Reading 2: Multiple Regression

Question 1

Adjusted R2

Answer 1

1 - ((n-1)/(n-k-1)*(1-R2))

• R2 increases as variables are added to the model. Helps with “overestimation of the regression”
• R2a will always be less than or equal to R2

Question 2

Dummy Variables

Answer 2

• binary (either on or off)

- n class = n-1 dummy variables

Question 3

Heteroskedasticity

Answer 3

• Occurs when the variance of the residuals is not constant across all observations
• Unconditional: not related to level of independent variables (causes no major problems)
• Conditional: related to level of independent variables and causes problems.

Question 4

Effects of heteroskedasticity on regression analysis

Answer 4

1) Standard errors are unreliable
2) Coefficients are unaffected
3) t-stats will be too big or too small
4) F-test is unreliable

Question 5

Detecting Heteroskedasticity

Answer 5

• examine the scatter plots of the residuals
• Breusch-Pagan chi-square test: n * (R2 from a second regression from the squared residuals of the first regression on independent variables)
• *one-tailed test because heteroskedasticity is only a problem if the R2 and BP test statistic are too large.

Question 6

Correcting Heteroskedasticity

Answer 6

Option 1: Calculate robust standard errors (White-corrected standard errors)
Option 2: Generalized least squares: eliminates heteroskedasticity by modifying the original equation

Question 7

Serial Correlation (autocorrelation)

Answer 7

-residual terms are correlated with one another

Positive: positive regression in one time period increases the probability of observing a positive regression error for the next time period.
Negative: negative regression in one time period increases the probability of observing a negative regression error for the next time period.

Question 8

Effect of Serial Correlation on Regression Analysis

Answer 8

• Results in standard errors that are too small
• Small standard errors will cause computed t-stats to be larger than they should be, which will cause too many Type I errors (rejection of null when it is actually true)
• F-test will also be unreliable because the MSE will be underestimated leading to too many Type I errors

Question 9

Detecting Serial Correlation

Answer 9

-Residual plots
-Durbin-Watson statistic: 2(1-r)
-r = correlation coefficient b/w residuals from one period and those from the
previous period

Rules:

• DW = 2 (homoskedastic and not serially correlated, r = 0)
• DW < 2 (positively serially correlated)
• DW > 2 (negatively serially correlated)

Question 10

Durbin Watson decision rule

Answer 10

Ho: Regression has no positive serial correlation

There are upper and lower critical DW-values:

• If DW < d1; the error terms are positively serially correlated (reject null)
• If dl < DW < du, test is inconclusive
• If DW > du, there is no evidence that the error terms are positively correlated.

Question 11

Correcting Serial Correlation

Answer 11

• Adjust the coefficient standard errors: Hansen method
  
  • Hansen method also correct for conditional heteroskedasticity (use if both are the
    issue)
• Improve specification of the model

Question 12

Multicollinearity

Answer 12

-refers to the condition when two or more independent variables or linear combinations of independent variables are highly correlated with each other

Question 13

Effects of multicollinearity on regression analysis

Answer 13

• unreliable coefficients
• standard errors are artificially inflated
• Greater probability of Type II error

Question 14

Detecting Multicollinearity

Answer 14

• t-tests are not significantly different from zero, while F-test is significant and R2 is high
• .7 is typically the level of correlation where multicollinearity is an issue

Question 15

Levels of Misspecification

Answer 15

1) Functional form can be misspecified
- important variables are omitted
- variables should be transformed
- data is improperly pooled: wrong time period chosen
2) Explanatory variables are correlated with the error term in time series models
- a lagged dependent variable is used as independent variable
- a function of the dependent variable is used as an independent variable (“forecasting the past”)
- Independent variables are measured with error
3) Other time-series misspecifications that result in nonstationary

Question 16

Unbiased estimator

Answer 16

-expected value of the estimator is equal to the parameter you are trying to estimate.

Question 17

Consistent estimator

Answer 17

• accuracy of the parameter estimate increases as the sample size increases.
• as sample size approaches infinity, standard error approaches zero