# ARMA_RESID - ARMA fitted values of standardized residuals

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

## Syntax

ARMA_RESID(X, Order, mean, sigma, phi, theta)

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 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).

## Remarks

1. The underlying model is described here.
2. Warning: ARMA_RESID() function is deprecated as of version 1.63: use ARMA_FIT function instead.
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 standardized residuals have a mean of zero and a variance of one (1).
6. The ARMA model's standardized residuals is 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 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.
7. The number of parameters in the input argument - phi - determines the order of the AR component.
8. The number of parameters in the input argument - theta - determines the order of the MA component.

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
A B C D E
Date Data
January 10, 2008 -0.30 0.032 ARMA
January 11, 2008 -1.28 -0.638 Mean -0.35
January 12, 2008 0.24 0.641 Sigma 1.3059
January 13, 2008 1.28 0.793 Phi_1 -0.4296
January 14, 2008 1.20 0.925 Theta_1 0.999897
January 15, 2008 1.73 1.167
January 16, 2008 -2.18 -1.715
January 17, 2008 -0.23 1.058 LLF stable?
January 18, 2008 1.10 0.137 -44 1
January 19, 2008 -1.09 -0.212
January 20, 2008 -0.69 -0.289
January 21, 2008 -1.69 -0.829
January 22, 2008 -1.85 -0.763
January 23, 2008 -0.98 -0.230
January 24, 2008 -0.77 -0.297
January 25, 2008 -0.30 0.183
January 26, 2008 -1.28 -0.851
January 27, 2008 0.24 0.951
January 28, 2008 1.28 0.503
January 29, 2008 1.20 1.205
January 30, 2008 1.73 0.905
January 31, 2008 -2.18 -1.570
February 1, 2008 -0.23 1.007
February 2, 2008 1.10 0.160
February 3, 2008 -1.09 -0.242
February 4, 2008 -0.69 -0.262
February 5, 2008 -1.69 -0.866
February 6, 2008 -1.85 -0.726
February 7, 2008 -0.98 -0.257