Stationarity Essentials: Mean, Variance, and Trend Insights Quiz Quiz

Explore the core concepts of stationarity in time series data, focusing on the roles of mean, variance, and trends. This quiz is designed to solidify your understanding of stationary processes, trend types, and key statistical properties essential for effective time series analysis.

1. Identifying Stationarity from Mean and Variance

   A time series is said to be stationary if its mean and variance remain constant over time. Which of the following best describes a stationary series?

   1. The variance fluctuates, but the mean is steady.
   2. The mean and variance do not change over time.
   3. The mean keeps increasing, but the variance is constant.
   4. Both the mean and variance change regularly each season.

   Explanation: A stationary series is defined by the fact that its mean and variance are both constant over time. Options where the mean or variance change, such as 'the mean keeps increasing' or 'the variance fluctuates', indicate non-stationary behavior. The choice where both change each season indicates the presence of trends or seasonality, which violates stationarity.

2. Understanding Trends in Time Series

   Which pattern in a time series would most clearly indicate a non-stationary series?

   1. Constant mean and variance throughout.
   2. Consistent variance with no apparent pattern.
   3. Random fluctuation around a constant average.
   4. A steadily increasing upward trend in the values.

   Explanation: A steadily increasing upward trend shows that the mean of the series is changing over time, which is a key sign of non-stationarity. Random fluctuations around a constant average, or constant mean and variance, indicate stationarity. Consistent variance alone is not enough; both mean and variance must be constant for stationarity.

3. Role of Variance in Stationarity

   If a monthly temperature time series shows increasing fluctuations (larger swings) as time goes on but maintains a stable average, what does this suggest?

   1. There is a deterministic trend in the data.
   2. The mean is not constant, implying non-stationarity.
   3. Both mean and variance are constant, implying stationarity.
   4. The variance is changing, indicating non-stationarity.

   Explanation: Increasing fluctuations mean the variance is not constant, which violates stationarity even if the mean is stable. The mean is constant in this scenario, so the second option is incorrect. Stationarity requires both mean and variance to be constant; thus, the third option is wrong. A deterministic trend refers to mean changes, not increasing variance.

4. Seasonality and Stationarity

   Consider a sales dataset where values consistently rise and fall at the same months each year but average out to the same mean over long periods. What aspect could make this series non-stationary?

   1. Perfectly flat values every month.
   2. A stable average with no trend or seasonality.
   3. Random noise with no periodic pattern.
   4. Presence of seasonality with regular fluctuations.

   Explanation: Regular seasonal fluctuations can violate strict stationarity, as the statistical properties (like mean and variance) change cyclically. A stable average with no trend or seasonality is likely stationary. Random noise is stationary if its mean and variance are constant, and perfectly flat values are stationary by definition. Only seasonality introduces periodic changes.

5. Mean Reversion in Time Series

   Which description best fits a mean-reverting process often seen in stationary time series?

   1. Values increase continuously over time without returning to the mean.
   2. Fluctuations amplify indefinitely as time passes.
   3. Values always stay equal and never change.
   4. Values tend to return to a constant long-term average after deviations.

   Explanation: Mean-reverting processes characteristic of stationarity see values returning to a stable average after being disturbed. Continuous increases without return to the mean indicate a non-stationary trend. Amplifying fluctuations suggest growing variance (non-stationarity), while values that never change do not represent realistic data series dynamics.

6. Unit Root and Stationarity

   A time series is said to contain a unit root. What does this imply about its stationarity?

   1. It is strictly stationary with constant mean and variance.
   2. It is seasonally stationary but not regularly stationary.
   3. It is non-stationary as its properties change over time.
   4. It shows cyclical trends only.

   Explanation: Having a unit root means the time series is non-stationary since its mean and/or variance depends on time, often requiring differencing. Strictly stationary series never contain a unit root. Seasonal stationarity involves cycles, but a unit root causes broader issues. Cyclical trends are not guaranteed with a unit root.

7. Differencing and Its Purpose

   If a time series is non-stationary due to a linear trend, what basic transformation is commonly applied to achieve stationarity?

   1. Differencing the series values.
   2. Applying a moving average smooth.
   3. Calculating the cumulative sum.
   4. Multiplying values by a constant.

   Explanation: Differencing the values (subtracting the previous from the current) is a standard way to remove linear trends and achieve stationarity. Cumulative sum accentuates trends and makes the series less stationary. Moving averages can reduce noise but may not remove a trend. Multiplying by a constant doesn't address changes in mean or trend.

8. Constant Variance Term

   What is the term for a stationary series whose variance does not change over time?

   1. Irregular
   2. Autoregressive
   3. Heteroscedastic
   4. Homoskedastic

   Explanation: Homoskedasticity means the variance remains steady throughout the series, a property of stationary data. Heteroscedastic describes changing variance, which is non-stationary. Autoregressive refers to relationships among time points, not variance. 'Irregular' does not specifically address variance characteristics.

9. Strict Stationarity vs. Weak Stationarity

   Which property distinguishes strict stationarity from weak stationarity in time series?

   1. Only the mean must remain constant over time.
   2. Only the variance must stay the same, regardless of mean.
   3. Only the second-order moments are constant.
   4. All statistical properties (not just mean and variance) are invariant over time.

   Explanation: Strict stationarity requires all statistical properties, including higher moments, to remain unchanged over time, whereas weak stationarity only needs the mean and variance to be constant. Options focusing solely on mean or variance describe neither strict nor weak stationarity correctly. Second-order moments apply to weak, not strict, stationarity.

10. Time Series Visualization

    When visually inspecting a time series plot, which feature is the clearest sign that it might not be stationary?

    1. An obvious upward or downward trend over time.
    2. Random variation about a fixed level.
    3. Constant spread of values throughout the plot.
    4. Flat horizontal pattern with no visible change.

    Explanation: A noticeable trend indicates changing mean, suggesting non-stationarity. Random variation about a fixed level, flat lines, and constant spread all suggest stationarity. Only pronounced trends reveal mean shifts or instability.