Have a Question?
Phone: +1 (888) 427-9486
+1 (312) 324-0367
Fax: +1 (312) 238-9092
Contact Us
ARMA_RESID
Returns an array of cells for the standardized residuals of a given ARMAi 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 of the data series (i.e. whether the first data point corresponds to the earliest or latest date (earliest date=1 (default), latest date=0)).
| Order | Description |
|---|---|
| 1 | ascending (the first data point corresponds to the earliest date) |
| 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
- 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 is defined as:
![\[ \epsilon_t = \frac{a_t}{\sigma_t} \]](/sites/all/files/tex/b4786b97f201ba14c24718e3cca5ef86aff846c0.png)
![\[ a_t = x_t - \hat x_t \]](/sites/all/files/tex/5a9cf3cd6f98c9f04d9b5105f43405d8bd9a869c.png)
![\[ \hat x_t = \mu + \sum_{i=1}^p \phi_i x_{t-i} + \sum_{j=1}^q \theta_j a_{t-j} \]](/sites/all/files/tex/9ac0626f1d2ca47381453ba0271b42177ae37f03.png)
Where:
-
is the ARMA model's standardized residual at time t.
-
is the ARMA model's residual at time t.
-
is the value of time series at time t.
-
is the fitted model value (i.e. conditional mean) at time t.
-
is the number of non-missing values in the data sample.
-
- The number of parameters in the input argument - phi - determines the order of the AR component.
- The number of parameters in the input argument - theta - determines the order of the MA component.
Examples
Example 1:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | Date | Data | |||
| 2 | January 10, 2008 | -0.30 | 0.032 |
ARMA |
|
| 3 | January 11, 2008 | -1.28 | -0.638 | Mean | -0.35 |
| 4 | January 12, 2008 | 0.24 | 0.641 | Sigma | 1.3059 |
| 5 | January 13, 2008 | 1.28 | 0.793 | Phi_1 | -0.4296 |
| 6 | January 14, 2008 | 1.20 | 0.925 | Theta_1 | 0.999897 |
| 7 | January 15, 2008 | 1.73 | 1.167 | ||
| 8 | January 16, 2008 | -2.18 | -1.715 | ||
| 9 | January 17, 2008 | -0.23 | 1.058 |
LLFi |
stable? |
| 10 | January 18, 2008 | 1.10 | 0.137 | -44 | 1 |
| 11 | January 19, 2008 | -1.09 | -0.212 | ||
| 12 | January 20, 2008 | -0.69 | -0.289 | ||
| 13 | January 21, 2008 | -1.69 | -0.829 | ||
| 14 | January 22, 2008 | -1.85 | -0.763 | ||
| 15 | January 23, 2008 | -0.98 | -0.230 | ||
| 16 | January 24, 2008 | -0.77 | -0.297 | ||
| 17 | January 25, 2008 | -0.30 | 0.183 | ||
| 18 | January 26, 2008 | -1.28 | -0.851 | ||
| 19 | January 27, 2008 | 0.24 | 0.951 | ||
| 20 | January 28, 2008 | 1.28 | 0.503 | ||
| 21 | January 29, 2008 | 1.20 | 1.205 | ||
| 22 | January 30, 2008 | 1.73 | 0.905 | ||
| 23 | January 31, 2008 | -2.18 | -1.570 | ||
| 24 | February 1, 2008 | -0.23 | 1.007 | ||
| 25 | February 2, 2008 | 1.10 | 0.160 | ||
| 26 | February 3, 2008 | -1.09 | -0.242 | ||
| 27 | February 4, 2008 | -0.69 | -0.262 | ||
| 28 | February 5, 2008 | -1.69 | -0.866 | ||
| 29 | February 6, 2008 | -1.85 | -0.726 | ||
| 30 | February 7, 2008 | -0.98 | -0.257 |
References
- Hamilton, J .D.; Time Series Analysis
, Princeton University Press (1994), ISBN 0-691-04289-6 - Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
