Adv Econometrics III

Key concepts & common STATA commands.

random variable

a numerical summary of a random outcome

outcome

the mutually exclusive result of a random process

variable

a measurable characteristic of a population

sample space

the set of all possible outcomes of a random process

event

a subset of the sample space

estimator

the function of the data in the sample derived to infer the estimand

estimand

the true value in the observable population which is to be estimated

target/structural parameter

the specific unknown population parameter that is to be estimated

central limit theorem

when N is sufficiently large, the distribution of the estimated mean becomes more normal

Gauss-Markov OLS Assumptions

1. Exogeneity

2. No multicolinearity

3. Linear relationship between dependent var & independent var

4. Homoskedasticity

5. No autocorrelation

6. Normally distributed error term

properties of OLS under Gauss-Markov assumptions

B-L-U-E

Best linear unbiased estimator

exogeneity

the x-variables and the error term are not correlated: E(εi | X) = 0. Therefore, neither the error term nor the dependent variable influence the explanatory variables since they are determined outside of the model.

multicollinearity

a breach in Gauss-Markov correlation between ≥2 explanatory variables, probably because they measure a similar trait

homoskedasticity

the variance (σ) of the error term (ε) is constant throughout the sample. Therefore, the dispersion of residuals is similar for all X

heteroskedasticity

the variance (σ) of the error term (ε) is not constant throughout the sample. Therefore, the dispersion of residuals is dissimilar for all X. This violates Gauss-Markov assumptions of OLS.

autocorrelation / serial correlation

the correlative relationship between an independent variable and its own past values. This violation of OLS assumptions often occurs in time series data.

endogeneity

correlation between explanatory vars and the error term such that there is a bilateral causal relationship between the X and Y variables.

residuals

the differences between observed (actual) values and the estimated values predicted by the model

grounds to reject the null hypothesis (Ho) and propose the alternative hypothesis (Ha) / statistical significance

p-value < critical value

insufficient grounds to reject the null hypothesis / statistical insignificance

p-value > critical value

p-value

the probability of observing a z-stat, t-stat, F-stat, etc. with an absolute value ≥ the observed results

Triple S method of analysing variable coefficients

sign, size & significance of the a variable’s estimated coefficient

dummy variable

a numerical var expressed as 0 or 1 to represent categorical data, often gender, race, union membership, etc.

elasticity

the % change of the dependent var due to 1% change in the independent var

linear-linear model (Y = f[X])

change Y = beta change X

linear-log model (Y=f[logX])

change Y = beta/100 % change X

log-linear model (logY = f[X])

% change Y = (100)beta change X

log-log model (logY = f[logX])

% change Y = beta% change X

internal validity

a regression that successfully yields inferences applicable to the chosen population

external validity

a regression whose inferences made from a sample can also be applied to other populations

Variance Inflation Factor (VIF)

a method of identifying multicollinearity by quantifying how much correlation between predictor variables inflates the variance of a regression coefficient. This index = 1/(1-R²). To run this in STATA, use command

Breusch-Pagan test for heteroskedasticity

Ho: constant variance/hetsked (σ1 = σ2, etc.)

Ha: inconstant variance/homosked (σ1 ≠ σ2, etc).

To run this test in Stata, use the command ESTAT HETTEST

robust standard errors

standard errors adjusted for heteroskedastiicity. HOW TO CALC IN STATA

standard errors

= variance / square root( # of observations)

Type I neoclassical measurement error

the error is uncorrelated with the true value of the variable (eg: independent inaccuracies in reporting one’s weight)

Type II neoclassical measurement error

the error is correlated with the true-value or with other variables (eg: many observations intentionally misrepresent a characteristic like income)

conditions for instrumented regression

1. relevance: the instrument must correlate with the problematic endogenous variable.

2. exclusive restriction: the instrument only affects the outcome through the endogenous x-variable

rule of thumb for weak instrument identification

F-stat < 10 for a significance test for

Hausman test

A test to determine if the estimator is consistent & efficient (adheres to BLUE)

Ho: the regressor is exogenous (E(Xiεi) = 0)

Ha: the regressor is endogenous (E(Xiεi) ≠ 0)

linear probability model

a OLS model following the binomial distribution that uses limited dependent variables. These models can suffer from issues like predicted probabilities outside the 0-1 range and heteroskedasticity.

probit model

the cumulative distribution function of independent variables which models the probability of an event’s occurrence. This model follows the standard normal distribution. Use STATA command LOGIT.

logit model

the log of the probability of an event’s occurrence. this model follows the logistic distribution and is interpreted as the “odds” of an event happening. Use STATA command LOGIT

latent variable

a variable that cannot be observed, but can be inferred from other observable variables

maximum likelihood estimation

estimating the parameters of an assumed probability distribution based on some observed data to maximise a likelihood function so that, under the assumed statistical model, the observed data is most probable.

MARGINS command

finds marginal effects

multinomial regressions

regressions for categorical data with no order/ranking

cross-sectional data

time-series data

panel data

a combination of cross-sectional data and time series data

Chow Test for structural change

Ho: coefficients are the same for every Y

Ha: coefficients are different for every Y

Difference-in-Diffferences

Causal estimator method of using control and treatments groups to examine trends in 2 groups pre- and post-intervention. This method addresses biases from pre-existing differences between the two groups and omitting time trends that would have occurred regardless of the intervention.

This method assumes:

• parallel trend

• exogeneity

• conditional independence

balanced panel data

panel data with an equal number of observations in each cross-section and time period

unbalanced panel data

panel data with an unequal number of observations in each cross-section and time period

Fixed Effect Model

STATA command XTSET

organises panel data properly