Outliers in two-step benchmarks

What happens without outliers

We use monthly, seasonally and working days adjusted (SA-WDA) turnover index (base 100 in 2015) as an indicator for monthly household actual final consumption in current M euros, benchmarked on the corresponding annual account, known up to 2021¹.

par(mfrow=c(1,2))
plot(turnover_catering)
plot(consumption_catering)

Let’s first use the most basic model, in levels and without constant :

benchmark <- twoStepsBenchmark(turnover_catering,consumption_catering)

We can compare the indicator and the estimated QNA series² :

plot(in_disaggr(benchmark, type="levels-rebased"), start=c(2010,1))

plot(in_disaggr(benchmark), start=c(2018,1))

The timing of Covid restrictions is clearly apparent on both the indicator (HF serie) and the benchmarked account (benchmark), with a first lockdown between March and May 2020, a second lockdown in November 2020, followed by a curfew that had an as strong impact on the consumptions of hotel and restaurant services, up to June 2021. In January 2022, the drop of activity is due to the Omicron variant of the Covid-19 virus, with no lockdown but a very large number of people affected and in quarantine at home.

The benchmark is quite good, and the account follows closely the indicator, with scaled-down evolutions. But we can clearly see in the account a decrease in consumption from mid-2019, that is not present in the indicator.

This is an artefact caused by the smoothing of the residual. There is a sharp 6 b€ drop in the annual residual between 2019 and 2020, and the smoothing procedure tries to compute a monthly series with the most gradual drop, thus starting to decrease in May 2019, which is clearly absurd in that case.

par(mfrow=c(1,2))
plot(residuals(benchmark))
plot(smoothed.part(benchmark))

There is thus a significant contribution of the smoothed part to the QNA evolution in 2019-2020 :

plot(in_disaggr(benchmark, type ="contributions"), start=2018)

Introduction of outliers

The idea behind the outlier is to capture this larger than usual residual and to affect it to the correct months and quarters. While we don’t know why there is a larger than usual gap between the indicator and the account, we are pretty sure that it is related to the Covid lockdowns. It is thus a better assumption to try to disaggregate this residual according to a measure of the stringency of the lockdowns rather than smoothing it agnostically over the year 2019 to 2022.

While there is a lot of ways to build this outlier indicator, a convenient way is to derive it from the usual indicator. In this example, we see that the turnover index is close to 130 before and after the lockdowns. Thus the distance to 130 is a good indicator of the intensity of the crisis :

outlier <- window(turnover_catering-130, c(2020,1), c(2021,12))
outlier[c(1,2, 20:24)] <- 0
plot(outlier)

The absolute value of the indicator is lower than 2 in January and February 2020 and after August 2021, so we set it to 0 for those months, restricting the outliers to the lockdowns period.

The idea is to add this outlier to the benchmark relation, as an additive outliers :

\[ C = I * a + out * b + u \]

Where:

• \(C\), an univariate time series, is the high-frequency account.
• \(I\), a matrix time series, combines the columns of the indicators and the constant.
• \(a\), a numeric vector, stands for the coefficients to be applied.
• \(out\) is the matrix time series of the ouliers. Here we have one additive outliers, completed with 0 outside the 2020-2021 period.
• \(b\), a numeric vector, stands for the coefficients to be applied to the outliers.
• \(u\), an univariate time series, is the smoothed part of the benchmark.

The coefficients are estimated at low-frequency with the annual accounts, the annualized turnover index and the aggregated outlier :

\[ C_{aggregated}' = I_{aggregated}' * a + out_{aggregated} * b + u' \]

These coefficients are applied at high-frequency, to obtain the fitted values of the benchmark:

\[ \text{fitted.values} = I * a + out * b \] u is smoothed as usual:

\[ u = smooth(extrapolation(C_{aggregated} - \text{fitted.values}_{aggregated})) \] This can be done using outliers in the twoStepsBenchmark function :

benchmark_out <- twoStepsBenchmark(turnover_catering,consumption_catering, 
                                    outliers = list(AO2020 = outlier))

coefficients(summary(benchmark_out))
#>              Estimate      StdErr   t.value      p.value
#> constant 24179.889019 1540.386318 15.697289 1.038088e-12
#> hfserie     46.720760    1.519744 30.742528 2.598318e-18
#> AO2020       7.964003    2.941739  2.707243 1.356125e-02

In this example, the indicator and the outlier are colinear in the lockdown period, so this is equivalent to estimated a specific a coefficient for the lockdown period, with an account that follow more closely the indicator during that period.

The resulting account reflects better the economic outlook, without contributions of the smoothed part in the end of 2019 :

plot(in_disaggr(benchmark_out, type ="contributions"), start=2018)
plot(in_disaggr(benchmark_out, type="levels-rebased"), start=c(2010,1))

The variation of the residual and the smoothed part are much lower :

par(mfrow=c(1,2))
plot(residuals(benchmark_out))
plot(smoothed.part(benchmark_out))