13A.3 Spurious regression · M13A Stat arb

Motivation

Regress the log price of one bank on another and you will get a slope with a $t$ -statistic in the hundreds and an $R^2$ above 0.8. This says almost nothing. Granger and Newbold (1974) showed it by simulation; Phillips (1986) explained it.

Setup

Let $y_t$ and $x_t$ be independent random walks with innovation variances $\sigma_y^2, \sigma_x^2$ . Run OLS of $y$ on a constant and $x$ . Write $\tilde W_x, \tilde W_y$ for independent demeaned Brownian motions.

Proposition 13A.3Phillips 1986

As $T\to\infty$ :

\begin{aligned} \hat\beta &\Rightarrow \frac{\sigma_y}{\sigma_x}\,\frac{\int \tilde W_x\tilde W_y}{\int\tilde W_x^2}, \qquad R^2 \Rightarrow \frac{\left(\int\tilde W_x\tilde W_y\right)^2}{\int\tilde W_x^2\int\tilde W_y^2}, \\ T^{-1/2}\,t_{\hat\beta} &\Rightarrow \text{a non-degenerate random variable}. \end{aligned} \tag{13A.8}

In particular $\hat\beta$ does not converge to the true value 0, $R^2$ does not converge to 0, and $|t_{\hat\beta}|\to\infty$ in probability.

Proof

Let $S_{xy} = \sum(x_t-\bar x)(y_t-\bar y)$ and similarly $S_{xx}, S_{yy}$ . As in Proposition 13A.1, $T^{-2}S_{xy} \Rightarrow \sigma_x\sigma_y\int\tilde W_x\tilde W_y$ , $T^{-2}S_{xx}\Rightarrow\sigma_x^2\int\tilde W_x^2$ , $T^{-2}S_{yy}\Rightarrow\sigma_y^2\int\tilde W_y^2$ , jointly. Then $\hat\beta = S_{xy}/S_{xx}$ and $R^2 = S_{xy}^2/(S_{xx}S_{yy})$ give the first two limits (the $T^{-2}$ factors cancel). For the $t$ -ratio, the residual sum of squares is $\text{SSR} = S_{yy} - S_{xy}^2/S_{xx}$ , so $T^{-2}\text{SSR} \Rightarrow \sigma_y^2\big[\int\tilde W_y^2 - (\int\tilde W_x\tilde W_y)^2/\int\tilde W_x^2\big] > 0$ . With $s^2 = \text{SSR}/(T-2)$ , $s^2 = O_p(T)$ . Then

t_{\hat\beta} = \frac{\hat\beta\,S_{xx}^{1/2}}{s} = \frac{O_p(1)\cdot O_p(T)}{O_p(T^{1/2})} = O_p(T^{1/2}),

and $T^{-1/2}t_{\hat\beta}$ converges to the ratio of the corresponding functionals, which is non-zero with probability one.∎

Simulation (spurious_regression in m13a_adf.py, 2,000 replications each). Share of $|t|>1.96$ on the slope, which should be 5%: 67.7% at $T=50$ , 84.2% at $T=250$ , 92.0% at $T=1{,}000$ . Median $R^2$ : 0.17, 0.17, 0.16. The rejection rate rises with $T$ because $t$ grows like $\sqrt T$ . The $R^2$ does not fall because its limit is a non-degenerate random variable.

On Indian data (computed for this course from Yahoo Finance adjusted closes, 4 Jan 2010 to 4 Nov 2026, 4,135 common trading days): regressing $\ln$ HDFC Bank on $\ln$ ICICI Bank gives $R^2 = 0.85$ and a slope $t$ -statistic of 151. The Durbin–Watson statistic is 0.003. Yet the Engle–Granger test of 13A.5 does not come close to rejecting "no cointegration" ( $p = 0.79$ ). Granger and Newbold's rule of thumb, $R^2 > \text{DW}$ means suspect a spurious regression, flags it at once.

Practitioner notes

Correlation of returns is a different object. HDFC Bank and ICICI Bank daily returns had correlation 0.54 over the same sample. High return correlation says the two move together day to day; it says nothing about whether the gap between their levels is anchored.
Screening pairs by price-level correlation selects exactly the spurious regressions above.