Statistics: Regression, R-squared, F-test, and Confidence Intervals

SSE

Sum(yihat-ybar)

SSR

Sum(yihat-yi)

SST

SSE+SSR

R^2

SSE/SST

Var(b)

(SSR/n-2)/SST

F test

((R^2ur - R^2r)/#tests)/((1-R^ur)/n-k-1)

95 conf int

[(b-1.96se(b)),(b+1.96se(b))]

T test

(Sample mean-pop mean)/(sd/sqrt n)

Homoskedasticity

Var(u∣x)=σ^2

How to test Homoskedasticity

F test using the equation given

test name for heteroskedacity

The Breusch-Pagan test

Weighted Least Squares (WLS)

Fix for heteroskedacity, you divide each term by the variance

var(u|x) = σ^2 h(x): divide function by sqrt(h(x))

direction of bias

if the other variables are greater than zero and x1 and x2 are positively correlated, or if the other variables are less than zero and x1 and x2 negatively correlated, upward bias. if not, downward bias

When do you use an IV estimator

if one variable is associated with other variables

conditions for z (iv)

Cov(z, x) != 0, cov(z, u) = 0, corr(x, u)> (corr(z, u)/corr(z,x))

Standard Error

SE(β^1)=sqrt(σ^2/SSTx)

classical errors in variables

x = x +u, cov(x, u)= 0

white test

A test for Heteroskedasticity where u^2 is regressed on independent variables(x), their cross product, and their squares

can be both linear and non linear

chow statistic

F=((SSRp−(SSR1+SSR2))/k(SSR1+SSR2))/(n1+n2−2k)

used when testing equality at regression parameters across different groups and time

endogeneity

a regressor correlated with U

var(ax+by)

a^2 var(x) + b^2 var (y) + 2ab cov(x,y)

2 stage least squares regression

Iv estimator where the fitted value from regressing the endogenous variables on all exogenous variables -> used when theres multiple IVs

level- log model

if y is a test score and x is funding, a 1% increase in funding leads to a beta1/100 point increase in the score

attenuation bias

occurs when an independent variable is measured with error, causing the OLS coefficient estimate to be biased toward zero. In other words, the estimated effect looks weaker than the true effect.

log-log model

ln(y) = β0 + β1ln(x) +u

if β1 = -.5, a 1% increase in price leads to a .5% decrease in quantity

measurement error

The difference between an observed variables & the variable that belongs in the regression equation

Dummy Variable Trap

including too many dummy variables among the independent variables; it occurs when an overall intercept is in a model and a dummy variable is in each group

Ex. dont include male and females as separate variables