Stationarity and Unit Root Testing
Eric Valenzuela
February 18, 2017
Introduction
This assignment involves an analysis of six different time series. These time series will be transformed and put through ADF, KPSS and ERS tests. It will then be determined if these time series are I(0), I(1) or I(2).
Data
Real Personal Consumption Expenditures Per Capita
Quarterly data of Real Personal Consumption Expenditures Per Capita from 1947 to 2016 can be found on the FRED website. A graph of this data, along with the log-transformed data, can be found below.
RPCEPC <- Quandl("FRED/A794RX0Q048SBEA", type="zoo")
plot(RPCEPC, main = "Real Personal Consumption Expenditures Per Capita")
RPCEPC_log <- log(RPCEPC)
plot(RPCEPC_log, main = "R. Personal Consumption Expenditures Per Capita: Log-Transformed")
For this series, it was determined to transform it using logarithms. The first difference of the log transformed series can be found below.
RPCEPC_d1 <- diff(RPCEPC_log)
plot(RPCEPC_d1, main = "Real Personal Consumption Expenditures Per Capita: 1st Diff")
S&P 500 Index
Daily data of the S&P 500 index from 1950 to 2017 can be found on Yahoo. A graph of this data, along with the log-transformed data, can be found below.
SPI_six <- Quandl("YAHOO/INDEX_GSPC", type="zoo")
SPI <- SPI_six$Close
plot(SPI, main = "S&P 500 Index")
SPI_log <- (log(SPI))
plot(SPI_log, main = "S&P 500 Index: Log-Transformed")
For this series, it was determined to transform it using logarithms. The first difference of the log transformed series can be found below.
SPI_d1 <- diff(SPI_log)
plot(SPI_d1, main = "S&P 500 Index: 1st Diff")
Personal Consumption Expenditures Chain Type Price Index
Quarterly data of the Personal Consumption Expenditures Chain Type Price Index from 1947 to 2016 can be found on the FRED website. A graph of this data, along with the log-transformed data, can be found below.
PCECTPI <- Quandl("FRED/PCECTPI", type="zoo")
plot(PCECTPI, main = "Personal Consumption Expenditures Chain Type Price Index")
PCECTPI_log <- log(PCECTPI)
plot(PCECTPI_log, main = "P. Consumption Expenditures Chain Type Price Index: Log-Transformed")
For this series, it was determined to transform it using logarithms. The first difference of the log transformed series can be found below.
PCECTPI_d1 <- diff(PCECTPI_log)
plot(PCECTPI_d1, main = "Personal Consumption Expenditures Chain Type Price Index: 1st Diff")
10-Year Treasury Constant Maturity Rate
Monthly data of the 10-Year Treasury Constant Maturity Rate from 1953 to 2017 can be found on the FRED website. A graph of this data, along with the log-transformed data, can be found below.
TCMR <- Quandl("FRED/GS10", type="zoo")
plot(TCMR, main = "10-Year Treasury Constant Maturity Rate")
TCMR_log <- log(TCMR)
plot(TCMR_log, main = "10-Year Treasury Constant Maturity Rate: Log-Transformed")
For this series, it was determined to transform it using logarithms. The first difference of the log transformed series can be found below.
TCMR_d1 <- diff(TCMR_log)
plot(TCMR_d1, main = "10-Year Treasury Constant Maturity Rate: 1st Diff")
Unemployment Rate
Monthly data of the Unemployment Rate from 1948 to 2017 can be found on the FRED website. A graph of this data, along with the log-transformed data, can be found below.
UERT <- Quandl("FRED/UNRATE", type="zoo")
plot(UERT, main = "Unemployment Rate")
UERT_log <- log(UERT)
plot(UERT_log, main = "Unemployment Rate: Log-Transformed")
For this series, it was determined to transform it using logarithms. The first difference of the log transformed series can be found below.
UERT_d1 <- diff(UERT_log)
plot(UERT_d1, main = "Unemployment Rate: 1st Diff")
Canada / U.S. Foreign Exchange Rate
Monthly data of the Canada / U.S. Foreign Exchange Rate from 1971 to 2017 can be found on the FRED website. A graph of this data, along with the log-transformed data, can be found below.
CUSFER <- Quandl("FRED/EXCAUS", type="zoo")
plot(CUSFER, main = "Canada / U.S. Foreign Exchange Rate")
CUSFER_log <- log(CUSFER)
plot(CUSFER_log, main = "Canada / U.S. Foreign Exchange Rate: Log-Transformed")
For this series, it was determined to transform it using logarithms. The first difference of the log transformed series can be found below.
CUSFER_d1 <- diff(CUSFER_log)
plot(CUSFER_d1, main = "Canada / U.S. Foreign Exchange Rate: 1st Diff")
Methodology
For each of the six time series, ADF, KPSS and ERS tests are performed.
Real Personal Consumption Expenditures Per Capita
For the log-transformed series, we can reject the null hypothesis for the KPSS test, but not the ADF test. This implies that the series has a unit root.
adf.test(RPCEPC_log)##
## Augmented Dickey-Fuller Test
##
## data: RPCEPC_log
## Dickey-Fuller = -1.2427, Lag order = 6, p-value = 0.8951
## alternative hypothesis: stationarykpss.test(RPCEPC_log, null="Trend")## Warning in kpss.test(RPCEPC_log, null = "Trend"): p-value smaller than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: RPCEPC_log
## KPSS Trend = 0.73917, Truncation lag parameter = 3, p-value = 0.01RPCEPC.urers1 <- ur.ers(RPCEPC_log, type="P-test", model="trend")
summary(RPCEPC.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 10.325
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89RPCEPC.urers2 <- ur.ers(RPCEPC_log, type="DF-GLS", model="trend")
summary(RPCEPC.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 10.325
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for the ADF test, but not the KPSS test. This implies that the series is stationary.
adf.test(RPCEPC_d1)## Warning in adf.test(RPCEPC_d1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: RPCEPC_d1
## Dickey-Fuller = -5.7821, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationarykpss.test(RPCEPC_d1, null="Trend")## Warning in kpss.test(RPCEPC_d1, null = "Trend"): p-value greater than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: RPCEPC_d1
## KPSS Trend = 0.094181, Truncation lag parameter = 3, p-value = 0.1RPCEPC.urers1 <- ur.ers(RPCEPC_d1, type="P-test", model="trend")
summary(RPCEPC.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.4007
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89RPCEPC.urers2 <- ur.ers(RPCEPC_d1, type="DF-GLS", model="trend")
summary(RPCEPC.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.4007
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89S&P 500 Index
For the log-transformed series, we can reject the null hypothesis for the KPSS test, but not the ADF test. This implies that the series has a unit root.
adf.test(SPI_log)##
## Augmented Dickey-Fuller Test
##
## data: SPI_log
## Dickey-Fuller = -2.2329, Lag order = 25, p-value = 0.4797
## alternative hypothesis: stationarykpss.test(SPI_log, null="Trend")## Warning in kpss.test(SPI_log, null = "Trend"): p-value smaller than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: SPI_log
## KPSS Trend = 4.4127, Truncation lag parameter = 29, p-value = 0.01SPI.urers1 <- ur.ers(SPI_log, type="P-test", model="trend")
summary(SPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 12.0101
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89SPI.urers2 <- ur.ers(SPI_log, type="DF-GLS", model="trend")
summary(SPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 12.0101
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for the ADF test, but not the KPSS test. This implies that the series is stationary.
adf.test(SPI_d1)## Warning in adf.test(SPI_d1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: SPI_d1
## Dickey-Fuller = -26.405, Lag order = 25, p-value = 0.01
## alternative hypothesis: stationarykpss.test(SPI_d1, null="Trend")## Warning in kpss.test(SPI_d1, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: SPI_d1
## KPSS Trend = 0.066239, Truncation lag parameter = 29, p-value =
## 0.1SPI.urers1 <- ur.ers(SPI_d1, type="P-test", model="trend")
summary(SPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.0134
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89SPI.urers2 <- ur.ers(SPI_d1, type="DF-GLS", model="trend")
summary(SPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.0134
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Personal Consumption Expenditures Chain Type Price Index
For the log-transformed series, we can reject the null hypothesis for the KPSS test, but not the ADF test. This implies that the series has a unit root.
adf.test(PCECTPI_log)##
## Augmented Dickey-Fuller Test
##
## data: PCECTPI_log
## Dickey-Fuller = -1.233, Lag order = 6, p-value = 0.8992
## alternative hypothesis: stationarykpss.test(PCECTPI_log, null="Trend")## Warning in kpss.test(PCECTPI_log, null = "Trend"): p-value smaller than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: PCECTPI_log
## KPSS Trend = 0.9172, Truncation lag parameter = 3, p-value = 0.01PCECTPI.urers1 <- ur.ers(PCECTPI_log, type="P-test", model="trend")
summary(PCECTPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 44.6187
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89PCECTPI.urers2 <- ur.ers(PCECTPI_log, type="DF-GLS", model="trend")
summary(PCECTPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 44.6187
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for both the KPSS test and the ADF test. This implies that both hypotheses are competent.
adf.test(PCECTPI_d1)##
## Augmented Dickey-Fuller Test
##
## data: PCECTPI_d1
## Dickey-Fuller = -3.5454, Lag order = 6, p-value = 0.0388
## alternative hypothesis: stationarykpss.test(PCECTPI_d1, null="Trend")## Warning in kpss.test(PCECTPI_d1, null = "Trend"): p-value smaller than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: PCECTPI_d1
## KPSS Trend = 0.72607, Truncation lag parameter = 3, p-value = 0.01PCECTPI.urers1 <- ur.ers(PCECTPI_d1, type="P-test", model="trend")
summary(PCECTPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 2.2022
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89PCECTPI.urers2 <- ur.ers(PCECTPI_d1, type="DF-GLS", model="trend")
summary(PCECTPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 2.2022
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for the ADF test, but not the KPSS test. This implies that the series is stationary.
PCECTPI_d2 <- diff(PCECTPI_d1)
adf.test(PCECTPI_d2)## Warning in adf.test(PCECTPI_d2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: PCECTPI_d2
## Dickey-Fuller = -9.1441, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationarykpss.test(PCECTPI_d2, null="Trend")## Warning in kpss.test(PCECTPI_d2, null = "Trend"): p-value greater than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: PCECTPI_d2
## KPSS Trend = 0.013372, Truncation lag parameter = 3, p-value = 0.1PCECTPI.urers1 <- ur.ers(PCECTPI_d2, type="P-test", model="trend")
summary(PCECTPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.0291
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89PCECTPI.urers2 <- ur.ers(PCECTPI_d2, type="DF-GLS", model="trend")
summary(PCECTPI.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.0291
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.8910-Year Treasury Constant Maturity Rate
For the log-transformed series, we can reject the null hypothesis for the KPSS test, but not the ADF test. This implies that the series has a unit root.
adf.test(TCMR_log)##
## Augmented Dickey-Fuller Test
##
## data: TCMR_log
## Dickey-Fuller = -1.5672, Lag order = 9, p-value = 0.7615
## alternative hypothesis: stationarykpss.test(TCMR_log, null="Trend")## Warning in kpss.test(TCMR_log, null = "Trend"): p-value smaller than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: TCMR_log
## KPSS Trend = 2.5123, Truncation lag parameter = 6, p-value = 0.01TCMR.urers1 <- ur.ers(TCMR_log, type="P-test", model="trend")
summary(TCMR.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 36.3134
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89TCMR.urers2 <- ur.ers(TCMR_log, type="DF-GLS", model="trend")
summary(TCMR.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 36.3134
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for the ADF test, but not the KPSS test. This implies that the series is stationary.
adf.test(TCMR_d1)## Warning in adf.test(TCMR_d1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: TCMR_d1
## Dickey-Fuller = -8.7766, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationarykpss.test(TCMR_d1, null="Trend")## Warning in kpss.test(TCMR_d1, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: TCMR_d1
## KPSS Trend = 0.022821, Truncation lag parameter = 6, p-value = 0.1TCMR.urers1 <- ur.ers(TCMR_d1, type="P-test", model="trend")
summary(TCMR.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.277
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89TCMR.urers2 <- ur.ers(TCMR_d1, type="DF-GLS", model="trend")
summary(TCMR.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.277
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Unemployment Rate
For the log-transformed series, we can reject the null hypothesis for both the KPSS test and the ADF test. This implies that both hypotheses are competent.
adf.test(UERT_log)## Warning in adf.test(UERT_log): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: UERT_log
## Dickey-Fuller = -4.2834, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationarykpss.test(UERT_log, null="Trend")## Warning in kpss.test(UERT_log, null = "Trend"): p-value smaller than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: UERT_log
## KPSS Trend = 0.55175, Truncation lag parameter = 6, p-value = 0.01UERT.urers1 <- ur.ers(UERT_log, type="P-test", model="trend")
summary(UERT.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 4.2513
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89UERT.urers2 <- ur.ers(UERT_log, type="DF-GLS", model="trend")
summary(UERT.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 4.2513
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for the ADF test, but not the KPSS test. This implies that the series is stationary.
adf.test(UERT_d1)## Warning in adf.test(UERT_d1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: UERT_d1
## Dickey-Fuller = -8.3573, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationarykpss.test(UERT_d1, null="Trend")## Warning in kpss.test(UERT_d1, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: UERT_d1
## KPSS Trend = 0.027839, Truncation lag parameter = 6, p-value = 0.1UERT.urers1 <- ur.ers(UERT_d1, type="P-test", model="trend")
summary(UERT.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 2.3914
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89UERT.urers2 <- ur.ers(UERT_d1, type="DF-GLS", model="trend")
summary(UERT.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 2.3914
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Canada / U.S. Foreign Exchange Rate
For the log-transformed series, we can reject the null hypothesis for the KPSS test, but not the ADF test. This implies that the series has a unit root.
adf.test(CUSFER_log)##
## Augmented Dickey-Fuller Test
##
## data: CUSFER_log
## Dickey-Fuller = -1.7645, Lag order = 8, p-value = 0.678
## alternative hypothesis: stationarykpss.test(CUSFER_log, null="Trend")## Warning in kpss.test(CUSFER_log, null = "Trend"): p-value smaller than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: CUSFER_log
## KPSS Trend = 1.3674, Truncation lag parameter = 5, p-value = 0.01CUSFER.urers1 <- ur.ers(CUSFER_log, type="P-test", model="trend")
summary(CUSFER.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 18.8285
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89CUSFER.urers2 <- ur.ers(CUSFER_log, type="DF-GLS", model="trend")
summary(CUSFER.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 18.8285
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89The test was repeated for the first difference series. Here, we can reject the null hypothesis for the ADF test, but not the KPSS test. This implies that the series is stationary.
adf.test(CUSFER_d1)## Warning in adf.test(CUSFER_d1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: CUSFER_d1
## Dickey-Fuller = -7.743, Lag order = 8, p-value = 0.01
## alternative hypothesis: stationarykpss.test(CUSFER_d1, null="Trend")## Warning in kpss.test(CUSFER_d1, null = "Trend"): p-value greater than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: CUSFER_d1
## KPSS Trend = 0.091534, Truncation lag parameter = 5, p-value = 0.1CUSFER.urers1 <- ur.ers(CUSFER_d1, type="P-test", model="trend")
summary(CUSFER.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.3408
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89CUSFER.urers2 <- ur.ers(CUSFER_d1, type="DF-GLS", model="trend")
summary(CUSFER.urers1)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.3408
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Conclusion
Based on the tests, it would appear that Real Personal Consumption Expenditures Per Capita, the S&P 500 Index, the 10-Year Treasury Constant Maturity Rate, the Unemployment Rate and the Canada/U.S. Foreign Exchange Rate are all I(1). There is some evidence pointing to the unemployment rate being I(0), though.
The Personal Consumption Expenditures Chain Type Price Index appears to be I(2), though there is some evidence pointing to this time series being I(1).