Econometrics: OLS and Regression Analysis Overview | Quizlet

P-value

The closer the value is to 0, the more confident that beta is not equal to 0

R²

The amount of variance in Y that's explained by the variance in X

Homoskedasticity

Constant variance of errors across observations.

Serial Correlation

Different error terms are not correlated with each other

Conditional Mean Independence

Error term is independent of Independent variables (controlling for x2, x1 can be estimated)

F-test (SSR)

(SSRr - SSRu/q)/(SSRu/(n-k-1))

F test R2

((R2u-R2r)/q)/((1-R2u)/(n-k-1))

Elasticity

Percentage change in dependent variable from independent variable.

Consistency

As n goes to infinity, estimator approaches the true value

Root MSE

Biased estimator of the standard error of the population

Population of Interest

Population to which results are inferred

Internal Validity

Inferences are valid for the population studied

External Validity

Generalizability of results to other populations.

Internal validity requirements

Estimator is unbiased and consistent, the distribution of test statistics is correct, and hypothesis tests have desired significance

Simultaneous Causality

X causes Y and Y causes X

Panel Data

Data collected over time for the same subjects.

Exogeneity

Variable uncorrelated with error term in model.

Endogeneity

Variable correlated with error term

IV Regression

Instrumental Variable Regression; solves OVB, measurement errors, and simulataneous causality

Instrument Validity

Cov(z,x) != 0 (relevance)

Cov(z,u) = 0 (independence)

Z isn't part of the model (Z doesn't explain y)

J Test

Can only be done in overidentification!! (m>k) Regress TSLS residuals on the Zs and Ws, then do an F test on significance of Zs.

J test distribution/statistic

J=mF ~ Chi squared dist

Maximum Likelihood

Estimation method maximizing probability of observed data.

Hausman Test

Ho: Xi exogenous, H1: Xi endogenous, see if t-test is significant

Pseudo R²

Alternative measure of fit for non-linear models.

Population studied

Population that was sampled

ESS

Sum (Yhat-Ybar)^2

TSS

Sum(Yi-Ybar)^2

SSR

Sum uhati^2

Efficiency

Smallest variance

SER

sqrt(SSR/(n-(k+1))

R^2

ESS/TSS = 1 - SSR/TSS

LSA 1

E(ui|Xi) = 0

LSA 2

(Xi, Yi) iid

LSA 3

Large outliers unlikely (finite 4th moments)

LSA 4

Var (ei | xi) = variance, homoskedasticity

LSA 5

Cov(ei, ej | xi, xj) = 0, no serial correlation (error terms are independent of other independent variables)

Type 1 error

Reject a true null

Type 2 error

Accept a false null

var(ax)

a^2 var(x)

Adjusted R2

1-((n-1/n-k-1)*(SSR/TSS))

Adjust R2 alternate formula

1- (s^2u/s^2y)

Cov(X+c, Y)

Cov(X,Y)

Cov(X+Y,Z)

Cov (X,Z) + Cov (Y,Z)

Problem with OVB

Violates LSA 1, leads to biased and inconsistent results

Positive bias

β > 0 & cov (x1, x2) >0 (same direction)

Negative Bias

β > 0 & cov (x1, x2) < 0

OVB formula

Bhat -> B1 + corr(x1,u) * Su/Sx (population correlation/std dev)

F test distribution

qF ~ chisquared(q) = qF (q, infinity)

F(a,b) in stata

F(df, n-df)

What do k and q mean?

k = number of regressors without the constant, k+1 parameters

q = number of restrictions in Ho

Requirements for OVB

Corr(omitted var, included var) != 0 , and omitted var is a determinant of Y

Conditional mean independence

Conditional expectation of ui given X1i and X2i does not depend on X1i. Controlling for x2i, x1i can be treated as random

Log-lin

1 unit inc in X = β *100% inc in Y

Lin-log

1% inc in X = .01* β inc in Y

Log-log

1% inc in X = β% inc in Y (elasticity)

E(rnormal())

0, in the normal distribution X is 0 and variance is 1

Threats to internal validity

OVB, functional form misspecification, measurement error, sample selection, simultaneous casusality

Other solutions than IV for OVB

Panel data deals with OVB, and RCTs can deal with OVB and simultaneous causality

Construct validity

How valid a test is according to theory

To see if an instrument is relevant

With q=1, look in the first regression and check if F=t^2 of the instrument>10

IV assumptions

E(ui | W) = 0

(X, W, Y, Z) iid

Large outliers unlikely

Valid instruments availble

Pros of linear probability model

Coefficients can be analyzed as normal, good linear approximation, unbiased & consistent

Cons of linear probability model

Probabilities aren't necessarily between (0,1), heteroskedastic, and error term can't be approximated by the normal distribution

Logit and probit errors

Homoskedastic

ML estimator

Has a consistent, asymptotically normal distribution

Best Linear Unbiased Estimator (BLUE)

LS is BLUE under the 5 gauss markov assumptions

Asymptotically normal

Distribution of estimators approaches normality as sample size increases.

Unbiased

E(estimator|X) = estimator

When analyzing the effect of coefficients

Remember to take the derivative, and always include CP!!!1

q = df true or false

true!

Unrestricted model

The model with all coefficients