Quantitative Methods

Question 1

Sum of Squared Errors (SSE)

Answer 1

Measures the unexplained variation in the dependent variable. Is the sum of the squared vertical differences between the actual values and the predicted values on the regression line.

Question 2

Breusch-Pagan (BP) Chi-Square Test

Answer 2

=n*residualR^2 with k degrees of freedom

If larger than the one-tailed critical value for a chi-square distribution, you should reject the null and conclude heteroskedasticity is present.

Question 3

Effect of Serial Correlation

Answer 3

• Positive serial correlation typically results in coefficient standard errors that are too small. This will cause the t-statistic to be too high and the null to be rejected too often.
• F-test will be unreliable because MSE will be underestimated

Question 4

F-Statistic

Answer 4

=MSR/MSE

If F-statistic > F-value, then you reject the null hypothesis, indicating at least one coefficient is statistically significant.

Question 5

Unconditional Heteroskedasticity

Answer 5

Occurs when the heteroskedasticity is not related to the level of independent variables. It usually causes no major problems with regression.

Question 6

Conditional Heteroskedasticity

Answer 6

Exists if the variance of the residual term increases as the value of the independent variable increases. Creates significant problems for regression and statistical inference.

Question 7

Adjusted R^2

Answer 7

=1 - { [ (n-1) / (n-k-1) ] x ( 1-R^2) }

Will always be equal to or less than R^2

Question 8

Root Mean Squared Error (RMSE)

Answer 8

Used to compare the accuracy of autoregressive models in forecasting out-of-sample values. The model with lower RMSE for the out-of-sample data will have lower forecast error and will be expected to have better predictive power in the future.

Question 9

Correcting Multicollinearity

Answer 9

Omit one or more of the correlated independent variables

Question 10

Random Walk with a Drift

Answer 10

The time series is expected to increase or decrease by a constant amount each period (b0).

Xt = b0 + b1*Xt-1 +εt

Question 11

Correcting Heteroskedasticity

Answer 11

• Calculate robust standard errors used to recalculate the t-statistics using the original regression coefficient if evidence of heteroskedasticity.
• Using generalized least squares, which attempts to eliminate heteroskedasticity, by modifying the original equation.

(If both heteroskedasticity and serial correlation, use the Hansen method)

Question 12

Detecting Multicollinearty

Answer 12

Fail to reject the null in a t-test, but fail to reject the null in an F-test while R^2 is high.

Question 13

Multicollinearty

Answer 13

The condition when. two or more independent variables are highly correlated with each other. This distorts the standard error of the estimate and the coefficient standard errors, leading to unreliable t-tests. Slope coefficients will be unreliable as well.

Question 14

Serial Correlation

Answer 14

The situation in which the residual terms are correlated with one another. Positive serial correlation exists when a positive regression error in one period increases the probability of observing a positive regression error in the next period. Negative serial correlation exists when a negative regression error in one period increases the probability of observing a negative regression error in the next period.

Question 15

Durbin-Watson Statistic

Answer 15

Detects the presence of serial correlation.

DW > 2 – Positive serial correlation
DW < 2 – Negative serial correlation
DW = 2 – No serial correlation
DW < dL – Reject null; positive serial correlation
DW > dU – Fail to reject null; no evidence of SC
dL < DW < dU – Test is inconclusive

Question 16

Confidence interval for Regression Coefficient

Answer 16

=b1 ± ( t x Standard Error)

t is from the t-table
Standard Error is = (coefficient / t-statistic) using n-k-1 degrees of freedom

Question 17

t-statistic

Answer 17

= ( Estimated Regression Coefficient - Hypothesized Value) / Standard Error of Coefficient
with n-k-1 degrees of freedom

Two-tailed when hypothesized value is = or ≠
One-tailed when hypothesized value is > or <

If t-statistic > t-value, then the coefficient is statistically significant and you reject the null hypothesis.

Question 18

Mean Square Error (MSE)

Answer 18

= SSE / (n-k-1)

Question 19

Regression Sum of Squares (RSS)

Answer 19

Measures the variation in the dependent variable that is explained by the independent variable. Is the sum of squared differences between predicted values and the mean.

Question 20

First Differencing

Answer 20

Used to transform a time-series with a random walk to a covariance stationary time series. Involves subtracting the value of the time series in the immediately preceding period from the current value of the time series to define a new dependent variable.

Question 21

p-value

Answer 21

The smallest level of significance for which the null hypothesis can be rejected. If the p-value is less than the significance level, the null hypothesis can be rejected. If the p-value is greater than the significance level the null hypothesis cannot be rejected (fail to reject).

Question 22

Coefficient of Determination (R^2)

Answer 22

The percentage of variation in the dependent variable that is collectively explained by all of the independent variables. R^2 will increase with more variables. √R^2 is the correlation.

= (SST - SSE) / SST

= RSS / SST

=Correlation Coefficient^2

Question 23

Standard Error of the Estimate (SEE)

Answer 23

The standard deviation of the residual terms in the regression. SEE will be low relative to total variability if the relationship between dependent and independent variables is strong, and will be high if the relationship its weak (the smaller the standard error, the better the fit)

=√MSE

=√[ SSE / (n-k-1) ]

Question 24

Cointegration

Answer 24

Two time-series are economically linked (related to the same macro variables) or follow the same trend and that relationship is not expected to change. If cointegrated, the error term from regressing one on the other is covariance stationary and the t-tests are reliable.

Question 25

Autoregressive Conditional Heteroskedasticity (ARCH)

Answer 25

Exists if the variance of the residuals in one period is dependent on the variance of the residuals in the previous period. When this condition exists, the standard errors of the regression coefficients in AR models and the hypothesis tests of these coefficients are invalid.

Question 26

Correcting Seasonality

Answer 26

An additional lag of the dependent variable (corresponding to the same period in the previous year) is added to the original model as another independent variable.

Question 27

Mean Reverting Level

Answer 27

Xt = b0 / (1 - b1)

If Xt > b0 / (1 - b1), the model predicts that Xt+1 < Xt
If Xt < b0 / (1 - b1), the model predicts that Xt+1 > Xt

Question 28

Type I Error

Answer 28

Incorrectly rejecting the null when it should not be rejected.

Question 29

Type II Error

Answer 29

Incorrectly failing to reject the null when out should be rejected.

Question 30

Unit Root

Answer 30

If the lag coefficient is equal to one. Will follow a random walk process. Since a time series follows a random walk is not covariance stationary, modeling such a time series in an AR model can lead to incorrect inferences.

Question 31

Linear Regression Assumptions

Answer 31

1. A linear relationship exists between the dependent and independent variable.
2. The variable is uncorrelated with the residuals.
3. Expected value of the residual terms is 0.
4. The variance of the residual term is constant for all observations.
5. The residual term is independently distributed; the residual term for one observation is not correlated with that of another observation.
6. The residual term is normally distributed.
7. Independent variables are not random, and there is no exact linear relationship between two or more independent variables.