ARIMA Differencing

Basic Concepts

In order to create a stationary process, differencing may be necessary. For example, the graph of closing Dow Jones indices for October 2015 (see Example 1 of Stationary Process), as shown in Figure 1, clearly shows an increasing trend. Differencing is a way to eliminate such trends.

Figure 1 – Dow Jones Indices for October 2015

First-order differencing addresses linear trends, and employs the transformation z_i = y_i – y_i-1. Second-order differencing addresses quadratic trends and employs a first-order difference of a first-order difference, namely z_i = (y_i – y_i-1) – (y_i-1 – y_i-2), which is equivalent to z_i = y_i – 2y_i-1+ y_i-2.

Taking first-order differences for the data in Figure 1 results in the chart on the right. The trend seems to have been eliminated.

ARIMA

An autoregressive integrated moving average (ARIMA) process (aka a Box-Jenkins process) adds differencing to an ARMA process. An ARMA(p,q) process with d-order differencing is called an ARIMA(p.d,q) process. Thus, for example, an ARIMA(2,1,0) process is an AR(2) process with first-order differencing.

It is important not to over-difference since this can cause you to use an incorrect model. Some rules-of-thumb indicating that you may have differenced too many times are:

• The autocorrelation of a differenced series is less than -.5
• Differencing increases the variance

Rules-of-thumb

An AR(p) or MA(q) process has a unit root if the sum of the non-constant coefficients is 1.

Additional rules-of-thumb:

• If an AR(p) process has a unit root then the level of differencing should be increased
• If an MA(q) process has a unit root then the level of differencing should be decreased

Assume that column A contains a time series of size n starting in cell A1. Now suppose we want to place the 1^st order differences (of size n-1) in column B starting in cell B2, the 2^nd order differences (of size n-2) in column C starting in cell C3, and so on with the 7^th order differences (of size n-7) in column H starting in cell H8, then the formulas that need to be used in these starting cells are as follows:

• B2: A2-A1
• C3: A3-2*A2+A1
• D4: A4-3*A3+3*A2-A1
• E5: A5-4*A4+6*A3-4*A2+A1
• F6: A6-5*A5+10*A4-10*A3+5*A2-A1
• G7: A7-6*A6+15*A5-20*A4+15*A3-6*A2+A1
• H8: A8-7*A7+21*A6-35*A5+35*A4-21*A3+7*A2-A1

For each column, you need to highlight the range down to the nth cell in that column and press Ctrl-D to get the other values. Note that for the 7^th order difference, the coefficients used are C(7,0) = 1, C(7,1) = 7, C(7,2) = 21, etc.

Worksheet Function

Real Statistics Function: The Real Statistics Resource Pack provides the following array function.

ADIFF(R1, d) – takes the time series in the n × 1 range R1 and outputs an n–d × 1 range containing the data in R1 differenced d times

Example 1: Find the 1^st, 2^nd, 3^rd, and 4^th differences for the data in column A of Figure 1.

Figure 1 – Differencing

Here cell B4 contains the formula =A5-A4, cell C4 contains the formula =B5-B4 (or A6-2*A5+A4), cell D4 contains the formula =C5-C4 (or A7-3*A6+3*A5-A4) and cell E4 contains the formula =D5-D4 (or A8-4*A7+6*A6-4*A5+A4).

Range G4:G22 contains the array formula =ADIFF($A$4:$A$23,G3). If we highlight the range G4:J22 and press Ctrl-R, we get the result shown on the right side of Figure 1.

References

Nau, R. (2020) Identifying the order of differencing in an ARIMA model
https://people.duke.edu/~rnau/411arim2.htm