Moving block bootstrap for time-series statistics

Puts uncertainty intervals on anything computed from a weekly panel – coefficients, contributions, average or marginal ROI – by resampling blocks of consecutive rows, refitting, and recomputing. Resampling single rows would destroy the autocorrelation that carryover creates and make the intervals too narrow; resampling blocks keeps it within each block (Kunsch, 1989).

Usage

block_bootstrap(
  data,
  statistic,
  times = 200L,
  block_length = NULL,
  by = NULL,
  level = 0.9,
  seed = NULL
)

Arguments

data: A data frame, rows in time order (within each group when by is supplied). It should hold everything statistic needs: typically the KPI, the transformed media, the controls and the raw spend.
statistic: A function of one data frame returning a named numeric vector. It is called once on data for the point estimate and once per replicate on a resampled copy.
times: Number of bootstrap replicates.
block_length: Rows per block. The default, ceiling(n^(1/3)), is the usual rate-optimal order for the moving block bootstrap; for weekly media data a block of at least the carryover's effective window is a sensible floor.
by: Optional character vector of grouping columns. Blocks are then drawn within each group, and each group keeps its own size, so a panel of geographies is resampled as a panel.
level: Confidence level of the percentile intervals.
seed: Optional random seed, for reproducible intervals.

Value

An object of class mm_bootstrap: a data frame with one row per element of the statistic – term, estimate (on the original data), lower, upper (percentile interval) and std_error – with the replicate matrix attached as the attribute "replicates" (one row per replicate, one column per term). Replicates in which statistic failed are dropped and counted in a message.

Details

Transform the media before bootstrapping and put the transformed columns in data. Resampled blocks are glued end to end, so re-running the adstock filter on resampled rows would carry spend across block boundaries that were never adjacent in time; the transformed columns already hold each week's correct carryover.

The interval reflects sampling variation given the model specification, including the transform parameters. It does not include the uncertainty in choosing those parameters unless statistic re-tunes them, which is possible but slow.

References

Kunsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. The Annals of Statistics, 17(3), 1217–1241.

Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer.

Examples

data(mm_weekly)
north <- mm_weekly[mm_weekly$geo == "north", ]
channels <- c("tv", "video", "search", "social", "display")
truth <- attr(mm_weekly, "truth")

# Transform once, on the full series, then bootstrap the model on top.
media <- as.data.frame(Map(function(x, d, h, s) media_transform(
  x, adstock = list(decay = d), saturation = list(half_max = h, shape = s)),
  north[channels], truth$decay[channels], truth$half_max[channels],
  truth$shape[channels]))
names(media) <- paste0(channels, "_t")
d <- cbind(north, media, week = seq_len(nrow(north)))

average_roi <- function(d) {
  fit <- lm(revenue ~ tv_t + video_t + search_t + social_t + display_t +
              week + price + seasonality + holiday, data = d)
  contrib <- colSums(sweep(as.matrix(d[paste0(channels, "_t")]), 2,
                           coef(fit)[paste0(channels, "_t")], `*`))
  stats::setNames(contrib / colSums(d[channels]), channels)
}

boot <- block_bootstrap(d, average_roi, times = 50, block_length = 13,
                        seed = 1)
boot
#> 
#> ── Block bootstrap 
#> 50 replicates, blocks of 13 rows, 90% percentile intervals
#>     term estimate    lower upper std_error
#>       tv    1.310  1.10874 1.882    0.2579
#>    video    1.924  0.01862 3.495    1.1227
#>   search    2.794  1.58295 4.258    0.8991
#>   social    3.614  0.51808 5.213    1.4353
#>  display    1.517 -0.03334 5.217    1.6852