Revise — Econometrics | Econometrics Atlas

Types of Economic Data

y_{it}, \quad i = 1,\dots,N, \;\; t = 1,\dots,T

Cross section: many units, one time period

3 pitfallsquiz

Probability and Statistics Review

\mathrm{Var}(X) = E\!\left[(X - \mu_X)^2\right], \quad \mathrm{Cov}(X,Y) = E\!\left[(X-\mu_X)(Y-\mu_Y)\right]

Expected value is the population mean; variance measures spread

The Population Regression Model

y = \beta_0 + \beta_1 x + u

Population model: $y = \beta_0 + \beta_1 x + u$

Correlation Versus Causality

E(u \mid x) = 0

Correlation is co-movement; causality is a true effect of changing $x$

The Applied Research Process

y = \beta_0 + \beta_1 x + u \;\Rightarrow\; H_0\!: \beta_1 = 0

Question, model, data, estimate, test, interpret

Doing Econometrics in Stata

t = \frac{\hat\beta_1}{\mathrm{se}(\hat\beta_1)}

Do-files make the whole analysis reproducible

The Multiple Regression Model

y=\beta_0+\beta_1 x_1+\beta_2 x_2+\dots+\beta_k x_k+u

MLR: $y=\beta_0+\beta_1 x_1+\dots+\beta_k x_k+u$, one error term for all unobservables.

OLS Estimation and Partialling Out

\min_{\hat{\beta}_0,\dots,\hat{\beta}_k}\ \sum_{i=1}^{n}\bigl(y_i-\hat{\beta}_0-\hat{\beta}_1 x_{i1}-\dots-\hat{\beta}_k x_{ik}\bigr)^2

OLS minimizes $\sum \hat{u}_i^2$ and solves all $\hat{\beta}_j$ jointly.

Omitted Variable Bias

E(\tilde{\beta}_1)=\beta_1+\beta_2\,\delta_1

OVB needs a relevant omitted variable that correlates with an included regressor.

Assumptions of Multiple Regression

E(u\mid x_1,x_2,\dots,x_k)=0

MLR.1 linear in parameters, MLR.2 random sampling, MLR.3 no perfect collinearity, MLR.4 zero conditional mean.

The Gauss-Markov Theorem

\text{Var}(u\mid x_1,\dots,x_k)=\sigma^2

MLR.5 adds homoskedasticity, $\text{Var}(u\mid x)=\sigma^2$.

Consistency and Asymptotic Normality

\text{plim}_{\,n\to\infty}\ \hat{\beta}_j=\beta_j

Consistency: $\text{plim}\,\hat{\beta}_j=\beta_j$, the estimate converges to the truth as $n$ grows.

Sampling Distribution and the Classical Linear Model

u\sim\mathcal{N}(0,\sigma^2)\ \text{independent of}\ (x_1,\dots,x_k)

MLR.6 adds normal errors and defines the classical linear model.

Hypothesis Testing with the t Statistic

t=\frac{\hat{\beta}_j-\beta_{j,0}}{\mathrm{se}(\hat{\beta}_j)}

t statistic: $t=(\hat{\beta}_j-\beta_{j,0})/\mathrm{se}(\hat{\beta}_j)$, usually with $\beta_{j,0}=0$.

Confidence Intervals for Coefficients

\hat{\beta}_j \pm c\cdot\mathrm{se}(\hat{\beta}_j)

CI: $\hat{\beta}_j \pm c\cdot\mathrm{se}(\hat{\beta}_j)$ with $c$ from the $t_{\,n-k-1}$ distribution.

Testing Multiple Restrictions with the F Test

F=\frac{(SSR_r-SSR_{ur})/q}{SSR_{ur}/(n-k-1)}

F test: $F=[(SSR_r-SSR_{ur})/q]\,/\,[SSR_{ur}/(n-k-1)]$ for $q$ joint restrictions.

Prediction and Prediction Intervals

\hat{y}^0 \pm c\cdot\mathrm{se}(\hat{y}^0)

Distinguish predicting the mean $E(y\mid x)$ from predicting a single new $y$.

Functional Form: Logs and Elasticities

\log(y)=\beta_0+\beta_1 \log(x)+u \quad\text{vs.}\quad \log(y)=\beta_0+\beta_1 x+u

Log-log slope is an **elasticity** (percent per percent).

Polynomials and Interaction Terms

\frac{\partial y}{\partial x}=\beta_1+2\beta_2 x,\qquad x^{*}=-\frac{\beta_1}{2\beta_2}

Quadratic slope is $\beta_1+2\beta_2 x$, not $\beta_1$.

Dummy Variables

y=\beta_0+\delta_0 d+\beta_1 x+u

A dummy is 0/1 and **shifts the intercept** by its coefficient.

Dummy Interactions and the Chow Test

y=\beta_0+\delta_0 d+\beta_1 x+\delta_1 (d\cdot x)+u

Dummy times continuous lets the **slope** differ by group.

The Linear Probability Model

P(y=1\mid \mathbf{x})=\beta_0+\beta_1 x_1+\dots+\beta_k x_k

LPM is OLS on a binary $y$; slopes are changes in $P(y=1)$.

Multicollinearity

\operatorname{Var}(\hat{\beta}_j)=\frac{\sigma^2}{\mathrm{SST}_j\,(1-R_j^2)}

High correlation among regressors inflates $\operatorname{Var}(\hat{\beta}_j)$ via the **VIF**.

Units of Measurement and Scaling

x_j^{*}=x_j/c \;\Rightarrow\; \hat{\beta}_j^{*}=c\,\hat{\beta}_j,\quad \operatorname{se}(\hat{\beta}_j^{*})=c\,\operatorname{se}(\hat{\beta}_j)

Dividing $x_j$ by $c$ multiplies $\hat{\beta}_j$ and its SE by $c$.

Functional Form Misspecification and RESET

y=\beta_0+\boldsymbol{\beta}\mathbf{x}+\delta_1 \hat{y}^{2}+\delta_2 \hat{y}^{3}+\text{error}

Wrong functional form biases coefficients and predictions.

Measurement Error and Proxy Variables

\operatorname{plim}\hat{\beta}_1=\beta_1\,\frac{\sigma_{x^{*}}^2}{\sigma_{x^{*}}^2+\sigma_e^2}

Classical error in $y$: more noise, larger SEs, still unbiased.

Missing Data and Influential Outliers

h_i=\frac{1}{n}+\frac{(x_i-\bar{x})^2}{\sum_{j}(x_j-\bar{x})^2}

Random missingness costs sample size; **systematic** missingness can bias.

The Nature of Time Series Data

\{y_t : t = 1, 2, \dots, n\}

A time series is **one realization** of a stochastic process

Static and Finite Distributed Lag Models

y_t = \alpha_0 + \delta_0 z_t + \delta_1 z_{t-1} + \dots + \delta_q z_{t-q} + u_t

Static model assumes an **immediate** effect of $z$ on $y$

Trending Time Series

y_t = \beta_0 + \beta_1 t + \beta_2 x_t + u_t

Trending series **drift** steadily over time

Seasonality in Time Series

y_t = \beta_0 + \gamma_2 Q2_t + \gamma_3 Q3_t + \gamma_4 Q4_t + \beta_1 x_t + u_t

Seasonality is a **repeating calendar pattern** within the year

The Time Series Assumptions (TS.1-TS.6)

E(u_t \mid X) = 0, \quad t = 1, \dots, n

TS.1-TS.5 are the **Gauss-Markov** conditions for time series; TS.6 adds normality

Serial Correlation in the Errors

u_t = \rho u_{t-1} + \varepsilon_t, \quad |\rho| < 1

Serial correlation = **errors correlated over time**

Testing for Serial Correlation

DW = \frac{\sum_{t=2}^{n} (\hat{u}_t - \hat{u}_{t-1})^2}{\sum_{t=1}^{n} \hat{u}_t^2} \approx 2(1 - \hat{\rho})

**Durbin-Watson** targets AR(1); $DW \approx 2(1-\hat{\rho})$

Correcting for Serial Correlation

y_t - \hat{\rho}\, y_{t-1} = \beta_0(1 - \hat{\rho}) + \beta_1 (x_t - \hat{\rho}\, x_{t-1}) + e_t

**Newey-West (HAC)** standard errors are the modern default fix

Stationarity and Unit Roots

y_t = y_{t-1} + \varepsilon_t

**Stationary + weakly dependent** is what OLS needs

Spurious Regression and Cointegration

y_t - \beta x_t = u_t \ \text{is} \ I(0), \quad y_t, x_t \sim I(1)

Regressing independent $I(1)$ series gives a **spurious regression**

Heteroskedasticity

\mathrm{Var}(u \mid x_1, \dots, x_k) = \sigma^2

Definition: $\mathrm{Var}(u\mid x)$ depends on the regressors, violating MLR.5

Heteroskedasticity-Robust Standard Errors

\widehat{\mathrm{Var}}(\hat{\beta}_1) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2 \, \hat{u}_i^2}{\left[\sum_{i=1}^{n} (x_i - \bar{x})^2\right]^2}

Replace $\sigma^2$ with $\hat{u}_i^2$ in the variance formula (sandwich form)

Testing for Heteroskedasticity: Breusch-Pagan and White

\hat{u}^2 = \delta_0 + \delta_1 x_1 + \delta_2 x_2 + \cdots + \delta_k x_k + \text{error}

Both tests use an auxiliary regression of $\hat{u}^2$ on explanatory terms

Weighted Least Squares and Feasible GLS

\min_{\beta} \; \sum_{i=1}^{n} \frac{\bigl(y_i - \beta_0 - \beta_1 x_{i1} - \cdots - \beta_k x_{ik}\bigr)^2}{\sigma_i^2}

WLS weights by $\tfrac{1}{\sigma_i^2}$, transforming the error to be homoskedastic

The Simple Linear Regression Model

y = \beta_0 + \beta_1 x + u

Model is $y = \beta_0 + \beta_1 x + u$, linear in the parameters

Deriving the OLS Estimates

\hat{\beta}_1 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}

OLS minimizes $\sum \hat{u}_i^2$, the sum of squared residuals

Fitted Values and Residuals

\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i, \qquad \hat{u}_i = y_i - \hat{y}_i

Fitted value $\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i$ lies on the line

Goodness of Fit and R-squared

SST = SSE + SSR

Decomposition $SST = SSE + SSR$ (total = explained + residual)

The Zero Conditional Mean Assumption

E(u \mid x) = 0

Assumption is $E(u \mid x) = 0$, the key identifying condition

Unbiasedness of OLS

E(\hat{\beta}_1) = \beta_1

Assumptions SLR.1 to SLR.4 give $E(\hat{\beta}_0) = \beta_0$ and $E(\hat{\beta}_1) = \beta_1$

Variance of the OLS Estimators

\mathrm{Var}(\hat{\beta}_1) = \frac{\sigma^2}{\sum_{i=1}^{n} (x_i - \bar{x})^2}

SLR.5 homoskedasticity: $\mathrm{Var}(u \mid x) = \sigma^2$, constant error variance