Autoregressive Integrated Moving Average with Exogenous Variables (ARIMAX) for Quantitative Forecasting

Introduction:

Autoregressive Integrated Moving Average with Exogenous Variables (ARIMAX) is a popular time series forecasting method that combines the power of ARIMA models with the inclusion of external variables, also known as exogenous variables. ARIMAX is particularly useful when there are external factors that influence the time series being analyzed and need to be accounted for in the forecasting process. In this section, we will explore the steps involved in implementing ARIMAX and provide an example to illustrate its application.

Steps to Follow:

  1. Data Collection and Preprocessing:

    • Gather the historical data for the time series you want to forecast.
    • Identify any exogenous variables that might impact the time series and collect relevant data for them.
    • Clean and preprocess the data by removing any outliers, missing values, or irregularities that could affect the accuracy of the forecast.

  2. Exploratory Data Analysis (EDA):

    • Perform an EDA to understand the patterns, trends, and seasonality present in the time series data.
    • Analyze the relationships between the exogenous variables and the time series of interest.

  3. Model Selection:

    • Conduct a stationarity test to determine if differencing is required to make the time series stationary.
    • If differencing is needed, calculate the order of differencing (d) required to achieve stationarity.
    • Identify the optimal values for the autoregressive (p) and moving average (q) parameters using techniques like autocorrelation function (ACF) and partial autocorrelation function (PACF) plots.
    • Consider the inclusion of exogenous variables and determine which variables to include in the model.

  4. Model Estimation:

    • Estimate the ARIMAX model using the identified values of p, d, q, and exogenous variables.
    • Use the maximum likelihood estimation (MLE) or least squares method to estimate the model parameters.
    • Evaluate the model’s goodness of fit using metrics like Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC).

  5. Diagnostic Checking:

    • Perform diagnostic checks on the model residuals to ensure they satisfy the assumptions of the ARIMAX model.
    • Analyze the residuals for autocorrelation, normality, and heteroscedasticity using statistical tests and visual inspection.

  6. Forecasting:

    • Once the ARIMAX model is validated, use it to make future predictions.
    • Incorporate the values of exogenous variables for the forecast period.
    • Calculate the forecasted values along with the corresponding confidence intervals.

Example:

Let’s consider a retail store that wants to forecast its monthly sales (time series) based on external factors such as advertising expenditure and holiday seasons (exogenous variables). The steps to implement ARIMAX for this scenario are as follows:

  1. Collect historical monthly sales data for several years and gather data on advertising expenditure and holiday seasons for the same time period.

  2. Preprocess the data by removing any outliers or missing values.

  3. Conduct an EDA to observe the patterns, trends, and seasonalities in the sales data. Also, analyze the relationships between sales, advertising expenditure, and holiday seasons.

  4. Perform a stationarity test on the sales data. If non-stationary, calculate the order of differencing required to achieve stationarity.

  5. Use ACF and PACF plots to identify the optimal values for the autoregressive (p) and moving average (q) parameters. Additionally, determine which exogenous variables (advertising expenditure and holiday seasons) to include in the model.

  6. Estimate the ARIMAX model by fitting the sales data to the identified values