RM - 2- Regression

Main Elements of a regression equation

1) Outcome variable
2) intercept (alpha)
3) treatment effect (beta)
4) Treatment/Explanatory variable
5) Control variables
6) error term

Formula omitted variable bias (bs=)

bs = bl + y*pi1

OLS meaning

ordinary least squares
unweight observations of which we minimize the prediction errors squared

Basic regression equation

Yi = a + b*Qi + yAi + ei

Dependent variable

Yi

Qi

treatment variable

Ai

Control variable

ei

error term

Intercept (a)

necessary to have
Outcome when treatment and controls are 0

treatment effect

causal effect we are after
if Q increases by one unit, Y changes by b units

y (gamma)

effect control variable

Why do we perform OLS

to get the parameters that give the best fit to our data

Results OLS

parameters that minimize the sum of the prediction errors squared

Error term

1) variable to capture everything that affects the outcome variable but is not included in the independent variable
2) difference between true population value and expected value predicted by regression line

Regression estimates interpretation

weighted average of group-specific differences

Omitted variable bias (statistical/mathematical definition, words)

difference between short and long regression coefficients

Def regression model

equation linking the treatment variable to the dependent variable while holding control variables fixed by including them in the model

OVB formula as tool

allows us to consider the impact of control for variables we wish we had
= make an educated guess as to the likely consequence of their omission

Difference between long and short regression

at least one more variable in the long regression

Assumption for OVB

about the relationship between omitted variable and treatment

Omitted variables

1) not an issue per se
2) residuals will be bigger (expectation of zero tho)
3) estimate is less precise, but unbiased

Presence of omitted variable bias when

1) omitted variable is correlated with treatment variable AND
2) omitted variable has a direct effect on the dependent variable

True effect of Treatment on Outcome

in the long regression

Omitted variable bias formula (words)

(relationship between OV and treatment) * effect of OV in the long regression

Assumption Relationship OV and Treatment Variable (mathematical expression) (𝛑 = pi)

Ai = pi0 + p1Qi + u

Steps to derive OVB

1) Assumption about relationship between OV and Treatment variable
2) Plugging assumption into long regression
3) reordering
4) we know in the long run short = long
5) short regression includes OVB

Final formula OVB (y= lambda) (𝛑 = pi)

Yi = (al + ypi0) + (bl +ypi1)*Qi+ (yui + eli)

OVB formula short (y= lambda) (𝛑 = pi)

OVB = Short - Long = bs - bl = pi * y