Question

Consider the linear regression model ? = ? +?? + ? Suppose the variance of e increases as X increases. What implications, if any, does this have for the OLS estimators and how would you proceed to estimate β in this case.

Answer #1

Suppose that your linear regression model includes a constant
term, so that in the linear regression model
Y = Xβ + ε
The matrix of explanatory variables X can be
partitioned as follows: X = [i X1]. The
OLS estimator of β can thus be partitioned
accordingly into b’ = [b0
b1’], where b0 is
the OLS estimator of the constant term and
b1 is the OLS estimator of the slope
coefficients.
a) Use partitioned regression to derive formulas for...

In the linear regression model ? = ?0 + ?1? + ?, let y be the
selling price of a house in dollars and x be its living area in
square feet. Define a new variable ? ∗ = ? − 1000 (that is, ? ∗ is
the square feet in excess of 1000), and estimate the model ? = ?0 ∗
+ ?1 ∗? ∗ + ?.
a] Show the relationship between the OLS estimators ?1̂∗ and ?̂1
....

CW 2
List 5 assumptions of the simple linear regression model.
You have estimated the following equation using OLS:
ŷ = 33.75 + 1.45 MALE
where y is annual income in thousands
and MALE is an indicator variable such that it is 1 for
males and 0 for females.
a) According to this model, what is the average income for
females?
b) According to this model, what is the average income for
females?
c) What does OLS stand for? How...

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]

Multiple Linear Regression
We consider the misspecification problem in multiple linear
regression. Suppose that the following model is adopted y = X1β1 +
ε while the true model is y = X1β1 + X2β2 + ε. For both models, we
assume E(ε) = 0 and V (ε) = σ^2I. Figure out conditions under which
the least squares estimate we obtained is unbiased.

Consider the simple linear regression model and let e = y
−y_hat, i = 1,...,n be the least-squares residuals, where y_hat =
β_hat + β_hat * x the fitted values.
(a) Find the expected value of the residuals, E(ei).
(b) Find the variance of the fitted values, V ar(y_hat ). (Hint:
Remember that y_bar i and β1_hat are uncorrelated.)

Consider the simple linear regression model for which the
population regression equation can be written in conventional
notation as: yi= Beta1(xi)+
Beta2(xi)(zi)2+ui
Derive the Ordinary Least Squares estimator (OLS) of beta
i.e(BETA)

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

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...

Consider the simple linear regression model y=10+30x+e where the
random error term is normally and independently distributed with
mean zero and standard deviation 1. Do NOT use
software. Generate a sample of eight observations, one each at the
levels x= 10, 12, 14, 16, 18, 20, 22, and 24.
Do NOT use software!
(a) Fit the linear regression model by least squares and find
the estimates of the slope and intercept.
(b) Find the estimate of ?^2 .
(c) Find...

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