 (0.043) |
| T | α 1 | α 2 | |||
| 250 | −0.525 (0.100) | 0.466 (0.097) | |||
| 500 | −0.513 (0.080) | 0.473 (0.078) | |||
| 1,000 | −0.501 (0.072) | 0.471 (0.070) | |||
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Note: We use 1,000 replications of the data-generating process given in Equation (20). pce denotes the percentages of correct estimation of the number of breaks m. The variance of the error terms is
Estimation of (multiple) structural breaks in the VECM using the group LASSO with BEA – Case 2 (including short-run dynamics).
| SB1: (τ = 0.5) | ||||||
| T | pce | τ | b 0,1 | b 1,1 | ||
| 100 | 84.8 | 0.479 (0.076) | −1.023 (0.169) | −1.992 (0.169) | ||
| 200 | 93.2 | 0.493 (0.042) | −1.011 (0.093) | −1.997 (0.107) | ||
| 400 | 96.9 | 0.500 (0.030) | −1.003 (0.088) | −1.997 (0.050) | ||
| T | α 0,1 | α 0,2 | α 1,1 | α 1,2 | ||
| 100 | −0.493 (0.166) | 0.036 (0.146) | −0.006 (0.090) | 0.387 (0.155) | ||
| 200 | −0.476 (0.123) | 0.007 (0.090) | −0.005 (0.056) | 0.385 (0.131) | ||
| 400 | −0.467 (0.101) | −0.008 (0.060) | −0.006 (0.042) | 0.393 (0.118) | ||
| SB2: (τ1 = 0.33, τ2 = 0.67) | ||||||
| T | pce | τ 1 | τ 2 | b 0,1 | b 1,1 | b 2,1 |
| 150 | 66.6 | 0.314 (0.055) | 0.649 (0.050) | −1.008 (0.105) | −1.981 (0.278) | −1.015 (0.098) |
| 300 | 81.8 | 0.325 (0.034) | 0.661 (0.028) | −1.007 (0.070) | −1.986 (0.094) | −1.009 (0.051) |
| 600 | 90.3 | 0.328 (0.026) | 0.667 (0.016) | −1.007 (0.039) | −1.981 (0.058) | −1.004 (0.024) |
| T | α 0,1 | α 0,2 | α 1,1 | α 1,2 | α 2,1 | α 2,2 |
| 150 | −0.531 (0.163) | 0.027 (0.160) | −0.053 (0.108) | 0.342 (0.155) | −0.476 (0.154) | −0.006 (0.108) |
| 300 | −0.503 (0.121) | −0.004 (0.096) | −0.035 (0.063) | 0.343 (0.128) | −0.469 (0.122) | −0.009 (0.076) |
| 600 | −0.489 (0.106) | −0.014 (0.060) | −0.036 (0.042) | 0.332 (0.119) | −0.459 (0.107) | −0.014 (0.052) |
| SB4: (τ1 = 0.2, τ2 = 0.4, τ3 = 0.6, τ4 = 0.8) | ||||||
| T | pce | τ 1 | τ 2 | τ 3 | τ 4 | |
| 250 | 40.2 | 0.199 (0.060) | 0.395 (0.065) | 0.587 (0.064) | 0.785 (0.059) | |
| 500 | 58.3 | 0.202 (0.040) | 0.396 (0.039) | 0.597 (0.035) | 0.792 (0.037) | |
| 1,000 | 61.1 | 0.203 (0.034) | 0.398 (0.029) | 0.598 (0.026) | 0.796 (0.022) | |
| T | b 0,1 | b 1,1 | b 2,1 | b 3,1 | b 4,1 | |
| 250 | −1.033 (0.173) | −1.904 (0.434) | −1.091 (0.729) | −1.894 (0.372) | −1.042 (0.164) | |
| 500 | −1.042 (0.359) | −1.959 (0.206) | −1.041 (0.249) | −1.965 (0.179) | −1.017 (0.080) | |
| 1,000 | −1.026 (0.121) | −1.959 (0.145) | −1.051 (0.399) | −1.976 (0.135) | −1.013 (0.061) | |
| T | α 0,1 | α 0,2 | α 1,1 | α 1,2 | α 2,1 | |
| 250 | −0.517 (0.188) | 0.036 (0.171) | −0.073 (0.149) | 0.313 (0.174) | −0.404 (0.199) | |
| 500 | −0.498 (0.141) | −0.003 (0.095) | −0.039 (0.089) | 0.341 (0.136) | −0.422 (0.157) | |
| 1,000 | −0.473 (0.128) | −0.014 (0.064) | −0.034 (0.065) | 0.346 (0.130) | −0.407 (0.143) | |
| T | α 2,2 | α 3,1 | α 3,2 | α 4,1 | α 4,2 | |
| 250 | 0.050 (0.167) | −0.071 (0.154) | 0.327 (0.170) | −0.471 (0.183) | 0.010 (0.130) | |
| 500 | 0.007 (0.098) | −0.034 (0.074) | 0.348 (0.141) | −0.454 (0.125) | −0.002 (0.079) | |
| 1,000 | −0.003 (0.063) | −0.026 (0.048) | 0.346 (0.120) | −0.445 (0.115) | −0.012 (0.053) | |
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Note: We use 1,000 replications of the data-generating process given in Equation (20). The variance of the error terms is
Estimation of (multiple) structural breaks in the VECM with N = 3 variables using the group LASSO with BEA.
| Case 1: | |||||
| SB1: (τ = 0.5) | |||||
| T | pce | τ | |||
| 100 | 93.4 | 0.495 (0.050) | |||
| 200 | 97.2 | 0.498 (0.027) | |||
| 400 | 98.8 | 0.498 (0.014) | |||
| SB2: (τ1 = 0.33, τ2 = 0.67) | |||||
| T | pce | τ 1 | τ 2 | ||
| 150 | 69.4 | 0.340 (0.072) | 0.654 (0.044) | ||
| 300 | 77.8 | 0.335 (0.039) | 0.663 (0.024) | ||
| 600 | 82.6 | 0.334 (0.027) | 0.666 (0.015) | ||
| SB4: (τ1 = 0.2, τ2 = 0.4, τ3 = 0.6, τ4 = 0.8) | |||||
| T | pce | τ 1 | τ 2 | τ 3 | τ 4 |
| 250 | 41.4 | 0.257 (0.086) | 0.429 (0.081) | 0.609 (0.068) | 0.782 (0.052) |
| 500 | 46.3 | 0.239 (0.074) | 0.420 (0.070) | 0.609 (0.040) | 0.791 (0.025) |
| 1,000 | 51.9 | 0.224 (0.056) | 0.408 (0.053) | 0.605 (0.025) | 0.794 (0.021) |
| Case 2: | |||||
| SB1: (τ = 0.5) | |||||
| T | pce | τ | |||
| 100 | 86.4 | 0.496 (0.044) | |||
| 200 | 94.8 | 0.499 (0.029) | |||
| 400 | 96.1 | 0.498 (0.020) | |||
| SB2: (τ1 = 0.33, τ2 = 0.67) | |||||
| T | pce | τ 1 | τ 2 | ||
| 150 | 75.8 | 0.324 (0.042) | 0.661 (0.031) | ||
| 300 | 84.0 | 0.329 (0.031) | 0.667 (0.023) | ||
| 600 | 87.6 | 0.332 (0.021) | 0.668 (0.011) | ||
| SB4: (τ1 = 0.2, τ2 = 0.4, τ3 = 0.6, τ4 = 0.8) | |||||
| T | pce | τ 1 | τ 2 | τ 3 | τ 4 |
| 250 | 46.8 | 0.212 (0.055) | 0.402 (0.055) | 0.593 (0.059) | 0.787 (0.049) |
| 500 | 51.5 | 0.209 (0.043) | 0.396 (0.046) | 0.598 (0.044) | 0.791 (0.043) |
| 1,000 | 62.5 | 0.207 (0.039) | 0.399 (0.037) | 0.605 (0.029) | 0.795 (0.029) |
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Note: We use 1,000 replications of the data-generating process given in Equation (20). The variance of the error terms is
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