ARMAX_FIT - ARMAX Model Fitted Values

Returns an array of cells for the in-sample model fitted values of the conditional mean, volatility or residuals.

Syntax

ARMAX_FIT ([y], [x], order, [β], µ, σ, [φ], [θ], return)

[Y]

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

[X]

Required. Is the independent variables (exogenous factors) time series data matrix, such that each column represents one variable.

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).

[β]

Optional. Is the coefficients array of the exogenous factors.

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).

Return

Optional. Is an integer switch to select the output type: (1 = Mean (default), 2 = Volatility, 3 = Raw Residuals, 4 = Standardized Residuals).

Value	Return
1	Fitted mean (default).
2	Fitted standard deviation or volatility.
3	Raw (non-standardized) residuals.
4	Standardized residuals.

Remarks

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. Each column in the explanatory factors input matrix (i.e., X) corresponds to a separate variable.
4. Each row in the explanatory factors input matrix (i.e., X) corresponds to an observation.
5. Observations (i.e., rows) with missing values in X or Y are assumed to be missing.
6. The number of rows of the explanatory variable (X) must be equal to the number of rows of the response variable (Y).
7. The time series is homogeneous or equally spaced.
8. The time series may include missing values (e.g., #N/A) at either end.
9. The long-run mean can take any value or be omitted, in which case a zero value is assumed.
10. The residuals/innovations standard deviation (σ) must be greater than zero.
11. For the input argument - ([β]):
• The input argument is optional and can be omitted, in which case no regression component is included (i.e., plain ARMA).
• The order of the parameters defines how the exogenous factor input arguments are passed.
• One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
12. For the input argument - ([φ]):
• The input argument is optional and can be omitted, in which case no AR component is included.
• The order of the parameters starts with the lowest lag.
• One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
• The order of the AR component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
13. For the input argument - ([θ]):
• The input argument is optional and can be omitted, in which case no MA component is included.
• The order of the parameters starts with the lowest lag.
• One or more values in the input argument can be missing or an error code (i.e., #NUM!, #VALUE!, etc.).
• The order of the MA component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
14. The function was added in version 1.63 SHAMROCK.

Files Examples

Related Links

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

References

• 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.