Cross-Lag Terms in Multiplicative Seasonal Models

Why Multiplication Creates Cross-Lags

A multiplicative seasonal model combines an ordinary time-series factor with a seasonal factor. Because the factors multiply, the resulting equation contains not only the ordinary lag and the seasonal lag, but also a cross-lag: a lag formed by adding their time offsets. For an ordinary lag \(p\) and a seasonal lag \(s\), that cross-lag is \(p+s\).

This is an algebraic consequence of multiplying lag polynomials. It does not mean that the model has acquired a separately specified ordinary autoregressive order of \(p+s\). The cross-lag coefficient is constrained by the coefficients of the factors being multiplied.

Definition: The backshift operator \(B\) acts on a time series by \(B x_t=x_{t-1}\), so \(B^k x_t=x_{t-k}\). A cross-lag is a lag term produced by multiplying ordinary and seasonal lag-polynomial terms. Multiplying \(B^i\) and \(B^{js}\) produces \(B^{i+js}\), corresponding to a lag of \(i+js\).

To make the signs explicit, use the following convention throughout this tutorial. An ordinary AR(1) factor is \(1-\phi B\), a seasonal AR(1) factor with period \(s\) is \(1-\Phi B^s\), an ordinary MA(1) factor is \(1+\theta B\), and a seasonal MA(1) factor is \(1+\Theta B^s\). Other texts may define MA parameters with a minus sign; the essential task is to expand the convention actually given.

If differencing is present, write \(w_t=(1-B)^d(1-B^s)^D y_t\) for the series to which the ARMA factors apply. The cross-lag expansion concerns the factors acting on \(w_t\), or on \(y_t\) when there is no differencing.

Deriving the Cross-Lag

Consider one ordinary AR term and one seasonal AR term. Their product is

(1-\phi B)(1-\Phi B^s) =1-\phi B-\Phi B^s+\phi\Phi B^{s+1}.

The product of the two lag terms is \(B \cdot B^s=B^{s+1}\). More generally, if the ordinary term is at lag \(p\), multiplication by a seasonal term at lag \(s\) creates a term at lag \(p+s\). The coefficient is the product of the two factor coefficients, with its sign determined by the signs in the factors.

This gives a useful general rule. If the ordinary factor is \(\sum_i a_i B^i\) and the seasonal factor is \(\sum_j A_j B^{js}\), their product is

\left(\sum_i a_i B^i\right) \left(\sum_j A_j B^{js}\right) =\sum_i\sum_j a_i A_j B^{i+js}.

Each pair of terms contributes a coefficient \(a_iA_j\) at lag \(i+js\). If different pairs produce the same lag, their contributions add. The same multiplication rule applies to MA factors; only the parameter signs in those factors differ under the convention used here.

Key result: In a multiplicative model, a cross-lag’s location comes from adding the lag indices, and its polynomial coefficient comes from multiplying the corresponding factor coefficients. To express an AR polynomial as an equation for the current series value, remember that nonzero-lag terms may change sign when moved to the right-hand side.

Worked AR Expansion

Worked Example: A Quarterly Seasonal AR(1) Cross-Lag

A hypothetical quarterly claim-count series has been differenced as required, producing \(w_t\). Its model is \((1-0.5B)(1-0.4B^4)w_t=\varepsilon_t\). Expand the equation, identify the cross-lag and coefficient, and write the equation with \(w_t\) isolated.

Step 1: Multiply the factors. The ordinary AR factor has a lag-1 coefficient of \(-0.5\); the seasonal factor has a lag-4 coefficient of \(-0.4\). Their product contributes \((-0.5)(-0.4)=0.20\) at lag \(1+4=5\):

$$ (1-0.5B)(1-0.4B^4) =1-0.5B-0.4B^4+0.20B^5. $$

Step 2: Apply the polynomial to \(w_t\). The expanded model is

w_t-0.5w_{t-1}-0.4w_{t-4}+0.20w_{t-5}=\varepsilon_t.

Step 3: Isolate \(w_t\). Moving the other terms to the right gives

w_t=0.5w_{t-1}+0.4w_{t-4}-0.20w_{t-5}+\varepsilon_t.

Check and mistake avoided. The cross-lag is lag 5, and its coefficient in the polynomial is \(+0.20\). Its coefficient in the isolated equation is \(-0.20\). A common error is to copy the polynomial sign onto the right-hand side; moving \(+0.20w_{t-5}\) across the equality changes its sign.

Cross-Lags in the MA Part

The same lag addition occurs in the MA part. With the stated plus-sign convention, multiplying an ordinary MA(1) factor by a seasonal MA(1) factor gives

(1+\theta B)(1+\Theta B^s) =1+\theta B+\Theta B^s+\theta\Theta B^{s+1}.

When this product acts on the innovations, the cross-lag is an earlier innovation, not an earlier value of the observed series. In particular, an MA cross-term at lag \(s+1\) is \(\theta\Theta\varepsilon_{t-s-1}\). Do not confuse this with the AR cross-term, which multiplies \(w_{t-s-1}\) in the polynomial equation.

Worked Example: A Seasonal MA Cross-Term and Its Sign

A hypothetical monthly service-volume model has MA factors \((1+0.3B)(1-0.5B^{12})\). Expand the factors and state the cross-term in the equation for \(w_t\).

Step 1: Identify the parameters and lags. Here \(\theta=0.3\), \(\Theta=-0.5\), and \(s=12\). The cross-lag is \(1+12=13\).

Step 2: Multiply the cross-term coefficients. Their product is \(0.3(-0.5)=-0.15\). Therefore,

(1+0.3B)(1-0.5B^{12}) =1+0.3B-0.5B^{12}-0.15B^{13}.

Step 3: Apply the MA polynomial to the innovations. The equation is

w_t=\varepsilon_t+0.3\varepsilon_{t-1} -0.5\varepsilon_{t-12}-0.15\varepsilon_{t-13}.

Check and mistake avoided. The lag-13 innovation coefficient is negative because \(0.3(-0.5)=-0.15\). Treating the seasonal coefficient as positive \(0.5\) would give the wrong sign. The cross-term is an innovation at \(t-13\), not an observed value \(w_{t-13}\).

Reading Mixed Multiplicative Models

A model may have both AR and MA factors. For example, the AR factors multiply one another, and the MA factors multiply one another, but the AR polynomial and MA polynomial remain on opposite sides of the model equation. They are not multiplied together to create additional AR–MA cross-terms.

Worked Example: Separating AR and MA Cross-Lags After Differencing

A hypothetical quarterly exposure series follows \((1-B)(1-B^4)(1-0.3B)(1-0.5B^4)y_t=(1+0.2B)(1+0.4B^4)\varepsilon_t\). Identify the differenced series, expand the AR and MA factors separately, and list their cross-lags and coefficients.

Step 1: Define the series being modeled by the ARMA factors.

$$ w_t=(1-B)(1-B^4)y_t. $$

Expanding only the differencing operators gives \(w_t=y_t-y_{t-1}-y_{t-4}+y_{t-5}\). The AR and MA equations below act on \(w_t\) and \(\varepsilon_t\), respectively.

Step 2: Expand the AR factors. The cross-lag is \(1+4=5\), with polynomial coefficient \((-0.3)(-0.5)=0.15\):

(1-0.3B)(1-0.5B^4)w_t =\left(1-0.3B-0.5B^4+0.15B^5\right)w_t.

Thus the AR side is \(w_t-0.3w_{t-1}-0.5w_{t-4}+0.15w_{t-5}\).

Step 3: Expand the MA factors. Their cross-lag is also 5, but its coefficient is \(0.2(0.4)=0.08\):

(1+0.2B)(1+0.4B^4)\varepsilon_t =\varepsilon_t+0.2\varepsilon_{t-1} +0.4\varepsilon_{t-4}+0.08\varepsilon_{t-5}.

Step 4: Combine the two sides. The full equation is

w_t-0.3w_{t-1}-0.5w_{t-4}+0.15w_{t-5} =\varepsilon_t+0.2\varepsilon_{t-1} +0.4\varepsilon_{t-4}+0.08\varepsilon_{t-5}.

Check and mistake avoided. Both sides contain a lag-5 term, but one multiplies \(w_{t-5}\) and the other multiplies \(\varepsilon_{t-5}\); their coefficients are \(0.15\) and \(0.08\), respectively. Do not combine them into one coefficient or create an extra term by multiplying the AR and MA polynomials together.

More Than One Ordinary Lag

Cross-lags need not be limited to \(p+s\). If an ordinary AR factor includes lags 1 and 2, and the seasonal factor includes lag 4, multiplication produces cross-lags \(1+4=5\) and \(2+4=6\), in addition to the original lags. With several seasonal terms, add each seasonal lag to each ordinary lag. If two combinations land on the same lag, add their signed coefficient contributions.

These terms are part of the algebraic structure of the multiplicative specification. They are not evidence by themselves that a model was separately selected with every resulting lag as an unconstrained parameter. The cross-coefficient is determined by a product of parameters, so it is linked to the coefficients in the original factors.

Write the factors exactly.
Preserve each sign and exponent in the ordinary and seasonal polynomials.

Add lag indices.
For each pair of terms, \(B^iB^{js}=B^{i+js}\); this locates the resulting term.

Multiply signed coefficients.
The product gives the contribution at the resulting lag; sum contributions if lags coincide.

Interpret the equation carefully.
AR polynomial signs may change when terms move across the equality. MA terms multiply innovations on the right-hand side.

Common Exam Traps

Using the wrong lag arithmetic. The ordinary and seasonal lag indices add: \(p+s\), not \(p\), \(s\), or \(ps\).
Ignoring the factor signs. A product of two negative AR coefficients in the polynomial is positive; a positive and a negative MA coefficient give a negative product.
Forgetting which series is lagged. AR terms multiply lagged observations of the modeled series; MA terms multiply lagged innovations.
Changing signs incorrectly when isolating the current value. First expand the polynomial equation, then move each nonzero-lag term across the equality.
Calling an implied cross-term a separate model order. Its coefficient is constrained by multiplication and is not an independently specified coefficient merely because it appears at a higher lag.
Applying the factors to the wrong series. When differencing is present, the ARMA factors act on the differenced series \(w_t\), not directly on \(y_t\) as if no differencing had occurred.

Key takeaway: Multiplicative seasonal factors create cross-lags because lag operators multiply by adding their exponents. Find the lag by addition, find the signed polynomial coefficient by multiplication, and then keep AR observations and MA innovations distinct when writing the full equation.

Check Your Understanding

Expand each factor or interpret the specified cross-lag before checking the final answer.

A seasonal AR(1) factor with \(s=6\) is multiplied by an ordinary AR(1) factor: \((1-0.25B)(1-0.8B^6)\). What is the cross-lag and its coefficient in the polynomial? (Answer: lag 7, coefficient \(+0.20\).)
Expand the MA factors \((1+0.2B)(1+0.5B^4)\). Give the cross-lag innovation and its coefficient. (Answer: lag 5, \(+0.10\varepsilon_{t-5}\).)
Expand \((1-0.3B+0.1B^2)(1-0.6B^4)\). List all nonzero lags beyond lag 0 and their polynomial coefficients. (Answer: lag 1: \(-0.3\); lag 2: \(+0.1\); lag 4: \(-0.6\); lag 5: \(+0.18\); lag 6: \(-0.06\).)
For \((1-0.4B)(1+0.25B^{12})\varepsilon_t\), identify the cross-term, including its sign. (Answer: \(-0.10\varepsilon_{t-13}\).)
A differenced quarterly series has AR factors \((1-0.2B)(1-0.5B^4)\) and MA factors \((1+0.3B)(1-0.2B^4)\). Give each side’s cross-lag term and explain why they are not combined. (Answer: AR polynomial cross-term \(+0.10B^5\), multiplying the lag-5 series value; MA cross-term \(-0.06\varepsilon_{t-5}\). They multiply different quantities and lie on opposite sides of the equation.)

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