Chaojun LI 李超君 5101209122
Campaign Expenditures
To study whether campaign expenditures affect election outcomes, we run a
regression and get the estimated equation,
voteA=45.08788+6.081356log(expendA)–6.615630log(expendB)+0.152014prtystra
(3.926796) (0.382114)
(0.378889)
(0.062026)
(1) To conduct the Breusch-Pagan, we regress the squared residuals on the three
independent variables, and get R2=0.054761, nR2=3R2=0.164283, degree of
freedom:3
H0: nR2=0, H0: nR2≠0
Critic value: 7.815, given α=0.05
3R2=0.164283<7.815, so we cannot reject the null hypothesis. Thus we can think
that the homoscedasticity holds.
(2) The White Procedure
voteA=45.08788+6.081356log(expendA)–6.615630log(expendB) +0.152014prtystra
(4.069780) (0.515030)
(0.331477)
(0.056016)
In the white procedure, the standard error for the coefficient of log(expendA) and
the intercept is larger, while that for the coefficient of log(expendB) and prtystra is
smaller.
(3) β1 means that when expendA increases by 1% ceteris paribus, voteA increases by
β1/100.
For example, for the equation we get β1 equals 6.081356, which means if expendA
increases by 1%, voteA will increase by 0.0608
(4) H0: β1+β2=0, H1: β1+β2≠0
̂1 and 𝛽
̂2 are separately 0.382114 and 0.378889 and
(5) Since the t-statistics for 𝛽
the p-value are separately 0.0000 and 0.0000. Thus we can think that both A’s and B’s
expenditure affect the outcome.
To test the hypothesis in (4), we first use t-test to get whether our conjecture is
right.
̂2 − 𝛽
̂1)=var(𝛽
̂2)+var(𝛽
̂1)-2cov(𝛽
̂2 , 𝛽
̂1)=0.14355+0.14601-2*(-0.002679)=0.29493
Var(𝛽
t-statistics: (6.081356-6.615630)/(0.29493)^0.5=-0.983794
critical point: (using standard normal approximation): ±1.96, given the level of
significance is 5%
Since t-statistics is larger than -1.96, we cannot reject the null hypothesis. Thus we
can think that 1% increase in A’s expenditure is offset by a 1% increase in B’s
expenditure.
(6) In order to estimate a model that directly gives the t statistic for testing the
hypothesis in (4), we first rearrange the model by plugging c=β1+β2 in and get,
voteA=β0+β1(log(expendA)–log(expendB))+c log(expendB)+β3prtystra+u
After we run the regression, we can get the function,
voteA=45.0879+6.08136(log(expendA)–log(expendB))
(3.926796) (0.382114)
-0.534275log(expendB)+0.15201prtystra
(0.533114)
(0.062026)
H0: c=0, H1: c≠0
t-statistics: -0.534275/0.533114=-1.00217
degree of freedom: 173-3-1=169
The critic value: ±2.261548, given the level of significance is 5%
Since t-statistics is -1.00217>-2.261548, thus we cannot reject the null
hypothesis. We can think that 1% increase in Candidate A’s expenditures is offset by a
1% increase in B’s expenditure, which is the same conclusion as we get in (5).
Price Discrimination
This article is to check whether fast-food restaurants charge higher prices in areas
with a larger concentration of blacks. We first run a regression and get the estimated
equation,
Log(psoda)=-1.463332+0.072807prpblck+0.136955log(income)+0.380360prppov obs: 401
(0.293711) (0.030676)
(0.026755)
(0.132790)
̂1=0, H1: 𝛽
̂1≠0
(1) H0: 𝛽
t-statistics: 2.373458
degree of freedom: 401-3-1=397
Given the level of significance is 5%, the critic value: ±1.965957
Given the level of significance is 1%, the critic value: ±2.58827
Since t-statistics=2.373458>1.965957 when the level of significance is 5%, we can
̂1 does not equal to zero.
reject Ho and think that 𝛽
However, when the level of significance is 1%, t-statistic is less than 2.58827. We
cannot reject the null hypothesis.
(2) The covariance between log(income) and prppov is 0.002802. The variance of
log(income) and prppov are 0.000716 and 0.017633, respectively. The correlation is
0.002802/(0.000716*0.017633)^0.5=0.788585
Variable
t-statistics
P-value
Prpblck
2.373458
0.018098
Log(income) 5.118780 4.80205E-07
prppov
2.864364
0.004400
(degree of freedom of the three variables are all 397)
Log(income) and prppov can be statistically significant in any cases. However,
prpblck is not significant given the level of significance is 1%.
(3) After we add log(hseval) into the model, the function we get is,
Log(psoda)=-0.841514+0.097550prpblck-0.052991log(income)
(0.292432) (0.029261)
(0.037526)
+0.052123prppov+0.121306log(hseval) obs: 401, RU2=0.183930
(0.134499)
(0.017684)
̂
𝐻0 : 𝛽log⁡̂
(ℎ𝑠𝑒𝑣𝑎𝑙) =0, H1: 𝛽log⁡(ℎ𝑠𝑒𝑣𝑎𝑙) ≠0
t-statistic for βlog(hseval): 6.859602
degree of freedom: 396
P-value: 2.66811E-11
Thus we can reject the null hypothesis and think log(hseval) is statistically
significant.
̂ = 𝛽prppov
̂ =0,
(4) H0: 𝛽log⁡(𝑖𝑛𝑐𝑜𝑚𝑒)
̂ ≠ 0, or 𝛽𝑝𝑟𝑝𝑝𝑜𝑣
̂ ≠0
H1: 𝛽log⁡(𝑖𝑛𝑐𝑜𝑚𝑒)
Using F test, we rewrite the regression model and get the new function,
Log(psoda)=-1.043086+0.128829prpblck+0.090341log(hseval), obs:401, RR2=0.169411
(0.126569) (0.022473)
(0.010618)
F-statistics:
2
2
(𝑅𝑈
−𝑅𝑅
)/2
2 )/(401−4−1)
(1−𝑅𝑈
=
(0.183930−0.169411)/2
(1−0.183930)/396
= 3.522690~F2,396
p-value: 0.030448
critical value: 3.01851, given the level of significance is 5%
Since the p-value we get is less than 5%, we cannot reject H0. Thus we can think
that the two variables are jointly significant.
(5) I will use the regression of (3), that is, the function I believe more is,
Log(psoda)=-0.841514+0.097550prpblck-0.052991log(income)
(0.292432) (0.029261)
(0.037526)
+0.052123prppov+0.121306log(hseval) obs: 401, RU2=0.183930
(0.134499)
(0.017684)
I believe this model first because it contains more valid variables. We have already
justify that log(hseval) is significant and log(income) and prppov are jointly significant.
Compared to the original model, this one takes log(hseval) out of the error term and
make the assumption that u is independent of regressors more valid and thus make
the model better.
Secondly, the adjusted R-squared also increases, which means the dependent
variable is better complained.
As a result, I think this model is better and once the proportion of black increase
by one, the price of the medium soda will increase by 9.755%.