Question

1.Consider a regression of y on two explanatory variables, x1 and x2, which are potentially correlated (though not perfectly). Say that x1 can take on any value between 1 and 100. A researcher draws a random sample of observations, with information on y, x1 and x2. She runs a regression on this sample, which we refer to as regression A. She then takes the subset of the data where x1 is restricted to only take values between 1 and 50, but there is no restriction on x2. She runs another regression, which we refer to as regression B. (a) [4] Do you expect the estimated coefficients to differ between regressions A and B? Explain. (b) [5] Do you expect any difference in the precision of the estimated coefficients between regressions A and B? Explain.

Answer #1

a)The estimated coefficients are expected to differ because the descriptive statistics of x1 and y have changed impacting the standard deviations and the mean of the data. The mean of x is expected to reduce and no restriction on x2 might not produce a similar result as in regression A.

b)As the data on x2 is not restricted, the model maynot produce a similar R^2. The new model can produce either a better R2 or a lower R^2 model. If one produces a model with a better R^2, the precision of the coefficients is expected to increase(lower SD) and similarly if the new model has a lower R^2, the precision is expected to reduce(higher SD).

Consider a regression of y on two explanatory variables, x1 and
x2, which are potentially correlated (though not perfectly). Say
that x1 can take on any value between 1 and 100. A researcher draws
a random sample of observations, with information on y, x1 and x2.
She runs a regression on this sample, which we refer to as
regression A.
She then takes the subset of the data where x1 is restricted to
only take values between 1 and 50,...

Consider a regression of y on x1,
x2 and x3. You are told
that x1 and x3 are
positively correlated but x2 is uncorrelated
with the other two variables.
[3] What, if anything, can you say about the relative
magnitudes of the estimated coefficients on each of the three
explanatory variables?
[6] What, if anything, can you say about the precision with
which we can estimate these coefficients?

A linear regression of a variable Y against the explanatory
variables X1 and X2 produced the following estimation model:
Y = 1615.495 + 9.957 X1 + 0.081 X2 +
e
(527.96) (6.32) (0.024)
The number in parentheses are the standard errors of each
coefficients
i. State the null and alternative hypothesis for the
coefficients
Select the appropriate test, compute the test statistic based on
the information above, and test the null hypothesis for each
coefficient by using a level of...

Consider the following data for a dependent variable y and two
independent variables, x1 and x2 . x 1 x 2 y 29 13 94 46 11 109 25
17 112 50 16 178 40 6 95 51 19 175 74 7 170 36 13 117 59 13 143 76
17 212 Round your all answers to two decimal places. Enter negative
values as negative numbers, if necessary. a. Develop an estimated
regression equation relating y to x1 . ŷ...

Consider the following data for a dependent variable y and two
independent variables, x1 and x2.
x1
x2
y
29
12
95
46
10
109
24
17
113
50
17
178
40
5
94
52
19
176
74
7
170
36
13
118
59
13
143
76
16
212
Round your all answers to two decimal places. Enter negative
values as negative numbers, if necessary.
a. Develop an estimated regression equation
relating y to x1.
y^=_____ + _____x1 (fill in...

6. Consider the following sample regression
results:
Y hat = 15.4 + 2.20 X1 +
48.14
X2
R2 = .355
(6.14)
(.42)
(5.21)
n = 27
The numbers in the parentheses are the estimated standard errors
of the sample regression coefficients.
6. (continued)
a. Construct a 95% confidence interval for
b1.
b. Is there evidence of a linear relationship
between X2 and Y at the 5% level of
significance?
c. If you were to use a global test...

2. Consider the data set has four variables which are Y, X1, X2
and X3. Construct a multiple regression
model using Y as response variable and other X variables as
explanatory variables.
(a) Write mathematics formulas (including the assumptions) and give
R commands to obtain linear
regression models for Y Xi, i =1, 2 and 3.
(b) Write several lines of R commands to obtain correlations
between Xi and Xj , i 6= j and i, j =
1, 2,...

1.
Suppose the variable x2 has been omitted from
the following regression equation, y = β0 +
β1x1 +β2x2 + u.
b1 is the estimator obtained when x2 is
omitted from the equation. The bias in b1 is positive
if
A.
β2<0 and x1 and x2 are
positive correlated
B.
β2=0 and x1 and x2 are negative
correlated
C.
β2>0 and x1 and x2 are
negative correlated
D.
β2>0 and x1 and x2 are
positive correlated
2.
Suppose the true...

8. Consider the following data for a dependent variable
y and two independent variables, x1 and
x2.
x1
x2
y
30
12
94
47
10
108
25
17
112
51
16
178
40
5
94
51
19
175
74
7
170
36
12
117
59
13
142
76
16
211
(a) Develop an estimated regression equation relating y
to x1. (Round your numerical values to one decimal
place.)
ŷ = ______
Predict y if x1 = 51. (Round
your answer...

A) Suppose you have modeled a linear program that includes the
decision variables x1 and x2. You wish to incorporate the
additional restriction that |x1 − x2| = 0, 5, or 12. Show how you
would formulate this problem as an ILP.
B) Suppose you have modeled an ILP that includes
includes a variable z, which is restricted to be at least 0, no
more than 20, and integer-valued. You wish to incorporate the
additional restriction that z does not take...

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