INNOVATIONS_MLE

This function estimates parameters of a SARIMA specification by maximizing the likelihood using the innovations algorithm.

The model order is controlled by non-seasonal (p,d,q) and seasonal (P,D,Q,s) components. The estimation method enforces stationarity and can enforce invertibility constraints for moving-average terms.

The result is returned as a row vector of estimated parameter values.

Excel Usage

=INNOVATIONS_MLE(endog, p, d, q, seasonal_p, seasonal_d, seasonal_q, seasonal_s, demean, enforce_invertibility)

endog (list[list], required): Time-series observations as a 2D range (data points).
p (int, optional, default: 0): Non-seasonal autoregressive order (lags).
d (int, optional, default: 0): Non-seasonal differencing order (differences).
q (int, optional, default: 0): Non-seasonal moving-average order (lags).
seasonal_p (int, optional, default: 0): Seasonal autoregressive order (lags).
seasonal_d (int, optional, default: 0): Seasonal differencing order (differences).
seasonal_q (int, optional, default: 0): Seasonal moving-average order (lags).
seasonal_s (int, optional, default: 0): Seasonal period length (periods).
demean (bool, optional, default: true): Whether to remove the mean before estimation (true/false).
enforce_invertibility (bool, optional, default: true): Whether to enforce invertibility constraints (true/false).

Returns (list[list]): Row vector containing estimated SARIMA parameter values.

Example 1: Innovations MLE for ARMA(1,1)

Inputs:

endog										p	d	q
10	10.5	11.2	10.9	11.5	12	11.8	12.3	12.1	12.7	1	0	1

Excel formula:

=INNOVATIONS_MLE({10,10.5,11.2,10.9,11.5,12,11.8,12.3,12.1,12.7}, 1, 0, 1)

Expected output:

Result
0.906275	-0.045126	0.231343

Example 2: Innovations MLE for AR(1)

Inputs:

endog										p	d	q
4	4.4	4.1	4.6	4.3	4.8	4.5	5	4.7	5.1	1	0	0

Excel formula:

=INNOVATIONS_MLE({4,4.4,4.1,4.6,4.3,4.8,4.5,5,4.7,5.1}, 1, 0, 0)

Expected output:

Result
0.239045	0.113806

Example 3: Innovations MLE for MA(1)

Inputs:

endog										p	d	q
7	6.9	7.2	7.1	7.4	7.3	7.6	7.5	7.7	7.8	0	0	1

Excel formula:

=INNOVATIONS_MLE({7,6.9,7.2,7.1,7.4,7.3,7.6,7.5,7.7,7.8}, 0, 0, 1)

Expected output:

Result
0.433657	0.0600301

Example 4: Innovations MLE without demeaning

Inputs:

endog										p	d	q	demean
2	2.2	2.5	2.7	2.9	3.1	3	3.3	3.5	3.6	1	0	1	false

Excel formula:

=INNOVATIONS_MLE({2,2.2,2.5,2.7,2.9,3.1,3,3.3,3.5,3.6}, 1, 0, 1, FALSE)

Expected output:

Result
0.996018	0.413066	0.0338986

Python Code

import numpy as np
from statsmodels.tsa.arima.model import innovations_mle as sm_innovations_mle

def innovations_mle(endog, p=0, d=0, q=0, seasonal_p=0, seasonal_d=0, seasonal_q=0, seasonal_s=0, demean=True, enforce_invertibility=True):
    """
    Estimate SARIMA parameters using innovations maximum likelihood.

    See: https://www.statsmodels.org/stable/generated/statsmodels.tsa.arima.model.innovations_mle.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        endog (list[list]): Time-series observations as a 2D range (data points).
        p (int, optional): Non-seasonal autoregressive order (lags). Default is 0.
        d (int, optional): Non-seasonal differencing order (differences). Default is 0.
        q (int, optional): Non-seasonal moving-average order (lags). Default is 0.
        seasonal_p (int, optional): Seasonal autoregressive order (lags). Default is 0.
        seasonal_d (int, optional): Seasonal differencing order (differences). Default is 0.
        seasonal_q (int, optional): Seasonal moving-average order (lags). Default is 0.
        seasonal_s (int, optional): Seasonal period length (periods). Default is 0.
        demean (bool, optional): Whether to remove the mean before estimation (true/false). Default is True.
        enforce_invertibility (bool, optional): Whether to enforce invertibility constraints (true/false). Default is True.

    Returns:
        list[list]: Row vector containing estimated SARIMA parameter values.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        endog = to2d(endog)

        if not isinstance(endog, list) or not all(isinstance(row, list) for row in endog):
            return "Error: Invalid input - endog must be a 2D list"

        series = []
        for row in endog:
            for value in row:
                try:
                    series.append(float(value))
                except (TypeError, ValueError):
                    continue

        if len(series) < 6:
            return "Error: endog must contain at least 6 numeric values"

        params, _ = sm_innovations_mle(
            series,
            order=(int(p), int(d), int(q)),
            seasonal_order=(int(seasonal_p), int(seasonal_d), int(seasonal_q), int(seasonal_s)),
            demean=bool(demean),
            enforce_invertibility=bool(enforce_invertibility)
        )

        if hasattr(params, "params"):
            values = np.asarray(params.params, dtype=float).ravel()
        elif hasattr(params, "to_array"):
            values = np.asarray(params.to_array(), dtype=float).ravel()
        else:
            values = np.asarray(params, dtype=float).ravel()

        return [values.tolist()]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

endog *

Time-series observations as a 2D range (data points).

Non-seasonal autoregressive order (lags).

Non-seasonal differencing order (differences).

Non-seasonal moving-average order (lags).

seasonal_p

Seasonal autoregressive order (lags).

seasonal_d

Seasonal differencing order (differences).

seasonal_q

Seasonal moving-average order (lags).

seasonal_s

Seasonal period length (periods).

demean

Whether to remove the mean before estimation (true/false).

enforce_invertibility

Whether to enforce invertibility constraints (true/false).