The document outlines a statistical analysis using regression models to examine the impact of various factors, including beauty, gender, and concealed weapon laws, on course evaluations and violent crimes. Key findings suggest that beauty affects course evaluations differently for men and women, and that 'shall-issue' laws may lead to significant declines in violent crime rates. Additionally, incorporating state fixed effects alters the estimated impact of concealed weapon laws, indicating that the initial results might overestimate their effect.

1. Problem 1 Use the data set TeachingRatings to carry out the following exercises:
Estimatearegressionof Course Eval on Beauty, Intro, OneCredit, Female, Minority,
and NNEnglish.
reg course_eval beauty intro onecredit female minority nnenglish, r
Linear regression Number of obs = 463
F( 6, 456) = 17.03
Prob > F = 0.0000
R-squared = 0.1546
Root MSE = .51351
------------------------------------------------------------------------------
| Robust
course_eval | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
beauty | .16561 .0315686 5.25 0.000 .1035721 .2276478
intro | .011325 .0561741 0.20 0.840 -.0990673 .1217173
onecredit | .6345271 .1080864 5.87 0.000 .4221178 .8469364
female | -.1734774 .0494898 -3.51 0.001 -.2707337 -.0762212
minority | -.1666154 .0674115 -2.47 0.014 -.2990912 -.0341397
nnenglish | -.2441613 .0936345 -2.61 0.009 -.42817 -.0601526
_cons | 4.068289 .0370092 109.93 0.000 3.995559 4.141019
------------------------------------------------------------------------------
b. Add Age and Age2
to the regression. Is there evidence that Age has a nonlinear
effect on Course Eval? Is there evidence that Age has any effect on Course Eval?
gen age2=age^2
reg course_eval beauty intro onecredit female minority nnenglish age age2, r
Linear regression Number of obs = 463
F( 8, 454) = 12.92
Prob > F = 0.0000
R-squared = 0.1573
Root MSE = .51383
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2. course_eval | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
beauty | .1596534 .0306439 5.21 0.000 .0994318 .2198749
intro | .0024414 .0564425 0.04 0.966 -.1084796 .1133623
onecredit | .6197589 .1085906 5.71 0.000 .4063564 .8331614
female | -.1881177 .0517023 -3.64 0.000 -.2897233 -.0865122
minority | -.1795689 .0692882 -2.59 0.010 -.3157342 -.0434036
nnenglish | -.2432153 .0959732 -2.53 0.012 -.4318221 -.0546085
age | .0195252 .0234711 0.83 0.406 -.0266002 .0656507
age2 | -.0002223 .0002442 -0.91 0.363 -.0007022 .0002576
_cons | 3.677032 .5497641 6.69 0.000 2.596634 4.75743
------------------------------------------------------------------------------
test age age2
( 1) age = 0
( 2) age2 = 0
F( 2, 454) = 0.63
Prob > F = 0.5339
(or Fhomo = R2
ur−R2
r
1−R2
ur
n−k−1
J
= 0.1573−0.1546
1−0.1573
463−9
2
≈ 0.7273 < 3 = F2,∞)
Neither the linear term nor the quadratic term is significant, so no evidence that Age has
a nonlinear effect on CourseEval. F test shows Age has no effect on the course evaluation.
c. Modifytheregressionin(a)sothattheeffectofBeautyon CourseEval is different
for men and women. Is the male-female difference in the effect of Beauty statistically
significant?
reg course_eval beauty beauty_female intro onecredit female minority nnenglish, r
Linear regression Number of obs = 463
F( 7, 455) = 15.09
Prob > F = 0.0000
R-squared = 0.1639
Root MSE = .51124
------------------------------------------------------------------------------
| Robust
course_eval | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
beauty | .2308198 .0477817 4.83 0.000 .1369195 .32472
beauty_fem~e | -.1407411 .0633588 -2.22 0.027 -.2652533 -.0162288
intro | -.0012302 .0555516 -0.02 0.982 -.1103998 .1079393
onecredit | .6565755 .1085514 6.05 0.000 .4432512 .8698999
female | -.1729451 .0493675 -3.50 0.001 -.2699617 -.0759285
minority | -.1347426 .0692342 -1.95 0.052 -.2708011 .0013159
nnenglish | -.2679069 .0928796 -2.88 0.004 -.4504331 -.0853808
_cons | 4.074949 .0373397 109.13 0.000 4.001569 4.148329
------------------------------------------------------------------------------
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3. Interaction term is significantly negative, which implies the effect of Beauty on CourseEval
differs across genders.
d. ProfessorSmithisaman.Hehascosmeticsurgerythatincreaseshisbeauty
index from one standard deviation below the average to one standard deviation above the
average. What is his value of Beauty before the surgery? After the surgery? Using the
regression in (c), construct a 95% confidence for the increase in his course evaluation.
sum beauty
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
beauty | 463 4.75e-08 .7886477 -1.450494 1.970023
dis r(mean)-r(sd)
-.78864762
dis r(mean)+r(sd)
.78864771
di .1369195*2*r(sd)
.21596249
di .32472*2*r(sd)
.51217934
His value of Beauty before the surgery is -.78864762 and the one after is .78864771. The
95% c.i. for the increase in his course evaluation, holding anything else constant, is the
95% c.i. of βmale
Beauty∆Beauty, which is [.21596249, .51217934].
e. Repeat(d)forProfessorJones,whoisawoman.
reg course_eval beauty beauty_male intro onecredit female minority nnenglish, r
Linear regression Number of obs = 463
F( 7, 455) = 15.09
Prob > F = 0.0000
R-squared = 0.1639
Root MSE = .51124
------------------------------------------------------------------------------
| Robust
course_eval | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
beauty | .0900787 .0399779 2.25 0.025 .0115144 .1686429
beauty_male | .1407411 .0633588 2.22 0.027 .0162288 .2652533
intro | -.0012302 .0555516 -0.02 0.982 -.1103998 .1079393
onecredit | .6565755 .1085514 6.05 0.000 .4432512 .8698999
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4. female | -.1729451 .0493675 -3.50 0.001 -.2699617 -.0759285
minority | -.1347426 .0692342 -1.95 0.052 -.2708011 .0013159
nnenglish | -.2679069 .0928796 -2.88 0.004 -.4504331 -.0853808
_cons | 4.074949 .0373397 109.13 0.000 4.001569 4.148329
------------------------------------------------------------------------------
di .0115144*2*r(sd)
.01816161
di .1686429*2*r(sd)
.26599966
The 95% c.i. for the increase in his course evaluation, holding anything else constant, is
the 95% c.i. of βfemale
Beauty ∆Beauty, which is [.01816161, .26599966].
f. Compare ¯R2 in (a) and (c). What value does the t-statistic of the coefficient on the
interaction of beauty and female have to exceed before ¯R2 in (c) exceeds ¯R2 in (a).
¯R2
XZ − ¯R2
X =(1 −
n − 1
n − 3
(1 − R2
XZ)) − (1 −
n − 1
n − 2
(1 − R2
X))
=
n − 1
n − 2
(R2
XZ − R2
X) + (1 − R2
XZ)(
n − 1
n − 2
−
n − 1
n − 3
) > 0
, which implies
FH0:β2=0 homoske =
R2
XZ − R2
X
1 − R2
XZ
n − 3
1
> 1.
Thus, we have t2
β2 > 1 and |tβ2| > 1.
Problem 2 Some U.S. states have enacted laws that allow citizens to carry concealed
weapons. These laws are known as ”shall-issue” laws because they instruct local author-
ities to issue a concealed weapons permit to all applicants who are citizens, are mentally
competent, and have not been convicted of a felony (some states have some additional
restrictions). Proponents argue that, if more people carry concealed weapons, crime will
decline because criminals are deterred from attacking other people. Opponents argue that
crime will increase because of accidental or spontanenous use of the weapon. In this ex-
ercise, you will analyze the effect of concealed weapons laws on violent crimes, using the
data set Guns. A detailed description is given in Guns Description.
a. Estimatearegressionofln( vio) against shall, incarc rate, density, avginc, pop,
pb1064, pw1064, pm1029. Interpret the coefficient on shall in the regression. Is this esti-
mate large or small in a ”real-world” sense?
reg lnvio shall incarc_rate density avginc pop pb1064 pw1064 pm1029, r
Linear regression Number of obs = 1173
F( 8, 1164) = 95.67
Prob > F = 0.0000
R-squared = 0.5643
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5. Root MSE = .42769
------------------------------------------------------------------------------
| Robust
lnvio | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
shall | -.3683869 .0347879 -10.59 0.000 -.436641 -.3001329
incarc_rate | .0016126 .0001807 8.92 0.000 .0012581 .0019672
density | .0266885 .0143494 1.86 0.063 -.0014651 .054842
avginc | .0012051 .0072778 0.17 0.869 -.013074 .0154842
pop | .0427098 .0031466 13.57 0.000 .0365361 .0488836
pb1064 | .0808526 .0199924 4.04 0.000 .0416274 .1200778
pw1064 | .0312005 .0097271 3.21 0.001 .012116 .0502851
pm1029 | .0088709 .0120604 0.74 0.462 -.0147917 .0325334
_cons | 2.981738 .6090198 4.90 0.000 1.786839 4.176638
------------------------------------------------------------------------------
The percentage decline of violence crimes associated with the introduction of shall is
36.84%. If the number of violence crimes for a state is about the average of our sam-
ple, which is 503.0747, then introducing the shall might cause a decline in the number of
violence crimes at 185, which is big in a “real world” sense.
b. Weextendtheregressionmodelwith50statedummiesasfollows:
ln(vio)it = β0 + β1shallit + Xitβ3 + Σ51
i=2βi
4Dit + it
Xit indexes a vector of controls in part (a). Do the results change when you add fixed state
effects? If so, which set of regression results is more credible, and why? Why don’t we add
51 state dummies?
. reg lnvio shall incarc_rate density avginc pop pb1064 pw1064 pm1029 s_2-s_51, r
Linear regression Number of obs = 1173
F( 58, 1114) = 364.90
Prob > F = 0.0000
R-squared = 0.9411
Root MSE = .16072
------------------------------------------------------------------------------
| Robust
lnvio | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
shall | -.0461415 .0199433 -2.31 0.021 -.0852721 -.007011
incarc_rate | -.000071 .0000973 -0.73 0.466 -.0002619 .0001199
density | -.1722901 .1048789 -1.64 0.101 -.3780725 .0334923
avginc | -.0092037 .0067335 -1.37 0.172 -.0224155 .004008
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6. pop | .0115247 .0097044 1.19 0.235 -.0075162 .0305655
pb1064 | .1042804 .0165552 6.30 0.000 .0717976 .1367633
pw1064 | .0408611 .0053859 7.59 0.000 .0302935 .0514287
pm1029 | -.0502725 .0077908 -6.45 0.000 -.0655588 -.0349863
s_2 | .0559649 .0788371 0.71 0.478 -.098721 .2106508
s_3 | .2404116 .0872338 2.76 0.006 .0692506 .4115727
By controlling the state specific time-invarying effect, the effect of shall decreases drasti-
cally. It might imply that states with lower violence crimes level tend to pass the shall,
and we will overestimate the effect of shall if this state fixed effect is omitted. Test below
shows that the state fixed effects are joint significant. It’s reasonable to believe the new
specification is more credible. To avoid perfect multicollinearity, we just add in 50 state
dummies.
F =
R2
ur − R2
r
1 − R2
ur
n − k
J
=
0.9411 − 0.5643
1 − 0.9411
1173 − 59
50
≈142.5 > 1.34 = F5%(50, ∞)
c. otheresultschangewhenyouaddfixedtimeeffects?Ifso,whichsetofregression
results is more credible, and why?
reg lnvio shall incarc_rate density avginc pop pb1064 pw1064 pm1029 s_2-s_51 y_2-y_23, r
Linear regression Number of obs = 1173
F( 80, 1092) = 479.78
Prob > F = 0.0000
R-squared = 0.9562
Root MSE = .14003
------------------------------------------------------------------------------
| Robust
lnvio | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
shall | -.0279935 .0193733 -1.44 0.149 -.0660066 .0100196
incarc_rate | .000076 .0000829 0.92 0.360 -.0000867 .0002387
density | -.091555 .0648682 -1.41 0.158 -.2188354 .0357254
avginc | .0009587 .0071969 0.13 0.894 -.0131627 .01508
pop | -.0047544 .0067029 -0.71 0.478 -.0179065 .0083976
pb1064 | .0291862 .021037 1.39 0.166 -.0120913 .0704637
pw1064 | .0092501 .0085181 1.09 0.278 -.0074637 .0259639
pm1029 | .0733254 .0187763 3.91 0.000 .0364837 .110167
s_2 | -.1473986 .0696406 -2.12 0.035 -.284043 -.0107542
s_3 | .1393569 .0716812 1.94 0.052 -.0012916 .2800054
s_4 | -.1565058 .0537734 -2.91 0.004 -.2620166 -.050995
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7. Effect of shall further declines and becomes insignificant. Year dummies pick up the non-
linear general year effects, and results above might imply introduction shall might correlate
with the year when, for all states, there’s a decline in violence crimes. Similar to part (c),
joint significance test implies it’s more credible to add in year fixed effects.
F =
R2
ur − R2
r
1 − R2
ur
n − k
J
=
0.9562 − 0.9411
1 − 0.9562
1173 − 81
22
≈17.11 > F5%(22, ∞) ≈ 1.57
d. Inyourview,whatarethemostimportantremainingthreatstotheinternalvalidity
of this regression analysis?
Unobserved social-economic shocks might be correlated both with the introduction of shall
and the violence crimes; the policy change can also be endogenous to crimes (simultane-
nous causality).
f. Basedonyouranalysis,whatconclusionswouldyoudrawabouttheeffectsof
concealed-weapon laws on these crime rates?
Convincing conclusion requires better research design. For example, we can check sub-
groups within state, which are categorized by the number of violence crimes in the pre-shall
period. Assume the other unobserved social-economic shocks do not affect the difference of
violence crime rates between these subgroups, pre and post the law change. If we observe
the difference changed from pre-period to post-period, then the only thing contributing to
this change is the introduction of shall. Also we can check the other kinds of crimes.
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