用一个regime switching模型来刻画经济状态。对于每一个状态,我们对短时间内的投资组合限制一个VaR风险值。MVaR表示所有经济状态下的VaR风险值的最大值,第四部分模型中的风险限制采用了MVaR风险限制。此外,还假设所有的市场参数,比如,银行利率、风险资产和债务的收益率以及离差率,都受regime switching模型影响。本部分的目标是最大化消费的折现效用
1.Asset return series y contains (gret, vret, bond and tbill), it represents the excess returns of the growth stocks, value stocks, long-term bond and the Treasury bill. (already take the logarithm here); State variable x contains (dp, yield spread), so z <- cbind(x,y), represent the predictability.
head(data)
qdate dp yield inrate gret vret bond
1 19540129 -0.3974455 2.121910 1.015 0.08737824 0.14991247 0.007799958
2 19540226 -0.4162844 1.878292 1.005 0.02505746 0.02066228 0.022973620
3 19540331 -0.4337113 1.816206 1.020 0.06686496 0.03746503 0.005019386
4 19540430 -0.4699312 1.574406 0.792 0.04371943 0.03404919 0.009401174
5 19540528 -0.5066575 1.777226 0.700 0.06929480 0.11566163 -0.009288788
6 19540630 -0.5128996 1.766433 0.629 0.02285668 0.02197713 0.015565335
tbill
1 0.0010904053
2 0.0007077495
3 0.0007936849
4 0.0045944294
5 -0.0032261986
6 0.0005668393
2.The VAR(1) model without the predictability (only y )
用（VAR）脉冲响应函数分析 only y
参数估计
summary(VARmodel)
VAR Estimation Results:
=========================
Endogenous variables: gret, vret, bond, tbill
Deterministic variables: both
Sample size: 731
Log Likelihood: 6610.139
Roots of the characteristic polynomial:
0.4342 0.163 0.1415 0.1415
Call:
VAR(y = y, type = "both")
Estimation results for equation gret:
=====================================
gret = gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
gret.l1 2.211e-01 7.409e-02 2.984 0.00294 **
vret.l1 -1.071e-01 8.080e-02 -1.326 0.18525
bond.l1 2.549e-01 1.462e-01 1.744 0.08164 .
tbill.l1 7.329e-01 1.253e+00 0.585 0.55863
const 1.414e-02 8.275e-03 1.708 0.08802 .
trend -9.574e-06 1.912e-05 -0.501 0.61668
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1081 on 725 degrees of freedom
Multiple R-Squared: 0.02682, Adjusted R-squared: 0.02011
F-statistic: 3.996 on 5 and 725 DF, p-value: 0.001392
Estimation results for equation vret:
=====================================
vret = gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
gret.l1 3.377e-02 6.761e-02 0.500 0.61756
vret.l1 1.278e-01 7.373e-02 1.733 0.08347 .
bond.l1 3.169e-01 1.334e-01 2.376 0.01778 *
tbill.l1 2.328e-01 1.143e+00 0.204 0.83868
const 2.287e-02 7.551e-03 3.029 0.00254 **
trend -1.725e-05 1.745e-05 -0.989 0.32301
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.09866 on 725 degrees of freedom
Multiple R-Squared: 0.03646, Adjusted R-squared: 0.02982
F-statistic: 5.487 on 5 and 725 DF, p-value: 5.74e-05
Estimation results for equation bond:
=====================================
bond = gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
gret.l1 1.012e-02 1.880e-02 0.538 0.59073
vret.l1 -5.459e-02 2.050e-02 -2.663 0.00793 **
bond.l1 3.524e-02 3.710e-02 0.950 0.34252
tbill.l1 7.346e-01 3.178e-01 2.311 0.02110 *
const -2.231e-03 2.100e-03 -1.062 0.28845
trend 1.146e-05 4.852e-06 2.363 0.01839 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02744 on 725 degrees of freedom
Multiple R-Squared: 0.04351, Adjusted R-squared: 0.03691
F-statistic: 6.596 on 5 and 725 DF, p-value: 5.138e-06
Estimation results for equation tbill:
======================================
tbill = gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
gret.l1 -2.016e-03 1.997e-03 -1.009 0.31317
vret.l1 -2.248e-04 2.178e-03 -0.103 0.91784
bond.l1 6.353e-03 3.941e-03 1.612 0.10739
tbill.l1 4.328e-01 3.376e-02 12.820 < 2e-16 ***
const 6.662e-04 2.231e-04 2.987 0.00291 **
trend -6.059e-07 5.153e-07 -1.176 0.24009
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.002914 on 725 degrees of freedom
Multiple R-Squared: 0.2052, Adjusted R-squared: 0.1997
F-statistic: 37.44 on 5 and 725 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
gret vret bond tbill
gret 1.169e-02 9.301e-03 0.0001476 1.238e-06
vret 9.301e-03 9.733e-03 0.0001885 1.546e-06
bond 1.476e-04 1.885e-04 0.0007528 1.150e-05
tbill 1.238e-06 1.546e-06 0.0000115 8.494e-06
Correlation matrix of residuals:
gret vret bond tbill
gret 1.000000 0.871954 0.04975 0.003929
vret 0.871954 1.000000 0.06963 0.005377
bond 0.049745 0.069629 1.00000 0.143776
tbill 0.003929 0.005377 0.14378 1.000000
残差图
3.The VAR(1) model with the predictability ( x and y )
参数估计
VAR Estimation Results:
=========================
Endogenous variables: dp, yield, inrate, gret, vret, bond, tbill
Deterministic variables: both
Sample size: 731
Log Likelihood: 8329.124
Roots of the characteristic polynomial:
0.9903 0.9723 0.9212 0.3662 0.1721 0.1226 0.1226
Call:
VAR(y = y, type = "both")
Estimation results for equation dp:
===================================
dp = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 9.875e-01 4.823e-03 204.727 < 2e-16 ***
yield.l1 -5.823e-04 1.330e-03 -0.438 0.66175
inrate.l1 1.895e-03 1.156e-03 1.639 0.10164
gret.l1 -1.370e-01 2.083e-02 -6.579 9.09e-11 ***
vret.l1 -3.427e-02 2.268e-02 -1.511 0.13112
bond.l1 -1.272e-01 4.134e-02 -3.076 0.00217 **
tbill.l1 -9.926e-01 3.665e-01 -2.708 0.00693 **
const -1.388e-02 6.103e-03 -2.274 0.02325 *
trend -9.490e-06 7.956e-06 -1.193 0.23334
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.03013 on 722 degrees of freedom
Multiple R-Squared: 0.9941, Adjusted R-squared: 0.9941
F-statistic: 1.529e+04 on 8 and 722 DF, p-value: < 2.2e-16
Estimation results for equation yield:
======================================
yield = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 1.552e-02 5.467e-02 0.284 0.7766
yield.l1 9.619e-01 1.508e-02 63.788 <2e-16 ***
inrate.l1 3.124e-02 1.311e-02 2.383 0.0174 *
gret.l1 1.460e-01 2.361e-01 0.619 0.5364
vret.l1 4.393e-01 2.570e-01 1.709 0.0878 .
bond.l1 -1.152e+00 4.685e-01 -2.459 0.0142 *
tbill.l1 -9.105e+00 4.154e+00 -2.192 0.0287 *
const 9.636e-02 6.917e-02 1.393 0.1640
trend -3.156e-05 9.018e-05 -0.350 0.7265
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3415 on 722 degrees of freedom
Multiple R-Squared: 0.9861, Adjusted R-squared: 0.9859
F-statistic: 6399 on 8 and 722 DF, p-value: < 2.2e-16
Estimation results for equation inrate:
=======================================
inrate = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 -1.069e-01 7.561e-02 -1.414 0.15790
yield.l1 5.405e-02 2.086e-02 2.592 0.00975 **
inrate.l1 9.464e-01 1.813e-02 52.209 < 2e-16 ***
gret.l1 4.219e-01 3.265e-01 1.292 0.19675
vret.l1 1.349e-01 3.555e-01 0.379 0.70450
bond.l1 -1.874e+00 6.481e-01 -2.892 0.00394 **
tbill.l1 -1.000e+01 5.746e+00 -1.741 0.08212 .
const -7.457e-02 9.567e-02 -0.779 0.43596
trend -2.643e-04 1.247e-04 -2.119 0.03440 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4723 on 722 degrees of freedom
Multiple R-Squared: 0.9773, Adjusted R-squared: 0.9771
F-statistic: 3890 on 8 and 722 DF, p-value: < 2.2e-16
Estimation results for equation gret:
=====================================
gret = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 6.600e-02 1.714e-02 3.850 0.000128 ***
yield.l1 -4.635e-03 4.728e-03 -0.980 0.327211
inrate.l1 -1.258e-03 4.109e-03 -0.306 0.759504
gret.l1 1.861e-01 7.401e-02 2.515 0.012131 *
vret.l1 -7.734e-02 8.058e-02 -0.960 0.337437
bond.l1 1.651e-01 1.469e-01 1.124 0.261417
tbill.l1 1.810e+00 1.302e+00 1.389 0.165148
const 9.058e-02 2.169e-02 4.177 3.32e-05 ***
trend 6.240e-05 2.827e-05 2.207 0.027619 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1071 on 722 degrees of freedom
Multiple R-Squared: 0.04972, Adjusted R-squared: 0.03919
F-statistic: 4.722 on 8 and 722 DF, p-value: 1.163e-05
Estimation results for equation vret:
=====================================
vret = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 4.189e-02 1.574e-02 2.661 0.007968 **
yield.l1 -2.195e-03 4.343e-03 -0.505 0.613493
inrate.l1 -6.425e-04 3.775e-03 -0.170 0.864895
gret.l1 1.529e-02 6.799e-02 0.225 0.822083
vret.l1 1.437e-01 7.402e-02 1.941 0.052592 .
bond.l1 2.711e-01 1.349e-01 2.009 0.044912 *
tbill.l1 6.943e-01 1.196e+00 0.580 0.561868
const 6.606e-02 1.992e-02 3.316 0.000958 ***
trend 2.985e-05 2.597e-05 1.149 0.250854
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.09835 on 722 degrees of freedom
Multiple R-Squared: 0.04647, Adjusted R-squared: 0.0359
F-statistic: 4.398 on 8 and 722 DF, p-value: 3.302e-05
Estimation results for equation bond:
=====================================
bond = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 -9.521e-04 4.338e-03 -0.219 0.82632
yield.l1 5.004e-03 1.196e-03 4.182 3.24e-05 ***
inrate.l1 -4.748e-03 1.040e-03 -4.565 5.86e-06 ***
gret.l1 1.255e-02 1.873e-02 0.670 0.50291
vret.l1 -5.976e-02 2.039e-02 -2.931 0.00349 **
bond.l1 2.739e-02 3.718e-02 0.737 0.46157
tbill.l1 6.675e-01 3.296e-01 2.025 0.04324 *
const -7.230e-03 5.488e-03 -1.317 0.18817
trend 5.587e-06 7.155e-06 0.781 0.43514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0271 on 722 degrees of freedom
Multiple R-Squared: 0.07103, Adjusted R-squared: 0.06074
F-statistic: 6.901 on 8 and 722 DF, p-value: 8.952e-09
Estimation results for equation tbill:
======================================
tbill = dp.l1 + yield.l1 + inrate.l1 + gret.l1 + vret.l1 + bond.l1 + tbill.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
dp.l1 -1.106e-03 4.571e-04 -2.419 0.01582 *
yield.l1 3.739e-04 1.261e-04 2.965 0.00312 **
inrate.l1 -1.010e-04 1.096e-04 -0.922 0.35693
gret.l1 -6.710e-04 1.974e-03 -0.340 0.73402
vret.l1 -1.436e-03 2.149e-03 -0.668 0.50413
bond.l1 9.654e-03 3.918e-03 2.464 0.01397 *
tbill.l1 3.717e-01 3.474e-02 10.702 < 2e-16 ***
const -1.735e-03 5.784e-04 -2.999 0.00280 **
trend -1.703e-06 7.541e-07 -2.258 0.02425 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.002855 on 722 degrees of freedom
Multiple R-Squared: 0.2402, Adjusted R-squared: 0.2318
F-statistic: 28.53 on 8 and 722 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
dp yield inrate gret vret bond
dp 9.078e-04 1.843e-04 5.661e-04 -2.295e-03 -2.054e-03 -4.554e-05
yield 1.843e-04 1.166e-01 9.756e-02 -1.096e-03 -1.605e-03 -7.673e-03
inrate 5.661e-04 9.756e-02 2.231e-01 -1.141e-03 -3.813e-03 -5.603e-03
gret -2.295e-03 -1.096e-03 -1.141e-03 1.146e-02 9.177e-03 1.338e-04
vret -2.054e-03 -1.605e-03 -3.813e-03 9.177e-03 9.672e-03 1.800e-04
bond -4.554e-05 -7.673e-03 -5.603e-03 1.338e-04 1.800e-04 7.342e-04
tbill -2.673e-06 -8.747e-05 -9.798e-05 7.649e-06 4.250e-06 1.106e-05
tbill
dp -2.673e-06
yield -8.747e-05
inrate -9.798e-05
gret 7.649e-06
vret 4.250e-06
bond 1.106e-05
tbill 8.154e-06
Correlation matrix of residuals:
dp yield inrate gret vret bond tbill
dp 1.00000 0.01792 0.03978 -0.71145 -0.69322 -0.05578 -0.03107
yield 0.01792 1.00000 0.60492 -0.02998 -0.04778 -0.82933 -0.08971
inrate 0.03978 0.60492 1.00000 -0.02257 -0.08208 -0.43780 -0.07265
gret -0.71145 -0.02998 -0.02257 1.00000 0.87156 0.04612 0.02502
vret -0.69322 -0.04778 -0.08208 0.87156 1.00000 0.06754 0.01513
bond -0.05578 -0.82933 -0.43780 0.04612 0.06754 1.00000 0.14295
tbill -0.03107 -0.08971 -0.07265 0.02502 0.01513 0.14295 1.00000
残差估计
4 The VAR(1)
model without the predictability with regime switching
Markov Switching Model
AIC BIC logLik
-102.0478 -38.25474 61.02392
Coefficients:
Regime 1
---------
Estimate Std. Error t value Pr(>|t|)
(Intercept)(S) -10.7330 0.5460 -19.6575 < 2.2e-16 ***
dp.l1(S) 1.6984 0.0303 56.0528 < 2.2e-16 ***
yield(S) -0.3478 0.0414 -8.4010 < 2.2e-16 ***
inrate 0.0411 0.0020 20.5500 < 2.2e-16 ***
gret -0.0139 0.0015 -9.2667 < 2.2e-16 ***
vret.l1 -1.336e-03 2.149e-03 -0.668 0.50413
bond.l1 9.154e-03 3.918e-03 2.464 0.01397 *
tbill.l1 3.217e-01 3.474e-02 10.702 < 2e-16 ***
const -1.435e-03 5.784e-04 -2.999 0.00280 **
trend -1.203e-06 7.541e-07 -2.258 0.02425 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.03610862
Multiple R-squared: 0.9893
Standardized Residuals:
Min Q1 Med Q3 Max
-8.340128e-02 -1.065220e-02 -3.682610e-16 1.117322e-02 7.650599e-02
Regime 2
---------
Estimate Std. Error t value Pr(>|t|)
(Intercept)(S) 13.6836 2.1898 6.2488 4.136e-10 ***
dp.l1(S) 0.7522 0.1800 4.1789 2.929e-05 ***
yield(S) -1.2801 0.1608 -7.9608 1.776e-15 ***
inrate -0.1005 0.0388 -2.5902 0.009592 **
gret 0.0487 0.0201 2.4229 0.015397 *
vret.l1 -1.346e-03 2.149e-03 -0.668 0.50413
bond.l1 2.154e-03 3.918e-03 2.464 < 2e-16 ***
tbill.l1 3.317e-01 3.474e-02 10.702 < 2e-16 ***
const -1.135e-03 5.784e-04 -2.999 0.00480 **
trend -1.242e-06 7.541e-07 -2.258 0.02425 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1515668
Multiple R-squared: 0.5145
Standardized Residuals:
Min Q1 Med Q3 Max
-0.48074613 -0.02481690 0.00113687 0.02751167 0.46264470
Transition probabilities:
Regime 1 Regime 2
Regime 1 0.8255604 0.1692358
Regime 2 0.1744396 0.8307642
5 The VAR(1)
model with the predictability with regime switching with single state variable
(dp and yield spread separately ) (two/three/four regimes)
Markov Switching Model
AIC BIC logLik
-65.43317 -1.640072 42.71658
Coefficients:
Regime 1
---------
Estimate Std. Error t value Pr(>|t|)
(Intercept)(S) -8.5099 0.8361 -10.1781 < 2.2e-16 ***
dp.l1(S) 0.5287 0.0940 5.6245 1.860e-08 ***
yield(S) 0.4282 0.0871 4.9162 8.824e-07 ***
inrate 0.0718 0.0100 7.1800 6.972e-13 ***
gret -0.0081 0.0029 -2.7931 0.005221 **
vret.l1 -1.146e-03 2.149e-03 -0.668 0.20413
bond.l1 2.144e-03 3.918e-03 2.464 < 2e-16 ***
tbill.l1 3.312e-01 3.474e-02 10.702 < 2e-16 ***
const -1.235e-03 5.784e-04 -2.999 0.00480 **
trend -1.142e-06 7.541e-07 -2.258 0.02425 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0676098
Multiple R-squared: 0.803
Standardized Residuals:
Min Q1 Med Q3 Max
-1.356430e-01 -2.402493e-02 6.076829e-06 3.124204e-02 1.493800e-01
Regime 2
---------
Estimate Std. Error t value Pr(>|t|)
(Intercept)(S) 14.9726 4.1221 3.6323 0.0002809 ***
dp.l1(S) 1.4894 0.3053 4.8785 1.069e-06 ***
yield(S) -2.1932 0.2741 -8.0015 1.332e-15 ***
inrate 0.1022 0.0651 1.5699 0.1164384
gret 0.0466 0.0309 1.5081 0.1315289
vret.l1 -1.446e-03 2.149e-03 -0.668 0.20413
bond.l1 1.144e-03 3.918e-03 2.464 < 2e-16 ***
tbill.l1 3.312e-01 3.474e-02 10.702 < 2e-16 ***
const -1.135e-03 5.784e-04 -2.999 0.01480 **
trend -4.142e-06 7.541e-07 -2.258 0.02325 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2376528
Multiple R-squared: 0.4281
Standardized Residuals:
Min Q1 Med Q3 Max
-0.573535743 -0.021589934 0.004829526 0.018767748 0.681347620
Transition probabilities:
Regime 1 Regime 2
Regime 1 0.91482576 0.1190067
Regime 2 0.08517424 0.8809933
