Choosing an Ordinary Difference Order
The difference order \(d\) in an ARIMA model records how many ordinary differences are applied before modeling the resulting series. Choosing it is a diagnostic decision, not a rule that every persistent-looking series must be differenced. Too little differencing can leave a changing level in the data; too much can add artificial dependence and discard useful information.
A time plot and an autocorrelation function (ACF) provide complementary evidence. The plot shows whether the series appears to fluctuate around a stable level or instead wanders, trends, or changes abruptly. The ACF summarizes the relationship between observations separated by different lags. Neither diagnostic, by itself, proves that a series is stationary or nonstationary. In particular, a slowly decaying ACF is a warning to investigate, not an automatic instruction to difference.
Reading Plots and ACFs Together
A series that fluctuates around a reasonably stable mean with roughly stable variability is a candidate for \(d=0\). Its ACF may have significant early lags, particularly when the series has an autoregressive pattern, but for a stationary process the correlations generally decay toward zero. The decay can be gradual when the process is highly persistent.
A series with a wandering level or a sustained trend may need differencing. A random walk is a standard example: its level accumulates shocks, while its first difference is the sequence of shocks. If a first difference has a stable-looking plot and an ACF that no longer shows persistent correlation, that is evidence in favor of \(d=1\). The evidence is stronger when the level plot and level ACF also suggest nonstationarity.
For a sample of size \(n\), a rough reference band often used to assess individual sample ACF values under a white-noise benchmark is \(\pm 1.96/\sqrt{n}\). It is a diagnostic aid, not a formal guarantee: the band is approximate, many lags are examined, and interpretation is affected by how the model and mean are estimated. A few values outside the band do not, by themselves, determine the difference order.
| Evidence in the level series | Evidence after one difference | Reasonable next step |
|---|---|---|
| Stable level and variability; ACF decays toward zero | Not needed for the initial decision | Consider \(d=0\) |
| Wandering level or sustained trend; ACF remains highly persistent | Plot and ACF are more stable, without persistent decay | Consider \(d=1\) |
| Abrupt shift, outlier, changing variability, or conflicting evidence | Difference may create a spike or leave unexplained structure | Investigate the cause and specification before choosing \(d\) |
These are evidence patterns, not mechanical tests. For example, a deterministic trend can make the level plot and ACF look persistent even though the source of the trend differs from a stochastic unit root. A break in level can also inflate apparent persistence. Such features call for scrutiny of the data and model assumptions; they should not be handled automatically by applying additional differences.
Why Differencing Changes the ACF
Differencing changes dependence, so an ACF observed after differencing must be interpreted as the ACF of a new series. The following derivation explains why differencing a stationary series can create negative lag-one correlation.
Suppose \(Y_t\) is a covariance-stationary AR(1) process with \(|\phi|<1\), mean \(\mu\), and autocovariance \(\gamma_k=\operatorname{Cov}(Y_t,Y_{t-k})=\gamma_0\phi^k\) for \(k\geq 0\). Set \(Z_t=\Delta Y_t=Y_t-Y_{t-1}\). By expanding the covariance and using stationarity,
Substituting \(\gamma_1=\gamma_0\phi\) and \(\gamma_2=\gamma_0\phi^2\), then dividing the lag-one covariance by the variance, gives
Because \(|\phi|<1\), this value is negative. Thus a negative lag-one ACF after differencing is not proof that a difference was wrong, but a pronounced negative value can be a warning when the original series already looked stationary. The calculation also shows why differencing can alter dependence even when it is unnecessary.
Worked Diagnostic Cases
Worked Example: A Persistent but Stable Series
A hypothetical portfolio’s monthly index is plotted over \(n=100\) months. The plot fluctuates around a stable level, with no sustained trend or obvious change in variability. Its sample ACF at lags 1 through 4 is \(0.52, 0.29, 0.17, 0.09\). The first-difference plot fluctuates around zero, and its lag-one sample ACF is \(-0.23\). Assess whether ordinary differencing is indicated.
Method. First calculate the rough reference band: \(1.96/\sqrt{100}=1.96/10=0.196\). In the level ACF, lag 1 is above the band, lag 2 is above it, and lags 3 and 4 are within it. The values decline toward zero rather than remaining near one across many lags.
The plot gives no clear indication of a changing level. The early ACF values show persistence, but the gradual decay is compatible with a stationary, serially dependent series. The lag-one ACF of the first difference, \(-0.23\), is also just below the rough lower reference value \(-0.196\). That negative value is a reason to be cautious about differencing, not evidence that differencing is required.
Conclusion. The available evidence favors \(d=0\): the level appears stable and its ACF decays. The level series may still require a dependence model; choosing \(d=0\) does not mean its observations are independent.
Common mistake avoided. Differencing just because the first two level ACF values are outside the band would ignore both the stable plot and the declining ACF. A stationary series can have significant autocorrelations.
Worked Example: A Wandering Level and a More Stable Difference
A hypothetical monthly claim-count index has \(n=80\) observations. Its level plot wanders upward and downward without returning consistently to a fixed level. The level ACF at lags 1 through 4 is \(0.91, 0.82, 0.74, 0.66\). The first differences calculated from consecutive observations are \(2,-1,3,0,-2,1,0,-1,2\), and the full first-difference plot fluctuates around zero without a visible trend. For the full differenced series, the sample ACF at lags 1 through 3 is \(0.08,-0.11,0.05\). Assess the ordinary difference order.
Method. The level reference band is approximately \(1.96/\sqrt{80}=1.96/8.944=0.219\). All four supplied level ACF values exceed \(0.219\), and they remain large across the displayed lags. This persistent ACF agrees with the wandering level plot.
The displayed consecutive differences can be checked directly: for example, if two successive levels are \(L_t\) and \(L_{t-1}\), the corresponding difference is \(L_t-L_{t-1}\); the problem supplies those differences for the full series. For the differenced series, the rough band is \(1.96/\sqrt{79}\), since differencing \(80\) levels leaves \(79\) differences. As \(\sqrt{79}\approx 8.888\), the band is approximately \(\pm0.221\). The supplied lag-one through lag-three values, \(0.08,-0.11,0.05\), all lie within it.
Conclusion. The wandering levels and highly persistent level ACF, followed by a stable-looking difference and small differenced ACF values, support \(d=1\). This is evidence from the combined diagnostics, not a proof based only on the reference band.
Common mistake avoided. Using the level-series sample size \(80\) for the differenced-series reference band overlooks that there are \(79\) first differences. The numerical difference here is small, but the sample size should match the series whose ACF is being interpreted.
Worked Example: An Abrupt Level Shift Calls for Investigation
A hypothetical quarterly service index is observed as \(5,6,5,6,11,12,11,12\). The level plot shows a clear shift between the fourth and fifth observations. Calculate the first differences and decide whether the plot alone justifies selecting \(d=1\) or a higher difference order.
Method. Subtract each observation from the one immediately before it. The seven differences are \(6-5=1\), \(5-6=-1\), \(6-5=1\), \(11-6=5\), \(12-11=1\), \(11-12=-1\), and \(12-11=1\). Therefore, the differenced series is \(1,-1,1,5,1,-1,1\).
The value \(5\) at the shift is unlike the other differences, which alternate between \(1\) and \(-1\). The original level plot’s change is localized at one boundary; it is not a sustained smooth trend. With only eight observations, sample ACFs would also be highly uncertain and should not be treated as decisive.
Conclusion. These data call for further investigation before choosing \(d\). The shift might reflect a real intervention, a measurement change, or another structural feature. The calculation does not establish that a second difference is appropriate: differencing again would transform the isolated jump into additional values and could obscure its cause.
Common mistake avoided. A large jump in a differenced plot is not, by itself, a reason to difference again. First determine whether the jump represents a structural change or data issue, and assess the model using adequate data.
A Practical Decision Process
Look for a stable center and variability, a sustained trend or wandering level, abrupt shifts, and unusual observations. Describe what the plot shows before choosing \(d\).
Ask whether correlations decay, remain persistently high, or show a pattern that conflicts with the plot. Treat approximate reference bands as aids rather than verdicts.
Calculate \(\Delta Y_t=Y_t-Y_{t-1}\), then inspect its plot and ACF. Check whether the changing level has been removed without creating an unexplained pattern.
Favor \(d=0\) when level behavior appears stable; favor \(d=1\) when nonstationary-looking levels become more stable after one difference; investigate further when breaks, outliers, or conflicting evidence dominate.
Common Traps
- “Significant ACF means difference.” A stationary series can have significant autocorrelations. Consider the ACF’s shape and the time plot, not just whether individual bars cross a reference band.
- “A slow decay proves a unit root.” A highly persistent stationary process can also decay slowly. Use the plot and other context to assess the evidence.
- “Difference until the ACF is small.” Differencing changes the dependence structure and can induce negative correlation. Select the smallest supported order, not the order that makes every sample ACF appear insignificant.
- “A break is just a trend.” A level shift can produce apparent persistence and a large first difference. Investigate the timing and meaning of the shift rather than automatically applying more differences.
Check Your Understanding
For each case, select \(d=0\), \(d=1\), or further investigation, and use the stated evidence rather than applying a rule mechanically.
- A series of \(n=100\) daily readings fluctuates around a stable level. Its ACF is \(0.40,0.18,0.07\) at lags 1, 2, and 3. What choice is best supported? (Answer: \(d=0\); the level is stable and the ACF decays.)
- A level series has \(n=64\), a wandering plot, and ACF values \(0.88,0.75,0.63\). After one difference, the plot fluctuates around zero and its first three ACF values are \(0.10,-0.08,0.12\). What is the best-supported order? (Answer: \(d=1\); the level is persistent and the differenced diagnostics are more stable.)
- For \(n=36\), calculate the rough reference band \(1.96/\sqrt{n}\). If a differenced series has a lag-one ACF of \(-0.31\), does that value alone establish that the original series was overdifferenced? (Answer: the band is approximately \(\pm0.327\); no, the ACF value alone does not establish overdifferencing.)
- A series is \(20,21,20,21,27,28,27,28\). Calculate its first differences and state the appropriate diagnostic conclusion. (Answer: \(1,-1,1,6,1,-1,1\); investigate the abrupt level shift before selecting a difference order.)
- A stable level plot has a steadily decaying ACF, with several early lags outside an approximate reference band. Its first-difference ACF has a strong negative lag-one value. Is differencing automatically justified? (Answer: no; the stable level and decaying ACF support considering \(d=0\), and the negative differenced lag calls for caution.)