Econometrics

Econometrics

statistical methods used to estimate/test economics relationships

Causal Effect

D on Y is Yi1 - Yi0

Problem of Causal Inference

can never observe opposite treatment for same individuals

Counterfactual

outcome for treatment you didn’t take. Can’t know but can estimate

Random variables

numerical summary of a random outcome

Probability distribution

prob of any event occurring. pdf(continous)

Cumulative distribution

probability of random variable<= some value (discrete). cdf: continous

Expected value

E(Y) = uY (mean)

RCTs

Randomized control trials

Law of Large Numbers

Larger n is, closer we get to true population mean with little variance

iid

independently and identically distributed

ind = one doesn’t affect other

ident = from same probability distribution

CLT

Central Limit Theorem

Data sets where n>=30 will be normally distributed, regardless of distribution for original set

Joint Probability

Prob that A and B happen

P(X=x, Y=y)

Conditional Probability

prob of Y happening, given X

P(Y=y|X=x)

formula on sheet

Bernoulli

random variable but with two possible outcomes

Estimator

Sample term

random variable

Formula to find estimate

Estimand

What we want to find, qualitative

Ex.: height of all students

Estimae

numerical value from estimator

Rules for a good estimator

Consistent

Unbiased

Efficient

Consistent

n is large. P that estimator is within small interval of mean is high

Unbiased

Expected value of estimator is the true value of the parameter

Efficient

Lowest variance

sample variance/deviation

spread of values of Y in our sample. dispersion

standard error

standard deviation of the sample mean

OLS

Ordinary Least Squares

Linear Regression Assumptions for Causal interpretation

1. E(ei|Xi) = 0 → Other determinants of Y outside of X are uncorrelated with X (violation is ommitted variable bias)

2. Observations are iid

3. Large outliars unlikely

4th assumption

Errors are homoskedastic

If the case, then BLUE

R and R²

R → correlation between x and y is positive/negative

R² → z% of the variability in y is explained by x

Hetero/homoskedastic and SE

SE for homo only valid if homo. SE for hetero-robust SE is always valid

Binary Ind Variable for Regression

Beta is average difference between Y=0 and Y=1

alpha is sample mean for 0

OVB Condition

X corr with omitted variable

Omitted variable is a determinant of Y

Biased Term

B1 + B2cov(X1X2)/var(X1)

+ = upward bias

- = downward bias

Multivariate regression interpretation

a 1 unit change oi X1 is associated with a B1 change in Y, holding all other variables constant (must list them out)

OLS Assumptions for Multivariate

E(ei|X1,X2,X3,…Xki)=0

Yi,X1, etc. are iid

Large outliers unlikely

No Perfect multicollinearity

Perfect multicollinearity

one of independent variables is a perfect linear function of other independent variables

for example: B3fracfemale + B4percfemale

fraction and percentage will be in each others formulas…

Dummy Variable Trap

multicollinearity condition applied to a specific set of outcome, liklihood of all add up to one

job happiness = a + Btransportation + ei

walk =1, bike = 2, car =3, train=4, bus=5

DVT is I make all 5 an individual regressor

Instead, n-1 of dummy varibales in regression, other one omitted will be base

Hypothesis testing for Multivariate

Can do the same way if testing one of them, same formula

For more than one, need to do Joint Hypothesis Test

Joint Hypothesis Test

H0: B1 = something and B2 = something and ….

Ha: one or more of the q restrictions do not hold

But, compute F-stat instead of T. At degrees of freedom and confidence level, is F-stat more extreme than given?

Adjusted R²

no matter what, R² will go up when you add another regressor, there will always be some sort of relationship calculated.

So, the adjusted version has a penalty for every additional regressor used

Quadratic Regression

sometimes a regression isn’t linear, so we can use a parabola to describe

Can check if linear by testing squared B against null that it’s 0

Quadratic interpretation

Y increasing at a decreasing rate. A 1 unit increase at mean X would cause XYZ change in Y

Linear Log Interpretation

1% increase in X is associated with 0.01B change in Y

Log Linear Interpretation

1 unit increase in X is associated with a 100B% change in Y

Log Log Interpretation

1 % increase in X associated with a B% change in Y

How can we compare Log regressions?

Can use R² to interpret log linear and log log since they are predicted the same log(Y).

Can’t compare to linear-log since it’s against just Y.

Between both those categories, just have to logic through what makes the most sense in terms of the intepretation

interaction term

when b1 and b2 have a relationship between each other that could affect their value, you account for that with the interaction term

B3(X1*X2)

Interaction term interpretation

Figure it out bru :(

Elasticity or holding something constant formula → non-linear model

constant goes on interaction term, other terms coeff (B) is added

Internal Validity

statistical inferences about causal effects are valid for the population being studied. 

• estimates are unbiased 

• SEs are correct

Threats:

• Violation of first OLS assumption → E[e,Xi] = 0

• sample selection bias

• simultaneous causality (if X causes Y and Y causes X)

Can mitigate OVB with a randomized experiment

External Validity

Inferences and conclusions can be generalized to outside populations

Binary Dependent Variables

“Did you vote?“

“Are you employed“

Linear Model for Binary Indep

• Y = a + B1X1 + B2X2 + … + ei

• B is the change in percentage points that Y=1 associated with a 1 unit change in X

Problem is that it doesn’t work well for very large numbers

Probit Model for Binary Indep

cumulative dist of standard normal func

P(Y=1|X) = phi(a+BX)

P(college=1|famincome) = phi(a+Bfamincome)

B is the change in z-score

IF B>0, a 1 unit increase in X is associated with a increased probability that Y=1 (and vice versa)

marginal change would be percentage point

Types of Data Sets

Cross-sectional → 1 time period, n entities

time series → 1 entity, t time periods

Panel (longitudinal) → data on same n entities over t time periods

Entity fixed effects

Controlling for things that change across entities but don’t change over time

Create dummy for each n-1 entities or just say oi.

• If oi, REMOVE INTERCEPT (it includes all dummies)

1 → B1: a 1 pp increase in UE is associated with a B1 increase in crime rate, controlling for state FE. B2: the difference in average crime rate between state 2 and state 1 during time period, controlling for UE 

2 → 1 pp increase in UE is associated with B1 increase in crime rate, controlling for state FE (o2 is the average crime rate in state 2 during the time period, controlling for UE) 

Time Fixed Effects

Change over time but not entities

yit

Same as Entity for incorporation

Assumptions for Entity FE

1. E(eit|Xi1,Xi1,…, ai) = 0

2. n are iid

3. large outliers unlikely

4. no perfect multicolinearity

Autocorrelation

often a problem with time-series data

Error terms often correlated across time within entity (high UE in state is highly correlated with a high one next year)

We use clustered SEs to fix this.

Endogeneity

independent variable is correlated with error term (violates first OLS assumptions)

Exogeneity

Not correlated

Instrumental Variables

Isolate the part of X that’s exogenous from error term. Meaning find variable Z, that is relevant and exogenous

• Corr(Z,X) not  = 0 (Relevance, must be correlated with X)

• Corr(Z,e) = 0 (Exogenous, only correlated with Y through X)

2 Stage Least Squares

Way of understanding instrumental variables:

Stage 1:

• Part that is correlated with e and predicted by Z

  • Xhati = ohat + pihatZi

Stage 2: Run OLS with this new predicted X

• Yi = a + BXhati + e

Difference in Differences

• Difference over time within each group (treatment/control) and then the differences of those differences

          1849     1854     Delta Yi        DiD

SV      135        147         12

La        85           19         -66             -78

Yit = a + B1Di + B2Postt + B3(Di * Postt) + eit

a = sample average in the control group in the pre treatment period

B1 = differences in pre treatment averages

B2 = differences in control group for post and pre treatment periods

B3 = DID estimator. Causal effect. difference of differences in the within-group average over time between the treated and control groups

Parallel Trends Assumption

Had treated group not been treated, they would’ve followed the trend of the control group