13A.5 The Engle–Granger two-step

Motivation

The simplest test for a pair: estimate the hedge ratio, then test the residual for a unit root. The catch is that the second step needs its own critical values.

13A.5.1Step 1: the cointegrating regression and superconsistency#

Run

p^1_t = \mu + b\,p^2_t + u_t \tag{13A.14}

by OLS. If the pair is cointegrated, $u_t$ is stationary.

Proposition 13A.5superconsistency

If $p^2$ is $I(1)$ and $u$ is $I(0)$ , then $\hat b - b = O_p(T^{-1})$ .

Proof sketch

$\hat b - b = \sum(p^2_t - \bar p^2)u_t\big/\sum(p^2_t-\bar p^2)^2$ . The denominator is $O_p(T^2)$ (Proposition 13A.3). The numerator is a sum of products of an $I(1)$ and an $I(0)$ variable: $T^{-1}\sum (p^2_t-\bar p^2)u_t$ converges to a functional of the form $\int \tilde W\,dB + \Lambda$ , where $B$ is the Brownian motion of $u$ 's partial sums and $\Lambda$ a one-sided long-run covariance. So the numerator is $O_p(T)$ and the ratio $O_p(T^{-1})$ .∎

Superconsistency makes the estimation error in $\hat b$ negligible compared with the spread itself. It does not make OLS $t$ -statistics on $\hat b$ valid: the limit is non-normal and has a second-order bias from $\Lambda$ (endogeneity and serial correlation). For inference on $b$ , use dynamic OLS (add leads and lags of $\Delta p^2$ ; Stock and Watson 1993) or fully modified OLS (Phillips and Hansen 1990).

13A.5.2Step 2: testing the residual, with the right critical values#

Apply the ADF regression to $\hat u_t$ without deterministic terms (they are absorbed in step 1, and $\hat u$ has mean zero by construction):

\Delta\hat u_t = \gamma\,\hat u_{t-1} + \sum_{j=1}^p\delta_j\Delta\hat u_{t-j} + e_t. \tag{13A.15}

$H_0$ : no cointegration ( $u$ has a unit root). Reject for large negative $t$ -ratios on $\gamma$ .

Why DF critical values are wrong here Under $H_0$ the step-1 regression is spurious: $\hat b$ is a random variable (Proposition 13A.3), not an estimate of anything. And OLS picks $\hat b$ to minimise $\sum\hat u_t^2$ , i.e. to make $\hat u$ look as stationary as possible. Both effects push the statistic to the left. Its limit is a functional of an $n$ -dimensional Brownian motion that depends on the number of variables $n$ and on the deterministic terms in step 1 (Phillips and Ouliaris 1990). MacKinnon (2010) gives response surfaces; statsmodels.tsa.stattools.coint uses them.

Engle–Granger, constant in step 1	1%	5%	10%
$n=2$ (a pair), asymptotic (MacKinnon 2010)	−3.90	−3.34	−3.04
$n=2$ , simulated $T=500$ , 5,000 draws (`m13a_adf.py`)	−3.87	−3.35	−3.06
$n=3$ , asymptotic (MacKinnon 2010)	−4.29	−3.74	−3.45
For comparison, DF with constant	−3.43	−2.86	−2.57

A residual statistic of −3.10 for a pair rejects at 10% and not at 5%. With DF critical values it would wrongly be called significant at 5%.

Exercise link engle_granger in m13a_adf.py must reproduce statsmodels.coint(trend="c", maxlag=1, autolag=None) exactly. A test checks that, for independent random walks, the EG critical value rejects about 5% of the time while the DF critical value rejects noticeably more.

13A.5.3Weaknesses#

Normalisation. Regressing $p^1$ on $p^2$ and $p^2$ on $p^1$ gives different residuals and can give different test conclusions in finite samples. Johansen's method (13A.6) is invariant to it.
One vector only. With $n\ge3$ assets, EG finds one combination and cannot tell whether there are more.
Multiple testing. Screening 50 bank and NBFC stocks gives $\binom{50}{2} = 1{,}225$ pairs. At a 5% level, about 61 will "cointegrate" by chance if none truly do. Apply the multiple-testing machinery of M04 (Bonferroni or Holm for family-wise error, Benjamini–Hochberg for false discoveries), and above all test out of sample.