Terms

Scope

Applies to any attempt to convert the gradient of a price-derived series into an angle, including oscillator slope and the divergence between two slopes computed by the tangent subtraction formula.

The objection

The argument of an inverse tangent must be dimensionless. The quantity $(MA_t - MA_{t-n})/n$ carries units of price per bar, so passing it to $\arctan$ is the same category of error as writing $\sin(5\,\text{metres})$.

The consequence is not subtle. For a pair quoted near $1.20$ a slope of $5\times10^{-5}$ returns about $0.003^\circ$. An equivalent trend on an instrument quoted near $50{,}000$ returns about $89.9^\circ$. The number reports the price scale, not the trend.

There is a second and independent problem. The visual angle of a line on a chart is a property of the axis scaling rather than of the data. Stretch the vertical axis and every line steepens while nothing about the series has changed.

What replaced it

Normalising by average true range restores dimensionlessness, since numerator and denominator are then both in price units. Kaufman's efficiency ratio is bounded on $[0,1]$ by construction and has no dimensional exposure at all.

Finding

References

• Cleveland, W. S. Visualizing Data, 1993, on the dependence of perceived line orientation on physical display units.

Terms

Scope

Holds at daily resolution and coarser, where $\alpha+\beta<1$ gives a stationary process with a defined unconditional variance. Does not hold as specified on hourly or finer data.

The objection

Intraday volatility carries a strong deterministic seasonal component. Activity rises on a daily cycle in almost every observed system, from traffic to electricity demand to markets, and again around scheduled events. The model has no term for that, so the seasonality is absorbed into $\alpha$ and $\beta$, and the fitted persistence is an artefact of the trading day rather than a property of the series.

Fitted to raw intraday returns it therefore produces a model that appears to forecast and is in fact reciting the timetable.

What would settle it

Deseasonalise first by estimating a diurnal factor $s_u$ for each intraday interval, then fit to $\varepsilon_t / s_u$. If the persistence survives that, it is real.

Finding

Terms

Scope

Applies to any single series long enough to be split into many windows. Says nothing about the relationship between two series, which is cointegration and a separate field.

Reading it

$H = 0.5$ is a random walk with no memory. $H > 0.5$ indicates persistence, where an up move is more likely to be followed by another. $H < 0.5$ indicates anti-persistence.

The objection

Classical rescaled-range estimation is biased upward in finite samples. A pure random walk of realistic length returns values comfortably above $0.5$, which means the estimator manufactures the appearance of memory in a series that has none. Short-range dependence and non-stationarity inflate it further.

The practical consequence is that an unqualified $H$ of, say, $0.58$ tells you nothing without knowing what the same estimator returns on synthetic noise of the same length.

What would settle it

Detrended fluctuation analysis with the Anis-Lloyd small-sample correction and a bootstrap confidence interval rather than a point value. If the interval excludes $0.5$, the memory is real.

Finding

Terms

Scope

Applies to every screen across instruments, parameters, lookbacks and sessions. It is the same problem as the trial-count problem in validation, arriving through hypothesis testing rather than through strategy search.

The arithmetic

At the conventional $\alpha = 0.05$, twenty independent tests give $\alpha_{\mathrm{fam}} \approx 0.64$. At one hundred tests it is $0.994$. Running a hundred comparisons and reporting the one that cleared the threshold is not a discovery. It is arithmetic behaving exactly as specified.

What is used instead

Controlling the family-wise rate outright is too conservative when hundreds of related signals are screened, because it destroys the power to find anything. The project controls the false discovery rate instead, which bounds the expected proportion of false positives among the rejected hypotheses.

Where the underlying series are autocorrelated, the standard errors feeding the tests are computed with a heteroskedasticity and autocorrelation consistent estimator, since otherwise the p-values are wrong before the correction is even applied.

Finding

Terms

Scope

Applies to any correlation or covariance matrix estimated from a finite sample, which in practice means every one. The result is exact in the limit where $N$ and $T$ both grow with $q$ fixed, and holds well enough at realistic sizes.

The claim

Take $N$ series of pure independent noise, each of length $T$, and compute the correlation matrix between them. It will not be the identity. It will have a spread of eigenvalues, and that spread is not an accident of your particular sample. It has a known shape, bounded above and below by $\lambda_{\pm}$.

The consequence is uncomfortable. Any eigenvalue of your real correlation matrix that falls inside that band is indistinguishable from noise. Not weak evidence of structure. No evidence at all.

The arithmetic that should worry you

Fifty instruments and two years of daily data gives $N = 50$, $T = 500$, so $q = 0.1$ and the noise band runs from about $0.47$ to about $1.73$. Most of the eigenvalues of a real correlation matrix over that sample sit inside it.

Push to $N = 200$ over the same period and $q = 0.4$, with the band running to roughly $2.65$. At $N > T$ the matrix is singular outright and cannot be inverted at all, which is a problem for anything requiring the inverse covariance, which is most of portfolio construction.

What it is used for here

Eigenvalues inside the band are clipped to their average before the matrix is reused, which leaves the handful of genuine modes and discards the rest. The largest eigenvalue almost always sits far above the band and corresponds to the market moving together, which is real and not noise.

Where it came from

Wigner introduced the approach to model energy levels in heavy atomic nuclei, where the exact Hamiltonian was unknown and unknowable, so he asked what a random one would look like. The question turns out to be the same question.

Finding

References

• Marchenko, V. and Pastur, L. Distribution of eigenvalues for some sets of random matrices, 1967.
• Laloux, L., Cizeau, P., Bouchaud, J.-P. and Potters, M. Noise dressing of financial correlation matrices, Physical Review Letters, 1999.
• Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions, 1955.

Terms

Scope

Applies wherever a distribution must be assumed and only partial information is available, which covers most of the modelling on this site. Does not licence assuming a distribution when the information to constrain it is absent.

The principle

Of all distributions consistent with what you actually know, choose the one that assumes the least beyond it. Any other choice smuggles in information you do not have.

The results fall out cleanly. Constrain nothing and you get the uniform distribution. Constrain the mean and variance on the real line and you get the Gaussian. Constrain the mean on the positive half line and you get the exponential. Each of these is the honest answer given the stated constraints, and each becomes a dishonest answer the moment the constraints are asserted rather than measured.

Why it matters here

The Gaussian is not a law of markets. It is what maximum entropy returns when you have only claimed to know a mean and a variance. Financial returns very obviously have more structure than that in the tails, so the Gaussian is not wrong because markets are strange. It is wrong because the constraints fed to it were too few.

What is under test

Whether adding tail constraints, or moving to a maximum entropy solution on a heavier-tailed family, produces a distribution that holds out of sample better than fitting a heavy-tailed family directly. Two routes to the same place, and they do not obviously agree.

Finding

References

• Jaynes, E. T. Information theory and statistical mechanics, Physical Review, 1957.

Terms

Scope

Applies to any forecast expressed as a probability of a binary event, which covers rainfall, elections, defaults, epidemics, equipment failure and match results identically. Does not apply to point forecasts, which cannot be scored honestly at all.

Why it exists

A forecast of seventy per cent is neither right nor wrong when the event happens. It can only be judged across many forecasts, and only against a rule that cannot be gamed. Glenn Brier proposed this one in 1950 for weather, and it remains the standard because of a property that is easy to state and easy to underrate.

The property that matters

The Brier score is $\textit{proper}$. Your expected score is best when you state exactly what you believe. Say ninety when you believe seventy and you will do worse in the long run, not better. Most intuitive scoring schemes fail this, and any scheme that fails it is quietly paying people to exaggerate.

What it decomposes into

Murphy showed the score splits into three parts. $\textit{Reliability}$ is whether seventy per cent means seventy per cent. $\textit{Resolution}$ is whether you discriminate between cases at all. $\textit{Uncertainty}$ is the difficulty of the question and belongs to nobody.

That decomposition settles a common confusion. A forecaster who says fifty per cent to everything is perfectly reliable and completely useless, because resolution is zero. Reliability alone is not skill.

The reference point

A raw score is meaningless without a baseline. Compare against the base rate. If a model cannot beat simply predicting the historical frequency, it has no skill regardless of how impressive the number looks.

Why it sits at the centre of this project

It is the only honest way to answer whether any prediction about anything was any good, and it applies unchanged across every domain listed on the front page. Almost nothing outside meteorology does this properly.

Finding

References

• Brier, G. W. Verification of forecasts expressed in terms of probability, Monthly Weather Review, 1950.
• Murphy, A. H. A new vector partition of the probability score, 1973.
• Gneiting, T. and Raftery, A. Strictly proper scoring rules, prediction and estimation, 2007.

Terms

Scope

Applies to every result on this site without exception, and retroactively, which is the uncomfortable part. A number recorded before the trial count was tracked cannot be deflated afterwards, because the count is unrecoverable.

The claim

If $N$ candidate models are tested and every one of them is pure noise with no true skill at all, the best will still show a positive Sharpe ratio, and the expected size of that false result grows with $\sqrt{2\log N}$.

The practical form is blunt. Testing roughly forty-five configurations on five years of daily observations is close to guaranteed to produce something that scores as well as a genuinely skilled forecaster in sample and no better than chance out of it.

Why the sample is smaller than it looks

Roughly twenty-five thousand hourly bars over four years sounds ample, but overlapping features destroy independence. A two-hundred-bar moving average means each observation shares one hundred and ninety-nine bars with its neighbour. Effective sample size falls to somewhere between five hundred and two and a half thousand, and it is the number of trades rather than bars that determines power.

What follows

Record the trial count before searching, not after. Deflate for trials, skewness, kurtosis and sample length. Use combinatorial purged cross-validation rather than a single split. Treat $t > 3.0$ rather than $t > 2.0$ as the threshold for a new discovery.

Finding

References

• Bailey, D. and López de Prado, M. The deflated Sharpe ratio, Journal of Portfolio Management, 2014.
• Harvey, C., Liu, Y. and Zhu, H. On the threshold for new factor discovery, 2016.
• López de Prado, M. Advances in Financial Machine Learning, 2018.