ARIMA Order Notation and Meaning

Introduction

The notation ARIMA(\(p,d,q\)) compresses three different modeling decisions into three integers. The first describes autoregressive terms, the second describes how many times the observed series is differenced, and the third describes moving-average terms involving past innovations. Reading the notation correctly means keeping these roles separate: the AR and MA orders apply to the series after differencing, not automatically to the original observations.

This distinction matters when interpreting a fitted time-series model. A value of \(d=1\), for example, says that the model is written for changes between consecutive observations. It does not say that the model has one moving-average term. The orders are useful labels, but they do not by themselves establish that a model is appropriate or that its assumptions hold.

Definition: An ARIMA(\(p,d,q\)) model applies \(d\) ordinary differences to an observed series \(y_t\). The resulting series, denoted \(w_t=\Delta^d y_t\), is modeled as an ARMA(\(p,q\)) process. Here \(p\) is the autoregressive order, \(d\) is the differencing order, and \(q\) is the moving-average order. The orders are nonnegative integers.

What Each Order Counts

The ordinary first difference is \(\Delta y_t=y_t-y_{t-1}\). Higher-order differences are formed by repeatedly differencing the result: \(\Delta^2 y_t=\Delta(\Delta y_t)\), and, in general, \(\Delta^d y_t\) means applying the difference operation \(d\) times. When \(d=0\), no differencing is applied, so \(w_t=y_t\).

After differencing, an ARMA(\(p,q\)) model relates the current value of \(w_t\) to its \(p\) most recent lagged values and the current and \(q\) most recent innovations. A standard zero-mean form is

w_t=\phi_1w_{t-1}+\cdots+\phi_pw_{t-p} +\varepsilon_t+\theta_1\varepsilon_{t-1}+\cdots+\theta_q\varepsilon_{t-q}.

The innovations \(\varepsilon_t\) are assumed to have mean zero, constant variance, and no autocorrelation; they are often modeled as independent and identically distributed, with a normal distribution used when that assumption is appropriate. The displayed equation uses a zero-mean or appropriately centered series. If a model includes a mean or other constant term, that feature must be specified separately; the three orders alone do not describe it.

Order	What it counts	Where it applies
\(p\)	Lagged values in the autoregressive part	The differenced series \(w_t\)
\(d\)	Repeated ordinary differences	The observed series \(y_t\)
\(q\)	Lagged innovations in the moving-average part	The equation for \(w_t\)

In particular, an MA(\(q\)) part uses past innovations, not past observed values. The number \(q\) is not the number of differences, and \(p\) does not count the total number of terms in the full equation. An ARIMA(\(2,1,1\)) has two autoregressive lags and one lagged innovation in the equation for the first-differenced series.

Reading the Orders Through the Model

A useful interpretation procedure is to identify the series named on the left side of the model equation first. If it is \(y_t\), the equation is for the undifferenced series and \(d=0\). If it is \(\Delta y_t\), then \(d=1\); if it is \(\Delta^2y_t\), then \(d=2\). Next, count the lagged values of that series on the right side to identify \(p\), and count the lagged innovations to identify \(q\).

Identify the modeled series.
Read whether the model is written for \(y_t\), \(\Delta y_t\), or a higher difference. This determines \(d\).

Count lagged series values.
Count the distinct lagged values of the modeled series included in its autoregressive part. The highest included AR lag gives \(p\) in the usual order notation.

Count lagged innovations.
Count the past innovation lags in the moving-average part. The highest included innovation lag gives \(q\).

Write the three orders in order.
Report ARIMA(\(p,d,q\)), keeping the differencing order in the middle.

Worked Example: Identifying All Three Orders From an Equation

A hypothetical insurer tracks a monthly index of claim-processing duration, \(y_t\). An analyst models its first difference \(w_t=\Delta y_t\) using

w_t=0.6w_{t-1}+\varepsilon_t-0.4\varepsilon_{t-1}.

Identify the ARIMA orders and explain what the equation says about the original series.

Method. First identify the series on the left: \(w_t=\Delta y_t\). One difference is applied, so \(d=1\).

The equation includes \(w_{t-1}\), a lagged value of the differenced series. There is one autoregressive lag, so \(p=1\). It also includes \(\varepsilon_{t-1}\), one lagged innovation, so \(q=1\). The current innovation \(\varepsilon_t\) is part of the model but does not count as a lagged innovation for the order.

Therefore the specification is ARIMA(\(1,1,1\)). It models the month-to-month changes in the index with one AR lag and one lagged innovation. It is not an ARMA(\(1,1\)) model for the original \(y_t\), because the equation is for \(\Delta y_t\).

Check and common mistake avoided. The left side explicitly contains one difference, and the right side contains one lagged \(w\) and one lagged innovation. Counting the current innovation as a lag would incorrectly give \(q=2\).

Why Repeated Differencing Sets \(d\)

The differencing order can be checked by expanding the differences. By definition, the second difference is the first difference of the first difference:

\begin{aligned} \Delta^2y_t &=\Delta y_t-\Delta y_{t-1}\\ &=(y_t-y_{t-1})-(y_{t-1}-y_{t-2})\\ &=y_t-2y_{t-1}+y_{t-2}. \end{aligned}

Thus a model for \(\Delta^2y_t\) has had two ordinary differences applied to the observed series. If that second-differenced series follows an ARMA(\(p,q\)) model, the definition of ARIMA gives the original series the order ARIMA(\(p,2,q\)). The AR and MA orders describe the equation for \(\Delta^2y_t\); they are not counted from the expanded expression in the original \(y\)'s.

More generally, the definition of an ARIMA model is the reason its notation works: it first transforms \(y_t\) into \(w_t=\Delta^d y_t\), then assigns AR and MA orders to the ARMA model for \(w_t\). This also explains why the original series may be nonstationary while the differenced series is modeled as stationary. The notation itself does not prove that a particular choice of \(d\) makes a series stationary; that requires evidence and appropriate model assessment.

Worked Example: Second Differences and an MA(1) Model

A hypothetical monthly portfolio index has recorded values \(y_0=100\), \(y_1=104\), \(y_2=111\), \(y_3=119\), and \(y_4=128\). An analyst specifies that the second-differenced series follows an MA(1) model:

\Delta^2y_t=\varepsilon_t+0.5\varepsilon_{t-1}.

Find the first and second differences available from the observations, then state the ARIMA order implied by the specified model.

Method. Compute successive first differences:

\begin{aligned} \Delta y_1&=y_1-y_0=104-100=4,\\ \Delta y_2&=y_2-y_1=111-104=7,\\ \Delta y_3&=y_3-y_2=119-111=8,\\ \Delta y_4&=y_4-y_3=128-119=9. \end{aligned}

Now difference the first differences:

\begin{aligned} \Delta^2y_2&=\Delta y_2-\Delta y_1=7-4=3,\\ \Delta^2y_3&=\Delta y_3-\Delta y_2=8-7=1,\\ \Delta^2y_4&=\Delta y_4-\Delta y_3=9-8=1. \end{aligned}

The specified model is for \(\Delta^2 y_t\), so \(d=2\). It has no lagged \(\Delta^2 y\) term, so \(p=0\). It has one lagged innovation, \(\varepsilon_{t-1}\), so \(q=1\). The implied order is ARIMA(\(0,2,1\)).

The three calculated second differences do not, by themselves, establish that an MA(1) model fits the series. The order follows from the stated model specification. This distinction avoids the common mistake of treating a short run of observed differences as sufficient evidence to identify AR and MA orders.

Separating AR Terms From MA Terms

Both the AR and MA parts use lags, but they lag different quantities. An autoregressive term uses past values of the modeled series. A moving-average term uses past innovations, which represent the unpredictable part of the series after accounting for the model. The symbols and their meanings should be read from the equation rather than inferred from the word “moving.”

Worked Example: A Model With Two AR Lags and One Innovation Lag

For a hypothetical risk index, let \(x_t\) be the centered observed series, and suppose its model is

x_t=1.2x_{t-1}-0.32x_{t-2}+\varepsilon_t+0.25\varepsilon_{t-1}.

Assume the model is being used without differencing. At a particular time, \(x_{t-1}=10\), \(x_{t-2}=8\), \(\varepsilon_t=2\), and \(\varepsilon_{t-1}=-1\). Identify the order and evaluate the model’s right-hand side for these supplied values.

Method. Since the equation is for \(x_t\) itself, no differencing is shown, and \(d=0\). The lagged series values are \(x_{t-1}\) and \(x_{t-2}\), so \(p=2\). The only lagged innovation is \(\varepsilon_{t-1}\), so \(q=1\). The model is ARIMA(\(2,0,1\)), equivalently an ARMA(\(2,1\)) model for the undifferenced series.

Substitute the values term by term:

\begin{aligned} x_t &=1.2(10)-0.32(8)+2+0.25(-1)\\ &=12-2.56+2-0.25\\ &=11.19. \end{aligned}

The computed right-hand side is \(11.19\). Check and common mistake avoided: the term \(-0.32x_{t-2}\) is an AR term, while \(0.25\varepsilon_{t-1}\) is an MA term. Counting both as the same kind of lag would obscure the order pair \(p=2,q=1\). The current innovation \(\varepsilon_t\) is included in the numerical calculation but does not increase \(q\).

Common Interpretive Traps

Putting the orders in the wrong positions. ARIMA is written \(p,d,q\): autoregressive order, differencing order, moving-average order. The middle value is not the MA order.
Counting the differencing operation as an AR lag. The expression \(\Delta y_t\) sets \(d=1\). It does not contribute to \(p\).
Counting innovations as observed-series lags. A term such as \(\varepsilon_{t-2}\) contributes to \(q\), not \(p\).
Counting the current innovation as a lag. \(\varepsilon_t\) appears in a standard MA equation, but only past innovations determine the positive MA order.
Assuming the notation guarantees a good model. ARIMA(\(p,d,q\)) identifies a model structure. It does not establish stationarity after differencing, suitable parameter values, or adequate residual behavior.
Applying nonseasonal orders to seasonal lags. The notation discussed here concerns ordinary differences and nonseasonal AR and MA lags. Seasonal orders and seasonal differences use additional notation.

Key takeaway: First identify the series being modeled to determine \(d\). Then count lagged values of that series for \(p\), and lagged innovations for \(q\). ARIMA(\(p,d,q\)) means that \(\Delta^d y_t\), not necessarily \(y_t\), follows an ARMA(\(p,q\)) model. This interpretation is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.

Check Your Understanding

For each item, identify the ARIMA order or calculate the requested differences. Read carefully whether the stated model is for the original series or for a differenced series.

A centered series \(z_t\) satisfies \(z_t=0.3z_{t-1}+\varepsilon_t-0.6\varepsilon_{t-1}+0.2\varepsilon_{t-2}\). What is its ARIMA order? (Answer: ARIMA(1,0,2))
A model for \(\Delta y_t\) contains \(\Delta y_{t-1}\), \(\Delta y_{t-3}\), \(\varepsilon_t\), and \(-0.4\varepsilon_{t-2}\), with no other lagged terms. What order is indicated by the highest included lags? (Answer: ARIMA(3,1,2))
Observed values are \(y_0=20\), \(y_1=23\), \(y_2=27\), and \(y_3=32\). Calculate \(\Delta y_1,\Delta y_2,\Delta y_3\) and \(\Delta^2y_2,\Delta^2y_3\). (Answer: First differences \(3,4,5\); second differences \(1,1\))
A specified model is \(\Delta^2 y_t=0.7\Delta^2 y_{t-1}+\varepsilon_t+0.1\varepsilon_{t-1}\). Identify \(p,d,q\), and state whether the model is for \(y_t\) or its second difference. (Answer: \(p=1,d=2,q=1\), ARIMA(1,2,1); the ARMA equation is for the second difference)
A model for \(\Delta y_t\) is \(\Delta y_t=0.5\Delta y_{t-1}+\varepsilon_t-0.2\varepsilon_{t-1}\). If \(\Delta y_{t-1}=6\), \(\varepsilon_t=1\), and \(\varepsilon_{t-1}=5\), evaluate the right-hand side and give the order. (Answer: \(0.5(6)+1-0.2(5)=3\); ARIMA(1,1,1))

What You'll Learn