ECON301 - Econometrics Fundamentals & Regression Analysis Flashcards

Question 1

Ordinary Least Squares (OLS)

Accepted Answer

A estimation method that calculates regression parameters by minimizing the sum of squared residuals: $$\min_{\hat{\beta}} \sum_{i=1}^n e_i^2 = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2$$

Question 2

Gauss-Markov Theorem

Accepted Answer

States that under the classical linear regression model (CLRM) assumptions, the OLS estimator is **BLUE**: **B**est **L**inear **U**nbiased **E**stimator (i.e., has minimum variance among all linear unbiased estimators).

Question 3

OLS Assumption: Exogeneity (Zero Conditional Mean)

Accepted Answer

Assumes $E(u_i | X_i) = 0$. The expected value of the error term $u_i$ conditional on explanatory variables $X_i$ is zero. Violation leads to **endogeneity** and biased parameter estimates.

Question 4

OLS Assumption: Homoskedasticity

Accepted Answer

Assumes $Var(u_i | X_i) = \sigma^2$. The variance of the error term remains constant across all values of the independent variables.

Question 5

Heteroskedasticity

Accepted Answer

A violation of homoskedasticity where error variance is non-constant ($Var(u_i | X_i) = \sigma_i^2$). **Impact**: OLS estimates remain unbiased, but standard errors are biased, rendering $t$ and $F$ statistics invalid.

Question 6

Detecting & Correcting Heteroskedasticity

Accepted Answer

**Detection**: Breusch-Pagan test, White test.
**Correction**: Use heteroskedasticity-robust standard errors (Huber-White) or Weighted Least Squares (WLS).

Question 7

Perfect vs. Imperfect Multicollinearity

Accepted Answer

**Perfect Multicollinearity**: One regressor is an exact linear function of another; OLS cannot be computed.
**Imperfect Multicollinearity**: High correlation between regressors; inflates standard errors and lowers $t$-statistics without causing bias.

Question 8

Coefficient of Determination ($R^2$)

Accepted Answer

Measures the proportion of total variance in the dependent variable explained by the regression model: $$R^2 = \frac{ESS}{TSS} = 1 - \frac{SSR}{TSS}$$ Ranges from 0 to 1.

Question 9

Adjusted $R^2$ ($\bar{R}^2$)

Accepted Answer

A modified version of $R^2$ that penalizes adding unnecessary explanatory variables ($k$): $$\bar{R}^2 = 1 - \left(1 - R^2\ight) \frac{n - 1}{n - k - 1}$$ Unlike $R^2$, adjusted $R^2$ can decrease if new variables add little explanatory power.

Question 10

Endogeneity

Accepted Answer

Occurs when an explanatory variable is correlated with the error term ($Corr(X, u) \
eq 0$). Causes: omitted variable bias, measurement error, or simultaneity/reverse causality.

Question 11

Omitted Variable Bias (OVB)

Accepted Answer

Bias caused by excluding a variable $Z$ that belongs in the true model and is correlated with an included regressor $X$. Direction of bias: $$Bias(\hat{\beta}_1) = \beta_2 \cdot \frac{Cov(X, Z)}{Var(X)}$$

Question 12

Instrumental Variable (IV) Conditions

Accepted Answer

An instrument $Z$ for endogenous regressor $X$ must satisfy two conditions:
1. **Inclusion Condition (Relevance)**: $Corr(Z, X) \
eq 0$
2. **Exclusion Restriction (Exogeneity)**: $Corr(Z, u) = 0$

Question 13

Two-Stage Least Squares (2SLS)

Accepted Answer

An IV estimation method used for endogenous regressors:
* **Stage 1**: Regress endogenous variable $X$ on instrument $Z$ to generate fitted values $\hat{X}$.
* **Stage 2**: Regress $Y$ on $\hat{X}$ to obtain unbiased coefficient estimates.

Question 14

p-value

Accepted Answer

The probability of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis ($H_0$) is true. Reject $H_0$ if $p \le \alpha$ (significance level).

Question 15

t-statistic

Accepted Answer

Measures how many standard errors an estimated coefficient $\hat{\beta}_j$ is away from the hypothesized value $\beta_{j,0}$: $$t = \frac{\hat{\beta}_j - \beta_{j,0}}{SE(\hat{\beta}_j)}$$ Used to test hypotheses about single parameters.

Question 16

F-statistic

Accepted Answer

Tests joint hypotheses that multiple coefficients equal zero simultaneously ($H_0: \beta_1 = \beta_2 = \dots = \beta_q = 0$): $$F = \frac{(SSR_r - SSR_{ur})/q}{SSR_{ur}/(n - k - 1)}$$

Question 17

Type I vs. Type II Error

Accepted Answer

* **Type I Error ($\alpha$)**: Incorrectly rejecting a true null hypothesis (False Positive).
* **Type II Error ($\beta$)**: Failing to reject a false null hypothesis (False Negative).

Question 18

Central Limit Theorem (CLT) in Econometrics

Accepted Answer

States that as sample size $n \	o \infty$, the sampling distribution of the OLS estimator approaches a normal distribution, regardless of the population error distribution. Enables asymptotic statistical inference.

Question 19

Standard Error of the Regression (SER)

Accepted Answer

Estimates the standard deviation of the error term $u$: $$SER = \hat{\sigma} = \sqrt{\frac{SSR}{n - k - 1}}$$ Measures the average distance that observed $Y$ values fall from the regression line.

Question 20

Dummy Variable Trap

Accepted Answer

A scenario where qualitative categories are fully encoded into $g$ dummy variables plus an intercept term, causing perfect multicollinearity. **Solution**: Include $g - 1$ dummy variables or drop the constant intercept.

ECON301 - Econometrics Fundamentals & Regression Analysis

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