Question

Consider the simple linear regression model for which the
population regression equation can be written in conventional
notation as: yi= **Beta**1(xi)+
Beta2(xi)(zi)^{2}+ui

Derive the Ordinary Least Squares estimator (OLS) of beta i.e(BETA)

Answer #1

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

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...

Suppose you have a cross-country dataset with values for GDP
(yi) and investment in research & development (xi). Describe
the method of ordinary least squares (OLS) to estimate the
following univariate linear regression model, i.e.
yi = β0 + β1 xi + εi
In particular, describe in your words which are the dependent
and the explanatory variables; how the OLS estimation method works;
how to interpret the estimates for the coefficients β0 and β1; what
is the coefficient of determination...

Multiple choice!
Consider the model Yi = B0 + B1X1i + B2X2i + B3X3i + B4X4i + Ui.
To test the null hypothseis of B2 = B3 = 0, the restricted
regression is:
A. Yi = B0 + B1X1i + B2X2i + B3X3i + B4X4i + Ui
B. Yi = B0 + Ui
C. Yi = B0 + B1X1i + B4X4i + Ui
D. Yi = B0 + B2X2i + B3X3i + Ui
Consider the model Yi = B0 +...

True or False: In the simple regression model, both ordinary
least squares (OLS) and Method of Moments estimators produce
identical estimates. Explain.

1. Consider the following linear regression model which
estimates only a constant:
Yi = β1 + ui
What will the value of ˆβ1 be? Remember we are minimizing the sum
of the squared residuals.
2. Consider the following regression model with K parameters:
Yi = β1 + β2X2i + β3X3i + ... + βKXKi + ui
Now consider the F-test of the null hypothesis that all slope
parameters (β2,β3,...,βK) are equal to zero. Using the equation
from class:
F =（(RSSk...

Question One
Consider the classical simple linear regression model:
?? = ?0 + ?1?? + ??, ?? ~ ?. ?. ?. (0, ?2)
(a) Provide, with details, the appropriate expressions for ?(??)
& ???(??). Assume
that ?(??) = ??, ???(??) = ?^2? & ???(??, ??) = 0; i.e., that
?? & ?? are
uncorrelated.
2.5 marks
(b) Suppose now that the explanatory variable ?? is correlated with
?? such that
???(??, ??) = ???. Under this scenario, derive ?(??) &...

Showing that residuals, , from the least squares fit of the
simple linear regression model sum to zero

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 following (generic) population regression
model:
Yi = β0 + β1X1,i + β2X2,i + β3X3,i + ui, i = 1,...,n . Transform
the regression to allow you to easily test the null hypothesis that
β1 + β3 = 1. State the new null hypothesis associated to this
transformed regression.

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