Formula Sheet

Exam Reference

Key formulas for ECMT5001 (USYD S1 2027). Organised by module. All notation follows Wooldridge Introductory Econometrics unless stated.

M01 — Statistics Review

Expectation and Variance

\[E(aX + bY) = aE(X) + bE(Y)\]

\[\text{Var}(X) = E(X^2) - [E(X)]^2\]

\[\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y) + 2ab\,\text{Cov}(X,Y)\]

\[\text{Cov}(X,Y) = E(XY) - E(X)E(Y)\]

\[\text{Corr}(X,Y) = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\,\text{Var}(Y)}} \in [-1, 1]\]

If \(X\) and \(Y\) are independent: \(\text{Cov}(X,Y) = 0\) (but not the converse in general).

Conditional Expectation

\[E(Y) = E[E(Y \mid X)] \quad \text{(Law of Iterated Expectations)}\]

\[\text{Var}(Y) = E[\text{Var}(Y \mid X)] + \text{Var}[E(Y \mid X)] \quad \text{(LTE variance decomposition)}\]

Standard Distributions

Distribution	PMF/PDF	\(E(X)\)	\(\text{Var}(X)\)
\(\text{Bernoulli}(p)\)	\(p^x(1-p)^{1-x}\)	\(p\)	\(p(1-p)\)
\(\text{Binomial}(n,p)\)	\(\binom{n}{x}p^x(1-p)^{n-x}\)	\(np\)	\(np(1-p)\)
\(\mathcal{N}(\mu,\sigma^2)\)	\(\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)	\(\mu\)	\(\sigma^2\)
\(\chi^2_k\)	—	\(k\)	\(2k\)
\(t_k\)	—	\(0\) (\(k>1\))	\(k/(k-2)\) (\(k>2\))
\(F_{k_1,k_2}\)	—	\(k_2/(k_2-2)\)	—

Hypothesis Testing

\(z\)-test for proportion \(p\):

\[z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}} \sim \mathcal{N}(0,1) \text{ under } H_0\]

One-sample \(t\)-test for \(\mu\):

\[t = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} \sim t_{n-1} \text{ under } H_0\]

\(95\%\) confidence interval for \(\mu\):

\[\bar{X} \pm t_{n-1, 0.025} \cdot \frac{s}{\sqrt{n}}\]

M02 — Simple Regression

The Simple Regression Model

\[y = \beta_0 + \beta_1 x + u, \quad E(u \mid x) = 0\]

OLS Estimators

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

\[\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}\]

Sums of Squares

\[SST = \sum_{i=1}^n (y_i - \bar{y})^2 \quad \text{(total)}\]

\[SSE = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2 \quad \text{(explained)}\]

\[SSR = \sum_{i=1}^n \hat{u}_i^2 \quad \text{(residual)}\]

\[SST = SSE + SSR\]

Goodness of Fit

\[R^2 = 1 - \frac{SSR}{SST} = \frac{SSE}{SST} \in [0,1]\]

For simple regression: \(R^2 = \hat{\text{Corr}}(x,y)^2\)

Variance and Standard Error of OLS

\[\text{Var}(\hat{\beta}_1) = \frac{\sigma^2}{SST_x}, \quad \text{Var}(\hat{\beta}_0) = \frac{\sigma^2 \sum x_i^2}{n \cdot SST_x}\]

\[\hat{\sigma}^2 = \frac{SSR}{n-2} \quad \text{(unbiased estimator of } \sigma^2\text{)}\]

\[SE(\hat{\beta}_1) = \frac{\hat{\sigma}}{\sqrt{SST_x}}\]

Inference in Simple Regression

\[t = \frac{\hat{\beta}_1 - \beta_1^0}{SE(\hat{\beta}_1)} \sim t_{n-2} \quad \text{(under MLR.1–MLR.6)}\]

\(95\%\) CI for \(\beta_1\): \(\hat{\beta}_1 \pm t_{n-2,0.025} \cdot SE(\hat{\beta}_1)\)

Algebraic Properties of OLS

\[\sum_{i=1}^n \hat{u}_i = 0, \quad \sum_{i=1}^n x_i \hat{u}_i = 0, \quad \bar{y} = \hat{\beta}_0 + \hat{\beta}_1\bar{x}\]

M03 — Multiple Regression

The Multiple Regression Model

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k + u\]

Matrix form: \(\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{u}\)

OLS estimator: \(\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}\) (requires \(\mathbf{X}'\mathbf{X}\) invertible)

Adjusted R-Squared

\[\bar{R}^2 = 1 - \frac{SSR/(n-k-1)}{SST/(n-1)} = 1 - \frac{n-1}{n-k-1}(1-R^2)\]

\(\bar{R}^2 \leq R^2\) always
\(\bar{R}^2\) can be negative
\(\bar{R}^2\) rises when adding a regressor iff its \(|t\text{-stat}| > 1\) (approx.)

MLR Assumptions Summary

Assumption	Statement	Consequence if Violated
MLR.1	Linear in parameters	Model misspecified
MLR.2	Random sampling	Generalisation invalid
MLR.3	No perfect collinearity	OLS cannot be computed
MLR.4	\(E(u\mid\mathbf{x})=0\)	OLS biased and inconsistent
MLR.5	\(\text{Var}(u\mid\mathbf{x})=\sigma^2\)	OLS inefficient; SEs wrong
MLR.6	\(u\mid\mathbf{x}\sim\mathcal{N}(0,\sigma^2)\)	Finite-sample \(t\)/\(F\) invalid

Gauss–Markov Theorem: Under MLR.1–MLR.5, OLS is BLUE (Best Linear Unbiased Estimator).

Omitted Variable Bias (OVB)

True model: \(y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u\)
Estimated (omitting \(x_2\)): \(y = \tilde{\beta}_0 + \tilde{\beta}_1 x_1 + v\)

\[\boxed{E(\tilde{\beta}_1) = \beta_1 + \beta_2\,\tilde{\delta}_1}\]

where \(\tilde{\delta}_1 = \dfrac{\sum(x_{i1}-\bar{x}_1)(x_{i2}-\bar{x}_2)}{\sum(x_{i1}-\bar{x}_1)^2}\) is the coefficient from regressing \(x_2\) on \(x_1\).

Bias direction:

	\(\tilde{\delta}_1 > 0\)	\(\tilde{\delta}_1 < 0\)
\(\beta_2 > 0\)	Upward bias	Downward bias
\(\beta_2 < 0\)	Downward bias	Upward bias

Variance of OLS in Multiple Regression

\[\text{Var}(\hat{\beta}_j) = \frac{\sigma^2}{SST_j(1 - R_j^2)}\]

where \(SST_j = \sum(x_{ij}-\bar{x}_j)^2\) and \(R_j^2\) is the \(R^2\) from regressing \(x_j\) on all other regressors.

Variance Inflation Factor

\[VIF_j = \frac{1}{1 - R_j^2}\]

\[\text{Var}(\hat{\beta}_j) = \frac{\sigma^2}{SST_j} \cdot VIF_j\]

Rule of thumb: \(VIF_j > 10\) indicates severe multicollinearity.

\[SE(\hat{\beta}_j) = \frac{\hat{\sigma}}{\sqrt{SST_j(1-R_j^2)}} = \frac{\hat{\sigma}\sqrt{VIF_j}}{\sqrt{SST_j}}\]

Inference in Multiple Regression

\(t\)-statistic for \(\beta_j\):

\[t_j = \frac{\hat{\beta}_j}{SE(\hat{\beta}_j)} \sim t_{n-k-1} \quad \text{under } H_0: \beta_j = 0 \text{ and MLR.1–MLR.6}\]

\(F\)-statistic (joint test of \(q\) restrictions):

\[F = \frac{(SSR_r - SSR_{ur})/q}{SSR_{ur}/(n-k-1)} \sim F_{q,\,n-k-1}\]

where \(r\) = restricted model, \(ur\) = unrestricted model, \(q\) = number of restrictions.

Equivalently using \(R^2\):

\[F = \frac{(R^2_{ur} - R^2_r)/q}{(1 - R^2_{ur})/(n-k-1)}\]

Quick Reference: Notation

Symbol	Definition
\(n\)	Sample size
\(k\)	Number of regressors (excluding intercept)
\(\hat{u}_i\)	OLS residual: \(y_i - \hat{y}_i\)
\(\sigma^2\)	Population error variance
\(\hat{\sigma}^2\)	\(SSR/(n-k-1)\) — estimated error variance
\(\hat{\sigma}\)	\(\sqrt{\hat{\sigma}^2}\) — root mean squared error (RMSE)
\(SST\)	\(\sum(y_i-\bar{y})^2\)
\(SSE\)	\(\sum(\hat{y}_i-\bar{y})^2\)
\(SSR\)	\(\sum\hat{u}_i^2\)
\(R^2\)	\(1 - SSR/SST\)
\(\bar{R}^2\)	Adjusted \(R^2\)
\(VIF_j\)	\(1/(1-R_j^2)\)