Formula for an ARIMA(1,1,1) solving for y

I have an Arima(1,1,1) model with predictors var1+var2+var3, but am struggling with how to write the equation. The problem is that on all of the sources I see a variation of the following is given.

$$\left( 1 - \sum_{i=1}^p \phi_i L^i\right) (1-L)^d y_t = \delta + \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t .$$

Is there a way to write the equation solving for $y_t$? I find that the above equation is difficult to understand. Though I have the model saved in R the users of the forecast want the coefficients so they can plug them into excel and I don't have a good way to explain the formula solving for $y_t$

• What do you mean by "predictors var1+var2+var3"? Do you have an ARIMAX model, or a regression with ARIMA errors? Or do you have simply a straightforward ARIMA(1,1,1) model? This earlier thread may be helpful, although it does not specifically contain an ARIMA(1,1,1) model: Interpret ARIMA models in plain english – Stephan Kolassa Dec 13 '17 at 17:05
• Let me give a slightly different example with some output from R. Lets say I have an ARIMA(5,1,0). With the following output: ar1=-.1,ar2=-.2,ar3=.01,ar4=.02,ar5=.03, var1=.5,var2=.3,var3=.6. How do I write that as an equation? – Alex Dec 13 '17 at 17:45
• What do you mean by "predictors var1+var2+var3"? – Stephan Kolassa Dec 13 '17 at 18:05
• Thanks for you patience, var1+var2+var3 would be coded as Arima(log(train[,"sales"]), xreg=cbind(var1,var2,var3),order = c(5,1,0)). Hyndman calls this "Dynamic Regression". – Alex Dec 13 '17 at 19:57
• Thank you, that clarifies matters. This is a case of regression with ARIMA errors, see Hyndman's blog post "The ARIMAX model muddle". I'll try to post an answer tomorrow and retract my close vote for now. Everybody: please leave open, thanks. – Stephan Kolassa Dec 13 '17 at 21:09

Arima() fits a so-called regression with ARIMA errors. Note that this is different from an ARIMAX model. In your particular case, you regress your focal variable on three predictors, with an ARIMA(1,1,1) structure on the residuals:

$$y_t = \beta_1x_{1t}+\beta_2x_{2t}+\beta_3x_{3t}+\epsilon_t$$

with $\epsilon_t\sim\text{ARIMA}(1,1,1)$.

To write down the formulas for $\epsilon_t$, we use the backshift operator. For ARIMA background, see here. A general ARIMA(1,1,1) model with AR parameter $\phi$ and MA parameter $\theta$ has the following form (note that some packages flip the sign of $\theta$, and that one sometimes fits a nonzero intercept $c$):

$$(1-\phi B)(1-B)\epsilon_t = (1+\theta B)\eta_t,$$

where $\eta_t\sim N(0,\sigma^2)$ are IID innovations. Some calculations show that

\begin{align*} & \epsilon_t-\phi\epsilon_{t-1}-\epsilon_{t-1}+\phi\epsilon_{t-2} \\ = & (1-\phi B)(\epsilon_t-\epsilon_{t-1}) \\ = & (1-\phi B)(1-B)\epsilon_t \\ = & (1+\theta B)\eta_t \\ = & \eta_t+\theta\eta_{t-1}, \end{align*}

or

$$\epsilon_t = (1+\phi)\epsilon_{t-1}-\phi\epsilon_{t-2}+\eta_t+\theta\eta_{t-1}.$$

• So in the end, this is the formula? $$y_t = \beta_1x_{1t}+\beta_2x_{2t}+\beta_3x_{3t}+ (1+\phi)\epsilon_{t-1}-\phi\epsilon_{t-2}+\eta_t+\theta\eta_{t-1}.$$ – Alex Dec 14 '17 at 14:34
• You can write it that way. I just don't like the way $\epsilon_t$ does not show up, but $\epsilon_{t-1}$ and $\epsilon_{t-2}$ do, which is why I stuck with writing down the regression and the ARIMA residual part separately. – Stephan Kolassa Dec 14 '17 at 14:41