The code that Î used to generate ARIMA summary is

`arimafore = forecast(auto.arima(sales), h = 12) summary(arimafore) `

For forecasting 12 months I got the summary below:

`Forecast method: ARIMA(2,1,0) with drift Model Information: Series: sales ARIMA(2,1,0) with drift Coefficients: ar1 ar2 drift -0.8102 -0.4819 2774.6233 s.e. 0.1750 0.1688 682.3571 sigma^2 estimated as 62683829: log likelihood=-248.34 AIC=504.67 AICc=506.78 BIC=509.39 Error measures: ME RMSE MAE MPE MAPE MASE ACF1 Training set 63.40175 7256.336 5466.688 -0.2247928 4.024713 0.1580417 -0.1562984 Forecasts: Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 Feb 2018 173223.9 163077.5 183370.4 157706.3 188741.6 Mar 2018 176272.6 165945.0 186600.2 160477.8 192067.3 Apr 2018 181744.1 170775.0 192713.2 164968.3 198519.9 May 2018 182201.7 169589.9 194813.4 162913.7 201489.6 Jun 2018 185553.8 172511.7 198595.9 165607.6 205500.0 Jul 2018 188977.2 175195.2 202759.2 167899.5 210054.9 Aug 2018 190947.9 176308.7 205587.0 168559.2 213336.5 Sep 2018 194061.2 178885.2 209237.1 170851.6 217270.8 Oct 2018 196948.9 181112.2 212785.5 172728.8 221168.9 Nov 2018 199468.7 182991.1 215946.3 174268.3 224669.0 Dec 2018 202395.3 185371.4 219419.1 176359.6 228430.9 Jan 2019 205169.6 187564.3 222774.9 178244.6 232094.6 `

Can you help me to interpret ARIMA(2,1,0) with drift? Is it more of a trend model?

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#### Best Answer

It's a model like this:

$$Delta x_{t} = b + sum_{i=1}^{m} phi(i) * Delta x_{t-i} + sum_{j=0}{m} theta(j) * epsilon(t-j)$$

"Integrating" this model gives you the original $x_{t}$ model. As you can see here by the integration you'll give $a + b*t$ member in you closed form for $x(t)$:

$$x(t) = a + b*t + sum_{i=1}{n+1} phi_{new}(i) x(t-i) + sum_{j=0}{m} theta_{new}(j) * epsilon(t-j)$$