Why ADF-Based Pair Hedge Selection Can Be Biased
Summary
The document examines a common pairs-trading heuristic: regress each asset on the other, construct a spread from each regression, and choose the hedge ratio associated with the stronger Augmented Dickey-Fuller result. It argues that this choice lacks a sound statistical justification. Ordinary least squares is asymmetric, so reversing the dependent and independent assets produces a different slope and residual series; selecting whichever spread looks more stationary can amount to data-driven selection bias or p-hacking.
As a more symmetric alternative, the answer proposes total least squares for estimating a hedge ratio between two assets. It notes that this idea has been suggested previously and mentions a possible concern about numerical stability, while acknowledging uncertainty about that criticism. A second response adds that the two regression directions produce different residual series, whose stationarity can also differ. The discussion motivates caution but provides no comparative backtest, formal derivation, or evidence that TLS is optimal; it presents TLS as a potentially more principled construction rather than a proven best method.
Key ideas
- Reversing the direction of an OLS regression changes the estimated hedge ratio.
- The two regression directions can produce residual spreads with different stationarity.
- Selecting the spread with the strongest ADF result may introduce statistical selection bias.
- Total least squares is suggested as a symmetric alternative for estimating a two-asset hedge ratio.
- The document does not establish TLS as optimal or resolve concerns about its numerical stability.
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# Answer by Alex C (score 1) # When constructing a cointegrating series, does choosing the linear regression with the lowest ADF test statistic yield the optimal hedging ratio? Multiple sources say that you should find the optimal hedging ratio between two stocks in a pairs trade by conducting 2 linear regressions (with each stock as the independent variable), and using whichever beta value yields a highest ADF [Augmented Dickey Fuller] test statistic . Does this actually give you the optimal hedging ratio? If so, what are the mathematics behind it? It seems like a very arbitrary procedure. ## Answer by Alex C (score 1) https://quant.stackexchange.com/a/46256 You are right that it is a "very arbitrary procedure". More charitably it is a "hack" that gives a practical solution without addressing the fundamental issue. The very fact that when you do an OLS regression of x vs y you get a different result than when you regress y vs x tells you that OLS regression is probably not the right tool to construct a hedged portfolio from 2 assets. How do you decide if x or y is to be considered the "independent asset"? In the early 2000's a quant at Merrill Lynch named Mary Ann Bartels suggested that it makes more sense to use a Total Least Squares (TLS) regression to find the hedge ratio. At least TLS regression is symmetric. I cannot find her published report right now, although the idea is still floating around and has apparently been re-discovered by at least one person (http://quantdevel.com/public/pdf/betterHedgeRatios.pdf) who does not acknowledge her prior work. It seems to me the idea has merit. (Someone told me that TLS will be less numerically stable than OLS but I don't know if that is a valid criticism). But you are right that choosing the regression with the better ADF is not justified and probably statistically biased (an example of "p-hacking"). ## Answer by Ashish Garg (score 0) https://quant.stackexchange.com/a/43680 The error series will be different in two cases and their stationarity will be different as well. Hence more likely than not, one of them will be less stationary than the other one.
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