Homework 5
Problem 1
(a) Bivariate Reduced Form VAR
Defining all the variables needed in this problem and then differentiate them to produce \(y_{1,t}\) and \(y_{2,t}\)
However for SP500 Data, since the data is daily, we need to change them into quarterly data
Estimating the VAR model from 1951Q1 to 2014Q4, looking at the smallest AIC, the proper lags will be equal to two
## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 1 2
##
## $criteria
## 1 2 3 4
## AIC(n) -1.465126e+01 -1.470486e+01 -1.469229e+01 -1.466603e+01
## HQ(n) -1.461704e+01 -1.464783e+01 -1.461245e+01 -1.456337e+01
## SC(n) -1.456626e+01 -1.456319e+01 -1.449395e+01 -1.441102e+01
## FPE(n) 4.335513e-07 4.109277e-07 4.161332e-07 4.272205e-07
## 5 6 7 8
## AIC(n) -1.464883e+01 -1.462725e+01 -1.461845e+01 -1.459013e+01
## HQ(n) -1.452337e+01 -1.447897e+01 -1.444736e+01 -1.439622e+01
## SC(n) -1.433716e+01 -1.425891e+01 -1.419344e+01 -1.410845e+01
## FPE(n) 4.346529e-07 4.441705e-07 4.481426e-07 4.610779e-07##
## VAR Estimation Results:
## =========================
## Endogenous variables: y1, y2
## Deterministic variables: const
## Sample size: 254
## Log Likelihood: 1154.493
## Roots of the characteristic polynomial:
## 0.4454 0.4454 0.304 0.304
## Call:
## VAR(y = y.z, p = 2, type = "const")
##
##
## Estimation results for equation y1:
## ===================================
## y1 = y1.l1 + y2.l1 + y1.l2 + y2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.2450402 0.0614771 3.986 8.84e-05 ***
## y2.l1 0.0230968 0.0064046 3.606 0.000375 ***
## y1.l2 0.0793493 0.0595434 1.333 0.183872
## y2.l2 0.0263004 0.0065720 4.002 8.29e-05 ***
## const 0.0046278 0.0007233 6.398 7.75e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.008038 on 249 degrees of freedom
## Multiple R-Squared: 0.2309, Adjusted R-squared: 0.2185
## F-statistic: 18.68 on 4 and 249 DF, p-value: 1.91e-13
##
##
## Estimation results for equation y2:
## ===================================
## y2 = y1.l1 + y2.l1 + y1.l2 + y2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.256054 0.609111 0.420 0.6746
## y2.l1 0.106501 0.063457 1.678 0.0945 .
## y1.l2 -0.819353 0.589953 -1.389 0.1661
## y2.l2 -0.040573 0.065115 -0.623 0.5338
## const 0.013836 0.007167 1.931 0.0547 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.07964 on 249 degrees of freedom
## Multiple R-Squared: 0.02087, Adjusted R-squared: 0.00514
## F-statistic: 1.327 on 4 and 249 DF, p-value: 0.2605
##
##
##
## Covariance matrix of residuals:
## y1 y2
## y1 6.46e-05 0.000087
## y2 8.70e-05 0.006342
##
## Correlation matrix of residuals:
## y1 y2
## y1 1.0000 0.1359
## y2 0.1359 1.0000(b) Granger Causality Test
To further test about the instantaneous causality existence, Granger Causality Test for both variable is conducted below:
## $Granger
##
## Granger causality H0: y1 do not Granger-cause y2
##
## data: VAR object varp
## F-Test = 0.96654, df1 = 2, df2 = 498, p-value = 0.3811
##
##
## $Instant
##
## H0: No instantaneous causality between: y1 and y2
##
## data: VAR object varp
## Chi-squared = 4.6072, df = 1, p-value = 0.03184## $Granger
##
## Granger causality H0: y2 do not Granger-cause y1
##
## data: VAR object varp
## F-Test = 15.762, df1 = 2, df2 = 498, p-value = 2.305e-07
##
##
## $Instant
##
## H0: No instantaneous causality between: y2 and y1
##
## data: VAR object varp
## Chi-squared = 4.6072, df = 1, p-value = 0.03184There is no indication of instantaneous causality, the predicting power is low for both variables.
(C) Restricted VAR
First, estimating the restricted VAR model that eliminates the lags
##
## VAR Estimation Results:
## =========================
## Endogenous variables: y1, y2
## Deterministic variables: const
## Sample size: 254
## Log Likelihood: 1139.086
## Roots of the characteristic polynomial:
## 0.4867 0.2014 0.2014 0.1565
## Call:
## VAR(y = y.z, p = 2, type = "const")
##
##
## Estimation results for equation y1:
## ===================================
## y1 = y1.l1 + y1.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.3301391 0.0628607 5.252 3.21e-07 ***
## y1.l2 0.0761755 0.0628199 1.213 0.226
## const 0.0044907 0.0007616 5.896 1.19e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.008497 on 251 degrees of freedom
## Multiple R-Squared: 0.4906, Adjusted R-squared: 0.4845
## F-statistic: 80.59 on 3 and 251 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation y2:
## ===================================
## y2 = y1.l1 + y2.l1 + y1.l2 + y2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.256054 0.609111 0.420 0.6746
## y2.l1 0.106501 0.063457 1.678 0.0945 .
## y1.l2 -0.819353 0.589953 -1.389 0.1661
## y2.l2 -0.040573 0.065115 -0.623 0.5338
## const 0.013836 0.007167 1.931 0.0547 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.07964 on 249 degrees of freedom
## Multiple R-Squared: 0.03656, Adjusted R-squared: 0.01721
## F-statistic: 1.89 on 5 and 249 DF, p-value: 0.09665
##
##
##
## Covariance matrix of residuals:
## y1 y2
## y1 7.278e-05 0.000087
## y2 8.700e-05 0.006342
##
## Correlation matrix of residuals:
## y1 y2
## y1 1.0000 0.1281
## y2 0.1281 1.0000## y1.l1 y2.l1 y1.l2 y2.l2 const
## y1 1 0 1 0 1
## y2 1 1 1 1 1## [[1]]
## y1.l1 y2.l1
## y1 0.3301391 0.0000000
## y2 0.2560542 0.1065012
##
## [[2]]
## y1.l2 y2.l2
## y1 0.07617549 0.00000000
## y2 -0.81935297 -0.04057338Second, estimating restricted VAR model eliminating all variables with t-statistics less than 2
##
## VAR Estimation Results:
## =========================
## Endogenous variables: y1, y2
## Deterministic variables: const
## Sample size: 254
## Log Likelihood: 1150.599
## Roots of the characteristic polynomial:
## 0.2734 0 0 0
## Call:
## VAR(y = y.z, p = 2, type = "const")
##
##
## Estimation results for equation y1:
## ===================================
## y1 = y1.l1 + y2.l1 + y2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.2734178 0.0577607 4.734 3.70e-06 ***
## y2.l1 0.0226323 0.0064051 3.533 0.000488 ***
## y2.l2 0.0266285 0.0065775 4.048 6.88e-05 ***
## const 0.0050198 0.0006618 7.585 6.55e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.00805 on 250 degrees of freedom
## Multiple R-Squared: 0.5446, Adjusted R-squared: 0.5374
## F-statistic: 74.76 on 4 and 250 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation y2:
## ===================================
## y2 = const
##
## Estimate Std. Error t value Pr(>|t|)
## const 0.01017 0.00501 2.03 0.0434 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.07984 on 253 degrees of freedom
## Multiple R-Squared: 0.01602, Adjusted R-squared: 0.01213
## F-statistic: 4.119 on 1 and 253 DF, p-value: 0.04344
##
##
##
## Covariance matrix of residuals:
## y1 y2
## y1 6.506e-05 8.224e-05
## y2 8.224e-05 6.477e-03
##
## Correlation matrix of residuals:
## y1 y2
## y1 1.0000 0.1267
## y2 0.1267 1.0000## y1.l1 y2.l1 y1.l2 y2.l2 const
## y1 1 1 0 1 1
## y2 0 0 0 0 1## [[1]]
## y1.l1 y2.l1
## y1 0.2734178 0.02263225
## y2 0.0000000 0.00000000
##
## [[2]]
## y1.l2 y2.l2
## y1 0 0.02662845
## y2 0 0.00000000(d) StargazerX
First, loading the source:
The ouput for all three models are
##
## =======================================================================================
## y
## VARP Y1 VARP Y2 VARPY1
## (1) (2) (3)
## ---------------------------------------------------------------------------------------
## y1.l1 0.245*** 0.256 0.330***
## (0.061) (0.609) (0.063)
##
## y2.l1 0.023*** 0.107*
## (0.006) (0.063)
##
## y1.l2 0.079 -0.819 0.076
## (0.060) (0.590) (0.063)
##
## y2.l2 0.026*** -0.041
## (0.007) (0.065)
##
## const 0.005*** 0.014* 0.004***
## (0.001) (0.007) (0.001)
##
## ---------------------------------------------------------------------------------------
## Observations 254 254 254
## R2 0.231 0.021 0.491
## Adjusted R2 0.219 0.005 0.485
## Residual Std. Error 0.008 (df = 249) 0.080 (df = 249) 0.008 (df = 251)
## F Statistic 18.685*** (df = 4; 249) 1.327 (df = 4; 249) 80.588*** (df = 3; 251)
## =======================================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01
##
## ====
## asis
## ----##
## ======================================================================================
## y
## VARPY2 VARPY1 VARPY2
## (1) (2) (3)
## --------------------------------------------------------------------------------------
## y1.l1 0.256 0.273***
## (0.609) (0.058)
##
## y2.l1 0.107* 0.023***
## (0.063) (0.006)
##
## y1.l2 -0.819
## (0.590)
##
## y2.l2 -0.041 0.027***
## (0.065) (0.007)
##
## const 0.014* 0.005*** 0.010**
## (0.007) (0.001) (0.005)
##
## --------------------------------------------------------------------------------------
## Observations 254 254 254
## R2 0.037 0.545 0.016
## Adjusted R2 0.017 0.537 0.012
## Residual Std. Error 0.080 (df = 249) 0.008 (df = 250) 0.080 (df = 253)
## F Statistic 1.890* (df = 5; 249) 74.756*** (df = 4; 250) 4.119** (df = 1; 253)
## ======================================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01
##
## ====
## asis
## ----(e) Impulse Response Function
First we have to plot the IRF and FEVD of the VAR model based on Choleski Decomposition 
And the FEVD
## y1 y2
## [1,] 1.0000000 0.0000000
## [2,] 0.8607464 0.1392536
## [3,] 0.8591922 0.1408078
## [4,] 0.8591882 0.1408118## y1 y2
## [1,] 0.01847382 0.9815262
## [2,] 0.02623100 0.9737690
## [3,] 0.02642786 0.9735721
## [4,] 0.02642813 0.9735719
Now if we reversed the order of the variable, there are changes in the result showing that order does matter.
## y2 y1
## [1,] 1.0000000 0.000000000
## [2,] 0.9935239 0.006476147
## [3,] 0.9934082 0.006591832
## [4,] 0.9934079 0.006592058## y2 y1
## [1,] 0.01847382 0.9815262
## [2,] 0.18059627 0.8194037
## [3,] 0.18207471 0.8179253
## [4,] 0.18207906 0.8179209