Classic Markov chain for distribution dynamics

Discretises a continuous longitudinal variable into ordered classes and estimates a discrete-time, first-order Markov transition probability matrix together with its ergodic (steady-state) distribution. This is the classic distribution-dynamics estimator of Quah (1993): the transition matrix summarises how units move between parts of the distribution from one period to the next, and the steady-state distribution describes where the process settles in the long run.

Input is tidy, long-format panel data with one row per unit-period. The estimator is agnostic to what the units are (regions, firms, households, ...) and to whether space is involved; spatial conditioning is added by a companion spatial_markov() function (available in a later release).

Usage

markov(data, id, time, value, k = 5, breaks = NULL, fixed = TRUE)

Arguments

data

A data frame in long format with one row per unit and period.

id

Name (character scalar) of the column identifying the unit.

time

Name (character scalar) of the column giving the period. Periods are compared in sorted order within each unit.

value

Name (character scalar) of the numeric column to analyse.

k

Integer number of classes to discretise value into. Default 5 (quintiles). Ignored if breaks is supplied.

breaks

Optional numeric vector of interior class boundaries. If supplied these fixed cut points are used instead of data-driven quantiles, giving length(breaks) + 1 classes.

fixed

Logical. If TRUE (default) a single set of quantile breaks is estimated from the pooled distribution and applied to every period (absolute mobility). If FALSE, breaks are re-estimated within each period (relative mobility). Ignored when breaks is supplied.

Value

An object of class sddr_markov: a list with elements

matrix

The k by k row-stochastic transition probability matrix P.

transitions

The k by k matrix of observed transition counts.

steady_state

The ergodic (stationary) distribution of P.

classes

Number of classes k.

breaks

The class boundaries used (when fixed).

n

Number of units contributing at least one transition.

n_transitions

Total number of observed transitions.

data

The input data augmented with the integer state column.

References

Quah, D. (1993). Empirical cross-section dynamics in economic growth. European Economic Review, 37(2-3), 426-434.

See also

steady_state()

Examples

set.seed(1)
df <- data.frame(
  id = rep(1:50, each = 4),
  time = rep(2000:2003, times = 50),
  value = rnorm(200)
)
m <- markov(df, id = "id", time = "time", value = "value", k = 4)
m
#> <sddr> classic Markov chain
#>   units: 50 | transitions: 150 | classes: 4 | breaks: fixed
#> 
#> Transition probability matrix (rows = from, cols = to):
#>       1     2     3     4
#> 1 0.225 0.225 0.275 0.275
#> 2 0.229 0.257 0.229 0.286
#> 3 0.306 0.333 0.111 0.250
#> 4 0.205 0.154 0.333 0.308
#> 
#> Ergodic (steady-state) distribution:
#>     1     2     3     4 
#> 0.240 0.239 0.241 0.281 
m$steady_state
#>         1         2         3         4 
#> 0.2396739 0.2387903 0.2408205 0.2807152