ARMAX_FORE - Forecasting for ARMAX Model

Calculates the out-of-sample conditional forecast (i.e., mean, error, and confidence interval)

Syntax

ARMAX_FORE(Y, X, Order, Beta, mean, sigma, phi, theta, T, Type, alpha)

Y

is the response or the dependent variable time series data array (a one-dimensional array of cells (e.g., rows or columns)).

X

is the independent variables (exogenous factors) time series data matrix, such that each column represents one variable.

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)).

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).

Beta

are the coefficients array of the exogenous factors.

mean

is the ARMA long-run mean (i.e., mu).

sigma

is the standard deviation of the model's residuals.

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).

T

is the simulation time/horizon (expressed in steps beyond the end of the time series).

Type

is an integer switch to select the forecast output type: (1 = mean (default), 2 = Std. Error, 3 = Term Struct, 4 = LL, 5 = UL)

Order	Description
1	Mean forecast value (default).
2	Forecast standard error (aka local volatility).
3	Volatility term structure.
4	The lower limit of the forecast confidence interval.
5	The upper limit of the forecast confidence interval.

alpha

is the statistical significance level (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

Remarks

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
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 long-run mean can take any value or be omitted, in which case a zero value is assumed.
6. The residuals/innovations standard deviation (sigma) must be greater than zero.
7. For the input argument (beta):
   • The input argument is optional and can be omitted, in which case no regression component is included (i.e., plain ARMA).
   • The order of the parameters defines how the exogenous factor input arguments are passed.
   • One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
8. For the input argument (phi):
   • The input argument is optional and can be omitted, in which case no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
   • The order of the AR component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
9. For the input argument (theta):
   • The input argument is optional and can be omitted, in which case 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 MA component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
10. 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

• 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.