Context
The 2026 upgrade adds uncertainty to figures that are a census. CBS reports every machine and every dollar, so there is no sampling error in the textbook sense. The useful questions are about variation: is a change bigger than month-to-month noise, is a council’s rate further from the state’s than its size would explain, did a trend really turn. Each method below had real alternatives, and the choices change the numbers.
Decision
Interrupted time series. Monthly NGR from July 2015 to June 2025 in average FY 2024/25 dollars, leaving out March to June 2020. The model has a linear trend, a level change and a slope change from July 2020, and eleven calendar-month effects. Standard errors are Newey–West with the rule-of-thumb lag floor(4 (n/100)^(2/9)) = 4, and intervals use t on n − 15 degrees of freedom. Three sensitivity specifications (nominal dollars, short lockdowns left out, NGR per machine) are shown next to it.
Councils. Each area’s NGR per machine is divided by the state rate of the same year, and the funnel limits are the state rate × exp(± z · c / √machines). The scale c is pooled from every area’s year-to-year variation, and a second set of limits adds Spiegelhalter’s between-area variance τ². Each area’s typical ratio is a geometric mean across years with a t interval whose standard error is widened for year-to-year dependence by √((1 + ρ) / (1 − ρ)), with ρ the pooled lag-1 autocorrelation of the log ratios within areas. The count of areas outside the limits is compared with the 5% the limits allow for by chance; it gets no confidence interval, because the areas are every published area, not a sample.
Concentration. A continuous two-segment line through the monthly HHI, with the break found by least squares (at least 24 months each side) and a moving-block bootstrap (1,000 resamples) for the break and both slopes. The block length is chosen from the residuals with Politis and White’s automatic rule (20 months), and the page shows the break’s interval for 6-, 12- and 24-month blocks as well. Annual HHI and NGR per machine get percentile bootstrap intervals over months (2,000 resamples).
Options considered
- Nominal or real dollars for the break. Nominal keeps the published figures, but post-2021 inflation then looks like a steeper post-COVID trend. Real is primary; nominal is shown as a sensitivity row.
- Bootstrap or Newey–West for the regression. A block bootstrap would also allow for autocorrelation, but Newey–West is standard, has a closed form, and can be checked exactly against statsmodels.
- Funnel variance: Poisson-like, additive or proportional. NGR isn’t a count, so Poisson limits don’t apply. A check across 44 areas showed that the standard deviation of an area’s log ratio falls with size at a slope of −0.51 (95% CI −0.68 to −0.34), close to the −0.5 that proportional (log-scale) variance implies, so the limits are on the log scale.
- Funnel scale from the cross-section or from years. Estimating c from the spread between councils would make about 5% of areas fall outside by construction. Estimating it from each area’s own year-to-year movement gives limits that mean something independent of the cross-section.
- Block length: a rule of thumb or the data. The usual n^(1/3) gives 6-month blocks, which is quick and needs no estimate, but it assumes the dependence dies out within a few months. The HHI residuals are far more persistent than that, so the block length is estimated from them (Politis and White 2004, with the 2009 correction; checked against the Python
archpackage). - Councils’ years: independent or dependent. A plain t interval across an area’s years is simple, but an area above the state rate one year tends to stay there, so its years are not independent. A Newey–West standard error per area would need more years than most areas have; one pooled lag-1 autocorrelation and the AR(1) widening factor is cruder but stable.
- Change point: mean shift (CUSUM, binary segmentation) or broken stick. The HHI series falls and then rises, so a mean-shift method would cut a trend into steps. A broken stick asks the actual question: when did the direction change.
Why
Each choice answers the question the page asks, can be explained in a sentence, and can be checked against a reference implementation (statsmodels for the regression, numpy for the funnel formulas and the grid search, R for STL).
What happened
The primary interrupted time series puts the level change at reopening at +$9.3m a month (95% CI +$5.8m to +$12.8m) and the trend change at +$3.4m a year (+$2.2m to +$4.6m). The direction holds in every specification, but the size moves: leaving out the two lockdown months raises the level change to +$11.0m, and in nominal dollars the trend change is +$5.5m a year. The residuals keep a lag-1 autocorrelation of 0.39 (0.60 when the lockdown months are left out), which is what Newey–West is for, but the lag-4 window may still be short for that much persistence.
In the councils, 39 of 48 areas sit outside the year-to-year 95% limits in FY 2024/25, so the funnel mostly shows that council differences are lasting rather than noise. With the between-area variance added, 9 areas are outside, all of them below the state rate. That is useful, but the second set of limits partly re-estimates what it then tests, so I describe it as “unusual among councils”, not as a significance test. The first version put a Wilson interval on the 39 of 48; review pointed out that the areas are a census, so the count is now compared with the 2.4 that chance alone would put outside the limits.
The first version also treated each area’s years as independent and found 15 areas consistently above the state rate and 37 below, out of 55 with three or more years. Within areas, the lag-1 autocorrelation of the log ratios is 0.43 (365 pairs of consecutive years in 44 areas with four or more years; the estimate from such short series is biased towards zero). Widening every standard error by 1.58 leaves 14 above, 34 below and 7 unclear. The paired calendar-month comparisons on the Trends page still treat their twelve monthly differences as independent, and the page says so.
The HHI breaks in December 2015 (95% CI August 2015 to June 2016): −224 points a year before, +36 after. The first version used 6-month blocks and reported September 2015 to March 2016. Review found that the residuals’ autocorrelation is 0.96 at lag 1, 0.72 at lag 6 and still 0.33 at lag 12, so 6-month blocks broke the dependence up and the interval was too narrow. With 12-, 20- and 24-month blocks the interval is 8, 10 and 11 months wide instead of 6. The break itself does not move. The interval also treats the two-line shape as given, so it is narrower than an honest “when did it turn” interval would be under a smoother model.
What I’d change
I would fit the interrupted time series with an explicit ARMA error model as a second check on the Newey–West intervals, given the residual autocorrelation. For councils I would add covariates CBS doesn’t publish but the ABS does (population, socio-economic index) to see how much of the spread they explain. For concentration I would compare the broken stick with a smooth spline to show how much the “turning point” depends on the two-line assumption.