Quantitative Methods

Question 1

Covariance

Answer 1

SUM[(Xbar-Xi)*(Ybar-Yi)] / (n-1)

Question 2

Correlation Coefficient

Answer 2

r = Cov(x,y) / [Sigma(x)*Sigma(y)]

where, -1 < r < 1

Question 3

T-test

Answer 3

t = [r * sqrt(n-2)] / sqrt(1-r^2)

Reject H0 if t+critical < t or t-critical>t.
t needs to be in between the two critical t’s

Question 4

Slope of regression line

Answer 4

b1 = Covariance / Variance

Question 5

Total Sum of Squares (SST)

Answer 5

SUM(Yi - Ybar)^2

Question 6

Sum of Squared Errors (SSE)

Answer 6

SUM(Yi-Yhat)^2

This is the unexplained variation in the regression

Question 7

Regression Sum of Squares (RSS)

Answer 7

SUM(Yhat-Ybar)^2

This is the explained variation in the regression

Question 8

Total Variation = Explained Variation + Unexplained Variation

Answer 8

SST = RSS + SSE

Question 9

Mean Regression Square of Sums (MSR)

Answer 9

=RSS/number of slope parameters

Question 10

Mean Squared Error (MSE)

Answer 10

MSE = SSE / (n-2)

Question 11

Regression degree of freedom = k
Error degree of freedom = n - k - 1

Where k = number of slope parameters

Answer 11

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Question 12

R^2 = (SST - SSE) / SST = RSS / SST

Answer 12

R^2 = (Total Variation - Unexplained Variation) / Total Variation = Explained Variation / Total Variation

Question 13

Standard Error of the Estimate (SEE)

Answer 13

SEE = SqRt (MSE) = SqRt(SSE/(n-2))

Smaller SEE means the regression has better fit

Question 14

F-Test tests the statistical significance of a regression

Answer 14

F = MSR / MSE = (Rss/k)/(SSE/[n-k-1])

Always a one tail test

Reject if F > Fcritical

Question 15

F-test hypothesis testing

Answer 15

1. H0 –> b1=b2=b3=b4=0 vs. Ha–> @ least 1 bi is not =0
2. Decision —> Reject H0 if F(test-statistic) > Fcritical
3. If rejected, Bj is significant

Question 16

Adjusted R^2

Answer 16

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

Question 17

t = coefficient / standard error

Answer 17

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Question 18

Breusch Pagan Chi-Square Test is used to detect heterskedasticity

Answer 18

ChiSquare = n * R^2residuals
with k degrees of freedom.

To correct heteroskedasticity, calculate the robust standard errors.

Question 19

Durbin-Watson Statistic is used to detect serial correlation (residual terms are correlated with each other)

Answer 19

DW = SUM[(EtHAT - Et-1HAT)^2] / SUM(EtHAT^2)

Decision Rule:
Dw < dl reject null, positive serial correlation
dl du fail to reject null, no evidence of serial correlation

Question 20

Use the Hansen method to correct for serial correlation

Answer 20

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Question 21

Detect Multicollinearity 1. t-test says no variable is statistically significant, and 2. F-test is statistically significant and R^2 is high.

Answer 21

Correct multicollinearity by omitting highly correlated variables

Question 22

Log-Linear Models

Answer 22

Yt = e^(b0+b1(t)) --->
ln(Yt) = b0 + b1(t)

Question 23

Autoregressive Models

Answer 23

Xt = b0 + b1Xt-1 + b2Xt-2+…+bpXt-p + ErrorTerm

Question 24

Chain rule of forecasting

Answer 24

Xt+1 = b0 + b1*Xt
Xt+2 = b0 + b2*Xt+1

Question 25

T-test

Answer 25

t = correlation of error terms / [1/SqRt(T)]
with T-2 degrees of freedom.

If the test is rejected, add more lag variables and re-test

Question 26

Mean Reverting Level is the tendency to move back to the mean

Answer 26

Xt = b0/(1-b1)

Question 27

Dickey Fuller Test

Answer 27

1. Xt = b0 + b1*Xt-1 + Error
2. Xt - Xt-1 = b0 + (b1-1)*Xt-1 + Error
3. Test the (b1-1) term using the t-test. If it is rejected, there is no unit root.

A unit root is when a time series is not covariance stationary.

Question 28

Auto Regressive Conditional Heteroskedasticity (ARCH) exists if the variance of the residuals in one period is dependent on the variance of the residuals in a previous period.

Answer 28

Errort^2 = A0 + A1*ErrorT-1^2 + Sigmat

Question 29

Variance of ARCH series

Answer 29

Sigmat+1^2 = A0 + A*Errort^2