Problem 1
Obtain monthly data for federal funds rate FRED/FEDFUNDS and for Moody’s AAA rated corporate bonds FED/RIMLPAAAR_N_M. Monetary transmission mechanism assumes that changes in Fed’s interest rate lead to changes in the interest rates at which the firms can borrow, which implies that these two time series should be cointegrated.
(a) Plot the two time series, in the same graph. Test both series and their first differences for unit root (e.g. using Elliott, Rothenberg and Stock test, or Zivot and Andrews’ test) to verify that they are I(1).
In general, the common null hypothesis for the Zivot-Andrews test is as follows:
\[{H_0}:{y_t} = \mu + {y_{t - 1}} + {\varepsilon _t}\]
This null is testing for the presence of a unit root process with drift that excludes exogenous structural change.
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## # Zivot-Andrews Unit Root / Cointegration Test #
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## The value of the test statistic is: -3.471##
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## # Zivot-Andrews Unit Root / Cointegration Test #
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## The value of the test statistic is: -18.4289By the above results it looks like ffRate is I(1), while differenced ffRate is I(0) as I would expect.
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## ################################################
## # Zivot-Andrews Unit Root / Cointegration Test #
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## The value of the test statistic is: -2.4141##
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## # Zivot-Andrews Unit Root / Cointegration Test #
## ################################################
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## The value of the test statistic is: -24.7895Again, as we see from the above results, mcbAAA is I(1) while differenced mcbAAA is I(0).
(b) Perform trace and max eigenvalue tests to determine whether the two series are cointegrated. Interpret the results.
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## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , with linear trend
##
## Eigenvalues (lambda):
## [1] 0.05651849 0.00623564
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 4.61 6.50 8.18 11.65
## r = 0 | 47.49 15.66 17.95 23.52
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ffRate.l1 mcbAAA.l1
## ffRate.l1 1.000000 1.00000
## mcbAAA.l1 -1.156885 69.15547
##
## Weights W:
## (This is the loading matrix)
##
## ffRate.l1 mcbAAA.l1
## ffRate.d -0.03650693 -1.530127e-04
## mcbAAA.d 0.01594947 -6.812011e-05##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , with linear trend
##
## Eigenvalues (lambda):
## [1] 0.05651849 0.00623564
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 4.61 6.50 8.18 11.65
## r = 0 | 42.88 12.91 14.90 19.19
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ffRate.l1 mcbAAA.l1
## ffRate.l1 1.000000 1.00000
## mcbAAA.l1 -1.156885 69.15547
##
## Weights W:
## (This is the loading matrix)
##
## ffRate.l1 mcbAAA.l1
## ffRate.d -0.03650693 -1.530127e-04
## mcbAAA.d 0.01594947 -6.812011e-05Based on the above results it does appear that these time series are cointegrated with order of integration of I(1). In both cases we can clearly see that out test statistic is far beyond the rejection region at r=0 so we Reject the Null that these time series are not cointegrate but we Fail to Reject the Null at the I(1) order meaning the time series are cointegrated with order I(1).
(c) Perform the test for the presence of a restricted constant rather than unrestricted constant in the model; based on the result of the test rerun the cointegration tests you performed in (b) if necessary.
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## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 5.656992e-02 6.236329e-03 1.476527e-18
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 4.61 7.52 9.24 12.97
## r = 0 | 47.53 17.85 19.96 24.60
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ffRate.l1 mcbAAA.l1 constant
## ffRate.l1 1.000000 1.00000 1.00000
## mcbAAA.l1 -1.156911 66.64856 -1.39404
## constant 3.118329 -471.27749 -54.94366
##
## Weights W:
## (This is the loading matrix)
##
## ffRate.l1 mcbAAA.l1 constant
## ffRate.d -0.03650126 -1.586843e-04 7.825856e-19
## mcbAAA.d 0.01595198 -7.063239e-05 -2.116698e-19##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: maximal eigenvalue statistic (lambda max) , without linear trend and constant in cointegration
##
## Eigenvalues (lambda):
## [1] 5.656992e-02 6.236329e-03 1.476527e-18
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 1 | 4.61 7.52 9.24 12.97
## r = 0 | 42.92 13.75 15.67 20.20
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## ffRate.l1 mcbAAA.l1 constant
## ffRate.l1 1.000000 1.00000 1.00000
## mcbAAA.l1 -1.156911 66.64856 -1.39404
## constant 3.118329 -471.27749 -54.94366
##
## Weights W:
## (This is the loading matrix)
##
## ffRate.l1 mcbAAA.l1 constant
## ffRate.d -0.03650126 -1.586843e-04 7.825856e-19
## mcbAAA.d 0.01595198 -7.063239e-05 -2.116698e-19## LR-test for no linear trend
##
## H0: H*2(r<=1)
## H1: H2(r<=1)
##
## Test statistic is distributed as chi-square
## with 1 degress of freedom
## test statistic p-value
## LR test 0 0.98(d) Test the restriction β2 = −1, explain what the results of the test suggest.
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## ######################
## # Johansen-Procedure #
## ######################
##
## Estimation and testing under linear restrictions on beta
##
## The VECM has been estimated subject to:
## beta=H*phi and/or alpha=A*psi
##
## [,1]
## [1,] 1
## [2,] -1
## [3,] 1
##
## Eigenvalues of restricted VAR (lambda):
## [1] 0.0415
##
## The value of the likelihood ratio test statistic:
## 11.71 distributed as chi square with 2 df.
## The p-value of the test statistic is: 0
##
## Eigenvectors, normalised to first column
## of the restricted VAR:
##
## [,1]
## [1,] 1
## [2,] -1
## [3,] 1
##
## Weights W of the restricted VAR:
##
## [,1]
## ffRate.d -0.0298
## mcbAAA.d 0.0106(e) Create a plot comparing the error term β′yt, showing the deviations from long run equilibrium, in the case where β is unrestricted and the case with restriction β2 = −1. Also add the mean values E(β′yt) into the graph.
## Response ffRate.d :
##
## Call:
## lm(formula = ffRate.d ~ ect1 + ffRate.dl1 + mcbAAA.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.9747 -0.1258 0.0068 0.1299 2.3016
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 -0.036501 0.009084 -4.018 6.47e-05 ***
## ffRate.dl1 0.327370 0.034986 9.357 < 2e-16 ***
## mcbAAA.dl1 0.592007 0.082526 7.174 1.79e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4555 on 734 degrees of freedom
## Multiple R-squared: 0.2137, Adjusted R-squared: 0.2104
## F-statistic: 66.48 on 3 and 734 DF, p-value: < 2.2e-16
##
##
## Response mcbAAA.d :
##
## Call:
## lm(formula = mcbAAA.d ~ ect1 + ffRate.dl1 + mcbAAA.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.16432 -0.08040 -0.00677 0.08703 1.11290
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.015952 0.004016 3.972 7.84e-05 ***
## ffRate.dl1 -0.008876 0.015470 -0.574 0.566
## mcbAAA.dl1 0.319787 0.036490 8.764 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2014 on 734 degrees of freedom
## Multiple R-squared: 0.1288, Adjusted R-squared: 0.1252
## F-statistic: 36.16 on 3 and 734 DF, p-value: < 2.2e-16## Response ffRate.d :
##
## Call:
## lm(formula = ffRate.d ~ ect1 + ffRate.dl1 + mcbAAA.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.0277 -0.1530 -0.0311 0.0941 2.2885
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 -0.029828 0.007917 -3.768 0.000178 ***
## ffRate.dl1 0.322410 0.034893 9.240 < 2e-16 ***
## mcbAAA.dl1 0.588813 0.082601 7.128 2.43e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4561 on 734 degrees of freedom
## Multiple R-squared: 0.2116, Adjusted R-squared: 0.2084
## F-statistic: 65.67 on 3 and 734 DF, p-value: < 2.2e-16
##
##
## Response mcbAAA.d :
##
## Call:
## lm(formula = mcbAAA.d ~ ect1 + ffRate.dl1 + mcbAAA.dl1 - 1, data = data.mat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.16237 -0.06712 0.00964 0.09822 1.13746
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## ect1 0.010645 0.003512 3.031 0.00252 **
## ffRate.dl1 -0.005324 0.015477 -0.344 0.73094
## mcbAAA.dl1 0.323001 0.036637 8.816 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2023 on 734 degrees of freedom
## Multiple R-squared: 0.1211, Adjusted R-squared: 0.1175
## F-statistic: 33.7 on 3 and 734 DF, p-value: < 2.2e-16
