Example 11.2: Mean and variance for stationary processes

We can use the ARMA(p, q) form given in (11.6) to arrive at general expres-sions for the mean and variance, although some simplifications are usually possible for specific models. Because the error terms have zero expectations, we see immediately that µ = λ / φ (1). Further, because the error terms are uncorrelated and each has variance σ ², we obtain for the variance:

∞

V(y_t) = ω ² = σ ² ∑ ψ_i ².

i =0

How does this definition of stationarity compare with that given in Sect. 3.3.2? The difference lies in the start-up conditions. In Chap. 3 we assumed a finite start-up, whereas here we are assuming that the series started in the infinite past. We can rewrite the AR(1) model as:

y_t = λ + φ ₁ y_{t− 1}+ ε _t, t =2, 3,...,

y ₁= λ + φ ₁ _ ₀+ ε ₁.

This form corresponds to the damped level model in Sect. 3.5.1 with α = 1. This correspondence enables us to see that the coefficients {k_j } in (3.8) are equivalent to the {ψ_j } in (11.8), and that the constant term is d_t = µ + φ ₁ ^t _ ₀, which converges to the mean when |φ ₁ | < 1. Similar equivalences may be established for more general models.

11.1.2 Invertibility

The representation in (11.7) enables us to recast any ARMA(p, q) process as an infinite order MA process. A similar manipulation enables us to rewrite

the model as:

[ φ (L)/ θ (L)](y_t − µ) = ε _t,

which may be represented as an infinite order AR process with operator π (L) = φ (L)/ θ (L)provided the series expansion of π (u)is absolutely con-vergent for |u | ≤ 1 or ∑^∞ _{i = 1} | π _i | < ∞. This requirement reduces to the condition that the roots of θ (u) = 0 should lie outside the unit circle. When this condition holds, we can write the model as:

π (L) y_t = µφ (1)/ θ (1) + ε _t or y_t = λ / θ (1) + π ₁ y_{t− 1} + π ₂ y_{t− 2} + · · · + ε _t.

When this representation is valid, we say the model is invertible. We relate this concept to our earlier discussion of forecastability in Sect. 11.4.

11.1 ARIMA Models

11.1.3 ARIMA Models

Our discussion in the previous section focused upon stationary models, yet nearly all of the discussion in earlier chapters has assumed the existence of trends or, at the very least, locally varying mean levels. In our experi-ence, most series in economics and business exhibit such behavior, so that stationary series are relatively rare. Indeed, in many cases where stationarity is observable, the time series has been transformed in some way; for exam-ple, a series of stock prices {y_t } typically follows a random walk, but the return on the stock, defined as r_t = (y_t − y_{t − 1})/ y_{t− 1} may well be stationary.

We extend the models under consideration to include nonstationary models by specifying the ARIMA class of models, which may be written as

φ (L)(1 − L) ^d yt = θ (L) ε t.

(11.9)

We drop the constant term in accordance with common usage. Note that we have partitioned the AR operator into two parts: the first term φ (L) is the standard AR polynomial and the second term (1 − L) ^d describes the differ-encing ¹operations. On occasion, it is convenient to represent this productby: η (L) = φ (L)(1 − L) ^d. Differencing once or twice is sufficient in most applications.

Once the appropriate order of differencing has been performed, the series z_t = (1 − L) ^d y_t may be modeled as an ARMA process, as in the previoussection. The full model for the original series is then referred to as an ARIMA(p, d, q) process. For full details of ARIMA processes, see Box et al. (1994, Chap. 4).

11.1.4 Seasonal Series

In order to complete the description of ARIMA models, we must also con-sider the existence of seasonal patterns in the data. If the series was purely seasonal, we could consider a model such as (11.9) but with each “month” relating back only to the same month in previous “years.” So, if there are m months, a purely seasonal model could be written as

Φ(L^m)[1 − L^m ] ^D yt = Θ(L^m) ε t.

(11.10)

The operator 1 − L^m defines a seasonal difference, whereas Φ(L^m) is the sea-sonal autoregressive polynomial and Θ(L^m) represents the seasonal moving average polynomial. Purely seasonal series may occur from time to time, but a far more common possibility is that there are both regular and seasonal effects to take into account. A natural way to do this is to combine (11.9) and

¹ Note that (1 − L) y_t = y_t − y_{t− 1} represents the difference between successive obser-vations.

168 11 Reduced Forms and Relationships with ARIMA Models

(11.10) to produce what is termed a “multiplicative” model in the literature (a confusing but now standard term):

φ (L)Φ(L^m)(1 − L) ^d [1 − L^m ] ^D yt = θ (L)Θ(L^m) ε t.

(11.11)

As before, we omit the constant term. Differencing may occur either at lag 1, or at lag m, or at both lags. Overall, the seasonal element may comprise P AR terms, Q MA terms and D seasonal differences. The full model is then denoted by ARIMA(p, d, q)(P, D, Q) _m.

An advantage of the multiplicative form is that the stationarity and invertibility conditions can be examined separately for the regular and seasonal components, by checking the roots of each polynomial in turn.

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