Quantitative Methods

Question 1

Linear Regression Equation

Answer 1

Linear regression equation: Y_i= b_0+ b_1 X_1+ ε_i,i=1,…,n where Yi – dependent, Xi – independent, bo – intercept, b1 – slope coefficient

Question 2

Confidence Interval

Answer 2

Confidence interval: (b_1 ) ̂± t_c s_(b_1 ) ̂ , tc – critical t-value, b1 hat – calculated t coefficient, sb – standard error

Question 3

Hypothesis Test (b1)

Answer 3

Hypothesis Test: = ((b_1 ) ̂-b_1)/s_(b_1 ) ̂ = (calculated t coefficient- H_0 Value)/(Standard Error) , where standard error= coefficient/(ANOVA t-stat), if t > tc, reject null hypothesis b1 = ___ (if testing for statistical significance from zero, b1 = 0)

Question 4

Standard Error of the Estimate (SEE)

Answer 4

Standard Error of the estimate of regression model: SEE= ((Unexplained Variation or sum of squares)/(n-1))^(1/2)

Question 5

Regression Degrees of Freedom

Answer 5

Regression degrees of freedom = # of independent variables

Question 6

Residual Degrees of Freedom

Answer 6

Residual degrees of freedom = total df – regression df = n – (k+1)

Question 7

MSS Regression

Answer 7

MSS Regression = Regression SS / Regression df

Question 8

MSS Residual

Answer 8

MSS Residual = Residual SS / Residual df

Question 9

F =

Answer 9

F = MSS Regression / MSS Residual

Question 10

Correlation =

Answer 10

Correlation = (Cov(X,Y))/(s_x s_y )=r where s is standard deviation = square root of variance

Question 11

T-Test (correlation is different from zero)

Answer 11

T-Test (correlation is different from zero): t=(r√(n-2))/√(1-r^2 ) where r is sample correlation, if t is greater than tc then reject the null hypothesis

Question 12

Multiple Linear Regression

Answer 12

Multiple Linear Regression: Y_i= b_0+b_1 X_1i+b_2 X_2i+⋯+b_k X_ki+ε_i,i=1,2,…,n

Question 13

Durbin Watson

Answer 13

Durbin Watson, if the DW stat is outside the critical values then fails to reject the null, if DW = 2 not serially correlates, if DW 2 then negatively correlated

Question 14

Multicollinearity (definition)

Answer 14

Multicollinearity – a regression assumption violation that occurs when two or more independent variables (or combinations of independent variables) are highly but not perfectly correlated with each other

Question 15

Heteroscedasticity (how to notice)

Answer 15

Heteroscedasticity – incorrect standard of errors

Question 16

Multicollinearity (hot to notice)

Answer 16

Multicollinearity – high R2 (f-stat related) and low t-stats

Question 17

Mean-Reverting Level =

Answer 17

Mean-reverting level = b_0/(1-b_1 )

Question 18

Unit Root

Answer 18

First use a unit root test for each of the two times series to determine whether either have a unit root: 1) if neither have unit root, can safely use linear regression to analyze relationship between two time series; 2) if one fails, cannot use linear regression to analyze relationship between two time series; 3) if both have unit root, need to establish whether the two time series are cointegrated before rely on regression

Question 19

Testing for ARCH

Answer 19

Test for autoaggressive conditional heteroscedasticity (ARCH): whether a1 is statistically different from 0 (a1 is like the b1 of the error regression function, where the error term is the independent variable)