Supplement to“Functional-Coe¢ cient Cointegration Models
in the Presence of Deterministic Trends”:
Monte Carlo Results
Masayuki Hirukawa
Setsunan UniversityMari Sakudo
Development Bank of Japanand Waseda UniversityMay 2015

Table S1 below presents simulation results in Section 6. While we provide only
sample medians of RMSEs, there is not much di¤erence in tendencies from their(unreported) sample averages. “Restricted”and “Unrestricted”denote the regression
with no linear trend (9) and the regression with a linear trend (10), respectively. For
each functional form of(z), the cases with=21=11= 0:4 are chosen as

benchmarks, the results of which are put in frames. Accordingly, each panel reports
the results when a parameter is changed with all others held constant. Furthermore,for each combination of the model (i.e., Restricted or Unrestricted),sample size, the smallest performance measure is typed in bold faces.

1(z) and theTable S1: Median RMSEs for LL and PLLR Estimators
Panel A: β(z) = 1
T = 100
Restricted
UnrestristedLLLLSPROT PLLRSPROT PLLR

(σ21 = φ11 = 0.4)

ρ-0.8-0.4
0.0
0.4
0.8

0.3303 0.5247 0.2233
0.0933 0.3550 0.1504
0.0801 0.3271 0.1386
0.1017 0.3652 0.1587
0.3926 0.5727 0.2613

0.4882 0.2843 0.2655
0.2314 0.1833 0.2312
0.2064 0.1829 0.2266
0.2339 0.1970 0.2348
0.6520 0.3312 0.2821

0.1171 0.3743 0.1698
0.1098 0.3740 0.1668
0.1036 0.3722 0.1615
0.1017 0.3652 0.1587
0.1073 0.3827 0.1626

0.2800 0.2197 0.2318
0.2425 0.2026 0.2361
0.2236 0.1905 0.2439
0.2339 0.1970 0.2348
0.2661 0.2226 0.2327

0.0912 0.0854 0.1467
0.0632 0.0693 0.0969
0.0607 0.0685 0.0805
0.0661 0.0703 0.0909
0.1016 0.0895 0.1577

0.0163 0.0647 0.0336
0.0161 0.0672 0.0339
0.0159 0.0620 0.0337
0.0160 0.0613 0.0334
0.0160 0.0615 0.0328

0.0789 0.0833 0.0917
0.0658 0.0728 0.0901
0.0626 0.0701 0.0908
0.0661 0.0703 0.0909
0.0776 0.0804 0.0932(ρ = σ21 = 0.4)

φ11
-0.8
-0.4
0.00.40.80.0267 0.2303 0.0588
0.0162 0.0573 0.0332
0.0148 0.0470 0.0289
0.0160 0.0613 0.0334
0.0324 0.2392 0.0646(ρ = φ11 = 0.4)

σ21-0.8-0.4
0.00.4
0.8

T = 250
Restricted
Unrestristed
LL
LL
SP
ROT PLLRSP
ROT PLLR

0.1400 0.4435 0.2388
0.0943 0.3810 0.1603
0.0953 0.3611 0.1424
0.1017 0.3652 0.1587
0.1599 0.4293 0.2272

0.3083 0.2627 0.3588
0.2159 0.1769 0.2441
0.2053 0.1746 0.2373
0.2339 0.1970 0.2348
0.3863 0.3119 0.3330

0.0220 0.0910 0.0498
0.0151 0.0561 0.0321
0.0141 0.0535 0.0303
0.0160 0.0613 0.0334
0.0265 0.0843 0.0512

0.0828 0.0962 0.1336
0.0584 0.0672 0.0960
0.0555 0.0644 0.0864
0.0661 0.0703 0.0909
0.1243 0.1295 0.1426

2Panel B: β(z) = 0.3 - 0.5exp(-1.25z )T = 100
Restricted
Unrestristed
LL
LL
SP
ROT PLLRSP
ROT PLLR(σ21 = φ11 = 0.4)

ρ-0.8
-0.4
0.00.40.80.3653 0.5906 0.4728
0.2015 0.4574 0.3447
0.1926 0.4476 0.3437
0.2183 0.4753 0.3665
0.3951 0.6544 0.5117

0.2004 0.4962 0.3699
0.2038 0.5205 0.3714
0.2026 0.4981 0.3789
0.2183 0.4753 0.3665
0.2133 0.5160 0.3913

0.1901 0.3983 0.3125
0.1531 0.2366 0.2098
0.1485 0.2311 0.2011
0.1501 0.2395 0.2078
0.1994 0.3730 0.2900

1.5740 1.8799 2.7853
1.1154 1.4865 1.9576
1.0507 1.4371 1.6559
1.0870 1.4601 1.8954
1.4396 1.5878 2.5937

1.4046 1.5102 1.9217
1.3727 1.5127 1.8618
1.4285 1.5241 1.8428
1.4481 1.5343 1.8642
1.4997 1.5806 1.8876

0.1495 0.2412 0.2086
0.1492 0.2418 0.2097
0.1503 0.2419 0.2102
0.1501 0.2395 0.2078
0.1518 0.2429 0.2121

1.0739 1.4432 1.8937
1.0565 1.4303 1.8621
1.0656 1.4336 1.8896
1.0870 1.4601 1.8954
1.0766 1.4471 1.8710(ρ = σ21 = 0.4)

φ11
-0.8
-0.4
0.0
0.4
0.8

2.1139 1.7938 1.4716
1.3191 1.4330 1.7825
1.2849 1.5045 1.8315
1.4481 1.5343 1.86422.8966 2.3439 1.7781(ρ = φ11 = 0.4)

σ21-0.8
-0.4
0.0
0.4
0.8

T = 250
Restricted
Unrestristed
LLLLSPROT PLLRSP
ROT PLLR

0.2299 0.5692 0.4378
0.2069 0.5031 0.3818
0.2034 0.5003 0.3772
0.2183 0.4753 0.3665
0.2390 0.5435 0.4223

1.4184 1.5584 1.9583
1.4304 1.5161 1.9683
1.4502 1.5098 1.9734
1.4481 1.5343 1.8642
1.4892 1.5467 1.9616

20.1523 0.2513 0.2140
0.1481 0.2394 0.2086
0.1491 0.2387 0.2082
0.1501 0.2395 0.2078
0.1532 0.2446 0.2118

1.0784 1.4450 1.9405
1.0850 1.4636 1.8931
1.0888 1.4598 1.8885
1.0870 1.4601 1.8954
1.0894 1.4600 1.8982

Table S1: Continued
Panel C: β(z) = 0.5/{1+exp(-4z)} - 0.75
T = 100
Restricted
Unrestristed
LL
LL
SP
ROT PLLRSP
ROT PLLR(σ21 = φ11 = 0.4)

ρ
-0.8
-0.4
0.0
0.4
0.8

0.5929 1.0702 0.7390
0.2920 0.6953 0.5228
0.2768 0.6542 0.4895
0.3114 0.7041 0.5475
0.6167 1.0291 0.7955

0.3235 0.6980 0.5330
0.3137 0.7253 0.5379
0.3162 0.7335 0.5343
0.3114 0.7041 0.5475
0.3128 0.7202 0.5544

0.2665 0.6255 0.4937
0.2149 0.3754 0.3121
0.2114 0.3316 0.2901
0.2215 0.3853 0.3219
0.3073 0.6649 0.4991

2.7205 3.0553 5.3516
1.7442 2.2647 3.0596
1.6012 2.1221 2.5243
1.8580 2.3553 3.0313
2.9097 3.0789 4.7178

2.6040 2.4923 3.1698
2.5161 2.5034 3.0659
2.5589 2.5125 3.0405
2.5738 2.5347 3.0350
2.5216 2.5086 3.2253

0.2226 0.3885 0.3210
0.2213 0.3814 0.3229
0.2207 0.3789 0.3187
0.2215 0.3853 0.3219
0.2228 0.3794 0.3203

1.8658 2.3659 2.9999
1.8313 2.3430 3.0091
1.8383 2.3231 2.9954
1.8580 2.3553 3.0313
1.8627 2.3925 3.0314(ρ = σ21 = 0.4)

φ11
-0.8
-0.4
0.0
0.4
0.8

5.0357 3.6707 3.3831
2.1158 2.2366 2.8554
1.8936 2.1101 2.7331
2.5738 2.5347 3.0350
5.8459 4.2715 3.5106(ρ = φ11 = 0.4)

σ21-0.8-0.4
0.00.4
0.8

T = 250
Restricted
Unrestristed
LL
LLSP
ROT PLLRSPROT PLLR

0.3248 0.7585 0.5914
0.3237 0.7532 0.5547
0.3248 0.7551 0.5414
0.3114 0.7041 0.5475
0.3577 0.8001 0.5825

2.5320 2.5217 3.0695
2.5783 2.6128 3.1558
2.5918 2.6151 3.1543
2.5738 2.5347 3.0350
2.5986 2.6180 3.2336

0.2243 0.3779 0.3195
0.2239 0.3748 0.3181
0.2236 0.3760 0.3176
0.2215 0.3853 0.3219
0.2226 0.3821 0.3242

1.8703 2.3538 2.9324
1.8588 2.3488 2.9180
1.8661 2.3451 2.9134
1.8580 2.3553 3.0313
1.8483 2.3537 3.0390

Panel D: β(z) = 0.25exp(-z2 )T = 100
Restricted
Unrestristed
LL
LL
SP
ROT PLLRSP
ROT PLLR(σ21 = φ11 = 0.4)

ρ
-0.8
-0.4
0.00.40.8

0.2604 0.4373 0.3173
0.1298 0.2820 0.2329
0.1232 0.2780 0.2170
0.1381 0.2999 0.2306
0.2679 0.4385 0.3477

0.1397 0.2984 0.2270
0.1358 0.3142 0.2234
0.1347 0.3020 0.2278
0.1381 0.2999 0.2306
0.1352 0.3220 0.2296

0.0934 0.1890 0.1425
0.0678 0.1183 0.0990
0.0638 0.1105 0.0920
0.0663 0.1141 0.0956
0.0843 0.2146 0.1551

0.6156 0.7427 1.2842
0.5311 0.6950 0.8623
0.5146 0.6820 0.8064
0.5092 0.6450 0.8304
0.7910 0.8805 1.4605

0.6902 0.7542 0.9615
0.7124 0.7532 0.9088
0.6698 0.7364 0.9120
0.7100 0.7397 0.9534
0.6860 0.7370 0.9395

0.0667 0.1145 0.0953
0.0669 0.1137 0.09620.0666 0.1125 0.0961
0.0663 0.1141 0.0956
0.0662 0.1128 0.0950

0.5195 0.6480 0.8263
0.5167 0.6513 0.8262
0.4952 0.6521 0.8666
0.5092 0.6450 0.8304
0.5110 0.6528 0.8379(ρ = σ21 = 0.4)

φ11-0.8
-0.4
0.0
0.4
0.8

1.2857 1.0293 0.9385
0.6415 0.6669 0.8858
0.6267 0.6562 0.8484
0.7100 0.7397 0.9534
1.5955 1.1008 0.9340(ρ = φ11 = 0.4)

σ21-0.8
-0.4
0.0
0.4
0.8

T = 250
Restricted
UnrestristedLL
LLSP
ROT PLLRSPROT PLLR

0.1559 0.4078 0.3091
0.1283 0.3026 0.2422
0.1285 0.2872 0.2205
0.1381 0.2999 0.2306
0.1818 0.3809 0.2967

0.7239 0.7604 1.0141
0.6825 0.7350 0.9461
0.6828 0.7377 0.9431
0.7100 0.7397 0.9534
0.7643 0.7804 0.9541

30.0681 0.1222 0.1003
0.0655 0.1121 0.0927
0.0650 0.1116 0.0917
0.0663 0.1141 0.0956
0.0692 0.1237 0.1025

0.5332 0.6787 0.8455
0.5313 0.6704 0.8429
0.5282 0.6703 0.8363
0.5092 0.6450 0.8304
0.5402 0.6881 0.8394