Factor diagnostics: monotonicity with Romano and Wolf, Patton and Timmerman

It is a snowy, blistering cold day in London. A perfect time to be introspective. So with your consent, I’d like to review a few diagnostic methods of factor premia – tests for monotonicity. There are sound theoretical reasons for expecting cross sectional differences in exposures to result in monotonic difference in returns. If one constructs n-tile portfolios, the lowest n-tile should under-perform the second n-tile and so on with the highest n-tile providing the highest performance.

The tests to confirm this outcome are necessarily complex. The first of these test is by Andrew Patton and Allen Timmerman. Another by Joseph P Romano and Michael Wolf. Both examine the differences in adjacent n-tile portfolios. How often do the returns of portfolio n-tile_{k-1} outperform n-tile_{k}? That should not happen too often. Patton and Timmerman asks if the relationship is monotonic increasing. Romano and Wolf respond with not so fast. There are cases where some n-tiles are larger than the previous but also larger than the next n-tile. Not strictly increasing. When this is the case, are the violations small enough in occurrence to still accept monotonicity?

The difference is that the Patton and Timmerman test allows for a wider class of outcomes to pass the monotonicity test. Romano Wolf is stricter. From the conclusion section of Romano and Wolf:

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In a recent paper, Patton and Timmermann (2010) propose a new test, taking all categories of the underlying characteristic into account. Compared to previous related proposals, they are, to our knowledge, the first ones to postulate a strictly increasing relation as the alternative hypothesis of the test, rather than as the null hypothesis. This is the correct formulation if the goal is to establish strict monotonicity with a quantifiable statistical significance. In addition, they postulate a weakly decreasing relation as the null hypothesis. Compared to allowing, more generally, also for a non-monotonic or weakly increasing relation under the null, this approach results in a smaller critical value and thereby in higher power of the test. But if a non-monotonic or weakly increasing relation is not ruled out, the test can break down and falsely ‘establish’ a strictly increasing relation with high probability. Therefore, the test of Patton and Timmermann (2010) should only be used as a test for the direction of (existing) monotonicity.

In this paper, we have proposed some alternative tests that also allow for a non-monotonic or weakly increasing relation under the null and that successfully control the probability of a type 1 error. These tests do not require modeling the dependence structure of the time series nor the covariance matrix of the observed return differentials. A suitable bootstrap method is used to implicitly account for these unknown features of the underlying data generating mechanism. As is unavoidable, such tests have lower power compared to the test of Patton and Timmermann (2010). This is the price one has to pay to safe-guard against the possibility of falsely ‘establishing’ strict monotonicity (beyond the nominal significance level of the test) in general settings.

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My code is available at my blog. I’ve altered Patton and Timmerman’s block bootstrap code with a version inspired by Kevin Sheppard’s code, it is much faster.

I've also added Jochen Heberle and Cristina Sattarhoff, A Fast Algorithm for the Computation of HAC Covariance Matrix Estimators, because who doesn't need a fast HAC estimator?

Andrew Patton and Allan Timmermann. Monotonicity in asset returns: New tests with applications to the term structure, the capm, and portfolio sorts. Journal of Financial Economics,98(3):605–625, 2010.

Romano, J.P. and Wolf, M. (2013). Testing for monotonicity in expected asset returns. Journal of Empirical Finance 23, 93-116.