So that equation (31) can be derived.
Appendix C. Griddy-Gibbs Sampler
The Griddy-Gibbs sampler is proposed by Ritter and Tanner (1992). This sampler can be im-plemented when the conditional posterior density is unknown to the researcher. The advantage of using this sampler over the importance sampler or the Metropolis-Hastings algorithm is that researcher does not have to provide an approximation of the function. The disadvantage is that this sampler demands more computing time. The procedure for implementing the Griddy-Gibbs sampler is as following:
1. Before we begin the chain, we must choose the range of the grid and the number of the grid.
The range should be chosen so that the generated numbers are not truncated.
2. Let vec(β)0= (β1,β2, . . . ,βm). With an arbitrary starting value (within the upper and the
5. Draw the random numbers from the uniform density with the lower bound as zeros and the upper bound as ΦU and invert cdf G by numerical interpolation to obtain a draw βi1 from ζ(β1|βi2,βi3, . . . ,βim,Y ).
6. Repeat steps 2-5 forβ2, . . . ,βm.
7. Set i = i + 1 (increment i by 1) and go to step 2.
Note that integration at the step 3 can be done by the deterministic approximation such as the Simpson’s rule or the Trapezoidal rule.
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Table 1: Monte Carlo Results for VAR Models: Average Posterior Probabilities DGP\#.of breaks m = 0 m = 1 m = 2 m = 3 m = 4
DGP 1 0.942 0.057 0.001 0.000 0.000
DGP 2 0.000 0.013 0.945 0.042 0.000
DGP 3 0.000 0.000 0.995 0.004 0.000
DGP 4 0.000 0.000 0.967 0.033 0.000
DGP 5 0.000 0.008 0.981 0.011 0.000
Table 2: Monte Carlo Mean of the Mode of the Posterior for the Break Points when m = 2: VAR Models
()=Monte Carlo standard deviation
DGP 2 DGP 3 DGP 4 DGP 5
1st break 99.571 (3.092) 100.06 (1.635) 99.987 (2.216) 100.03 (1.504) 2nd break 200.94 (2.237) 200.97 (1.403) 200.85 (3.093) 201.02 (1.883)
The true value of the first break is at t = 100, and the second is at t = 200.
Table 3: Parameters Given in DGPs for Monte Carlo Simulations of VECM with Two Structural Breaks
DGP 2 DGP 3 DGP 4 DGP 5
µ1 (-0.0551, -0.0370) (-0.0072, 0.0250) (-0.0389, -0.0862) (-0.0342, 0.0355) µ2 (0.0746, 0.1578) (0.0045, -0.1671) (0.0378, 0.1123) (-0.0017, -0.0535) µ3 (-0.0448, -0.2355) (0.0421, -0.1282) (-0.0558, -0.1406) (-0.0227, -0.0997) α1 (0.0004, 0.1766) (-0.0511, 0.0983) (-0.0333, 0.1530) (-0.0503, 0.1413)
α2 (-0.2292, 0.1425) (-0.0525, 0.0331)
α3 (0.0118, 0.0884) (0.0018, 0.0746)
β (1, -0.9902) (1, -0.9894) (1, -0.9807) (1, -0.9744)
Table 4: Monte Carlo Results for VECM Models: Average Posterior Probabilities
DGP\ #.of breaks m = 0 m = 1 m = 2 m = 3 m = 4
DGP 1 0.946 0.054 0.000 0.000 0.000
DGP 2 0.084 0.030 0.875 0.030 0.000
DGP 3 0.000 0.000 0.902 0.098 0.000
DGP 4 0.085 0.108 0.807 0.000 0.000
DGP 5 0.000 0.000 0.944 0.056 0.000
Table 5: Monte Carlo Mean of the Mode of the Posterior for the Break Points when m = 2:
VECM
()=Monte Carlo standard deviation
DGP 2 DGP 3 DGP 4 DGP 5
1st break 91.231 (21.24) 99.090 (3.335) 100.96 (17.43) 100.27 (0.793) 2nd break 188.21 (23.74) 200.33 (1.085) 203.01 (13.54) 200.82 (2.170)
The true value of the first break is at t = 100, and the second is at t = 200.
Table 6: Model Selection: Application 1
p\m m = 0 m = 1 m = 2 m = 3 m = 4 p = 3 0.0000 0.0000 0.0000 0.0002 0.0000 p = 4 0.0000 0.0000 0.0000 0.9315 0.0120 p = 5 0.0000 0.0000 0.0000 0.0344 0.0219
Note: Each element shows the posterior probability in () using Chib’s (1995) method.
p: the number of the lag in a VAR m: the number of the structural breaks
Table 7: Posterior Results of a VAR Model with m = 3 and p = 4 for Application 1
(): standard deviation
(a) Estimates of Break Points
Posterior Mode 95% HPDR 1st break 1974:07 (0.5580) 1973:09, 1975:07 2nd break 1978:11 (0.5602) 1977:10, 1979:10 3rd break 1983:01 (0.1637) 1982:09, 1983:03
(b) Estimates of Other Parameters (Mean of the Posterior)
Parameters sp ip Parameters sp ip
µ1 -0.0123 (0.0170) 0.1283 (0.0573) sp(−2) -0.2612 (0.0228) 0.0212 (0.0217) µ2 0.0543 (0.0302) 0.2197 (0.0748) ip(−2) 0.0590 (0.0081) 0.0762 (0.0221) µ3 -0.0488 (0.0952) 0.0823 (0.0442) sp(−3) 0.0410 (0.0272) -0.1705 (0.0213) µ4 0.0773 (0.0071) 0.1058 (0.l0133) ip(−3) 0.0113 (0.0067) 0.0129 (0.0114) sp(−1) 1.1967 (0.0131) 0.0675 (0.0119) sp(−4) -0.0292 (0.0157) 0.1290 (0.0133) ip(−1) 0.0049 (0.0071) 1.0700 (0.0163) ip(−4) -0.0718 (0.0049) -0.2303 (0.0075)
Ω1=
Table 8: Selection of the Number of the Breaks for Application 2 by Model 1
p\m m = 0 m = 1 m = 2 m = 3 m = 4 p = 2 0.0000 0.0000 0.0000 0.0000 0.0000 p = 3 0.0000 0.0000 0.0000 0.9973 0.0018 p = 4 0.0000 0.0000 0.0000 0.0007 0.0000
Note: Each element shows the posterior probability in ().
p: the number of the lag in a VAR m: the number of the structural breaks
Table 9: Selection of the Number of Ranks of Model 1 with m = 3 and p = 3 rank\i i = 1 i = 2 i = 3 i = 4
r = 0 0.0000 0.0311 0.0000 0.0742 r = 1 1.0000 0.9699 0.6121 0.9258 r = 2 0.0000 0.0000 0.3880 0.0000
Note: Each element shows the posterior probability in ().
r: the number of rank
i: regime divided by the structural breaks
Table 10: Posterior Results of Model 1 with m = 3 and p = 3, and r = 1 for all regimes for Application 2
(a) Estimates of Break Points Posterior Mode 95% HPDR 1st break 1979:10 (0.0548) 1979:09, 1979:10 2nd break 1983:01 (0.0601) 1982:10, 1983:02 3rd break 1988:06 (0.1226) 1988:04, 1988:08
(b) Estimates of Other Parameters (Mean of the Posterior)
Parameters rl rs Parameters rl rs
µ1 0.0219 (0.0053) -0.0043 (0.0167) α3 -0.0265 (0.0049) 0.0755 (0.0093) µ2 -0.0005 (0.0763) 0.2087 (0.0819) α4 -0.0006 (0.0010) 0.0171 (0.0011)
µ3 0.0238 (0.0114) -0.1297 (0.0132) β1 1 -0.9911 (0.0636)
µ4 -0.0194 (0.0017) -0.0289 (0.0055) β2 1 -0.9915 (0.0793)
α1 -0.0100 (0.0011) 0.0368 (0.0023) β3 1 -1.0023 (0.0193)
α2 -0.0831 (0.0189) 0.1610 (0.0679) β4 1 -0.9935 (0.0473)
Ω1=
Table 11: Selection of the Number of the Breaks for Application 2 by Model 2
p\m m = 0 m = 1 m = 2 m = 3 m = 4 p = 2 0.0000 0.0000 0.0000 0.0000 0.0000 p = 3 0.0000 0.0000 0.0000 0.9719 0.0263 p = 4 0.0000 0.0000 0.0000 0.0009 0.0009
Note: Each element shows the posterior probability.
p: the number of the lag in a VAR m: the number of the structural breaks
Table 12: Posterior Results of Model 2 with m = 3 and p = 3, and r = 1 for all regimes for Application 2
(a) Estimates of Break Points Posterior Mode 95% HPDR 1st break 1979:10 (0.0552) 1979:09, 1979:10 2nd break 1983:01 (0.0610) 1982:11, 1983:02 3rd break 1988:06 (0.1376) 1988:04, 1988:08
(b) Estimates of Other Parameters (Mean of the Posterior)
Parameters rl rs
µ1 0.0225 (0.0053) -0.0061 (0.0162) µ2 0.0000 (0.0644) 0.2117 (0.0762) µ3 0.0263 (0.0173) -0.1228 (0.0165) µ4 -0.0189 (0.0024) -0.0301 (0.0071) α1 -0.0102 (0.0012) 0.0371 (0.0022) α2 -0.0846 (0.0181) 0.1672 (0.0640) α3 -0.0257 (0.0051) 0.0709 (0.0216) α4 -0.0003 (0.0013) 0.0173 (0.0014)
β 1 -0.9844 (0.0610)
Figure 1: The Interest Rates Spread (sp) and the US Industrial Production Growth Rate (ip)
solid line - the interest rates spread, dotted line - the US industrial production growth rate
1970 1975 1980 1985 1990 1995 2000 2005
−12.5
−10.0
−7.5
−5.0
−2.5 0.0 2.5 5.0 7.5 10.0
Figure 2: Posterior Probability Mass of the Break Dates for Application 1
1972.5 1973.0 1973.5 1974.0 1974.5 1975.0 1975.5 1976.0 1976.5 0.25
0.50 0.75
Density
1st Estimated Dates
1977.0 1977.5 1978.0 1978.5 1979.0 1979.5 1980.0 1980.5 1981.0 0.25
0.50 0.75
Density
2nd Estimated Dates
1982.8 1982.9 1983.0 1983.1 1983.2 1983.3 1983.4 1983.5 1983.6 1
2 3
Density
3rd Estimated Dates
Figure 3: US Long- and Short-Term Interest Rates
solid line - 10-year TB rate, dotted line - 3-month TB rate
1970 1975 1980 1985 1990 1995 2000 2005
2.5 5.0 7.5 10.0 12.5 15.0 17.5
Long− and short rate
Figure 4: Posterior Probability Mass of the Break Dates for Application 2 - Model 1
1970 1975 1980 1985 1990 1995 2000 2005
10 20 30 40 50 60 70 80
Figure 5: Posterior Density ofαfor Application 2 - Model 1
−0.014 −0.012 −0.010 −0.008 −0.006 100
−0.05 −0.04 −0.03 −0.02 −0.01 0.00 25
50 75
Density
alpha−31
0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 20
40 Density
alpha−32
−0.006 −0.004 −0.002 0.000 0.002 200
400 Density
alpha−41
0.014 0.016 0.018 0.020 0.022 100
200 300 400 Density
alpha−42
Figure 6: Posterior Probability Mass of the Break Dates for Application 2 - Model 2
1970 1975 1980 1985 1990 1995 2000 2005
10 20 30 40 50 60 70 80 90
Figure 7: Posterior Density ofαfor Application 2 - Model 2
−0.014 −0.012 −0.010 −0.008 −0.006 100
−0.04 −0.03 −0.02 −0.01 0.00 25
50 75
Density
alpha−31
0.025 0.050 0.075 0.100 0.125 0.150 10
0.014 0.016 0.018 0.020 0.022 100
200 300 Density
alpha−42