Effect of Right-To-Carry Law on Crime Rate

The plots below show the trends crime rate over time. The line plots are changed to blue when the right-to-carry law was enacted for a given state. In general, there is no differentiation in the trends between states that enacted the right-to-carry and states that did not. While all the states appear to have the same pattern over time with sharp increase in crime in the 1990’s and then a gradual decline, State 11 had the highest crime rate overall. To ensure the patterns are visible, the data was subsetted before plotting.

Effect of Incarceration Rate on Crime Rate

Before exploring the effect of incarceration rate on various crime rates, the trend of incarceration rate over time was explored. There was a steady increasing trend over time irrespective of the implementation of the right-to-carry law. State 11 stands out as having a very high incarceration rate, which is expected, because it also had very high crime rates.

Since the data is skewed, logs of the rates are used for the pairplots. Incarceration rates have a positive association with the various crime rates, implying incarceration rates increase as crime rates increase. An interpretation based on logic is that incarceration rate (independent variable) is a direct consequence of crime rate (dependent variable), but since incarceration rate is the independent variable, there could be a case of reverse causality. An implication for model building is that incaceration rate could be an endogenous variable (when an independent variable is correlated with the error term of a model).

Also there is a positive association between the crime rates, indicating that the crime rates are associated, for example, higher violence rates are associated with high murder rates.

Data Cleaning

Density plots of the continuous variables show that most of them are skewed. This is because they are mostly rates and strictly positive, therefore log transformations were applied to most of them (except pw1064) to resolve skewness. To resolve skewness in pw1064 Box-Cox transformation was first applied, but because the range of values was about 100 times more than that of the other variables, min-max scaling was applied so that the variable does not have an undue influence on the regression estimates because of its numerical range.

Introduction To Panel Regression

Panel Data can be described as cross sectional data over time; outcomes and factors of entities, are collected over time. In this case, the outcomes will be crime rates, the entities will be the fifty states and six territories, and all the other independent variables including incarceration rate and the right-to-carry law will be the factors. Ordinary Least Squares (OLS) is typically used to determine the relationship between a dependent variable y and independent variable x. For panel data, data on all the entities is pooled together in OLS providing a common intercept and slope for all entities and in effect disregarding differences or heterogeneity between entities.

\[ y_{it} = \beta_1 + \beta_2x_{2it} + \beta_3x_{3it} + ... + \beta_nx_{nit} + e_{it} \] where,

• x are the independent variables
• n = 1,…,n are the number of indepenent variables
• \(\beta\) are the coefficients of the independent variables

This model is called a Pooled Least Squares model and it is ineffective for analyzing panel data. One reason is that the observations are dependent, and therefore correlated within entity. This is a violation of the requirement of independence for OLS. Two other reasons for the inapplicability of the Pooled Least Squares model, are discussed below.

Individual Heterogeneity:
If OLS is used to estimate this data, the assumption is that there are no differences or heterogeneity between states, consequently, a common intercept and slope will be estimated for all the states. This is resolved by utilizing different intercepts called fixed effects for each of the states. The fixed effects for the states account for heterogeneity between states so that estimated effects for each outcome is unique to each state. In other words, the slopes for the factors remain constant across entities while the intercepts are different producing entity specific estimates. The estimator utilized for this type of model is a Least Squares Dummy estimator, because dummy variables for each entity (the intercepts) is coded into the model. The Least Squares Dummy estimator is given below.

\[ y_{it} = \beta_{11}D_1 + \beta_{12}D_2 + .... + \beta_{1(i-1)}D_{i-1} + \beta_2x_{2it} + \beta_2x_{3it} + ... + \beta_nx_{nit} + e_{it} \] where,

• x are the independent variables
• n = 1,…,n are the number of independent variables
• \(\beta\) are the coefficients of the independent variables
• D are the entity binary dummy variables where \(D_{i-1} = \begin{cases} 1 & \quad \text{ } i \text{ is a given entity}\\ 0 & \quad \text{reference entity } \end{cases}\)
  
• i = 1,…,i are the number of entities

Note: a dummy variable is always dropped to avoid exact collinearity, so for i variables, i-1 dummy variables will be included in the model

Omitted Variable Bias:
Another challenge is the effect of unobserved factors on the regression, in OLS the effect of such variables on the coefficients is that they would either inflate or decrease the true value of the coefficients. Panel regression has two models that can control for both observed and unobserved factors:

• Entity Fixed Effect Models: If an omitted variable does not vary over time, the Entity Fixed Effects Model obtains consistent estimates by getting deviations from the mean of the variables overtime. The model is a fixed effects estimator, the derivation is shown below;

  For data on an entity i :

\[ y_{it} = \beta_1 + \beta_2x_{2it} + \beta_3x_{3it} + ... + \beta_nx_{nit} + e_{it} \]

Average the data across time on both sides, by summing both sides and dividing by total time T:

\[ {1\over T}\sum_{t=1}^{T}(y_{it} = \beta_1 + \beta_2x_{2it} + \beta_3x_{3it} + ... + \beta_nx_{nit} + e_{it}) \]

Since the coefficients do not change the equation simplifies to, (the bar refers to the parameters average) \[ \overline{y_i} = {1\over T}\sum_{t=1}^{T}y_{it} = \beta_1 + \beta_2{1\over T}\sum_{t=1}^{T}x_{2it} + \beta_3{1\over T}\sum_{t=1}^{T}x_{3it} + ... + \beta_n{1\over T}\sum_{t=1}^{T}x_{nit} + {1\over T}\sum_{t=1}^{T}e_{it} \\ = \beta_1 + \beta_2\overline {x}_{2i} + \beta_3\overline{x}_{3i} + ... + \beta_n\overline{x}_{ni} + \overline{e}_{i} \]

Finally, subtract term by term to get deviation from the mean for each entity \[ y_{it} = \beta_1 + \beta_2x_{2it} + \beta_3x_{3it} + ... + \beta_nx_{nit} + e_{it} \\ (\overline{y}_i = \beta_1 + \beta_2\overline {x}_{2it} + \beta_3\overline{x}_{3it} + ... + \beta_n\overline{x}_{nit} + \overline{e}_{it}) \\ \overline {y_{it} - \overline{y}_{i} = \beta_2(x_{2it} - \overline{x}_{2i}) + \beta_3(x_{3it} - \overline{x}_{3i}) + ... + \beta_n(x_{nit} - \overline{x}_{ni} ) + (e_{it}-\overline{e}_{i}})\]

Deviation from mean model across entities can be summarized as; \[ \tilde{y}_{it} = \beta_2\tilde{x}_{2it} + \beta_3\tilde{x}_{3it} + ... + \beta_n\tilde{x}_{nit} +\tilde{e}_{it} \]

This eliminates all time-invariant features implying that, if a variable is constant or slow changing over time, it has no effect on the Entity Fixed Effect Model even if it is unobserved. Another implication is that the variation studied is within an entity, not across entities. Therefore, the model will be consistent even with endogenous variables because unobserved factors that do not vary are dropped from the model, and their correlation with other variables is of no effect as it is not captured by the error term.

• Time Entity Fixed Effect Model: If a variable is constant across entities but changes over time, the effect is captured by adding time dummy variables and entity fixed effects to the same model as shown below :

\[ y_{it} = \beta_1 + \beta_2x_{2it} + ... + \beta_nx_{nit} + \gamma_{1}D_{1} + ... + \gamma_{i-1}D_{i-1} + \delta_{1}T_{1} +....+ \delta_{t-1}T_{t-1} + e_{it} \]

where,

• x are the independent variables
• n = 1,…,n is the number of independent variables
• \(\beta\) are the coefficients of the independent variables
• D are the entity binary dummy variables where \(D_{i-1} = \begin{cases} 1 & \quad \text{ } i \text{ is a given entity}\\ 0 & \quad \text{reference entity } \end{cases}\)
• i = 1, …, i are the number of entities
• \(\gamma\) are the coefficients of the entity binary dummy variable
• T is time binary dummy variable to represent each time period where \(T_{i-1} = \begin{cases} 1 & \quad \text{ } i \text{ is a given entity}\\ 0 & \quad \text{reference entity } \end{cases}\)
• t = 1, …, t is the time period in the dataset
  
• \(\delta\) is the coefficient of the time binary dummy variable
  

Note: a dummy variable is always dropped to avoid exact collinearity, so for i variables, i-1 dummy variables will be included in the model

The result is that, omitted variables that are either constant over time within an entity (e.g Race) or, constant across entities but vary over time (e.g right-to-carry law), have no effect on the Time Entity Fixed Effect Model. Please note that the examples given are for clarification. There is a caveat, if the omitted variables vary over time and are correlated with other independent variables the model will be baised.

It should also be noted that Panel Regression also includes a Random Effects Model, which is most suited to entities that are randomly sampled. In this case, the Time Fixed Effects model will be more efficient because;

• All the states in the United States were utilized (not randomly sampled).
• The right-to-carry law changes in some of the states over time.

The models were first tested for time fixed effect if statistically significant, then the Time Fixed Effects model is retained. They were then tested for serial correlation, and heteroskedacity (the results are presented under the “Model and Test” section). If the results are significant, heteroskedastic-consistent covariance estimates are printed using the Arellano method (this is presented under the “Interpretation of Heteroskedastic Consistent Coefficients” section). After which each model was interpreted considering the central question and statistically significant effects, values are reported approximately to 2 decimal places. A discussion of the results is provided under the “Analysis of Model Results”.

Variable Selection

To test the theory that incarceration rate is a consequence of crime rate, the Granger causality test was utilized. It was significant for all rates, this implies that incarceration rate is dependent on crime rate, and based on logical reasoning this is correct. People are arrested after they commit a crime, and not vice versa. Therefore, using incareation rate as an independent variable could cause the estimates to be biased because as stated earlier there will be reverse bias and as such the incarceration rate variable could be endogenous.

If there was access to more data, then an instrumental variable would have been used inplace of incarceration rate. For a variable to be an instrumental variable it must have these properties;

• It does not have a direct effect on the dependent variable, in this case crime rate.
• It is exogenous. This is a direct consequence of the first requirement, an exogenous variable is not correlated with the error term.
• It is strongly correlated with the endogenous variable. This means that variation in the endogenous variable can be explained by the instrumental variable as such, the instrumental variable can be used inplace of endogenous variable.

Since the dataset does not provide any more data, the only solution was to drop incarceration rate as a variable from the models investigating the effects of the right-to-carry law on crime rates. The log of the other variables will be utilized in building the crime models.

Granger Test for Relationship Between Violence Rate and Incarceration Rate

Granger causality test

Model 1: lincarc_rate ~ Lags(lincarc_rate, 1:3) + Lags(lvio, 1:3)
Model 2: lincarc_rate ~ Lags(lincarc_rate, 1:3)
  Res.Df Df      F    Pr(>F)    
1   1163                        
2   1166 -3 15.243 9.847e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Granger Test for Relationship Between Murder Rate and Incarceration Rate

Granger causality test

Model 1: lincarc_rate ~ Lags(lincarc_rate, 1:3) + Lags(lmur, 1:3)
Model 2: lincarc_rate ~ Lags(lincarc_rate, 1:3)
  Res.Df Df      F    Pr(>F)    
1   1163                        
2   1166 -3 16.764 1.151e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Granger Test for Relationship Between Robbery Rate and Incarceration Rate

Granger causality test

Model 1: lincarc_rate ~ Lags(lincarc_rate, 1:3) + Lags(lrob, 1:3)
Model 2: lincarc_rate ~ Lags(lincarc_rate, 1:3)
  Res.Df Df     F    Pr(>F)    
1   1163                       
2   1166 -3 9.047 6.358e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Violence Rate Model

    F test for twoways effects

data:  lvio ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +  ...
F = 26.087, df1 = 22, df2 = 1093, p-value < 2.2e-16
alternative hypothesis: significant effects

    Lagrange Multiplier Test - time effects (Breusch-Pagan) for balanced
    panels

data:  lvio ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +  ...
chisq = 8.5464, df = 1, p-value = 0.003462
alternative hypothesis: significant effects

    Breusch-Godfrey/Wooldridge test for serial correlation in panel models

data:  lvio ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +     lpm1029
chisq = 682.41, df = 23, p-value < 2.2e-16
alternative hypothesis: serial correlation in idiosyncratic errors

    Breusch-Pagan test

data:  lvio ~ lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +     lpm1029 + shall
BP = 322.87, df = 7, p-value < 2.2e-16

t test of coefficients:

               Estimate Std. Error t value Pr(>|t|)   
shall         -0.032365   0.034032 -0.9510 0.341796   
lpop           3.112136   2.508121  1.2408 0.214937   
lavginc        0.344901   0.242535  1.4221 0.155292   
ldensity      -3.400211   2.545159 -1.3360 0.181843   
lpb1064       -0.436247   0.172586 -2.5277 0.011621 * 
lpw1064_scale -0.400930   0.399808 -1.0028 0.316176   
lpm1029        1.732230   0.542702  3.1919 0.001454 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    Estimate Std. Error t-value  Pr(>|t|)    
1  -11.24940    4.96173 -2.2672 0.0235706 *  
2  -20.30368    8.05931 -2.5193 0.0119011 *  
4  -14.23654    5.98117 -2.3802 0.0174727 *  
5  -11.80990    5.00181 -2.3611 0.0183944 *  
6  -14.30187    6.35310 -2.2512 0.0245731 *  
8  -14.32091    5.86542 -2.4416 0.0147808 *  
9   -3.92975    2.03743 -1.9288 0.0540172 .  
10  -0.82367    0.99442 -0.8283 0.4076825    
11  12.55240    3.58325  3.5031 0.0004785 ***
12 -10.57052    5.02925 -2.1018 0.0357985 *  
13 -11.53351    5.12519 -2.2504 0.0246246 *  
15  -5.17436    2.40698 -2.1497 0.0317956 *  
16 -14.83520    5.60113 -2.6486 0.0081989 ** 
17 -11.04311    5.06457 -2.1805 0.0294358 *  
18 -10.65214    4.52146 -2.3559 0.0186535 *  
19 -13.26613    5.09364 -2.6044 0.0093271 ** 
20 -13.77628    5.57357 -2.4717 0.0135982 *  
21 -11.31572    4.64960 -2.4337 0.0151050 *  
22 -10.43551    4.78730 -2.1798 0.0294827 *  
23 -12.11009    4.36225 -2.7761 0.0055953 ** 
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---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

   Estimate Std. Error t-value Pr(>|t|)  
77 -11.4352     4.5896 -2.4916  0.01287 *
78 -11.3751     4.5900 -2.4782  0.01335 *
79 -11.2540     4.5897 -2.4520  0.01436 *
80 -11.1693     4.5881 -2.4344  0.01507 *
81 -11.1570     4.5880 -2.4318  0.01518 *
82 -11.1639     4.5878 -2.4334  0.01512 *
83 -11.1937     4.5881 -2.4397  0.01486 *
84 -11.1628     4.5890 -2.4325  0.01515 *
85 -11.1051     4.5894 -2.4197  0.01569 *
86 -11.0193     4.5899 -2.4008  0.01653 *
87 -11.0086     4.5902 -2.3983  0.01664 *
88 -10.9345     4.5905 -2.3820  0.01739 *
89 -10.8675     4.5908 -2.3672  0.01810 *
90 -10.6997     4.5865 -2.3329  0.01984 *
91 -10.6246     4.5861 -2.3167  0.02071 *
92 -10.5807     4.5864 -2.3070  0.02124 *
93 -10.5414     4.5862 -2.2985  0.02172 *
94 -10.5430     4.5862 -2.2988  0.02170 *
95 -10.5360     4.5863 -2.2973  0.02179 *
96 -10.5806     4.5865 -2.3069  0.02125 *
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---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Murder Rate Model

    F test for twoways effects

data:  lmur ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +  ...
F = 8.7321, df1 = 22, df2 = 1093, p-value < 2.2e-16
alternative hypothesis: significant effects

    Lagrange Multiplier Test - time effects (Breusch-Pagan) for balanced
    panels

data:  lmur ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +  ...
chisq = 3.8923, df = 1, p-value = 0.04851
alternative hypothesis: significant effects

    Breusch-Godfrey/Wooldridge test for serial correlation in panel models

data:  lmur ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +     lpm1029
chisq = 180.32, df = 23, p-value < 2.2e-16
alternative hypothesis: serial correlation in idiosyncratic errors

    Breusch-Pagan test

data:  lmur ~ lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +     lpm1029 + shall
BP = 801.7, df = 7, p-value < 2.2e-16

t test of coefficients:

                Estimate Std. Error t value  Pr(>|t|)    
shall         -0.0310143  0.0358548 -0.8650 0.3872290    
lpop          -6.2191299  1.7570799 -3.5395 0.0004178 ***
lavginc        0.9752524  0.2782946  3.5044 0.0004762 ***
ldensity       5.8682291  1.7422097  3.3683 0.0007828 ***
lpb1064       -0.0351460  0.2091153 -0.1681 0.8665594    
lpw1064_scale -0.0076034  0.3326009 -0.0229 0.9817657    
lpm1029        0.7638855  0.4346935  1.7573 0.0791472 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    Estimate Std. Error t-value  Pr(>|t|)    
1   21.49916    7.68766  2.7966 0.0052552 ** 
2   34.32512   12.48702  2.7489 0.0060788 ** 
4   25.78526    9.26718  2.7824 0.0054883 ** 
5   21.32883    7.74977  2.7522 0.0060178 ** 
6   28.41469    9.84344  2.8867 0.0039701 ** 
8   24.67088    9.08784  2.7147 0.0067378 ** 
9    6.36946    3.15678  2.0177 0.0438656 *  
10   0.61376    1.54074  0.3984 0.6904454    
11 -17.53012    5.55185 -3.1575 0.0016347 ** 
12  22.06737    7.79228  2.8320 0.0047114 ** 
13  22.34773    7.94093  2.8142 0.0049769 ** 
15   7.71535    3.72936  2.0688 0.0387981 *  
16  22.58422    8.67835  2.6024 0.0093836 ** 
17  22.00398    7.84700  2.8041 0.0051346 ** 
18  19.00026    7.00552  2.7122 0.0067890 ** 
19  20.07040    7.89205  2.5431 0.0111238 *  
20  23.24433    8.63565  2.6917 0.0072181 ** 
21  19.57799    7.20405  2.7176 0.0066790 ** 
22  20.99480    7.41740  2.8305 0.0047330 ** 
23  16.48264    6.75884  2.4387 0.0148994 *  
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---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

   Estimate Std. Error t-value Pr(>|t|)   
77  18.9027     7.1110  2.6582 0.007970 **
78  18.8901     7.1117  2.6562 0.008018 **
79  18.9526     7.1112  2.6652 0.007808 **
80  18.9969     7.1087  2.6723 0.007645 **
81  19.0048     7.1086  2.6735 0.007618 **
82  18.9236     7.1082  2.6622 0.007877 **
83  18.8580     7.1087  2.6528 0.008098 **
84  18.7326     7.1101  2.6346 0.008542 **
85  18.7697     7.1107  2.6396 0.008418 **
86  18.8320     7.1115  2.6481 0.008211 **
87  18.8063     7.1120  2.6443 0.008303 **
88  18.8107     7.1125  2.6447 0.008293 **
89  18.8060     7.1130  2.6439 0.008313 **
90  18.8629     7.1062  2.6544 0.008060 **
91  18.9088     7.1057  2.6611 0.007903 **
92  18.8685     7.1061  2.6553 0.008040 **
93  18.9544     7.1058  2.6675 0.007756 **
94  18.8413     7.1059  2.6515 0.008129 **
95  18.8533     7.1060  2.6532 0.008090 **
96  18.7835     7.1062  2.6432 0.008329 **
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---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Robbery Rate Model

    F test for twoways effects

data:  lrob ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +  ...
F = 19.626, df1 = 22, df2 = 1093, p-value < 2.2e-16
alternative hypothesis: significant effects

    Lagrange Multiplier Test - time effects (Breusch-Pagan) for balanced
    panels

data:  lrob ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +  ...
chisq = 48.761, df = 1, p-value = 2.891e-12
alternative hypothesis: significant effects

    Breusch-Godfrey/Wooldridge test for serial correlation in panel models

data:  lrob ~ shall + lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +     lpm1029
chisq = 687.62, df = 23, p-value < 2.2e-16
alternative hypothesis: serial correlation in idiosyncratic errors

    Breusch-Pagan test

data:  lrob ~ lpop + lavginc + ldensity + lpb1064 + lpw1064_scale +     lpm1029 + shall
BP = 229.77, df = 7, p-value < 2.2e-16

t test of coefficients:

                Estimate Std. Error t value Pr(>|t|)   
shall         -0.0025759  0.0395756 -0.0651 0.948116   
lpop           0.8245808  3.6153122  0.2281 0.819627   
lavginc        0.8124509  0.3607253  2.2523 0.024503 * 
ldensity      -0.7360935  3.6305596 -0.2027 0.839369   
lpb1064       -0.6859220  0.2312863 -2.9657 0.003086 **
lpw1064_scale -0.7799880  0.5041266 -1.5472 0.122103   
lpm1029        2.2082523  0.7178287  3.0763 0.002148 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

   Estimate Std. Error t-value  Pr(>|t|)    
1  -4.58217    6.74343 -0.6795 0.4969639    
2  -6.97255   10.95330 -0.6366 0.5245379    
4  -5.44409    8.12894 -0.6697 0.5031795    
5  -4.99441    6.79790 -0.7347 0.4626804    
6  -5.09463    8.63441 -0.5900 0.5552871    
8  -5.89494    7.97162 -0.7395 0.4597676    
9  -3.09650    2.76905 -1.1183 0.2637033    
10 -2.16112    1.35150 -1.5991 0.1100971    
11  2.79842    4.86995  0.5746 0.5656595    
12 -4.01458    6.83519 -0.5873 0.5570966    
13 -4.36928    6.96559 -0.6273 0.5306154    
15 -2.86149    3.27130 -0.8747 0.3819159    
16 -7.68418    7.61243 -1.0094 0.3129944    
17 -4.18846    6.88319 -0.6085 0.5429791    
18 -5.04954    6.14507 -0.8217 0.4114144    
19 -6.87875    6.92271 -0.9937 0.3206129    
20 -5.83243    7.57497 -0.7700 0.4414900    
21 -5.32656    6.31922 -0.8429 0.3994601    
22 -3.99079    6.50636 -0.6134 0.5397606    
23 -7.13261    5.92868 -1.2031 0.2292105    
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---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

   Estimate Std. Error t-value Pr(>|t|)
77  -5.4008     6.2376 -0.8658   0.3868
78  -5.3807     6.2382 -0.8625   0.3886
79  -5.2607     6.2378 -0.8434   0.3992
80  -5.1121     6.2356 -0.8198   0.4125
81  -5.0732     6.2355 -0.8136   0.4161
82  -5.1127     6.2352 -0.8200   0.4124
83  -5.2069     6.2356 -0.8350   0.4039
84  -5.2665     6.2368 -0.8444   0.3986
85  -5.2303     6.2374 -0.8385   0.4019
86  -5.1507     6.2381 -0.8257   0.4092
87  -5.1738     6.2385 -0.8293   0.4071
88  -5.1287     6.2389 -0.8221   0.4112
89  -5.0640     6.2393 -0.8116   0.4172
90  -4.9158     6.2334 -0.7886   0.4305
91  -4.7738     6.2330 -0.7659   0.4439
92  -4.7697     6.2333 -0.7652   0.4443
93  -4.7371     6.2330 -0.7600   0.4474
94  -4.7154     6.2331 -0.7565   0.4495
95  -4.7034     6.2332 -0.7546   0.4507
96  -4.7537     6.2334 -0.7626   0.4459
97  -4.8206     6.2338 -0.7733   0.4395
98  -4.9268     6.2345 -0.7902   0.4296
99  -5.0014     6.2346 -0.8022   0.4226

Analysis of the Results

The results of the three models where consistent in finding that the right-to-carry law did not have a statistically significant impact on crime rates from 1977 - 1999, meaning that conclusions cannot be drawn on the association of the right-to-carry law on crime rates for this time period.

An interesting finding was the statistically significant association between an increase in average income, robbery rates, and murder rates. This notion does not readily lend itself to common logic, it would be expected that a higher standard of living would reduce crime. A reason for this problem might be the actual factor in use. An average will tend to be skewed if the underlying data is skewed, in this instance it is expected that incomes would be right skewed, with very few individuals earning extravagant incomes, as such average income will not be a true representative of the socio-economic factor of each state. This is further buttressed by the fact that the feature, larger population density (which is usually associated with poor socio-economic factors) had a statistical association with increased murder rates. A solution would be to utilize the median income as it would provide a better picture of what income on average would be.

Also, the models might be biased because of omitted variables. For example, size of the population is only useful when other socio-economic indicators like educational levels, unemployment rates of the public are taken into consideration. Consequently, more factors would be needed for better analysis.

Lastly, an increase in the black population had a significant association with lower violence and murder rates, while a higher young male population (ages 10 - 29) had a significant association with high murder and robbery rates. Since crime is a character flaw and not a cultural phenomenon, race and gender are inefficient at providing indicators for crime. Consequently, as has been stated earlier more socio-economic factors would need to be included in the models.