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dbc

Dynamic bias correction for panel data estimators

This package implements the dynamic biases correction estimator as per Klosin, S, Dynamic Biases of Static Panel Data Estimators. This allows the estimation of treatment effects in panel fixed effects models where the outcome is dynamic. See below for usage and examples.

Installation

# Install the latest version on github
pak::pak("github::klosins/dbc")

Usage

library(dbc)

The follows shows example usage of the DBC under three general types of model. The variables of interest are the outcome variable, $Y_{it}$ for individual $i$ and time $t$, and the treatment variable $D_{it}$.

The first is the exogenous treatment model, where the outcome is:

$$Y_{it} = \alpha_{i} + \rho_{1} Y_{t-1} + \tau D_{t} + \beta_{1} X_{1} +\varepsilon_{it}$$

and with the treatment equation:

$$D_{it} = u_{it}$$

The following shows the simulation of this model using the dbc::DGP data generating process function, and then the estimation of this model.

summary(fit_exog) shows the bias-corrected estimates. The reported standard errors used underlying iid errors, and are not clustered.

The biased, OLS estimates of the model can be shown as fit_exog$biased_coefficients. str(fit_exog) shows the full list of objects returned with the dbc model.

set.seed(42)

# Exogenous treatment (no lag_y in treatment equation)
data_exog <- DGP(N = 500, N_T = 4, rho1 = 0.2, rho2 = 0, tau = 0.5, n_X1 = 1, n_X2 = 0)

fit_exog <- dbc(
    outcome_fml = y ~ lag_y + D + X1_1,
    lag_y       = "lag_y",
    treatment   = "D",
    panel_id    = "panel_id",
    time_id     = "time",
    data        = data_exog
)


summary(fit_exog)
#> Dynamic Bias-Corrected Estimator
#> 
#> Call:
#> dbc(outcome_fml = y ~ lag_y + D + X1_1, lag_y = "lag_y", treatment = "D",     panel_id = "panel_id", time_id = "time", data = data_exog)
#> 
#> Outcome equation:
#>       Estimate Std. Error z value Pr(>|z|)    
#> lag_y  0.21457    0.02129   10.08   <2e-16 ***
#> D      0.46353    0.03051   15.19   <2e-16 ***
#> X1_1   0.95184    0.03370   28.25   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> N = 500  T = 3  phi = 0.2146  GMM: (converged)

The second, is the endogenous treatment model, where the treatment effect is also a function of the previous values of the outcome:

$$Y_{it} = \alpha_{i} + \rho_{1} Y_{t-1} + \tau D_{t} \beta_{1}X_{1} +\varepsilon_{it}$$

$$D_{it} = c_{i} + \rho_{2} Y_{t-1} + \beta_{2} X_{2} + u_{it}$$

# Endogenous treatment (lag_y in treatment equation)
data_endo <- DGP(N = 500, N_T = 4, rho1 = 0.2, rho2 = 0.3, tau = 0.5,n_X1 = 1, n_X2 = 1)

fit_endo <- dbc(
    outcome_fml   = y ~ lag_y + D,
    treatment_fml = D ~ lag_y,
    lag_y         = "lag_y",
    treatment     = "D",
    panel_id      = "panel_id",
    time_id       = "time",
    data          = data_endo
)

summary(fit_endo)
#> Dynamic Bias-Corrected Estimator
#> 
#> Call:
#> dbc(outcome_fml = y ~ lag_y + D, treatment_fml = D ~ lag_y, lag_y = "lag_y",     treatment = "D", panel_id = "panel_id", time_id = "time",     data = data_endo)
#> 
#> Outcome equation:
#>       Estimate Std. Error z value Pr(>|z|)    
#> lag_y  0.19601    0.02423    8.09 5.99e-16 ***
#> D      0.47204    0.03146   15.00  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Treatment equation:
#>       Estimate Std. Error z value Pr(>|z|)    
#> lag_y  0.28360    0.02233    12.7   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> N = 500  T = 3  phi = 0.3299  GMM: (converged)

Last, the interaction model, which allows for interactions between the treatment variable and exogenous variables, $W$:

$$Y_{it} = \alpha_{i} + \rho_{1} Y_{t-1} + \tau_{1} W_{1} \times D_{t} + \tau_{2} W_{2} \times D_{t} \beta_{1}X_{1} +\varepsilon_{it}$$

$$D_{it} = c_{i} + \rho_{2} Y_{t-1} + \beta_{2} X_{2} + u_{it}$$

# With moderators/interactions
data_interact <- DGP(
    N = 500, N_T = 4,
    rho1 = 0.2, rho2 = 0.3,
    tau = c(0.8, 1.2),
    n_X1 = 1, n_X2 = 1, n_W = 2
)


fit_interact <- dbc(
    outcome_fml   = y ~ lag_y + D:W_1 + D:W_2 + X1_1,
    treatment_fml = D ~ lag_y + X2_1,
    lag_y         = "lag_y",
    treatment     = "D",
    panel_id      = "panel_id",
    time_id       = "time",
    data          = data_interact
)

summary(fit_interact)
#> Dynamic Bias-Corrected Estimator
#> 
#> Call:
#> dbc(outcome_fml = y ~ lag_y + D:W_1 + D:W_2 + X1_1, treatment_fml = D ~     lag_y + X2_1, lag_y = "lag_y", treatment = "D", panel_id = "panel_id",     time_id = "time", data = data_interact)
#> 
#> Outcome equation:
#>       Estimate Std. Error z value Pr(>|z|)    
#> lag_y 0.201906   0.003954   51.07   <2e-16 ***
#> D:W_1 0.800101   0.004293  186.36   <2e-16 ***
#> D:W_2 1.201314   0.003830  313.65   <2e-16 ***
#> X1_1  1.017534   0.032734   31.09   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Treatment equation:
#>       Estimate Std. Error z value Pr(>|z|)    
#> lag_y  0.30295    0.00296  102.35   <2e-16 ***
#> X2_1   1.00565    0.03086   32.58   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> N = 500  T = 3  phi = 0.2051  GMM: (converged)

S3 methods

Standard model sumamry functions on the dbc output are provided.

Returning the bias-corrected coefficients:

coef(fit_endo)
#>          lag_y              D treat_eq:lag_y 
#>      0.1960087      0.4720423      0.2836029

Returning the variance-convariance matrix on the bias-corrected coefficients:

vcov(fit_endo)
#>                        lag_y             D treat_eq:lag_y
#> lag_y           0.0005870888 -0.0001764293   0.0000619551
#> D              -0.0001764293  0.0009897852   0.0001792766
#> treat_eq:lag_y  0.0000619551  0.0001792766   0.0004988017

Standard confidence intervals:

confint(fit_endo, level = 0.90)
#>                       5%       95%
#> lag_y          0.1561540 0.2358634
#> D              0.4202938 0.5237908
#> treat_eq:lag_y 0.2468670 0.3203389

The number of observations:

nobs(fit_endo)
#> [1] 1500

broom / modelsummary

The package has methods to interface with the broom and modelsummary packages.

Create a tidy version of the bias-corrected coefficients:

library(broom)

tidy(fit_endo, conf.int = TRUE, conf.level = 0.95)
#>       group  term  estimate  std.error statistic      p.value  conf.low conf.high
#> 1   Outcome lag_y 0.1960087 0.02422991  8.089533 5.989376e-16 0.1485189 0.2434985
#> 2   Outcome     D 0.4720423 0.03146085 15.004117 6.900337e-51 0.4103802 0.5337044
#> 3 Treatment lag_y 0.2836029 0.02233387 12.698333 6.040256e-37 0.2398293 0.3273765

glance shows other model information, like number of observations.

glance(fit_endo)
#>         N N_T nobs  phi converged
#> lag_y 500   3 1500 0.33      TRUE

We can produce tables from modelsummary. Using the shape = group + term ~ model argument groups the table by whether the coefficients are in the outcome or treatment equation.

library(modelsummary)

# Multiple models side-by-side
modelsummary(
    list(Exogenous = fit_exog, Endogenous = fit_endo),
    shape = group + term ~ model
)
Exogenous Endogenous
Outcome lag_y 0.215 0.196
(0.021) (0.024)
D 0.464 0.472
(0.031) (0.031)
X1_1 0.952
(0.034)
Treatment lag_y 0.284
(0.022)
Num.Obs. 1500 1500
N 500 500
N_T 3 3
phi 0.215 0.33

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