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).

Warning

ARMA_LLF(.) function is deprecated as of version 1.63: use ARMA_GOF(.) function instead.

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.