ARMA_LLF - Log Likelihood Function of an ARMA Model

Computes the log-likelihood function (LLF) of the estimated ARMA model.

Syntax

ARMA_LLF ([x], order, µ, σ, [φ], [θ])

[X]

Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).

Order

Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).

Value	Order
1	Ascending (the first data point corresponds to the earliest date) (default).
0	Descending (the first data point corresponds to the latest date).

µ

Optional. Is the ARMA model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.

σ

Required. Is the standard deviation value of the model's residuals/innovations.

[φ]

Optional. Are the parameters of the AR(p) component model: [φ1, φ2 … φp] (starting with the lowest lag).

[θ]

Optional. Are the parameters of the MA(q) component model: [θ1, θ2 … θq] (starting with the lowest lag).

Remarks

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. The time series is homogeneous or equally spaced.
4. The time series may include missing values (e.g., #N/A) at either end.
5. The residuals/innovations standard deviation (σ) must be greater than zero.
6. The ARMA model has independent and normally distributed residuals with constant variance. The ARMA log-likelihood function becomes:$$\ln L^* = -T\left(\ln 2\pi \hat \sigma^2+1\right)/2 $$

   Where:

   • $\hat \sigma$ is the standard deviation of the residuals.
7. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
8. The number of parameters in the input argument - ([φ]) - determines the order of the AR component.
9. The number of parameters in the input argument - ([θ]) - determines the order of the MA component.

Files Examples

Related Links

• Wikipedia - Likelihood function.
• Wikipedia - Likelihood principle.
• Wikipedia - Autoregressive moving average model.

References

• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
• James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
• Box, Jenkins and Reinsel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
• Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.