# ARIMA(1,1,1) Model - Forecast

How does one

1. write the mathematical equation for the ARIMA(1,1,1) model with the estimated coefficients below and
2. use the ARIMA(1,1,1) model and time series points below to produce a forecast value for time = 41 (e.g. the forecast value for t = 41 is 18)?

auto.arima(deseasonal_cnt, seasonal=FALSE)

Series: deseasonal_cnt
ARIMA(1,1,1)

Coefficients:

ar1     ma1
0.5510  -0.2496

s.e. 0.0751 0.0849

sigma^2 estimated as 26180: log likelihood=-4708.91
AIC=9423.82 AICc=9423.85 BIC=9437.57

airplane[35:40]
19.2981 17.1230 16.1230 18.1352 19.3401 18.1414

fitted(Arima(airplane, c(1,1,1)))[35:40]
18.3492 18.3752 17.8322 18.7924 19.8253 18.8198

• Check out similar questions on this site and you will find a dozen answers. Apr 27, 2018 at 7:22

## 1 Answer

All ARIMA models can be expressed in a similar form to this

• What do your variables $Y$, $X$, $a$ refer to? They don't seem to connect with anything named in the question, so some explanation is needed.
– whuber
May 14, 2018 at 21:40
• Y is the observed dependent series (endogenous) and the X's are possible exogenous series either user-specified or analytically developed (pulses/level shiftts/trends.seasonal pulses)... none in this case. May 14, 2018 at 22:22