Fit a Spatial-X (SLX) model

Estimates a Spatial-X regression of the form $$y = X\beta + WX\theta + \varepsilon,$$ where W is a spatial weights matrix and X includes both the directly-entering regressors and a user-specified subset that is spatially lagged. SLX estimation is OLS on the augmented design matrix, which makes estimation, interpretation, and effects decomposition much simpler than for SAR-family models (Wimpy, Whitten, and Williams 2021).

Usage

slx(
  formula,
  data,
  W = NULL,
  lag = NULL,
  order = 1L,
  spatial = NULL,
  time_lag = 0L,
  id = NULL,
  time = NULL,
  na.action = stats::na.omit
)

Arguments

formula: A standard model formula, e.g. y ~ x1 + x2 + x3.
data: A data frame (or sf object) whose rows correspond, in order, to the rows/columns of W (cross-sectional mode) or contain id and time columns (panel mode).
W: An slx_W object produced by slx_weights(), a numeric or sparse matrix, or - in panel mode - a named list of such objects keyed by time value. Required when spatial is not supplied.
lag: Character vector of variable names from formula that should be spatially lagged. If NULL (default) and W is supplied, all right-hand-side variables are lagged.
order: Integer vector giving the orders of W to include for each lagged variable (e.g. 1:2 adds both Wx and W^2 x). Default 1.
spatial: Optional named list for variable-specific weights matrices. Names must match variables in formula. Each element is either an slx_W, a list of slx_Ws (multiple channels for one variable), a time-keyed list of slx_Ws (single channel, time- varying), or a named list of channels whose values are themselves single or time-keyed slx_Ws.
time_lag: Integer, number of time periods to lag the spatial terms (TSLS). Requires id and time. Default 0.
id, time: Column names identifying the panel unit and period. Required for panel mode or TSLS. time must be numeric.
na.action: How to handle missing values. Defaults to stats::na.omit in cross-sectional mode. In panel mode missing values on formula variables are dropped; id and time must never be missing.

Value

An object of class slx with elements:

fit: The underlying lm object.
formula: The expanded model formula.
call: The original call.
W: The weights matrix (or list thereof) used.
lag_terms: A data frame mapping spatial-lag terms to their source variable, W matrix, order, and time lag.
data: The (possibly augmented) model frame.
panel: Logical flag.

Cross-sectional and panel modes

When id and time are NULL, slx() treats the data as a single cross-section: rows of data must correspond, in order, to the rows and columns of W. When both id and time are supplied, slx() enters panel mode and performs block-wise spatial lagging period by period. In panel mode the weights matrices must have dimnames matching the unit identifiers in data[[id]]; the same units need not appear in every period (unbalanced panels are supported).

Time-varying weights

In panel mode, a W argument can be either (a) a single slx_W object applied to every period or (b) a named list of slx_W objects keyed by the stringified time value (e.g. list("1990" = W_90, "1991" = W_91)). The same options apply to any entry inside spatial.

Temporally-lagged spatial lag (TSLS)

The time_lag argument implements equation 7 of Wimpy, Whitten, and Williams (2021): it lags every constructed spatial term W_t x by time_lag periods within each unit so that the regressor becomes W_t x_{t - time_lag}. The first time_lag periods per unit drop out, mirroring any temporally-lagged variable.

References

Wimpy, C., Whitten, G. D., & Williams, L. K. (2021). X Marks the Spot: Unlocking the Treasure of Spatial-X Models. Journal of Politics, 83(2), 722-739.

Vega, S. H., & Elhorst, J. P. (2015). The SLX Model. Journal of Regional Science, 55(3), 339-363.

Examples

data(defense_burden)
W_c <- slx_weights(style = "custom", matrix = defense_burden$W_contig,
                   row_standardize = FALSE)
fit <- slx(ch_milex ~ milex_tm1 + log_pop_tm1 + civilwar_tm1,
           data = defense_burden$data, W = W_c, lag = "civilwar_tm1")
slx_effects(fit)
#> # A tibble: 3 × 8
#>   variable     w_name type     estimate std.error conf.low conf.high p.value
#>   <chr>        <chr>  <chr>       <dbl>     <dbl>    <dbl>     <dbl>   <dbl>
#> 1 civilwar_tm1 NA     direct     -1.21      0.573    -2.34   -0.0774  0.0364
#> 2 civilwar_tm1 W      indirect    0.133     0.838    -1.52    1.79    0.874 
#> 3 civilwar_tm1 W      total      -1.08      1.02     -3.08    0.930   0.291