Empirical Methods Test 1: Classes 9-13

Comprehensive vocabulary flashcards reviewing key econometric concepts from Empirical Methods Test 1 (Classes 9 to 13), covering bivariate OLS regression, goodness of fit, hypothesis testing, multiple regression, omitted variable bias, dummy variables, interaction terms, and nonlinear regression functions.

Bivariate Regression

A statistical tool used for measuring the relationship between one independent variable ($X$) and one dependent variable ($Y$) by fitting a regression line through the data.

Intercept Parameter ($\alpha$)

The predicted value of the dependent variable ($Y$) when the independent variable ($X$) is equal to zero ($X = 0$).

Slope Parameter ($\beta$)

The parameter that indicates how much the dependent variable ($Y$) changes in response to a one-unit increase in the independent variable ($X$).

Residual ($\hat{u}_i$)

The random component of the dependent variable, equal to the vertical difference between the actual observed value of the dependent variable ($Y_i$) and its predicted value ($\hat{Y}_i$) from the regression line: $\hat{u}_i = Y_i - \hat{Y}_i$.

Population Regression Model

The theoretical model expressing the relationship in the underlying population: $Y_i = \alpha + \beta X_i + u_i$.

Sample Regression Model

The empirical regression model estimated from sample data: $Y_i = \hat{\alpha} + \hat{\beta}X_i + \hat{u}_i$, where hats denote estimated parameters.

Ordinary Least Squares (OLS)

A method of fitting the best regression line through sample data by minimizing the sum of squared estimated residuals.

Residual Sum of Squares (RSS)

The sum of the squared estimated residuals from a regression model, which Ordinary Least Squares (OLS) minimizes to fit the regression line.

OLS Slope Estimator ($\hat{\beta}$)

The formula used to calculate the estimated slope parameter in bivariate OLS regression: $\hat{\beta} = \frac{\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^n (X_i - \bar{X})^2}$.

OLS Intercept Estimator ($\hat{\alpha}$)

The formula used to compute the estimated intercept parameter in bivariate OLS regression: $\hat{\alpha} = \bar{Y} - \hat{\beta}\bar{X}$.

R-squared ($r^2$)

The most widely used indicator of overall goodness of fit, ranging between $0$ and $1$, which measures the proportion of variation in the dependent variable explained by the regression model: $r^2 = \frac{\text{MSS}}{\text{TSS}}$.

Total Sum of Squares (TSS)

A measure of the total variation of the dependent variable around its sample mean, defined by the formula: $\text{TSS} = \sum_{i=1}^n (Y_i - \bar{Y})^2$.

Model Sum of Squares (MSS)

A measure of the explained variation of the dependent variable predicted by the regression line, defined by the formula: $\text{MSS} = \sum_{i=1}^n (\hat{Y}_i - \bar{Y})^2$.

Adjusted R-squared

A goodness-of-fit statistic similar to R-squared that accounts for the number of variables in the model; it rises if an added variable improves the model more than expected by random chance and falls if it improves it less.

Root Mean-Squared Error (Root MSE)

The standard error of the regression model, representing the average distance between the data points and the regression line.

Degrees of Freedom (Regression)

The number of observations minus the number of parameters estimated in the model, expressed as $n - k$.

t-Statistic (for Slope Coefficient)

A test statistic used to evaluate the statistical significance of a regression slope parameter, calculated as: $t_{n-k} = \frac{\hat{\beta}}{\text{se}(\hat{\beta})}$.

Confidence Interval (for Regression Slope)

An interval estimate around the estimated slope coefficient within which the true parameter is expected to fall at a given confidence level, calculated as: $\hat{\beta} \pm [t \times \text{se}(\hat{\beta})]$.

Multiple Regression

A regression tool used to measure the relationship between an independent variable ($X$) and a dependent variable ($Y$) while controlling for confounding variables ($Z$).

Omitted Variable Bias (OVB)

A bias that occurs when a model fails to include a confounding variable ($Z$) that is correlated with both the independent variable ($X$) and the dependent variable ($Y$).

Upward Bias

An omitted variable bias where the estimated coefficient is too large, occurring when $Z$ has a positive effect on $Y$ and is positively correlated with $X$, or when $Z$ has a negative effect on $Y$ and is negatively correlated with $X$.

Downward Bias

An omitted variable bias where the estimated coefficient is too small, occurring when $Z$ has a positive effect on $Y$ and is negatively correlated with $X$, or when $Z$ has a negative effect on $Y$ and is positively correlated with $X$.

Dummy Variable

A categorical variable that takes on one of two possible numerical values: $0$ or $1$.

Dummy-Variable Trap (Perfect Multicollinearity)

An exact linear relationship between two or more independent variables that arises when all categories of a dummy variable set are included along with an intercept, which is solved by omitting one category.

Interaction Effect

A phenomenon where the effect of one independent variable is contingent upon the value of another independent variable, modeled by including the product of both variables as an additional regressor.

Linear Regression Model

A regression model whose slope parameter is constant for all values of the independent variable ($X$).

Nonlinear Regression Model

A regression model whose slope is not constant across all values of the independent variable ($X$).

Quadratic Regression Model

A nonlinear model that includes both $X$ and $X^2$ as independent variables ($Y_i = \hat{\alpha} + \hat{\beta}_1 X_i + \hat{\beta}_2 X_i^2 + \hat{u}_i$), allowing the slope to vary depending on the level of $X$.

Level-Level Regression

A regression model of the form $Y = \alpha + \beta X$, where a one-unit change in $X$ leads to a $\beta$-unit change in $Y$.

Log-Linear Regression

A regression model of the form $\log(Y) = \alpha + \beta X$, where a one-unit change in $X$ leads to a $100 \times \beta$ percent change in $Y$.

Linear-Log Regression

A regression model of the form $Y = \alpha + \beta \log(X)$, where a one percent change in $X$ leads to a $\frac{\beta}{100}$-unit change in $Y$.

Log-Log Regression

A regression model of the form $\log(Y) = \alpha + \beta \log(X)$, where a one percent change in $X$ leads to a $\beta$ percent change in $Y$.