CSAD herding, step by stepCSAD হার্ডিং, ধাপে ধাপে

What CSAD measures

On any trading day, some stocks go up, some go down, some barely move. Cross-sectional absolute deviation is one number for the whole market that day: how far, on average, each stock's return sat from that day's own market return. Take the market's return for the day, subtract it from every stock's own return, ignore the sign, and average the result across all the stocks in the sample. A small CSAD means the market moved as one; a large CSAD means today was a day of very different stories across different names.

The regression, and a negative squared term

The standard test regresses the day's CSAD on the day's absolute market return and on that same number squared: CSAD equals a constant, plus one slope on the absolute market return, plus a second slope on its square, plus noise. In an ordinary market, where each stock reacts to a big market move roughly in proportion to its own beta, dispersion should keep growing, at least as fast as the market's move, as that move gets bigger: a high-beta stock and a low-beta stock have more room to pull apart on a nine per cent day than on a two per cent one. Herding says the opposite happens. On the days the market swings hardest, if enough investors are following the index, or following each other, rather than their own read of a stock's own news, returns bunch up around the market's own number instead of spreading further apart. CSAD then grows more slowly than the straight-line part of the regression predicts, or turns down altogether, at the largest market moves. That shows up as a negative coefficient on the squared term, and it has to be a coefficient that clears the usual bar for significance before it is read as herding rather than as noise.

A negative squared term says dispersion grew more slowly than the size of the market's move predicts. It does not, by itself, say why: crowding, a shared piece of news reaching every stock at once, and a data problem in how the market return was built can all leave the same sign behind.

1. Collect every stock's own return for each day, and enough stocks to average across.
2. Compute the market's return for that day as the average across the same stocks.
3. Compute that day's CSAD: the average absolute distance of each stock's return from the market's.
4. Regress the day-by-day CSAD on the day's absolute market return and on its square.
5. Read the sign of the squared term, and whether it clears the bar for significance.

Worked example: five stocks, three days

Day 1: the market return is the average of the five, 2 plus 3 minus 1 plus 4 plus 2, which is 10, divided by 5, which is 2.0 per cent. Each stock's distance from that, ignoring sign: 0, 1, 3, 2 and 0. Add them: 6. Divide by 5: CSAD is 1.2 per cent.

Day 2: the average of -3, -5, -2, -4 and -6 is -20 divided by 5, which is -4.0 per cent. Distances: 1, 1, 2, 0 and 2. Sum 6, divide by 5: CSAD is again 1.2.

Day 3: the average of 8, 10, 9, 11 and 7 is 45 divided by 5, which is 9.0 per cent. Distances: 1, 1, 0, 2 and 2. Sum 6, divide by 5: CSAD is 1.2 a third time.

The market moved four times as far on day 2 as on day 1, and it moved more than four times as far again on day 3, yet dispersion did not budge. In an ordinary market that would be strange: a bigger market move should give a high-beta stock and a low-beta stock more room to disagree, not the same room every time. Three flat readings against a market move that keeps growing is exactly the shape a negative and significant squared term is built to catch: stocks huddled around the market's own number on the days it would have made most sense for them to spread out.

Up days and down days

Herding is not necessarily the same size, or even the same sign, in a rising market and a falling one. Investors crowding into the same trades out of fear on the way down is a different behaviour from crowding into the same winners on the way up, and a regression run on the whole sample together can average the two into something neither day actually looked like. Split the sample by the sign of that day's market return and run the same regression on each half separately: an up-day squared term and a down-day squared term, each with its own sign and its own significance, rather than one number standing in for both.

Thin market cautions

On a market where several counters go a whole session without a trade, a stale price is recorded as a zero return whether the true price moved or not. On a day the market itself swings hard, the stocks that did not trade sit at zero while the ones that did trade move with the market, which narrows the measured spread for a reason that has nothing to do with anybody following anybody. That narrowing can manufacture the same negative squared term a real herding test is looking for, out of missing trades rather than crowd behaviour. Checking how many of the day's stocks actually traded, and reporting it alongside the regression, is part of the test, not an afterthought to it.

Where it goes wrong

• Reading a negative squared term as herding with no check on how many stocks traded that day.
• Running the test on too few days and reporting a coefficient as though it had cleared a bar for significance.
• Skipping the up-day and down-day split and reporting one number for a market that behaves differently in each direction.
• Computing the market return from a different set of stocks than the ones CSAD is measured against.
• Reporting the sign of the squared term with no mention of how large the coefficient is.

Checklist

• CSAD and the market return are computed from the same set of stocks.
• The regression has enough days behind it for a standard error to mean something.
• The up-day and down-day halves are reported separately, not only the whole sample.
• The share of stocks that actually traded that day is checked before a thin day is trusted.
• Herding is claimed only where the squared term is both negative and significant, not one without the other.

The lab runs the CSAD regression from the method list, on daily returns for as many stocks as are entered, and reads its own squared term's sign as herding or no herding in the summary line; its market button and the workshop's Returns tool turn a run of daily prices into the returns this test needs before the first CSAD can be computed.