ARIMAX Model and Forecast

An ARMAX model (i.e. an ARIMA model with an exogenous variable) without constant takes the form

ARMAX model

This is simply an ARMA model with an extra independent variable (covariant) on the right side of the equation. Using the lag operator, this is equivalent to

ARMAX using lag functionorARMAX model aternatuve equation

One way to deal with such a model is to reinterpret it as a linear regression plus ARMA errors:

ARMAX equation simplifiedwhereEquation for u terms

This model is equivalent to
ARMAX model equation

Example 1: Create an ARIMAX model for the data on the left side of Figure 1 where X1 and X2 are exogenous variables and Y is a time series. Create a forecast for the next 3 elements based on this model.

ARIMAX model initialization

Figure 1 – Initialization of the ARIMAX model

Real Statistics Data Analysis Tool: You can use the ARIMAX data analysis tool to do this. Press Ctrl-m, select ARIMAX from the Time S tab and fill in the dialog box that appears as shown in Figure 2.

ARIMAX dialog box

Figure 2 – ARIMAX dialog box

The results are shown on the right side of Figure 1 as well as in Figure 3 and 4.

In Figure 1, range G4:G22 contains the array formula =ADIFF(B4:B23,1), range H5:H22 contains =ADIFF(C4:C23,1) and I5:I22 contains =ADIFF(D:D23,1).

The left side of Figure 3 contains the usual regression analysis of X1 and X2 on Y, which results in the regression model

ARIMAX regression formula

The residuals are calculated by

ARIMAX residuals

where we are expecting the residuals to follow an ARIMA(0,0,1) model. These residuals are shown in range J5:J22 of Figure 1, as calculated by the array formula

=I4:I22-TREND(I4:I22,G4:H22,,TRUE)

ARIMAX regression analysis

Figure 3 – OLS Regression model

The residuals from the OLS regression model now become the data elements for the ARIMA model, as shown in Figure 4. Note that the constant term is subsumed in the regression model and so is not included in the ARIMA model. Similarly, the differencing has already been accounted for and so is not part of the ARIMA model. Thus, we are assuming that the residuals follow an MA(1) model.

ARMAmodel for residuals

Figure 4 – ARIMA(0,0,1) model for the residuals

The forecast for the model shown in Figure 4 is shown in Figure 5. Note that the zero forecast values shown in cells AV24 and AV25 wouldn’t necessarily be zero if we had used a different ARIMA model for the residuals.

ARIMAX residuals forecast

Figure 5 – Residuals forecast 

The forecast in Figure 5 is only for the residuals times series. We now need to create a forecast for the original time series at times t = 21, 22 and 23, based on the values we expect for the exogenous variables X1 and X2 at those times.

Suppose that these exogenous variables take the values shown in range B24:C26 of Figure 6. Note that this figure shows the bottom part of corresponding columns from Figure 1, where the added rows correspond to the three forecast values.

The added entries in range D24:D26 show the forecasted values for the original time series at times t = 21, 22 and 24 corresponding to the X1 and X2 values shown in B24:C26. These forecasted values are calculated as shown in Figure 6.

ARIMAX time series forecast

Figure 6 – Time series forecast

Place the formula =B24-B23 in cell G23, highlight range G23:H25 and press Ctrl-R and Ctrl-D. This differences the new X1 and X2 values. Next, place the array formula =TREND(I4:I22,G4:H22,G23:H25) in range I23:I25. This calculates the differenced Y forecast values.

Now place the formula =AV23 in cell J23, highlight the range J23:J25 and press Ctrl-D, to show the forecasted residual values. Finally, insert the formula =D23+I23+J23 in cell D24, highlight the range D24:D26 and press Ctrl-D to obtain the requested forecast for Y.

17 thoughts on “ARIMAX Model and Forecast”

  1. Hello, Charles. Thank you for the tools. When i was forecasting with ARIMAX model, there was a message that said “a run time error was ocured. The analysis tool will be aborted. Application defined-or object defined error”. Could you help me?
    and why does the forecast use Arimax residual values, why not use the original data?

    Reply

  2. Hi Charles..
    i am Riana, from Indonesia. in the material you wrote it says that “use ARIMAX data analysis tool to do this,Press Ctrl-m”, but I’m having trouble with ms excel, I can’t use the shortcut ctrl-m. what should i do, charles?

    thanks, Charles
    I hope you reply to this comment,

    Reply

  3. Good evening Carlos!

    I’m doing a college project on demand forecasting and we should use a method that we didn’t learn in class. My group chose the ARIMAX model, but as we haven’t seen anything in class about this model, we’re having some difficulties. Could you send me this spreadsheet that you used in the example?
    My email is kevintiscoski@gmail.com

    I would be very grateful. Hugs from Brazil!

    Reply

  4. Charles, thanks for your tool. I am learning a lot about time series analyses and getting a lot out of your tool.
    I’ve tried to replicate your above example to ensure I understand the last steps to predict the next values. However, I get a different result than you do in cells I24 and I25. Should the formula in those two cells also be exactly “=TREND(I4:I22,G4:H22,G23:H25)”?
    Thanks.

    Reply

          • I see the problem now. On my computer, running Excel 365, the formula in range I23:I25 is written as =@TREND(I4:I22,G4:H22,G23:H25). This yields -1.3552 in each cell.
            The formula should be =TREND(I4:I22,G4:H22,G23:H25), without the @. This yields the values shown on the webpage. I still need to explore why the @ has been added incorrectly.
            Thank you for bringing this issue to my attention.
            Charles

        • After following up on another comment that identified the sample problem, I see what is going wrong.
          On my computer, running Excel 365, the formula in range I23:I25 is written as =@TREND(I4:I22,G4:H22,G23:H25). This yields -1.3552 in each cell.
          The formula should be =TREND(I4:I22,G4:H22,G23:H25), without the @. This yields the values shown on the webpage. I still need to explore why the @ has been added incorrectly.
          Thank you for bringing this issue to my attention.
          Charles

          Reply

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