A. Formula Sheet
Use this sheet for quick review. Focus on interpretation first, then calculation.
Descriptive Statistics
Mean
\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \]
Average value of a variable.
Variance
\[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2 \]
Average squared distance from the mean.
Standard Deviation
\[ s = \sqrt{s^2} \]
Typical distance from the mean.
Correlation
\[ r_{xy} = \frac{\operatorname{cov}(x,y)}{s_x s_y} \]
Measures linear association from -1 to 1. Correlation is not causation.
Regression
Simple Regression
\[ Y_i = \beta_0 + \beta_1 X_i + u_i \]
\(\beta_1\) measures the expected change in \(Y\) for a one-unit increase in \(X\).
Multiple Regression
\[ Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \cdots + \beta_k X_{ki} + u_i \]
Each coefficient is interpreted holding the other included variables constant.
Residual
\[ \hat{u}_i = Y_i - \hat{Y}_i \]
Difference between actual and predicted value.
R-squared
\[ R^2 = 1 - \frac{SSR}{SST} \]
Share of variation in \(Y\) explained by the model.
Adjusted R-squared
\[ \bar{R}^2 = 1 - (1-R^2)\frac{n-1}{n-k-1} \]
Penalizes adding extra variables that do not improve the model enough.
Inference
t-statistic
\[ t = \frac{\hat{\beta}_j - \beta_{j,0}}{SE(\hat{\beta}_j)} \]
Used to test whether a coefficient differs from a hypothesized value.
Confidence Interval
\[ \hat{\beta}_j \pm t^* SE(\hat{\beta}_j) \]
Range of plausible values for the coefficient.
p-value Rule
If \(p < 0.05\), reject the null hypothesis at the 5% significance level.
Always interpret economic significance as well as statistical significance.
F-test
\[ H_0:\beta_1=\beta_2=\cdots=\beta_q=0 \]
Rejecting \(H_0\) means the variables are jointly significant.
Prediction Accuracy
MAE
\[ MAE = \frac{1}{n}\sum_{i=1}^{n}|Y_i-\hat{Y}_i| \]
Average absolute prediction error.
MSE
\[ MSE = \frac{1}{n}\sum_{i=1}^{n}(Y_i-\hat{Y}_i)^2 \]
Average squared prediction error.
RMSE
\[ RMSE = \sqrt{MSE} \]
Typical prediction error measured in the units of \(Y\).
Functional Forms
Log-linear Model
\[ \ln(Y_i) = \beta_0 + \beta_1 X_i + u_i \]
A one-unit increase in \(X\) is associated with approximately \(100\beta_1\%\) change in \(Y\).
Linear-log Model
\[ Y_i = \beta_0 + \beta_1 \ln(X_i) + u_i \]
A 1% increase in \(X\) is associated with approximately \(\beta_1/100\) unit change in \(Y\).
Log-log Model
\[ \ln(Y_i) = \beta_0 + \beta_1 \ln(X_i) + u_i \]
\(\beta_1\) is an elasticity.
A 1% increase in \(X\) is associated with a \(\beta_1\%\) change in \(Y\).
Categorical Variables
Dummy Variable
The coefficient compares the included category to the omitted reference category, holding other variables constant.
Interaction Model
\[ Y_i = \beta_0 + \beta_1 X_i + \beta_2 D_i + \beta_3(X_iD_i) + u_i \]
\[ \frac{\partial Y_i}{\partial X_i} = \beta_1 + \beta_3 D_i \]
\(\beta_3\) shows how the slope of \(X\) changes when \(D=1\).
Diagnostics
Variance Inflation Factor (VIF)
VIF > 5 suggests possible multicollinearity.
VIF > 10 is often considered serious.
Durbin-Watson
Values near 2 suggest little autocorrelation.
Values substantially below 2 suggest positive autocorrelation.
Breusch-Pagan Test
Tests for heteroskedasticity.
A small p-value suggests nonconstant error variance.
Ramsey RESET Test
Tests for functional form problems.
A small p-value suggests the model may be misspecified.