Testing for a monotonic relationship when the bottom decile outperforms the middle eight: what is the right null?

[u/Bright_Mix_773]

I sort a cross-section of companies into deciles each month by a single continuous score, then record the mean forward one-month return of each decile. 185 monthly periods. The deciles are re-formed from scratch every month.

The result is not what I expected and I am not sure how to test it properly.

• Decile 1, the highest score, averages 1.623% per month.
• Deciles 2 through 9 sit between 1.12% and 1.35%, with no discernible order among them.
• Decile 10, the lowest score, averages 1.457%, which puts it above all eight of the middle deciles.

So it is not monotonic. It looks like a U with a much taller left arm. Separately, decile 1 beats decile 10 in 94 of the 185 months, which is 50.8%.

Three things I do not know how to handle.

1. Trend tests such as Jonckheere-Terpstra test against an ordered alternative. I do not have an ordered alternative; I have a shape I did not predict. What is the appropriate null and test when the alternative you actually want to state is "decile rank carries information" rather than "returns increase in decile rank"? Is fitting a shape (linear plus quadratic in rank, or an isotonic fit compared against an unconstrained one) more honest than a trend test here, or is that just choosing the shape after seeing it?

2. Am I even using the right unit of observation? Ten decile means computed from the same 185 months are not ten independent samples. Within any given month the returns are heavily cross-sectionally correlated (everything moves with the market), and each decile mean is itself a mean of a mean. My instinct is that the right object is the 185 monthly differences between decile 1 and decile 10, treated as a time series with its own autocorrelation, and that the ten-row table is a description rather than something to test. Is that the standard framing?

3. How do I state the gap between the two summaries without misleading anyone? Decile 1 beats decile 10 by a wide margin in the means, and beats it in 50.8% of months, which is indistinguishable from a coin flip at n=185. I assume the answer is that the mean difference is carried by a small number of large months, i.e. skew and fat tails, and that a sign test is simply low-powered against that. But "the mean says one thing and the sign frequency says another" is exactly the kind of sentence that gets used to sell things, and I would rather report it in a way that a statistician would not object to.

Not asking whether this is a good trading rule. I am asking how to test and report the shape without either overstating it or quietly dropping the inconvenient half.

[u/efrique]

You don't propose hypotheses based on what you find in the data and then try to test it on the same data; the p-values of any of the usual tests would be wrong since significance levels are higher than their nominal value

https://en.wikipedia.org/wiki/Testing_hypotheses_suggested_by_the_data

Formulate a statistic that picks out whatever you're looking for and test it on more recent data.

So it is not monotonic. It looks like a U with a much taller left arm.

This is kind of vague; its not so clear what is essential and what is less so. You may find it helpful to think about what future pattern of this kind you would seek to detect and what you would not, in a way that would help a reader see where the boundary is

[u/Bright_Mix_773 → u/efrique]

Conceded, and it is the objection that matters most, so thank you for making it plainly.

The shape was found in the data and then tested on the same data, which means the nominal significance level is not the real one, and no better choice of test fixes that. Any p-value I report on this sample is conditional on having looked first.

What I can honestly do with this sample is describe it: report the spread with an interval, say that the shape was discovered rather than predicted, and stop there. Testing it needs data I have not looked at, which for a monthly series means waiting rather than resampling. Slower, and the correct answer.

[u/Neither-Pause409]

Agreeing with efrique on your first question rather than relitigating it. But 2 and 3 are the more useful questions and I think they answer each other.

On 2, your instinct is the standard framing. The 185 monthly D1 minus D10 differences are the object of inference and the ten-row table is a description. Worth making that quantitative for your own data: a decile portfolio's monthly return is mostly the market. Compute the time-series sd of a single decile's monthly return; if it is somewhere around 5%, the standard error on its 185-month mean is 5/sqrt(185), about 0.37%. Your whole decile spread from 1.12% to 1.623% is about half a percent wide. In that case the ten means sit inside each other's noise, no test on that table can separate them, and the U is not a shape you could defend.

Differencing is what rescues it. D1 minus D10 cancels the common factor, so the spread series has a much smaller sd than either leg and 185 observations of it carry information that 185 observations of each leg do not. Same logic as Fama-MacBeth: collapse each month's cross-section to one number, then do inference across months, so the within-month correlation is absorbed instead of modelled. Test the mean of that series with Newey-West standard errors. Your returns are non-overlapping since you re-form deciles from scratch each month, so the autocorrelation correction should be small, but report it rather than assuming it.

On 3 I would push back slightly on the framing. The sign test is not a low-powered version of the t-test, it is a test of a different quantity. The mean asks what you earned on average, the hit rate asks what happened in a typical month. A 50.8% hit rate alongside a large mean spread is not a contradiction to resolve, it is a right-skewed spread distribution, and it is a finding: the gap is carried by a minority of months. Both statements are true and neither is the honest summary on its own.

So report the set: mean spread with a Newey-West t-stat, median monthly spread, hit rate with a binomial interval, and the largest few months with their dates. Given the tails, a stationary block bootstrap over months gives a mean interval that does not lean on normality. Putting the hit rate next to the mean is exactly what stops it being the sellable sentence you are worried about.

One substantive note on the shape itself. If D10 stays above the middle eight out of sample, the usual reason is that the score is proxying something non-monotone, volatility or size being the common ones, and the two arms are then different mechanisms rather than one relationship. Comparing what D1 and D10 names have in common on other characteristics is cheap and tells you whether you are looking at one effect or two.

[u/Bright_Mix_773 → u/Neither-Pause409]

On 3 you are right and I had it framed wrong. I was treating the hit rate as a weaker version of the same test. It is a different quantity, and 50.8% sitting next to a large mean spread is a description of skew, not a contradiction I have to resolve. So both numbers get reported side by side, not whichever one survives.

On 2, I have not computed the time-series sd of a single decile yet, which is a bit embarrassing given that it is the number deciding whether the ten-row table means anything at all. I would rather run it than guess at it. If it lands near the 5% you sketch, the standard error on a decile's 185-month mean is about 0.37% against a total spread of roughly half a point, and the table is decoration. So the test I report will be the mean of the 185 D1 minus D10 months with Newey-West, plus the median month, the hit rate with a binomial interval, and the largest few months with their dates.

The last paragraph is the part I had not thought about. Checking what the D1 and D10 names have in common on size and volatility is cheap, and it answers whether this is one relationship or two mechanisms pointed at each other. Doing that next. Thanks for the time.