3.2 Testing for stationarity (visual and statistical methods)

Visual inspection techniques are crucial for assessing stationarity in time series data. By plotting the series and analyzing autocorrelation functions, we can identify trends, seasonality, and changes in variance that indicate non-stationarity.

Statistical tests provide a quantitative approach to confirm or reject visual inspection findings. The Augmented Dickey-Fuller and KPSS tests offer formal hypotheses for determining stationarity, helping us make objective decisions about appropriate modeling techniques.

Visual Inspection Techniques for Stationarity

Visual inspection for stationarity

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  time series - How to interpret these acf and pacf plots - Cross Validated View original

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  Interpretación de la estacionalidad en gráficos ACF y PACF View original

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  ARIMA 时间序列1: 差分, ACF, PACF - 灰信网（软件开发博客聚合） View original

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  time series - How to interpret these acf and pacf plots - Cross Validated View original

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Top images from around the web for Visual inspection for stationarity

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  time series - How to interpret these acf and pacf plots - Cross Validated View original

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  Interpretación de la estacionalidad en gráficos ACF y PACF View original

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  ARIMA 时间序列1: 差分, ACF, PACF - 灰信网（软件开发博客聚合） View original

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  time series - How to interpret these acf and pacf plots - Cross Validated View original

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  Interpretación de la estacionalidad en gráficos ACF y PACF View original

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• Plotting the time series
  • Identifies trends, seasonality, and changes in variance over time
  • Non-stationary series may exhibit clear trends (upward or downward) or changing variance (increasing or decreasing spread)
• Analyzing the autocorrelation function (ACF)
  • Measures the correlation between a time series and its lagged values
  • For a stationary series, ACF should decay quickly to zero (within a few lags)
  • Slow decay or significant correlations at large lags (beyond the 95% confidence interval) suggest non-stationarity
• Analyzing the partial autocorrelation function (PACF)
  • Measures the correlation between a time series and its lagged values, while controlling for the effects of intermediate lags
  • For a stationary series, PACF should have significant correlations only at a few lags (typically one or two)
  • Persistent significant correlations at multiple lags (beyond the 95% confidence interval) suggest non-stationarity

Interpretation of visual methods

• Plotting the time series
  • If the series appears to have a constant mean (no clear trend) and variance (consistent spread) over time, it may be stationary
  • If the series exhibits trends (increasing or decreasing), seasonality (regular patterns), or changing variance (increasing or decreasing spread), it is likely non-stationary
• Analyzing the ACF
  • If the ACF decays quickly to zero (within a few lags), the series is likely stationary
  • If the ACF decays slowly or has significant correlations at large lags (beyond the 95% confidence interval), the series is likely non-stationary
• Analyzing the PACF
  • If the PACF has significant correlations only at a few lags (typically one or two), the series is likely stationary
  • If the PACF has persistent significant correlations at multiple lags (beyond the 95% confidence interval), the series is likely non-stationary
• Combining visual inspection results
  • If all visual inspection methods suggest stationarity (constant mean and variance, quick ACF decay, few significant PACF lags), the series is likely stationary
  • If one or more methods suggest non-stationarity (trends, seasonality, changing variance, slow ACF decay, persistent significant PACF lags), further investigation or statistical tests may be necessary

Statistical Tests for Stationarity

Importance of stationarity tests

• Statistical tests provide a formal, quantitative approach to assessing stationarity
• They help to confirm or reject the conclusions drawn from visual inspection methods
• Statistical tests are important for making objective decisions about stationarity and selecting appropriate modeling techniques (ARIMA, SARIMA, etc.)

Hypotheses in stationarity tests

• Augmented Dickey-Fuller (ADF) test
  • Null hypothesis: The time series has a unit root and is non-stationary
  • Alternative hypothesis: The time series does not have a unit root and is stationary
  • If the p-value is less than the chosen significance level (0.05), reject the null hypothesis and conclude that the series is stationary
• Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test
  • Null hypothesis: The time series is trend-stationary (stationary around a deterministic trend)
  • Alternative hypothesis: The time series is not trend-stationary and has a unit root or is non-stationary
  • If the p-value is greater than the chosen significance level (0.05), fail to reject the null hypothesis and conclude that the series is trend-stationary