ECN 102: Analysis of Economics Data

Midterm Formula Sheet – Beauregard

Univariate Data

\[\bar{x}=\frac1n\sum_{i=1}^nx_i\quad s_x^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2\] \[\bar{x}\pm t^*_{\alpha/2;n-1}\times(s_x/\sqrt{n})\quad t=\frac{\bar{x}-\mu_0}{s_x/\sqrt{n}}\] \[ttail(df,t)=P[T>t]\text{ where }T\sim T(df)\] \[invttail(df,p)\rightarrow t^*:P[T>t^*]=p\text{ where }T\sim T(df)\]

Bivariate Data

\[r_{xy}=\frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2\times\sum_{i=1}^n(y_i-\bar{y})^2}} = \frac{s_{xy}}{s_x\times s_y}\] \[\hat{y}=b_1+b_2x\] \[b_1=\bar{y}-b_2\bar{x}\] \[b_2=\frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2}= r_{xy}\frac{s_y}{s_x}\] \[TSS=\sum_{i=1}^n(y_i-\bar{y})^2\quad ResSS=\sum_{i=1}^n(y_i-\hat{y}_i)^2\quad ExpSS=\sum_{i=1}^n(\hat{y}_i-\bar{y})^2\] \[R^2=\frac{ExpSS}{TSS}=1-\frac{ResSS}{TSS}\] \[b_2\pm t^*_{\alpha/2;n-2}\times s_{b_2}\quad t=\frac{b_2-\beta_2}{s_{b_2}}\] \[s_{b_2}=\frac{s_e}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2}}\quad s_e=\sqrt{\frac{1}{n-2}\sum_{i=1}^n(y_i-\hat{y}_i)^2}\]