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Unit Root Tests |
Ashiq Zaman |
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now |
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The main condition for any time series regressions is the data must be stationary. Any non-stationary time series result gives a spurious result . So, for any time series analysis stationarity test is prerequisite. Unit Root test is one of the tests of stationarity which becomes popular over the last few years. There are some ways of doing Unit Root test and each way has its own advantages and disadvantages. In this study two tests is used to avoid each other disadvantages. The tests are conducted in log form and first difference log form of individual market indices. If the result gives the rejection of null hypothesis then it can be said that the selected time series data are stationary. The ADF test investigate the existence of unit root in an Autoregressive (AR) model. If any explanatory variable is the lagged values of dependant variable, then it is known as AR model. The most commonly used AR model is yt=β+ρyt-1+εt where yt represents variable of interest, t represents time index and ρ and ε are coefficient and disturbance terms accordingly. In this study the Augmented Dickey-Fuller (ADF) test is conducted based on the following regressions
where Δ represents the first difference operator, m represents the number of lags, εt represents pure white noise error term and Yt represents time series. Equation (4.1a) represents the pure random walk model without constant and trend and equation (4.1b) represents random walk with constant but without time trend and (4.1c) represents random walk with constant and time trend.
Before doing ADF test it assumes that the error terms of the model are statistically significant, and it has a constant variance.
Import Pilot data from a CSV file (preloaded data from Thomson Reuters datastream)
price<-read.csv("price.csv")
Changes the dates as factors to actual dates
library(zoo)
price.zoo=zoo(price[,-1], order.by=as.Date(strptime(as.character(price[,1]), "%d/%m/%Y")))
head(price.zoo)
STOXX50 DAX30 CAC40 Russell1000
2010-01-01 7.994619 8.692394 8.278004 7.070286
2010-01-04 8.012283 8.707533 8.297536 7.086295
2010-01-05 8.010479 8.704811 8.297272 7.089636
2010-01-06 8.009582 8.705220 8.298457 7.090651
2010-01-07 8.008811 8.702736 8.300230 7.094652
2010-01-08 8.012300 8.705764 8.305271 7.097789
Given each series a name
stoxx<- price.zoo[,1]
dax30<- price.zoo[,2]
cac40<- price.zoo[,3]
russell<- price.zoo[,4]
Load urca to run ADF test
library(urca)
Run ADF test with one lag with lag selection AIC neither an intercept nor a trend is included in the test regression
adf.stoxx.none<-ur.df(stoxx,type = c("none"),lags = 1, selectlags = "AIC")
summary(adf.stoxx.none)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.132369 -0.005705 0.000096 0.006212 0.098561
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 -4.057e-06 3.132e-05 -0.130 0.897
z.diff.lag 1.461e-03 1.926e-02 0.076 0.940
Residual standard error: 0.01304 on 2695 degrees of freedom
Multiple R-squared: 8.361e-06, Adjusted R-squared: -0.0007337
F-statistic: 0.01127 on 2 and 2695 DF, p-value: 0.9888
Value of test-statistic is: -0.1295
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62
Run ADF test with one lag with lag selection AIC: with drift
adf.stoxx.drift<-ur.df(stoxx,type = c("drift"), lags = 1, selectlags = "AIC")
summary(adf.stoxx.drift)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.132557 -0.005590 0.000417 0.006275 0.097817
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.034597 0.014392 2.404 0.0163 *
z.lag.1 -0.004319 0.001795 -2.406 0.0162 *
z.diff.lag 0.003630 0.019263 0.188 0.8506
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01303 on 2694 degrees of freedom
Multiple R-squared: 0.002146, Adjusted R-squared: 0.001405
F-statistic: 2.897 on 2 and 2694 DF, p-value: 0.05537
Value of test-statistic is: -2.4058 2.8977
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.43 -2.86 -2.57
phi1 6.43 4.59 3.78
Run ADF test with one lag with lag selection AIC: with trend
adf.stoxx.trend<-ur.df(stoxx,type = c("trend"), lags = 1, selectlags = "AIC")
summary(adf.stoxx.trend)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression trend
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-0.134032 -0.005497 0.000497 0.006237 0.098413
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.509e-02 2.001e-02 3.252 0.00116 **
z.lag.1 -8.291e-03 2.551e-03 -3.250 0.00117 **
tt 1.003e-06 4.580e-07 2.190 0.02858 *
z.diff.lag 5.842e-03 1.928e-02 0.303 0.76186
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01302 on 2693 degrees of freedom
Multiple R-squared: 0.00392, Adjusted R-squared: 0.002811
F-statistic: 3.533 on 3 and 2693 DF, p-value: 0.01422
Value of test-statistic is: -3.2503 3.5338 5.2968
Critical values for test statistics:
1pct 5pct 10pct
tau3 -3.96 -3.41 -3.12
phi2 6.09 4.68 4.03
phi3 8.27 6.25 5.34
In Phillips-Perron test it makes the error terms weakly dependent and another properties of this test is the error terms are heterogeneously distributed.
The Phillips-Perron (PP) test is conducted based on the equation (4.2)
where Δ represents first difference operator, β represents constant and µt represents error term. Equation (4.2a), (4.2b) and (4.2c) represents PP without constant and time trend, with constant and with constant and time trend accordingly.
Run PP test with Z alpha type and model constant
pp.stoxx.constant<-ur.pp(stoxx, type = c("Z-alpha"), model = c("constant"), lags = c("short"))
summary(pp.stoxx.constant)
##################################
# Phillips-Perron Unit Root Test #
##################################
Test regression with intercept
Call:
lm(formula = y ~ y.l1)
Residuals:
Min 1Q Median 3Q Max
-0.132568 -0.005588 0.000428 0.006275 0.097654
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.034534 0.014376 2.402 0.0164 *
y.l1 0.995690 0.001793 555.284 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01303 on 2696 degrees of freedom
Multiple R-squared: 0.9913, Adjusted R-squared: 0.9913
F-statistic: 3.083e+05 on 1 and 2696 DF, p-value: < 2.2e-16
Value of test-statistic, type: Z-alpha is: -11.1338
aux. Z statistics
Z-tau-mu 2.3502
Run PP test with Z alpha type and model trend
pp.stoxx.trend<-ur.pp(stoxx, type = c("Z-alpha"), model = c("trend"), lags = c("short"))
summary(pp.stoxx.trend)
##################################
# Phillips-Perron Unit Root Test #
##################################
Test regression with intercept and trend
Call:
lm(formula = y ~ y.l1 + trend)
Residuals:
Min 1Q Median 3Q Max
-0.133993 -0.005497 0.000497 0.006216 0.098137
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.514e-02 2.039e-02 3.195 0.00142 **
y.l1 9.919e-01 2.544e-03 389.950 < 2e-16 ***
trend 9.665e-07 4.569e-07 2.115 0.03449 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01302 on 2695 degrees of freedom
Multiple R-squared: 0.9913, Adjusted R-squared: 0.9913
F-statistic: 1.544e+05 on 2 and 2695 DF, p-value: < 2.2e-16
Value of test-statistic, type: Z-alpha is: -21.3576
aux. Z statistics
Z-tau-mu 3.3191
Z-tau-beta 2.0785
The KPSS test, short for, Kwiatkowski-Phillips-Schmidt-Shin (KPSS), is a type of Unit root test that tests for the stationarity of a given series around a deterministic trend.
Read data as CSV file
kpss<-read.csv("kpss_before.csv")
head(kpss)
## DATE UK USA GERMANY FRANCE CANADA JAPAN
## 1 03/01/2000 8.843644 7.282912 8.817410 8.685647 9.037623 9.848732
## 2 04/01/2000 8.804754 7.243813 8.792846 8.643301 9.012206 9.852345
## 3 05/01/2000 8.785065 7.245734 8.779876 8.608806 9.002014 9.827823
## 4 06/01/2000 8.771407 7.246689 8.775692 8.603391 9.001376 9.807432
## 5 07/01/2000 8.780288 7.273419 8.821874 8.619679 9.039483 9.808815
## 6 10/01/2000 8.795992 7.284547 8.842968 8.638724 9.059808 9.808815
## HONG.KONG KOREA BRAZIL RUSSIA INDIA CHINA SOUTH.AFRICA
## 1 9.762479 6.935439 9.736842 5.166271 8.589534 7.282912 9.030878
## 2 9.745243 6.965118 9.670988 5.166271 8.610867 7.243813 9.028508
## 3 9.670718 6.893971 9.695540 5.187442 8.586159 7.245734 9.017965
## 4 9.625969 6.867756 9.686947 5.223163 8.598133 7.246689 9.019450
## 5 9.642488 6.855040 9.699472 5.223163 8.596832 7.273419 9.044496
## 6 9.670808 6.894913 9.742262 5.258797 8.615841 7.284547 9.077054
Changes the dates as factors to actual dates
install.packages("zoo")
library(zoo)
d.zoo=zoo(kpss[,-1], order.by=as.Date(strptime(as.character(kpss[,1]), "%d/%m/%Y")))
head(d.zoo)
## UK USA GERMANY FRANCE CANADA JAPAN HONG.KONG
## 2000-01-03 8.843644 7.282912 8.817410 8.685647 9.037623 9.848732 9.762479
## 2000-01-04 8.804754 7.243813 8.792846 8.643301 9.012206 9.852345 9.745243
## 2000-01-05 8.785065 7.245734 8.779876 8.608806 9.002014 9.827823 9.670718
## 2000-01-06 8.771407 7.246689 8.775692 8.603391 9.001376 9.807432 9.625969
## 2000-01-07 8.780288 7.273419 8.821874 8.619679 9.039483 9.808815 9.642488
## 2000-01-10 8.795992 7.284547 8.842968 8.638724 9.059808 9.808815 9.670808
## KOREA BRAZIL RUSSIA INDIA CHINA SOUTH.AFRICA
## 2000-01-03 6.935439 9.736842 5.166271 8.589534 7.282912 9.030878
## 2000-01-04 6.965118 9.670988 5.166271 8.610867 7.243813 9.028508
## 2000-01-05 6.893971 9.695540 5.187442 8.586159 7.245734 9.017965
## 2000-01-06 6.867756 9.686947 5.223163 8.598133 7.246689 9.019450
## 2000-01-07 6.855040 9.699472 5.223163 8.596832 7.273419 9.044496
## 2000-01-10 6.894913 9.742262 5.258797 8.615841 7.284547 9.077054
given each series a name
uk <- d.zoo[,1]
uk_diff<-diff(uk)
usa <- d.zoo[,2]
usa_diff<-diff(usa)
ger<-d.zoo[,3]
ger_diff<-diff(ger)
fra<-d.zoo[,4]
fra_diff<-diff(fra)
can<-d.zoo[,5]
can_diff<-diff(can)
jap<-d.zoo[,6]
jap_diff<-diff(jap)
hk<-d.zoo[,7]
hk_diff<-diff(hk)
kor<-d.zoo[,8]
kor_diff<-diff(kor)
bra<-d.zoo[,9]
bra_diff<-diff(bra)
rus<-d.zoo[,10]
rus_diff<-diff(rus)
ind<-d.zoo[,11]
ind_diff<-diff(ind)
chi<-d.zoo[,12]
chi_diff<-diff(chi)
sa<-d.zoo[,13]
sa_diff<-diff(sa)
KPSS test for UK before crisis
KPSS test with a drift
library(urca)
kpss.uk.drift<- ur.kpss(uk, type="mu", lags="short")
summary(kpss.uk.drift)
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: mu with 8 lags.
##
## Value of test-statistic is: 6.195
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.347 0.463 0.574 0.739
kpss.uk.drift.diff<- ur.kpss(uk_diff, type="mu", lags="short")
summary(kpss.uk.drift.diff)
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: mu with 8 lags.
##
## Value of test-statistic is: 0.2521
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.347 0.463 0.574 0.739
*KPSS test with a trend *
kpss.uk.trend<- ur.kpss(uk, type="tau", lags="short")
summary(kpss.uk.trend)
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 8 lags.
##
## Value of test-statistic is: 5.0498
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216
kpss.uk.trend.diff<- ur.kpss(uk_diff, type="tau", lags="short")
summary(kpss.uk.trend.diff)
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 8 lags.
##
## Value of test-statistic is: 0.1315
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216
will be added soon