Category 1

Real personal consumption expenditures per capita

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.numeric
library(zoo)
library(xts)
library(dygraphs)
library(knitr)
library(forecast)
library(tseries)
library(urca)
Quandl.api_key('CEFP3eWxEJwr_uUP9a2D')
RPCE <- Quandl("FRED/A794RX0Q048SBEA", type="ts")

Original Time Series Graph

plot(RPCE, xlab=" Years", ylab="", main="Log Real personal consumption expenditures per capita" )

Log-transformed index Graph

l.RPCE <- log(RPCE)
plot(l.RPCE, xlab=" Years", ylab="", main="Real personal consumption expenditures per capita" )

Original and Log-transformed time series grows steadily and shows increasing variability over time.

dl.RPCE <- 100*diff(l.RPCE,lag=1)
plot(dl.RPCE, xlab=" Years", ylab="", main="Diff Log Real personal consumption expenditures per capita" )

ADF test

adf.test(l.RPCE)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  l.RPCE
## Dickey-Fuller = -1.169, Lag order = 6, p-value = 0.91
## alternative hypothesis: stationary

By ADF test we cannot reject the null hypothesis that the time series has a unit root.

KPSS test

kpss.test(l.RPCE, null="Trend")
## Warning in kpss.test(l.RPCE, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  l.RPCE
## KPSS Trend = 0.75764, Truncation lag parameter = 3, p-value = 0.01
lRPCE.urkpss <- ur.kpss(l.RPCE, type="tau", lags="short")
summary(lRPCE.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 5 lags. 
## 
## Value of test-statistic is: 0.5183 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

By KPSS test we reject the null hypothesis that the time series is stationary.

ERS test

lRPCE.urers1 <- ur.ers(l.RPCE, type="P-test", model="trend")
summary(lRPCE.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: 10.5264 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
lRPCE.urers2 <- ur.ers(l.RPCE, type="DF-GLS", model="trend")
summary(lRPCE.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.044935 -0.003521  0.000139  0.004139  0.035773 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.01701    0.01027  -1.656   0.0988 .  
## yd.diff.lag1  0.06119    0.05988   1.022   0.3078    
## yd.diff.lag2  0.37417    0.05976   6.261 1.49e-09 ***
## yd.diff.lag3  0.03304    0.06034   0.548   0.5845    
## yd.diff.lag4 -0.13376    0.06018  -2.223   0.0271 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.007614 on 271 degrees of freedom
## Multiple R-squared:  0.1397, Adjusted R-squared:  0.1238 
## F-statistic: 8.802 on 5 and 271 DF,  p-value: 9.37e-08
## 
## 
## Value of test-statistic is: -1.6562 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

Both ERS tests (P-test and DF-GLS test) suggest that our time series is not stationary.

First differece tests of ADF, KPSS and ERS

Now we perform ADF, KPSS and ERS tests for the Real personal consumption expenditures per capita in first differences.

ADF test

adf.test(dl.RPCE)
## Warning in adf.test(dl.RPCE): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  dl.RPCE
## Dickey-Fuller = -5.7979, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationary

KPSS test

kpss.test(dl.RPCE, null="Level")
## Warning in kpss.test(dl.RPCE, null = "Level"): p-value greater than printed
## p-value
## 
##  KPSS Test for Level Stationarity
## 
## data:  dl.RPCE
## KPSS Level = 0.17227, Truncation lag parameter = 3, p-value = 0.1
lRPCE.urkpss <- ur.kpss(dl.RPCE, type="tau", lags="short")
summary(lRPCE.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 5 lags. 
## 
## Value of test-statistic is: 0.0878 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

ERS test

dlRPCE.urers1 <- ur.ers(dl.RPCE, type="P-test", model="trend")
summary(dlRPCE.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: 0.747 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
dlRPCE.urers2 <- ur.ers(dl.RPCE, type="DF-GLS", model="trend")
summary(dlRPCE.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9583 -0.3908  0.0216  0.3550  3.2568 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.55507    0.09130  -6.080 4.09e-09 ***
## yd.diff.lag1 -0.35260    0.08819  -3.998 8.24e-05 ***
## yd.diff.lag2  0.04955    0.08623   0.575    0.566    
## yd.diff.lag3  0.09053    0.08093   1.119    0.264    
## yd.diff.lag4 -0.03140    0.06027  -0.521    0.603    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7791 on 270 degrees of freedom
## Multiple R-squared:  0.5073, Adjusted R-squared:  0.4982 
## F-statistic:  55.6 on 5 and 270 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -6.0799 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

All tests confirm that first difference logarithmic Real personal consumption expenditures per capita is stationary.

Category B

S&P 500 Index

SNP.all <- Quandl("YAHOO/INDEX_GSPC", type="zoo") 
SNP <- ts(SNP.all$Close)
l.SNP <- log(SNP)
plot(SNP, xlab=" Years", ylab="", main="S&P 500 Index" )

plot(l.SNP, xlab="Years", ylab="", main="Log S&P 500 Index" )

dl.SNP <- 100*diff(l.SNP,lag=1)
plot(dl.SNP, xlab=" 1950-2017", ylab="", main="Diff of Log S&P 500 Index"  )

ADF test

adf.test(l.SNP)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  l.SNP
## Dickey-Fuller = -2.2359, Lag order = 25, p-value = 0.4784
## alternative hypothesis: stationary

ADF test shoes that we cannot reject the null hypothesis that the time series has a unit root.

KPSS test

kpss.test(l.SNP, null="Trend")
## Warning in kpss.test(l.SNP, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  l.SNP
## KPSS Trend = 4.2602, Truncation lag parameter = 30, p-value = 0.01
lSNP.urkpss <- ur.kpss(l.SNP, type="tau", lags="short")
summary(lSNP.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 14 lags. 
## 
## Value of test-statistic is: 8.7771 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

KPSS test suggests to reject the null hypothesis that the time series is stationary.

ERS test

lSNP.urers1 <- ur.ers(l.SNP, type="P-test", model="trend")
summary(lSNP.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: 11.9736 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
lSNP.urers2 <- ur.ers(l.SNP, type="DF-GLS", model="trend")
summary(lSNP.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.228557 -0.004316  0.000253  0.004710  0.105621 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.0004540  0.0002344  -1.937 0.052814 .  
## yd.diff.lag1  0.0286647  0.0076852   3.730 0.000192 ***
## yd.diff.lag2 -0.0422727  0.0076883  -5.498 3.89e-08 ***
## yd.diff.lag3  0.0031772  0.0076882   0.413 0.679427    
## yd.diff.lag4 -0.0081289  0.0076849  -1.058 0.290176    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.009673 on 16930 degrees of freedom
## Multiple R-squared:  0.002805,   Adjusted R-squared:  0.00251 
## F-statistic: 9.524 on 5 and 16930 DF,  p-value: 4.375e-09
## 
## 
## Value of test-statistic is: -1.9366 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

According to ERS tests our time series is not stationary.

First differece tests of ADF, KPSS and ERS

ADF test

adf.test(dl.SNP)
## Warning in adf.test(dl.SNP): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  dl.SNP
## Dickey-Fuller = -26.454, Lag order = 25, p-value = 0.01
## alternative hypothesis: stationary

KPSS test

kpss.test(dl.SNP, null="Level")
## Warning in kpss.test(dl.SNP, null = "Level"): p-value greater than printed
## p-value
## 
##  KPSS Test for Level Stationarity
## 
## data:  dl.SNP
## KPSS Level = 0.070918, Truncation lag parameter = 30, p-value =
## 0.1
lSNP.urkpss <- ur.kpss(dl.SNP, type="tau", lags="short")
summary(lSNP.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 14 lags. 
## 
## Value of test-statistic is: 0.0653 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

ERS tests

dlSNP.urers1 <- ur.ers(dl.SNP, type="P-test", model="trend")
summary(dlSNP.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: 0.0132 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
dlSNP.urers2 <- ur.ers(dl.SNP, type="DF-GLS", model="trend")
summary(dlSNP.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.6921  -0.6338  -0.1500   0.3615  13.1977 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.393329   0.011314  -34.76   <2e-16 ***
## yd.diff.lag1 -0.448796   0.011593  -38.71   <2e-16 ***
## yd.diff.lag2 -0.366503   0.011095  -33.03   <2e-16 ***
## yd.diff.lag3 -0.233428   0.009846  -23.71   <2e-16 ***
## yd.diff.lag4 -0.116634   0.007633  -15.28   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.029 on 16929 degrees of freedom
## Multiple R-squared:  0.4202, Adjusted R-squared:   0.42 
## F-statistic:  2454 on 5 and 16929 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -34.7644 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

S&P 500 Index is stationary according to all the tests.

Category C

Gross Domestic Product: Implicit Price Deflator

IPD <- Quandl("FRED/GDPDEF", type="ts")

We are going to plot the original and log-transformed GDP : Implicit Price Deflator.

plot(IPD, xlab="Years", ylab="", main="GDP : Implicit Price Deflator" )

l.IPD <- log(IPD)
plot(l.IPD, xlab=" Years", ylab="", main="Log GDP : Implicit Price Deflator" )

dl.IPD <- 100*diff(l.IPD,lag=1)
plot(dl.IPD, xlab="Years", ylab="", main="Diff Log GDP : Implicit Price Deflator"  )

ADF test

adf.test(l.IPD)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  l.IPD
## Dickey-Fuller = -1.0433, Lag order = 6, p-value = 0.9303
## alternative hypothesis: stationary

ADF test shows that we cannot reject the null hypothesis that the time series has a unit root.

KPSS test

kpss.test(l.IPD, null="Trend")
## Warning in kpss.test(l.IPD, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  l.IPD
## KPSS Trend = 0.98785, Truncation lag parameter = 3, p-value = 0.01
lIPD.urkpss <- ur.kpss(l.IPD, type="tau", lags="short")
summary(lIPD.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 5 lags. 
## 
## Value of test-statistic is: 0.6633 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

KPSS test shows that we reject the null hypothesis that the time series is stationary.

ERS test

lIPD.urers1 <- ur.ers(l.IPD, type="P-test", model="trend")
summary(lIPD.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: 30.7793 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
lIPD.urers2 <- ur.ers(l.IPD, type="DF-GLS", model="trend")
summary(lIPD.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0213074 -0.0017538 -0.0003115  0.0013199  0.0192671 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.001905   0.001645  -1.158   0.2480    
## yd.diff.lag1  0.585564   0.059548   9.834   <2e-16 ***
## yd.diff.lag2  0.153612   0.066944   2.295   0.0225 *  
## yd.diff.lag3  0.153397   0.067009   2.289   0.0228 *  
## yd.diff.lag4 -0.040672   0.058979  -0.690   0.4910    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.003752 on 271 degrees of freedom
## Multiple R-squared:  0.6418, Adjusted R-squared:  0.6352 
## F-statistic: 97.12 on 5 and 271 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -1.1578 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

Both ERS tests shows that our time series is not stationary.

First differece tests of ADF, KPSS and ERS

ADF test

adf.test(dl.IPD)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  dl.IPD
## Dickey-Fuller = -3.5512, Lag order = 6, p-value = 0.03824
## alternative hypothesis: stationary

KPSS test

kpss.test(dl.IPD, null="Level")
## Warning in kpss.test(dl.IPD, null = "Level"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Level Stationarity
## 
## data:  dl.IPD
## KPSS Level = 0.9464, Truncation lag parameter = 3, p-value = 0.01
dl.IPD.urkpss <- ur.kpss(dl.IPD, type="tau", lags="short")
summary(dl.IPD.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 5 lags. 
## 
## Value of test-statistic is: 0.5093 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

ERS tests

dl.IPD.urers1 <- ur.ers(dl.IPD, type="P-test", model="trend")
summary(dl.IPD.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: 3.3529 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
dl.IPD.urers2 <- ur.ers(dl.IPD, type="DF-GLS", model="trend")
summary(dl.IPD.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.25753 -0.17889 -0.02241  0.13887  1.82827 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.14755    0.03999  -3.690 0.000271 ***
## yd.diff.lag1 -0.26984    0.06514  -4.142  4.6e-05 ***
## yd.diff.lag2 -0.09394    0.06570  -1.430 0.153929    
## yd.diff.lag3  0.06550    0.06404   1.023 0.307347    
## yd.diff.lag4  0.04604    0.05905   0.780 0.436208    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3767 on 270 degrees of freedom
## Multiple R-squared:   0.17,  Adjusted R-squared:  0.1547 
## F-statistic: 11.06 on 5 and 270 DF,  p-value: 1.045e-09
## 
## 
## Value of test-statistic is: -3.6898 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

All tests confirm that the first difference log of GDP Implicit Price Deflator is stationary.

Category 4

10-Year Treasury with wConstant Maturity Rate

TYTSMR <- Quandl("FRED/DGS10", type="zoo") 
plot(TYTSMR, xlab="Years", ylab="", main="10-Year Treasury with Constant Maturity Rate" )

l.TYTSMR <- log(TYTSMR)
plot(l.TYTSMR, xlab="Years", ylab="", main="Log 10-Year Treasury with Constant Maturity Rate" )

d.TYTSMR <- 100*diff(TYTSMR,lag=1)
plot(d.TYTSMR, xlab="Years", ylab="", main="Diff Log 10-Year Treasury with Constant Maturity Rate"  )

ADF test

adf.test(TYTSMR)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  TYTSMR
## Dickey-Fuller = -2.1993, Lag order = 23, p-value = 0.4939
## alternative hypothesis: stationary

ADF test shows us that we cannot reject the null hypothesis that the time series has a unit root.

KPSS tests

kpss.test(TYTSMR, null="Trend")
## Warning in kpss.test(TYTSMR, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  TYTSMR
## KPSS Trend = 9.0346, Truncation lag parameter = 27, p-value = 0.01
TYTSMR.urkpss <- ur.kpss(TYTSMR, type="tau", lags="short")
summary(TYTSMR.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 13 lags. 
## 
## Value of test-statistic is: 18.0225 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

KPSS test shows us that we will reject the null hypothesis that the time series is stationary.

ERS tests

TYTSMR.urers1 <- ur.ers(TYTSMR, type="P-test", model="trend")
summary(TYTSMR.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: 32.3186 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
TYTSMR.urers2 <- ur.ers(TYTSMR, type="DF-GLS", model="trend")
summary(TYTSMR.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.74002 -0.02917 -0.00002  0.02896  0.63769 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.0001711  0.0001657  -1.033   0.3018    
## yd.diff.lag1  0.0692932  0.0085098   8.143 4.19e-16 ***
## yd.diff.lag2  0.0117515  0.0085302   1.378   0.1683    
## yd.diff.lag3 -0.0006357  0.0085304  -0.075   0.9406    
## yd.diff.lag4 -0.0181333  0.0085102  -2.131   0.0331 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.06622 on 13805 degrees of freedom
## Multiple R-squared:  0.00544,    Adjusted R-squared:  0.00508 
## F-statistic:  15.1 on 5 and 13805 DF,  p-value: 8.01e-15
## 
## 
## Value of test-statistic is: -1.0326 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

Both ERS tests suggest that our time series is not stationary.

First differece tests of ADF, KPSS and ERS

ADF test

adf.test(d.TYTSMR)
## Warning in adf.test(d.TYTSMR): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  d.TYTSMR
## Dickey-Fuller = -22.406, Lag order = 23, p-value = 0.01
## alternative hypothesis: stationary

KPSS tests

kpss.test(d.TYTSMR, null="Level")
## Warning in kpss.test(d.TYTSMR, null = "Level"): p-value greater than
## printed p-value
## 
##  KPSS Test for Level Stationarity
## 
## data:  d.TYTSMR
## KPSS Level = 0.20953, Truncation lag parameter = 27, p-value = 0.1
d.TYTSMR.urkpss <- ur.kpss(d.TYTSMR, type="tau", lags="short")
summary(d.TYTSMR.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 13 lags. 
## 
## Value of test-statistic is: 0.0469 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

ERS tests

d.TYTSMR.urers1 <- ur.ers(d.TYTSMR, type="P-test", model="trend")
summary(d.TYTSMR.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: 0.0158 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
d.TYTSMR.urers2 <- ur.ers(d.TYTSMR, type="DF-GLS", model="trend")
summary(d.TYTSMR.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -73.743  -2.333   1.149   3.707  62.830 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.726240   0.016274 -44.625  < 2e-16 ***
## yd.diff.lag1 -0.164396   0.014996 -10.963  < 2e-16 ***
## yd.diff.lag2 -0.116506   0.013400  -8.694  < 2e-16 ***
## yd.diff.lag3 -0.081771   0.011363  -7.197 6.49e-13 ***
## yd.diff.lag4 -0.065545   0.008493  -7.717 1.27e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.747 on 13804 degrees of freedom
## Multiple R-squared:  0.4449, Adjusted R-squared:  0.4447 
## F-statistic:  2213 on 5 and 13804 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -44.6249 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

All tests confirm that the first difference of log 10-Year Treasury with Constant Maturity Rate is stationary.

Category E

Employment Population Ratio

EPR <- Quandl("FRED/EMRATIO", type="ts")
plot(EPR, xlab="Years", ylab="", main="EPR" )

l.EPR <- log(EPR)
plot(l.EPR, xlab="Years", ylab="", main="Log EPR" )

d.EPR <- 100*diff(EPR,lag=1)
plot(d.EPR, xlab="Years", ylab="", main="Diff Log EPR"  )

ADF test

adf.test(EPR)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  EPR
## Dickey-Fuller = -2.4233, Lag order = 9, p-value = 0.3991
## alternative hypothesis: stationary

ADF test shows that we will not reject the null hypothesis that the time series has a unit root.

KPSS test

kpss.test(EPR, null="Trend")
## Warning in kpss.test(EPR, null = "Trend"): p-value smaller than printed p-
## value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  EPR
## KPSS Trend = 1.2292, Truncation lag parameter = 6, p-value = 0.01
EPR.urkpss <- ur.kpss(EPR, type="tau", lags="short")
summary(EPR.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 6 lags. 
## 
## Value of test-statistic is: 1.2292 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

KPSS test shows that we’d reject the null hypothesis that the time series is stationary.

ERS tests

EPR.urers1 <- ur.ers(EPR, type="P-test", model="trend")
summary(EPR.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: 12.5701 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
EPR.urers2 <- ur.ers(EPR, type="DF-GLS", model="trend")
summary(EPR.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.78175 -0.10833  0.00144  0.11638  0.80890 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.006589   0.003534  -1.864 0.062610 .  
## yd.diff.lag1 -0.141486   0.034532  -4.097 4.60e-05 ***
## yd.diff.lag2  0.132831   0.034452   3.855 0.000125 ***
## yd.diff.lag3  0.146381   0.034386   4.257 2.31e-05 ***
## yd.diff.lag4  0.127873   0.034288   3.729 0.000205 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1994 on 821 degrees of freedom
## Multiple R-squared:  0.06848,    Adjusted R-squared:  0.06281 
## F-statistic: 12.07 on 5 and 821 DF,  p-value: 2.666e-11
## 
## 
## Value of test-statistic is: -1.8645 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

Both ERS tests suggest that our time series is not stationary.

First differece tests of ADF, KPSS and ERS

ADF test

adf.test(d.EPR)
## Warning in adf.test(d.EPR): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  d.EPR
## Dickey-Fuller = -6.926, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationary

KPSS test

kpss.test(d.EPR, null="Level")
## Warning in kpss.test(d.EPR, null = "Level"): p-value greater than printed
## p-value
## 
##  KPSS Test for Level Stationarity
## 
## data:  d.EPR
## KPSS Level = 0.12379, Truncation lag parameter = 6, p-value = 0.1
d.EPR.urkpss <- ur.kpss(d.EPR, type="tau", lags="short")
summary(d.EPR.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 6 lags. 
## 
## Value of test-statistic is: 0.098 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

ERS tests

d.EPR.urers1 <- ur.ers(d.EPR, type="P-test", model="trend")
summary(d.EPR.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: 1.2622 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
d.EPR.urers2 <- ur.ers(d.EPR, type="DF-GLS", model="trend")
summary(d.EPR.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -81.932 -12.614  -1.301   9.992  82.680 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.54673    0.06602  -8.282 4.91e-16 ***
## yd.diff.lag1 -0.58224    0.06354  -9.163  < 2e-16 ***
## yd.diff.lag2 -0.45006    0.05995  -7.508 1.57e-13 ***
## yd.diff.lag3 -0.28528    0.05123  -5.568 3.48e-08 ***
## yd.diff.lag4 -0.11873    0.03405  -3.487 0.000515 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.92 on 820 degrees of freedom
## Multiple R-squared:  0.5816, Adjusted R-squared:  0.579 
## F-statistic:   228 on 5 and 820 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -8.2818 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

All tests confirm that the first difference log EPR is stationary.

Category F

U.S. / U.K. Foreign Exchange Rate

EXUSUK <- Quandl("FRED/EXUSUK", type="ts") 
plot(EXUSUK, xlab=" Years", ylab="", main="U.S. / U.K. FER" )

l.EXUSUK <- log(EXUSUK)
plot(l.EXUSUK, xlab=" Years", ylab="", main="U.S. / U.K. FER" )

d.EXUSUK <- 100*diff(EXUSUK,lag=1)
plot(d.EXUSUK, xlab="Years", ylab="", main="Diff Log U.S. / U.K. FER"  )

ADF test

adf.test(EXUSUK)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  EXUSUK
## Dickey-Fuller = -2.8386, Lag order = 8, p-value = 0.2233
## alternative hypothesis: stationary

ADF test shows that we cannot reject the null hypothesis that the time series has a unit root.

KPSS test

kpss.test(EXUSUK, null="Trend")
## Warning in kpss.test(EXUSUK, null = "Trend"): p-value smaller than printed
## p-value
## 
##  KPSS Test for Trend Stationarity
## 
## data:  EXUSUK
## KPSS Trend = 0.88408, Truncation lag parameter = 5, p-value = 0.01
EXUSUK.urkpss <- ur.kpss(EXUSUK, type="tau", lags="short")
summary(EXUSUK.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 6 lags. 
## 
## Value of test-statistic is: 0.7648 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

KPSS test shows that we will reject the null hypothesis that the time series is stationary.

ERS tests

EXUSUK.urers1 <- ur.ers(EXUSUK, type="P-test", model="trend")
summary(EXUSUK.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: 8.4934 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
EXUSUK.urers2 <- ur.ers(EXUSUK, type="DF-GLS", model="trend")
summary(EXUSUK.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.155650 -0.023298 -0.000354  0.024026  0.121032 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.012440   0.005427  -2.292  0.02226 *  
## yd.diff.lag1  0.409620   0.042626   9.610  < 2e-16 ***
## yd.diff.lag2 -0.135449   0.045930  -2.949  0.00332 ** 
## yd.diff.lag3  0.108469   0.045874   2.365  0.01840 *  
## yd.diff.lag4 -0.011065   0.042837  -0.258  0.79627    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03818 on 546 degrees of freedom
## Multiple R-squared:  0.1538, Adjusted R-squared:  0.146 
## F-statistic: 19.85 on 5 and 546 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -2.2924 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

Both ERS tests suggest that our time series is not stationary.

First differece tests of ADF, KPSS and ERS

ADF test

adf.test(d.EXUSUK)
## Warning in adf.test(d.EXUSUK): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  d.EXUSUK
## Dickey-Fuller = -7.6658, Lag order = 8, p-value = 0.01
## alternative hypothesis: stationary

KPSS test

kpss.test(d.EXUSUK, null="Level")
## Warning in kpss.test(d.EXUSUK, null = "Level"): p-value greater than
## printed p-value
## 
##  KPSS Test for Level Stationarity
## 
## data:  d.EXUSUK
## KPSS Level = 0.059692, Truncation lag parameter = 5, p-value = 0.1
d.EXUSUK.urkpss <- ur.kpss(d.EXUSUK, type="tau", lags="short")
summary(d.EXUSUK.urkpss)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 6 lags. 
## 
## Value of test-statistic is: 0.05 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216

ERS tests

d.EXUSUK.urers1 <- ur.ers(d.EXUSUK, type="P-test", model="trend")
summary(d.EXUSUK.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: 0.4452 
## 
## Critical values of P-test are:
##                 1pct 5pct 10pct
## critical values 3.96 5.62  6.89
d.EXUSUK.urers2 <- ur.ers(d.EXUSUK, type="DF-GLS", model="trend")
summary(d.EXUSUK.urers2)
## 
## ############################################### 
## # Elliot, Rothenberg and Stock Unit Root Test # 
## ############################################### 
## 
## Test of type DF-GLS 
## detrending of series with intercept and trend 
## 
## 
## Call:
## lm(formula = dfgls.form, data = data.dfgls)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.7951  -2.5567  -0.3266   2.1186  11.4357 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## yd.lag       -0.61916    0.06622  -9.350   <2e-16 ***
## yd.diff.lag1  0.04066    0.06330   0.642    0.521    
## yd.diff.lag2 -0.08909    0.05793  -1.538    0.125    
## yd.diff.lag3  0.02120    0.04948   0.428    0.669    
## yd.diff.lag4  0.02648    0.04287   0.618    0.537    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.867 on 545 degrees of freedom
## Multiple R-squared:  0.3246, Adjusted R-squared:  0.3184 
## F-statistic: 52.38 on 5 and 545 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -9.3504 
## 
## Critical values of DF-GLS are:
##                  1pct  5pct 10pct
## critical values -3.48 -2.89 -2.57

All tests confirm that the first difference U.S. / U.K. Foreign Exchange Rate is stationary.

All the analyzed time series appear to be I(1)