Supplemental Material for
”Testing a Large Set of Zero Restrictions in Regression Models,
with an Application to Mixed Frequency Granger Causality”
Eric Ghysels∗
Jonathan B. Hill†
Kaiji Motegi‡
March 10, 2017
∗
Department of Economics and Department of Finance, Kenan-Flagler Business School, University of North
Carolina at Chapel Hill. E-mail: [email protected]
†
Department of Economics, University of North Carolina at Chapel Hill. E-mail: [email protected]
‡
Graduate School of Economics, Kobe University. E-mail: [email protected]
1
Contents
A Omitted Figures
4
B Omitted Tables
4
List of Figures
F.1 Visual Explanation of Mixed Frequency Time Series . . . . . . . . . . . . . . . .
4
List of Tables
T.1 Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(1) - IID
Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . . . . .
5
T.2 Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(1) GARCH Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . .
7
T.3 Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(2) - IID
Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . . . . .
9
T.4 Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(2) GARCH Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . .
11
T.5 Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(1) - IID
Error and Non-Robust Covariance Matrix . . . . . . . . . . . . . . . . . . . . . .
13
T.6 Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(1) - GARCH
Error and Non-Robust Covariance Matrix . . . . . . . . . . . . . . . . . . . . . .
14
T.7 Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(2) - IID
Error and Non-Robust Covariance Matrix . . . . . . . . . . . . . . . . . . . . . .
15
T.8 Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(2) - GARCH
Error and Non-Robust Covariance Matrix . . . . . . . . . . . . . . . . . . . . . .
16
T.9 Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(1) - IID
Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . . . . .
17
T.10 Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(1) GARCH Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . .
22
T.11 Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(2) - IID
Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . . . . .
27
T.12 Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(2) GARCH Error and Robust Covariance Matrix for Max Tests . . . . . . . . . . .
30
T.13 Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(1) - IID
Error and Non-Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . .
33
T.14 Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(1) - GARCH
Error and Non-Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . .
2
36
T.15 Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(2) - IID
Error and Non-Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . .
39
T.16 Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(2) - GARCH
Error and Non-Robust Covariance Matrix for Max Tests . . . . . . . . . . . . . .
42
T.17 Ljung-Box Q Tests of Residual Whiteness with Double Blocks-of-Blocks Bootstrap 45
3
A
Omitted Figures
Figure F.1: Visual Explanation of Mixed Frequency Time Series
...
Note: This figure explains a standard notation used in the Mixed Data Sampling (MIDAS) literature. Assume
there are only one high frequency variable xH and only one low frequency variable xL . In low frequency period
τL , we sequentially observe xH (τL , 1), xH (τL , 2), . . . , xH (τL , m), xL (τL ).
B
Omitted Tables
4
Table T.1: Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(1) - IID
Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: b = 012×1
hM F
4
8
12
24
MF
Max, Wald
.056, .049
.053, .050
.045, .050
.043, .044
hM F
4
8
12
24
MF
Max, Wald
.059, .050
.053, .041
.046, .052
.046, .050
A.1. d = 0.2 (low persistence in xH )
A.1.1. TL = 80
A.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.059, .040
.055, .055
4
.050, .056
1
.046, .051
2
.049, .042
.056, .038
8
.051, .044
2
.055, .041
3
.049, .046
.047, .054
12
.051, .053
3
.051, .046
4
.055, .049
.060, .044
24
.045, .051
4
.052, .050
A.2. d = 0.8 (high persistence in xH )
A.2.1. TL = 80
A.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.058, .049
.056, .058
4
.052, .050
1
.055, .047
2
.052, .045
.058, .049
8
.052, .047
2
.055, .054
3
.051, .038
.051, .038
12
.048, .056
3
.051, .052
4
.050, .045
.059, .044
24
.046, .044
4
.053, .055
LF (stock)
Max, Wald
.053, .052
.050, .048
.049, .048
.052, .050
LF (stock)
Max, Wald
.056, .054
.054, .044
.050, .050
.047, .050
B. Decaying Causality: bj = (−1)j−1 0.3/j for j = 1, . . . , 12
B.1. d = 0.2 (low persistence in xH )
hM F
4
8
12
24
MF
Max, Wald
.450, .521
.344, .436
.298, .340
.215, .222
hM F
4
8
12
24
MF
Max, Wald
.761, .727
.677, .604
.621, .444
.497, .278
B.1.1. TL = 80
B.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.083, .062
.639, .592
4
.836, .914
1
.100, .086
2
.065, .057
.519, .517
8
.751, .848
2
.090, .076
3
.067, .051
.474, .437
12
.705, .768
3
.076, .074
4
.056, .059
.423, .407
24
.618, .588
4
.068, .072
B.2. d = 0.8 (high persistence in xH )
B.2.1. TL = 80
B.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.297, .273
.862, .855
4
.977, .978
1
.539, .535
2
.225, .230
.783, .761
8
.967, .937
2
.440, .422
3
.185, .171
.728, .694
12
.957, .889
3
.383, .379
4
.170, .150
.693, .627
24
.920, .750
4
.353, .340
LF (stock)
Max, Wald
.913, .910
.869, .843
.826, .809
.810, .731
LF (stock)
Max, Wald
.992, .992
.985, .982
.976, .965
.970, .953
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs:
cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated have two low frequency lags of xL (i.e. q = 2).
The max-test p-value is computed using 5000 draws from the null limit distribution. The Wald test p-value is
computed using the parametric bootstrap based on Gonçalves and Kilian (2004), with 1000 bootstrap replications. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000 for max tests and 1000 for Wald
tests.
5
Table T.1: Continued
C. Lagged Causality: bj = 0.3 × I(j = 12) for j = 1, . . . , 12
hM F
4
8
12
24
MF
Max, Wald
.057, .054
.047, .047
.383, .267
.287, .196
hM F
4
8
12
24
MF
Max, Wald
.081, .070
.205, .132
.917, .709
.876, .469
C.1. d = 0.2 (low persistence in xH )
C.1.1. TL = 80
C.1.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.121, .111
.058, .036
4
.050, .049
1
2
.089, .081
.070, .061
8
.051, .050
2
3
.081, .091
.066, .062
12
.809, .635
3
4
.072, .074
.061, .048
24
.733, .487
4
C.2. d = 0.8 (high persistence in xH )
C.2.1. TL = 80
C.2.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.575, .566
.064, .062
4
.117, .102
1
2
.588, .572
.835, .806
8
.422, .294
2
3
.493, .501
.790, .732
12
1.00, .995
3
4
.464, .437
.745, .717
24
.998, .947
4
= 160
LF (flow)
Max, Wald
.185, .183
.148, .148
.133, .119
.111, .110
LF (stock)
Max, Wald
.057, .045
.090, .090
.085, .078
.076, .074
= 160
LF (flow)
Max, Wald
.874, .867
.893, .908
.861, .888
.831, .829
LF (stock)
Max, Wald
.072, .076
.992, .993
.985, .983
.980, .954
D. Sporadic Causality: (b3 , b7 , b10 ) = (0.2, 0.05, −0.3) and bj = 0 for other j’s
hM F
4
8
12
24
MF
Max, Wald
.221, .215
.163, .172
.397, .424
.312, .271
hM F
4
8
12
24
MF
Max, Wald
.424, .304
.387, .418
.742, .810
.638, .535
D.1. d = 0.2 (low persistence in xH )
D.1.1. TL = 80
D.1.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.058, .041
.050, .056
4
.478, .423
1
2
.055, .045
.054, .050
8
.396, .338
2
3
.050, .053
.058, .064
12
.845, .830
3
4
.053, .046
.058, .046
24
.779, .669
4
D.2. d = 0.8 (high persistence in xH )
D.2.1. TL = 80
D.2.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.068, .051
.257, .213
4
.754, .616
1
2
.118, .084
.357, .442
8
.748, .831
2
3
.100, .094
.299, .337
12
.990, .995
3
4
.090, .076
.266, .316
24
.976, .970
4
= 160
LF (flow)
Max, Wald
.054, .061
.055, .057
.044, .043
.052, .058
LF (stock)
Max, Wald
.055, .055
.046, .054
.054, .049
.053, .058
= 160
LF (flow)
Max, Wald
.089, .081
.184, .182
.161, .138
.138, .125
LF (stock)
Max, Wald
.487, .463
.664, .740
.587, .691
.550, .657
= 160
LF (flow)
Max, Wald
.188, .190
.140, .139
.113, .107
.113, .102
LF (stock)
Max, Wald
.064, .068
.064, .051
.056, .056
.055, .054
= 160
LF (flow)
Max, Wald
.919, .917
.881, .888
.850, .823
.823, .772
LF (stock)
Max, Wald
.459, .440
.486, .530
.425, .446
.374, .393
E. Uniform Causality: bj = 0.02 for j = 1, . . . , 12
hM F
4
8
12
24
MF
Max, Wald
.058, .049
.057, .049
.056, .057
.050, .065
hM F
4
8
12
24
MF
Max, Wald
.388, .247
.485, .239
.480, .217
.350, .142
E.1. d = 0.2 (low persistence in xH )
E.1.1. TL = 80
E.1.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.111, .110
.058, .054
4
.072, .060
1
2
.088, .097
.059, .044
8
.077, .081
2
3
.073, .081
.054, .047
12
.072, .073
3
4
.077, .053
.053, .064
24
.062, .055
4
E.2. d = 0.8 (high persistence in xH )
E.2.1. TL = 80
E.2.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.657, .629
.253, .260
4
.691, .534
1
2
.556, .524
.250, .280
8
.802, .609
2
3
.493, .441
.205, .209
12
.823, .533
3
4
.442, .403
.182, .184
24
.735, .375
4
6
Table T.2: Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(1) GARCH Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: b = 012×1
hM F
4
8
12
24
MF
Max, Wald
.055, .046
.046, .045
.048, .041
.043, .046
hM F
4
8
12
24
MF
Max, Wald
.056, .036
.054, .040
.047, .034
.043, .041
A.1. d = 0.2 (low persistence in xH )
A.1.1. TL = 80
A.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.057, .054
.059, .057
4
.052, .053
1
.052, .063
2
.056, .050
.051, .039
8
.052, .045
2
.053, .048
3
.050, .051
.051, .041
12
.045, .046
3
.042, .053
4
.054, .038
.054, .047
24
.041, .031
4
.049, .045
A.2. d = 0.8 (high persistence in xH )
A.2.1. TL = 80
A.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.057, .043
.054, .050
4
.051, .032
1
.054, .052
2
.053, .038
.054, .040
8
.049, .050
2
.049, .040
3
.050, .039
.049, .039
12
.052, .061
3
.054, .048
4
.058, .051
.056, .036
24
.040, .034
4
.049, .040
LF (stock)
Max, Wald
.053, .060
.053, .045
.049, .051
.051, .046
LF (stock)
Max, Wald
.048, .041
.052, .046
.046, .056
.050, .047
B. Decaying Causality: bj = (−1)j−1 0.3/j for j = 1, . . . , 12
B.1. d = 0.2 (low persistence in xH )
hM F
4
8
12
24
MF
Max, Wald
.523, .624
.410, .481
.361, .434
.270, .250
hM F
4
8
12
24
MF
Max, Wald
.533, .631
.443, .487
.413, .434
.311, .251
B.1.1. TL = 80
B.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.089, .083
.681, .649
4
.878, .920
1
.118, .109
2
.078, .073
.574, .540
8
.809, .868
2
.103, .094
3
.068, .054
.518, .481
12
.770, .826
3
.090, .078
4
.072, .065
.493, .445
24
.692, .639
4
.078, .081
B.2. d = 0.8 (high persistence in xH )
B.2.1. TL = 80
B.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.073, .058
.676, .679
4
.881, .906
1
.088, .079
2
.068, .055
.583, .553
8
.833, .879
2
.070, .070
3
.059, .050
.515, .529
12
.816, .826
3
.070, .067
4
.065, .050
.482, .426
24
.733, .675
4
.071, .054
LF (stock)
Max, Wald
.932, .919
.880, .859
.852, .846
.832, .790
LF (stock)
Max, Wald
.930, .899
.885, .866
.861, .835
.837, .790
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs:
cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated have two low frequency lags of xL (i.e. q = 2).
The max-test p-value is computed using 5000 draws from the null limit distribution. The Wald test p-value is
computed using the parametric bootstrap based on Gonçalves and Kilian (2004), with 1000 bootstrap replications. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000 for max tests and 1000 for Wald
tests.
7
Table T.2: Continued
C. Lagged Causality: bj = 0.3 × I(j = 12) for j = 1, . . . , 12
hM F
4
8
12
24
MF
Max, Wald
.066, .054
.059, .063
.327, .275
.235, .161
hM F
4
8
12
24
MF
Max, Wald
.073, .060
.094, .098
.346, .430
.276, .237
C.1. d = 0.2 (low persistence in xH )
C.1.1. TL = 80
C.1.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.135, .107
.066, .047
4
.071, .058
1
2
.101, .108
.073, .087
8
.063, .058
2
3
.089, .093
.065, .053
12
.763, .610
3
4
.085, .078
.068, .071
24
.685, .434
4
C.2. d = 0.8 (high persistence in xH )
C.2.1. TL = 80
C.2.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.477, .443
.066, .043
4
.079, .069
1
2
.387, .353
.407, .371
8
.115, .140
2
3
.351, .284
.347, .337
12
.854, .862
3
4
.323, .242
.315, .258
24
.788, .703
4
= 160
LF (flow)
Max, Wald
.198, .184
.143, .159
.135, .126
.130, .114
LF (stock)
Max, Wald
.066, .068
.098, .073
.079, .074
.071, .062
= 160
LF (flow)
Max, Wald
.778, .749
.699, .677
.660, .607
.623, .546
LF (stock)
Max, Wald
.063, .062
.713, .699
.667, .614
.624, .575
D. Sporadic Causality: (b3 , b7 , b10 ) = (0.2, 0.05, −0.3) and bj = 0 for other j’s
hM F
4
8
12
24
MF
Max, Wald
.218, .231
.170, .174
.373, .386
.276, .255
hM F
4
8
12
24
MF
Max, Wald
.195, .183
.160, .152
.363, .427
.268, .266
D.1. d = 0.2 (low persistence in xH )
D.1.1. TL = 80
D.1.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.069, .058
.076, .068
4
.451, .451
1
2
.062, .055
.053, .056
8
.374, .346
2
3
.057, .053
.059, .049
12
.809, .812
3
4
.065, .057
.056, .044
24
.721, .644
4
D.2. d = 0.8 (high persistence in xH )
D.2.1. TL = 80
D.2.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.138, .110
.071, .068
4
.395, .378
1
2
.105, .099
.139, .118
8
.331, .325
2
3
.099, .094
.115, .103
12
.807, .844
3
4
.096, .079
.102, .094
24
.723, .674
4
= 160
LF (flow)
Max, Wald
.075, .061
.063, .071
.060, .042
.057, .060
LF (stock)
Max, Wald
.064, .059
.065, .054
.059, .045
.062, .055
= 160
LF (flow)
Max, Wald
.215, .206
.174, .155
.156, .148
.132, .136
LF (stock)
Max, Wald
.074, .067
.226, .234
.202, .186
.167, .165
= 160
LF (flow)
Max, Wald
.148, .128
.107, .101
.093, .089
.085, .087
LF (stock)
Max, Wald
.062, .054
.057, .052
.056, .044
.057, .047
= 160
LF (flow)
Max, Wald
.247, .241
.186, .180
.165, .146
.156, .138
LF (stock)
Max, Wald
.064, .067
.126, .114
.101, .113
.104, .098
E. Uniform Causality: bj = 0.02 for j = 1, . . . , 12
hM F
4
8
12
24
MF
Max, Wald
.061, .049
.053, .048
.056, .043
.048, .042
hM F
4
8
12
24
MF
Max, Wald
.066, .058
.064, .046
.069, .050
.064, .054
E.1. d = 0.2 (low persistence in xH )
E.1.1. TL = 80
E.1.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.103, .107
.065, .056
4
.071, .049
1
2
.075, .063
.053, .042
8
.061, .055
2
3
.071, .079
.056, .054
12
.061, .060
3
4
.062, .060
.056, .042
24
.052, .063
4
E.2. d = 0.8 (high persistence in xH )
E.2.1. TL = 80
E.2.2. TL
LF (flow)
LF (stock)
MF
hLF
Max, Wald Max, Wald
hM F
Max, Wald hLF
1
.154, .132
.065, .051
4
.067, .069
1
2
.106, .072
.089, .056
8
.077, .071
2
3
.103, .094
.075, .051
12
.089, .081
3
4
.094, .070
.073, .072
24
.083, .061
4
8
Table T.3: Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(2) - IID
Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: b = 024×1
A.1. d = 0.2 (low persistence in xH )
hM F
16
20
24
MF
Max, Wald
.043, .066
.040, .046
.039, .050
hM F
16
20
24
MF
Max, Wald
.050, .048
.041, .046
.049, .045
A.1.1. TL = 80
A.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.058, .058
.055, .057
16
.051, .043
1
.051, .055
2
.050, .047
.054, .042
20
.047, .040
2
.050, .055
3
.051, .052
.051, .046
24
.041, .044
3
.050, .062
A.2. d = 0.8 (high persistence in xH )
A.2.1. TL = 80
A.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.054, .055
.056, .043
16
.048, .051
1
.054, .050
2
.050, .053
.056, .059
20
.045, .040
2
.056, .046
3
.055, .058
.057, .043
24
.047, .037
3
.053, .054
LF (stock)
Max, Wald
.053, .038
.048, .053
.049, .042
LF (stock)
Max, Wald
.052, .051
.047, .046
.055, .050
B. Decaying Causality: bj = (−1)j−1 0.3/j for j = 1, . . . , 24
B.1. d = 0.2 (low persistence in xH )
hM F
16
20
24
MF
Max, Wald
.257, .309
.233, .245
.206, .221
hM F
16
20
24
MF
Max, Wald
.571, .403
.524, .325
.517, .291
B.1.1. TL = 80
B.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.079, .073
.632, .650
16
.650, .718
1
.100, .105
2
.067, .064
.533, .507
20
.622, .638
2
.091, .087
3
.062, .059
.451, .452
24
.620, .596
3
.079, .082
B.2. d = 0.8 (high persistence in xH )
B.2.1. TL = 80
B.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.325, .297
.853, .832
16
.946, .845
1
.573, .541
2
.236, .239
.770, .770
20
.937, .787
2
.469, .441
3
.207, .182
.744, .674
24
.929, .747
3
.407, .387
LF (stock)
Max, Wald
.919, .925
.856, .859
.834, .787
LF (stock)
Max, Wald
.991, .989
.984, .974
.979, .963
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs:
cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated have two low frequency lags of xL (i.e. q = 2).
The max-test p-value is computed using 5000 draws from the null limit distribution. The Wald test p-value is
computed using the parametric bootstrap based on Gonçalves and Kilian (2004), with 1000 bootstrap replications. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000 for max tests and 1000 for Wald
tests.
9
Table T.3: Continued
C. Lagged Causality: bj = 0.3 × I(j = 24) for j = 1, . . . , 24
hM F
16
20
24
hM F
16
20
24
C.1. d = 0.2 (low persistence in xH )
C.1.1. TL = 80
C.1.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.042, .050
1
.054, .052
.057, .048
16
.047, .037
1
.056, .035
.047, .035
2
.101, .101
.056, .054
20
.047, .049
2
.172, .157
.248, .184
3
.090, .093
.067, .058
24
.729, .511
3
.151, .131
C.2. d = 0.8 (high persistence in xH )
C.2.1. TL = 80
C.2.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.058, .049
1
.055, .045
.061, .049
16
.074, .068
1
.058, .046
.141, .102
2
.497, .471
.054, .052
20
.353, .177
2
.840, .839
.871, .507
3
.534, .536
.791, .732
24
1.00, .944
3
.878, .875
LF (stock)
Max, Wald
.056, .046
.046, .059
.085, .078
LF (stock)
Max, Wald
.052, .051
.064, .058
.984, .976
D. Sporadic Causality: (b5 , b12 , b17 , b19 ) = (−0.2, 0.1, 0.2, −0.35) and bj = 0 for other j’s
hM F
16
20
24
hM F
16
20
24
D.1. d = 0.2 (low persistence in xH )
D.1.1. TL = 80
D.1.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.121, .128
1
.066, .060
.056, .050
16
.291, .285
1
.079, .082
.442, .464
2
.065, .064
.054, .049
20
.901, .930
2
.091, .100
.410, .430
3
.060, .059
.054, .059
24
.891, .913
3
.081, .076
D.2. d = 0.8 (high persistence in xH )
D.2.1. TL = 80
D.2.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.355, .218
1
.263, .226
.150, .136
16
.734, .567
1
.470, .466
.580, .685
2
.272, .270
.134, .128
20
.957, .994
2
.540, .506
.555, .638
3
.237, .234
.138, .160
24
.951, .985
3
.485, .468
LF (stock)
Max, Wald
.058, .053
.048, .056
.052, .058
LF (stock)
Max, Wald
.266, .256
.237, .253
.241, .285
E. Uniform Causality: bj = 0.02 for j = 1, . . . , 24
hM F
16
20
24
hM F
16
20
24
E.1. d = 0.2 (low persistence in xH )
E.1.1. TL = 80
E.1.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.059, .050
1
.125, .102
.063, .052
16
.074, .086
1
.185, .190
.051, .069
2
.124, .125
.066, .079
20
.076, .097
2
.226, .240
.051, .051
3
.111, .109
.058, .049
24
.074, .089
3
.196, .196
E.2. d = 0.8 (high persistence in xH )
E.2.1. TL = 80
E.2.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.651, .366
1
.778, .773
.242, .236
16
.959, .807
1
.979, .971
.679, .381
2
.865, .876
.529, .560
20
.966, .862
2
.998, .999
.669, .341
3
.824, .838
.482, .544
24
.970, .833
3
.994, .991
10
LF (stock)
Max, Wald
.068, .072
.067, .054
.064, .057
LF (stock)
Max, Wald
.450, .421
.866, .899
.832, .894
Table T.4: Rejection Frequencies of High-to-Low Causality Tests Based on MF-VAR(2) GARCH Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: b = 024×1
A.1. d = 0.2 (low persistence in xH )
hM F
16
20
24
MF
Max, Wald
.046, .043
.045, .040
.045, .048
hM F
16
20
24
MF
Max, Wald
.046, .037
.047, .048
.041, .043
A.1.1. TL = 80
A.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.060, .054
.055, .044
16
.043, .052
1
.054, .049
2
.060, .055
.060, .044
20
.053, .040
2
.049, .053
3
.051, .038
.046, .037
24
.047, .040
3
.047, .055
A.2. d = 0.8 (high persistence in xH )
A.2.1. TL = 80
A.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.054, .046
.054, .050
16
.052, .037
1
.054, .048
2
.054, .042
.056, .046
20
.043, .046
2
.048, .048
3
.054, .042
.049, .042
24
.046, .042
3
.050, .048
LF (stock)
Max, Wald
.049, .037
.059, .055
.052, .031
LF (stock)
Max, Wald
.055, .042
.056, .056
.053, .046
B. Decaying Causality: bj = (−1)j−1 0.3/j for j = 1, . . . , 24
B.1. d = 0.2 (low persistence in xH )
hM F
16
20
24
MF
Max, Wald
.312, .337
.273, .314
.259, .239
hM F
16
20
24
MF
Max, Wald
.354, .324
.312, .280
.315, .262
B.1.1. TL = 80
B.1.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.094, .087
.680, .651
16
.736, .785
1
.117, .100
2
.082, .063
.564, .553
20
.707, .716
2
.097, .100
3
.074, .057
.514, .507
24
.697, .654
3
.093, .069
B.2. d = 0.8 (high persistence in xH )
B.2.1. TL = 80
B.2.2. TL = 160
LF (flow)
LF (stock)
MF
LF (flow)
hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
1
.080, .069
.689, .671
16
.771, .780
1
.099, .091
2
.064, .076
.571, .563
20
.741, .724
2
.087, .092
3
.073, .045
.523, .469
24
.731, .679
3
.081, .088
LF (stock)
Max, Wald
.930, .931
.890, .883
.855, .810
LF (stock)
Max, Wald
.928, .923
.897, .879
.860, .829
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs:
cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated have two low frequency lags of xL (i.e. q = 2).
The max-test p-value is computed using 5000 draws from the null limit distribution. The Wald test p-value is
computed using the parametric bootstrap based on Gonçalves and Kilian (2004), with 1000 bootstrap replications. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000 for max tests and 1000 for Wald
tests.
11
Table T.4: Continued
C. Lagged Causality: bj = 0.3 × I(j = 24) for j = 1, . . . , 24
hM F
16
20
24
hM F
16
20
24
C.1. d = 0.2 (low persistence in xH )
C.1.1. TL = 80
C.1.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.051, .061
1
.056, .046
.055, .044
16
.051, .053
1
.056, .053
.048, .050
2
.115, .107
.064, .043
20
.056, .054
2
.179, .165
.205, .158
3
.099, .077
.061, .062
24
.682, .530
3
.164, .143
C.2. d = 0.8 (high persistence in xH )
C.2.1. TL = 80
C.2.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.058, .046
1
.061, .060
.055, .040
16
.060, .040
1
.065, .056
.067, .051
2
.433, .402
.056, .041
20
.096, .112
2
.760, .752
.206, .281
3
.376, .362
.352, .311
24
.770, .740
3
.729, .709
LF (stock)
Max, Wald
.049, .048
.056, .054
.088, .077
LF (stock)
Max, Wald
.055, .051
.057, .065
.673, .640
D. Sporadic Causality: (b5 , b12 , b17 , b19 ) = (−0.2, 0.1, 0.2, −0.35) and bj = 0 for other j’s
hM F
16
20
24
hM F
16
20
24
D.1. d = 0.2 (low persistence in xH )
D.1.1. TL = 80
D.1.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.113, .117
1
.067, .063
.062, .055
16
.290, .276
1
.077, .057
.414, .440
2
.078, .064
.058, .057
20
.883, .886
2
.085, .070
.391, .373
3
.071, .055
.065, .058
24
.858, .881
3
.077, .079
D.2. d = 0.8 (high persistence in xH )
D.2.1. TL = 80
D.2.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.111, .124
1
.068, .055
.061, .051
16
.256, .284
1
.069, .061
.429, .454
2
.088, .063
.078, .050
20
.877, .888
2
.105, .104
.415, .376
3
.078, .062
.072, .082
24
.883, .875
3
.093, .091
LF (stock)
Max, Wald
.062, .053
.066, .063
.060, .068
LF (stock)
Max, Wald
.057, .057
.087, .091
.097, .084
E. Uniform Causality: bj = 0.02 for j = 1, . . . , 24
hM F
16
20
24
hM F
16
20
24
E.1. d = 0.2 (low persistence in xH )
E.1.1. TL = 80
E.1.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.051, .061
1
.091, .078
.054, .055
16
.062, .064
1
.152, .140
.050, .053
2
.107, .103
.062, .054
20
.060, .052
2
.166, .158
.049, .042
3
.091, .070
.054, .041
24
.055, .072
3
.140, .139
E.2. d = 0.8 (high persistence in xH )
E.2.1. TL = 80
E.2.2. TL = 160
MF
LF (flow)
LF (stock)
MF
LF (flow)
Max, Wald hLF Max, Wald Max, Wald
hM F Max, Wald hLF Max, Wald
.079, .077
1
.173, .175
.060, .051
16
.116, .130
1
.312, .292
.072, .070
2
.198, .187
.111, .107
20
.118, .092
2
.380, .391
.077, .065
3
.165, .148
.116, .104
24
.112, .121
3
.319, .311
12
LF (stock)
Max, Wald
.063, .060
.060, .047
.058, .056
LF (stock)
Max, Wald
.063, .061
.187, .176
.198, .215
13
3
4
.055
.054
.052
8
12
24
2
3
.049
.053
.050
8
12
24
MF
4
3
2
1
hLF
4
3
2
1
hLF
.052
.056
.059
.052
4
8
12
24
4
3
2
1
hLF
3
4
.053
.050
.048
8
12
24
.055, .053
.059, .051
.058, .047
.049, .053
Flow, Stock
.058, .058
.059, .055
.047, .059
.055, .051
24
12
8
4
hM F
24
12
8
4
4
3
2
1
hLF
.925
.962
.969
.983
MF
4
3
2
1
hLF
B.2.2 TL = 160
.546
.651
.690
.777
MF
.370, .968
.394, .980
.445, .985
.536, .994
Flow, Stock
.190, .727
.204, .758
.233, .793
.320, .866
Flow, Stock
.079, .802
.079, .835
.089, .869
.098, .920
Flow, Stock
.063, .453
.064, .490
.075, .552
.087, .660
Flow, Stock
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
4
3
2
1
hLF
4
3
2
1
hLF
4
3
2
1
hLF
.999
1.00
.462
.123
MF
4
3
2
1
hLF
C.2.2 TL = 160
.901
.931
.240
.088
MF
C.2.1 TL = 80
C.2 d = 0.8
.750
.820
.056
.050
MF
C.1.2 TL = 160
.329
.406
.055
.059
MF
C.1.1 TL = 80
C.1 d = 0.2
.850, .985
.861, .987
.917, .994
.884, .074
Flow, Stock
.494, .771
.540, .809
.625, .860
.601, .061
Flow, Stock
.118, .083
.129, .082
.150, .096
.209, .049
Flow, Stock
.095, .073
.091, .064
.095, .073
.127, .055
Flow, Stock
C. Lagged Causality
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
4
3
2
1
hLF
4
3
2
1
hLF
4
3
2
1
hLF
.985
.994
.770
.772
MF
4
3
2
1
hLF
D.2.2 TL = 160
.693
.791
.421
.465
MF
D.2.1 TL = 80
D.2 d = 0.8
.802
.862
.420
.485
MF
D.1.2 TL = 160
.352
.455
.179
.237
MF
D.1.1 TL = 80
D.1 d = 0.2
.146, .569
.167, .614
.193, .686
.084, .476
Flow, Stock
.097, .298
.103, .326
.135, .402
.073, .275
Flow, Stock
.057, .052
.056, .057
.058, .056
.063, .056
Flow, Stock
.060, .059
.057, .053
.059, .052
.063, .055
Flow, Stock
D. Sporadic Causality
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
4
3
2
1
hLF
4
3
2
1
hLF
4
3
2
1
hLF
.759
.839
.826
.708
MF
4
3
2
1
hLF
E.2.2 TL = 160
.412
.524
.511
.411
MF
E.2.1 TL = 80
E.2 d = 0.8
.065
.082
.084
.072
MF
E.1.2 TL = 160
.056
.067
.068
.064
MF
E.1.1 TL = 80
E.1 d = 0.2
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from the null limit distribution. Nominal size is α = 0.05. The
number of Monte Carlo samples is 5000.
.834, .396
.858, .431
.889, .507
.930, .470
Flow, Stock
.482, .210
.517, .225
.579, .276
.680, .257
Flow, Stock
.116, .057
.125, .061
.135, .064
.189, .066
Flow, Stock
.088, .056
.084, .060
.094, .061
.124, .063
Flow, Stock
E. Uniform Causality
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs: cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated
2
1
4
hLF
MF
.052
hM F
A.2.2 TL = 160
MF
hM F
B.2.1 TL = 80
.640
.714
.762
.836
MF
B.1.2 TL = 160
.243
.322
.376
.483
A.2.1 TL = 80
hM F
24
12
8
4
hM F
24
12
8
4
hM F
B.2 d = 0.8
Flow, Stock
.057, .052
.055, .054
.054, .049
.049, .054
Flow, Stock
.056, .056
.058, .045
.047, .048
.056, .061
Flow, Stock
A.2 d = 0.8
4
1
4
hLF
MF
.053
hM F
A.1.2 TL = 160
2
1
hLF
MF
.055
B.1.1 TL = 80
A.1.1 TL = 80
4
B.1 d = 0.2
A.1 d = 0.2
hM F
B. Decaying Causality
A. Non-Causality
Table T.5: Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(1) - IID Error and Non-Robust Covariance Matrix
14
3
4
.053
.054
.048
8
12
24
2
3
.057
.050
.052
8
12
24
MF
4
3
2
1
hLF
4
3
2
1
hLF
.056
.052
.058
.049
4
8
12
24
4
3
2
1
hLF
3
4
.053
.054
.055
8
12
24
.056, .051
.056, .057
.058, .056
.048, .053
Flow, Stock
.060, .051
.056, .056
.051, .050
.053, .052
24
12
8
4
hM F
24
12
8
4
4
3
2
1
hLF
.761
.845
.840
.897
MF
4
3
2
1
hLF
B.2.2 TL = 160
.351
.433
.479
.563
MF
.073, .845
.082, .875
.080, .888
.098, .937
Flow, Stock
.071, .512
.069, .540
.070, .610
.089, .696
Flow, Stock
.090, .852
.090, .873
.101, .899
.120, .935
Flow, Stock
.070, .528
.076, .554
.078, .602
.096, .703
Flow, Stock
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
4
3
2
1
hLF
4
3
2
1
hLF
4
3
2
1
hLF
.816
.868
.138
.085
MF
4
3
2
1
hLF
C.2.2 TL = 160
.318
.366
.102
.083
MF
C.2.1 TL = 80
C.2 d = 0.8
.713
.785
.076
.075
MF
C.1.2 TL = 160
.269
.353
.070
.073
MF
C.1.1 TL = 80
C.1 d = 0.2
.645, .649
.670, .692
.728, .743
.783, .066
Flow, Stock
.350, .341
.362, .355
.421, .445
.486, .073
Flow, Stock
.129, .084
.137, .089
.157, .094
.212, .066
Flow, Stock
.090, .070
.088, .076
.104, .078
.139, .070
Flow, Stock
C. Lagged Causality
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
4
3
2
1
hLF
4
3
2
1
hLF
4
3
2
1
hLF
.753
.818
.355
.412
MF
4
3
2
1
hLF
D.2.2 TL = 160
.298
.400
.171
.207
MF
D.2.1 TL = 80
D.2 d = 0.8
.751
.820
.396
.483
MF
D.1.2 TL = 160
.305
.397
.179
.230
MF
D.1.1 TL = 80
D.1 d = 0.2
.144, .171
.157, .194
.179, .225
.224, .073
Flow, Stock
.092, .109
.100, .120
.117, .135
.131, .072
Flow, Stock
.060, .061
.064, .058
.065, .063
.067, .073
Flow, Stock
.066, .062
.062, .064
.068, .066
.077, .072
Flow, Stock
D. Sporadic Causality
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
24
12
8
4
hM F
4
3
2
1
hLF
4
3
2
1
hLF
4
3
2
1
hLF
.094
.097
.086
.071
MF
4
3
2
1
hLF
E.2.2 TL = 160
.066
.076
.069
.068
MF
E.2.1 TL = 80
E.2 d = 0.8
.054
.062
.065
.068
MF
E.1.2 TL = 160
.056
.061
.061
.061
MF
E.1.1 TL = 80
E.1 d = 0.2
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from the null limit distribution. Nominal size is α = 0.05. The
number of Monte Carlo samples is 5000.
.172, .108
.177, .111
.195, .121
.252, .064
Flow, Stock
.104, .081
.105, .079
.120, .091
.148, .058
Flow, Stock
.091, .051
.093, .058
.108, .058
.143, .067
Flow, Stock
.075, .059
.073, .058
.086, .061
.102, .061
Flow, Stock
E. Uniform Causality
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs: cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated
2
1
4
hLF
MF
.055
hM F
A.2.2 TL = 160
MF
hM F
B.2.1 TL = 80
.723
.800
.832
.900
MF
B.1.2 TL = 160
.307
.404
.458
.561
A.2.1 TL = 80
hM F
24
12
8
4
hM F
24
12
8
4
hM F
B.2 d = 0.8
Flow, Stock
.052, .053
.050, .055
.050, .059
.051, .050
Flow, Stock
.056, .054
.058, .046
.058, .054
.059, .057
Flow, Stock
A.2 d = 0.8
4
1
4
hLF
MF
.055
hM F
A.1.2 TL = 160
2
1
hLF
MF
.056
B.1.1 TL = 80
A.1.1 TL = 80
4
B.1 d = 0.2
A.1 d = 0.2
hM F
B. Decaying Causality
A. Non-Causality
Table T.6: Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(1) - GARCH Error and Non-Robust Covariance Matrix
15
2
.051
.050
20
24
2
.051
.052
20
24
MF
3
2
1
hLF
3
2
1
hLF
.048
.053
.051
16
20
24
3
2
1
hLF
.055
.049
16
20
24
3
2
1
hLF
.054, .047
.049, .051
.051, .055
Flow, Stock
.056, .053
.058, .053
.052, .059
24
20
16
hM F
24
20
16
3
2
1
hLF
.929
.941
.949
MF
3
2
1
hLF
B.2.2 TL = 160
.537
.570
.609
MF
.423, .976
.470, .984
.578, .993
Flow, Stock
.217, .750
.249, .810
.329, .876
Flow, Stock
.079, .843
.089, .872
.099, .928
Flow, Stock
.066, .491
.075, .547
.084, .665
Flow, Stock
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
3
2
1
hLF
3
2
1
hLF
3
2
1
hLF
.999
.369
.088
MF
3
2
1
hLF
C.2.2 TL = 160
.894
.163
.067
MF
C.2.1 TL = 80
C.2 d = 0.8
.746
.049
.056
MF
C.1.2 TL = 160
.272
.050
.054
MF
C.1.1 TL = 80
C.1 d = 0.2
.889, .988
.848, .065
.056, .046
Flow, Stock
.579, .803
.519, .058
.061, .061
Flow, Stock
.156, .076
.167, .053
.055, .052
Flow, Stock
.096, .070
.103, .059
.056, .057
Flow, Stock
C. Lagged Causality
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
3
2
1
hLF
3
2
1
hLF
3
2
1
hLF
.956
.963
.758
MF
3
2
1
hLF
D.2.2 TL = 160
.623
.634
.382
MF
D.2.1 TL = 80
D.2 d = 0.8
.911
.916
.328
MF
D.1.2 TL = 160
.455
.482
.139
MF
D.1.1 TL = 80
D.1 d = 0.2
.498, .251
.542, .251
.477, .275
Flow, Stock
.267, .149
.303, .143
.267, .162
Flow, Stock
.086, .053
.094, .058
.079, .057
Flow, Stock
.068, .051
.067, .057
.068, .059
Flow, Stock
D. Sporadic Causality
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
3
2
1
hLF
3
2
1
hLF
3
2
1
hLF
.975
.977
.972
MF
3
2
1
hLF
E.2.2 TL = 160
.742
.748
.727
MF
E.2.1 TL = 80
E.2 d = 0.8
.084
.084
.089
MF
E.1.2 TL = 160
.068
.072
.068
MF
E.1.1 TL = 80
E.1 d = 0.2
number of Monte Carlo samples is 5000.
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from the null limit distribution. Nominal size is α = 0.05. The
.993, .839
.997, .872
.983, .475
Flow, Stock
.848, .512
.896, .567
.799, .267
Flow, Stock
.185, .070
.228, .070
.194, .060
Flow, Stock
.123, .063
.139, .064
.118, .057
Flow, Stock
E. Uniform Causality
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs: cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated
MF
.057
hM F
A.2.2 TL = 160
MF
hM F
B.2.1 TL = 80
.644
.656
.684
MF
B.1.2 TL = 160
.261
.264
.289
A.2.1 TL = 80
hM F
24
20
16
hM F
24
20
16
hM F
B.2 d = 0.8
Flow, Stock
.057, .046
.054, .049
.054, .052
Flow, Stock
.051, .056
.050, .055
.061, .048
Flow, Stock
A.2 d = 0.8
3
1
16
hLF
MF
.053
hM F
A.1.2 TL = 160
3
1
hLF
MF
.056
B.1.1 TL = 80
A.1.1 TL = 80
16
B.1 d = 0.2
A.1 d = 0.2
hM F
B. Decaying Causality
A. Non-Causality
Table T.7: Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(2) - IID Error and Non-Robust Covariance Matrix
16
2
.048
.049
20
24
2
.054
.057
20
24
MF
3
2
1
hLF
3
2
1
hLF
.050
.056
.053
16
20
24
3
2
1
hLF
.053
.053
16
20
24
3
2
1
hLF
.056, .054
.053, .054
.057, .049
Flow, Stock
.054, .059
.055, .053
.055, .056
24
20
16
hM F
24
20
16
3
2
1
hLF
.761
.770
.790
MF
3
2
1
hLF
B.2.2 TL = 160
.335
.350
.390
MF
.097, .870
.094, .903
.114, .938
Flow, Stock
.071, .543
.076, .607
.088, .695
Flow, Stock
.095, .868
.107, .891
.123, .936
Flow, Stock
.077, .558
.083, .601
.095, .702
Flow, Stock
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
3
2
1
hLF
3
2
1
hLF
3
2
1
hLF
.785
.110
.072
MF
3
2
1
hLF
C.2.2 TL = 160
.238
.084
.064
MF
C.2.1 TL = 80
C.2 d = 0.8
.705
.060
.060
MF
C.1.2 TL = 160
.237
.058
.057
MF
C.1.1 TL = 80
C.1 d = 0.2
.741, .680
.770, .066
.063, .057
Flow, Stock
.403, .368
.459, .067
.061, .053
Flow, Stock
.160, .088
.181, .060
.053, .046
Flow, Stock
.109, .072
.115, .058
.059, .050
Flow, Stock
C. Lagged Causality
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
3
2
1
hLF
3
2
1
hLF
3
2
1
hLF
.890
.900
.297
MF
3
2
1
hLF
D.2.2 TL = 160
.455
.480
.137
MF
D.2.1 TL = 80
D.2 d = 0.8
.901
.904
.309
MF
D.1.2 TL = 160
.447
.470
.136
MF
D.1.1 TL = 80
D.1 d = 0.2
.102, .104
.100, .092
.072, .065
Flow, Stock
.085, .085
.080, .074
.074, .060
Flow, Stock
.084, .071
.097, .072
.071, .062
Flow, Stock
.078, .068
.080, .074
.063, .062
Flow, Stock
D. Sporadic Causality
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
24
20
16
hM F
3
2
1
hLF
3
2
1
hLF
3
2
1
hLF
.131
.131
.124
MF
3
2
1
hLF
E.2.2 TL = 160
.086
.089
.076
MF
E.2.1 TL = 80
E.2 d = 0.8
.072
.068
.073
MF
E.1.2 TL = 160
.055
.057
.060
MF
E.1.1 TL = 80
E.1 d = 0.2
number of Monte Carlo samples is 5000.
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from the null limit distribution. Nominal size is α = 0.05. The
.340, .215
.389, .190
.312, .064
Flow, Stock
.180, .130
.213, .114
.171, .054
Flow, Stock
.144, .068
.169, .063
.154, .066
Flow, Stock
.091, .060
.106, .066
.093, .059
Flow, Stock
E. Uniform Causality
There is weak persistence in xL (a = 0.2) and low-to-high decaying causality with alternating signs: cj = (−1)j−1 × 0.4/j for j = 1, . . . , 12. The models estimated
MF
.052
hM F
A.2.2 TL = 160
MF
hM F
B.2.1 TL = 80
.719
.741
.766
MF
B.1.2 TL = 160
.296
.325
.352
A.2.1 TL = 80
hM F
24
20
16
hM F
24
20
16
hM F
B.2 d = 0.8
Flow, Stock
.056, .051
.053, .050
.061, .061
Flow, Stock
.051, .055
.047, .058
.052, .056
Flow, Stock
A.2 d = 0.8
3
1
16
hLF
MF
.053
hM F
A.1.2 TL = 160
3
1
hLF
MF
.053
B.1.1 TL = 80
A.1.1 TL = 80
16
B.1 d = 0.2
A.1 d = 0.2
hM F
B. Decaying Causality
A. Non-Causality
Table T.8: Rejection Frequencies of High-to-Low Max Tests Based on MF-VAR(2) - GARCH Error and Non-Robust Covariance Matrix
Table T.9: Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(1) - IID
Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 012×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
A.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.063, .053
.051, .043
.054, .054
.051, .055
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.051, .048
.053, .050
.042, .061
.050, .048
8
.069, .036
.060, .068
.064, .055
.066, .048
8
.062, .041
.057, .050
.061, .052
.059, .046
12
.082, .058
.074, .069
.087, .052
.076, .049
12
.064, .051
.063, .046
.061, .054
.063, .045
24
.141, .059
.154, .052
.171, .039
.186, .042
24
.078, .048
.087, .047
.085, .045
.087, .054
A.1.2 Mixed Frequency Tests with MIDAS
A.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.058, .048
.046, .037
.052, .053
.046, .054
4
.053, .054
.052, .052
.044, .063
.046, .050
8
.054, .037
.047, .059
.048, .050
.050, .046
8
.057, .045
.047, .054
.048, .053
.050, .046
12
.056, .052
.049, .069
.051, .052
.035, .051
12
.048, .057
.051, .044
.046, .056
.046, .043
.056, .059
.054, .046
.057, .047
.045, .040
24
.052, .041
.050, .056
.046, .053
.050, .053
24
A.1.3 Low Frequency Tests (Flow Sampling)
A.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.051, .039
.050, .046
.049, .048
.043, .055
1
.054, .065
.053, .047
.050, .064
.048, .054
2
.049, .041
.049, .051
.050, .052
.049, .035
2
.050, .058
.053, .048
.055, .054
.051, .037
3
.055, .064
.056, .056
.052, .054
.051, .039
3
.052, .039
.054, .043
.050, .054
.052, .038
4
.058, .055
.057, .046
.050, .050
.057, .046
4
.055, .045
.059, .058
.057, .063
.052, .052
A.1.4 Low Frequency Tests (Stock Sampling)
A.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.055, .060
.052, .063
.047, .041
.043, .053
1
.048, .047
.049, .039
.051, .042
.043, .048
2
.062, .063
.051, .055
.055, .054
.043, .053
2
.054, .048
.046, .054
.047, .053
.050, .050
3
.058, .046
.056, .052
.055, .043
.052, .051
3
.053, .044
.052, .063
.047, .047
.045, .042
4
.057, .055
.055, .050
.053, .065
.047, .053
4
.053, .045
.053, .050
.050, .064
.052, .051
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated have
two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from the null
limit distribution. The Wald test p-value is computed using the parametric bootstrap based on Gonçalves and
Kilian (2004), with 1000 bootstrap replications. We implement mixed frequency tests with and without an Almon
MIDAS polynomial of dimension s = 3. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000
for max tests and 1000 for Wald tests.
17
Table T.9: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 12
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS
B.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.484, .585
.368, .538
.304, .389
8
.484, .556
.378, .413
.314, .358
.220, .256
4
.863, .934
.797, .885
.735, .825
.642, .637
.234, .234
8
.858, .912
.786, .880
.742, .808
.651, .635
12
.496, .511
.401, .418
24
.533, .444
.470, .321
.349, .337
.263, .200
12
.856, .914
.780, .878
.727, .811
.647, .642
.414, .264
.353, .153
24
.852, .893
.794, .796
.745, .737
.655, .580
B.1.2 Mixed Frequency Tests with MIDAS
B.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.501, .588
.372, .542
.308, .404
.218, .256
4
.869, .939
.806, .902
.745, .843
.654, .648
8
.493, .610
.376, .497
.311, .418
.214, .283
8
.879, .935
.801, .910
.762, .847
.665, .679
12
.496, .603
.387, .538
.323, .423
.227, .273
12
.882, .945
.798, .907
.756, .857
.671, .704
.492, .650
.377, .516
.311, .436
.219, .301
24
.872, .957
.812, .875
.771, .836
.663, .707
24
B.1.3 Low Frequency Tests (Flow Sampling)
B.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.087, .080
.069, .071
.059, .062
.058, .045
2
.080, .085
.067, .070
.061, .062
3
.086, .079
.073, .059
.073, .054
.097, .066
.079, .064
.069, .058
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.112, .132
.091, .086
.080, .092
.077, .075
.056, .058
2
.120, .100
.095, .085
.082, .080
.074, .061
.061, .069
3
.113, .100
.093, .089
.084, .080
.073, .075
.065, .051
4
.118, .102
.103, .089
.082, .084
.081, .075
B.1.4 Low Frequency Tests (Stock Sampling)
B.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.057, .042
.051, .054
.044, .059
.047, .054
1
.060, .051
.057, .055
.052, .054
.050, .051
2
.060, .049
.052, .057
.048, .059
.056, .047
2
.061, .065
.055, .052
.051, .048
.051, .051
3
.062, .040
.062, .051
.047, .051
.050, .053
3
.060, .060
.056, .058
.057, .051
.052, .056
4
.063, .071
.054, .059
.046, .052
.051, .036
4
.060, .062
.059, .056
.052, .061
.056, .054
18
Table T.9: Continued
C. Lagged Causality: cj = 0.25 × I(j = 12) for j = 1, . . . , 12
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS
C.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.061, .051
.050, .044
.182, .198
8
.065, .045
.063, .054
.213, .182
.134, .115
4
.051, .051
.055, .050
.514, .444
.421, .280
.157, .129
8
.064, .052
.060, .046
.507, .409
.410, .274
12
.085, .053
.091, .044
24
.133, .045
.157, .051
.234, .165
.177, .092
12
.071, .054
.063, .043
.523, .374
.432, .297
.309, .115
.280, .088
24
.086, .056
.086, .049
.527, .346
.446, .257
C.1.2 Mixed Frequency Tests with MIDAS
C.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.058, .045
.049, .045
.188, .206
.130, .132
4
.051, .048
.051, .044
.521, .460
.431, .287
8
.053, .051
.047, .046
.201, .195
.134, .146
8
.055, .053
.056, .041
.519, .445
.422, .301
12
.061, .059
.054, .040
.193, .199
.133, .122
12
.056, .046
.051, .046
.537, .426
.434, .324
.050, .045
.057, .044
.190, .179
.131, .122
24
.051, .057
.051, .049
.533, .436
.445, .327
24
C.1.3 Low Frequency Tests (Flow Sampling)
C.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.085, .066
.073, .070
.065, .058
.056, .058
2
.087, .073
.077, .082
.064, .072
3
.087, .088
.076, .058
.070, .070
.083, .081
.070, .071
.074, .082
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.119, .091
.098, .087
.082, .090
.076, .089
.066, .075
2
.121, .118
.097, .105
.084, .085
.082, .075
.062, .058
3
.117, .099
.090, .086
.084, .082
.082, .097
.064, .067
4
.111, .121
.105, .092
.095, .076
.075, .081
C.1.4 Low Frequency Tests (Stock Sampling)
C.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.544, .513
.444, .450
.369, .377
.321, .306
1
.845, .848
.792, .768
.738, .717
.705, .645
2
.537, .522
.440, .447
.378, .392
.324, .312
2
.851, .836
.781, .775
.725, .704
.693, .661
3
.535, .519
.433, .409
.383, .350
.337, .320
3
.838, .853
.781, .767
.737, .697
.692, .642
4
.531, .527
.424, .393
.364, .345
.331, .277
4
.838, .839
.774, .744
.729, .693
.693, .644
19
Table T.9: Continued
D. Sporadic Causality: (c3 , c7 , c10 ) = (0.3, 0.15, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS
D.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.450, .422
.361, .414
.469, .582
8
.459, .418
.381, .384
.480, .537
.338, .384
4
.843, .828
.789, .807
.911, .952
.858, .880
.361, .348
8
.845, .827
.801, .802
.921, .954
.859, .868
12
.444, .385
.404, .365
24
.491, .312
.463, .260
.505, .490
.398, .304
12
.834, .784
.801, .768
.919, .957
.852, .863
.558, .372
.466, .205
24
.837, .772
.802, .733
.911, .928
.856, .801
D.1.2 Mixed Frequency Tests with MIDAS
D.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.456, .452
.368, .432
.482, .600
.341, .389
4
.854, .838
.799, .820
.917, .957
.868, .881
8
.463, .468
.379, .437
.468, .597
.350, .404
8
.864, .857
.816, .836
.929, .970
.875, .887
12
.446, .481
.388, .431
.489, .601
.362, .398
12
.860, .836
.822, .814
.936, .970
.872, .905
.462, .461
.393, .404
.492, .607
.342, .399
24
.865, .851
.825, .844
.931, .970
.873, .890
24
D.1.3 Low Frequency Tests (Flow Sampling)
D.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.070, .079
.057, .052
.047, .052
.053, .051
2
.078, .060
.055, .045
.062, .058
3
.072, .064
.063, .062
.056, .071
.071, .063
.061, .052
.062, .057
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.082, .084
.069, .067
.067, .066
.062, .054
.055, .057
2
.094, .089
.076, .076
.072, .063
.067, .066
.060, .060
3
.088, .082
.079, .083
.068, .072
.070, .080
.059, .062
4
.085, .071
.077, .080
.066, .075
.061, .057
D.1.4 Low Frequency Tests (Stock Sampling)
D.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.056, .059
.048, .045
.050, .053
.049, .049
1
.056, .051
.051, .043
.049, .059
.051, .039
2
.055, .045
.051, .052
.053, .049
.053, .053
2
.052, .059
.051, .060
.052, .045
.052, .047
3
.054, .048
.057, .036
.050, .053
.060, .043
3
.057, .041
.056, .038
.056, .053
.053, .052
4
.059, .040
.052, .048
.057, .046
.054, .048
4
.057, .058
.052, .043
.058, .052
.052, .055
20
Table T.9: Continued
E. Uniform Causality: cj = 0.07 for j = 1, . . . , 12
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS
E.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.127, .125
.136, .148
.151, .190
8
.142, .120
.164, .146
.173, .167
.107, .137
4
.223, .222
.283, .296
.309, .417
.212, .261
.123, .121
8
.240, .203
.281, .324
.311, .382
.227, .236
12
.167, .115
.183, .135
24
.222, .096
.265, .110
.194, .151
.146, .093
12
.244, .198
.298, .307
.312, .371
.234, .262
.289, .097
.263, .081
24
.281, .191
.331, .288
.374, .328
.285, .242
E.1.2 Mixed Frequency Tests with MIDAS
E.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.126, .130
.131, .147
.151, .192
.100, .154
4
.225, .238
.286, .323
.307, .420
.215, .277
8
.128, .131
.150, .164
.149, .170
.097, .115
8
.231, .228
.279, .348
.301, .423
.215, .274
12
.131, .126
.143, .162
.147, .195
.103, .114
12
.231, .209
.282, .339
.298, .420
.216, .279
.127, .101
.141, .165
.138, .174
.097, .113
24
.242, .221
.281, .336
.310, .418
.212, .319
24
E.1.3 Low Frequency Tests (Flow Sampling)
E.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.539, .510
.440, .443
.378, .363
.340, .333
2
.540, .510
.443, .426
.382, .385
3
.534, .499
.430, .410
.374, .372
.519, .524
.433, .387
.373, .347
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.853, .841
.776, .776
.738, .697
.698, .665
.333, .308
2
.857, .852
.791, .763
.740, .702
.705, .637
.331, .299
3
.846, .839
.779, .774
.733, .707
.692, .669
.338, .278
4
.847, .831
.771, .762
.722, .711
.689, .648
E.1.4 Low Frequency Tests (Stock Sampling)
E.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.119, .103
.091, .082
.076, .076
.074, .094
1
.195, .178
.141, .142
.126, .144
.108, .108
2
.116, .111
.091, .099
.077, .087
.073, .063
2
.196, .188
.139, .130
.122, .136
.109, .095
3
.121, .106
.090, .089
.079, .072
.076, .078
3
.185, .173
.132, .145
.114, .141
.115, .112
4
.114, .100
.097, .064
.088, .083
.080, .063
4
.193, .178
.146, .146
.123, .130
.110, .110
21
Table T.10: Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(1) GARCH Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 012×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
A.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.055, .054
.051, .039
.048, .049
.046, .046
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.055, .058
.051, .050
.050, .048
.045, .038
8
.068, .047
.064, .045
.068, .047
.061, .050
8
.053, .044
.055, .049
.050, .033
.058, .036
12
.080, .047
.079, .041
.080, .047
.076, .046
12
.060, .054
.061, .044
.065, .047
.057, .039
24
.139, .044
.144, .062
.157, .046
.167, .052
24
.079, .052
.083, .037
.077, .046
.082, .046
A.1.2 Mixed Frequency Tests with MIDAS
A.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.051, .054
.051, .045
.046, .044
.042, .043
4
.054, .063
.050, .047
.047, .052
.041, .042
8
.056, .047
.049, .044
.051, .051
.042, .047
8
.048, .046
.046, .055
.046, .034
.050, .043
12
.056, .061
.051, .044
.049, .045
.043, .044
12
.049, .056
.048, .053
.049, .043
.044, .043
.055, .052
.049, .060
.046, .039
.043, .035
24
.051, .058
.049, .048
.046, .046
.046, .047
24
A.1.3 Low Frequency Tests (Flow Sampling)
A.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.056, .044
.053, .050
.052, .050
.043, .049
1
.053, .046
.051, .043
.054, .049
.042, .044
2
.063, .042
.053, .047
.047, .039
.050, .055
2
.054, .051
.054, .049
.049, .047
.051, .045
3
.060, .058
.056, .051
.052, .045
.055, .040
3
.053, .050
.054, .048
.049, .041
.052, .044
4
.060, .064
.057, .041
.053, .051
.050, .043
4
.053, .053
.058, .059
.055, .035
.049, .052
A.1.4 Low Frequency Tests (Stock Sampling)
A.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.054, .057
.050, .041
.044, .049
.044, .050
1
.047, .029
.050, .048
.049, .056
.051, .045
2
.057, .039
.052, .042
.051, .046
.047, .051
2
.052, .054
.050, .057
.049, .053
.048, .052
3
.062, .034
.055, .048
.051, .046
.048, .047
3
.055, .057
.055, .036
.052, .061
.049, .047
4
.057, .044
.054, .043
.060, .049
.056, .046
4
.050, .042
.049, .052
.053, .032
.054, .039
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated have
two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from the null
limit distribution. The Wald test p-value is computed using the parametric bootstrap based on Gonçalves and
Kilian (2004), with 1000 bootstrap replications. We implement mixed frequency tests with and without an Almon
MIDAS polynomial of dimension s = 3. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000
for max tests and 1000 for Wald tests.
22
Table T.10: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 12
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS
B.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.459, .522
.353, .401
.290, .371
8
.471, .481
.357, .417
.295, .335
.208, .231
4
.823, .893
.739, .825
.695, .735
.578, .588
.220, .208
8
.818, .846
.744, .803
.687, .733
.585, .570
12
.464, .445
.370, .397
24
.508, .354
.430, .280
.320, .324
.234, .189
12
.824, .851
.740, .786
.677, .724
.596, .555
.393, .228
.333, .156
24
.813, .831
.751, .753
.702, .688
.614, .508
B.1.2 Mixed Frequency Tests with MIDAS
B.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.470, .548
.360, .431
.293, .381
.206, .250
4
.837, .901
.757, .835
.708, .747
.590, .623
8
.477, .523
.358, .464
.293, .392
.208, .232
8
.835, .881
.766, .844
.705, .778
.602, .626
12
.472, .534
.361, .465
.306, .396
.202, .252
12
.840, .894
.759, .847
.705, .804
.611, .634
.479, .540
.361, .470
.309, .380
.201, .237
24
.838, .909
.766, .853
.723, .799
.620, .623
24
B.1.3 Low Frequency Tests (Flow Sampling)
B.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.096, .103
.083, .073
.065, .070
.066, .072
2
.102, .081
.082, .070
.070, .058
3
.107, .089
.084, .085
.072, .071
.107, .085
.086, .082
.073, .068
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.159, .137
.125, .125
.093, .096
.090, .076
.061, .063
2
.155, .142
.119, .123
.108, .102
.093, .094
.069, .077
3
.145, .142
.116, .123
.107, .103
.094, .107
.069, .069
4
.150, .146
.119, .122
.102, .108
.086, .092
B.1.4 Low Frequency Tests (Stock Sampling)
B.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.056, .054
.055, .052
.049, .049
.047, .040
1
.055, .065
.056, .059
.054, .049
.052, .058
2
.057, .038
.052, .035
.046, .058
.050, .043
2
.062, .054
.057, .051
.054, .050
.055, .047
3
.057, .053
.055, .045
.049, .048
.052, .057
3
.062, .053
.057, .062
.051, .057
.058, .049
4
.063, .059
.058, .049
.057, .046
.051, .044
4
.056, .055
.061, .060
.053, .062
.057, .036
23
Table T.10: Continued
C. Lagged Causality: cj = 0.25 × I(j = 12) for j = 1, . . . , 12
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS
C.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.054, .045
.054, .039
.211, .205
8
.068, .048
.065, .043
.208, .182
.139, .143
4
.054, .044
.053, .051
.513, .476
.429, .343
.167, .101
8
.056, .050
.054, .046
.522, .469
.425, .289
12
.081, .047
.080, .049
24
.132, .053
.144, .040
.240, .169
.181, .106
12
.066, .055
.058, .044
.519, .409
.428, .268
.309, .122
.271, .094
24
.083, .050
.084, .045
.528, .380
.450, .272
C.1.2 Mixed Frequency Tests with MIDAS
C.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.055, .036
.052, .047
.210, .213
.140, .155
4
.051, .041
.051, .048
.521, .478
.442, .343
8
.055, .042
.046, .045
.199, .196
.149, .121
8
.048, .048
.048, .042
.537, .476
.439, .309
12
.055, .047
.050, .050
.210, .205
.146, .126
12
.053, .060
.044, .047
.538, .472
.428, .318
.053, .055
.050, .035
.215, .213
.145, .129
24
.052, .046
.053, .044
.536, .487
.442, .356
24
C.1.3 Low Frequency Tests (Flow Sampling)
C.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.093, .082
.088, .080
.072, .063
.063, .078
2
.103, .099
.086, .076
.075, .071
3
.108, .088
.087, .067
.077, .064
.101, .091
.088, .084
.078, .087
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.156, .157
.126, .114
.111, .103
.104, .113
.068, .063
2
.150, .162
.128, .132
.110, .132
.096, .107
.075, .049
3
.153, .164
.124, .129
.110, .107
.099, .085
.077, .063
4
.151, .132
.133, .111
.110, .125
.100, .103
C.1.4 Low Frequency Tests (Stock Sampling)
C.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.552, .553
.455, .439
.402, .398
.363, .340
1
.847, .870
.787, .785
.746, .752
.720, .705
2
.557, .525
.457, .450
.397, .374
.355, .304
2
.848, .863
.795, .775
.742, .735
.707, .650
3
.554, .521
.453, .415
.402, .378
.364, .342
3
.845, .836
.784, .763
.745, .714
.696, .651
4
.538, .501
.456, .415
.394, .344
.356, .283
4
.844, .848
.787, .778
.743, .740
.698, .662
24
Table T.10: Continued
D. Sporadic Causality: (c3 , c7 , c10 ) = (0.3, 0.15, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS
D.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.471, .445
.386, .454
.488, .675
8
.464, .444
.385, .436
.495, .649
.378, .445
4
.834, .832
.810, .835
.913, .968
.860, .900
.382, .400
8
.831, .831
.791, .820
.908, .958
.853, .901
12
.464, .422
.410, .372
24
.510, .314
.471, .299
.510, .587
.391, .365
12
.834, .808
.794, .815
.908, .972
.855, .887
.574, .465
.492, .246
24
.825, .762
.787, .790
.908, .928
.854, .832
D.1.2 Mixed Frequency Tests with MIDAS
D.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.473, .467
.402, .490
.502, .698
.384, .467
4
.842, .843
.821, .839
.920, .974
.870, .911
8
.476, .493
.388, .489
.509, .700
.380, .473
8
.852, .857
.806, .840
.920, .972
.864, .929
12
.463, .487
.403, .468
.508, .697
.374, .476
12
.853, .850
.816, .858
.921, .987
.869, .920
.503, .464
.403, .496
.506, .698
.385, .445
24
.852, .864
.821, .852
.933, .971
.868, .927
24
D.1.3 Low Frequency Tests (Flow Sampling)
D.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.084, .067
.061, .056
.065, .065
.055, .043
2
.080, .070
.069, .061
.059, .055
3
.079, .075
.071, .067
.062, .055
.091, .062
.068, .060
.060, .056
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.111, .088
.091, .092
.079, .071
.075, .075
.059, .066
2
.112, .097
.087, .068
.083, .076
.080, .090
.060, .055
3
.110, .102
.093, .093
.083, .071
.073, .064
.066, .047
4
.103, .088
.089, .098
.081, .071
.083, .054
D.1.4 Low Frequency Tests (Stock Sampling)
D.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.058, .058
.046, .039
.050, .043
.050, .041
1
.058, .052
.046, .051
.049, .048
.048, .047
2
.061, .050
.054, .040
.051, .052
.049, .033
2
.061, .047
.054, .039
.053, .042
.052, .051
3
.055, .040
.060, .046
.045, .049
.052, .044
3
.052, .044
.049, .046
.051, .045
.053, .053
4
.057, .063
.053, .036
.055, .041
.054, .048
4
.062, .050
.055, .059
.052, .054
.055, .057
25
Table T.10: Continued
E. Uniform Causality: cj = 0.07 for j = 1, . . . , 12
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS
E.2.1 Mixed Frequency Tests without MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.131, .171
.150, .210
.164, .269
8
.140, .162
.179, .191
.179, .260
.103, .165
4
.247, .321
.290, .497
.319, .622
.220, .460
.126, .166
8
.245, .315
.302, .481
.319, .597
.243, .425
12
.166, .155
.180, .185
24
.221, .106
.282, .144
.203, .229
.158, .147
12
.254, .278
.313, .464
.340, .590
.252, .435
.302, .200
.261, .122
24
.286, .252
.346, .397
.383, .571
.296, .397
E.1.2 Mixed Frequency Tests with MIDAS
E.2.2 Mixed Frequency Tests with MIDAS
rM F = 4
rM F = 8
rM F = 12
rM F = 24
rM F = 4
rM F = 8
rM F = 12
rM F = 24
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hM F
Max, Wald
Max, Wald
Max, Wald
Max, Wald
4
.133, .168
.151, .219
.159, .287
.106, .169
4
.250, .332
.296, .527
.322, .635
.216, .463
8
.130, .169
.160, .230
.157, .304
.108, .185
8
.242, .337
.297, .528
.316, .658
.232, .471
12
.140, .172
.144, .226
.157, .294
.109, .183
12
.252, .309
.293, .487
.321, .683
.235, .505
.136, .152
.156, .221
.156, .307
.100, .197
24
.256, .305
.301, .506
.319, .677
.230, .512
24
E.1.3 Low Frequency Tests (Flow Sampling)
E.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.703, .680
.610, .592
.552, .520
.512, .475
2
.705, .701
.617, .590
.563, .530
3
.681, .701
.604, .580
.556, .527
.684, .667
.596, .554
.529, .498
4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.951, .944
.911, .907
.891, .875
.867, .850
.510, .473
2
.946, .952
.914, .906
.881, .870
.867, .826
.505, .472
3
.947, .934
.919, .909
.892, .867
.862, .822
.491, .439
4
.942, .935
.911, .896
.888, .866
.854, .820
E.1.4 Low Frequency Tests (Stock Sampling)
E.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 4
rLF = 1
rLF = 2
rLF = 3
rLF = 4
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
hLF
Max, Wald
Max, Wald
Max, Wald
Max, Wald
1
.120, .118
.099, .095
.076, .080
.071, .076
1
.195, .180
.145, .156
.133, .115
.109, .116
2
.119, .112
.096, .092
.084, .067
.071, .077
2
.198, .180
.146, .143
.133, .130
.109, .117
3
.115, .128
.095, .089
.083, .071
.078, .071
3
.197, .187
.151, .158
.128, .127
.108, .126
4
.120, .111
.105, .089
.079, .071
.080, .077
4
.186, .172
.151, .149
.129, .118
.117, .103
26
Table T.11: Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(2) - IID
Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 024×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.117, .056
.108, .054
.104, .048
20
.140, .045
.145, .065
.134, .047
24
.181, .044
.181, .041
.178, .052
A.1.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.049, .048
.045, .049
.042, .064
20
.051, .050
.047, .060
.045, .049
24
.046, .053
.049, .054
.045, .053
A.1.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.054, .030
.051, .054
.050, .059
2
.051, .054
.051, .051
.044, .048
3
.060, .039
.057, .055
.047, .047
A.1.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.057, .043
.052, .064
.049, .045
2
.053, .039
.051, .055
.055, .066
3
.062, .047
.050, .051
.052, .051
A.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.076, .036
.068, .058
.069, .045
20
.081, .035
.081, .049
.080, .057
24
.091, .047
.093, .051
.089, .051
A.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.052, .046
.044, .050
.046, .037
20
.048, .030
.046, .048
.050, .042
24
.049, .043
.049, .055
.045, .048
A.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.054, .056
.048, .045
.048, .056
2
.056, .049
.049, .044
.048, .055
3
.061, .051
.052, .039
.053, .044
A.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.055, .050
.053, .049
.051, .047
2
.056, .047
.051, .049
.048, .058
3
.052, .052
.056, .059
.050, .051
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from
the null limit distribution. The Wald test p-value is computed using the parametric bootstrap based
on Gonçalves and Kilian (2004), with 1000 bootstrap replications. We implement mixed frequency tests
with and without an Almon MIDAS polynomial of dimension s = 3. Nominal size is α = 0.05. The
number of Monte Carlo samples is 5000 for max tests and 1000 for Wald tests.
27
Table T.11: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 24
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS
B.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.326, .256
.304, .230
.287, .182
16
.712, .764
.670, .706
.657, .629
20
.354, .209
.327, .201
.319, .171
20
.722, .759
.683, .682
.658, .639
24
.399, .216
.373, .165
.356, .130
24
.715, .709
.682, .634
.668, .620
B.1.2 Mixed Frequency Tests with MIDAS
B.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.270, .355
.248, .341
.218, .268
16
.731, .839
.686, .791
.674, .716
20
.269, .359
.243, .353
.227, .296
20
.742, .857
.694, .767
.665, .746
24
.264, .363
.245, .319
.226, .271
24
.730, .827
.695, .769
.678, .737
B.1.3 Low Frequency Tests (Flow Sampling)
B.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.089, .071
.070, .076
.064, .055
1
.120, .115
.088, .097
.081, .084
2
.089, .084
.076, .084
.057, .055
2
.125, .142
.101, .088
.078, .077
3
.079, .069
.070, .067
.068, .061
3
.110, .111
.087, .081
.086, .080
B.1.4 Low Frequency Tests (Stock Sampling)
B.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.058, .051
.053, .061
.052, .052
1
.056, .064
.065, .052
.055, .051
2
.055, .056
.054, .060
.055, .052
2
.061, .065
.060, .061
.052, .063
.059, .059
.060, .053
.055, .048
3
.058, .066
.060, .048
.057, .050
3
C. Lagged Causality: cj = 0.25 × I(j = 24) for j = 1, . . . , 24
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS
C.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.109, .047
.107, .039
.208, .109
16
.071, .045
.075, .065
.436, .292
.145, .051
.139, .046
.231, .089
20
.077, .042
.083, .040
.435, .264
20
24
.179, .044
.189, .046
.278, .079
24
.085, .038
.091, .038
.452, .236
C.1.2 Mixed Frequency Tests with MIDAS
C.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.049, .043
.046, .047
.134, .118
16
.051, .044
.045, .055
.446, .320
.049, .057
.048, .049
.131, .120
20
.050, .041
.046, .043
.438, .319
20
24
.050, .041
.048, .049
.127, .127
24
.049, .041
.048, .052
.441, .315
C.1.3 Low Frequency Tests (Flow Sampling)
C.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.051, .049
.067, .062
.064, .063
1
.051, .055
.097, .114
.084, .083
.054, .061
.073, .082
.062, .059
2
.054, .052
.092, .075
.092, .089
2
3
.054, .043
.072, .069
.073, .070
3
.056, .050
.095, .097
.086, .092
C.1.4 Low Frequency Tests (Stock Sampling)
C.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.055, .039
.423, .416
.368, .362
1
.052, .055
.780, .776
.730, .708
2
.054, .058
.421, .407
.367, .357
2
.055, .040
.769, .762
.743, .700
3
.062, .043
.413, .414
.356, .352
3
.055, .040
.778, .759
.733, .726
28
Table T.11: Continued
D. Sporadic Causality: (c5 , c12 , c17 , c19 ) = (−0.15, 0.15, 0.25, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS
D.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.174, .101
.438, .325
.406, .300
16
.254, .244
.870, .886
.850, .838
.203, .086
.453, .305
.429, .262
20
.273, .232
.866, .881
.866, .834
20
24
.242, .086
.503, .247
.472, .210
24
.279, .221
.867, .839
.855, .818
D.1.2 Mixed Frequency Tests with MIDAS
D.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.105, .113
.384, .466
.347, .422
16
.236, .287
.887, .933
.871, .888
20
.100, .124
.377, .455
.342, .429
20
.246, .245
.882, .939
.882, .909
24
.103, .126
.376, .490
.350, .418
24
.231, .270
.892, .920
.875, .908
D.1.3 Low Frequency Tests (Flow Sampling)
D.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.051, .043
.051, .049
.049, .060
1
.055, .055
.051, .054
.050, .052
2
.059, .044
.050, .057
.054, .051
2
.060, .043
.057, .061
.053, .053
3
.060, .044
.056, .051
.053, .048
3
.057, .055
.060, .048
.051, .063
D.1.4 Low Frequency Tests (Stock Sampling)
D.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.240, .232
.177, .196
.133, .141
1
.446, .430
.358, .347
.299, .281
.243, .219
.181, .180
.166, .138
2
.446, .405
.352, .340
.304, .272
2
3
.228, .219
.184, .151
.162, .145
3
.445, .407
.368, .333
.307, .273
E. Uniform Causality: cj = 0.05 for j = 1, . . . , 24
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS
E.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.175, .100
.184, .112
.192, .116
16
.225, .271
.243, .290
.267, .288
.209, .103
.221, .110
.226, .099
20
.246, .207
.264, .251
.290, .294
20
24
.256, .081
.283, .086
.282, .095
24
.268, .217
.274, .255
.307, .265
E.1.2 Mixed Frequency Tests with MIDAS
E.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.098, .112
.102, .117
.099, .124
16
.189, .293
.202, .306
.216, .355
20
.100, .119
.109, .140
.093, .142
20
.193, .283
.197, .315
.225, .354
24
.097, .132
.094, .128
.105, .141
24
.201, .256
.209, .332
.219, .359
E.1.3 Low Frequency Tests (Flow Sampling)
E.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.324, .327
.435, .493
.371, .434
1
.574, .588
.793, .838
.749, .778
2
.308, .309
.439, .488
.370, .420
2
.576, .542
.790, .847
.732, .803
3
.311, .292
.439, .492
.375, .428
3
.575, .573
.784, .811
.737, .792
E.1.4 Low Frequency Tests (Stock Sampling)
E.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.089, .094
.097, .105
.090, .078
1
.121, .108
.146, .158
.128, .139
2
.082, .095
.098, .080
.080, .095
2
.119, .123
.151, .181
.132, .139
3
.091, .110
.102, .089
.088, .082
3
.130, .119
.158, .150
.123, .150
29
Table T.12: Rejection Frequencies of Low-to-High Causality Tests Based on MF-VAR(2) GARCH Error and Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 024×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.107, .032
.100, .048
.107, .041
20
.130, .034
.143, .043
.130, .031
24
.156, .053
.168, .038
.181, .034
A.1.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.044, .043
.047, .047
.042, .046
20
.047, .046
.048, .052
.040, .042
24
.050, .058
.046, .045
.041, .038
A.1.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.056, .049
.051, .043
.045, .048
2
.056, .064
.047, .050
.056, .047
3
.058, .052
.053, .040
.052, .033
A.1.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.050, .045
.053, .040
.049, .061
2
.057, .049
.045, .046
.049, .053
3
.056, .050
.060, .055
.050, .040
A.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.072, .051
.071, .034
.062, .037
20
.077, .039
.078, .050
.075, .051
24
.085, .046
.091, .051
.088, .049
A.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
16
.043, .051
.046, .032
.043, .035
20
.050, .047
.045, .047
.045, .053
24
.047, .037
.044, .050
.048, .057
A.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.053, .051
.047, .036
.048, .044
2
.048, .054
.051, .045
.046, .057
3
.057, .046
.048, .044
.052, .056
A.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
hLF Max, Wald Max, Wald Max, Wald
1
.053, .049
.050, .040
.046, .048
2
.055, .062
.055, .045
.049, .048
3
.057, .045
.047, .058
.050, .046
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from
the null limit distribution. The Wald test p-value is computed using the parametric bootstrap based
on Gonçalves and Kilian (2004), with 1000 bootstrap replications. We implement mixed frequency tests
with and without an Almon MIDAS polynomial of dimension s = 3. Nominal size is α = 0.05. The
number of Monte Carlo samples is 5000 for max tests and 1000 for Wald tests.
30
Table T.12: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 24
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS
B.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.310, .251
.295, .203
.272, .187
16
.655, .694
.615, .627
.583, .544
20
.328, .225
.323, .179
.304, .182
20
.661, .647
.641, .608
.608, .518
24
.370, .215
.362, .194
.343, .134
24
.668, .692
.643, .606
.617, .527
B.1.2 Mixed Frequency Tests with MIDAS
B.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.254, .355
.239, .279
.205, .253
16
.671, .771
.639, .714
.600, .644
20
.262, .329
.230, .310
.215, .266
20
.682, .748
.649, .699
.615, .655
24
.271, .330
.242, .317
.215, .282
24
.678, .781
.645, .718
.614, .652
B.1.3 Low Frequency Tests (Flow Sampling)
B.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.104, .096
.088, .078
.063, .069
1
.146, .143
.118, .129
.101, .110
2
.104, .070
.085, .088
.074, .082
2
.149, .148
.122, .126
.096, .101
3
.100, .103
.089, .071
.075, .072
3
.148, .153
.123, .108
.103, .103
B.1.4 Low Frequency Tests (Stock Sampling)
B.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.053, .036
.054, .066
.051, .047
1
.062, .048
.055, .054
.048, .057
2
.054, .058
.055, .046
.048, .048
2
.060, .071
.055, .048
.052, .061
.057, .049
.054, .061
.052, .060
3
.059, .055
.061, .046
.058, .064
3
C. Lagged Causality: cj = 0.25 × I(j = 24) for j = 1, . . . , 24
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS
C.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.098, .048
.102, .060
.200, .102
16
.068, .038
.069, .044
.430, .255
20
.127, .030
.125, .046
.229, .092
20
.075, .045
.080, .043
.432, .236
.161, .047
.157, .042
.276, .077
24
.091, .055
.090, .046
.434, .243
24
C.1.2 Mixed Frequency Tests with MIDAS
C.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.047, .041
.047, .054
.144, .137
16
.047, .039
.046, .042
.434, .304
20
.046, .037
.043, .046
.144, .122
20
.050, .044
.047, .050
.431, .313
24
.049, .041
.045, .053
.149, .138
24
.052, .052
.049, .044
.442, .309
C.1.3 Low Frequency Tests (Flow Sampling)
C.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.049, .054
.078, .073
.075, .065
1
.053, .040
.117, .129
.109, .118
.050, .051
.079, .069
.074, .077
2
.053, .039
.122, .107
.108, .114
2
3
.053, .053
.075, .083
.069, .062
3
.052, .051
.119, .106
.099, .095
C.1.4 Low Frequency Tests (Stock Sampling)
C.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.046, .044
.444, .432
.382, .364
1
.059, .044
.788, .772
.739, .723
2
.056, .044
.436, .438
.382, .384
2
.050, .040
.782, .766
.733, .720
3
.054, .034
.447, .404
.386, .364
3
.051, .047
.770, .772
.737, .687
31
Table T.12: Continued
D. Sporadic Causality: (c5 , c12 , c17 , c19 ) = (−0.15, 0.15, 0.25, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS
D.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.181, .093
.447, .364
.403, .320
16
.261, .240
.857, .890
.836, .885
.205, .095
.471, .313
.440, .265
20
.287, .229
.863, .875
.847, .863
20
24
.233, .091
.487, .303
.488, .252
24
.297, .262
.863, .882
.843, .823
D.1.2 Mixed Frequency Tests with MIDAS
D.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.110, .108
.409, .498
.380, .477
16
.255, .290
.880, .939
.853, .917
20
.109, .135
.412, .501
.369, .485
20
.263, .291
.876, .941
.866, .933
24
.116, .130
.401, .541
.372, .472
24
.251, .297
.878, .943
.858, .921
D.1.3 Low Frequency Tests (Flow Sampling)
D.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.054, .041
.049, .046
.048, .043
1
.055, .054
.060, .057
.062, .040
2
.053, .050
.061, .053
.055, .051
2
.052, .057
.063, .070
.056, .056
3
.060, .045
.059, .052
.052, .052
3
.051, .049
.060, .065
.055, .050
D.1.4 Low Frequency Tests (Stock Sampling)
D.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.258, .229
.191, .169
.158, .167
1
.458, .462
.380, .354
.319, .338
2
.255, .258
.191, .190
.155, .155
2
.457, .453
.380, .373
.331, .329
3
.254, .241
.188, .191
.164, .132
3
.452, .451
.361, .357
.319, .265
E. Uniform Causality: cj = 0.05 for j = 1, . . . , 24
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS
E.2.1 Mixed Frequency Tests without MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.177, .143
.199, .136
.186, .137
16
.226, .315
.251, .415
.270, .512
.208, .113
.214, .122
.230, .139
20
.248, .344
.252, .446
.284, .462
20
24
.244, .115
.263, .113
.286, .108
24
.263, .333
.301, .422
.307, .428
E.1.2 Mixed Frequency Tests with MIDAS
E.2.2 Mixed Frequency Tests with MIDAS
rM F = 16
rM F = 20
rM F = 24
rM F = 16
rM F = 20
rM F = 24
hM F Max, Wald Max, Wald Max, Wald
hM F Max, Wald Max, Wald Max, Wald
16
.104, .177
.119, .177
.108, .202
16
.188, .387
.217, .492
.237, .590
20
.098, .159
.102, .203
.116, .229
20
.195, .415
.212, .528
.228, .579
24
.102, .176
.114, .177
.118, .208
24
.193, .423
.228, .525
.228, .546
E.1.3 Low Frequency Tests (Flow Sampling)
E.2.3 Low Frequency Tests (Flow Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.445, .432
.613, .663
.526, .601
1
.742, .735
.910, .924
.872, .921
2
.436, .452
.598, .655
.545, .611
2
.734, .739
.913, .946
.887, .923
3
.430, .407
.585, .607
.537, .585
3
.741, .722
.912, .933
.883, .909
E.1.4 Low Frequency Tests (Stock Sampling)
E.2.4 Low Frequency Tests (Stock Sampling)
rLF = 1
rLF = 2
rLF = 3
rLF = 1
rLF = 2
rLF = 3
hLF
Max, Wald Max, Wald Max, Wald
hLF
Max, Wald Max, Wald Max, Wald
1
.091, .081
.097, .099
.080, .092
1
.113, .139
.161, .164
.138, .135
2
.084, .083
.098, .110
.084, .097
2
.125, .108
.153, .178
.140, .165
3
.091, .073
.106, .089
.087, .086
3
.116, .124
.161, .154
.142, .146
32
Table T.13: Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(1) - IID Error
and Non-Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 012×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.064
.068
.066
.071
8
.074
.080
.082
.084
12
.087
.098
.101
.102
24
.162
.184
.196
.238
A.1.2 Mixed Frequency Tests with MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.065
.062
.063
.065
8
.059
.067
.066
.063
12
.056
.063
.062
.057
24
.062
.059
.065
.064
A.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.049
.054
.057
.058
2
.060
.057
.058
.053
3
.058
.057
.056
.057
4
.064
.058
.052
.060
A.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.051
.050
.056
.051
2
.061
.056
.056
.049
3
.069
.056
.060
.062
4
.065
.057
.061
.064
A.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.059
.058
.063
.056
8
.065
.070
.061
.061
12
.068
.064
.074
.070
24
.089
.098
.097
.108
A.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.054
.052
.056
.055
8
.058
.058
.055
.051
12
.054
.051
.055
.057
24
.056
.056
.059
.058
A.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.054
.049
.055
.049
2
.058
.056
.052
.052
3
.049
.060
.049
.050
4
.060
.058
.054
.057
A.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.050
.055
.055
.047
2
.054
.053
.048
.054
3
.056
.055
.049
.052
4
.055
.053
.052
.061
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from
the null limit distribution. We implement mixed frequency tests with and without an Almon MIDAS
polynomial of dimension s = 3. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000.
33
Table T.13: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 12
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.525
.406
.354
.275
8
.519
.436
.382
.291
12
.530
.444
.384
.318
24
.585
.500
.466
.438
B.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.532
.414
.357
.277
8
.534
.425
.368
.269
12
.533
.429
.364
.271
24
.543
.417
.354
.274
B.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.094
.073
.067
.063
2
.087
.076
.067
.070
3
.087
.083
.069
.071
4
.091
.084
.075
.073
B.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.061
.056
.053
.047
.066
.062
.059
.058
2
3
.066
.064
.059
.063
4
.065
.066
.056
.061
B.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.880
.808
.759
.677
8
.875
.807
.764
.667
12
.879
.810
.756
.678
24
.869
.811
.755
.680
B.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.882
.820
.772
.688
8
.889
.826
.778
.685
12
.893
.826
.782
.695
24
.893
.831
.768
.694
B.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.115
.095
.092
.084
2
.118
.100
.088
.082
3
.118
.095
.092
.075
4
.119
.101
.089
.084
B.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.057
.062
.049
.048
2
.058
.059
.055
.052
3
.063
.060
.058
.058
4
.066
.067
.060
.056
C. Lagged Causality: cj = 0.25 × I(j = 24) for j = 1, . . . , 12
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.064
.065
.225
.179
8
.074
.081
.241
.197
.097
.093
.263
.222
12
24
.162
.182
.345
.330
C.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.061
.059
.222
.178
.057
.059
.220
.177
8
12
.066
.064
.231
.169
24
.060
.065
.221
.166
C.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.087
.082
.070
.068
2
.082
.080
.072
.074
3
.093
.084
.088
.072
.093
.089
.080
.076
4
C.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.561
.475
.410
.371
2
.569
.462
.418
.363
.561
.451
.416
.354
3
4
.568
.460
.415
.365
C.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.057
.057
.537
.451
8
.058
.066
.531
.447
12
.069
.071
.528
.460
24
.090
.091
.546
.474
C.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.057
.059
.550
.458
8
.055
.057
.538
.458
12
.055
.056
.544
.463
24
.057
.055
.557
.466
C.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.118
.111
.089
.080
2
.128
.102
.097
.085
3
.129
.102
.088
.086
4
.123
.103
.095
.085
C.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.862
.804
.758
.721
2
.858
.792
.744
.717
3
.862
.792
.744
.709
4
.855
.792
.739
.702
34
Table T.13: Continued
D. Sporadic Causality: (c3 , c7 , c10 ) = (0.3, 0.15, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.473
.404
.530
.413
8
.495
.425
.556
.445
12
.494
.438
.561
.455
24
.533
.503
.626
.559
D.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.474
.416
.537
.422
8
.501
.428
.553
.429
12
.493
.419
.554
.417
24
.492
.427
.545
.425
D.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.075
.068
.062
.063
2
.082
.067
.067
.063
3
.077
.068
.060
.067
4
.078
.072
.061
.070
D.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.057
.050
.059
.054
.054
.061
.056
.058
2
3
.059
.057
.055
.054
4
.058
.061
.060
.062
D.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.862
.826
.937
.878
8
.856
.824
.929
.877
12
.845
.809
.930
.875
24
.841
.813
.928
.875
D.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.866
.836
.940
.889
8
.870
.839
.941
.890
12
.861
.828
.941
.894
24
.864
.831
.948
.887
D.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.088
.082
.067
.066
2
.091
.082
.068
.064
3
.097
.076
.070
.070
4
.098
.082
.068
.070
D.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.055
.049
.055
.053
2
.057
.061
.051
.052
3
.059
.050
.058
.054
4
.054
.059
.053
.049
E. Uniform Causality: cj = 0.07 for j = 1, . . . , 12
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.152
.168
.180
.140
8
.164
.193
.201
.159
.196
.220
.230
.205
12
24
.256
.327
.345
.340
E.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.147
.161
.179
.137
.150
.167
.179
.127
8
12
.162
.175
.185
.134
24
.149
.174
.183
.135
E.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.557
.472
.400
.358
2
.578
.457
.412
.369
3
.569
.463
.405
.368
.569
.460
.403
.377
4
E.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.125
.107
.087
.085
2
.123
.105
.087
.084
.125
.093
.094
.081
3
4
.125
.094
.092
.084
E.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.235
.305
.329
.247
8
.252
.296
.352
.257
12
.254
.329
.355
.266
24
.278
.364
.403
.331
E.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.239
.308
.329
.246
8
.254
.292
.337
.235
12
.243
.309
.334
.244
24
.247
.309
.340
.253
E.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.866
.796
.751
.726
2
.866
.803
.752
.713
3
.860
.797
.734
.702
4
.853
.780
.743
.714
E.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.206
.142
.117
.111
2
.196
.152
.123
.107
3
.191
.152
.122
.111
4
.201
.153
.127
.121
35
Table T.14: Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(1) - GARCH
Error and Non-Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 012×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.067
.068
.065
.070
8
.071
.076
.087
.073
12
.088
.096
.106
.101
24
.155
.169
.197
.213
A.1.2 Mixed Frequency Tests with MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.066
.066
.063
.065
8
.056
.061
.066
.049
12
.067
.064
.065
.058
24
.057
.064
.064
.060
A.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.058
.055
.058
.050
2
.058
.051
.055
.059
3
.055
.068
.062
.058
4
.066
.060
.060
.067
A.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.053
.061
.056
.050
2
.054
.058
.056
.052
3
.062
.057
.058
.061
4
.063
.059
.059
.062
A.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.062
.058
.063
.064
8
.065
.064
.068
.066
12
.079
.071
.074
.072
24
.088
.097
.097
.101
A.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.060
.059
.062
.062
8
.061
.059
.060
.056
12
.063
.056
.055
.057
24
.059
.059
.058
.057
A.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.057
.054
.048
.057
2
.055
.062
.054
.050
3
.061
.060
.050
.052
4
.059
.059
.056
.052
A.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.057
.059
.051
.051
2
.060
.052
.046
.053
3
.054
.056
.054
.055
4
.055
.054
.063
.058
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from
the null limit distribution. We implement mixed frequency tests with and without an Almon MIDAS
polynomial of dimension s = 3. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000.
36
Table T.14: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 12
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.500
.413
.333
.267
8
.501
.404
.352
.264
12
.512
.413
.380
.301
24
.558
.479
.452
.398
B.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.511
.409
.338
.268
8
.508
.399
.339
.254
12
.506
.389
.356
.261
24
.524
.404
.358
.251
B.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.104
.093
.077
.076
2
.113
.088
.087
.080
3
.108
.089
.082
.082
4
.114
.095
.082
.086
B.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.062
.056
.054
.053
.058
.065
.058
.054
2
3
.061
.057
.056
.060
4
.065
.069
.061
.066
B.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.843
.775
.714
.631
8
.844
.768
.711
.633
12
.843
.772
.724
.642
24
.845
.769
.732
.641
B.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.850
.784
.726
.644
8
.859
.787
.732
.646
12
.861
.787
.744
.659
24
.861
.790
.741
.650
B.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.156
.125
.105
.093
2
.162
.121
.116
.101
3
.154
.133
.111
.108
4
.159
.131
.113
.100
B.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.056
.060
.052
.053
2
.064
.053
.051
.054
3
.066
.062
.061
.056
4
.071
.063
.065
.058
C. Lagged Causality: cj = 0.25 × I(j = 12) for j = 1, . . . , 12
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.064
.066
.239
.183
8
.074
.085
.255
.206
.086
.097
.268
.234
12
24
.155
.173
.352
.334
C.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.060
.064
.240
.181
.059
.065
.238
.181
8
12
.059
.064
.234
.185
24
.066
.057
.244
.185
C.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.099
.091
.079
.073
2
.109
.090
.084
.073
3
.104
.090
.086
.083
.120
.099
.092
.085
4
C.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.586
.508
.445
.397
2
.590
.494
.440
.382
.577
.491
.430
.389
3
4
.563
.483
.418
.377
C.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.061
.057
.550
.456
8
.061
.066
.543
.468
12
.072
.072
.560
.462
24
.090
.093
.562
.483
C.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.058
.056
.555
.467
8
.056
.059
.554
.478
12
.064
.057
.573
.468
24
.061
.057
.568
.476
C.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.154
.132
.118
.100
2
.166
.135
.114
.103
3
.160
.136
.113
.108
4
.162
.129
.117
.103
C.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.873
.823
.765
.731
2
.866
.805
.763
.722
3
.872
.813
.770
.723
4
.860
.800
.762
.718
37
Table T.14: Continued
D. Sporadic Causality: (c3 , c7 , c10 ) = (0.3, 0.15, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.508
.434
.568
.442
8
.502
.443
.562
.448
12
.505
.459
.581
.468
24
.536
.518
.631
.577
D.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.518
.436
.576
.443
8
.520
.437
.572
.444
12
.513
.454
.569
.444
24
.518
.448
.576
.463
D.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.085
.072
.068
.064
2
.094
.078
.069
.071
3
.091
.078
.073
.063
4
.086
.077
.076
.066
D.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.058
.054
.051
.053
.061
.057
.058
.051
2
3
.058
.066
.058
.053
4
.068
.058
.062
.058
D.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.855
.821
.928
.871
8
.851
.807
.934
.877
12
.841
.808
.929
.880
24
.849
.817
.928
.881
D.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.867
.834
.942
.880
8
.873
.828
.937
.893
12
.860
.830
.941
.897
24
.875
.837
.943
.899
D.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.109
.097
.083
.071
2
.109
.089
.081
.081
3
.120
.095
.079
.076
4
.116
.093
.089
.075
D.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.056
.056
.054
.054
2
.059
.060
.049
.056
3
.067
.054
.049
.061
4
.061
.056
.051
.060
E. Uniform Causality: cj = 0.07 for j = 1, . . . , 12
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.169
.187
.198
.149
8
.173
.212
.221
.187
.200
.227
.252
.210
12
24
.276
.324
.367
.334
E.1.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.168
.187
.200
.145
.163
.190
.193
.163
8
12
.165
.182
.208
.143
24
.167
.190
.210
.137
E.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.742
.640
.577
.545
2
.740
.644
.589
.560
3
.716
.642
.594
.550
.713
.646
.578
.532
4
E.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.131
.105
.087
.083
2
.129
.107
.095
.083
.138
.106
.093
.092
3
4
.125
.120
.101
.090
E.2.1 Mixed Frequency Tests without MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.260
.324
.357
.257
8
.278
.342
.376
.281
12
.290
.354
.392
.283
24
.294
.390
.445
.338
E.2.2 Mixed Frequency Tests with MIDAS
hM F
rM F = 4 rM F = 8 rM F = 12 rM F = 24
4
.266
.333
.361
.262
8
.277
.343
.369
.274
12
.279
.339
.361
.263
24
.270
.343
.379
.270
E.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.958
.925
.906
.879
2
.954
.928
.898
.876
3
.954
.920
.893
.874
4
.949
.920
.892
.881
E.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
rLF = 4
1
.199
.152
.133
.120
2
.202
.159
.135
.118
3
.197
.158
.132
.119
4
.209
.151
.132
.125
38
Table T.15: Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(2) - IID Error
and Non-Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 024×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.148
.142
.146
20
.166
.193
.185
.219
.231
.230
24
A.1.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.067
.061
.059
20
.054
.071
.061
24
.062
.064
.063
A.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.058
.059
.049
2
.054
.063
.058
3
.059
.060
.060
A.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.056
.055
.054
2
.060
.067
.056
3
.057
.056
.055
A.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.083
.087
.084
20
.101
.093
.100
24
.102
.110
.107
A.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.054
.054
.054
20
.060
.060
.058
24
.055
.056
.058
A.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.054
.049
.055
2
.049
.054
.051
3
.055
.054
.051
A.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.056
.053
.050
2
.059
.051
.052
3
.059
.052
.056
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from
the null limit distribution. We implement mixed frequency tests with and without an Almon MIDAS
polynomial of dimension s = 3. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000.
39
Table T.15: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 24
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS B.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.380
.355
.361
16
.728
.713
.680
20
.415
.401
.403
20
.735
.704
.676
.455
.437
.429
24
.732
.712
.706
24
B.1.2 Mixed Frequency Tests with MIDAS
B.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.322
.283
.273
16
.743
.727
.701
20
.322
.295
.274
20
.754
.718
.687
24
.324
.287
.271
24
.745
.722
.719
B.1.3 Low Frequency Tests (Flow Sampling)
B.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.092
.078
.065
1
.120
.098
.086
2
.086
.071
.068
2
.119
.095
.093
3
.082
.084
.070
3
.118
.092
.089
B.1.4 Low Frequency Tests (Stock Sampling)
B.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.059
.056
.057
1
.064
.060
.062
2
.065
.064
.055
2
.061
.058
.061
3
.062
.060
.061
3
.067
.060
.059
C. Lagged Causality: cj = 0.25 × I(j = 24) for j = 1, . . . , 24
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS C.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.138
.144
.243
16
.081
.086
.451
.173
.181
.290
20
.093
.093
.469
20
24
.227
.220
.331
24
.099
.103
.466
C.1.2 Mixed Frequency Tests with MIDAS
C.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.060
.066
.163
16
.057
.057
.460
20
.063
.058
.169
20
.057
.056
.458
24
.063
.065
.165
24
.056
.052
.451
C.1.3 Low Frequency Tests (Flow Sampling)
C.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.058
.074
.077
1
.057
.097
.089
2
.062
.077
.075
2
.049
.099
.093
.064
.082
.074
3
.057
.093
.095
3
C.1.4 Low Frequency Tests (Stock Sampling)
C.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.054
.458
.400
1
.056
.778
.734
.060
.469
.401
2
.058
.775
.743
2
3
.057
.437
.393
3
.055
.787
.739
40
Table T.15: Continued
D. Sporadic Causality: (c5 , c12 , c17 , c19 ) = (−0.15, 0.15, 0.25, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS D.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.213
.433
.400
16
.292
.797
.779
20
.246
.466
.444
20
.304
.801
.781
.297
.511
.500
24
.314
.808
.782
24
D.1.2 Mixed Frequency Tests with MIDAS
D.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.126
.358
.318
16
.263
.804
.791
20
.115
.360
.329
20
.261
.806
.784
24
.130
.368
.322
24
.261
.810
.791
D.1.3 Low Frequency Tests (Flow Sampling)
D.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.059
.059
.056
1
.056
.054
.049
2
.059
.057
.056
2
.054
.057
.054
3
.061
.054
.054
3
.051
.054
.052
D.1.4 Low Frequency Tests (Stock Sampling)
D.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.249
.202
.157
1
.461
.373
.316
2
.249
.202
.166
2
.459
.363
.311
3
.256
.186
.177
3
.454
.357
.312
E. Uniform Causality: cj = 0.05 for j = 1, . . . , 24
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS E.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.224
.235
.246
16
.249
.282
.308
.266
.277
.296
20
.269
.301
.325
20
24
.325
.336
.356
24
.303
.315
.343
E.1.2 Mixed Frequency Tests with MIDAS
E.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.124
.135
.142
16
.211
.231
.262
20
.128
.136
.134
20
.208
.240
.254
24
.126
.141
.150
24
.223
.232
.249
E.1.3 Low Frequency Tests (Flow Sampling)
E.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.336
.472
.410
1
.596
.805
.758
2
.338
.490
.422
2
.589
.800
.761
.333
.464
.414
3
.595
.806
.751
3
E.1.4 Low Frequency Tests (Stock Sampling)
E.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.092
.102
.088
1
.127
.160
.132
.095
.106
.097
2
.122
.169
.139
2
3
.088
.110
.092
3
.115
.162
.134
41
Table T.16: Rejection Frequencies of Low-to-High Max Tests Based on MF-VAR(2) - GARCH
Error and Non-Robust Covariance Matrix for Max Tests
A. Non-Causality: c = 024×1
A.1 TL = 80
A.2 TL = 160
A.1.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.124
.133
.141
20
.161
.159
.175
.200
.215
.220
24
A.1.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.059
.057
.065
20
.061
.060
.057
24
.064
.062
.064
A.1.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.058
.061
.052
2
.056
.052
.057
3
.053
.057
.060
A.1.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.056
.056
.054
2
.057
.057
.053
3
.059
.057
.057
A.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.080
.080
.084
20
.094
.096
.093
24
.102
.100
.113
A.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
16
.056
.054
.054
20
.059
.059
.057
24
.062
.051
.055
A.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.053
.053
.050
2
.054
.052
.047
3
.059
.055
.054
A.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
1
.054
.052
.048
2
.051
.058
.057
3
.055
.053
.056
The AR(1) coefficient for xL is a = 0.2, while the AR(1) coefficient for xH is d = 0.2. There is high-to-low
decaying causality with alternating signs: bj = (−1)j−1 × 0.2/j for j = 1, . . . , 12. The models estimated
have two low frequency lags of xL (i.e. q = 2). The max-test p-value is computed using 5000 draws from
the null limit distribution. We implement mixed frequency tests with and without an Almon MIDAS
polynomial of dimension s = 3. Nominal size is α = 0.05. The number of Monte Carlo samples is 5000.
42
Table T.16: Continued
B. Decaying Causality: cj = (−1)j−1 × 0.3/j for j = 1, . . . , 24
B.1 TL = 80
B.2 TL = 160
B.1.1 Mixed Frequency Tests without MIDAS B.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.371
.350
.341
16
.702
.666
.634
20
.401
.390
.368
20
.695
.670
.649
.437
.429
.397
24
.710
.669
.646
24
B.1.2 Mixed Frequency Tests with MIDAS
B.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.316
.285
.265
16
.723
.683
.653
20
.316
.281
.255
20
.716
.687
.649
24
.315
.290
.250
24
.714
.688
.653
B.1.3 Low Frequency Tests (Flow Sampling)
B.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.100
.095
.080
1
.157
.125
.112
2
.105
.096
.084
2
.168
.125
.106
3
.107
.088
.085
3
.155
.134
.107
B.1.4 Low Frequency Tests (Stock Sampling)
B.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.057
.056
.057
1
.063
.065
.057
2
.063
.061
.051
2
.071
.063
.062
3
.062
.062
.061
3
.063
.065
.065
C. Lagged Causality: cj = 0.25 × I(j = 24) for j = 1, . . . , 24
C.1 TL = 80
C.2 TL = 160
C.1.1 Mixed Frequency Tests without MIDAS C.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.127
.131
.261
16
.078
.087
.465
.150
.176
.285
20
.095
.098
.463
20
24
.195
.211
.334
24
.089
.108
.479
C.1.2 Mixed Frequency Tests with MIDAS
C.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.060
.061
.182
16
.057
.058
.471
20
.059
.061
.178
20
.057
.059
.469
24
.057
.063
.180
24
.053
.058
.459
C.1.3 Low Frequency Tests (Flow Sampling)
C.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.056
.090
.079
1
.049
.125
.116
2
.061
.084
.085
2
.055
.125
.116
.057
.084
.081
3
.054
.127
.116
3
C.1.4 Low Frequency Tests (Stock Sampling)
C.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.061
.465
.413
1
.054
.801
.764
.050
.469
.423
2
.056
.793
.752
2
3
.061
.461
.416
3
.051
.798
.750
43
Table T.16: Continued
D. Sporadic Causality: (c5 , c12 , c17 , c19 ) = (−0.15, 0.15, 0.25, −0.3) and cj = 0 for other j’s
D.1 TL = 80
D.2 TL = 160
D.1.1 Mixed Frequency Tests without MIDAS D.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.216
.443
.415
16
.304
.802
.780
20
.248
.487
.448
20
.326
.810
.788
.288
.510
.513
24
.346
.808
.794
24
D.1.2 Mixed Frequency Tests with MIDAS
D.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.136
.385
.352
16
.294
.818
.790
20
.139
.391
.354
20
.295
.819
.796
24
.148
.378
.360
24
.297
.819
.799
D.1.3 Low Frequency Tests (Flow Sampling)
D.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.060
.053
.052
1
.054
.056
.055
2
.065
.053
.060
2
.060
.055
.057
3
.059
.065
.065
3
.063
.058
.060
D.1.4 Low Frequency Tests (Stock Sampling)
D.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.259
.207
.182
1
.472
.387
.332
2
.276
.209
.181
2
.466
.394
.335
3
.268
.212
.183
3
.472
.393
.339
E. Uniform Causality: cj = 0.05 for j = 1, . . . , 24
E.1 TL = 80
E.2 TL = 160
E.1.1 Mixed Frequency Tests without MIDAS E.2.1 Mixed Frequency Tests without MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.224
.253
.253
16
.273
.302
.321
.270
.279
.306
20
.293
.325
.332
20
24
.309
.337
.347
24
.322
.336
.358
E.1.2 Mixed Frequency Tests with MIDAS
E.2.2 Mixed Frequency Tests with MIDAS
hM F rM F = 16 rM F = 20
rM F = 24
hM F rM F = 16 rM F = 20
rM F = 24
16
.145
.143
.162
16
.235
.251
.276
20
.135
.146
.149
20
.248
.269
.276
24
.140
.143
.153
24
.243
.275
.273
E.1.3 Low Frequency Tests (Flow Sampling)
E.2.3 Low Frequency Tests (Flow Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.464
.646
.594
1
.754
.929
.901
2
.470
.648
.582
2
.761
.925
.897
.478
.634
.592
3
.755
.923
.899
3
E.1.4 Low Frequency Tests (Stock Sampling)
E.2.4 Low Frequency Tests (Stock Sampling)
hLF
rLF = 1
rLF = 2
rLF = 3
hLF
rLF = 1
rLF = 2
rLF = 3
1
.088
.107
.098
1
.129
.167
.139
.092
.106
.091
2
.129
.160
.145
2
3
.094
.103
.097
3
.127
.169
.139
44
Table T.17: Ljung-Box Q Tests of Residual Whiteness with Double Blocks-of-Blocks Bootstrap
A. Rolling Window Analysis: Number of Rejections out of 129 Windows
A.1. Mixed Frequency High-to-Low Regression Model
b=4
b = 10
b = 20
k = 4 k = 8 k = 12 k = 4 k = 8 k = 12 k = 4 k = 8 k = 12
α = 0.01
2
0
0
3
0
0
0
0
0
α = 0.05
13
5
1
9
3
10
6
0
0
α = 0.10
24
9
32
26
14
32
21
18
6
A.2. Low Frequency High-to-Low Regression Model
b=4
b = 10
b = 20
k = 4 k = 8 k = 12 k = 4 k = 8 k = 12 k = 4 k = 8 k = 12
α = 0.01
15
0
2
11
0
9
0
0
0
α = 0.05
51
23
33
47
8
23
4
0
3
α = 0.10
79
46
54
77
20
40
18
6
16
A.3. Mixed Frequency Low-to-High Regression Model (without MIDAS Polynomial)
b=4
b = 10
b = 20
k = 4 k = 8 k = 12 k = 4 k = 8 k = 12 k = 4 k = 8 k = 12
α = 0.01
0
4
0
1
1
1
1
0
0
α = 0.05
4
8
5
4
5
3
3
0
1
α = 0.10
14
23
12
10
9
8
14
7
2
A.4. Mixed Frequency Low-to-High Regression Model (with MIDAS Polynomial)
b=4
b = 10
b = 20
k = 4 k = 8 k = 12 k = 4 k = 8 k = 12 k = 4 k = 8 k = 12
α = 0.01
0
0
0
0
0
0
0
0
0
α = 0.05
25
14
16
1
2
1
2
0
0
α = 0.10
41
28
32
2
5
7
4
2
2
A.5. Low Frequency Low-to-High Regression Model
b=4
b = 10
b = 20
k = 4 k = 8 k = 12 k = 4 k = 8 k = 12 k = 4 k = 8 k = 12
α = 0.01
0
0
2
0
0
0
0
0
0
α = 0.05
31
17
26
1
0
2
2
1
2
α = 0.10
67
36
45
2
6
3
8
8
6
B. Full Sample Analysis: Bootstrapped p-values
MF High-to-Low
LF High-to-Low
MF Low-to-High (w/o MIDAS)
MF Low-to-High (w/ MIDAS)
LF Low-to-High
k=4
.107
.021∗∗
.076∗
.035∗∗
.043∗∗
b=4
k=8
.180
.066∗
.280
.179
.041∗∗
k = 12
.084∗
.024∗∗
.054∗
.019∗∗
.001∗∗∗
k=4
.097∗
.018∗∗
.148
.046∗∗
.011∗∗
b = 10
k=8
.264
.061∗
.366
.203
.045∗∗
k = 12
.160
.007∗∗∗
.133
.037∗∗
.003∗∗∗
k=4
.089∗
.053∗
.539
.718
.582
b = 20
k=8
.300
.082∗
.397
.594
.506
k = 12
.191
.017∗∗
.343
.720
.433
In the empirical application, we implement the Ljung-Box Q test with lag k = 4, 8, 12 in order to test for residual
whiteness. p-values are computed based on the double blocks-of-blocks bootstrap of Horowitz, Lobato, Nankervis,
and Savin (2006) with block size b ∈ {4, 10, 20}, M1 = 999 first-stage samples, and M2 = 249 second-stage
samples. In Panel A we conduct rolling window analysis with 129 windows (80 quarters in each window). We
report in how many windows out of 129 the null hypothesis of whiteness is rejected at level α = 0.01, 0.05, 0.10. In
Panel B we conduct full sample analysis, and we report the bootstrapped p-values for residual whiteness. Three
asterisks (∗∗∗ ) indicates rejection at 1%, two asterisks (∗∗ ) indicates rejection at 5%, and one asterisk (∗ ) indicates
rejection at 10%.
45
References
Gonçalves, S., and L. Kilian (2004): “Bootstrapping Autoregressions with Conditional
Heteroskedasticity of Unknown Form,” Journal of Econometrics, 123, 89–120.
Horowitz, J., I. N. Lobato, J. C. Nankervis, and N. E. Savin (2006): “Bootstrapping
the Box-Pierce Q Test: A Robust Test of Uncorrelatedness,” Journal of Econometrics, 133,
841–862.
46