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, 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).

Remarks

1. The underlying model is described here.
2. The time series is homogeneous or equally spaced.
3. The time series may include missing values (e.g., #N/A) at either end.
4. The standardized residuals have a mean of zero and a variance of one (1).
5. 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.
6. The number of parameters in the input argument - ([φ]) - determines the order of the AR component.
7. The number of parameters in the input argument - ([θ]) - determines the order of the MA component.

Files Examples

Related Links

• 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.