Question

1. Consider the following linear regression model which
estimates only a constant:

Yi = β1 + ui

What will the value of ˆβ1 be? Remember we are minimizing the sum
of the squared residuals.

2. Consider the following regression model with K parameters:

Yi = β1 + β2X2i + β3X3i + ... + βKXKi + ui

Now consider the F-test of the null hypothesis that all slope
parameters (β2,β3,...,βK) are equal to zero. Using the equation
from class:

F =（(RSSk −RSSm)/(m−k)） RSSm/(N −m)

Prove that the F statistic for this test is equivalent to the
following expression:

F =（ESSm/(m−1) RSSm）/(N −m)

Hint: Your answer from question one will be helpful, make sure you correctly identify the restricted and unrestricted models, and make sure that you remember all of the relevant equations relating to RSS and R2.

Answer #1

7)
Consider the following regression model
Yi = β0 + β1X1i + β2X2i + β3X3i + β4X4i + β5X5i + ui
This model has been estimated by OLS. The Gretl output is
below.
Model 1: OLS, using observations 1-52
coefficient
std. error
t-ratio
p-value
const
-0.5186
0.8624
-0.6013
0.5506
X1
0.1497
0.4125
0.3630
0.7182
X2
-0.2710
0.1714
-1.5808
0.1208
X3
0.1809
0.6028
0.3001
0.7654
X4
0.4574
0.2729
1.6757
0.1006
X5
2.4438
0.1781
13.7200
0.0000
Mean dependent var
1.3617
S.D. dependent...

Consider the following (generic) population regression
model:
Yi = β0 + β1X1,i + β2X2,i + β3X3,i + ui, i = 1,...,n . Transform
the regression to allow you to easily test the null hypothesis that
β1 + β3 = 1. State the new null hypothesis associated to this
transformed regression.

1. Consider the bivariate model: Yi = β0+β1Xi+ui . Explain what
it means for the OLS estimator, βˆ 1, to be consistent. (You may
want to draw a picture.)
2. (Circle all that applies) Which of the following regression
functions is/are linear in the parameters a) Yi = β1 + β2 1 Xi b)
Yi = β1 + β 3 2Xi c) Yi = β1 + β2Xi

Consider the following (generic) population regression model: Yi
= β0 + β1X1,i + β2X2,i + β3X3,i + ui, i = 1, ..., n (∗) Transform
the regression to allow you to easily test the null hypothesis that
β1 + β3 = 1. State the new null hypothesis associated to this
transformed regression. Would you expect to reject or accept the
null hypothesis? Why?

1. Consider the model Ci= β0+β1 Yi+ ui. Suppose you run this
regression using OLS and get the following results: b0=-3.13437;
SE(b0)=0.959254; b1=1.46693; SE(b1)=21.0213; R-squared=0.130357;
and SER=8.769363. Note that b0 and b1 the OLS estimate of b0 and
b1, respectively. The total number of observations is
2950.According to these results the relationship between C and Y
is:
A. no relationship
B. impossible to tell
C. positive
D. negative
2. Consider the model Ci= β0+β1 Yi+ ui. Suppose you run this...

Multiple choice!
Consider the model Yi = B0 + B1X1i + B2X2i + B3X3i + B4X4i + Ui.
To test the null hypothseis of B2 = B3 = 0, the restricted
regression is:
A. Yi = B0 + B1X1i + B2X2i + B3X3i + B4X4i + Ui
B. Yi = B0 + Ui
C. Yi = B0 + B1X1i + B4X4i + Ui
D. Yi = B0 + B2X2i + B3X3i + Ui
Consider the model Yi = B0 +...

Consider the following linear regression model:
yi=β1+β2x2i+β3x3i+ei
σ2i=α1+α2x2i+α3x22i
What is the form of the auxiliary regression of the LM test for
heteroskedasticity?
Select one:
a. ^e2i=α1+α2x2i+α3x22i
b. ^e2i=α1+α2x2i
c. ^e2i=α1+α2x2i+α3x3i
d. ^e2i=α1+α2x22i

1.
Consider regression through the origin,
y=β1x1+β2x2+u, which of
the following statements is wrong?
a.
The degree of freedom for estimating the variance of error term
is n−2.
b.
The sum of residuals equals to 0.
c.
If the true intercept parameter doesn’t equal to 0, all slope
estimators are biased.
d.
The residual is uncorrelated with the independent variable.
2.
Which of the following statements is true of hypothesis
testing?
a.
OLS estimates maximize the sum of squared residuals....

Consider the following model:
Yi = β0 + β1(X1)i + β2(X2)i + β3(X3)i + β4(X4)i + ui
Where:
Y = Score in Standardized Test
X1 = Student IQ
X2 = School District
X3 = Parental Education
X4 = Parental Income
The data for 5,000 students was collected via a simple random
sample of all 8th graders in New Jersey. Suppose you
want to test the hypothesis that parental attributes have no impact
on student achievement. Which of the following is most
accurate?...

Consider the following model:
Yi = β0 +
β1(X1)i +
β2(X2)i +
β3(X3)i +
β4(X4)i + ui
Where:
Y = Score in Standardized Test
X1 = Student IQ
X2 = School District
X3 = Parental Education
X4 = Parental Income
The data for 5,000 students was collected via a simple random
sample of all 8th graders in New Jersey. Suppose you
want to test the hypothesis that parental attributes have no impact
on student achievement. Which of the following is...

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