BASIC ECONOMETRICS - causal inference and non-stationarity

interpretation of the estimate of β₁ - DiD (treatment)

The causal effect of [intervention] is to [increase/decrease] [outcome] in the treatment group, between the before and after periods, by (a2 + B1) − a2 = units, ceteris paribus.

Conclusion of a hypothesis test - DiD |t| > tc

Since |t| > tc, we reject H₀. There is sufficient evidence that [variable] has a statistically significant effect on [outcome], ceteris paribus.

Conclusion of a hypothesis test - DiD |t| < tc

Since |t| > tc, we fail to reject H₀. There is insufficient evidence that [variable] has a statistically significant effect on [outcome], ceteris paribus.

ADF - Under H0 statement

The test statistic follows a non-standard Dickey-Fuller distribution

ADF test - p-value > 0.05

Since p-value > 0.05, we fail to reject H₀. There is insufficient evidence to conclude that the series is stationary, thus we fail to reject non-stationarity.

ADF test - p-value < 0.05

Since p-value < 0.05, we reject H₀. There is sufficient evidence to conclude that the series is stationary.

Define ATE

E[Y1 - Y0]

Interpretation of ATE

it represents the expected difference in the outcome between receiving and not receiving the treatment for the entire population.

Define ATT

E[Y1-Y0 | D = 1]

Interpretation of ATT

It represents the expected difference in the outcome for individuals who actually received the treatment.

Define simple difference in average outcomes

E[Y|D=1] - E[Y|D=0] - the difference in mean outcomes between treated and control groups

Decomposed simple diff

diff = ATT + selection bias, where selection bias arises from difference in untreated potential outcomes between treated and control individuals

Condition for an ATT estimation

if the conditional independence assumption holds, then selection bias is zero and the simple difference identifies ATT

Definition of classical measurement error

occurs when the observed explanatory variable (Xi) differs from the true variable (X*i) due to a random error term (vi)

Two conditions for classical measurement error

E[vi​]=0 (the error has zero mean) ; Cov(X*i,vi)=0 (the measurement error is uncorrelated with the true variable)

why is it called attenuation bias

because the OLS estimator is biased towards zero, meaning the estimated coefficient is systematically shrunk in magnitude relative to the true parameter B1.

why does measurement error cause attenuation bias

This occurs because the measurement error in the explanatory variable weakens the observed relationship between X and Y, so OLS attributes part of the variation in X to noise rather than the true signal.

What happens to magnitude of attenuation bias as σ²_υ increases

the amount of measurement error increases, strengthening this attenuation effect, so the estimated coefficient moves further toward zero and the magnitude of the bias increases.