Question

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, 3.

(c) Give R commands to construct the correlation matrix for all X
variables

(d) Give commands to do the multiple regression for the full model
Y X1 + X2 + X3.

(e) Give the simple R command to do model selection automatically
for the reduced model

Answer #1

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?

•List three variables (X1, X2, X3) you’d include in a Multiple
Regression Model in order to better predict an outcome (Y)
variable. For example, you might list three variables that could be
related to how long a person will live (Y). Or you might list three
variables that contribute to a successful restaurant. Your
Regression Model should have three variables that will act as
“predictors” (X1, X2, X3) of a “criterion” (Y’). Note that the
outcome or criterion variable (e.g....

Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

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 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,...

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,...

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...

1. (a) Construct a Pearson’s χ 2 test for H0 : (X1, X2, X3, X4)
has multinomial distribution with parameters (θ1, 3θ1, θ2, 1 − 4θ1
− θ2) against HA : (X1, X2, X3, X4) has some other multinomial
distribution, at the significance level α = 0.05. (b) Apply the
test in (a) to the data X = (26, 52, 34, 18

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...

You see scatter plots of three pairs: Y versus X1, Y versus X2,
and Y versus X3. If each of the first two scatters is fairly close
to a straight line, but the third is more a shapeless swarm of
points, which of the following can you conclude?
a. It is possible that a regression with all three X's could
have an appreciably higher R-square value than a regression with
only X1 and X2, due to the correlation structure among...

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