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

## Syntax

**ARMA_FIT**(**X**, Order, **Mean**, **sigma**, **phi**, **theta**, Type)

- X
- is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
- Order
- is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Order Description 1 Ascending (the first data point corresponds to the earliest date) ( **default**).0 Descending (the first data point corresponds to the latest date). - mean
- is the ARMA model mean (i.e., mu).
- sigma
- is the standard deviation of the model's residuals/innovations.
- phi
- are the parameters of the AR(p) component model (starting with the lowest lag).
- theta
- are the parameters of the MA(q) component model (starting with the lowest lag).
- Type
- is an integer switch to select the output type: (1 = Mean (default), 2 = Volatility, 3 = Raw Residuals, 4 = Standardized Residuals).
Order Description 1 Fitted mean ( **default**).2 Fitted standard deviation or volatility. 3 Raw (non-standardized) residuals. 4 Standardized residuals.

## Remarks

- The underlying model is described here.
- The Log-Likelihood Function (LLF) is described here.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The integration order argument (d) must be a positive integer.
- The long-run mean can take any value or may be omitted, in which case a zero value is assumed.
- The residuals/innovations standard deviation (sigma) must be greater than zero.
- For the input argument (phi):
- 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).

- For the input argument (theta):
- 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).

- 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 Reisel; 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.

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