An event study by handহাতে হাতে ইভেন্ট স্টাডি

Two windows, with a gap between them

An event study needs two stretches of time that do different jobs. The estimation window is a run of ordinary trading days, long enough to fit a market model on, typically well over a hundred sessions, chosen precisely because nothing unusual was happening to the stock during it. The event window is short, often no more than a week either side of the date being tested, and it is where the question actually gets asked. Between the two sits a gap of a few weeks: if the estimation window ran right up to the event, any rumour or run-up in the days before the announcement would already be baked into the fitted alpha and beta, and the model would then judge the event against a version of the stock that had already half-priced it in.

Fitting the market model, in the quiet window

Fitting the model itself is the same regression as any market model: the stock's return on the market's return, over the estimation window rather than the event window, giving a slope, beta, and an intercept, alpha, exactly as worked through on a full example elsewhere in this room. What that regression also leaves behind is the spread of its own residuals, the standard deviation of the gap between what the model predicted and what actually happened, on the ordinary days when nothing was going on. That number is carried forward and used again below, once the event window is reached. Here it is supplied rather than refitted: alpha of 0.05 per cent a day, beta of 1.2, and a residual standard deviation of 0.8 per cent a day, all taken as the estimation window's own output.

Alpha, beta and the residual standard deviation are only as good as the estimation window they came from. A window that itself sat inside a takeover rumour, a previous scandal or an unusually calm quarter for the whole market will hand the event window a model that was never really quiet.

Abnormal and cumulative abnormal returns

An abnormal return is what is left over once the market's own contribution is taken out: the stock's actual return on a day, minus what the fitted line predicts for that day given the market's own return. A predicted return of its own is alpha plus beta times that day's market return, the same market model equation as always, just evaluated one day at a time inside the event window rather than fitted there. Add the day's abnormal returns across the whole event window, running total by running total, and the sum at the end is the cumulative abnormal return, CAR, one number for whatever the event window was worth altogether.

Worked example: a five-day event window

Day -2: predicted return is 0.05 plus 1.2 times 0.5, which is 0.05 plus 0.6, which is 0.65. The actual return was 1.0, so the abnormal return is 1.0 minus 0.65, which is 0.35.

Day -1: predicted return is 0.05 plus 1.2 times -0.3, which is 0.05 minus 0.36, which is -0.31. Actual was 0.0, so abnormal is 0.0 minus -0.31, which is 0.31. Running total, the cumulative abnormal return so far: 0.35 plus 0.31, which is 0.66.

Day 0, the event itself: predicted return is 0.05 plus 1.2 times 0.2, which is 0.05 plus 0.24, which is 0.29. Actual was 3.0, a jump the market's own move of 0.2 per cent does not begin to explain, so abnormal is 3.0 minus 0.29, which is 2.71. Cumulative: 0.66 plus 2.71, which is 3.37.

Day +1: predicted is 0.05 plus 1.2 times 0.4, which is 0.53. Actual 1.5, abnormal 0.97. Cumulative: 3.37 plus 0.97, which is 4.34.

Day +2: predicted is 0.05 plus 1.2 times -0.1, which is -0.07. Actual 0.2, abnormal 0.27. Cumulative, at the end of the window: 4.34 plus 0.27, which is 4.61 per cent.

The t test on CAR

The cumulative abnormal return above is a number; the question is whether it is a large one. The test compares it with the spread the estimation window's own residuals already showed: divide the cumulative abnormal return by the residual standard deviation from that quiet window, times the square root of how many days are in the event window. Here that is 4.61 divided by 0.8 times the square root of 5. The square root of 5 is about 2.236. 0.8 times 2.236 is about 1.789. 4.61 divided by 1.789 is about 2.58. Against the usual two-sided five per cent line of about 1.96, this cumulative abnormal return clears the bar: the stock did more, around this announcement, than five days of ordinary market exposure would explain, by more than chance alone is likely to produce.

Confounding events

The test above assumes the event window's only unusual news is the one being studied. A dividend declared the same week as the earnings figure, or an index rebalancing that happens to fall inside the window, will move the abnormal return too, and the test cannot tell the two apart. The fix is not statistical: check the news wire for every day inside the window, not only the date an announcement was formally dated, before crediting the CAR to the event the study set out to test.

Clustered event dates

The t test above treats the estimation window's residual spread as the whole story of how much a CAR like this one varies by chance, which holds for one firm on one date. It stops holding the moment many firms share the same event date, a national budget, a policy rate decision, a sector-wide reporting deadline, because those firms' abnormal returns then share one market-wide shock and stop being independent draws. Averaging across such a cluster as though each firm were a separate, unrelated event overstates how confident the result should be. Treat the clustered date as one observation, not many: build one CAR for the whole cluster, or adjust the test's variance for the correlation the shared date introduces, rather than running the simple test once per firm and counting each as independent evidence.

Where it goes wrong

• Letting the estimation window run up to the event, so alpha and beta already carry the run-up.
• Reporting a CAR with no t statistic beside it.
• Treating several firms that share one event date as several independent tests.
• Crediting the whole CAR to the event without checking for other news in the same window.
• Widening the event window after seeing the result, until the number looks the way it was expected to.

Checklist

• The estimation window and the event window are separated by a gap, stated in the write-up.
• Alpha, beta and the residual standard deviation all come from the estimation window, never the event window.
• The cumulative abnormal return carries a t statistic, not just a sign.
• The news wire has been checked for the whole event window, not only the announcement's own date.
• Firms sharing one event date are treated as one clustered observation, not several independent ones.

The lab runs an event study from the method list: fit the market model over the estimation window, then feed it the event window's own returns for the abnormal returns, the CAR and its t statistic, written straight into the APA table; its market button fetches the daily series both windows need, so neither is typed in by hand.