hypothesis testing, OLS, assumptions

hypothesis testing, t-tests and errors, OLS regression with dummy variable, assumption

hypothesis testing

state H0 and H1, compute t statistic (how many SE is Y^ from H0), find p-value, compare p to a, if p < a → reject H0

what does SE(Y^) mean

how much does sample mean vary

which test to use

2-sided is used by default (µ≠µ0, p=P(T>|t|)), 1-sided when theory predicts a direction (µ>µ0, p=P(T>t))

2 ways to be wrong

type I - false positive and type II - false negative

type I

false positive - reject H0 when it is true (probability a)

type II

false negative - fail to reject H0 when it is false (probability ß)

lowering a

leads to increasing ß (trade off)

Ordinary least squares

minimises the sum of the squared residuals

Step 1 - intercept only

Yi = ß0 + Ei → ß^0 = Y^

Step 2 - Dummy variable

Yi = ß0 + ß1Di + Ei

when D=0

E(Y) = ß0 = Y^B

when D=1

E(Y) = ß0 +ß1 = Y^A

step 3 - continuous X (what if X is not a dummy)

ß^1 = Cov(X,Y) / Var(X), ß0 = Y^ - ß^1*X^

assumptions - if they hold ß^1 is unbiased, consistent, efficient

zero conditional mean, i.i.d, no large outliers

Assumption 1- zero conditional mean

E(E|X) = 0, all factors in E that affect Y must be on average unrelated to X

Assumption 2 - i.i.d. observations

independent and identically distributed

Assumption 3 - No large outliers

because squaring magnifies large errors

main threat - OVB

when Z affects both X and Y - true model: Yi = ß0 + ß1Xi + ß2Zi + Ei

Bias(ß^1) = ß2*Cov(X,Z)/Var(X)

same sign = bias is upward, opposite sign = bias is downward

other violations

reverse causality, selection bias, measurement error

3 reasons why correlation arises

X causes Y, Y causes X, Z cause both (confounding)

multivariate model - Yi = ß0 + ß1Xi + ß2Zi + Ei

adding controls to remove omitted variable bias, ß^1 = effect of X on Y holding Z constant, ß^2 = effect of Z on Y holding X constant

limits of multivariate regression

removes bias from observed confounders included, does not fix reverse causality, does not fix selection bias, does not remove unobserved confounders