Spurious Regression

Definition

A spurious regression is a regression between two or more independently generated nonstationary (I(1)) time series that produces apparently significant statistical relationships — high $R^2$ , low p-values — purely as an artifact of their shared random-walk behavior, with no genuine causal or equilibrium relationship between them.

Key Ideas

The problem arises because standard ordinary least squares (OLS) asymptotics (central limit theorem (CLT), law of large numbers (LLN)) break down when regressors are nonstationary. The t-statistic does not converge to a standard normal; instead its magnitude grows without bound as $T \to \infty$ .
The characteristic symptom is strongly autocorrelated residuals (Durbin-Watson statistic (DW) near zero) alongside apparently high fit — a diagnostic alert in applied work.
The solution is to test for unit roots in each series, then test for cointegration. If the series are not cointegrated, work in first differences. If they are cointegrated, use the vector error correction model (VECM) which produces valid inference.
Cointegration is the exception that restores meaningful levels regression: if the residual $\varepsilon_t$ is I(0), the regression is genuine rather than spurious.

How It Works

The Granger-Newbold (1974) Simulation Finding

Granger and Newbold (1974) generated pairs of independent random walks:

$\xi_t = \xi_{t-1} + \varepsilon_t, \qquad \varepsilon_t \overset{iid}{\sim}(0, \sigma^2) \tag{RW}$

and regressed one on the other by OLS. Despite the series being independent, they found:

The t-statistic for the slope coefficient was significant far more often than the nominal size predicts (e.g., $\gg 5\%$ rejections at the 5% level).
$R^2$ was often high (the random walks drift together by chance).
The Durbin-Watson statistic was close to zero, indicating extreme residual autocorrelation.

The conclusion: in the presence of independent stochastic trends, standard inference about regression coefficients is invalid.

Phillips (1986) Asymptotic Theory

Phillips (1986) provided the rigorous asymptotic theory explaining the simulation findings. The root cause is not serial correlation per se — it is the non-ergodicity of integrated processes: for an I(1) series, $T^{-2}\sum x_t^2 \Rightarrow \sigma_w^2\int_0^1 W(t)^2\,dt$ , a random variable, not a constant. The usual LLN and CLT fail at their foundation.

Theorem 1 (simple spurious regression, $m=1$ regressor; $y_t, x_t$ independent random walks):

Result	Statement
(a) Slope	$\hat\beta \Rightarrow (\sigma_v/\sigma_w)\,\zeta$ — non-degenerate random limit, not zero
(b) Intercept	$\hat\alpha \to_p \infty$ — diverges
(c) $t_\beta$	$t_\beta \sim O_p(T^{1/2})$ — diverges; no limiting distribution
(d) $t_\alpha$	$t_\alpha \sim O_p(T^{1/2})$ — diverges
(e) $R^2$	$R^2 \Rightarrow$ non-degenerate limit strictly in $(0,1)$
(f) DW	$\mathrm{DW} \to_p 0$ ; $T\cdot\mathrm{DW}$ has non-degenerate limit
(g) $r_s$	$r_s \to_p 1$ — residual autocorrelation approaches one
(h) $Q_k$	$Q_k \sim O_p(T)$ — Box-Ljung diverges at rate $T$

Theorem 2 (multiple spurious regression, $m$ I(1) regressors): the joint $F$ -statistic diverges at $O(T)$ , faster than individual $t$ -stats at $O(T^{1/2})$ . This explains the Granger-Newbold pattern: 76% rejection at 5% nominal size with $m=1$ , rising to 96% with $m=5$ .

The limiting distributions are ratios of stochastic integrals of Brownian motions — qualitatively different from the usual normal/chi-squared limits. The Functional Central Limit Theorem (Billingsley 1968) replaces the classical CLT as the key technical tool. Results extend to general $\alpha$ -mixing, heterogeneous autoregressive integrated moving average (ARIMA) $(p,1,q)$ innovations.

Cointegration Connection (Phillips 1986)

Phillips (1986) shows that when the innovation variance matrix $\Sigma$ is singular — i.e., $y_t$ and $x_t$ are cointegrated — OLS 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: a significant regression with $\mathrm{DW}\approx 0$ is a signal of spuriousness, not of a genuine relationship.

Why Cointegration Is the Exception

When $y_t$ and $x_t$ are genuinely cointegrated — $y_t - \beta x_t = u_t \sim I(0)$ — the OLS estimator $\hat\beta$ does converge to $\beta$ , and in fact at the superconsistent rate $T^{-1}$ (Stock 1987), rather than the usual $T^{-1/2}$ . The regression is not spurious: the stationary residual $u_t$ is the cointegrating error, and the estimated relationship is a genuine long-run equilibrium.

Diagnostics and Prevention

Symptom	Cause	Remedy
High R², significant t-stats	Both series trend together by chance	Test for unit roots
DW $\approx 0$	Strongly autocorrelated residuals from I(1) errors	Test residuals for stationarity (augmented Dickey-Fuller (ADF), Kwiatkowski-Phillips-Schmidt-Shin (KPSS))
Contradictory coefficient signs	Spurious equilibrium	Difference the series, or use VECM if cointegrated

The standard workflow: (1) ADF/KPSS unit-root tests on each series; (2) if all I(1), test residuals for I(0) (Engle-Granger cointegration test) or use Johansen's full-system test; (3) if cointegrated, use VECM; if not, first-difference VAR.

The Stochastic Unit Root (Granger-Swanson 1994)

A key limitation of standard unit-root testing is that many processes that are not I(1) in the pure sense are also not rejected by Dickey-Fuller tests. Granger and Swanson (1994) introduce the stochastic unit root process:

$X_t = \alpha_t X_{t-1} + \varepsilon_t, \qquad \alpha_t \text{ stationary, exogenous to } X, \quad E(\alpha_t) = 1$

where $\alpha_t$ is a random coefficient with mean one. This process is:

sometimes explosive ( $\alpha_t > 1$ ), sometimes stationary ( $|\alpha_t| < 1$ ), and occasionally exactly unit root ( $\alpha_t = 1$ ).
Rejected by Dickey-Fuller (DF) tests at roughly nominal size — it mimics I(1) behavior in Dickey-Fuller simulations.
Fundamentally different from a pure unit root: the variance of a pure I(1) random walk grows proportionally to $t$ , whereas the variance of a stochastic unit root process grows exponentially with $t$ .

The practical implication (Granger 1997): a process that does not reject an I(1) null may be any of:

pure I(1) (random walk)
stochastic unit root
I(0) with a broken linear trend (Perron 1989)
fractionally integrated with $d$ near 1
near-unit-root stationary AR

The I(1) vs. broken-trend debate (Perron 1989 and response literature) asked whether macro series are "better described" as unit root or stationary with broken trend. Granger (1997) dismisses this as "fatuous" — the question excludes too many plausible alternatives, including all of the above. The practically relevant question is simply whether the series exhibits persistence, for which an I(1) null provides a useful operational approximation even if not a literal description.

A related observation: if $\log x_t$ has a pure unit root, then $x_t$ itself has a stochastic unit root (Granger-Swanson 1994). This means conclusions about integration order are not invariant to monotone transformations.

Why It Matters

The spurious regression problem explains why decades of published macroeconomic empirical work that ignored nonstationarity potentially produced invalid conclusions.
Recognizing the problem was the catalyst for the entire cointegration literature and the development of VAR/VECM as the standard macro modeling framework.
The Granger-Newbold finding (and Engle-Granger 1987 solution) is one of the most influential contributions in applied econometrics — central to Granger's Nobel Prize.

Open Questions

In finite samples, distinguishing spurious from genuine cointegration is non-trivial, especially when the cointegrating relationship is weak (near the unit circle) — tests have low power.
The problem extends to I(2) variables (two rounds of differencing needed), fractionally integrated I(d) variables, and seasonally integrated variables.
The stochastic unit root problem means that even a "correct" diagnosis of I(1) is potentially wrong: the true data-generating process (DGP) may have a stochastic coefficient rather than a fixed unit root. This distinction matters for long-run forecasting but is difficult to test.