Returns an array of cells for the standardized residuals of a given ARMA model.

## Syntax

**ARMA_RESID** (**[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_RESID(.) function is deprecated as of version 1.63: use the ARMA_FIT(.) function instead.

## Remarks

- The underlying model 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 standardized residuals have a mean of zero and a variance of one (1).
- The ARMA model's standardized residuals are defined as:$$\epsilon_t = \frac{a_t}{\sigma_t}$$ $$a_t = x_t - \hat x_t$$ $$\hat x_t = \mu + \sum_{i=1}^p \phi_i x_{t-i} + \sum_{j=1}^q \theta_j a_{t-j}$$
Where:

- $\epsilon$ is the ARMA model's standardized residual at time $t$.
- $a_t$ is the ARMA model's residual at time $t$.
- $x_t$ is the value of the time series at time $t$.
- $\hat x_t$ is the fitted model value (i.e., conditional mean) at time $t$. $$1\leq t \leq T$$
- $T$ is the number of non-missing values in the data sample.

- The number of parameters in the input argument - ([φ]) - determines the order of the AR component.
- The number of parameters in the input argument - ([θ]) - determines the order of the MA component.

## Files Examples

## Related Links

## 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.

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