Assignment 6: Regression Fundamentals

Practice flashcards covering the fundamentals of regression analysis, including OLS estimation, bias, and inference based on Assignment 6 lecture notes.

Population regression coefficients

The parameters $\beta_0$ and $\beta_1$ that solve the population least-squares problem.

Population Regression Function (PRF)

A function that provides the best linear approximation to the CEF (Conditional Expectation Function).

Coefficient $\beta_1$

A measure of the change in $y$ associated with a unit change in $x$, holding all of the unobservables constant.

Population least-squares problem

A problem solved by finding values that minimize the expected value of the squared difference between the dependent variable and the CEF.

OLS estimator for $\beta_1$

An estimator obtained by plugging in the sample covariance between $x$ and $y$ and the sample variance for $x$.

Frisch-Waugh-Lovell theorem

A theorem stating that one can control for other explanatory variables by either including them directly or regressing on the residuals from a regression of $x$ on the other variables.

Partial effect

The effect of $x_1$ measured by $\beta_1$ when the PRF includes more than one $x$, without necessarily giving a causal interpretation.

Unbiased

The property of the estimator $\hat{\beta}_1$ when the sampling error equals $0$ on average, which occurs if $E(u_i | x_{i1}) = 0$.

Consistent

A property of the estimator $\hat{\beta}_1$ where the sampling error converges to $0$ as the sample size increases.

Upward Bias

A condition occurring in the estimator $\hat{\beta}_1$ when an omitted variable $x_{i2}$ is left out and $\beta_2$ and $\text{cov}(x_{i1}, x_{i2})$ have the same sign.

Measurement Error (Biased Down)

The effect on the estimator $\hat{\beta}_2$ if the variable $x_{i2}$ is not omitted but is measured with error.

$$R^2$$

A statistic that measures how much of the variance of the dependent variable is accounted for by the independent variables.

Sampling Distribution (CLT)

The distribution of the OLS estimator which, according to the Central Limit Theorem, can be regarded as approximately normal for large samples.

Heteroscedasticity

A condition where the variance of the errors depends on the explanatory variables; the R function lm ignores this, leading to incorrect standard errors.

Robust standard errors

Standard errors and test statistics that the modern approach to regression inference requires to be reported to account for heteroscedasticity.

Test statistic

The ratio of an explanatory variable's coefficient estimate to its standard error, used to determine if there is a statistically significant association.

$p$-values

Statistics where smaller values indicate stronger evidence against the null hypothesis.