How do you choose parameters for an ARIMA model?

To choose parameters for an ARIMA model, you need to determine three values: p (autoregressive order), d (degree of differencing), and q (moving average order). Start by analyzing your time series data for trends and stationarity. If the data has a trend or varying mean/variance, apply differencing (parameter d) to make it stationary. For example, if the data becomes stationary after one differencing operation, set d=1. Tools like the Augmented Dickey-Fuller (ADF) test can help confirm stationarity. Avoid over-differencing, as it can introduce unnecessary noise.

Next, identify p and q using autocorrelation (ACF) and partial autocorrelation (PACF) plots. For p (AR term), look at the PACF plot: if significant spikes occur up to lag 2 before dropping off, this suggests p=2. For q (MA term), check the ACF plot: spikes up to lag 3 might indicate q=3. However, real-world data is often messy, so use these plots as a starting point. For example, if the ACF decays slowly and the PACF has a sharp cutoff at lag 1, an AR(1) model (p=1) might be appropriate. Combining this with grid search (testing multiple p/q combinations) and metrics like AIC or BIC helps refine choices.

Finally, validate the model by testing residuals for randomness. After fitting ARIMA(p,d,q), check if residuals resemble white noise (no autocorrelation). For instance, use the Ljung-Box test: a p-value >0.05 suggests residuals are random. If not, adjust p or q and retest. Tools like auto_arima (from Python’s pmdarima library) automate parameter selection but understanding the manual process ensures better troubleshooting. Always split data into train/test sets to evaluate forecast accuracy—e.g., a model with p=1, d=1, q=1 might perform poorly on test data, prompting a reevaluation of parameters.