ARMA_FIT - ARMA Model Fitted Values

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

Syntax

ARMA_FIT ([x], order, µ, σ, [φ], [θ], return)

[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).

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. 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 integration order argument (d) must be a positive integer.
6. The long-run mean can take any value or may be omitted, in which case a zero value is assumed.
7. The residuals/innovations standard deviation (σ) must be greater than zero.
8. For the input argument – ([φ]):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters can be missing or an error code (i.e., #NUM!, #VALUE!, etc.).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - ([θ]):
   • The input argument is optional and can be omitted, so 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 last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. 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

• 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.