Unit Root Tests: Dickey-Fuller and KPSS Fundamentals Quiz Quiz

Assess your understanding of key concepts in time series analysis with this quiz focused on Dickey-Fuller and KPSS unit root tests. Explore the differences, objectives, assumptions, and interpretation of these essential statistical tools for testing stationarity in data.

1. Purpose of Dickey-Fuller Test

   What is the primary purpose of the Dickey-Fuller test when analyzing a time series dataset?

   1. To determine the autocorrelation function
   2. To estimate the mean of the series
   3. To detect outliers in the dataset
   4. To test whether the series has a unit root

   Explanation: The Dickey-Fuller test is mainly used to determine if a time series possesses a unit root, which indicates non-stationarity. Detecting outliers, estimating the mean, or calculating the autocorrelation function are different statistical tasks not addressed by this test. Outliers and mean estimation require separate analyses, and autocorrelation calculations are not the test's objective.

2. Null Hypothesis in Dickey-Fuller

   What is the null hypothesis for the standard Dickey-Fuller test on a time series?

   1. The mean is zero
   2. The series is trend stationary
   3. The time series has a unit root
   4. The time series is stationary

   Explanation: In the Dickey-Fuller test, the null hypothesis is that the time series has a unit root, implying non-stationarity. A stationary series is the alternative hypothesis. The test does not directly assess if the mean is zero or if the series is trend stationary; those involve different hypotheses.

3. Interpreting Significant Dickey-Fuller Result

   If the result of a Dickey-Fuller test is statistically significant, what does this imply about the time series?

   1. The series is trending upwards
   2. The series has autocorrelation only at lag 1
   3. The series does not have a unit root
   4. The mean is changing over time

   Explanation: A significant result in the Dickey-Fuller test means rejecting the null hypothesis of a unit root, indicating the series is stationary. A trending series or changing mean are not directly concluded from this result, and autocorrelation at lag 1 is not specific to the test outcome.

4. KPSS Test Null Hypothesis

   What is the null hypothesis of the KPSS (Kwiatkowski-Phillips-Schmidt-Shin) test when applied to a time series?

   1. The time series has a unit root
   2. The mean is non-zero
   3. The series is white noise
   4. The time series is stationary

   Explanation: The KPSS test's null hypothesis states that the series is stationary around a deterministic trend or mean. Unlike the Dickey-Fuller test, a unit root is the alternative, not the null. The mean being non-zero or series being white noise are not what the KPSS test directly tests for.

5. Opposite Hypotheses

   How do the null hypotheses of the Dickey-Fuller and KPSS tests fundamentally differ?

   1. Both null hypotheses claim the series is non-stationary
   2. Both tests check for autocorrelation only
   3. Dickey-Fuller is for outliers, KPSS is for missing data
   4. Dickey-Fuller tests for unit roots, KPSS tests for stationarity

   Explanation: The Dickey-Fuller test posits the series has a unit root, while KPSS asserts the series is stationary—making their null hypotheses opposites. The other options are incorrect: neither test is designed only for autocorrelation, outlier detection, or missing data.

6. Testing Trend Stationarity

   Which test would you typically use to check if a time series is trend stationary rather than having a unit root?

   1. KPSS test
   2. Spectral analysis
   3. Variance ratio test
   4. Ljung-Box test

   Explanation: The KPSS test can test for trend stationarity under one version of its implementation. Variance ratio and Ljung-Box focus on other aspects (random walk and autocorrelation, respectively), while spectral analysis examines frequency components.

7. Augmented Dickey-Fuller Extension

   What does the 'augmented' part of the Augmented Dickey-Fuller (ADF) test refer to?

   1. Inclusion of lagged difference terms to handle autocorrelation
   2. Calculating averages over moving windows
   3. Subtracting trend components
   4. Addition of seasonal averages

   Explanation: The ADF test augments the basic Dickey-Fuller test by adding lagged differenced terms to account for higher-order autocorrelation. It does not add seasonal averages, calculate moving averages, or subtract trends as part of the augmentation.

8. Stationary Series Example

   Given a time series with constant mean and variance over time, which unit root test result would most likely occur?

   1. Fail to reject the Dickey-Fuller null; fail to reject the KPSS null
   2. Fail to reject both nulls
   3. Reject both nulls
   4. Reject the Dickey-Fuller null; fail to reject the KPSS null

   Explanation: A stationary series should lead to rejection of the Dickey-Fuller null (no unit root) and failure to reject the KPSS null (stationarity). The other choices either conflate the hypotheses or incorrectly suggest both nulls would be rejected or retained together.

9. Non-Stationarity Detection

   Which combination of test results indicates strong evidence that a time series is non-stationary?

   1. Reject the Dickey-Fuller null; fail to reject the KPSS null
   2. Reject both nulls
   3. Fail to reject both nulls
   4. Fail to reject the Dickey-Fuller null; reject the KPSS null

   Explanation: Failing to reject the Dickey-Fuller null suggests a unit root (non-stationarity), and rejecting the KPSS null implies non-stationarity, supporting this conclusion. The other combinations do not provide clear evidence of non-stationarity or may contradict one another.

10. Use in Model Choice

    Why are unit root tests like Dickey-Fuller and KPSS important before modeling time series data with ARIMA?

    1. They automatically select lag lengths
    2. They estimate seasonal adjustment factors directly
    3. They help confirm stationarity, a key requirement of ARIMA models
    4. They visualize time series trends

    Explanation: ARIMA models require stationary time series, and unit root tests verify this crucial assumption. Estimating seasonal adjustments, selecting lag lengths, or trend visualization are not direct purposes of these tests, though they are related to broader modeling tasks.