library(tseries)
library(urca)
library(readr)
library(zoo)


Setup

data <- read.csv("M3b_stationary/module3b_data_PE_ratios.csv")
data$date <- as.Date(data$date)

subset_data <- subset(data, date >= as.Date("2000-04-01") & date <= as.Date("2015-02-01"))

ts_pe_aus <- ts(subset_data$pe_aus, start = c(2000, 4), end = c(2015, 2), frequency = 12)


Plot Data

plot(ts_pe_aus, 
     main = "Price-Earnings Ratio, Australia", 
     ylab = "PE Ratio", 
     xlab = "Year")


Compared to the PE-Ratio for India, Australia’s series tend to be more stable except for the sharp anomaly during the global financial crisis.
The P-E ratio spike (structural break) in late-2009 to mid-2010 is likely due to a temporary collapse in earnings while stock prices stayed flat or fell more slowly.

Check Stationarity


Augmented Dickey-Fuller (ADF) test

Null Hypothesis (H₀): The series has a unit root → it is non-stationary.
p < 0.05 -> Reject H₀. Series is stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is non-stationary at 5%.

adf_none     <- ur.df(ts_pe_aus, type = "none", selectlags = "AIC")
adf_drift    <- ur.df(ts_pe_aus, type = "drift", selectlags = "AIC")
adf_trend    <- ur.df(ts_pe_aus, type = "trend", selectlags = "AIC")

summary(adf_none)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -47.748  -0.296   0.466   1.024  19.316 
## 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)  
## z.lag.1    -0.02039    0.01401  -1.455   0.1475  
## z.diff.lag  0.15506    0.07468   2.076   0.0393 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.49 on 175 degrees of freedom
## Multiple R-squared:  0.03273,    Adjusted R-squared:  0.02167 
## F-statistic:  2.96 on 2 and 175 DF,  p-value: 0.0544
## 
## 
## Value of test-statistic is: -1.455 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.58 -1.95 -1.62
summary(adf_drift)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression drift 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -45.231  -0.831  -0.151   0.496  19.665 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  1.63843    0.65536   2.500  0.01334 * 
## z.lag.1     -0.07901    0.02721  -2.904  0.00417 **
## z.diff.lag   0.18452    0.07452   2.476  0.01424 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.424 on 174 degrees of freedom
## Multiple R-squared:  0.06627,    Adjusted R-squared:  0.05553 
## F-statistic: 6.174 on 2 and 174 DF,  p-value: 0.002567
## 
## 
## Value of test-statistic is: -2.9035 4.2153 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1  6.52  4.63  3.81
summary(adf_trend)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression trend 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -45.262  -0.855  -0.135   0.522  19.633 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  1.462352   0.851831   1.717  0.08782 . 
## z.lag.1     -0.079744   0.027375  -2.913  0.00405 **
## tt           0.002127   0.006548   0.325  0.74572   
## z.diff.lag   0.184832   0.074723   2.474  0.01434 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.435 on 173 degrees of freedom
## Multiple R-squared:  0.06684,    Adjusted R-squared:  0.05065 
## F-statistic:  4.13 on 3 and 173 DF,  p-value: 0.007379
## 
## 
## Value of test-statistic is: -2.913 2.8309 4.2464 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -3.99 -3.43 -3.13
## phi2  6.22  4.75  4.07
## phi3  8.43  6.49  5.47

Intercept appears to be most appropriate, since trend (tt) p-value is not significant.
Test-stat check against tau also confirms that the intercept level is significant. Test Statistic < Critical Value -> Reject H₀ → evidence of stationarity. Test Statistic ≥ Critical Value -> Fail to reject H₀ → series is likely non-stationary.

Phillips-Perron (PP) test

Null Hypothesis (H₀): The series has a unit root → it is non-stationary.
p < 0.05 -> Reject H₀. Series is stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is non-stationary at 5%.

pp_drift <- ur.pp(ts_pe_aus, type = "Z-alpha", model = "constant")
pp_trend <- ur.pp(ts_pe_aus, type = "Z-alpha", model = "trend")

summary(pp_drift)
## 
## ################################## 
## # Phillips-Perron Unit Root Test # 
## ################################## 
## 
## Test regression with intercept 
## 
## 
## Call:
## lm(formula = y ~ y.l1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -46.883  -0.761  -0.131   0.502  20.285 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.38428    0.65436   2.115   0.0358 *  
## y.l1         0.93331    0.02707  34.483   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.476 on 176 degrees of freedom
## Multiple R-squared:  0.8711, Adjusted R-squared:  0.8703 
## F-statistic:  1189 on 1 and 176 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic, type: Z-alpha  is: -17.4956 
## 
##          aux. Z statistics
## Z-tau-mu            2.5588
summary(pp_trend)
## 
## ################################## 
## # Phillips-Perron Unit Root Test # 
## ################################## 
## 
## Test regression with intercept and trend 
## 
## 
## Call:
## lm(formula = y ~ y.l1 + trend)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -46.913  -0.832  -0.104   0.519  20.258 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.395754   0.657441   2.123   0.0352 *  
## y.l1        0.932717   0.027225  34.259   <2e-16 ***
## trend       0.001788   0.006567   0.272   0.7858    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.488 on 175 degrees of freedom
## Multiple R-squared:  0.8711, Adjusted R-squared:  0.8697 
## F-statistic: 591.5 on 2 and 175 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic, type: Z-alpha  is: -17.6488 
## 
##            aux. Z statistics
## Z-tau-mu              0.9079
## Z-tau-beta            0.3183


Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test

Null Hypothesis (H₀): The series is stationary.
p < 0.05 -> Reject H₀. Series is non-stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is Series is stationary at 5%.

kpss_level   <- kpss.test(ts_pe_aus)
## Warning in kpss.test(ts_pe_aus): p-value greater than printed p-value
kpss_trend   <- kpss.test(ts_pe_aus, null = "Trend")
## Warning in kpss.test(ts_pe_aus, null = "Trend"): p-value greater than printed
## p-value
print(kpss_level)
## 
##  KPSS Test for Level Stationarity
## 
## data:  ts_pe_aus
## KPSS Level = 0.15229, Truncation lag parameter = 4, p-value = 0.1
print(kpss_trend)
## 
##  KPSS Test for Trend Stationarity
## 
## data:  ts_pe_aus
## KPSS Trend = 0.11185, Truncation lag parameter = 4, p-value = 0.1



Pre-Global Crisis

subset_preGFC <- subset(data, date >= as.Date("2000-04-01") & date <= as.Date("2009-05-01"))

ts_pe_aus_preGFC <- ts(subset_preGFC$pe_aus, start = c(2000, 4), end = c(2009, 5), frequency = 12)

plot(ts_pe_aus_preGFC, 
     main = "Price-Earnings Ratio, Australia", 
     ylab = "PE Ratio", 
     xlab = "Year")


Augmented Dickey-Fuller (ADF) test

Null Hypothesis (H₀): The series has a unit root → it is non-stationary.
p < 0.05 -> Reject H₀. Series is stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is non-stationary at 5%.

adf_none_pre    <- ur.df(ts_pe_aus_preGFC, type = "none", selectlags = "AIC")
adf_drift_pre   <- ur.df(ts_pe_aus_preGFC, type = "drift", selectlags = "AIC")
adf_trend_pre   <- ur.df(ts_pe_aus_preGFC, type = "trend", selectlags = "AIC")

summary(adf_none_pre)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.1696 -0.4818  0.1898  0.5409  6.2797 
## 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)
## z.lag.1    -0.00381    0.00821  -0.464    0.644
## z.diff.lag -0.07849    0.10047  -0.781    0.436
## 
## Residual standard error: 1.528 on 106 degrees of freedom
## Multiple R-squared:  0.007802,   Adjusted R-squared:  -0.01092 
## F-statistic: 0.4168 on 2 and 106 DF,  p-value: 0.6603
## 
## 
## Value of test-statistic is: -0.4641 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.58 -1.95 -1.62
summary(adf_drift_pre)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression drift 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9928 -0.6348 -0.0167  0.4880  5.5515 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  1.87104    0.80470   2.325   0.0220 *
## z.lag.1     -0.10659    0.04493  -2.372   0.0195 *
## z.diff.lag  -0.03031    0.10060  -0.301   0.7638  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.497 on 105 degrees of freedom
## Multiple R-squared:  0.05638,    Adjusted R-squared:  0.03841 
## F-statistic: 3.137 on 2 and 105 DF,  p-value: 0.04752
## 
## 
## Value of test-statistic is: -2.3724 2.8153 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1  6.52  4.63  3.81
summary(adf_trend_pre)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression trend 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9082 -0.5769 -0.0215  0.5346  5.5762 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  2.895705   1.422264   2.036   0.0443 *
## z.lag.1     -0.146548   0.064128  -2.285   0.0243 *
## tt          -0.005772   0.006602  -0.874   0.3840  
## z.diff.lag  -0.005885   0.104522  -0.056   0.9552  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.498 on 104 degrees of freedom
## Multiple R-squared:  0.06326,    Adjusted R-squared:  0.03624 
## F-statistic: 2.341 on 3 and 104 DF,  p-value: 0.07755
## 
## 
## Value of test-statistic is: -2.2852 2.1274 3.1898 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -3.99 -3.43 -3.13
## phi2  6.22  4.75  4.07
## phi3  8.43  6.49  5.47


Phillips-Perron (PP) test

Null Hypothesis (H₀): The series has a unit root → it is non-stationary.
p < 0.05 -> Reject H₀. Series is stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is non-stationary at 5%.

pp_drift_pre    <- ur.pp(ts_pe_aus_preGFC, type = "Z-alpha", model = "constant")
pp_trend_pre    <- ur.pp(ts_pe_aus_preGFC, type = "Z-alpha", model = "trend")
summary(pp_drift_pre)
## 
## ################################## 
## # Phillips-Perron Unit Root Test # 
## ################################## 
## 
## Test regression with intercept 
## 
## 
## Call:
## lm(formula = y ~ y.l1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.0584 -0.5844  0.0085  0.5204  5.5725 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.8688     0.7771   2.405   0.0179 *  
## y.l1          0.8941     0.0433  20.648   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.486 on 107 degrees of freedom
## Multiple R-squared:  0.7994, Adjusted R-squared:  0.7975 
## F-statistic: 426.3 on 1 and 107 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic, type: Z-alpha  is: -10.4331 
## 
##          aux. Z statistics
## Z-tau-mu            2.2906
summary(pp_trend_pre)
## 
## ################################## 
## # Phillips-Perron Unit Root Test # 
## ################################## 
## 
## Test regression with intercept and trend 
## 
## 
## Call:
## lm(formula = y ~ y.l1 + trend)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9238 -0.5641 -0.0064  0.5254  5.5916 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.598900   1.071187   2.426   0.0169 *  
## y.l1         0.852873   0.060052  14.202   <2e-16 ***
## trend       -0.006214   0.006274  -0.990   0.3242    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.486 on 106 degrees of freedom
## Multiple R-squared:  0.8012, Adjusted R-squared:  0.7975 
## F-statistic: 213.6 on 2 and 106 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic, type: Z-alpha  is: -15.3423 
## 
##            aux. Z statistics
## Z-tau-mu              2.5867
## Z-tau-beta           -0.9440



Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test

Null Hypothesis (H₀): The series is stationary.
p < 0.05 -> Reject H₀. Series is non-stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is Series is stationary at 5%.

kpss_level_pre  <- kpss.test(ts_pe_aus_preGFC)
kpss_trend_pre  <- kpss.test(ts_pe_aus_preGFC, null = "Trend")

print(kpss_level_pre)
## 
##  KPSS Test for Level Stationarity
## 
## data:  ts_pe_aus_preGFC
## KPSS Level = 1.3587, Truncation lag parameter = 4, p-value = 0.01
print(kpss_trend_pre)
## 
##  KPSS Test for Trend Stationarity
## 
## data:  ts_pe_aus_preGFC
## KPSS Trend = 0.15853, Truncation lag parameter = 4, p-value = 0.03956

Seems to have conflicting results in ADF and PP tests, but KPSS confirms that trend specification appears to be most appropriate.


Breakpoint Unit-Root Test


Zivot-Andrews (ZA) test

Null Hypothesis (H₀): The time series has a unit root.
p-value < 0.05: reject the null → The series is stationary around a broken trend.
p-value ≥ 0.05: fail to reject the null → The series is non-stationary, even accounting for a structural break.

Note: Based on the p-values, the test implies that the series is non-stationary, and the breakpoint reported is 2009-09 (trend) and 2010-05 (intercept). 

za_intercept <- ur.za(ts_pe_aus, model = "intercept", lag = NULL)
za_trend <- ur.za(ts_pe_aus, model = "trend", lag = NULL)
summary(za_intercept)
## 
## ################################ 
## # Zivot-Andrews Unit Root Test # 
## ################################ 
## 
## 
## Call:
## lm(formula = testmat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -43.049  -1.394   0.241   1.297  18.661 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.18574    0.86010   0.216 0.829282    
## y.l1         0.89725    0.02753  32.589  < 2e-16 ***
## trend        0.03925    0.01119   3.507 0.000577 ***
## du          -5.01143    1.23780  -4.049 7.74e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.302 on 174 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.8822, Adjusted R-squared:  0.8802 
## F-statistic: 434.5 on 3 and 174 DF,  p-value: < 2.2e-16
## 
## 
## Teststatistic: -3.7321 
## Critical values: 0.01= -5.34 0.05= -4.8 0.1= -4.58 
## 
## Potential break point at position: 122
summary(za_trend)
## 
## ################################ 
## # Zivot-Andrews Unit Root Test # 
## ################################ 
## 
## 
## Call:
## lm(formula = testmat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -47.014  -0.864  -0.153   0.660  19.854 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.89986    0.92602   0.972    0.333    
## y.l1         0.92697    0.02790  33.223   <2e-16 ***
## trend        0.01058    0.01137   0.930    0.354    
## dt          -0.02830    0.02991  -0.946    0.345    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.489 on 174 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.8718, Adjusted R-squared:  0.8696 
## F-statistic: 394.4 on 3 and 174 DF,  p-value: < 2.2e-16
## 
## 
## Teststatistic: -2.6173 
## Critical values: 0.01= -4.93 0.05= -4.42 0.1= -4.11 
## 
## Potential break point at position: 114
format(as.Date(as.yearmon(time(ts_pe_aus)[122])), "%Y-%m")
## [1] "2010-05"
format(as.Date(as.yearmon(time(ts_pe_aus)[114])), "%Y-%m")
## [1] "2009-09"




Pre-Global Crisis


subset_postGFC <- subset(data, date >= as.Date("2010-12-01") & date <= as.Date("2015-02-01"))

ts_pe_postGFC <- ts(subset_postGFC$pe_aus, start = c(2010, 12), end = c(2015, 2), frequency = 12)

plot(ts_pe_postGFC, 
     main = "Price-Earnings Ratio, Australia", 
     ylab = "PE Ratio", 
     xlab = "Year")



Augmented Dickey-Fuller (ADF) test

Null Hypothesis (H₀): The series has a unit root → it is non-stationary.
p < 0.05 -> Reject H₀. Series is stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is non-stationary at 5%.

adf_none_post     <- ur.df(ts_pe_postGFC, type = "none", selectlags = "AIC")
adf_drift_post    <- ur.df(ts_pe_postGFC, type = "drift", selectlags = "AIC")
adf_trend_post    <- ur.df(ts_pe_postGFC, type = "trend", selectlags = "AIC")

summary(adf_none_post)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.0261 -0.5258  0.1244  0.7281  3.0568 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## z.lag.1    -0.001351   0.009348  -0.145    0.886
## z.diff.lag -0.076502   0.170543  -0.449    0.656
## 
## Residual standard error: 1.165 on 47 degrees of freedom
## Multiple R-squared:  0.005088,   Adjusted R-squared:  -0.03725 
## F-statistic: 0.1202 on 2 and 47 DF,  p-value: 0.887
## 
## 
## Value of test-statistic is: -0.1446 
## 
## Critical values for test statistics: 
##      1pct  5pct 10pct
## tau1 -2.6 -1.95 -1.61
summary(adf_drift_post)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression drift 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8636 -0.5882  0.0311  0.8506  3.1648 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  1.35208    1.03833   1.302    0.199
## z.lag.1     -0.07608    0.05813  -1.309    0.197
## z.diff.lag  -0.03606    0.17212  -0.209    0.835
## 
## Residual standard error: 1.156 on 46 degrees of freedom
## Multiple R-squared:  0.04045,    Adjusted R-squared:  -0.001272 
## F-statistic: 0.9695 on 2 and 46 DF,  p-value: 0.3869
## 
## 
## Value of test-statistic is: -1.3087 0.8584 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau2 -3.51 -2.89 -2.58
## phi1  6.70  4.71  3.86
summary(adf_trend_post)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression trend 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.0214 -0.6424  0.0379  0.6965  3.1814 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  1.78367    1.23596   1.443    0.156
## z.lag.1     -0.11717    0.08587  -1.365    0.179
## tt           0.01130    0.01729   0.654    0.517
## z.diff.lag  -0.02314    0.17433  -0.133    0.895
## 
## Residual standard error: 1.163 on 45 degrees of freedom
## Multiple R-squared:  0.04947,    Adjusted R-squared:  -0.01389 
## F-statistic: 0.7807 on 3 and 45 DF,  p-value: 0.5109
## 
## 
## Value of test-statistic is: -1.3646 0.7076 1.0594 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau3 -4.04 -3.45 -3.15
## phi2  6.50  4.88  4.16
## phi3  8.73  6.49  5.47


Phillips-Perron (PP) test

Null Hypothesis (H₀): The series has a unit root → it is non-stationary.
p < 0.05 -> Reject H₀. Series is stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is non-stationary at 5%.

pp_drift_post     <- ur.pp(ts_pe_postGFC, type = "Z-alpha", model = "constant")
pp_trend_post     <- ur.pp(ts_pe_postGFC, type = "Z-alpha", model = "trend")

summary(pp_drift_post)
## 
## ################################## 
## # Phillips-Perron Unit Root Test # 
## ################################## 
## 
## Test regression with intercept 
## 
## 
## Call:
## lm(formula = y ~ y.l1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8736 -0.5677 -0.0004  0.7861  3.1930 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.37160    0.99779   1.375    0.176    
## y.l1         0.92232    0.05578  16.536   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.133 on 48 degrees of freedom
## Multiple R-squared:  0.8507, Adjusted R-squared:  0.8476 
## F-statistic: 273.4 on 1 and 48 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic, type: Z-alpha  is: -3.6343 
## 
##          aux. Z statistics
## Z-tau-mu            1.3297
summary(pp_trend_post)
## 
## ################################## 
## # Phillips-Perron Unit Root Test # 
## ################################## 
## 
## Test regression with intercept and trend 
## 
## 
## Call:
## lm(formula = y ~ y.l1 + trend)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.0355 -0.6076  0.0133  0.6307  3.1976 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.11606    1.42979    1.48    0.146    
## y.l1         0.87983    0.08079   10.89  1.9e-14 ***
## trend        0.01175    0.01608    0.73    0.469    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.138 on 47 degrees of freedom
## Multiple R-squared:  0.8524, Adjusted R-squared:  0.8461 
## F-statistic: 135.7 on 2 and 47 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic, type: Z-alpha  is: -5.7712 
## 
##            aux. Z statistics
## Z-tau-mu              1.5691
## Z-tau-beta            0.6849


Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test

Null Hypothesis (H₀): The series is stationary.
p < 0.05 -> Reject H₀. Series is non-stationary at 5%.
p ≥ 0.05 -> Fail to reject H₀. Series is Series is stationary at 5%.

kpss_level_post   <- kpss.test(ts_pe_postGFC, null = "Level")
## Warning in kpss.test(ts_pe_postGFC, null = "Level"): p-value smaller than
## printed p-value
kpss_trend_post   <- kpss.test(ts_pe_postGFC, null = "Trend")

print(kpss_level_post)
## 
##  KPSS Test for Level Stationarity
## 
## data:  ts_pe_postGFC
## KPSS Level = 0.85134, Truncation lag parameter = 3, p-value = 0.01
print(kpss_trend_post)
## 
##  KPSS Test for Trend Stationarity
## 
## data:  ts_pe_postGFC
## KPSS Trend = 0.18051, Truncation lag parameter = 3, p-value = 0.02331

Again, the results seem to be conflicting in ADF and PP tests, but KPSS confirms that trend specification appears to be most appropriate.

IMF remarks: This exercise illustrates the challenges that can arise in real world data with limited observations. While the entire sample spanned 15 years of monthly data, we also observed a major break 10 years in. To complicate matters, the break did not seem to be a one-time level shift but rather two structural breaks, the first sending the P-E ratio from the 15-20 range to 65-75 for about one year and then down again to the 15-20 range. Our best guess would be that pe_aus is nonstationary, as this is consistent with our results for preferred specifications in the pre-crisis sample and the breakpoint unit root test. As for the post-crisis sample, we must bear in mind that there are only 5 years of data, and the unit root tests gave results that conflicted with those of the KPSS test.