Canadian Hare Abundance Series

Example

Canadian Hare Abundance Series illustrates parameter estimation methods for a real time series dataset.

Dataset Description

The Canadian hare abundance series records annual hare population counts in Canada over multiple decades. This series typically exhibits:

• Non-stationarity
• Possible cyclical patterns
• Need for transformation (often log)

Model Identification

1. Visual inspection: Plot series, check for trend/seasonality
2. Stationarity test: Apply differencing if needed
3. ACF/PACF analysis: Identify candidate ARIMA(p,d,q) models

Parameter Estimation Example

After identifying a candidate model (e.g., AR(1) or ARIMA(1,1,0)):

Method of Moments

• Compute sample autocorrelation $r_1$
• For AR(1): ${\hat{\varphi }}_1=r_1$

Conditional Least Squares

• Minimize $S_c(\varphi )={\sum }_{t=2}^n(Y_t-\hat{\mu }-\varphi (Y_{t-1}-\hat{\mu }){)}^2$
• Solve numerically or use closed form

Maximum Likelihood

• Specify likelihood assuming normal errors
• Estimate $\hat{\varphi }$ by maximizing $\log L$
• Estimate ${\hat{\sigma }}_e^2=S(\hat{\varphi })/n$

Model Comparison

Compare fitted models using:

• AIC/BIC criteria
• Residual diagnostics (Ljung-Box test)
• Forecast accuracy

Key Lessons

• Real data often requires preprocessing (transformation, differencing)
• Multiple estimation methods provide similar results for large samples
• Model diagnostics are essential for validating estimates

Related

• AIC Criterion
• Ljung-Box Test
• Maximum Likelihood Method