12.2 Spurious regression and the asymptotics of a unit root
Granger and Newbold (1974) simulated pairs of independent random walks and regressed one on the other. The \(t\) test of a zero slope at the nominal \(5\%\) level rejected about three-quarters of the time, \(R^2\) was often large, and the Durbin–Watson statistic was very small. Software 12.6.2 repeats their experiment, and shows that matters get worse, not better, as \(n\) grows. The explanation needs a different limit theory, because sums of a random walk grow faster than \(n\).
Definition 12.2.1. A standard Brownian motion (Wiener process) \(W(r)\), \(0 \leq r \leq 1\), is a random function with \(W(0) = 0\), continuous paths, and independent normal increments \(W(r) - W(q) \sim N(0, r - q)\) for \(q < r\). In particular \(W(1) \sim N(0, 1)\).
Theorem 12.2.2 (Functional central limit theorem). Let \(u_t\) be independent and identically distributed with mean zero and variance \(\sigma ^2\), and \(Y_t = u_1 + \dots + u_t\). Then the step function \(X_n(r) = Y_{\lfloor nr\rfloor }/(\sigma \sqrt n)\) converges in distribution, as a random function on \([0, 1]\), to \(W(r)\). Consequently, by the continuous mapping theorem, \(g(X_n) \xrightarrow {d} g(W)\) for every continuous functional \(g\), such as \(\int _0^1X_n(r)^2\,dr\).
The theorem, due to Donsker, is the central limit theorem applied to the whole path rather than to its end point; at \(r = 1\) it is the ordinary central limit theorem. We take it as given. The results that follow need only a few of its consequences.
Lemma 12.2.3. With \(Y_t = Y_{t-1} + u_t\), \(Y_0 = 0\) and \(u_t\) as in Theorem 12.2.2,
- (i)
- \(\displaystyle \sum _{t=1}^nY_{t-1}u_t = \frac 12\Bigl (Y_n^2 - \sum _{t=1}^nu_t^2\Bigr )\) exactly, and hence \(\displaystyle \frac {1}{\sigma ^2n}\sum _{t=1}^nY_{t-1}u_t \xrightarrow {d} \frac 12\bigl (W(1)^2 - 1\bigr )\);
- (ii)
- \(\displaystyle E\sum _{t=1}^nY_{t-1}^2 = \frac {\sigma ^2n(n - 1)}{2}\), and \(\displaystyle \frac {1}{\sigma ^2n^2}\sum _{t=1}^nY_{t-1}^2 \xrightarrow {d} \int _0^1W(r)^2\,dr\);
the two limits holding jointly.
Proof. (i) Squaring \(Y_t = Y_{t-1} + u_t\) gives \(Y_t^2 - Y_{t-1}^2 = 2Y_{t-1}u_t + u_t^2\). Summing over \(t\), the left side telescopes to \(Y_n^2 - Y_0^2 = Y_n^2\). Dividing by \(\sigma ^2n\), \(Y_n^2/(\sigma ^2n) = X_n(1)^2 \xrightarrow {d} W(1)^2\) by the central limit theorem, and \(\frac 1n\sum u_t^2 \xrightarrow {p} \sigma ^2\) by the law of large numbers; Slutsky’s theorem gives the limit.
(ii) \(E(Y_{t-1}^2) = (t - 1)\sigma ^2\), and \(\sum _{t=1}^n(t - 1) = n(n - 1)/2\). For the limit, \(Y_{t-1}^2/(\sigma ^2n) = X_n(r)^2\) for \(r \in [(t - 1)/n, t/n)\), so \(\frac {1}{\sigma ^2n^2}\sum _tY_{t-1}^2 = \int _0^1X_n(r)^2\,dr\) exactly, and Theorem 12.2.2 applies. Both statistics are functionals of the same path \(X_n\), which gives joint convergence. □
Compare the orders of magnitude with the stationary case. There, \(\sum Y_{t-1}^2\) grows like \(n\) and \(\sum Y_{t-1}u_t\) like \(\sqrt n\). With a unit root, \(\sum Y_{t-1}^2\) grows like \(n^2\) and \(\sum Y_{t-1}u_t\) like \(n\). The limits are not normal: \(W(1)^2\) is \(\chi ^2_1\), and \(\int W^2\) is a random variable with no textbook name.
Theorem 12.2.4 (Dickey–Fuller). Let \(\hat \rho \) be the least squares estimate in the regression \(Y_t = \rho Y_{t-1} + u_t\), with \(t\) ratio \(\tau = (\hat \rho - 1)/\text {se}(\hat \rho )\) for \(H_0\colon \rho = 1\). If in fact \(\rho = 1\), \[ n(\hat \rho - 1) \xrightarrow {d} \frac {\frac 12\bigl (W(1)^2 - 1\bigr )}{\int _0^1W(r)^2\,dr}, \qquad \tau \xrightarrow {d} \frac {\frac 12\bigl (W(1)^2 - 1\bigr )}{\bigl (\int _0^1W(r)^2\,dr\bigr )^{1/2}} . \] In particular \(\hat \rho \) is superconsistent: \(\hat \rho - 1\) is of order \(1/n\), not \(1/\sqrt n\). And \(P(\hat \rho < 1) \to P(\chi ^2_1 < 1) = 0.683\).
Proof. \(\hat \rho - 1 = \sum Y_{t-1}u_t/\sum Y_{t-1}^2\), so \[ n(\hat \rho - 1) = \frac {\frac {1}{\sigma ^2n}\sum Y_{t-1}u_t}{\frac {1}{\sigma ^2n^2}\sum Y_{t-1}^2}, \] and the first result follows from the joint limits of Lemma 12.2.3 and the continuous mapping theorem. For \(\tau \), \(\text {se}(\hat \rho ) = s/\sqrt {\sum Y_{t-1}^2}\) with \(s^2 \xrightarrow {p} \sigma ^2\), so \[ \tau = \frac {\sum Y_{t-1}u_t}{s\sqrt {\sum Y_{t-1}^2}} = \frac {\sigma }{s}\cdot \frac {\frac {1}{\sigma ^2n}\sum Y_{t-1}u_t}{\bigl (\frac {1}{\sigma ^2n^2}\sum Y_{t-1}^2\bigr )^{1/2}} , \] which converges to the stated limit. Since the denominators are positive, the sign of \(\hat \rho - 1\) is the sign of \(Y_n^2 - \sum u_t^2\), whose limit is the sign of \(W(1)^2 - 1\). □
The distribution of \(\tau \) is skewed to the left: about two-thirds of the time \(\hat \rho < 1\) even when the truth is \(\rho = 1\). Using the normal critical value \(-1.645\) for a one-sided test would reject the unit root far too often. Figure 12 compares the distributions.
Theorem 12.2.5 (Spurious regression; Phillips 1986). Let \(Y_t\) and \(X_t\) be independent random walks, and let \(b\), \(t_b\), \(R^2\) and \(d\) be the slope, its \(t\) ratio, the coefficient of determination and the Durbin–Watson statistic from the least squares regression of \(Y_t\) on a constant and \(X_t\). Then, as \(n \to \infty \),
- (i)
- \(b\) converges in distribution to a non-degenerate random variable, not to the true value \(0\);
- (ii)
- \(t_b/\sqrt n\) converges in distribution to a non-degenerate random variable, so \(|t_b| \to \infty \) in probability and the \(t\) test rejects with probability tending to one;
- (iii)
- \(R^2\) converges in distribution to a non-degenerate random variable;
- (iv)
- \(d \xrightarrow {p} 0\), at rate \(1/n\).
Sketch of proof. All the sums of squares and cross products of the levels are of order \(n^2\), and after division by \(n^2\) they converge, by Theorem 12.2.2 applied to the two independent walks, to integrals of two independent Brownian motions \(W_Y\) and \(W_X\) (demeaned, because of the constant). The slope is a ratio of two such sums, which gives (i), and \(R^2\) is a ratio of such limits, which gives (iii). The residuals \(e_t = Y_t - a - bX_t\) are themselves of order \(\sqrt n\), so \(s^2 = \sum e_t^2/(n - 2)\) is of order \(n\), while \(\sum (X_t - \bar X)^2\) is of order \(n^2\). Hence \(\text {se}(b) = s/\sqrt {\sum (X_t - \bar X)^2}\) is of order \(n^{-1/2}\), while \(b\) is of order one, and \(t_b\) is of order \(\sqrt n\), which is (ii). Finally, \(\Delta e_t = \Delta Y_t - b\Delta X_t\) is stationary-like, so \(\sum (\Delta e_t)^2\) is of order \(n\), while \(\sum e_t^2\) is of order \(n^2\): \(d\) is of order \(1/n\), which is (iv). □
The regression is spurious because the residuals are \(I(1)\): there is no value of the slope that makes \(Y_t - bX_t\) stationary, and least squares picks whichever value happens to track the two stochastic trends best over the sample. The Durbin–Watson statistic is the giveaway, and Granger and Newbold’s rule of thumb, “be suspicious when \(R^2 > d\)”, is a practical form of (iii) and (iv). The remedy is not to abandon regression on levels, but to test first whether the series have unit roots, and then whether some linear combination of them is stationary.
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