Unit roots, spurious regression and the basis of pairs trading

Prices wander; returns do not. Almost every mistake in applied time series comes from running a regression on the first kind of series while reasoning as though it were the second. Cointegration is the precise condition under which a regression on levels *is* legitimate, and it is the statistical content of a pairs trade.

Definition 3.1

Stationarity

Weak stationarity, $\mathbb{E}[x_t] = \mu, \quad \mathrm{Cov}(x_t, x_{t+h}) = \gamma(h)$ — The mean is constant and the covariance depends only on the lag, not on where you are in the sample. A random walk fails this: its variance grows without bound, so it has no mean to revert to and no fixed scale.

Figure 3.2 · What a unit root does to a shock. A shock to a stationary series dies; a shock to a random walk never does. That single difference is why two unrelated walks appear to be related, why the regression of one on the other is spurious, and why the whole of pairs trading rests on finding a combination that decays.

Equation 3.3

The unit root

Stationary when $|\phi| < 1$ ; a random walk when $\phi = 1$ . That single boundary separates a series that mean-reverts from one that does not.

x_t = \phi\,x_{t-1} + \varepsilon_t

$\phi = 1$: The unit root. Shocks are permanent and the variance grows linearly in time.
$|\phi| < 1$: Shocks decay with half-life $\ln(0.5)/\ln\phi$ .

Why spurious regression happens. Take two independent random walks and regress one on the other. You will routinely find a t-statistic above 2 and an $R^2$ of 0.3 or more, on data with no relationship whatever. The reason is that the residual is itself a random walk, so it is massively autocorrelated, and the usual standard-error formula assumes it is not — it therefore understates the true uncertainty by an enormous factor. The regression is not detecting a relationship; it is detecting that both series have long, slow swings.

Aspect	Augmented Dickey–Fuller	KPSS
Null hypothesis	There is a unit root	The series is stationary
Rejecting means	The series is stationary	There is a unit root
Known weakness	Low power near $\phi = 1$	Sensitive to the lag choice
Good practice	Run both — agreement is more informative than either alone	—

Table 3.4 · Testing for a unit root. The nulls are opposite, which is exactly why running both is worthwhile: "ADF fails to reject and KPSS rejects" is a much stronger statement than either result on its own.

Unit roots, spurious regression and the basis of pairs trading

After this lesson you should be able to

Stationarity

The unit root

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