Two systems sit in your notes, both tested, both sound. One buys breakouts and needs moves to continue. One buys pull-backs and needs moves to reverse. Eleven names sit in your watchlist, and the way you have been assigning systems to names is by opening the chart and forming an impression — which, as an earlier lesson in this track demonstrated with a coin, is a procedure that cannot distinguish a trending stock from a random one. Both systems are bets on the same property of a price series, and that property has a name and can be counted in about ten minutes.
In some places a wet day makes the next day more likely to be wet — the system has parked overhead and it rains for four days. In others a heavy day empties the sky and tomorrow is clear. Both places have the same annual rainfall and the same average day. If you are deciding whether to carry an umbrella tomorrow, the average is worthless and the question "does yesterday tell me anything about tomorrow" is the only one that matters. You would answer it by counting pairs of days, not by looking at a photograph of the sky.
Counting pairs of sessions is exactly the measurement. It says nothing about whether the stock goes up, which is a different question and a harder one. It says whether the moves stick together — and that is precisely what decides which of your two systems has any chance in this name.
Measurement one: count the pairs
For every session, note the sign of that day’s return and the sign of the next day’s. There are four combinations, and the count fits in a spreadsheet column and a pivot. Continuations are up-then-up plus down-then-down. Reversals are the two mixed cells. This is the sign version of [[Autocorrelation]] at a lag of one day, and it is more honest than a correlation coefficient because you can see how many observations produced it.
| Today ↓ / tomorrow → | Up | Down | Row total |
|---|---|---|---|
| Up | 141 | 121 | 262 |
| Down | 118 | 120 | 238 |
| Column total | 259 | 241 | 500 |
Measurement two: the variance ratio
The second measurement uses more of the data and is the one worth running properly. If daily moves are independent, variance accumulates in proportion to time, so the standard deviation of a five-day move should be √5 — about 2.24 — times the standard deviation of a one-day move. That is not a law; it is the assumption inside every √252 you have ever used. So compute the five-day standard deviation directly and compare it with the prediction. The [[Variance ratio]] is the square of the ratio of those two standard deviations.
Three ways this measurement lies, all of them avoidable
- Overlapping windows destroy your sample without changing your row count. If you compute a five-day return starting on every single day, consecutive observations share four of their five days, so they are very nearly the same observation twice. Five hundred sessions give you 100 genuinely independent five-day blocks, not 496 — and at a twenty-day horizon they give you 25, from which nothing may be concluded about anything. Use non-overlapping blocks when you want to count observations honestly, and remember that a longer horizon costs you sample size at exactly the rate it lengthens the horizon.
- The bid-ask bounce fakes a negative answer. From the first lesson in this module: in a wide-spread name, consecutive prints alternate sides, which manufactures reversal whatever the stock is doing. Any thin scrip will report a variance ratio below one and a reversal-heavy sign table. Run this on liquid names, on official closes, or you are measuring the spread and calling it behaviour.
- The answer describes the window, and windows end. Persistence is not a permanent attribute of a company. A name that trended through one bull phase reverted through the range that followed, and the measurement taken across both reports the average of two behaviours, which is a third behaviour that never occurred. Recompute on a rolling basis, and treat a change in the answer as information rather than as noise.
- 1Get clean daily closes, and check the spread first
Official closes, at least 500 sessions, and a look at the best bid and offer. If the spread is a large fraction of a daily range, stop here — the measurement will come back "reverts" and it will be telling you about the market maker, not the stock. And note from the first lesson that in a name thin enough to worry about, the official close may itself be nothing more than the last print, so it is the spread check rather than the choice of close that protects you.
- 2Build the four-cell sign table and compute the standard error
The continuation percentage, and √(0.25 ÷ n) beside it. If the continuation rate is inside two standard errors of 50%, write "no signal" and mean it, rather than reporting the number to one decimal place as though it were a result.
- 3Compute the variance ratio at 2, 5, 10 and 20 days, non-overlapping
Four numbers, one line each. Record the number of independent blocks next to each, because that number is what tells you whether to believe the last one.
- 4Write the shape, not the verdict
A single line in your notes — "reverts at 1–2 days, mildly trending at 20, thin sample beyond that" — is worth more than a label, because it tells you which system belongs on the name and at which horizon the entry has to be taken.
- 5Recompute quarterly, and keep the old answers
The series of answers is itself information: a name whose ratio has been drifting from 1.4 towards 0.9 over four quarters is telling you something the current number alone cannot.
Someone reports that a stock continued its direction on 52.2% of 500 consecutive sessions and concludes that it is a trending stock suited to a breakout system. What is wrong with that?
Kuch jagah ek geela din agle din ko bhi geela bana deta hai — badal sar pe aake baith gaye aur chaar din barasta hai. Kuch jagah bhaari barsaat aasman khaali kar deti hai aur agla din saaf. Dono ki saal bhar ki barsaat ek jaisi ho sakti hai. Kal chhatri leni hai ya nahi — us sawaal mein average bekaar hai; matlab sirf itna rakhta hai ki aaj kal ke baare mein kuch batata hai ya nahi. Aur uska jawab jodiyan gin ke milta hai, aasman ki photo dekh ke nahi. Chart pe wahi: aaj ke return ka sign, kal ke return ka sign, chaar khaane. 500 session pe ek midcap — 141 upar-upar, 121 upar-neeche, 118 neeche-upar, 120 neeche-neeche. Continuation = 141 + 120 = 261, yaani 52.2%, toh lagta hai stock trending hai. Ab imaandari: aadhe ke aas-paas kisi proportion ka standard error √(0.25 ÷ n) hota hai — n = 500 pe 2.2 percentage point. Matlab 52.2% sikke se sirf ek standard error door hai, aur bina kisi persistence waali series bhi aisa nateeja teen mein ek baar de degi. Sach mein do point ka edge saabit karna ho toh n 2,500 se oopar chahiye — das saal ek hi share ka, aur das saal mein woh share ek hi tarah behave nahi karega. Doosra naap zyada kaam ka hai: agar din aazaad hote toh paanch din ka σ rozana ka √5 = 2.24 guna hona chahiye. Rozana 1.4% pe woh 3.13% banta hai. Naap ke 3.90% mila? Ratio 1.25, variance ratio 1.55 — din ek doosre ko aage badha rahe the. 2.50% mila? Ratio 0.80, variance ratio 0.64 — din ek doosre ko kaat rahe the. Do cheezein dhyaan mein: patle share pe yeh naap hamesha "reverts" bolega, kyunki woh spread ka bounce hai; aur overlapping window mein 500 session ke 496 nahi, sirf 100 aazaad paanch-din block hote hain.
- A breakout system bets that persistence is positive; a pull-back system bets it is negative. Persistence is measurable.
- The sign table at lag one is the readable version: continuations against reversals, with the count visible.
- If independence held, a five-day σ would be √5 times the daily one. Compare them: above 1 means moves reinforced, below 1 means they cancelled.
- The standard error near a half is √(0.25 ÷ n) — 2.2 points at 500 sessions, so most results of this kind are indistinguishable from a coin.
- Overlapping windows leave your row count intact and gut your sample: 500 sessions give 100 independent five-day blocks, not 496.
- Use the answer to rule a system out of a name, never to promise that a name will keep behaving as it did.
Mark it done to track your progress through the curriculum.