AppliedMultiple choiceFree solution

You regress one stock’s price on another’s and get R2=0.95R^2 = 0.95 with a t-statistic of 4040. What is your first move?

  1. ATrade the relationship, since it is overwhelmingly significant
  2. BTest whether the residual is stationary
  3. CAdd more control variables to the specification
  4. DReport the result, since R2R^2 above 0.9 is conclusive
Show the answer and worked solution

Answer: B – Test whether the residual is stationary

Both series are close to random walks, and a regression of one integrated series on another produces inflated R2R^2 and t-statistics even when the two are entirely unrelated: the residual inherits the unit root, so the usual standard errors are wrong by an order of magnitude. The diagnostic is to test the residual for stationarity with an augmented Dickey and Fuller test. If it is stationary the two series are cointegrated and the relationship is real and tradable; if it is not, you have a spurious regression and the statistics mean nothing. Adding controls in levels compounds the problem rather than fixing it.

Worked solution

  1. Formula
    spurious if the residual has a unit root\text{spurious if the residual has a unit root}
  2. Substitute
    R2=0.95, t=40 on two integrated seriesR^2 = 0.95, \ t = 40 \text{ on two integrated series}
  3. Solve
    such figures arise between unrelated random walks\text{such figures arise between unrelated random walks}
    The standard errors are wrong by an order of magnitude.
  4. ADF on the residual settles it\text{ADF on the residual settles it}
  5. Answer
    Test the residual for stationarity\text{Test the residual for stationarity}

Sanity check. A stationary residual means cointegration and a real relationship; a unit root means the statistics mean nothing.

  • A. Those figures arise routinely between two unrelated random walks.
  • B. Correct. If the residual has a unit root this is a spurious regression; if it is stationary the series are cointegrated.
  • C. More regressors on levels compounds the problem rather than fixing it.
  • D. R2R^2 is the least informative statistic in a regression on levels.

Takeaway: A high R2R^2 between two price series usually means nothing at all.

Answer it in practice – your answer is marked and recorded.

Reported in interviews at

Citadel, Two Sigma, D. E. Shaw, AQR

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You regress one stock’s price on another’s and get R² = 0.95 with a t-statistic of 40. · QuantMax