Economics Quantitative Methods and Econometrics Syllabus

State the three axioms (Kolmogorov) of probability.

(1) $P(A)\geq 0$ for any event $A$; (2) $P(S)=1$ for the sample space $S$; (3) for mutually exclusive events, $P(A\cup B)=P(A)+P(B)$.

Define conditional probability and Bayes' theorem.

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Bayes' theorem: $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$, updating prior beliefs given evidence.

Give the mean and variance of a binomial random variable with parameters $n$ and $p$.

$E[X]=np$ and $\operatorname{Var}(X)=np(1-p)$, for $X\sim\text{Binomial}(n,p)$ counting successes in $n$ independent trials.

State the key properties of the normal distribution $N(\mu,\sigma^{2})$.

It is symmetric and bell-shaped about $\mu$; about $68\%$, $95\%$, and $99.7\%$ of values lie within $1$, $2$, and $3$ standard deviations. Standardised via $z=\frac{x-\mu}{\sigma}$.

When is the Poisson distribution used, and what are its mean and variance?

It models counts of independent events in a fixed interval at constant rate $\lambda$. Its mean and variance are both equal to $\lambda$: $E[X]=\operatorname{Var}(X)=\lambda$.

State the Central Limit Theorem.

For i.i.d. samples with mean $\mu$ and finite variance $\sigma^{2}$, the distribution of the sample mean $\bar{X}$ approaches $N\!\left(\mu,\frac{\sigma^{2}}{n}\right)$ as $n\to\infty$, regardless of the population's shape.

Define the standard error of the sample mean and how it changes with sample size.

$SE(\bar{X})=\frac{\sigma}{\sqrt{n}}$ (or $\frac{s}{\sqrt{n}}$ estimated). It shrinks proportionally to $\frac{1}{\sqrt{n}}$, so quadrupling $n$ halves the standard error.

What properties define a good point estimator?

Unbiasedness ($E[\hat{\theta}]=\theta$), efficiency (minimum variance among unbiased estimators), consistency ($\hat{\theta}\to\theta$ as $n\to\infty$), and sufficiency (uses all sample information).

Write a confidence interval for the population mean when $\sigma$ is unknown.

$\bar{x} \pm t_{\alpha/2,\,n-1}\cdot \frac{s}{\sqrt{n}}$, using the $t$-distribution with $n-1$ degrees of freedom.

How do you interpret a $95\%$ confidence interval?

If the sampling procedure were repeated many times, about $95\%$ of the constructed intervals would contain the true parameter. It is not a $95\%$ probability that this specific interval contains it.

Distinguish the null hypothesis $H_0$ from the alternative $H_1$.

$H_0$ is the default claim of no effect or no difference (e.g. $\beta=0$); $H_1$ is what we conclude if $H_0$ is rejected (e.g. $\beta\neq 0$). Tests seek evidence against $H_0$.

Define Type I and Type II errors and their associated probabilities.

Type I error: rejecting a true $H_0$, probability $\alpha$ (significance level). Type II error: failing to reject a false $H_0$, probability $\beta$. Power $=1-\beta$.

What is a p-value, and how is it used with significance level $\alpha$?

The p-value is the probability of observing a test statistic at least as extreme as the one obtained, assuming $H_0$ is true. Reject $H_0$ when p-value $<\alpha$.

When do you use a $t$-test versus a $z$-test for a mean?

Use a $z$-test when the population variance $\sigma^{2}$ is known (or $n$ large); use a $t$-test when $\sigma$ is unknown and estimated by $s$, especially with small samples, using $n-1$ degrees of freedom.

State the simple linear regression population model and name each term.

$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$, where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\varepsilon_i$ is the random error term.

Write the OLS slope estimator for simple regression.

$\hat{\beta}_1 = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^{2}} = \frac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)}$, and $\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}$.

What does OLS minimise, and what is a residual?

OLS minimises the residual sum of squares $\sum \hat{\varepsilon}_i^{2}=\sum (y_i-\hat{y}_i)^{2}$. A residual $\hat{\varepsilon}_i = y_i - \hat{y}_i$ is the difference between observed and fitted values.

Define $R^{2}$ and state its interpretation and range.

$R^{2} = 1-\frac{SS_{res}}{SS_{tot}} = \frac{SS_{exp}}{SS_{tot}}$, the proportion of variation in $y$ explained by the model. It ranges $0\leq R^{2}\leq 1$.

How do you interpret a slope coefficient $\hat{\beta}_1$ in multiple regression?

It is the expected change in $y$ for a one-unit increase in $x_1$, holding all other explanatory variables constant (the ceteris paribus / partial effect).

Why is adjusted $R^{2}$ preferred to $R^{2}$ when comparing models?

Adjusted $R^{2}=1-\frac{(1-R^{2})(n-1)}{n-k-1}$ penalises adding regressors, so it does not automatically rise with more variables; it can fall if a variable adds little explanatory power.

What does an $F$-test assess in multiple regression?

It tests the joint null that all slope coefficients are zero ($H_0:\beta_1=\cdots=\beta_k=0$), i.e. overall model significance, comparing explained to unexplained variance.

How is a dummy (indicator) variable interpreted in a regression?

A dummy takes values $0$ or $1$; its coefficient gives the difference in the expected value of $y$ between the category coded $1$ and the omitted baseline category, ceteris paribus.

List the classical (Gauss-Markov) assumptions of the linear regression model.

Linearity in parameters, random sampling, no perfect multicollinearity, zero conditional mean $E[\varepsilon\mid X]=0$ (exogeneity), homoskedasticity $\operatorname{Var}(\varepsilon\mid X)=\sigma^{2}$, and no autocorrelation. Normality is added for exact inference.

State the Gauss-Markov theorem.

Under the classical assumptions, the OLS estimator is BLUE: the Best (minimum-variance) Linear Unbiased Estimator of the regression coefficients.

What is heteroskedasticity, what does it harm, and how is it addressed?

Heteroskedasticity is non-constant error variance $\operatorname{Var}(\varepsilon_i)\neq\sigma^{2}$. OLS stays unbiased but standard errors are biased, invalidating inference. Fix with robust (White) standard errors or GLS/WLS.

Define multicollinearity and describe its consequences and a diagnostic.

Multicollinearity is high correlation among regressors. It inflates coefficient standard errors, making estimates imprecise and unstable, though OLS stays unbiased. Diagnosed via the Variance Inflation Factor, $VIF_j=\frac{1}{1-R_j^{2}}$.

What is autocorrelation, and which test commonly detects first-order serial correlation?

Autocorrelation is correlation between error terms across observations, $\operatorname{Cov}(\varepsilon_t,\varepsilon_{t-1})\neq 0$, common in time series. The Durbin-Watson statistic (near $2$ means none) detects first-order serial correlation.

Define endogeneity and name a common remedy.

Endogeneity occurs when a regressor correlates with the error term, $E[\varepsilon\mid X]\neq 0$ (from omitted variables, simultaneity, or measurement error), biasing OLS. A common remedy is instrumental variables (2SLS).

What does stationarity mean for a time series, and why does it matter?

A (weakly) stationary series has constant mean, constant variance, and autocovariance depending only on lag, not time. Stationarity is required for reliable estimation; non-stationary series risk spurious regression.

Distinguish an AR(1) from an MA(1) process.

AR(1): $y_t=\phi y_{t-1}+\varepsilon_t$ depends on its own past value. MA(1): $y_t=\varepsilon_t+\theta\varepsilon_{t-1}$ depends on the current and past error (shock) terms.

What is a unit root, and which test checks for it?

A unit root ($\phi=1$ in $y_t=\phi y_{t-1}+\varepsilon_t$) makes a series non-stationary (a random walk). The Augmented Dickey-Fuller (ADF) test checks for it; failing to reject implies a unit root. Differencing induces stationarity.

What is the key advantage of panel data over pure cross-section or time series?

Panel data track the same units over time, allowing control for unobserved time-invariant individual heterogeneity, more variation and degrees of freedom, and better handling of omitted-variable bias.

Compare fixed-effects and random-effects panel models, and name the choosing test.

Fixed effects allow unit-specific intercepts correlated with regressors (differences them out, controlling for time-invariant confounders). Random effects assume those effects are uncorrelated with regressors, giving more efficiency. The Hausman test guides the choice.

What is the fundamental problem of causal inference?

For any unit we observe only one potential outcome (treated or untreated), never both, so the individual treatment effect $Y_i(1)-Y_i(0)$ is never directly observed; we estimate averages via comparable groups.

Why is randomised assignment the gold standard for identifying causal effects?

Randomisation makes treatment independent of potential outcomes, balancing observed and unobserved confounders on average, so the difference in group mean outcomes is an unbiased estimate of the average treatment effect.

Explain the difference-in-differences estimator.

DiD compares the change in outcomes over time for a treated group with the change for a control group: $\widehat{ATT}=(\bar{Y}_{T,post}-\bar{Y}_{T,pre})-(\bar{Y}_{C,post}-\bar{Y}_{C,pre})$. It relies on the parallel-trends assumption.

Distinguish correlation from causation with a practical caution for economic data.

Correlation is statistical association; causation means one variable produces a change in another. Association can arise from reverse causality, confounding, or coincidence, so correlation alone does not establish a causal (policy) effect.

What is the difference between nominal and real economic values?

Nominal values are measured at current prices; real values are adjusted for inflation to a base year, allowing comparison of purchasing power or output over time. Real $=$ nominal deflated by a price index.

How do you interpret coefficients in a log-log versus a log-linear regression?

In log-log ($\ln y=\beta_0+\beta_1\ln x$), $\beta_1$ is an elasticity (a $1\%$ rise in $x$ changes $y$ by $\beta_1\%$). In log-linear ($\ln y=\beta_0+\beta_1 x$), a one-unit rise in $x$ changes $y$ by about $100\beta_1\%$.

What is an index number, and how is a simple price index computed relative to a base year?

An index number expresses a value relative to a base period set to $100$. A simple price index is $\frac{P_t}{P_0}\times 100$, so a value of $115$ means a $15\%$ increase from the base year.