Pooled panel regression turns eight shows and five years into forty rows

Every show in the portfolio has five annual numbers and every analyst who has tried to model one of them has run into the same wall. Five points, no degrees of freedom, no interval worth publishing.

Pooled panel regression is the move that changes the arithmetic. Stack every show-year as its own row and eight shows with five editions each becomes a table of forty observations, with a column for the show, a column for the year and however many covariates you can defend. The estimation problem stops being hopeless and becomes merely tight.

Forty rows, and what the structure costs

The table is the easy part. What people underestimate is how much of it the fixed effects consume before any question gets answered.

A two-way specification has a show effect for each of the eight shows and a year effect for each of the five years. Written with a grand mean, that is seven show deviations and four year deviations, so twelve parameters in total. Forty rows minus twelve leaves twenty eight residual degrees of freedom.

Twenty eight is workable. It is enough that a t statistic behaves roughly the way people expect and enough that a standard error means something. It is not enough to support a specification with a dozen covariates and a set of interactions, and the count is the discipline: four or five substantive covariates is the ceiling on a portfolio this size, and every one past that widens every interval in the model.

Baltagi's Econometric Analysis of Panel Data, first published by Wiley in 1995 and revised through several editions since, is the standard textbook treatment of the estimator and of the variants that sit around it. The mechanics are not in dispute. What matters for a show portfolio is which of them survives at forty rows.

What can a show fixed effect actually absorb?

Everything about a show that does not change between editions, and nothing about a show that does.

That is the whole rule, and it disposes of most of the covariates a team wants to put in the model. Hall capacity, if the show runs in the same hall every year, is a constant within that show and therefore perfectly collinear with the show's own dummy. The regression will drop it, or the software will report it as not estimable, and someone will spend an afternoon assuming they made a coding error.

The same applies to the vertical, the country, the ownership history, the brand age and whether the show is co-located with a conference. All of them are absorbed. The show effect is a single number that says this show is bigger or smaller than the portfolio average by a fixed amount, and it swallows every permanent explanation of why.

What survives is variation within a show over time. Whether this edition changed hall. How many weeks the registration campaign ran. Whether the show moved in the calendar. The list price of a delegate pass. Whether the venue was under refurbishment. These are the covariates a fixed effects panel can actually speak about, and there are fewer of them than most model specifications assume.

The compensation is that the show effect is doing real work. A portfolio containing a 4,000 attendee show and a 40,000 attendee one has an enormous amount of variation that has nothing to do with anything you want to measure, and absorbing it into eight numbers is what makes the remaining twenty eight degrees of freedom informative.

The two-way model, worked by hand

Take the portfolio means by year: 14,200 registrations in 2022, 15,100 in 2023, 14,800 in 2024, 16,000 in 2025 and 16,700 in 2026. The grand mean of those five is 76,800 divided by 5, which is 15,360.

The year effects are the deviations: minus 1,160 for 2022, minus 260 for 2023, minus 560 for 2024, plus 640 for 2025 and plus 1,340 for 2026. That sequence is the shared cycle, and it is the thing no single show can estimate on its own because a single show has one observation per year.

Now take show A, whose five editions ran at 12,100, 12,800, 12,400, 13,600 and 14,300. Its own mean is 65,200 divided by 5, which is 13,040, so its show effect is 13,040 minus 15,360, which is minus 2,320.

The fitted value for show A in 2024 is the grand mean plus the show effect plus the year effect: 15,360 minus 2,320 minus 560, which is 12,480. The actual was 12,400, so the residual is minus 80.

That is the two-way fixed effects prediction in three additions, and it is worth doing by hand once because it shows what the model believes. It believes show A is 2,320 registrations below the portfolio average in every year, and that 2024 was 560 below trend for everybody. A residual of minus 80 says nothing else happened to show A in 2024 that the portfolio did not also experience.

Any covariate you add is then explaining residuals of that size. If your candidate variable cannot plausibly move a show by more than a hundred registrations, it will not be distinguishable from noise at twenty eight degrees of freedom, and putting it in costs you a degree of freedom for nothing.

Should you cluster the standard errors on eight shows?

You should want to, because show-year errors are correlated within a show, and you should be uneasy about the result.

A clustered variance estimator is itself estimated from the number of clusters. With eight shows there are eight cluster-level quantities feeding it, and the estimate will typically come back too small, which makes every t statistic in the output look better than it is. The failure is silent, and it is the single most common way a panel result on a small portfolio ends up overstated.

The defensible options are to report the conventional standard errors alongside the clustered ones and note that both are suspect, or to use the number of shows as the degrees of freedom for the t distribution rather than the number of rows. Eight clusters gives seven degrees of freedom, where the two-sided 95 per cent multiplier is 2.36 instead of the 2.05 you get at twenty eight. That widens every interval by about 15 per cent, which is a modest price for not misleading the room.

What breaks when the panel has five years

The temptation with a forecasting objective is to add last year's registrations as a regressor. This is where the small sample stops being a nuisance and becomes a bias.

Nickell showed in Econometrica in 1981 that the within-group estimator applied to a model with a lagged dependent variable and fixed individual effects is inconsistent as the number of individuals grows, if the number of time periods is held fixed, and that the inconsistency is of order one over T. With T of 5, one over T is 0.2. A coefficient on the lag that you would want to interpret as persistence is being estimated with a bias of roughly that order, and adding more shows does not help, because the bias is in the time dimension.

The practical consequence is that a dynamic panel on five editions is not a forecasting model you should defend. Either drop the lag and accept a static specification, or accept that the persistence estimate is directional and stop reading its size.

The other thing that breaks is the year effect itself when you want to forecast forward. Year effects are estimated for years you have observed. The next edition has no year effect, and the model has no opinion about what it will be. You have to supply it, either by extrapolating the sequence, which on five points is the short series problem all over again and O11's subject, or by treating it as a scenario input that the commercial team sets.

Where this stops

A panel model with show fixed effects assumes the show effects are constant. Over five editions, in an events business, they frequently are not.

If a show moved venue in year three, its true effect is one number for the first two editions and a different number for the last three, and the fitted single effect is an average that describes neither period. The residuals will show it, as a run of positive residuals followed by a run of negative ones, and the fix is either a break dummy, which costs a degree of freedom, or a shorter estimation window, which costs rows. That choice belongs to O31 and O32.

The deeper limit is that fixed effects treat the show effects as things to be estimated separately, with no relationship between them. A show with an unusual five years gets an unusual effect and nothing pulls it back. Treating those effects instead as draws from a distribution, so that each one is shrunk toward the portfolio in proportion to how little data supports it, is the multilevel alternative, and borrowing strength across shows covers it in O14. On forty rows the difference between the two is usually small for the big shows and large for the newest one.

If what you want is a forecasting engine rather than an interpretable set of coefficients, the panel is also the wrong shape, because it has one row per show-year and throws away every weekly snapshot inside the campaign. Fitting across all of that at once is a global forecasting model and O15's territory.

Build the table this week. One row per show per edition, columns for show, year, final registrations, and any variable you can name that changed within a show between editions. Then count how many of your candidate covariates actually vary inside a show. If the answer is one or two, the panel will still be worth running, and you now know in advance what it can and cannot tell you. What a system needs to hold to keep that table current sits with the forecasting methods behind it.