Phillips (1986) Understanding Spurious Regressions in Econometrics

Summary

Phillips (1986) provides the rigorous asymptotic theory explaining the Granger-Newbold (1974) simulation finding that ordinary least squares (OLS) regressions between independent random walks produce spuriously significant results. Working in a general framework that allows $\alpha$-mixing innovations and heterogeneous ARIMA(p,1,q) error structures, Phillips shows that the sample moments of integrated processes do not obey the usual law of large numbers (LLN): instead, $T^{-2}\sum x_t^2 \Rightarrow \sigma_w^2 \int_0^1 W(t)^2\,dt$ (a random variable). Consequently all standard OLS diagnostics — t-statistics, F-statistics, $R^2$, Durbin-Watson (DW) — have non-standard or divergent limits, and Granger-Newbold's "spurious regression" is explained not by serial correlation per se but by the fundamental non-ergodicity of I(1) processes. The paper also makes the cointegration connection explicit: when the series are cointegrated, OLS remains consistent (the cointegration case is the exception that validates levels regression), and DW $\to$ 0 under the null of no cointegration — the analytic basis for the Engle-Granger (1987) residual-based test.

Key Claims

• Functional central limit theorem (CLT) foundation. Standard OLS asymptotics rest on the LLN ($T^{-1}\sum x_t^2 \to \sigma^2$). For I(1) regressors, $T^{-2}\sum x_t^2 \Rightarrow \sigma_w^2 \int_0^1 W(t)^2\,dt$: sample second moments converge to random variables, not constants, so all classical limit theory fails.
• Theorem 1 — simple spurious regression ($y_t$, $x_t$ independent random walks; $m=1$ regressor):
  • (a) $\hat\beta \Rightarrow (\sigma_v/\sigma_w)\,\zeta$ — a non-degenerate random variable (ratio of Wiener functional integrals), not zero.
  • (b) $\hat\alpha \to_p \infty$ — intercept estimate diverges.
  • (c) $t_\beta \sim O_p(T^{1/2})$ — the $t$-ratio for the slope diverges; no limiting distribution.
  • (d) $t_\alpha \sim O_p(T^{1/2})$ — the intercept $t$-ratio diverges likewise.
  • (e) $R^2 \Rightarrow$ non-degenerate limit strictly in $(0,1)$ — consistently positive even under independence.
  • (f) $\mathrm{DW} \to_p 0$; $T\cdot\mathrm{DW}$ has a non-degenerate limit — DW is a consistent detector of the spurious case.
  • (g) $r_s \to_p 1$ — residual serial correlation approaches one.
  • (h) $Q_k \sim O_p(T)$ — Box-Ljung portmanteau statistic diverges at rate $T$.
• Theorem 2 — multiple spurious regression ($m$ independent I(1) regressors): the $F$-statistic diverges at $O(T)$, faster than the $O(T^{1/2})$ of individual $t$-tests; this explains why Granger-Newbold observed 76% rejection at the 5% level with $m=1$ and 96% rejection with $m=5$.
• Root cause. Granger-Newbold attributed spurious regression to serial correlation in residuals; Phillips shows the true cause is the non-ergodicity of integrated processes — $T^{-1}\sum x_t^2 \not\to$ constant — which breaks all moment convergence that classical asymptotics requires.
• Cointegration connection. If the variance matrix $\Sigma$ of the innovation vector is singular — i.e., $y_t$ and $x_t$ are cointegrated — then $\hat\beta$ is consistent for the cointegrating vector. Under no cointegration, $\mathrm{DW}\to_p 0$, which is the analytic foundation for the Engle-Granger (1987) residual-based DW cointegration test.
• Generality. Results extend beyond white noise to general $\alpha$-mixing, heterogeneous innovations; the Functional Central Limit Theorem (Billingsley 1968) is the key technical tool replacing the classical CLT.

Concepts Introduced or Extended

• Spurious Regression
• Cointegration
• Unit Root Tests

Entities Mentioned

• Peter C. B. Phillips
• Clive Granger

Quotes

"The main conclusion to be drawn from the analysis is that the spurious regression phenomenon is not simply a consequence of highly autocorrelated residuals but is a manifestation of the non-ergodic behavior of integrated processes."

"When the system is cointegrated... the DW statistic converges to zero... providing the basis for Granger and Engle's [1985] test for cointegration."

My Take

The paper's key insight — replacing Granger-Newbold's residual-autocorrelation explanation with the deeper non-ergodicity of integrated processes — reframes spurious regression from a pathology of OLS inference into a fundamental failure of moment convergence. The functional CLT apparatus (Wiener process limits for normalized sample moments) became the foundational toolkit for the entire subsequent literature on integrated-process asymptotics: Phillips-Durlauf (1986), Phillips (1991), the Park-Phillips inference theory, and Toda-Phillips (1993) all extend this framework. The DW-to-zero result elegantly closes the loop with Engle-Granger (1987): spurious regression detection and cointegration testing reduce to the same diagnostic.