3. MLR - Estimation

Question 1

What are the advantages of a multiple regression over a simple regression?

Answer 1

• Incorporate more explanatory factors into the model
• Explicitly hold fixed other factors that otherwise would be in the error term u
• Allow for more flexible functional forms

Question 2

What are slope parameters?

Answer 2

Parameters other than the intercept

Question 3

How do the methods used change for a regression that has a quadratic function?

Answer 3

Mechanically there will be no difference in using the mehtod of OLS to estimate the parameters but there is a difference in the interpretation of the coefficients

Question 4

How can a model be more flexible?

Answer 4

A model can contain logs and quadratic forms suggesting different relationships between the explanatory variable and the dependent variable

Question 5

What is the partial effect?

Answer 5

Where you fix one variable and see what happens to y when the other variable changes

Question 6

What do residuals measure?

Answer 6

The difference between the true value and the fitted value

Question 7

What is Frisch-Waugh theorem?

Answer 7

1) Regress the explanatory variable x on all other explanatory variables and obtain the residuals r^
2) Regress the dependent variable y on the resulas r^ to obtain B^

Question 8

Why does this procedure work?

Answer 8

• The residuals from the first regression is the part of the explanatory variable that is uncorrelated with the other explanatory variables
  The slope coefficient of the second regression therefore represents the isolated effect of the explanatory variable

Question 9

What are the standard assumptions for the multiple regression model?

Answer 9

MLR.1 - Linear in parameters
MLR.2 - Random sampling
MLR.3 - No Perfect collinearity
MLR.4 - Zero conditional mean

Question 10

What is perfect collinearity?

Answer 10

If an independent variable is an exact linear combination of the other independent variables then we say the model suffers from perfect collinearity and it cannot be estimated by OLS. Some exact relationship between the regressors.

Question 11

What is key to remember about the MLR.3 assumption - no perfect collinearity?

Answer 11

Assumption MLR.3 does allow the independent variables to be correlated; they just cannot be perfectly correlated

Question 12

Why, in a multiple regression model is the likelihood of the assumption of zero conditional mean more likely to hold?

Answer 12

Because fewer things end up in the error term u

Question 13

What does over-specifying the model mean?

Answer 13

One (or more) independent variable is included in the model even though it has no partial effect on y in the population

Question 14

Why do we use OLS estimates?

Answer 14

Because they are good enough on average that they will be equal to the true value

Question 15

How do OLS estimates relate to population parameters?

Answer 15

The OLS estimators are unbiased estimators of the population parameters

Question 16

How do you interpret unbiasedness?

Answer 16

The estimated coefficients may be smaller or larger depending on the sample that as a result of it being random however on average they will be equal to the values that characterise the true relationship

Question 17

What is the issue regarding overspecifying the model?

Answer 17

Including irrelevant variables may increase sampling variance thus reducing the precision of the estimates but it has no effect on the level of unbiasedness

Question 18

Alongside poor specification of our model, when else can MLR.3 Perfect collinearity fail?

Answer 18

Assumption MLR.3 also fails if the sample size, n, is too small in relation to the number of parameters being estimated.

Question 19

What is underspecifying the model and why is it problematic?

Answer 19

Excluding a relevant variable or underspecifying the model causes the OLS to be biased

Question 20

Between simple and multiple regression analysis, which one is more likely to have omitted variable bias?

Answer 20

Simple

Question 21

What are EXOGENOUS explanatory vairables?

Answer 21

When assumption MLR.4 (Zero conditional mean) holds

Question 22

What are ENDOGENOUS explanatory variables?

Answer 22

If xj is correlated with u for any reason, then xj is said to be an endogenous explanatory variable. The term “endogenous explanatory variable” has evolved to cover any case in which an explanatory variable may be correlated with the error term.

Question 23

What is the difference between MLR.3 and MLR.4?

Answer 23

Assumption MLR.3 rules out certain relationships among the independent or explanatory variables and has nothing to do with the error, u. You will know immediately when carrying out OLS estimation whether or not Assumption MLR.3 holds. On the other hand, Assumption MLR.4—the much more important of the two—restricts the relationship between the unobserved factors in u and the explanatory variables. Unfortunately, we will never know for sure whether the average value of the unobserved factors is unrelated to the explanatory variables.

Question 24

What notation change do we observe when we know we are underspecifying our model?

Answer 24

We use the symbol “~” rather than “^” to emphasise that ߘ1 comes from an underspecified model.

Question 25

In the context of omitting a variable from your analysis, when do we say a sample has upward bias?

Answer 25

In the context of omitting a variable if E(ߘ1) > ß1, then we say that ߘ1 has an upward bias

Question 26

In the context of omitting a variable from your analysis, when do we say a sample has downward bias?

Answer 26

When E(ߘ1) < ß1, ߘ1 has a downward bias. These definitions are the same whether ß1 is positive or negative.

Question 27

In the context of omitting a variable from your analysis, when do we say a sample is biased towards zero?

Answer 27

The phrase biased toward zero refers to cases where E(ߘ1) is closer to zero than is ß1.

Question 28

What type of bias do we identify if ß1 is positive and biased towards 0?

Answer 28

If ß1 is positive, then ߘ1 is biased toward zero if it has a downward bias.

Question 29

What type of bias do we identify if ß1 is negative and biased towards 0?

Answer 29

On the other hand, if ß1 < 0, then ߘ1 is biased toward zero if it has an upward bias.

Question 30

What does correlation between a single explanatory variable and the error usually suggest?

Answer 30

Correlation between a single explanatory variable and the error generally results in all OLS estimators being biased. The only exception to this is when x1 and x2 are also uncorrelated.

Question 31

What is homoskedasticity?

Answer 31

The error u has the same variance given any value of the explanatory variables. In other words, Var (u|x1, …, xk) = σ^2.

Question 32

For a given dependent variable, what is the only way to reduce error variance?

Answer 32

For a given dependent variable y, there is really only one way to reduce the error variance, and that is to add more explanatory variables to the equation (take some factors out of the error term). Unfortunately, it is not always possible to find additional legitimate factors that affect y.

Question 33

What is multicollinearity?

Answer 33

High (but not perfect) correlation between two or more independent variables is called multicollinearity

Question 34

What is the systematic part of a model?

Answer 34

The part consisting of explanatory variables

Question 35

What is imperfect multicollinearity?

Answer 35

The sampling variance of the slope estimator for xj will be higher when xj can be better explained by the other independent variables

Question 36

What does omitted variable bias result in?

Answer 36

You will end up being very precise to something that isn’t true.

Question 37

What happens to the slope estimator under perfect multicollinearity?

Answer 37

The variance of the slope estimator will approach infinity

Question 38

How can multicollinearity be reduced?

Answer 38

Grouping similar variables or by dropping some independent variables however this may lead to omitted variable bias

Question 39

What is the choice to include a specific variable motivated by?

Answer 39

The choice can be made by analysing the tradeoff between bias and variance

Question 40

What does OLS are BLUE mean?

Answer 40

Best Linear Unbiased Estimator

Question 41

What assumptions have to be satisfied for OLS to be BLUE?

Answer 41

The Gauss-Markov assumptions

Question 42

How do you find which unbiased estimator has the smallest variance?

Answer 42

In order to answer this question one usually limits oneself to linear estimators, i.e estimators linear in the dependent variable

Question 43

What will be the best prediction of y?

Answer 43

It’s conditional expectation

Question 44

What happens in the model if a relevant variable is excluded from the model?

Answer 44

This induces a bias in B^~1 unless x1 and x2 are uncorrelated

Question 45

What is the cost of including an irrelevant variable in the model?

Answer 45

A higher variance for the estimator of B1

Question 46

How do you calculate the degrees of freedom?

Answer 46

Count the number of parameters, including the intercept, and subtract this amount from the number of observations. (In the rare case that an intercept is not estimated, the number of parameters decreases by one.)

Question 47

How can we detect multicollinearity?

Answer 47

Using variance inflation factors (vif)