Activity6

Stationarity and Unit Root Tests

What is a Unit Root?

A unit root is a characteristic of a time series that makes it non-stationary. If a series has a unit root, it means that shocks to the system have a permanent effect, and the series does not revert to a long-term mean. Examples include random walks and random walks with drift.

Augmented Dickey-Fuller (ADF) Test

The ADF test is used to test for the presence of a unit root in a time series. The null hypothesis (\(H_0\)) is that the series has a unit root (i.e., it is non-stationary). The alternative hypothesis (\(H_1\)) is that the series is stationary. If the p-value from the ADF test is less than a significance level (e.g., 0.05), we reject the null hypothesis and conclude that the series is stationary.

KPSS Test

The KPSS test is another test for stationarity, but it has a different null hypothesis. The null hypothesis (\(H_0\)) for the KPSS test is that the series is stationary (around a deterministic trend). The alternative hypothesis (\(H_1\)) is that the series has a unit root (i.e., it is non-stationary). If the p-value is less than the significance level, we reject the null hypothesis and conclude that the series is non-stationary.

Key Differences Between ADF and KPSS Tests

  1. Null Hypothesis:

    • ADF: \(H_0\) = Series has a unit root (non-stationary).
    • KPSS: \(H_0\) = Series is stationary (around a deterministic trend).

  2. Interpretation:

    • ADF: Rejecting \(H_0\) implies stationarity.
    • KPSS: Rejecting \(H_0\) implies non-stationarity.

  3. Use Case:

    • ADF is more commonly used to test for unit roots.
    • KPSS is often used as a complementary test to confirm stationarity.

Step 1: Simulate an AR(1) Process

# Simulate AR(1) process
set.seed(123)
n <- 1000
phi <- 0.9
ar1 <- tsibble(t = 1:n, y = arima.sim(model = list(ar = phi), n = n), index = t)

# Plot the AR(1) process
autoplot(ar1, y) + labs(title = "Simulated AR(1) Process")

# Perform ADF test
adf_test <- adf.test(ar1$y)
adf_test

    Augmented Dickey-Fuller Test

data:  ar1$y
Dickey-Fuller = -5.8012, Lag order = 9, p-value = 0.01
alternative hypothesis: stationary
# Perform KPSS test
kpss_test <- ar1 %>%
  features(y, unitroot_kpss)
kpss_test
# A tibble: 1 × 2
  kpss_stat kpss_pvalue
      <dbl>       <dbl>
1     0.195         0.1

Step 2: Simulate a Random Walk with Drift

# Simulate Random Walk with Drift
set.seed(123)
n <- 100
drift <- 0.1
rw_drift <- tsibble(t = 1:n, y = cumsum(rnorm(n, mean = drift)), index = t)

# Plot the Random Walk with Drift
autoplot(rw_drift, y) + labs(title = "Simulated Random Walk with Drift")

# Perform ADF test
adf_test_rw <- adf.test(rw_drift$y)
adf_test_rw

    Augmented Dickey-Fuller Test

data:  rw_drift$y
Dickey-Fuller = -1.8871, Lag order = 4, p-value = 0.6234
alternative hypothesis: stationary
# Perform KPSS test
kpss_test_rw <- rw_drift %>%
  features(y, unitroot_kpss)
kpss_test_rw
# A tibble: 1 × 2
  kpss_stat kpss_pvalue
      <dbl>       <dbl>
1      1.86        0.01

Step 3: Interpret the Results

  • AR(1) Process: The ADF test indicates stationarity (reject \(H_0\)), and the KPSS test indicates stationarity (fail to reject \(H_0\)) as well.
  • Random Walk with Drift: The ADF test indicates non-stationarity (fail to reject \(H_0\)), and the KPSS test indicates non-stationarity (reject \(H_0\)) as well.

Step 4: Apply to Real-World Data (S&P 500 Prices)

# Get S&P 500 data in tidy format
sp500 <- tq_get("^GSPC", 
                from = "2010-01-01", 
                to = "2023-01-01",
                get = "stock.prices") %>%
  select(date, adjusted) %>%  # Use adjusted closing prices
  rename(price = adjusted) %>% 
  as_tsibble(index = date)  # Convert to tsibble with date index

# Plot the S&P 500 series
autoplot(sp500, price) + labs(title = "S&P 500 Price Series")

# Perform ADF test on the original series
adf_test_sp500 <- adf.test(sp500$price)
adf_test_sp500

    Augmented Dickey-Fuller Test

data:  sp500$price
Dickey-Fuller = -2.8302, Lag order = 14, p-value = 0.2268
alternative hypothesis: stationary
# Perform KPSS test on the original series
kpss_test_sp500 <- sp500 %>%
  features(price, unitroot_kpss)
kpss_test_sp500
# A tibble: 1 × 2
  kpss_stat kpss_pvalue
      <dbl>       <dbl>
1      30.1        0.01
# Difference the series and perform ADF test
diff_sp500 <- sp500 %>% mutate(price = difference(price)) %>% tidyr::drop_na()
adf_test_diff_sp500 <- adf.test(diff_sp500$price)
adf_test_diff_sp500

    Augmented Dickey-Fuller Test

data:  diff_sp500$price
Dickey-Fuller = -15.29, Lag order = 14, p-value = 0.01
alternative hypothesis: stationary
# Perform KPSS test on the differenced series
kpss_test_diff_sp500 <- diff_sp500 %>%
  features(price, unitroot_kpss)
kpss_test_diff_sp500
# A tibble: 1 × 2
  kpss_stat kpss_pvalue
      <dbl>       <dbl>
1    0.0471         0.1

Lab Activity

Repeat the stationarity and unit root analysis (using ADF and KPSS tests) for NVIDIA (NVDA) stock prices from 2010-01-01 to 2023-01-01. Apply differencing or other transformations as needed to achieve stationarity, and interpret the results.

Solution:

# Step 1: Get NVIDIA (NVDA) stock data
nvda <- tq_get("NVDA", 
               from = "2010-01-01", 
               to = "2023-01-01",
               get = "stock.prices") %>%
  select(date, adjusted) %>%  # Use adjusted closing prices
  rename(price = adjusted) %>% 
  as_tsibble(index = date)  # Convert to tsibble with date index

# Step 2: Plot the NVIDIA stock price series
autoplot(nvda, price) + 
  labs(title = "NVIDIA (NVDA) Stock Price Series", 
       x = "Date", 
       y = "Adjusted Closing Price")

# Step 3: Perform ADF test on the original series
adf_test_nvda <- adf.test(nvda$price)
adf_test_nvda

    Augmented Dickey-Fuller Test

data:  nvda$price
Dickey-Fuller = -2.3584, Lag order = 14, p-value = 0.4266
alternative hypothesis: stationary
# Step 4: Perform KPSS test on the original series
kpss_test_nvda <- nvda %>%
  features(price, unitroot_kpss)
kpss_test_nvda
# A tibble: 1 × 2
  kpss_stat kpss_pvalue
      <dbl>       <dbl>
1      22.2        0.01
# Step 5: Difference the series to achieve stationarity
diff_nvda <- nvda %>% mutate(price = difference(price))

# Step 6: Plot the differenced series
autoplot(diff_nvda, price) + 
  labs(title = "Differenced NVIDIA (NVDA) Stock Price Series", 
       x = "Date", 
       y = "Differenced Price")

# Step 7: Perform ADF test on the differenced series
adf_test_diff_nvda <- adf.test(na.omit(diff_nvda$price))
adf_test_diff_nvda

    Augmented Dickey-Fuller Test

data:  na.omit(diff_nvda$price)
Dickey-Fuller = -14.601, Lag order = 14, p-value = 0.01
alternative hypothesis: stationary
# Step 8: Perform KPSS test on the differenced series
kpss_test_diff_nvda <- diff_nvda %>%
  features(price, unitroot_kpss)
kpss_test_diff_nvda
# A tibble: 1 × 2
  kpss_stat kpss_pvalue
      <dbl>       <dbl>
1    0.0934         0.1