Question

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) Given the “good” instrument variable Z for X, what are the two steps to get the estimator of β1. (2SLS: two stage least square, just write out the two regression models)

Answer #1

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)...

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?

True or False:
( ) Consider the model y=α1+β1x+u, where x is an
endogenous covariate and z is a valid instrument for
x. The model x=α2+β2z+u2 can be said to be a reduced-form
equation for x.
( ) Valid instrumental variables are usually very difficult to
find.
( ) 2SLS is (under certain assumptions) the most efficient IV
estimator.
( ) The normal distribution of the error term is a crucial
assumption for making inferences using the 2SLS estimator.

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

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