Managerial Economics: Ch. 4 and Appendix E

Population Regression Line or True

Regression Line

Yi = A + B(Xi) + ei

• Line resulting from regressing the dependent

variable on the independent variable where

the entire population of values of the

Error term

a random amount that is added or subtracted from the population regression line

Estimated Regression Line or sample regression line

the line resulting from

regressing the dependent variable on the

independent variable where only a

sample of the variables is used.

• General Expression is:

Y(hat) = a + b(X)

Y(hat)

the value of the dependent

variable predicted by the regression line

a and b are estimators of A and B, respectively, of the population regression line

• a measures the intercept of the regression line. measures the value of the dependent variable when the independent variables have a value of zero

• b measures its slope. measures the change in the predicted value of Y, associated with a one-unit change in the value of X

Method of Least Squares is often used to determine

values of a and b

Sum of Squared Residuals

Σ [Yi - Ŷi]^2 = Σ [Yi -

a - bXi]^2

Method of Least Squares: a -->

Ȳ - bx̄

Method of Least Squares: b -->

Σ [Yi - Ȳ][Xi - x̄] / Σ [Xi - x̄]2

The coefficient of determination is denoted by

R^2

• Its value lies between 0 and 1. The closer it is to 1, the better the fit; the closer to zero, the worse

the fit.

• In a simple linear regression, R2 is the square of the correlation coefficient, r

total sum of squares

Σ [Yi - Ȳ]^2 = Σ [Yi -Ŷi] ^2 + Σ [Ŷi - Ȳ]^2

R^2 =

variation explained by regression /

total variation

Standard Error of the Estimate

a measure of the amount of scatter of individual observations around the regression line

F Statistic

Answers the question of whether any of

the independent variables really

influences the dependent variable

• Large values of F tend to imply that at

least one of the independent variables

has an effect on the dependent variables.

t statistic

Used to determine which of the independent

variables influences the dependent variable

multicollinearity

A condition where two or more

independent variables are very highly correlated

Root Mean squared error

E = (Σ [Yi - Ŷi]2/n)^0.5

Y = TxSxCxI

• Trend (T)

• Seasonal variation (S)

• Cyclical variation (C)

• Irregular Movements

Which is better fit, linear or quadratic?

• Answer: if B2 is statistically significant, then

quadratic is the best fit.

• Answer: if B2 is not statistically significant,

and B1 is significant than the linear model is

best.

alternative way of writing this: Yt = αβt

• Yt = α(1 + r)t = Y0(1 + r)t, where β = (1+ r), r =β -1

- compound growth, This is nonlinear

lnYt = ln(α) + ln(βt)

• = ln(α) + t ln(β)

• = A + Bt, where A = ln (α) and B = ln(β),

• so β =eB

• and r = β -1

• This is a simple linear regression

dummy variable

A variable for which all cases falling into a specific category assume the value of 1, and all cases not falling into that category assume a value of 0.