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Decompose the change of a share into within-source and composition

Source: R/source_composition.R
decompose_share_change.Rd

Applies the dynamic Olley-Pakes decomposition of Melitz and Polanec (2015) to the aggregate share of each key between two periods (or two windows of periods), with sources in the role of firms.

Usage

decompose_share_change(
  counts,
  key_cols,
  source_col = "source",
  time_col,
  t0,
  t1,
  x_col = "x",
  n_col = "n",
  source_totals = NULL
)

Arguments

counts

data.frame or data.table with one row per key, source and period, holding the count x_col and, when source_totals is NULL, the source denominator n_col.

key_cols

Character vector of key columns.

source_col

Name of the source column (default "source").

time_col

Name of the period column.

t0, t1

Values of time_col of the base and the comparison period; vectors of several values define windows, over which counts and denominators are summed by source. They must not overlap.

x_col, n_col

Names of the count and denominator columns (defaults "x", "n").

source_totals

Optional data.frame with one row per source and period and columns source_col, time_col, n_col (postings of the source in the period, also where the key has no posting). Without it the denominators are read from counts and must be the same for all keys of a source and period.

Value

A data.table with one row per key of counts and columns key_cols, p0, p1, delta, entro, fra, ingressi, uscite, composizione. The attribute fonti lists the sources with their denominators n0, n1 and class classe ("C" continuing, "E" entering, "X" exiting).

Details

The aggregate share is P = sum_s w_s p_s with source weights w_s = n_s / sum(n) and source shares p_s = x_s / n_s. Sources are continuing (C: n > 0 in both periods), entering (E: only in t1) or exiting (X: only in t0). With W_G the weight of group G and P_G its aggregate share, Melitz and Polanec (2015) write

P_1 - P_0 = (P_C1 - P_C0) + W_E1 (P_E1 - P_C1) + W_X0 (P_C0 - P_X0)

and split the change among continuing sources as in Olley and Pakes (1996), with weights renormalised within C (omega_s = n_s / sum_C n) and averages between the two periods (bars):

  • entro = sum_C mean(omega_s) (p_s1 - p_s0): change of the shares within sources;

  • fra = sum_C mean(p_s) (omega_s1 - omega_s0): reallocation of postings among continuing sources;

  • ingressi = W_E1 (P_E1 - P_C1), uscite = W_X0 (P_C0 - P_X0).

The four components sum to delta = p1 - p0 up to rounding error; composizione = fra + ingressi + uscite. Groups without sources contribute 0; when there is no continuing source P_C is set to 0, so that ingressi = p1 and uscite = -p0.

References

Melitz, M. J. and Polanec, S. (2015). Dynamic Olley-Pakes productivity decomposition with entry and exit. RAND Journal of Economics, 46(2), 362-375.

Olley, G. S. and Pakes, A. (1996). The dynamics of productivity in the telecommunications equipment industry. Econometrica, 64(6), 1263-1297.

Examples

counts <- data.frame(
  code = "a",
  source = c("s1", "s2", "s1", "s2", "s3"),
  mese = c(1, 1, 2, 2, 2),
  x = c(10, 5, 12, 20, 30)
)
totals <- data.frame(
  source = c("s1", "s2", "s1", "s2", "s3"),
  mese = c(1, 1, 2, 2, 2),
  n = c(100, 100, 100, 200, 100)
)
decompose_share_change(counts, "code", time_col = "mese", t0 = 1,
                       t1 = 2, source_totals = totals)
#>      code    p0    p1 delta  entro          fra   ingressi uscite composizione
#>    <char> <num> <num> <num>  <num>        <num>      <num>  <num>        <num>
#> 1:      a 0.075 0.155  0.08 0.0375 -0.005833333 0.04833333      0       0.0425