SARIMA Modeling for CO2 Levels

Example

This example demonstrates modeling monthly CO2 levels in Alert, NWT, Canada (1994-2004) using a Seasonal ARIMA model.

1. Identification

• The original time series plot shows a clear upward trend and a strong seasonal pattern.
• The ACF of the original series confirms this with strong correlations at lags 12, 24, 36, etc.
• A first difference $\nabla Y_t=Y_t-Y_{t-1}$ removes the general upward trend, but strong seasonality remains.
• A seasonal difference ${\nabla }_{12}(\nabla Y_t)$ removes the remaining seasonality. The new series appears stationary.
• The ACF of the doubly differenced series shows very little remaining autocorrelation, suggesting a simple model incorporating lag 1 and lag 12 autocorrelations might be adequate.

2. Model Specification

Based on the ACF, a multiplicative seasonal $\text{ARIMA}(0,1,1)\times (0,1,1{)}_{12}$ is specified.

3. Model Fitting

Let $X_t=\nabla Y_t=Y_t-Y_{t-1}$ and $W_t={\nabla }_{12}{X}_t=X_t-X_{t-12}$. Then $W_t=Y_t-Y_{t-1}-Y_{t-12}+Y_{t-13}$.

The fitted model is:

$$Y_t-Y_{t-1}-Y_{t-12}+Y_{t-13}=e_t-\theta e_{t-1}-\Theta e_{t-12}+\theta \Theta e_{t-13}$$

4. Diagnostic Checking

• Residual analysis (ACF, PACF) is performed to ensure errors resemble white noise.
• Normality of residuals is checked (e.g., via QQ plots or histograms).
• Overfitting is considered by trying a more complex model like $\text{ARIMA}(0,1,2)\times (0,1,1{)}_{12}$ to see if it significantly improves the fit.