Multiple Regression

Question 1

We can use multiple regression models to:

Answer 1

1 - Identify relationships between variables
2 - Forecast variables
3 - Test existing theories

Question 2

The general multiple linear regression model is:

Answer 2

Yi = b0 + b1X1i + b2X2i + … + bkXki + εi

Question 3

The residual, ε is

Answer 3

the difference between the observed value, Yi, and the predicted value from the regression, Y

Question 4

The p-value is

Answer 4

the smallest level of significance for which the null hypothesis can be rejected

Question 5

If the p-value is less than the significance level

Answer 5

the null hypothesis can be rejected

Question 6

If the p-value is greater than the significance level

Answer 6

the null hypothesis cannot be rejected.

Question 7

intercept term

Answer 7

is the value of the dependent variable when the independent variables are all equal to zero.

Question 8

Assumptions underlying a multiple regression model include

Answer 8

• linear relationship exists between x + y.
• residuals are normally distributed.
• variance of the error terms is constant.
• residual are not correlated.
• variables are not random, and no exact linear relation between variables.

Question 9

R2

Answer 9

evaluates the overall effectiveness of the entire set of independent variables in explaining the dependent variable

R2 = total variation-unexplained variation / total variation

R2 = SST-SSE / SST

R2 = explained variation / total variation

R2 = RSS / SST

Question 10

Adjusted R2

Answer 10

R2a =1- [(n-1/n-k-1)×(1-R2)]

Question 11

AIC

Answer 11

better forecast

Question 12

BIC

Answer 12

goodness of fit

Question 13

nested models

Answer 13

one model, called the full model or unrestricted model, has a higher number of independent variables

Question 14

Restricted model

Answer 14

subset of the independent variables

Question 15

F-statistic

Answer 15

F= (SSER-SSEU)/q / (SSEU)/(n-k-1)

=(RSSU)/k / (SSEU)/(n-k-1)

Question 16

reject H0 if

Answer 16

F (test-statistic) > Fc (critical value)

Question 17

F-test evaluates whether

Answer 17

the relative decrease in SSE due to the inclusion of q additional variables is statistically justified.

Question 18

Regression model specification

Answer 18

selection of the explanatory (independent) variables to be included in a model

Question 19

Examples of Misspecification of Functional Form

Answer 19

Misspecification #1: Omitting a Variable

Misspecification #2: Variable Should Be Transformed

Misspecification #3: Inappropriate Scaling of the Variable

Misspecification #4: Incorrectly Pooling Data

Question 20

Omission of important independent variable(s) effect

Answer 20

Biased and inconsistent regression parameters

serial correlation or heteroskedasticity

Question 21

Inappropriate variable form effect

Answer 21

heteroskedasticity

Question 22

Inappropriate variable scaling effect

Answer 22

heteroskedasticity or multicollinearity

Question 23

Data improperly pooled effect

Answer 23

heteroskedasticity or serial correlation

Question 24

Omission of important independent variable(s)

Answer 24

one or more variables that should have been included are omitted.

Question 25

Inappropriate variable form

Answer 25

The relationship between the dependent and independent variables may be non-linear.

Question 26

Inappropriate variable scaling

Answer 26

Variables may need to be transformed

Question 27

Data improperly pooled

Answer 27

Sample has periods of dissimilar economic environments

Question 28

Heteroskedasticity

Answer 28

variance of the residuals is not the same across all observations

Question 29

Unconditional heteroskedasticity

Answer 29

heteroskedasticity is not related to the level of the independent variables

Question 30

Conditional heteroskedasticity

Answer 30

related to (i.e., conditional on) the level of the independent variables

Question 31

Effect of Heteroskedasticity on Regression Analysis

Answer 31

1- standard errors are unreliable estimates. (Type I errors) 2- F-test is unreliable.

Question 32

Detecting Heteroskedasticity

Answer 32

scatter plots Breusch Pagan chi-square (χ2) test

Question 33

BP chi-square test statistic=

Answer 33

n × R2resid with k degrees of freedom n=the number of observations k=the number of independent variables

Question 34

To correct for conditional heteroskedasticity

Answer 34

robust standard errors, used to recalculate the t-statistics

Question 35

Serialcorrelation / auto correlation

Answer 35

regression residual terms are correlated with one another

Question 36

Effect of Serial Correlation

Answer 36

Consider a model that employs a lagged value of the dependent variable as one of the independent variables. Residual autocorrelation in such a model causes the estimates of the slope coefficients to be inconsistent. If the model does not have any lagged dependent variables, then the estimates of the slope coefficient will be consistent.

Question 37

Effect on Standard Errors (Serial Correlation)

Answer 37

Positive serial correlation results in coefficient standard errors that are too small, causing t-statistics (and F-statistic) to be larger than they should be, leads to Type I errors.

Question 38

Detecting Serial Correlation

Answer 38

Durbin--Watson (DW) statistic Breusch--Godfrey (BG) test

Question 39

Breusch--Godfrey (BG) test

Answer 39

regresses the regression residuals against the original set of independent variables, plus one or more additional variables representing lagged residual(s)

Question 40

Correction for Serial Correlation

Answer 40

robust standard errors used to recalculate the t-statistics using the original regression coefficients.

Question 41

Multicollinearity

Answer 41

two or more of the independent variables in a multiple regression are highly correlated with each other.

Question 42

Effect of Serial Correlation on Model Parameters

Answer 42

slope coefficients are imprecise and unreliable inflates standard errors and lowers t-stats.

Question 43

Effect on Standard Errors (Multicollinearity)

Answer 43

Standard errors are too high. This leads to Type II errors.

Question 44

Detection of Multicollinearity

Answer 44

variance inflation factor (VIF)

Question 45

variance inflation factor (VIF) formula

Answer 45

VIFj = 1 / (1 – R2j)

Question 46

VIF values

Answer 46

greater than 5 (i.e., R2 > 80%) warrant further investigation, above 10 (i.e., R2 > 90%) indicates severe multicollinearity.

Question 47

Correction for Multicollinearity

Answer 47

omit one or more of the correlated independent variables increase sample size

Question 48

We can identify outliers using

Answer 48

studentized residuals

Question 49

studentized residuals steps

Answer 49

extreme observations that, when excluded, cause a significant change to model coefficients.

Question 50

Detecting Influential Data Points

Answer 50

Cook's distance

Question 51

Cook's distance

Answer 51

composite metric (i.e., it takes into account both the leverage and outliers) for evaluating if a specific observation is influential.

Question 52

Cooks Distance Formula

Answer 52

D = e2/k×MSE x [h(1-h2)] k= number of independent variables MSE = mean square error of the regression model h= leverage value

Question 53

Cooks Distance (Di values)

Answer 53

greater than √k/n indicate influential data point

Question 54

Logistic regression (logit) models

Answer 54

ln(p1−p) = b0+b1X1+b2X2+...+ε

Question 55

likelihood ratio (LR)

Answer 55

LR = –2 (log likelihood restricted model – log likelihood unrestricted model)

Question 56

likelihood ratio (LR) values

Answer 56

Negative . (closer to 0) indicate a better-fitting restricted model.

Question 57

high-leverage points

Answer 57

extreme observations of the independent or 'X' variables

Question 58

Influential data points

Answer 58

extreme observations that when excluded cause a significant change in model coefficients

Question 59

Qualitative independent variables (dummy variables)

Answer 59

effect of a binary independent variable