Statistical Models

Statistical Models (ARIMA, ETS, SARIMAX, ARAR)

When to Use

Use statistical models when:

• The series is short (< 200 observations)
• Interpretability is important (ARIMA coefficients, ETS components)
• You need built-in prediction intervals without residual bootstrapping
• As a baseline to compare against ML models

Related skills

• Prerequisite: autocorrelation-and-lag-selection (read ACF/PACF to identify ARIMA orders p, d, q before fitting)
• Alongside: baseline-forecasting (the other benchmark family: naive equivalent-date baselines)
• Next: prediction-intervals (ForecasterStats provides built-in parametric intervals via the interval_method argument)
• Next: hyperparameter-optimization (tune ARIMA order / seasonal_order via grid search)

Stop Conditions

Scan before writing code. Each row lists a rule, the symptom when it is broken, and the recovery. Full pitfall catalog: the troubleshooting-common-errors skill.

Available Models

Complete Workflow: ARIMA

import pandas as pd
from skforecast.recursive import ForecasterStats
from skforecast.stats import Arima
from skforecast.model_selection import backtesting_stats, TimeSeriesFold

# 1. Load data
data = pd.read_csv('data.csv', index_col='date', parse_dates=True)
data = data.asfreq('MS')  # Monthly Start frequency

# 2. Manual ARIMA: specify order and seasonal_order
arima_model = Arima(
    order=(1, 1, 1),              # (p, d, q)
    seasonal_order=(1, 1, 1),     # (P, D, Q)
    m=12,                         # Seasonal period
)
forecaster = ForecasterStats(estimator=arima_model)
forecaster.fit(y=data['target'])
predictions = forecaster.predict(steps=12)

# 3. Prediction intervals (all stat models support this natively,
#    no bootstrapping needed). Accepts both `interval` and `alpha`.
predictions_interval = forecaster.predict_interval(
    steps=12,
    interval=[0.1, 0.9],  # quantiles (0-1). Or use alpha=0.2 for 80% interval
)

Auto-ARIMA (Automatic Order Selection)

# Set order=None and seasonal_order=None to enable automatic order selection
auto_arima = Arima(order=None, seasonal_order=None, m=12)
forecaster = ForecasterStats(estimator=auto_arima)
forecaster.fit(y=data['target'])

# Check selected order
print(forecaster.estimator.best_params_['order'])
print(forecaster.estimator.best_params_['seasonal_order'])

predictions = forecaster.predict(steps=12)

ETS (Exponential Smoothing)

from skforecast.stats import Ets

# Model string: 1st=Error, 2nd=Trend, 3rd=Seasonal
# A=Additive, M=Multiplicative, N=None, Z=Auto-select
ets_model = Ets(model='AAA', m=12)
forecaster = ForecasterStats(estimator=ets_model)
forecaster.fit(y=data['target'])
predictions = forecaster.predict(steps=12)

Auto-ETS (Automatic Model Selection)

# Use model='ZZZ' (or model=None) to let ETS automatically select
# the best Error, Trend, and Seasonal components
auto_ets = Ets(model='ZZZ', m=12)
forecaster = ForecasterStats(estimator=auto_ets)
forecaster.fit(y=data['target'])

# Check the selected model configuration
print(forecaster.estimator.best_params_)

predictions = forecaster.predict(steps=12)

SARIMAX (with Exogenous Variables)

from skforecast.stats import Sarimax

sarimax_model = Sarimax(
    order=(1, 1, 1),
    seasonal_order=(1, 1, 1, 12),  # (P, D, Q, seasonal_period)
)
forecaster = ForecasterStats(estimator=sarimax_model)
forecaster.fit(y=data['target'], exog=exog_train)

# For prediction, exog must cover the forecast horizon
predictions = forecaster.predict(steps=12, exog=exog_test)

ARAR

from skforecast.stats import Arar

arar_model = Arar()
forecaster = ForecasterStats(estimator=arar_model)
forecaster.fit(y=data['target'])
predictions = forecaster.predict(steps=12)

Backtesting Statistical Models

cv = TimeSeriesFold(
    steps=12,
    initial_train_size=len(data) - 60,
    refit=False,
)

metric, predictions_bt = backtesting_stats(
    forecaster=forecaster,
    y=data['target'],
    cv=cv,
    metric='mean_absolute_error',
    freeze_params=True,  # Params from first fit reused in refits (avoids re-running auto selection)
)
# If freeze_params=False, auto selection runs independently each fold and output
# includes an extra 'estimator_params' column with the parameters selected per fold.

Multiple Models Simultaneously

# ForecasterStats accepts a list of models — fits each independently
from skforecast.stats import Arima, Ets

models = [
    Arima(order=(1, 1, 1), seasonal_order=(1, 1, 1), m=12),
    Ets(model='AAA', m=12),
]
forecaster = ForecasterStats(estimator=models)
forecaster.fit(y=data['target'])

# predict returns DataFrame with one column per model
predictions = forecaster.predict(steps=12)

Common Mistakes

1. Using deprecated Ets(error=, trend=, seasonal=) syntax: Use Ets(model='AAA', m=12) with a model string instead.
2. Forgetting m parameter: ARIMA and ETS seasonal models require m (seasonal period).
3. Not using backtesting_stats: Use backtesting_stats() for statistical models, NOT backtesting_forecaster().
4. Using grid_search_forecaster for stats: Use grid_search_stats() or random_search_stats() instead.
5. Passing seasonal_order=(1,1,1,12) to Arima: Arima uses a 3-tuple seasonal_order=(P,D,Q) plus a separate m=12 parameter. The 4-tuple (P,D,Q,s) format is only for Sarimax.

References

See references/model-parameters.md for complete constructor signatures of all statistical models (Arima, Sarimax, Ets, Arar), the Ets model string format, Auto-ARIMA parameters, seasonal_order differences between Arima and Sarimax, and grid search param_grid examples.