Decomposing an Explosive Autoregression#
Note
This section is an addition to the book, with no source outside it. It applies the root-sorting device of Chapter IX to the simplest unstable process. It connects to the martingale and bubble representations of Bubbles and to Exercise 36 of Exercises.
Throughout this chapter the device of solving stable roots backward and unstable roots forward (introduced in Chapter IX and used in A Difficulty in Interpreting Vector Autoregressions, Exact Linear Rational Expectations Models, and Chapter XIV) has been a way to pick out a bounded or stationary solution of an unstable difference equation. This short section applies the same device to the simplest possible unstable process — a scalar explosive autoregression — and reads the result as a statement about how to represent a nonstationary process.
An explosive AR(1) and its two solutions#
Consider the explosive first-order autoregression
Iterating (498) backward from an initial condition \(x_0\) gives the causal solution
which expresses \(x_t\) through current and past shocks but explodes as \(t\to\infty\). Because \(a>1\), the root sits outside the region that would make (499) a stationary Wold representation. Following the chapter’s recipe, we instead solve the unstable root forward. Rewrite (498) as \(x_t = a^{-1}x_{t+1} - a^{-1}\varepsilon_{t+1}\) and define the discounted process \(y_t \equiv a^{-t}x_t\), which obeys the stable recursion \(y_t = y_{t-1} + a^{-t}\varepsilon_t\). Since \(a>1\), the sum \(\sum_{j\ge1} a^{-j}\varepsilon_j\) converges in mean square, so
is a well-defined, time-invariant random variable — the discounted value, from the vantage point of the infinite future, of the initial condition and the entire shock sequence. Writing \(y_t = Z - \sum_{j>t} a^{-j}\varepsilon_j\) and multiplying by \(a^{t}\) yields the decomposition
Reading the decomposition#
The two terms in (501) separate the process into an explosive part with a random amplitude and a stationary part that looks only forward:
Explosive component \(a^{t}Z\). All of the nonstationarity is carried by a single deterministic explosive path \(a^{t}\) scaled by the time-invariant random amplitude \(Z\) of (500). Once the realization is fixed, \(Z\) is a number; the process diverges along \(a^{t}Z\) exactly as the homogeneous solution of Chapter IX diverges along \(\lambda^{t}\) times an arbitrary constant, except that here the “constant” is pinned down by the whole shock history.
Stationary anticipative component \(\tilde x_t = -\sum_{k\ge1} a^{-k}\varepsilon_{t+k}\). This is the unstable root solved forward: a geometric sum of future shocks, discounted by the stable factor \(a^{-1}<1\). It is covariance stationary, with autocovariance
(502)#\[\operatorname{Cov}(\tilde x_t,\tilde x_{t+h}) = \frac{\sigma^2}{a^2-1}\,a^{-|h|},\]which is precisely the covariogram of a stable AR(1) with root \(a^{-1}\). Unlike the one-sided moving averages that dominate this book, \(\tilde x_t\) is anticipative — it depends on shocks dated later than \(t\) — so it is exactly the kind of non-realizable representation that Chapter IX, Section 13 and Chapter XIV contrast with nonanticipative ones.
Two connections#
To solving unstable roots forward. The stationary anticipative term is the forward solution of the unstable root, and it is exactly the object that Exercise 36 asks the reader to construct and then push one step further. There the deviation-from-trend \(s_t = \tilde x_t\) is shown to admit a fundamental, one-sided Wold representation \(s_t = (1-a^{-1}L)^{-1}u_t\) in a new white noise \(u_t \ne \varepsilon_t\) with the smaller variance \(\sigma_u^2 = \sigma^2/a^2\). Thus “solving the unstable root forward” trades the original explosive innovation \(\varepsilon_t\) for a fundamental innovation, at the cost of shrinking the innovation variance by \(a^{-2}\) — the scalar analogue of the innovation-relabeling that reappears whenever a vector system is factored into stable and unstable roots.
To representing a nonstationary process. Decomposition (501) is the explosive-root cousin of the unit-root decompositions studied in Exercise 57 and its solution. When \(a=1\), Beveridge and Nelson (1981) split a difference- stationary series into a random-walk permanent component built from past shocks plus a stationary transitory part; when \(a>1\), the permanent piece becomes the explosive \(a^{t}Z\) and — tellingly — its amplitude \(Z\) depends on the entire shock sequence, while the stationary remainder looks purely forward. The direction in which the “trend” accumulates information flips as the root crosses the unit circle. The same forward-looking, martingale-like object \(Z\) underlies the nonstationary bubble solutions of Section 36, where an arbitrary martingale can be added to a stationary particular solution.
Note
Explosive autoregressions were studied early by Quenouille (1957) and Chow (1983); their representation theory is developed by McCabe and Tremayne (1989), and their estimation theory by Phillips and Magdalinos (2007). Textbook treatments of the forward (anticipative) solution appear in Hamilton (1994). The permanent/transitory reading is due to Beveridge and Nelson (1981).
References#
Stephen Beveridge and Charles R. Nelson. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the `business cycle'. Journal of Monetary Economics, 7(2):151–174, 1981.
Gregory C. Chow. Econometrics. McGraw-Hill, New York, 1983.
James D. Hamilton. Time Series Analysis. Princeton University Press, Princeton, NJ, 1994.
B. P. M. McCabe and A. R. Tremayne. On the representations of explosive autoregressions. Journal of Time Series Analysis, 10(4):353–362, 1989.
Peter C. B. Phillips and Tassos Magdalinos. On the treatment of explosive autoregressions in econometrics. Advances in Econometrics, 20:257–294, 2007.
Maurice H. Quenouille. The Analysis of Multiple Time-Series. Charles Griffin, London, 1957.