Econometrics || with 100% Accurate Solutions.
2a) Let income denote the person's income and age denote the persons age. We would like to run
an OLS regression on the equation BMI=B0+B1fast+B2income+B3age+U. Write down all the
assumptions that guarantee that the OLS estimators of the coefficients of this equation are
unbiased correct answers 1)Linearity- E[U|fast, income, age]=0, The expected value of y is linear
in X1,X2,X3
2)Exogeneity- E[U|fast, income, age]=0, unobservables are constant and do not effect BMI
3)Random Sampling- The data was randomly collected
4)No Perfect Collinearity- No regressor is constant and don't perfectly predict each other
2B) If the model is true, is the OLS regression a good method for discovering the value of the
coefficients? Why? correct answers If assumptions in a) are true then OLS is a good way to
estimate linear relationship between BMI, fast, income, and age
2C) This model is far from realistic. At the very least, we should also include information about
whether the person is employed or not (say work=1 if the person is employed, work=0 if the
person is not). Describe in as much depth as you can what is the bias resulting of omitting work
from the model correct answers Some variation in BMI explained by the variable fast could need
to be attributed to work. Where BetaOneHat equals BetaOne plus BetaHatWork times
ThetaHatOne where BetaHatWork is the slope of work in BMI regression and ThetaOneHat is
the slope of fast on work where work=Theta0+Theta1work+Theta2income+Theta3age+U
BetaHatWork effect on BMI: Positive, people who work are more healthy
ThetaHatOne effect on BMI: Positive, people who work might eat out more
B1hat>B1
Is B1hat > B1
Is B1hat < B1
I think B(omit)hat is positive/negative because
I think ThetaOneHat is positive/negative because
If B(omit)hat and B1Hat have the same sign,
B1Hat > B1
If B(omit)hat and ThetaOneHat have the same sign,
B1Hat < B1
2D) Suppose that our data was collected inside of a country which has 4 towns (t1, t2, t3, t4). We
have information about exactly where the person lives (which town). Write the new model which
incorporates the employment and living location information, and interpret B0. Do you expect it
to be higher or lower than B0 in the model in item a) correct answers New Model: y =
B0+B1fast+B2income+B3age+B4work+B5t2+B6t3+B7t4+U
, In a) B0 assumed that fast=income=age=0
In d) B0 assumed that fast=income=age=work=t2=t3=t4=0
Since t2=t3=t4=0 t1 has to equal 1 which means that B0 would be higher in d) if you thought that
people in t1 had higher BMIs once controlling for income, fast, and age
2E) What is the variance of BetaOneHat in the new model from item (d)? Do you expect that it
will be bigger or smaller than in the original model in item (a)? Explain. correct answers
Variance of B1Hat is (sigma squared divided by n) over ((the expected variance of X1) times (1
minus r-squared i)) where sigma squared is the variance(U) and (1 minus r-squared i) is the
regression of X1 onto other controls
By adding d variables, n and the expected variance of X1 remain unchanged
If variables predict y, then sigma squared will decrease and will tend to lower variance of B1Hat
If variables predict X1, (1 minus r-squared i) will increase and will tend to increase variance of
B1Hat
Which do I think is more important
2F) Should we incorporate the interactions between work and the locations where people live?
Why? Incorporate the interactions, and write the new model. Interpret B2. Do you expect it to be
higher or lower than B2 in the model you wrote in item (d)? correct answers Including this
would create superfluous variables so they should not be included. The interpretation of B2
remains unchanged
y=B0+B1X1+B2X2+B3X3+B4X4+B5d2+B6d3+B7d2X4+B8d2X4+U
2G) If we really wanted to write a causal model of the effect of fast food consumption on
obesity, what else should we include? Write down as many variables as you can up to 20. The
variables should be different enough that redundancy is unlikely, and the association should be
obvious to anyone. Choose one of them, and defend why it should have been included among the
controls in the model in item (f) correct answers Create 10 BS variables. Choose one that fits
best. This variable should be included because
-It is associated with the treatment
-It is associated with the outcome
-It is not redundant
2H) If BetaOneHat = .5 and SE(BetaFiveHat) = .2, would you say that fast food consumption
impacts a person's BMI? Why? correct answers Test stat = absolute value of B1Hat divided by
Standard error of B1Hat.
Fail to reject if T-stat < crit value, x does not influence y
Reject null if T-stat > crit value, x has influence on y
2a) Let income denote the person's income and age denote the persons age. We would like to run
an OLS regression on the equation BMI=B0+B1fast+B2income+B3age+U. Write down all the
assumptions that guarantee that the OLS estimators of the coefficients of this equation are
unbiased correct answers 1)Linearity- E[U|fast, income, age]=0, The expected value of y is linear
in X1,X2,X3
2)Exogeneity- E[U|fast, income, age]=0, unobservables are constant and do not effect BMI
3)Random Sampling- The data was randomly collected
4)No Perfect Collinearity- No regressor is constant and don't perfectly predict each other
2B) If the model is true, is the OLS regression a good method for discovering the value of the
coefficients? Why? correct answers If assumptions in a) are true then OLS is a good way to
estimate linear relationship between BMI, fast, income, and age
2C) This model is far from realistic. At the very least, we should also include information about
whether the person is employed or not (say work=1 if the person is employed, work=0 if the
person is not). Describe in as much depth as you can what is the bias resulting of omitting work
from the model correct answers Some variation in BMI explained by the variable fast could need
to be attributed to work. Where BetaOneHat equals BetaOne plus BetaHatWork times
ThetaHatOne where BetaHatWork is the slope of work in BMI regression and ThetaOneHat is
the slope of fast on work where work=Theta0+Theta1work+Theta2income+Theta3age+U
BetaHatWork effect on BMI: Positive, people who work are more healthy
ThetaHatOne effect on BMI: Positive, people who work might eat out more
B1hat>B1
Is B1hat > B1
Is B1hat < B1
I think B(omit)hat is positive/negative because
I think ThetaOneHat is positive/negative because
If B(omit)hat and B1Hat have the same sign,
B1Hat > B1
If B(omit)hat and ThetaOneHat have the same sign,
B1Hat < B1
2D) Suppose that our data was collected inside of a country which has 4 towns (t1, t2, t3, t4). We
have information about exactly where the person lives (which town). Write the new model which
incorporates the employment and living location information, and interpret B0. Do you expect it
to be higher or lower than B0 in the model in item a) correct answers New Model: y =
B0+B1fast+B2income+B3age+B4work+B5t2+B6t3+B7t4+U
, In a) B0 assumed that fast=income=age=0
In d) B0 assumed that fast=income=age=work=t2=t3=t4=0
Since t2=t3=t4=0 t1 has to equal 1 which means that B0 would be higher in d) if you thought that
people in t1 had higher BMIs once controlling for income, fast, and age
2E) What is the variance of BetaOneHat in the new model from item (d)? Do you expect that it
will be bigger or smaller than in the original model in item (a)? Explain. correct answers
Variance of B1Hat is (sigma squared divided by n) over ((the expected variance of X1) times (1
minus r-squared i)) where sigma squared is the variance(U) and (1 minus r-squared i) is the
regression of X1 onto other controls
By adding d variables, n and the expected variance of X1 remain unchanged
If variables predict y, then sigma squared will decrease and will tend to lower variance of B1Hat
If variables predict X1, (1 minus r-squared i) will increase and will tend to increase variance of
B1Hat
Which do I think is more important
2F) Should we incorporate the interactions between work and the locations where people live?
Why? Incorporate the interactions, and write the new model. Interpret B2. Do you expect it to be
higher or lower than B2 in the model you wrote in item (d)? correct answers Including this
would create superfluous variables so they should not be included. The interpretation of B2
remains unchanged
y=B0+B1X1+B2X2+B3X3+B4X4+B5d2+B6d3+B7d2X4+B8d2X4+U
2G) If we really wanted to write a causal model of the effect of fast food consumption on
obesity, what else should we include? Write down as many variables as you can up to 20. The
variables should be different enough that redundancy is unlikely, and the association should be
obvious to anyone. Choose one of them, and defend why it should have been included among the
controls in the model in item (f) correct answers Create 10 BS variables. Choose one that fits
best. This variable should be included because
-It is associated with the treatment
-It is associated with the outcome
-It is not redundant
2H) If BetaOneHat = .5 and SE(BetaFiveHat) = .2, would you say that fast food consumption
impacts a person's BMI? Why? correct answers Test stat = absolute value of B1Hat divided by
Standard error of B1Hat.
Fail to reject if T-stat < crit value, x does not influence y
Reject null if T-stat > crit value, x has influence on y
