Call:
lm(formula = lcrmrte ~ lprbarr, data = Crime)
Residuals:
Min 1Q Median 3Q Max
-1.87702 -0.31597 0.05811 0.35574 1.33322
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.39869 0.06579 -66.86 <2e-16 ***
lprbarr -0.61954 0.04909 -12.62 <2e-16 ***
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.512 on 628 degrees of freedom
Multiple R-squared: 0.2023, Adjusted R-squared: 0.2011
F-statistic: 159.3 on 1 and 628 DF, p-value: < 2.2e-16
According to the result of OLS, if the probability of arrest increases by 1%, then the number of crimes committed per person will decrease by 0.62%. It is statistically significant and matches the previous expectation.
Although the regression results are as expected, variables may still be omitted. Some omitted variables vary across the county:
Income Level. Variations in average income levels across regions can influence crime rates, with lower-income areas typically experiencing higher crime rates.
Education Levels. Differences in educational attainment and resources across regions may affect the likelihood of youth engaging in criminal behavior. Areas with fewer educational opportunities often have higher crime rates.
Legal and Policy Environment. Variations in the severity of legal punishments for certain crimes may affect the probability of crime.
Other omitted variables that vary over time and thus cause a bias:
Comparing the OLS coefficients, the coefficient for the FE Model decreases from -0.61954 to -0.092, which means the relationship between the probability of arrest and Crime is still negative but weakened. On the other hand, it also indicates that the OLS coefficients receive some bias from omitted variables and the fixed effect model controls for some biases that will not be observed.
3.6 One way and Two Way FE Implementation ( county and year )
fatal_fe_lm_mod2 <-lm(data = Crime,formula = lcrmrte ~ lprbarr +as.factor(county) +as.factor(year), )fatal_fe_lm_mod_time <-lm( data = Crime,formula = lcrmrte ~ lprbarr +as.factor(year), )# estimate the fixed effects regression with plm()fatal_fe_mod2 <-plm(formula = lcrmrte ~ lprbarr +as.factor(year), data = Crime,index =c("county", "year"), # declaring data to be panelmodel ="within"# fixed effects mdodel )fatal_fe_mod_time <-plm(formula = lcrmrte ~ lprbarr +as.factor(year), data = Crime,index =c("year"), # declaring data to be panelmodel ="within"# fixed effects mdodel )stargazer(fatal_fe_lm_mod,# County - One Way OLS (FE) fatal_fe_mod_time, # County - One Way OLS (FE) fatal_fe_lm_mod2, # County - Time (Two Way) OLS fatal_fe_mod2, # County - Time (Two Way) FEtype ="text",column.labels =c("FE - One way (county)","FE - One way (time)","FE - Two way (county-time) OLS","FE - Two way (county-time) PLM" ),add.lines =list(c('County Fixed effects', "Yes", "No", "Yes", "Yes"),c('Time Fixed effects', "No", "Yes", "Yes", "Yes") ),keep =c("lprbarr"))
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Dependent variable:
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lcrmrte
OLS panel OLS panel
linear linear
FE - One way (county) FE - One way (time) FE - Two way (county-time) OLS FE - Two way (county-time) PLM
(1) (2) (3) (4)
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lprbarr -0.092*** -0.617*** -0.078** -0.078**
(0.033) (0.049) (0.031) (0.031)
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County Fixed effects Yes No Yes Yes
Time Fixed effects No Yes Yes Yes
Observations 630 630 630 630
R2 0.914 0.202 0.923 0.114
Adjusted R2 0.900 0.193 0.909 -0.046
Residual Std. Error 0.181 (df = 539) 0.173 (df = 533)
F Statistic 63.715*** (df = 90; 539) 157.559*** (df = 1; 622) 66.338*** (df = 96; 533) 9.803*** (df = 7; 533)
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Note: *p<0.1; **p<0.05; ***p<0.01
In a fixed effects model, the fixed effects control for specific characteristics based on the specification used. For my model, it is mainly controlled by time-invariant characteristics.
Compared to the OLS coefficients, the coefficients from the individual county-fixed effects model show a significant decrease. Additionally, the R² of the county-fixed effects model reaches 0.914, which is much higher than that of the time-fixed effects model. The R² of the model with both time and entity fixed effects only improves by 0.009 compared to the county-fixed effects model, indicating that the county-fixed effects control for a significant portion of the bias.
The coefficient of the FE model depends on how the fixed effects are defined and what they control for. As shown in section 3.6, the one-way and two-way fixed effects models have different coefficients. Because the one-way fixed effect only controls for either time or county, while the two-way fixed effect controls for both. If you use different methods to fix the same conditions, such as demeaned regression and county-fixed effect model, the coefficients will be the same.