Problem 1
Use Zivot-Andrews test to analyze the presence of a unit root and structural breaks in the following series:
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.
Depending on the model variant, the alternative hypothesis is a trend stationary process that allows for a one time break in the level, the trend or both.
The following three models represent the alternative hypothesis for the Zivot-Andrews test:
\[\eqalign{
& {\rm{model A: }}{H_A}:{y_t} = {\mu _1} + ({\mu _2} - {\mu _1})D{U_t} + \beta t + {\varepsilon _t} \cr
& {\rm{model B: }}{H_A}:{y_t} = \mu + {\beta _1}t + ({\beta _2} - {\beta _1})D{T_t}(\hat \lambda ) + {\varepsilon _t} \cr
& {\rm{model C: }}{H_A}:{y_t} = {\mu _1} + ({\mu _2} - {\mu _1})D{U_t}(\hat \lambda ) + {\beta _1}t + {\beta _1}t + ({\beta _2} - {\beta _1})D{T_t}(\hat \lambda ) + {\varepsilon _t} \cr} \]
We will use this model to test various time series for the presence of a one time break in the level, the trend or both.
(a) Log transformed Retail and Food Services Sales FRED/RSAFS
# Model A
za_lnRSAFS_all.1 <- ur.za(lnRSAFS_all, model="intercept", lag=10)
summary(za_lnRSAFS_all.1)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.030923 -0.004594 0.000230 0.005347 0.056017
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.698e+00 2.769e-01 6.133 3.14e-09 ***
## y.l1 8.594e-01 2.298e-02 37.400 < 2e-16 ***
## trend 5.728e-04 9.561e-05 5.991 6.80e-09 ***
## y.dl1 -1.001e-01 5.756e-02 -1.739 0.0833 .
## y.dl2 1.414e-02 5.744e-02 0.246 0.8057
## y.dl3 2.180e-02 5.731e-02 0.380 0.7040
## y.dl4 7.519e-02 5.733e-02 1.312 0.1908
## y.dl5 3.753e-03 5.729e-02 0.066 0.9478
## y.dl6 9.203e-02 5.727e-02 1.607 0.1092
## y.dl7 2.125e-02 5.747e-02 0.370 0.7119
## y.dl8 -5.253e-02 5.750e-02 -0.914 0.3618
## y.dl9 1.307e-01 5.759e-02 2.270 0.0240 *
## y.dl10 7.355e-02 5.774e-02 1.274 0.2039
## du -2.615e-02 4.648e-03 -5.626 4.71e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.009217 on 264 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.9989, Adjusted R-squared: 0.9988
## F-statistic: 1.787e+04 on 13 and 264 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -6.1179
## Critical values: 0.01= -5.34 0.05= -4.8 0.1= -4.58
##
## Potential break point at position: 199Acf(za_lnRSAFS_all.1@res, type="correlation", lag.max = 24)
plot(za_lnRSAFS_all.1)
#Model B
za_lnRSAFS_all.2 <- ur.za(lnRSAFS_all, model="trend", lag=10)
summary(za_lnRSAFS_all.2)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.040601 -0.004724 0.000413 0.005441 0.056928
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.603e-01 2.262e-01 3.361 0.000891 ***
## y.l1 9.371e-01 1.885e-02 49.725 < 2e-16 ***
## trend 2.648e-04 8.659e-05 3.058 0.002457 **
## y.dl1 -8.784e-02 6.064e-02 -1.449 0.148658
## y.dl2 3.489e-02 6.055e-02 0.576 0.564959
## y.dl3 3.983e-02 6.037e-02 0.660 0.509948
## y.dl4 8.946e-02 6.042e-02 1.481 0.139896
## y.dl5 1.713e-02 6.058e-02 0.283 0.777620
## y.dl6 1.194e-01 6.054e-02 1.973 0.049583 *
## y.dl7 4.541e-02 6.096e-02 0.745 0.456979
## y.dl8 -1.890e-02 6.102e-02 -0.310 0.756990
## y.dl9 1.599e-01 6.100e-02 2.621 0.009288 **
## y.dl10 8.836e-02 6.115e-02 1.445 0.149667
## dt -1.275e-04 5.176e-05 -2.464 0.014373 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.009644 on 264 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.9988, Adjusted R-squared: 0.9987
## F-statistic: 1.632e+04 on 13 and 264 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -3.3389
## Critical values: 0.01= -4.93 0.05= -4.42 0.1= -4.11
##
## Potential break point at position: 147Acf(za_lnRSAFS_all.2@res, type="correlation", lag.max = 24)
plot(za_lnRSAFS_all.2)
# Model C
za_lnRSAFS_all.3 <- ur.za(lnRSAFS_all, model="both", lag=10)
summary(za_lnRSAFS_all.3)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.031223 -0.004365 0.000117 0.005373 0.056227
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.867e+00 3.135e-01 5.954 8.35e-09 ***
## y.l1 8.453e-01 2.605e-02 32.445 < 2e-16 ***
## trend 6.369e-04 1.108e-04 5.749 2.48e-08 ***
## y.dl1 -8.998e-02 5.820e-02 -1.546 0.1233
## y.dl2 2.512e-02 5.820e-02 0.432 0.6664
## y.dl3 3.219e-02 5.799e-02 0.555 0.5793
## y.dl4 8.588e-02 5.805e-02 1.479 0.1402
## y.dl5 1.580e-02 5.821e-02 0.271 0.7863
## y.dl6 1.036e-01 5.813e-02 1.783 0.0758 .
## y.dl7 3.348e-02 5.843e-02 0.573 0.5672
## y.dl8 -4.038e-02 5.844e-02 -0.691 0.4902
## y.dl9 1.422e-01 5.843e-02 2.434 0.0156 *
## y.dl10 8.439e-02 5.848e-02 1.443 0.1502
## du -2.675e-02 4.675e-03 -5.722 2.86e-08 ***
## dt -5.584e-05 4.884e-05 -1.143 0.2539
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.009212 on 263 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.9989, Adjusted R-squared: 0.9988
## F-statistic: 1.661e+04 on 14 and 263 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -5.9357
## Critical values: 0.01= -5.57 0.05= -5.08 0.1= -4.82
##
## Potential break point at position: 199Acf(za_lnRSAFS_all.3@res, type="correlation", lag.max = 24)
plot(za_lnRSAFS_all.3)
From the above results we can see that we Reject the Null and it is reasonable to assume we have a change in the intercept but not a change in the trend.
By plotting the t-statistics for the lagged endogenous variable we can more plainly see that we accept a change in level but reject a change in trend.
The above plot shows a plot of the Log Transformed data with an indicator where the Zivot-Andrews test tells us we have a structural break in the level, which seems correct by viewing the data plot.
(b) Log transformed Personal consumption expenditures: Nondurable goods (chain-type price index)
# Model A
za_lnDNDG_all.1 <- ur.za(lnDNDG_all, model="intercept", lag=12)
summary(za_lnDNDG_all.1)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.036287 -0.002285 -0.000083 0.002380 0.022527
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.853e-02 6.689e-03 4.265 2.29e-05 ***
## y.l1 9.907e-01 2.284e-03 433.685 < 2e-16 ***
## trend 1.499e-05 5.295e-06 2.831 0.00478 **
## y.dl1 4.606e-01 3.818e-02 12.066 < 2e-16 ***
## y.dl2 -1.369e-01 4.167e-02 -3.285 0.00107 **
## y.dl3 1.384e-02 4.198e-02 0.330 0.74179
## y.dl4 2.268e-02 4.195e-02 0.541 0.58896
## y.dl5 -1.219e-02 4.204e-02 -0.290 0.77201
## y.dl6 8.493e-03 4.205e-02 0.202 0.84000
## y.dl7 1.741e-02 4.202e-02 0.414 0.67872
## y.dl8 -7.532e-03 4.203e-02 -0.179 0.85782
## y.dl9 5.523e-02 4.216e-02 1.310 0.19062
## y.dl10 2.130e-03 4.231e-02 0.050 0.95987
## y.dl11 1.820e-01 4.205e-02 4.329 1.73e-05 ***
## y.dl12 -1.681e-01 3.833e-02 -4.387 1.34e-05 ***
## du 5.821e-03 1.176e-03 4.951 9.42e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004953 on 656 degrees of freedom
## (13 observations deleted due to missingness)
## Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999
## F-statistic: 5.954e+05 on 15 and 656 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -4.0504
## Critical values: 0.01= -5.34 0.05= -4.8 0.1= -4.58
##
## Potential break point at position: 168Acf(za_lnDNDG_all.1@res, type="correlation", lag.max = 24)
plot(za_lnDNDG_all.1)
# Model B
za_lnDNDG_all.2 <- ur.za(lnDNDG_all, model="trend", lag=12)
summary(za_lnDNDG_all.2)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.036143 -0.002398 -0.000183 0.002354 0.023001
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.634e-02 6.861e-03 3.839 0.000136 ***
## y.l1 9.905e-01 2.525e-03 392.236 < 2e-16 ***
## trend 5.049e-05 1.245e-05 4.054 5.65e-05 ***
## y.dl1 4.680e-01 3.821e-02 12.247 < 2e-16 ***
## y.dl2 -1.325e-01 4.181e-02 -3.169 0.001600 **
## y.dl3 1.756e-02 4.212e-02 0.417 0.676833
## y.dl4 2.836e-02 4.207e-02 0.674 0.500464
## y.dl5 -8.011e-03 4.218e-02 -0.190 0.849428
## y.dl6 1.392e-02 4.217e-02 0.330 0.741491
## y.dl7 2.092e-02 4.217e-02 0.496 0.619890
## y.dl8 -3.588e-03 4.217e-02 -0.085 0.932214
## y.dl9 5.774e-02 4.232e-02 1.364 0.172913
## y.dl10 3.125e-03 4.248e-02 0.074 0.941380
## y.dl11 1.838e-01 4.221e-02 4.354 1.55e-05 ***
## y.dl12 -1.675e-01 3.849e-02 -4.351 1.57e-05 ***
## dt -3.664e-05 8.420e-06 -4.352 1.56e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004974 on 656 degrees of freedom
## (13 observations deleted due to missingness)
## Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999
## F-statistic: 5.905e+05 on 15 and 656 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -3.7546
## Critical values: 0.01= -4.93 0.05= -4.42 0.1= -4.11
##
## Potential break point at position: 260Acf(za_lnDNDG_all.2@res, type="correlation", lag.max = 24)
plot(za_lnDNDG_all.2)
# Model C
za_lnDNDG_all.3 <- ur.za(lnDNDG_all, model="both", lag=12)
summary(za_lnDNDG_all.3)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.036092 -0.002359 -0.000174 0.002346 0.022436
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.712e-02 6.726e-03 4.033 6.16e-05 ***
## y.l1 9.908e-01 2.281e-03 434.392 < 2e-16 ***
## trend 3.066e-05 1.032e-05 2.972 0.003067 **
## y.dl1 4.566e-01 3.818e-02 11.959 < 2e-16 ***
## y.dl2 -1.393e-01 4.163e-02 -3.347 0.000864 ***
## y.dl3 1.035e-02 4.196e-02 0.247 0.805174
## y.dl4 1.949e-02 4.192e-02 0.465 0.642197
## y.dl5 -1.532e-02 4.201e-02 -0.365 0.715411
## y.dl6 5.472e-03 4.202e-02 0.130 0.896418
## y.dl7 1.406e-02 4.199e-02 0.335 0.737861
## y.dl8 -1.044e-02 4.199e-02 -0.249 0.803802
## y.dl9 5.177e-02 4.213e-02 1.229 0.219654
## y.dl10 -1.172e-03 4.228e-02 -0.028 0.977901
## y.dl11 1.795e-01 4.200e-02 4.273 2.21e-05 ***
## y.dl12 -1.721e-01 3.833e-02 -4.491 8.40e-06 ***
## du 4.850e-03 1.296e-03 3.742 0.000198 ***
## dt -1.653e-05 9.346e-06 -1.769 0.077362 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004945 on 655 degrees of freedom
## (13 observations deleted due to missingness)
## Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999
## F-statistic: 5.6e+05 on 16 and 655 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -4.0493
## Critical values: 0.01= -5.57 0.05= -5.08 0.1= -4.82
##
## Potential break point at position: 168Acf(za_lnDNDG_all.3@res, type="correlation", lag.max = 24)
plot(za_lnDNDG_all.3)
In all three of the above Zivot_Andrews tests (Models A-C) it would appear that we Fail to Reject the Null which could indicate we are dealing with the presence of a unit root process with drift.
(c) Log transformed real wage per hour in manufacturing
# Model A
za_lnRWPHM_all.1 <- ur.za(lnRWPHM_all, model="intercept", lag=4)
summary(za_lnRWPHM_all.1)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0170725 -0.0038458 -0.0000097 0.0038727 0.0216554
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.765e-02 1.080e-02 -4.413 1.49e-05 ***
## y.l1 9.757e-01 4.849e-03 201.227 < 2e-16 ***
## trend 4.546e-05 1.678e-05 2.709 0.007201 **
## y.dl1 3.152e-02 5.771e-02 0.546 0.585373
## y.dl2 1.457e-02 5.733e-02 0.254 0.799648
## y.dl3 8.821e-02 5.648e-02 1.562 0.119540
## y.dl4 -1.919e-01 5.602e-02 -3.426 0.000711 ***
## du -4.147e-03 1.613e-03 -2.571 0.010704 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.006028 on 263 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.9991, Adjusted R-squared: 0.999
## F-statistic: 4.013e+04 on 7 and 263 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -5.0191
## Critical values: 0.01= -5.34 0.05= -4.8 0.1= -4.58
##
## Potential break point at position: 142Acf(za_lnRWPHM_all.1@res, type="correlation", lag.max = 24)
plot(za_lnRWPHM_all.1)
# Model B
za_lnRWPHM_all.2 <- ur.za(lnRWPHM_all, model="trend", lag=4)
summary(za_lnRWPHM_all.2)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0175199 -0.0038520 -0.0001165 0.0041902 0.0225317
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.532e-02 9.777e-03 -3.613 0.000362 ***
## y.l1 9.808e-01 4.458e-03 220.014 < 2e-16 ***
## trend 1.233e-05 1.086e-05 1.135 0.257504
## y.dl1 4.475e-02 5.808e-02 0.770 0.441702
## y.dl2 2.640e-02 5.775e-02 0.457 0.647955
## y.dl3 1.033e-01 5.677e-02 1.819 0.070010 .
## y.dl4 -1.813e-01 5.646e-02 -3.212 0.001484 **
## dt NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.006092 on 264 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.999, Adjusted R-squared: 0.999
## F-statistic: 4.585e+04 on 6 and 264 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -4.2987
## Critical values: 0.01= -4.93 0.05= -4.42 0.1= -4.11
##
## Potential break point at position: 1Acf(za_lnRWPHM_all.2@res, type="correlation", lag.max = 24)
plot(za_lnRWPHM_all.2)
# Model C
za_lnRWPHM_all.3 <- ur.za(lnRWPHM_all, model="both", lag=4)
summary(za_lnRWPHM_all.3)##
## ################################
## # Zivot-Andrews Unit Root Test #
## ################################
##
##
## Call:
## lm(formula = testmat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0173172 -0.0041552 -0.0001391 0.0039140 0.0219687
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.840e-02 1.044e-02 -3.678 0.000285 ***
## y.l1 9.795e-01 4.718e-03 207.603 < 2e-16 ***
## trend 1.714e-05 1.244e-05 1.378 0.169367
## y.dl1 3.626e-02 5.808e-02 0.624 0.532927
## y.dl2 2.134e-02 5.761e-02 0.370 0.711336
## y.dl3 9.851e-02 5.662e-02 1.740 0.083087 .
## y.dl4 -1.855e-01 5.633e-02 -3.294 0.001125 **
## du -5.705e-03 2.897e-03 -1.969 0.049987 *
## dt 3.699e-04 2.206e-04 1.677 0.094814 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.00607 on 262 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.9991, Adjusted R-squared: 0.999
## F-statistic: 3.463e+04 on 8 and 262 DF, p-value: < 2.2e-16
##
##
## Teststatistic: -4.3498
## Critical values: 0.01= -5.57 0.05= -5.08 0.1= -4.82
##
## Potential break point at position: 255Acf(za_lnRWPHM_all.3@res, type="correlation", lag.max = 24)
plot(za_lnRWPHM_all.3)
In this time series of Log transformed real wage per hour in manufacturing it looks like the Zivot-Andrews test is suggesting that we Reject the Null in the first test where Model A is our alternative which we are accepting at the greater that 2.5% c.v. for a trend stationary time series with a structural break in the level at position 142 which is April 1982. This seems reasonable but just by eyeing the data I would have said we had a change in the trend around this point. Interesting results!