HW4
Mai AlSwayan
Question
The Task is to choose a specific time series from each category and for each of the time series set chosen: 1.determine whether to transform it using logarithm. 2.plot the original and transformed time series. 3.Plot first differences of either original or log-transformed time series. 4.Perform ADF,KPSS and ERS tests on either original or log-transformed time series: - If these tests suggest the presence of a unit root also perform these tests for series in first differences, and if necessary also on second differences. 5.Summarize the results of the tests - determine which time series appear to be I(0), I(1), I(2).
The answer: The chosen time series: Category A: Real Personal Consumption Expenditures Per Capita. Category B: Shanghai Composite Index Category C: Consumer Price Index for All Urban Consumers: All Items. Category D: 3-Month Treasury Bills Secondary Market Rate. Category E: Unemployment Rate. Category F: Canada / U.S. Foreign Exchange Rate.
Catogry A
overview
library(Quandl)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(forecast)
library(urca)
rpce <- Quandl("FRED/A794RX0Q048SBEA", type="zoo")
str(rpce)## 'zooreg' series from 1947 Q1 to 2016 Q4
## Data: num [1:280] 8378 8479 8468 8431 8437 ...
## Index: Class 'yearqtr' num [1:280] 1947 1947 1948 1948 1948 ...
## Frequency: 4summary(rpce)## Index rpce
## Min. :1947 Min. : 8378
## 1st Qu.:1964 1st Qu.:11720
## Median :1982 Median :17762
## Mean :1982 Mean :19920
## 3rd Qu.:1999 3rd Qu.:27782
## Max. :2017 Max. :35987head(rpce)## 1947 Q1 1947 Q2 1947 Q3 1947 Q4 1948 Q1 1948 Q2
## 8378 8479 8468 8431 8437 8501tail(rpce)## 2015 Q3 2015 Q4 2016 Q1 2016 Q2 2016 Q3 2016 Q4
## 35015 35147 35236 35550 35743 35987part 1 ,2
Oroginal Data
plot(rpce, xlab="Years", ylab="Thousand $", main="quarterly Real Personal Consumption Expenditures Per Capita")
Logarithm
lrpce <- log(rpce)plot
plot(lrpce, xlab="Years", ylab="Thousand $", main="Log of quarterly Real Personal Consumption Expenditures Per Capita")
##Part 3 #first differences of original data
drpce <- diff(rpce)
summary(drpce)## Index drpce
## Min. :1947 Min. :-470.00
## 1st Qu.:1965 1st Qu.: 24.00
## Median :1982 Median : 101.00
## Mean :1982 Mean : 98.96
## 3rd Qu.:1999 3rd Qu.: 182.00
## Max. :2017 Max. : 414.00plot(drpce, xlab="Years", ylab="Thousand $", main="first differences of quarterly Real Personal Consumption Expenditures Per Capita")
2nd differences of original data.
drpce2 <-diff(drpce)
summary(drpce2)## Index drpce2
## Min. :1948 Min. :-734.0000
## 1st Qu.:1965 1st Qu.:-102.0000
## Median :1982 Median : -3.5000
## Mean :1982 Mean : 0.5144
## 3rd Qu.:1999 3rd Qu.: 86.7500
## Max. :2017 Max. : 582.0000plot(drpce2, type= "l", xlab= "Years", ylab="drpce", main=" Second differences of quarterly Real Personal Consumption Expenditures/Capita", major.format="%Y Q%q")## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter
Part 4
ADF,KPSS,ERS tests original data
1.ADF
library(tseries)
adf.test(rpce)##
## Augmented Dickey-Fuller Test
##
## data: rpce
## Dickey-Fuller = -2.1827, Lag order = 6, p-value = 0.4991
## alternative hypothesis: stationaryadf.test(drpce)## Warning in adf.test(drpce): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: drpce
## Dickey-Fuller = -4.9702, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationaryadf.test(drpce2)## Warning in adf.test(drpce2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: drpce2
## Dickey-Fuller = -8.1753, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationary2.KPSS test
kpss.test(rpce, null="Trend")## Warning in kpss.test(rpce, null = "Trend"): p-value smaller than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: rpce
## KPSS Trend = 1.327, Truncation lag parameter = 3, p-value = 0.01kpss.test(drpce, null="Trend")## Warning in kpss.test(drpce, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: drpce
## KPSS Trend = 0.098054, Truncation lag parameter = 3, p-value = 0.1kpss.test(drpce2, null="Trend")## Warning in kpss.test(drpce2, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: drpce2
## KPSS Trend = 0.0092178, Truncation lag parameter = 3, p-value =
## 0.1ERS test
rpce.ers <- ur.ers(rpce, type="P-test", model="trend")
summary(rpce.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 39.9552
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89drpce.ers <- ur.ers(drpce, type="P-test", model="trend")
summary(drpce.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 1.3946
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89drpce2.ers <- ur.ers(drpce2, type="P-test", model="trend")
summary(drpce2.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.1519
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Catogry B
overview
library(Quandl)
library(forecast)
library(urca)
SHci <- Quandl("YAHOO/INDEX_SSEC", type="zoo")
str(SHci)## 'zoo' series from 1997-07-02 to 2017-04-18
## Data: num [1:4799, 1:6] 1256 1195 1139 1162 1093 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:6] "Open" "High" "Low" "Close" ...
## Index: Date[1:4799], format: "1997-07-02" "1997-07-03" "1997-07-04" "1997-07-07" ...summary(SHci)## Index Open High Low
## Min. :1997-07-02 Min. :1008 Min. :1020 Min. : 998.2
## 1st Qu.:2002-06-27 1st Qu.:1495 1st Qu.:1508 1st Qu.:1481.2
## Median :2007-06-15 Median :2066 Median :2079 Median :2053.0
## Mean :2007-06-04 Mean :2235 Mean :2256 Mean :2211.9
## 3rd Qu.:2012-05-19 3rd Qu.:2843 3rd Qu.:2875 3rd Qu.:2811.3
## Max. :2017-04-18 Max. :6057 Max. :6124 Max. :6040.7
## Close Volume Adjusted Close
## Min. :1011 Min. : 0 Min. :1011
## 1st Qu.:1495 1st Qu.: 0 1st Qu.:1495
## Median :2068 Median : 51400 Median :2068
## Mean :2236 Mean : 82775 Mean :2236
## 3rd Qu.:2847 3rd Qu.: 114400 3rd Qu.:2847
## Max. :6092 Max. :10065200 Max. :6092head(SHci)## Open High Low Close Volume Adjusted Close
## 1997-07-02 1255.909 1261.571 1147.331 1199.061 0 1199.061
## 1997-07-03 1194.676 1194.676 1149.940 1150.623 0 1150.623
## 1997-07-04 1138.921 1163.249 1124.776 1159.342 0 1159.342
## 1997-07-07 1161.707 1163.447 1085.572 1096.819 0 1096.819
## 1997-07-08 1092.799 1115.433 1066.044 1109.666 0 1109.666
## 1997-07-09 1113.641 1122.674 1104.086 1120.841 0 1120.841tail(SHci)## Open High Low Close Volume Adjusted Close
## 2017-04-11 3266.222 3290.389 3244.405 3288.966 281300 3288.966
## 2017-04-12 3283.837 3284.934 3262.277 3273.830 269400 3273.830
## 2017-04-13 3265.218 3281.137 3261.490 3275.960 207300 3275.960
## 2017-04-14 3276.138 3276.711 3238.896 3244.483 208100 3244.483
## 2017-04-17 3229.949 3229.949 3199.912 3222.167 212700 3222.167
## 2017-04-18 3215.396 3225.055 3196.488 3196.713 188700 3196.713part 1 ,2
Oroginal Data
plot(SHci, xlab="Years", ylab="", main="Shanghai Composite daily Index")
Logarithm
lSHci <- log(SHci)plot
plot(lSHci, xlab="Years", ylab="", main="Log of Shanghai Composite daily Index")
##Part 3 #first differences of original data
dSHci <- diff(SHci)
summary(dSHci)## Index Open High
## Min. :1997-07-03 Min. :-417.3838 Min. :-382.9287
## 1st Qu.:2002-06-28 1st Qu.: -15.4878 1st Qu.: -11.6674
## Median :2007-06-16 Median : 0.7080 Median : 1.3391
## Mean :2007-06-05 Mean : 0.4084 Mean : 0.4092
## 3rd Qu.:2012-05-20 3rd Qu.: 17.4638 3rd Qu.: 13.3383
## Max. :2017-04-18 Max. : 423.4558 Max. : 296.4800
## Low Close Volume
## Min. :-379.3320 Min. :-354.6841 Min. :-10058200
## 1st Qu.: -13.3188 1st Qu.: -13.8298 1st Qu.: -4000
## Median : 0.1230 Median : 1.1175 Median : 0
## Mean : 0.4271 Mean : 0.4164 Mean : 39
## 3rd Qu.: 15.8318 3rd Qu.: 15.9780 3rd Qu.: 3200
## Max. : 371.7500 Max. : 351.4028 Max. : 10060800
## Adjusted Close
## Min. :-354.6841
## 1st Qu.: -13.8298
## Median : 1.1175
## Mean : 0.4164
## 3rd Qu.: 15.9780
## Max. : 351.4028plot(dSHci, xlab="Years", ylab="", main="first differences of Shanghai Composite daily Index")
2nd differences of original data.
dSHci2 <-diff(dSHci)
summary(dSHci2)## Index Open High
## Min. :1997-07-04 Min. :-501.9382 Min. :-396.2495
## 1st Qu.:2002-07-01 1st Qu.: -24.7910 1st Qu.: -17.8320
## Median :2007-06-18 Median : -1.3881 Median : -1.7361
## Mean :2007-06-05 Mean : 0.0097 Mean : 0.0129
## 3rd Qu.:2012-05-21 3rd Qu.: 24.4680 3rd Qu.: 16.7539
## Max. :2017-04-18 Max. : 634.3257 Max. : 419.6191
## Low Close Volume
## Min. :-466.2334 Min. :-447.7144 Min. :-20119000
## 1st Qu.: -20.9869 1st Qu.: -23.2788 1st Qu.: -5200
## Median : 0.4279 Median : -1.6161 Median : 0
## Mean : -0.0013 Mean : 0.0048 Mean : -5
## 3rd Qu.: 20.1458 3rd Qu.: 22.6301 3rd Qu.: 6800
## Max. : 533.5911 Max. : 497.9800 Max. : 10061400
## Adjusted Close
## Min. :-447.7144
## 1st Qu.: -23.2788
## Median : -1.6161
## Mean : 0.0048
## 3rd Qu.: 22.6301
## Max. : 497.9800plot(dSHci2, type= "l", xlab= "Years", ylab="dSHci", main="Second differences of Shanghai Composite daily Index", major.format="%Y Q%q")## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in axis(side = 2, xpd = NA, major.format = "%Y Q%q"):
## "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in axis(side = 2, xpd = NA, major.format = "%Y Q%q"):
## "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in axis(side = 2, xpd = NA, major.format = "%Y Q%q"):
## "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in axis(side = 2, xpd = NA, major.format = "%Y Q%q"):
## "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in axis(side = 2, xpd = NA, major.format = "%Y Q%q"):
## "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in axis(side = 2, xpd = NA, major.format = "%Y Q%q"):
## "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter
Part 4
ADF,KPSS,ERS tests original data
1.ADF Couldn’t preform
2.KPSS test
SHci.urkpss <- ur.kpss(SHci, type="tau", lags="short")
summary(SHci.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 16 lags.
##
## Value of test-statistic is: 3.2268
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216dSHci.urkpss <- ur.kpss(dSHci, type="tau", lags="short")
summary(dSHci.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 16 lags.
##
## Value of test-statistic is: 4e-04
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216dSHci2.urkpss <- ur.kpss(dSHci2, type="tau", lags="short")
summary(dSHci2.urkpss)##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: tau with 16 lags.
##
## Value of test-statistic is: 3e-04
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.119 0.146 0.176 0.216ERS test
SHci.ers <- ur.ers(SHci, type="P-test", model="trend")
summary(SHci.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.0603
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dSHci.ers <- ur.ers(dSHci, type="P-test", model="trend")
summary(dSHci.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.0512
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dSHci2.ers <- ur.ers(dSHci2, type="P-test", model="trend")
summary(dSHci2.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.7694
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Catogry C
overview
library(Quandl)
library(forecast)
library(urca)
cpi <- Quandl("FRED/CPIAUCSL", type="zoo")
str(cpi)## 'zooreg' series from Jan 1947 to Mar 2017
## Data: num [1:843] 21.5 21.6 22 22 21.9 ...
## Index: Class 'yearmon' num [1:843] 1947 1947 1947 1947 1947 ...
## Frequency: 12summary(cpi)## Index cpi
## Min. :1947 Min. : 21.48
## 1st Qu.:1965 1st Qu.: 31.04
## Median :1982 Median : 94.70
## Mean :1982 Mean :104.58
## 3rd Qu.:2000 3rd Qu.:167.45
## Max. :2017 Max. :244.46head(cpi)## Jan 1947 Feb 1947 Mar 1947 Apr 1947 May 1947 Jun 1947
## 21.48 21.62 22.00 22.00 21.95 22.08tail(cpi)## Oct 2016 Nov 2016 Dec 2016 Jan 2017 Feb 2017 Mar 2017
## 241.694 242.199 242.821 244.158 244.456 243.752part 1 ,2
Oroginal Data
plot(cpi, xlab="Years", ylab="", main="consumer Price Index for All Urban Consumers")
Logarithm
lcpi <- log(cpi)plot
plot(lcpi, xlab="Years", ylab="", main="Log of consumer Price Index for All Urban Consumers")
##Part 3 #first differences of original data
dcpi <- diff(cpi)
summary(dcpi)## Index dcpi
## Min. :1947 Min. :-3.842
## 1st Qu.:1965 1st Qu.: 0.057
## Median :1982 Median : 0.200
## Mean :1982 Mean : 0.264
## 3rd Qu.:2000 3rd Qu.: 0.400
## Max. :2017 Max. : 2.700plot(dcpi, xlab="Years", ylab="", main="first differences of consumer Price Index for All Urban Consumers")
2nd differences of original data.
dcpi2 <-diff(dcpi)
summary(dcpi2)## Index dcpi2
## Min. :1947 Min. :-2.400000
## 1st Qu.:1965 1st Qu.:-0.100000
## Median :1982 Median : 0.000000
## Mean :1982 Mean :-0.001004
## 3rd Qu.:2000 3rd Qu.: 0.100000
## Max. :2017 Max. : 2.290000plot(dcpi2, type= "l", xlab= "Years", ylab="dcpi", main="Second differences of consumer Price Index for All Urban Consumers", major.format="%Y Q%q")## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter
Part 4
ADF,KPSS,ERS tests original data
1.ADF
library(tseries)
adf.test(cpi)##
## Augmented Dickey-Fuller Test
##
## data: cpi
## Dickey-Fuller = -2.8415, Lag order = 9, p-value = 0.2221
## alternative hypothesis: stationaryadf.test(dcpi)## Warning in adf.test(dcpi): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dcpi
## Dickey-Fuller = -6.4061, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationaryadf.test(dcpi2)## Warning in adf.test(dcpi2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dcpi2
## Dickey-Fuller = -16.132, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationary2.KPSS test
kpss.test(cpi, null="Trend")## Warning in kpss.test(cpi, null = "Trend"): p-value smaller than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: cpi
## KPSS Trend = 2.593, Truncation lag parameter = 6, p-value = 0.01kpss.test(dcpi, null="Trend")## Warning in kpss.test(dcpi, null = "Trend"): p-value smaller than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: dcpi
## KPSS Trend = 0.53688, Truncation lag parameter = 6, p-value = 0.01kpss.test(dcpi2, null="Trend")## Warning in kpss.test(dcpi2, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: dcpi2
## KPSS Trend = 0.0092016, Truncation lag parameter = 6, p-value =
## 0.1ERS test
cpi.ers <- ur.ers(cpi, type="P-test", model="trend")
summary(cpi.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 430.6597
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dcpi.ers <- ur.ers(dcpi, type="P-test", model="trend")
summary(dcpi.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.7125
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dcpi2.ers <- ur.ers(dcpi2, type="P-test", model="trend")
summary(dcpi2.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.6564
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Catogry D
overview
library(Quandl)
library(forecast)
library(urca)
TBsmr <- Quandl("FRED/TB3MS", type="zoo")
str(TBsmr)## 'zooreg' series from Jan 1934 to Mar 2017
## Data: num [1:999] 0.72 0.62 0.24 0.15 0.16 0.15 0.15 0.19 0.21 0.27 ...
## Index: Class 'yearmon' num [1:999] 1934 1934 1934 1934 1934 ...
## Frequency: 12summary(TBsmr)## Index TBsmr
## Min. :1934 Min. : 0.010
## 1st Qu.:1955 1st Qu.: 0.380
## Median :1976 Median : 3.040
## Mean :1976 Mean : 3.521
## 3rd Qu.:1996 3rd Qu.: 5.355
## Max. :2017 Max. :16.300head(TBsmr)## Jan 1934 Feb 1934 Mar 1934 Apr 1934 May 1934 Jun 1934
## 0.72 0.62 0.24 0.15 0.16 0.15tail(TBsmr)## Oct 2016 Nov 2016 Dec 2016 Jan 2017 Feb 2017 Mar 2017
## 0.33 0.45 0.51 0.51 0.52 0.74part 1 ,2
Oroginal Data
plot(TBsmr, xlab="Years", ylab="", main="3-Month Treasury Bills Secondary Market Rate")
Logarithm
lTBsmr <- log(TBsmr)plot
plot(lTBsmr, xlab="Years", ylab="", main="Log of 3-Month Treasury Bills Secondary Market Rate")
##Part 3 #first differences of original data
dTBsmr <- diff(TBsmr)
summary(dTBsmr)## Index dTBsmr
## Min. :1934 Min. :-4.62000
## 1st Qu.:1955 1st Qu.:-0.05000
## Median :1976 Median : 0.00000
## Mean :1976 Mean : 0.00002
## 3rd Qu.:1996 3rd Qu.: 0.09000
## Max. :2017 Max. : 2.61000plot(dTBsmr, xlab="Years", ylab="", main="first differences of 3-Month Treasury Bills Secondary Market Rate")
2nd differences of original data.
dTBsmr2 <-diff(dTBsmr)
summary(dTBsmr2)## Index dTBsmr2
## Min. :1934 Min. :-4.340000
## 1st Qu.:1955 1st Qu.:-0.080000
## Median :1976 Median : 0.000000
## Mean :1976 Mean : 0.000321
## 3rd Qu.:1996 3rd Qu.: 0.090000
## Max. :2017 Max. : 3.110000plot(dTBsmr2, type= "l", xlab= "Years", ylab="dTBsmr", main="Second differences of 3-Month Treasury Bills Secondary Market Rate", major.format="%Y Q%q")## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter
Part 4
ADF,KPSS,ERS tests original data
1.ADF
library(tseries)
adf.test(TBsmr)##
## Augmented Dickey-Fuller Test
##
## data: TBsmr
## Dickey-Fuller = -2.1747, Lag order = 9, p-value = 0.5044
## alternative hypothesis: stationaryadf.test(dTBsmr)## Warning in adf.test(dTBsmr): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dTBsmr
## Dickey-Fuller = -9.4505, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationaryadf.test(dTBsmr2)## Warning in adf.test(dTBsmr2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dTBsmr2
## Dickey-Fuller = -18.521, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationary2.KPSS test
kpss.test(TBsmr, null="Trend")## Warning in kpss.test(TBsmr, null = "Trend"): p-value smaller than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: TBsmr
## KPSS Trend = 2.311, Truncation lag parameter = 7, p-value = 0.01kpss.test(dTBsmr, null="Trend")## Warning in kpss.test(dTBsmr, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: dTBsmr
## KPSS Trend = 0.034017, Truncation lag parameter = 7, p-value = 0.1kpss.test(dTBsmr2, null="Trend")## Warning in kpss.test(dTBsmr2, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: dTBsmr2
## KPSS Trend = 0.0047924, Truncation lag parameter = 7, p-value =
## 0.1ERS test
TBsmr.ers <- ur.ers(TBsmr, type="P-test", model="trend")
summary(TBsmr.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 11.6282
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dTBsmr.ers <- ur.ers(dTBsmr, type="P-test", model="trend")
summary(dTBsmr.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.1397
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dTBsmr2.ers <- ur.ers(dTBsmr2, type="P-test", model="trend")
summary(dTBsmr2.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.1808
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Catogry E
overview
library(Quandl)
library(forecast)
library(urca)
Unr <- Quandl("FRED/UNRATE", type="zoo")
str(Unr)## 'zooreg' series from Jan 1948 to Mar 2017
## Data: num [1:831] 3.4 3.8 4 3.9 3.5 3.6 3.6 3.9 3.8 3.7 ...
## Index: Class 'yearmon' num [1:831] 1948 1948 1948 1948 1948 ...
## Frequency: 12summary(Unr)## Index Unr
## Min. :1948 Min. : 2.500
## 1st Qu.:1965 1st Qu.: 4.700
## Median :1983 Median : 5.600
## Mean :1983 Mean : 5.806
## 3rd Qu.:2000 3rd Qu.: 6.900
## Max. :2017 Max. :10.800head(Unr)## Jan 1948 Feb 1948 Mar 1948 Apr 1948 May 1948 Jun 1948
## 3.4 3.8 4.0 3.9 3.5 3.6tail(Unr)## Oct 2016 Nov 2016 Dec 2016 Jan 2017 Feb 2017 Mar 2017
## 4.8 4.6 4.7 4.8 4.7 4.5part 1 ,2
Oroginal Data
plot(Unr, xlab="Years", ylab="", main="Unemployment Rate")
Logarithm
lUnr <- log(Unr)plot
plot(lUnr, xlab="Years", ylab="", main="Log of Unemployment Rate")
##Part 3 #first differences of original data
dUnr <- diff(Unr)
summary(dUnr)## Index dUnr
## Min. :1948 Min. :-1.500000
## 1st Qu.:1965 1st Qu.:-0.100000
## Median :1983 Median : 0.000000
## Mean :1983 Mean : 0.001325
## 3rd Qu.:2000 3rd Qu.: 0.100000
## Max. :2017 Max. : 1.300000plot(dUnr, xlab="Years", ylab="", main="first differences of Unemployment Rate")
2nd differences of original data.
dUnr2 <-diff(dUnr)
summary(dUnr2)## Index dUnr2
## Min. :1948 Min. :-2.8000000
## 1st Qu.:1965 1st Qu.:-0.2000000
## Median :1983 Median : 0.0000000
## Mean :1983 Mean :-0.0007238
## 3rd Qu.:2000 3rd Qu.: 0.2000000
## Max. :2017 Max. : 1.7000000plot(dUnr2, type= "l", xlab= "Years", ylab="dUnr", main="Second differences of Unemployment Rate", major.format="%Y Q%q")## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter
Part 4
ADF,KPSS,ERS tests original data
1.ADF
library(tseries)
adf.test(Unr)##
## Augmented Dickey-Fuller Test
##
## data: Unr
## Dickey-Fuller = -3.9387, Lag order = 9, p-value = 0.01206
## alternative hypothesis: stationaryadf.test(dUnr)## Warning in adf.test(dUnr): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dUnr
## Dickey-Fuller = -8.2844, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationaryadf.test(dUnr2)## Warning in adf.test(dUnr2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dUnr2
## Dickey-Fuller = -12.179, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationary2.KPSS test
kpss.test(Unr, null="Trend")## Warning in kpss.test(Unr, null = "Trend"): p-value smaller than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: Unr
## KPSS Trend = 0.50543, Truncation lag parameter = 6, p-value = 0.01kpss.test(dUnr, null="Trend")## Warning in kpss.test(dUnr, null = "Trend"): p-value greater than printed p-
## value##
## KPSS Test for Trend Stationarity
##
## data: dUnr
## KPSS Trend = 0.031636, Truncation lag parameter = 6, p-value = 0.1kpss.test(dUnr2, null="Trend")## Warning in kpss.test(dUnr2, null = "Trend"): p-value greater than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: dUnr2
## KPSS Trend = 0.0073992, Truncation lag parameter = 6, p-value =
## 0.1ERS test
Unr.ers <- ur.ers(Unr, type="P-test", model="trend")
summary(Unr.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 3.8756
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dUnr.ers <- ur.ers(dUnr, type="P-test", model="trend")
summary(dUnr.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 2.1915
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dUnr2.ers <- ur.ers(dUnr2, type="P-test", model="trend")
summary(dUnr2.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.3631
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89Catogry F
overview
library(Quandl)
library(forecast)
library(urca)
CUSAfex <- Quandl("FRED/EXCAUS", type="zoo")
str(CUSAfex)## 'zooreg' series from Jan 1971 to Apr 2017
## Data: num [1:556] 1.01 1.01 1.01 1.01 1.01 ...
## Index: Class 'yearmon' num [1:556] 1971 1971 1971 1971 1971 ...
## Frequency: 12summary(CUSAfex)## Index CUSAfex
## Min. :1971 Min. :0.9553
## 1st Qu.:1983 1st Qu.:1.0576
## Median :1994 Median :1.1988
## Mean :1994 Mean :1.2170
## 3rd Qu.:2006 3rd Qu.:1.3508
## Max. :2017 Max. :1.5997head(CUSAfex)## Jan 1971 Feb 1971 Mar 1971 Apr 1971 May 1971 Jun 1971
## 1.0118 1.0075 1.0064 1.0077 1.0087 1.0213tail(CUSAfex)## Nov 2016 Dec 2016 Jan 2017 Feb 2017 Mar 2017 Apr 2017
## 1.3434 1.3339 1.3183 1.3109 1.3387 1.3358part 1 ,2
Oroginal Data
plot(CUSAfex, xlab="Years", ylab="", main="Canada / U.S. Foreign Exchange Rate")
Logarithm
lCUSAfex <- log(CUSAfex)plot
plot(lCUSAfex, xlab="Years", ylab="", main="Log of Canada / U.S. Foreign Exchange Rate")
##Part 3 #first differences of original data
dCUSAfex <- diff(CUSAfex)
summary(dCUSAfex)## Index dCUSAfex
## Min. :1971 Min. :-0.0742000
## 1st Qu.:1983 1st Qu.:-0.0090000
## Median :1994 Median : 0.0001000
## Mean :1994 Mean : 0.0005838
## 3rd Qu.:2006 3rd Qu.: 0.0105500
## Max. :2017 Max. : 0.1265000plot(dCUSAfex, xlab="Years", ylab="", main="first differences of Canada / U.S. Foreign Exchange Rate")
2nd differences of original data.
dCUSAfex2 <-diff(dCUSAfex)
summary(dCUSAfex2)## Index dCUSAfex2
## Min. :1971 Min. :-9.410e-02
## 1st Qu.:1983 1st Qu.:-1.130e-02
## Median :1994 Median : 2.000e-04
## Mean :1994 Mean : 2.530e-06
## 3rd Qu.:2006 3rd Qu.: 1.140e-02
## Max. :2017 Max. : 1.218e-01plot(dCUSAfex2, type= "l", xlab= "Years", ylab="dCUSAfex", main="Second differences of Canada / U.S. Foreign Exchange Rate", major.format="%Y Q%q")## Warning in plot.window(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "major.format" is not a graphical
## parameter## Warning in axis(side, at = z, labels = labels, ...): "major.format" is not
## a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "major.format"
## is not a graphical parameter## Warning in box(...): "major.format" is not a graphical parameter## Warning in title(...): "major.format" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "major.format" is
## not a graphical parameter
Part 4
ADF,KPSS,ERS tests original data
1.ADF
library(tseries)
adf.test(CUSAfex)##
## Augmented Dickey-Fuller Test
##
## data: CUSAfex
## Dickey-Fuller = -1.7258, Lag order = 8, p-value = 0.6944
## alternative hypothesis: stationaryadf.test(dCUSAfex)## Warning in adf.test(dCUSAfex): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dCUSAfex
## Dickey-Fuller = -7.6633, Lag order = 8, p-value = 0.01
## alternative hypothesis: stationaryadf.test(dCUSAfex2)## Warning in adf.test(dCUSAfex2): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: dCUSAfex2
## Dickey-Fuller = -14.707, Lag order = 8, p-value = 0.01
## alternative hypothesis: stationary2.KPSS test
kpss.test(CUSAfex, null="Trend")## Warning in kpss.test(CUSAfex, null = "Trend"): p-value smaller than printed
## p-value##
## KPSS Test for Trend Stationarity
##
## data: CUSAfex
## KPSS Trend = 1.3198, Truncation lag parameter = 5, p-value = 0.01kpss.test(dCUSAfex, null="Trend")## Warning in kpss.test(dCUSAfex, null = "Trend"): p-value greater than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: dCUSAfex
## KPSS Trend = 0.098258, Truncation lag parameter = 5, p-value = 0.1kpss.test(dCUSAfex2, null="Trend")## Warning in kpss.test(dCUSAfex2, null = "Trend"): p-value greater than
## printed p-value##
## KPSS Test for Trend Stationarity
##
## data: dCUSAfex2
## KPSS Trend = 0.0054261, Truncation lag parameter = 5, p-value =
## 0.1ERS test
CUSAfex.ers <- ur.ers(CUSAfex, type="P-test", model="trend")
summary(CUSAfex.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 18.2746
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dCUSAfex.ers <- ur.ers(dCUSAfex, type="P-test", model="trend")
summary(dCUSAfex.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.3213
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89dCUSAfex2.ers <- ur.ers(dCUSAfex2, type="P-test", model="trend")
summary(dCUSAfex2.ers)##
## ###############################################
## # Elliot, Rothenberg and Stock Unit Root Test #
## ###############################################
##
## Test of type P-test
## detrending of series with intercept and trend
##
## Value of test-statistic is: 0.0012
##
## Critical values of P-test are:
## 1pct 5pct 10pct
## critical values 3.96 5.62 6.89