Box and Jenkins promoted a three-step approach for choosing a suitable model to estimate and forecast univariate time series data. This method comprises three main stages:
Estimation Phase: Each proposed model is fitted, and various coefficients, such as α and β, are examined for significance. The objective here is to select a model that fits the data well and is stable.
Diagnostic Checking: Ensuring that the model's residuals mimic a white noise process.
Model Selection Principles:
1. Parsimony: Economists should aim for simplicity. Adding more parameters may enhance fit but reduces degrees of freedom. Several models may share similar properties, requiring careful consideration.
2. Stationarity and Invertibility: t-statistics and Q-statistics assume data stationarity. Autoregressive coefficients should align with this assumption. If the estimated value of a1 is close to unity, it suggests an AR(1) model.
3. Goodness of Fit: Measures like R2 and the average residual sum of squares are common in ordinary least squares. Parsimony recommends using AIC or SBC for overall model fit. Be cautious of unstable estimates that do not converge quickly.
4. Postestimation Evaluation: Plotting residuals to detect outliers and periods where the model poorly fits the data. Standardized residuals should have no more than 5% lying outside the -2 to +2 range if residuals are normally distributed. Check for serial correlation in residuals using ACF and PACF. Be wary of marginally significant correlations or barely significant Q-statistics.
Model Adequacy: For forecasting, the sum of squared forecast errors can compare different models' adequacy. Walter Enders' work provides a comprehensive guide for time series analysis.