ARIMA Model Parameters to Know for Intro to Time Series

Understanding ARIMA model parameters is key in time series analysis. These parameters—AR, I, and MA—help capture patterns, ensure stationarity, and improve forecasting accuracy. Mastering these concepts will enhance your ability to analyze and predict time-dependent data effectively.

1. AR (Autoregressive) parameter (p)

   • Represents the number of lagged observations included in the model.
   • Indicates how past values influence the current value of the time series.
   • Higher values of p can capture more complex patterns but may lead to overfitting.

2. I (Integrated) parameter (d)

   • Represents the number of differences needed to make the time series stationary.
   • A difference is the subtraction of the previous observation from the current observation.
   • Typically, d is set to 0, 1, or 2; higher values are rare and indicate more complex trends.

3. MA (Moving Average) parameter (q)

   • Represents the number of lagged forecast errors in the prediction equation.
   • Captures the relationship between an observation and a residual error from a moving average model.
   • Helps to smooth out short-term fluctuations and highlight longer-term trends.

4. Stationarity requirement

   • A stationary time series has constant mean and variance over time.
   • Non-stationary data can lead to unreliable and spurious results in ARIMA modeling.
   • Techniques like differencing or transformation can be used to achieve stationarity.

5. ACF (Autocorrelation Function)

   • Measures the correlation between a time series and its lagged values.
   • Helps identify the presence of seasonality and the appropriate q parameter.
   • ACF plots can indicate how many lags are significant for the model.

6. PACF (Partial Autocorrelation Function)

   • Measures the correlation between a time series and its lagged values after removing the effects of intervening lags.
   • Useful for determining the appropriate p parameter in the ARIMA model.
   • PACF plots help identify the number of significant lags to include in the model.

7. Model selection criteria (AIC, BIC)

   • AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are used to compare different ARIMA models.
   • Both criteria penalize model complexity to avoid overfitting.
   • Lower values of AIC or BIC indicate a better-fitting model.

8. Residual analysis

   • Involves examining the residuals (errors) of the fitted ARIMA model to ensure they behave like white noise.
   • Residuals should be normally distributed with constant variance and no autocorrelation.
   • Helps validate the model's adequacy and identify potential improvements.

9. Forecasting with ARIMA models

   • ARIMA models can be used to make future predictions based on past data.
   • Forecasts are generated using the estimated parameters and the most recent observations.
   • Confidence intervals can be constructed to assess the uncertainty of the forecasts.

10. Seasonal ARIMA (SARIMA) parameters

    • Extends ARIMA to account for seasonality in the data by adding seasonal parameters (P, D, Q).
    • P represents seasonal autoregressive terms, D represents seasonal differences, and Q represents seasonal moving average terms.
    • SARIMA models are denoted as ARIMA(p, d, q)(P, D, Q)s, where s is the seasonal period.