# ARMA_FORE - Forecasting for ARMA Model

Calculates the out-of-sample conditional mean forecast.

## Syntax

ARMA_FORE(X, Order, mean, sigma, phi, theta, T, Type, alpha)

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 ARMA model long-run 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).

T is the forecast time/horizon (expressed in terms of 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 Lower limit of the forecast confidence interval
5 Upper limit of the forecast confidence interval

alpha is the statistical significance level. If missing, a default of 5% is assumed.

## 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 long-run mean can take any value or be omitted, in which case a zero value is assumed.
5. The residuals/innovations standard deviation (sigma) must be greater than zero.
6. 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 an error code (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).
7. 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).

## Examples

Example 1:

A B C D
1 Date Data
2 1/1/2008 -0.30 ARMA
3 1/2/2008 -1.28 Mean -0.00258
4 1/3/2008 0.24 Sigma 0.14
5 1/4/2008 1.28 Phi_1 -0.236
6 1/5/2008 1.20 Theta_1 -5.60E-05
7 1/6/2008 1.73
8 1/7/2008 -2.18
9 1/8/2008 -0.23
10 1/9/2008 1.10
11 1/10/2008 -1.09
12 1/11/2008 -0.69
13 1/12/2008 -1.69
14 1/13/2008 -1.85
15 1/14/2008 -0.98
16 1/15/2008 -0.77
17 1/16/2008 -0.30
18 1/17/2008 -1.28
19 1/18/2008 0.24
20 1/19/2008 1.28
21 1/20/2008 1.20
22 1/21/2008 1.73
23 1/22/2008 -2.18
24 1/23/2008 -0.23
25 1/24/2008 1.10
26 1/25/2008 -1.09
27 1/26/2008 -0.69
28 1/27/2008 -1.69
29 1/28/2008 -1.85
30 1/29/2008 -0.98

Formula Description (Result)
=ARMA_FORE(\$B\$2:\$B\$30,1,\$D\$3,\$D\$4,\$D\$5,\$D\$6,1) The conditional mean forecast value at T+1 (0.228)
=ARMA_FORE(\$B\$2:\$B\$30,1,\$D\$3,\$D\$4,\$D\$5,\$D\$6,2) The conditional mean forecast value at T+2 (-0.057)
=ARMA_FORE(\$B\$2:\$B\$30,1,\$D\$3,\$D\$4,\$D\$5,\$D\$6,3) The conditional mean forecast value at T+3 (0.010)
=ARMA_FORE(\$B\$2:\$B\$30,1,\$D\$3,\$D\$4,\$D\$5,\$D\$6,4) The conditional mean forecast value at T+4 (-0.006)