Stop-crime slopes and cross-sectional dependence

Estimate the count mean on an explicit scale

A recorded count and a conditional mean are different quantities. The fixed-effects method defaults to Poisson pseudo-likelihood, modelling expected recorded counts, with clustered covariance. The stop covariate is log(1 + stops). Its coefficient is a log-mean slope per unit of that covariate; at intensity S, the conventional stops elasticity is the slope multiplied by S / (1 + S).

sim <- lamp_simulate(n_areas = 40, n_months = 36, design = "continuous",
                     effect = -0.2, base_rate = 30, seed = 5)
fit <- lamp_elasticity(sim, "affected_crime", lags = 0:3, method = "fe")
fit
#> 
#> ── streetlamp estimate: lamp_elasticity
#> Outcome: affected_crime; family: Poisson pseudo-likelihood on counts
#> Treatment: continuous; clustered by area
#> Identifying assumption: The association between searching and recorded crime,
#> net of area and month effects. Causal only if the variation in searching has a
#> source outside the crime process; see lamp_allocation().
#> Sample: 1320 area-months in 40 areas; 0 rows dropped for coverage.
#>   Term              Estimate  Std. error              95% CI       p
#>   ────────────────  ────────  ──────────  ──────────────────  ──────
#>   Log stops, lag 0    −0.192      0.0157    [−0.223, −0.161]  <0.001
#>   Log stops, lag 1   −0.0239      0.0166  [−0.0564, 0.00865]   0.150
#>   Log stops, lag 2  −0.00125      0.0157    [−0.032, 0.0295]   0.936
#>   Log stops, lag 3  −0.00859       0.016   [−0.0399, 0.0227]   0.591
#> Sum over lags: -0.226 [-0.305, -0.146]
#> Pesaran CD: 13.2 (p = 5.05e-40), mean pairwise correlation 0.083
lamp_effect_summary(fit)
#> # A tibble: 1 × 6
#>   estimate std_error conf_low conf_high level definition                        
#>      <dbl>     <dbl>    <dbl>     <dbl> <dbl> <chr>                             
#> 1   -0.226    0.0405   -0.305    -0.146  0.95 1 * .log_s_lag0 + 1 * .log_s_lag1…

The lag sum describes a persistent shift in log-one-plus stop intensity. Calendar months define the lags: an absent month cannot turn a two-month-old observation into last month’s value. The fitted sample and exclusions are reported. Poisson pseudo-likelihood includes zeros and uses robust clustered covariance rather than imposing Poisson variance for inference. It still requires a correctly specified mean and suitable exogeneity.

Common correlated effects have a different response scale

The CCE implementations currently model log(1 + observed crime). That is not log(expected crime), especially with sparse counts. For a comparison between the three implementations on that response scale, explicitly select legacy OLS for the FE row:

legacy_fe <- lamp_elasticity(sim, "crime_total", lags = 0:1, family = "ols_log")
pooled <- lamp_elasticity(sim, "crime_total", lags = 0:1, method = "cce_pooled")
mg <- lamp_elasticity(sim, "crime_total", lags = 0:1, method = "cce_mg")
rbind(fe = lamp_effect_summary(legacy_fe),
      cce_pooled = lamp_effect_summary(pooled),
      cce_mg = lamp_effect_summary(mg))
#> # A tibble: 3 × 6
#>   estimate std_error conf_low conf_high level definition                       
#> *    <dbl>     <dbl>    <dbl>     <dbl> <dbl> <chr>                            
#> 1   -0.199    0.0239   -0.247    -0.150  0.95 1 * .log_s_lag0 + 1 * .log_s_lag1
#> 2   -0.197    0.0234   -0.244    -0.149  0.95 1 * .log_s_lag0 + 1 * .log_s_lag1
#> 3   -0.199    0.0315   -0.263    -0.136  0.95 1 * .log_s_lag0 + 1 * .log_s_lag1

CCE uses cross-sectional averages to proxy common factors. Mean-group inference averages area slopes and uses their cross-area dispersion. Force clustering for that method is not implemented and is rejected.

The auxiliary Pesaran CD statistic uses an OLS regression of log-one-plus outcomes with area effects only. A large value describes shared movements. It does not establish that the clustered intervals from the primary count model are invalid, or that adding common-factor proxies identifies a cause.

Reverse allocation remains an identification question

allocation <- lamp_allocation(sim, crime_lags = 1:3)
allocation$diagnostics$interpretation
#> [1] "No clear allocation response was detected in this panel. This does not establish exogeneity or rule out reverse allocation and confounding."

Police deployment can respond to recent or anticipated crime. A detected allocation response must be considered when interpreting the stop-crime association. A non-significant response does not establish exogeneity or rule out reverse allocation through other channels. Neither coefficient sign supplies an intervention assignment mechanism.

Converting the estimate requires further assumptions

lamp_crimes_prevented(fit, sim)
#> 
#> ── Crimes prevented per 1000 searches
#> 648 [420, 876] on affected_crime
#> 
#> ── Assumption chain
#> 1. The log-one-plus stops slope is -0.226, summed over lags 0, 1, 2, 3.
#> 2. The denominator is mean stops plus one; legacy OLS also uses mean crime plus
#> one.
#> 3. The effect is proportional, so it scales with the mean level of crime.
#> 4. At the sample mean of 21.9 crimes and 6.63 searches per area-month.
#> 5. The association between searching and recorded crime, net of area and month
#> effects. Causal only if the variation in searching has a source outside the
#> crime process; see lamp_allocation().
#> 6. Recorded crime only: unreported crime and recording changes are not
#> separated.
#> 7. The average effect is applied at the margin, which ignores diminishing
#> returns.

This is a local arithmetic approximation at fitted-sample means. Because the covariate is log-one-plus stops, its denominator is mean stops plus one. A count-mean slope uses mean crime in the numerator. Legacy OLS requires an approximation on mean crime plus one and cannot be treated as an expected count effect without further retransformation assumptions. The output retains the assumption chain and does not certify prevention.

See vignette("research-validation", package = "streetlamp") for heterogeneous effects, simulation truth, complete robustness grids, conditional trend sensitivity and the limits of the London empirical case.