Question

Consider the complementary log log model log(− log(1 − π(x))) = β0 + β1x.

Show the greatest rate of change of π(x) occurs at x = −β0/β1.

Answer #1

Consider the complementary log log model log(− log(1 − π(x))) =
β0 + β1x.
Show the greatest rate of change of π(x) occurs at x =
−β0/β1.

For the model ln(y) = β0 + β1x + ε, β1 is the approximate
percentage change in Y when x increases by 1%. True or false?

Suppose y is determined by the true model
y=β0+β1x+β2z+ε, and that
β2 >0 and COV(z,x) < 0. If someone were interested
in estimating β1, and did so by using OLS to estimate y
= β0 + β1x + u, would the OLS estimator of β1
be biased or not? If it is unbiased, explain why. If it is biased,
is the bias positive or negative? Why?

Consider the model ln(Yi)=β0+β1Xi+β2Ei+β3XiEi+ui, where Y is an
individual's annual earnings in dollars, X is years of work
experience, and E is years of education. Consider an individual
with a high-school degree (E=12yrs) who has been working for 20
years. The expected increase in log earnings next year (when
X=21yrs) compared to this year is, dropping units,
β1
β1+12β3
β1+β3
β0+21β1+12β2+252β3

Suppose you are given the following
simple dataset, regress Y on X:
y=β0+β1x+u
X
Y
1
2
2
4
6
6
Calculate β0 andβ1Show algebraic
steps.
Interpret β0 and β1
Calculate the predicted(fitted)value of each observation
Calculate the residual ofeach observation
When x=3, what is the predicted value of Y?
Calculate SSR, SST, and then SSE.
How much of the variation in Y is explained by X?
8)Calculate the variance estimator

Model: Y = β0 + β1X + u If E(u|X) 6= 0, then we know the OLS
estimator will be biased and all further inference like Hypothesis
test and Confidence interval will be invalid. In presence of such
violation, we can go to Instrument variable estimation/regression
method to rebuild the valid empirical study. Assume there is a
“good” instrument variable Z for X. (1) How would you argue this is
a valid instrument variable?(Hint: validity condition and relevance
condition) (2)...

Model: Y = β0 + β1X + u If E(u|X) 6= 0, then we know the OLS
estimator will be biased and all further inference like Hypothesis
test and Confidence interval will be invalid. In presence of such
violation, we can go to Instrument variable estimation/regression
method to rebuild the valid empirical study. Assume there is a
“good” instrument variable Z for X. (1) How would you argue this is
a valid instrument variable?(Hint: validity condition and relevance
condition) (2)...

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

The statistical model
for simple linear regression is written as μy
= β0 +
β1*x, where μy
represents the mean of a Normally distributed response variable and
x represents the explanatory variable. The parameters
β0 and β1 are estimated,
giving the linear regression model defined by
μy = 70 + 10*x , with standard
deviation σ = 5.
(multiple choice
question)
What is the
distribution of the test statistic used to test the null hypothesis
H0 : β1 =
0...

Consider the multiple linear regression model
y = β0 +β1x1 +β2x2 +β3x3 +β4x4 +ε
Using the procedure for testing a general linear hypothesis, show
how to test
a. H 0 : β 1 = β 2 = β 3 = β 4 = β
b. H 0 : β 1 = β 2 , β 3 = β 4
c. H0: β1-2β2=4β3
β1+2β2=0

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