########################################################################
# Stochastic volatility with leverage
#
# Dependent variable:y_t=log(p_t) -
log(p_{t-1}) (return)
#
where p_t is topix at time t
#
# Model
# y_t =
eps_t*exp(h_t/2), eps_t ~ N(0,1)
# h_{t+1}= mu + phi
(h_t-mu)+eta_t, eta_t ~N(0,sigma.eta^2)
# Corr(eps_t,eta_t)= rho
# ->
Given y_t,
# h_{t+1}=
mu+phi*(h_t-mu)+rho*sigma.eta*exp(-h_t/2)*y_t+u_t
# u_t ~ N(0,
sigma.eta^2(1-rho^2))
########################################################################
model
{
#
Compute returns in percentage for t=1,2,..,1232
for(t
in 1:n-1)
{
y[t]
<- log(topix[t+1]) - log(topix[t])
}
#
h_1 ~ N(mu, sigma2/(1-phi^2))
h[1] ~ dnorm(mu, tau20)
tau20
<-
tau2.eta*(1-phi1*phi1)
tau21 <-
tau2.eta/(1-rho*rho)
for(t
in 1:n-1)
{
#
measurement equation
y[t]
~ dnorm(0, tau2[t])
tau2[t]
<- exp(-h[t])
#
state equation
h[t+1] ~ dnorm(mu.h[t], tau21)
mu.h[t]
<- mu+phi1*(h[t]-mu)+rho*sigma.eta*exp(-h[t]/2)*y[t]
v[t] <-
exp((h[t])/2)
time[t]
<- t
}
#priors
mu ~ dnorm(-10, 1)
rho ~ dunif(-1, 1)
tau2.eta
~ dgamma(2.5, 0.025)
sigma.eta <- sqrt(1/tau2.eta)
sigma.eps <- exp(mu/2)
phi0 ~ dbeta(20, 1.5)
phi1 <- 2*phi0-1 #
phi0=(phi+1)/2
}
# Initial values. Load below values for mu and lsigma and
"gen inits" for
list(
mu = -9, tau2.eta = 100, phi0 =
0.95, rho = -0.4)
# Data
list(n
= 1233)
# Data
topix[]
1152.34
1175.93
1167.96
1159.33
1165.19
1159.23
1141.24
1124.95
1128.36
1154.1
1208.2
1227.78
1239.22
1242.58
1235.37
1255.08
1293.23
1297.82
1296.97
1285.12
1269.8
1270.82
1281.83
1264.18
1275.94
1282.24
1297.11
1301.92
1297.1
1277.44
1270.49
1269.84
1258.55
1258.21
1259.07
1254.66
1232.84
1241.56
1258.75
1276.85
1296.64
1293.7
1286.62
1276.88
1291.7
1277.51
1273.07
1259.33
1250.86
1276.05
1267.29
1270.15
1251.88
1245.67
1258.06
1246.05
1241.2
1238.98
1261.04
1264.21
1243.62
1247.02
1228.64
1192.53
1186.73
1196.81
1210.41
1247.89
1251.34
1244.84
1242.41
1245.4
1246.05
1214.92
1211.19
1217.39
1221.7
1215.86
1223.23
1235.43
1205.55
1206.83
1221.96
1216.74
1195.48
1188.13
1191.11
1207.2
1202.24
1205.74
1205.88
1208.77
1209.92
1222.58
1227.74
1235.59
1228.85
1226.43
1228.18
1219.37
1222.82
1221.14
1201.39
1211.51
1202.99
1207.52
1199.12
1197.92
1207.54
1195.23
1175.33
1174.04
1160.98
1161.11
1166.02
1194.79
1188.76
1187.81
1177.52
1182.88
1184.59
1194.62
1208.2
1232.04
1277.31
1263.26
1263.78
1258.65
1271.95
1267.55
1261.94
1235.47
1258.56
1273.79
1274.47
1282.2
1280.84
1270.57
1255.61
1245.86
1253.64
1236.74
1239.98
1244.01
1254.69
1256.9
1238.99
1233.11
1238.63
1228.32
1221.65
1202.41
1179.29
1181.81
1179.56
1169.23
1157.09
1172.4
1187.16
1174.71
1168.95
1163.11
1165.12
1141.35
1109.01
1090.11
1097.96
1112.86
1118.74
1105.88
1089.37
1129.97
1134.02
1118.12
1104.14
1075.24
1086.3
1088.02
1064.72
1066.89
1049.9
1057.27
1067.91
1050.61
1065.48
1065.67
1040.83
1011.78
1013.44
994.82
1001.54
1044.16
995.29
988.65
1023.02
999.17
987.38
988.57
1005.22
1025.51
1050.99
1077.11
1075.67
1063.26
1047.98
1041.96
1031.4
1042.54
1041.82
1074.63
1113.65
1101.62
1090
1082.15
1080.04
1096.04
1085.59
1098
1108.77
1106.57
1112.75
1110.79
1139.34
1152.44
1154.81
1162.25
1159.42
1139.55
1141.63
1139.65
1125.59
1123.55
1128.21
1129.28
1139.68
1137.65
1113.17
1097.89
1094.16
1092.2
1095.65
1099.38
1101.59
1082.35
1077.44
1087.32
1083.59
1087.26
1083.17
1065.95
1050.35
1066.46
1067.64
1055.71
1054.36
1056.5
1055.28
1071.69
1077.81
1076.19
1091.2
1104.74
1096.04
1101.76
1112.73
1117.4
1115.79
1127.72
1119.61
1106.59
1096.55
1087.66
1083.81
1091.8
1080.4
1084.4
1091.36
1096.58
1105.13
1096.5
1094.89
1098.1
1110.59
1118.06
1120.3
1124.12
1120.89
1109.8
1091.91
1102.46
1109.05
1149.52
1151.4
1165.72
1189.95
1201.5
1196.45
1209.84
1242.81
1259.81
1237.24
1273.07
1259.45
1238.79
1258.88
1267.68
1275.9
1270.13
1269.52
1304.76
1307.5
1308.66
1318.89
1325.5
1350.98
1340.36
1323.7
1331.56
1341.75
1340.37
1344.54
1327.88
1330.31
1315.27
1318.48
1336.3
1349.44
1353.33
1350.2
1340.56
1382.07
1371.18
1373.89
1357.17
1361.41
1353.32
1343.09
1323.4
1321.93
1301.54
1303.52
1309.37
1313.28
1305.69
1307.85
1296.6
1293.54
1297.21
1319.71
1326.37
1318.34
1317.18
1330.8
1336.81
1344.25
1367.89
1381.61
1383.54
1387.16
1387.62
1402.42
1399.86
1421.42
1422.53
1410.35
1410.51
1411.59
1420.74
1430.22
1420.22
1452.25
1464.16
1487.35
1479.95
1478.1
1468.08
1465.65
1484.03
1484.76
1500.36
1514.1
1506.22
1510.91
1497.8
1452.68
1437.87
1436.71
1438.43
1453.29
1467.72
1475.43
1470.01
1479.42
1456.02
1430.75
1419.4
1424.31
1422.25
1425.47
1438.61
1447.33
1472.07
1476.16
1474.77
1478.69
1502.64
1520.09
1504.81
1496.58
1484.24
1480.72
1497.01
1461.43
1498.22
1495.78
1501.16
1512.71
1504.62
1506.48
1507.75
1518.28
1533.33
1518.21
1494.87
1504.95
1518.79
1522.38
1474.34
1432.16
1433.98
1470.6
1462.65
1506.33
1516.48
1518.51
1512.47
1518.57
1528.39
1535.25
1540.03
1528.43
1537.37
1504.18
1464.41
1478.28
1511.37
1506
1519.78
1534.65
1542.34
1527.68
1534.05
1566.61
1561.44
1575
1596.98
1598
1573.72
1570.34
1594.83
1584.77
1595.31
1586.26
1584.49
1592.74
1628.5
1646.25
1660.82
1672.28
1654.85
1668.28
1683.92
1638.56
1614.94
1606.8
1615.81
1634.98
1630.15
1620.51
1611.83
1624.29
1615.6
1604.81
1614.08
1621.51
1636.5
1620.74
1624.8
1657.03
1670.85
1657.96
1705.92
1713.24
1726.21
1709.57
1654.14
1607.33
1607.12
1673.1
1653.11
1664.25
1655.18
1681.86
1661.53
1631.68
1634.08
1627.94
1642.6
1638.28
1666.68
1692.26
1699.68
1710.52
1705.02
1714.1
1734.14
1745.52
1756.64
1742.65
1741.34
1727.34
1721.2
1690.01
1696.04
1724.46
1714.33
1682.77
1648.12
1676.29
1694.23
1725.92
1726.89
1724.81
1726.9
1715.65
1695.18
1648.87
1657.62
1659.81
1643.2
1629.92
1559.1
1581.93
1605.86
1653.23
1675.01
1672.54
1658.85
1652.76
1643.41
1655.04
1686.23
1736.35
1707.58
1702.55
1732.69
1709.13
1693.95
1659.93
1685.37
1712.4
1696.21
1699.33
1659.39
1650.41
1559.25
1600.44
1616.07
1633.32
1637.08
1661.24
1671.53
1667.52
1648.92
1652.08
1701.48
1706.58
1690.53
1662.28
1651.29
1608.55
1633.2
1630.2
1649.59
1632.67
1598.65
1570.48
1519.51
1513.06
1512.42
1523.47
1520.55
1526.76
1522.03
1521.37
1543.43
1559.31
1592
1585.87
1591.78
1586.04
1587.2
1583.37
1570.06
1541.78
1507.47
1511.71
1539.4
1560.77
1587.7
1572.14
1555.03
1551.28
1575.32
1581.99
1595.2
1595.57
1609.56
1606.79
1596.86
1590.81
1600.45
1614.89
1612.75
1602.74
1579.43
1581.65
1571.57
1539.73
1543.95
1520.03
1499.97
1502.65
1498.15
1473.99
1451.87
1458.54
1481.13
1481.29
1459.08
1451.18
1480.96
1470.74
1483.45
1480.65
1490.79
1495.37
1500.45
1495.64
1477.35
1484.68
1475.39
1495.5
1500.46
1515.25
1531.23
1543.03
1538.87
1516.63
1516.05
1508.4
1498.41
1488.24
1484.09
1489.69
1506.3
1475.32
1474.75
1484.9
1481.05
1478.14
1492.89
1516.21
1510.28
1469.73
1476.03
1470.15
1448.98
1442.95
1468.77
1490.75
1498.71
1512.65
1504.26
1499.13
1477.65
1456.36
1449.99
1443.56
1452.41
1434.36
1408.97
1408.51
1433.14
1423.07
1415.92
1402
1407.74
1400.57
1383.34
1381.59
1416.87
1427.37
1454.97
1445.38
1443.5
1420.45
1401.54
1393.23
1401.35
1403.43
1373.6
1376.68
1360.34
1360.17
1340.77
1348.13
1356.51
1355.93
1342.58
1355.89
1387.83
1392.81
1379.37
1374.31
1364.39
1372.99
1391.92
1391.17
1385.71
1364.32
1344.43
1340.89
1306.83
1276.18
1261.98
1274.63
1292.78
1291.95
1292.14
1289.57
1294.61
1286.53
1284.19
1267
1250.54
1233.81
1244.09
1254.42
1269.73
1287.87
1315.01
1311.45
1307.54
1310.71
1301.94
1294.58
1292.72
1301.34
1299.51
1291.38
1294.87
1276.51
1257.37
1258.92
1263.18
1246.96
1269.88
1254.71
1253.85
1262.02
1239.47
1236.14
1242.97
1235.01
1241.2
1261.73
1266.36
1248.86
1236.16
1222
1198.2
1200.08
1228.06
1235.05
1234.23
1226.94
1193.1
1177.63
1148.36
1188.99
1196.07
1203.37
1269.82
1259.84
1291.45
1327.34
1336.45
1323.57
1296.24
1281.17
1270.71
1285.98
1315
1329.31
1307.56
1280.23
1276.34
1294.71
1307.86
1296.69
1289.34
1293.27
1333.76
1334.88
1334.98
1323.82
1332.94
1354.17
1367.72
1379.5
1418.56
1425.51
1430.53
1404.11
1383.37
1382.43
1375.37
1359.49
1376.41
1370.14
1378.64
1374.96
1392.78
1373.14
1362.48
1364.86
1353.48
1349.77
1345.37
1318.97
1320.31
1315.67
1311.11
1316.02
1302.31
1316.13
1319.17
1300.52
1275.68
1269.32
1260.17
1262.2
1261.9
1253.85
1265.54
1296.9
1304.63
1288.28
1292.67
1289.58
1292.89
1300.07
1295.51
1291.42
1273.46
1263.94
1242.02
1239.4
1237.7
1237.31
1252.29
1247.1
1231.59
1222.67
1197.14
1194.18
1163.53
1176.75
1191.95
1191.27
1189.54
1174.4
1196.84
1216.86
1228.95
1214.55
1209.88
1222.54
1205.31
1184.85
1180.39
1175.39
1194.24
1184.49
1175.94
1161.22
1159.59
1154.83
1169.14
1152.1
1154.86
1155.69
1138.27
1117.01
1103.45
1104.14
1070
1092.32
1084.47
1080.59
1069.82
1058.29
1042.28
994.2
1007.75
1022.57
1004.65
1014.55
1028.72
1012.18
1005.02
1004.6
995.47
1010.65
1026.8
1046.05
1073.64
1060.83
1069.51
1062.27
1041.63
1041.71
1074.03
1076.57
1068.64
1083.02
1077.89
1064.73
1071.24
1083.03
1093.74
1104.46
1113.45
1097.82
1073.36
1061.49
1065.24
1065.75
1056.87
1058.54
1061.94
1040.53
1045.62
1029.8
1016.29
1024.06
1021.78
1044.17
1055.24
1070.76
1051.16
1055.14
1071.4
1086.07
1072.52
1048.26
1050.37
1052.66
1028.85
1035.36
1057.53
1054.04
1041.73
1017.99
1013.78
1029.24
1007.38
1004.05
997.06
990.89
1000.42
1007.85
1007.91
1003.21
996.48
1019.15
1040.02
1051.54
1048.28
1025.63
1022.98
1010.36
991.41
979.15
985.99
989.38
1001.91
1001.4
982.75
978.57
984.86
988.08
993.03
969.68
968.71
975.56
957.8
935.57
924.81
925.01
938.99
961.6
974.34
986.99
983.76
979.07
981.72
953.53
963.56
983.27
992.16
994.24
987.14
1014.48
1018.62
1041.42
1086.78
1072.41
1082.84
1098.45
1116.12
1120.17
1092.01
1076.34
1090.04
1106.24
1097.2
1116.2
1092.56
1077.98
1067.68
1067.2
1080.52
1085.83
1065.14
1060.72
1060.78
1090.27
1090.94
1086.8
1088.97
1069.51
1084.85
1063.18
1056.51
1065.77
1085.6
1088.66
1088.13
1096.56
1098.96
1104.65
1100.52
1098.79
1088.36
1085.52
1089.82
1082.23
1074.65
1092.15
1085.77
1081.81
1082.12
1082.47
1091.56
1111.62
1114.91
1114.52
1114.04
1134.01
1140.35
1138.58
1130.72
1124.5
1122.03
1123.77
1123.7
1129.98
1120.91
1120.45
1099.51
1103.24
1093.54
1096.35
1094.74
1072.99
1047.37
1037.42
1035.42
1010.32
1012.88
993.64
1013.03
1005.63
991.71
1005.3
1029.49
1020.7
1020.14
1041.67
1036.75
1055.48
1039.1
1042.76
1028.6
1024.82
1015.62
994.46
985.61
996.27
1003.49
978.86
984.18
987.41
986.67
964.9
948.04
962.93
972.81
967.14
954.18
950.17
949.2
949.29
963.82
969.06
976.77
952.76
950.45
959.77
966.74
959.48
951.03
940.14
947.57
968.01
959.24
973.98
964.72
946.87
940.57
937.77
925.29
895.09
894.3
894.71
902.99
913.42
918.33
915.02
912.64
916.38
922.98
938.41
936.64
920.91
905.02
909.83
923.54
926.88
913.25
911.92
895.47
879.85
881.14
861.01
856.4
837.57
844.77
858.73
885.85
881.66
892.33
897.4
888.74
857.35
867.76
859.44
867.3
867.27
857.86
876.44
864.76
875.33
886.95
880.21
872.44
855.21
830.74
843.24
834.27
832.01
836.85
820.03
817
837.19
855.81
863.29
876.06
859.28
876.75
888.35
893.81
893.05
883.11
872.81
864.27
858.48
849.31
860.17
852.59
848.9
833.1
833.31
828.06
812.22
823.73
825.15
826.97
833.09
843
841.96
END