Pen & Paper

a) !" = !" + ! + !" (1) with !" = !,!!! + !" and (!|) = !!"

If he error term is serially correlated over time, then standard statistical inference as normally applied to regressions is invalid because standard errors are estimated with bias. To avoid this problem, the residuals must be modeled. If the process generating the residuals is found to be a stationary first-order autoregressive structure, !" = ! ,!!!+ !" , < 1, with the errors {!"} being white noise, then the CochraneOrcutt procedure can be used to transform the model by taking a quasi-difference:

!" !,!!! = ! 1 + !" !,!!! + !" (2)

We can then rewrite equation (1) in this way !" = ! 1 + !" + !"

The term is typically unknown, and along with we may wish to estimate it. If is not known, then it is estimated by first regressing the untransformed model and obtaining the residuals !", and regressing !" on !,!!!, leading to an estimate of and making the transformed regression (2) feasible. The resulting estimator for is given by:

= !,!!!

!

!!!

!

!!!

!!

!"!,!!!

!

!!!

!

!!!

Solving this problem we can then start the within transformation to estimate the fixed effects estimator:

First, we calculate the sample average of !" across periods and we denote it as ! =

!!

!"!!!!

So, ! = ! 1 + ! + !

Second from the transformed regression:

!" ! = ! 1 + !" + !" ! 1 + ! + !

!" ! = !" ! + !" !

!" = !" + !"

Now we stack by observation for t=1, , T giving the giant regression.

! = ! + !

The within-group FE estimator is pooled OLS on the transformed regression !(stacked by observation).

!" = ()!!

!" = (!!)!! !

!

!!!

!

!!!

!

b) We have, !" = + !" + ! + !"

Assumption: ! and !" are uncorrelated with !".

Hence, the model can be rewritten as:

!" = !" + !", Where !" = ! + !"

!" Is treated as an error term consisting in two components: an individual specific component that does not vary overtime, and a reminder component that is assumed to be uncorrelated overtime.

!" = ! + !" = (! + !,!!! + !")

!" = !! + !!

We need to find !!. And as we already know !" is an AR (1), hence:

!! =!! !!!!

(1)

With the estimation of as in question a) we can now calculate !" :

!" = !! + !

1 !

Now we need to find the covariance between the unobserved terms within periods.

!" = ! + !"

!,!!! = ! + !,!!!

!,!!! = ! + !,!!! () Since ! do not change overtime.

The covariance can be calculated by:

(!" ,!,!!!) = (! + !" ,! + !,!!!)

(!" ,!,!!!) = (! + !,!!! + !" ,! + !,!!!)

(!" ,!,!!!) = !! + !!

(!" ,!,!!!) = !! + !

1 !

In the same way,

(!" ,!,!!!) = (! + ! ,! + !,!!! ) (!" ,!,!!!) = (! + !,!!! + !" ,! + !,!!!)

(!" ,!,!!!) = (! + !,!!! + !,!!! + !" , ! + !,!!!) (!" ,!,!!!) = !! + !!!

(!" ,!,!!!) = !! + !!

1 !

Iterating forward we have: (!" ,!,!!!) = !! + !

!! !!!!

With j = 1,2T

Hence,

(!" ,!,!!!) = !! + !!! !!!!

With j = 1,2T

Now we can show that the variance covariance matrix will be:

! =

!! +!

1 ! !! + !

! 1 !

!! + !!

1 ! !! +

! 1 !

TxT matrix

Using the GLS estimator we can estimate our Random Effects estimator.

!"# = (!!!)!!!!!

DATA 1

a) i) ii)

First of all we perform an estimation of the regression using lvio as dependent variable, which is the logarithm of violent crime rate against shall.

In this case we find a coefficient of -.443 (with a significant level of 99%), which suggests us that the shall-issue laws reduce the violent crime rate by 44%. Table 1.

We find this value surprisingly high from an economic point of view. To our knowledge we think that is odd a law like this, which allows you to carry a weapon, to have such an impact on reducing the violent crime rate.

Nevertheless, looking at our R we can deduct that this model is still too simple and with a really low explanatory power (R = 0.0866), which let us conclude that we are probably over-valuating the effect of the variable shall on our model.

For this reason we add new control variables to eliminate this overvaluation.

From the second regression (Table 2) we can conclude that adding the control variables results in a small drop in the estimated coefficient (now the coefficient for shall is -.368 with the same significance level of 99% as before), still too small to have a reasonable interpretation from an economic point of view (the decrease of the violent crime rate by 36% after the enactment of the shall law seems a too large effect).

Even if this result do not seem logic from an economic point of view, we have to highlight that, adding control variables, our model achieves a higher explanatory power (R = 0.5643), as we can see also from the likelihood ratio test (really high value = 990.02).

Consequentially, we can deduct that we have a problem of omitted variables in our regressions (e.g.: the quality of security forces, the level of poverty, etc.).

b) Now we analyze the changes if we add fixed state effects in our regression. The results are presented on Table 3.

We can notice that this model is more credible than the other one studied at point a) ii).

Indeed, we can see that applying the FE (fixed-effects) estimator using the within transformation our coefficient falls deeply from -0.3687 to -.046, which is a more credible value from an economic point of view, even if our explanatory power is not so high anymore (R within= 0.218).

Our regression model with fixed effects is more credible because applying this method we are controlling for heterogeneity that is fixed in time because each state is different from the other and the different states are not all included in our control variables.

For this reason our model with fixed-effects is more credible: it is more specific, it highlights the mismatching between the different states and it confirms our previous theory that in a) ii) there was important omitted variable bias.

Moreover, the estimate of the effect of shall issue laws on the violent crime rate is no longer statistically different from 0.

c) Now we analyze how our results change when we add fixed state effects and fixed time effects: this means that we are considering every variable fixed in time in each different state (Results presented on Table 4).

In this way we have the possibility to see how is changed the rate of violent crime after the introduction of the shall-issue law, keeping fixed the time variables of each state.

From an economic point of view this approach is the most useful and realistic because it helps us to not over-evaluate our values extrapolating them from the historic connection [for example we could have years of crisis, war, civil war, political changing () which could generate an increase in the violent crime rate].

We can notice that adding fixed effects on years our coefficient falls further to -.0279935 (not significantly different from 0) and there is also an increase in the value of R from the previous regression with just fixed effects on states [in point (b) R was = 21.8%; with fixed effects also on years R = 0.4180]. The higher value of R is to interpret as a more significant statistic explanation of our model.

As consequence we can see that the coefficient is not significant anymore and the violent crime rate is less affected by the introduction of the shall law.

d) In this point we are going to study how our model changes adding random state effects and fixed time effects (Results presented on Table 5). Its easy to notice that our results are not so different from what we obtained at the previous point (we still have insignificant coefficients): even if in the OLS regression the coefficient were significant applying the Breusch-Pagan test for all three regressions we have to reject H0 in all cases. The rejection of the null hypothesis deletes the possibility to do the Pooled OLS.

For this reason we decided to do the Haussmann test, which evaluates the consistency of an estimator when compared to an alternative, less efficient, estimator, which is already known to be consistent, and it helps us to evaluate if a statistical model corresponds to the data.

We did the Haussmann test between the random state effect and the fixed-effect models and we arrived at the conclusion that we cannot reject H0 (this means that we can take in account the most efficient estimator because there isnt a huge difference between the FE and the RE).

Then we arrived at the conclusion that the RE is not consistent and, consequentially, we have to use the FE estimator as we did in the previous point [see point (c)].

In all three cases (violent crime rates, robbery rates and murder rates), we reject the Haussmann test, which means we cannot use the random effect estimators, as they are not consistent. Therefore, one has to apply the fixed effect estimators as seen in part (c). On a economically point of view, there is little evidence that the individual heterogeneity is randomly distributed between different states, which could make sense as we assume that the state specific intercepts are not independent from the explanatory variables.

e) Now we are going to use the fixed effect and time effect model to test lagged effects of shall-issue laws and to evaluate how change the response between the short and long run (Results presented on Table 6).

Its easy to notice that all our shall dummy variables and all the shall-lagged dummy variables are in the reality non significant. The only difference in the trend of our three variables is in the robbery rate, which becomes negative after the addition of the lagged variable (its equal to -0.040).

Then, we analyzed the Arellano-Bond dynamic panel-data estimation.

This test is in relation to two-step robust estimation and it is useful to study the autocorrelation.

From this test we arrive at the conclusion that the lagged variable estimates for the robbery rate and the violence rate are both positive and with a really high value, instead the murder rate is not so affected from the rate of murders calculated for the previous year (lvio= 0.599; lrob = 0.441; lmur = 0.029).

The short-term effect is characterized by the sum of all b coefficients. b is a scalar of all the independent explanatory variables and in our analysis it assumes a value of -0.061 for lvio, of 0.011 for lrob and of 0.25 for lmur (p-value lmur= 0.08).

On the other hand, the long-run effect is the sum of all the b coefficients divided by (1-a), where a represents a scalar for the coefficient of the lagged dependent variable.

The values that we obtained for the long-run are: 0.153 for lvio, 0.019 for lrob and 0.257 for lmur and similarly to the short-run only the lmur is significant and as before with a p-value around 0.08.

f) Using the robbery rate instead of the violent crime rate.

We can notice that our results are not very different from what we obtained using the violent crime rate in point a) and b).

Indeed, in the OLS model the regression coefficients of the shall-issue law on the log of robbery rates (lrob) is equal to -0.773 (we can interpret it as a decreasing of the robbery rates of 77% after the introduction of this law) with a very low R = 0.12 ;

adding control variables in the OLS model our values are lower but not so significantly: the coefficient is equal to -0.529 ( it is still to high to have a logic economic interpretation) , even if R gives an higher explanatory power to our model (R = 0.60 ).

In the fixed-effects model in which is kept constant the states, we can see that the coefficient of robbery rates falls deeply showing that the enactment of the shall law generates only a decreasing of the 0.8% in the robbery rate, although our R has a really low value = 0.04 (its in any case a plausible value taking in account the restrictions generated by the fixed effect).

When we fix also the years, our results are pretty different from what we obtained studying the violent crime rate: the robbery rate has a positive coefficient = 0.027, which we can interpret as an increase of 2.7% of the robbery rate after the introduction of the shall law (R= 0.24, not so statistically significant).

Addressing the murder crime rate;

We can see when we perform OLS on lmur as dependent variable and shall as explanatory (Table 1) a highly significant negative impact of shall on the murder crime rate. In states with a shall law in motion we can see that the murder rates decrease by 47%. By the same justification given for the violent crime rates on question a) this value seems to high. For the second part of question a) adding the control variables does not really change the outcome on an economic sense. The value it is still high.

On question b) applying fixed state effects makes the value of the coefficient for shall drop massively. Now the effect of shall law on murder rates is only -4.6% (Table 3). With the fixed effects estimator the effect of the law in the murder rate seems to be more credible.

Regarding question c) when applying fixed state and time effects the shall dummy variables has a negative value for the murder rate. But in any case this value is not significant as one can see in Table 4.

For question d) when using random state effects, combined with fixed time effects we get negative effects on murder rates but the coefficients have no significance (Table 5).

g) When we use in first place pooled OLS, it seems that the effect of shall law variable is enormous when we address the problem of crime reduction. While using other methods and analyzing if they work better the influence and the significance of the shall variable on crime statistics takes a noose dive. From a political point of view it seems demagogic to introduce such a law to reduce crime because the effect of such option may be exaggerated and lightly interpreted.

Table 1. OLS

Variables lvio lrob lmur

shall -0.443*** (0.048) -0.773***

(0.069) -0.473***

(0.049)

_cons 6.135*** (0.019) 4.873*** (0.028)

1.898*** (0.022)

R2 0.087 0.121 0.083 N 1,173 1,173 1,173

Robust standard errors in parentheses *** p

Table 4. Fixed effects and fixed state effects

Variables lvio lrob lmur shall -0.028

(0.041) 0.027

(0.052) -0.015 (0.038)

Incarc_rate 0.000 (0.000)

0.000 (0.000)

-0.000 (0.000)

density 0.001 (0.016)

-0.045 (0.198)

-0.544* (0.319)

avginc -0.005 (0.015)

0.014 (0.025)

0.057*** (0.017)

pop 0.012 (0.014)

0.000 (0.026)

-0.032 (0.021)

pb1064 0.029 (0.050)

0.014 (0.084)

0.022 (0.076)

pw1064 0.009 (0.024)

-0.013 (0.033)

-0.000 (0.020)

pm1029 0.073 (0.052)

0.105 (0.073)

0.069 (0.042)

_Iyear_78 0.059*** (0.016)

0.033 (0.022)

-0.001 (0.032)

_Iyear_99 0.433

(0.286) 0.189

(0.385) -0.255 (0.242)

_cons 3.766*** (1.152)

3.279* (1.677)

0.188 (1.057)

R2 0.418 0.236 0.291 N 1,173 1,173 1,173

Robust standard errors in parentheses *** p

Table 6. Arellano-Bond linear dynamic panel-data estimation

Variables lvio lrob lmur lvio(-1) 0.599***

(0.020)

lrob(-1) 0.441*** (0.023)

lmur(-1) 0.029 (0.030)

shall -0.026* (0.014)

0.026 (0.021)

-0.061* (0.034)

Incarc_rate -0.000*** (0.000)

-0.000*** (0.000)

-0.000* (0.000)

density 0.038 (0.065)

0.036 (0.094)

0.381** (0.166)

avginc -0.005 (0.004)

-0.039*** (0.006)

0.001 (0.012)

pop -0.005 (0.008)

0.030** (0.012)

-0.014 (0.023)

pb1064 -0.027 (0.017)

-0.026 (0.023)

-0.071 (0.045)

pw1064 0.025*** (0.003)

0.040*** (0.004)

0.026*** (0.009)

pm1029 -0.061*** (0.005)

-0.055*** (0.006)

-0.012 (0.012)

_cons 2.196*** (0.247)

1.601*** (0.340)

0.704 (0.657)

N 1,071 1,071 1,071 Robust standard errors in parentheses

*** p

DATA 2

a) OLS

When using POLS we get the estimator:

Hence, plugging in the model for !":

It is easy to see that the estimator is biased since the term on the right does not have zero expectation due to the correlation between the individual effect, , which is part of , and the explanatory variable, !,!!!.

The Fixed Effect estimator (within transformation):

It is easy to see that it is also biased.!,!!! !,!! is correlated with the error term

!" ! because the averages include all time values of both variables.

The First Differences estimator: `

Being !,!!! !,!!! correlate with !" !,!!! this estimator is also biased. To solve this kind of problems is useful to use instrumental variables with GMM method.

b) We estimated the parameters from the model by three different methods (pooled OLS, fixed effects and first difference) the results are presented on Table 1. We can see from the table that we can achieve significant coefficients on all the variables with all three methods. As we know, Pooled OLS is consistent and asymptotic normal only if it and it are not correlated, but it is not efficient. Indeed, we can notice from our table that we cannot use the OLS standard error estimates because there is clustering (autocorrelation). Consequentially, these are biased and not useful for testing hypothesis.

In our table the first coulmn shows the Pooled OLS, the second the Fixed Effect, the third the First Difference Estimators, and the fourth is the GMM.

We can see that for lagged employment (L.n) the highest coefficient is found using the POLS method (0.936) and it decreases if we use the FE (0.538) or FD method (0.185). Logically, it means that more employment today is probably caused by more employment yesterday.

Estimates indicate that there are differences and that the OLS estimator attributes more of the difference in wages to employment level while FE and FD are less optimistic in that respect.

We can interpret the different results that we obtain using one procedure or the others as a sign that the assumptions justifying Pooled OLS are perhaps not valid (the estimates are biased).

On the other hand, if we look at the wage, our results differ less (POLS= -0.459; FE= -0.524; FD= -0.510) and if we look at the lag it is possible to notice that its relevance decreases from POLS to FD.

This result is logic: if I have an higher current product wage, this will reduce my current labor demand. On the other hand my current employment is positively affected by the previous wages.

Studying the capitalwe should higlight that the effect is the opposite: its value is smaller using the POLS (0.058), while it increases with FE (0.298) and it reaches its higher value with the FD method (0.360).

We can see the same increase, even if it is not so significant in this case, when we study the industry output: POLS= 0.328; FE= 0.354; FD= 0.360. Indeed labor and industry output are complements: if I increase the labor, I will increase consequentially also the industry output.

We obtain more unexpected results for labor and capital. These two values are substitutes, which means that I would expect a negative value for the capital coefficient, instead my result is positive. If we lag the capital we find a negative coefficient between the variables: this means that the capital of yesterday impacts negatively on the labor of today, as we could expect

c) As we know the general GMM approach does not impose that uit is i.i.d. over individuals and time and the optimal weighting matrix is thus estimated without imposing these restrictions. However, the absence of autocorrelation is needed to guarantee the validity of moment conditions, which means that weve to impose the absence of autocorrelation in uit, combined with a homoskedasticity assumption.

Consequentially, when we want to estimate the model with GMM method we cannot assume homoskedasticity.

The empirical implementation of the first-difference GMM estimator quite often suffers from poor small sample properties, mostly attributable to the large number of,

potentially weak, instruments. When instruments are weak, they provide only very little information about the parameters of interest, which leads to poor small sampke properties of the GMM estimator. Another problem arises when the number of instruments (moment conditions) is too large relatively the sample size.

This means that GMM estimator has misspecification tests and tend to be misleading. This may be particularly the case for the two-step estimator, which relies upon the estimation of a potentially high dimensional optimal weighting matrix. This means that the twostep approach could be not sufficient to solve this problem (standard errors can be biased downwards), but we probably need to apply the robust option to correct it.

As shown by Roodsman, reducing and collapsing instruments help sto reduce the bias in first-difference and system GMM estimators and to increase the the ability of the overidentifying restrictions tests to detect misspecification.

From our table we can see that in the GMM column that there is an overestimation of the lagged employment coefficient if we use OLS or FE (GMM= 0.388; FE=0.538 ; OLS= 0.936) and, even if FD coefficienti s not overestimated it is biased downward.

In any case the Arellano and Bond test tells us that there isnt autocorrelation (H0: no-autocorrelation, its not rejected, p-value=0.083) , which means that the GMM estimatori s consistent.

We wont reject the null hypothesis in the Arellano and Bond test in the second-order (correlation=0), instead in first-order-differences our errors will be correlated (our p-value=0.454 for the first-order-p-value rejects the H0).

d) The main difference between short and long-run is that the long-run effect shows also the fact that wages affect past value of employment, which, consequentially, has effects on the current level of employment.

Indeed, as we know, the short-run effect of wages is given by the sum of the coefficients: 1+2 = 0.262, which is significant at the 2% level.

We can explain this value as the impact that the wages have on employment. It seems clear that this negative value means that this impact happens immediatly.

On the other hand, the long-run effect is: because, as we said above, the total long-run effect the impact of wages on past values of employment.

This negative values that characterized the long-run effect tells us that there is a negative impact on labor demand caused by real product wages, which is higher than in the short run.

Table 1

POLS FE FD GMM

LD.Ln employment 0.185*** (0.030) 0.538*** (0.028)

0.388*** (0.106)

LD.Ln employment 0.185*** (0.030)

Ln wage -0.459*** (0.048) -0.524***

(0.051) -0.450***

(0.120)

L.Ln wage 0.373*** (0.048) 0.236*** (0.050)

0.189* (0.088)

D.Ln wage -0.510*** (0.045)

LD.Ln wage 0.134** (0.046)

Ln capital 0.058*** (0.006) 0.298*** (0.023)

0.272*** (0.054)

Ln industry output 0.328*** (0.046) 0.354*** (0.049)

0.608*** (0.081)

D.Ln capital 0.360*** (0.025)

D.Ln industry output

0.555*** (0.066)

Constant -1.196*** (0.219) -0.143 (0.297)

-1.251* (0.534)

R-squared 0.992 0.716 0.490 N 891 891 751 751

* p