Question

β_hat is the OLS estimator which is the vector form of all β in the regression

Assume Cov(X,U) = 0. Show that (e^xβ_hat)−1 is a biased estimator for (e^xβ) -1 . Show that e^x̂β −1 is a consistent estimator for e^xβ−1.

Answer #1

a. If the OLS estimator is unbiased for the true population
parameter, is the OLS estimate necessarily equal to the population
parameter? Explain your answer in
detail.
b. Suppose that the true population regression (data generating
process) is given by Y i = B 0 + B 1 X i +u i .
Further suppose that the population covariance between X i and u
i is equal to some positive value A , rather than zero: COV(X i ,u
i...

Based on the definition of the linear regression model in its
matrix form, i.e., y=Xβ+ε, the assumption that
ε~N(0,σ2I), and
the general formula for the point estimators for the parameters of
the model
(b=XTX-1XTy);
show:
how to derivate the formula for the point estimators for the
parameters of the models by means of the Least Square Estimation
(LSE). [Hint: you must minimize
ete]
that the LSE estimator, i.e.,
b=XTX-1XTy,
is unbiased. [Hint:
E[b]=β]

1) Which of the following does not generally decrease the
variance of the OLS estimator of slope βˆ1?
a) Increasing the variance of the error term, b) Increasing the
sample size, c) None of the above, d) Increasing the variance of
the independent variable
2) If the independent variable is a binary variable then which
of the following is true?
a)β0 is a population mean for the group with a value of 1 for
the independent variable,
b) β1 is...

If the errors in the CLR model are not normally distributed,
although the OLS estimator is no longer BLUE, it is still
unbiased.
In the CLR model, βOLS is biased if explanatory
variables are endogenous.
The value of R2 in a multiple regression cannot be
high if all the estimates of the regression coefficients are shown
to be insignificantly different from zero based on individual
t tests.
Suppose the CNLR applies to a simple linear regression y =
β1 +...

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

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)...

Lecture 9 we examined robust, regression-based Lagrange
Multiplier test statistics for testing the functional form of a
conditional mean where
H0 : m(xi, θ) = m(xiβ)
HA : m(xi, θ) = m(xiβ + δ1(xiβ)2 + δ2(xiβ)3)
Suppose that we want to perform a similar test for the
functional form of a probit model, i.e.,
H0 : G(xiθ) = Φ(xiβ)
HA : G(xiθ) = Φ(xiβ + δ1(xiβ)2 + δ2(xiβ)3)
Using the method from Lecture 9, explain which regression you
would run...

a)
In a regression, if you have used a variable which is in the form
of a dummy variable (0= not white, 1= white) such as; lnw= alpha +
beta white + u, does the regression in excel also show the value of
non white? If so, how?
b) Also, is an r squared value of 0.19 considered good or not?
What does it mean?
Plese provide easy to understand and clear answers which
perfectly answer the question only.

Based on the definition of the linear regression model in its
matrix form, i.e., y=Xβ+ε, the assumption that
ε~N(0,σ2I), the
general formula for the point estimators for the parameters of the
model
(b=XTX-1XTy),
and the definition of
varb=Eb-Ebb-EbT
Show that
varb=σ2XTX-1
Note: the derivations in here need to be done in matrix form.
Simple algebraic method will not be allowed.

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