ECN 102: Analysis of Economics Data
Final Exam Formula Sheet – Beauregard
Univariate Data
\[\bar{x}=\frac1n\sum_{i=1}^nx_i\quad s_x^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2\quad se_x=\frac{s_x}{\sqrt{n}}\]
\[\bar{x}\pm t^*_{\alpha/2;n-1}\times(s_x/\sqrt{n})\quad t=\frac{\bar{x}-\mu_0}{s_x/\sqrt{n}}\]
\[ttail(df,t)=P[T>t]\quad invttail(df,p)\rightarrow t^*:P[T>t^*]=p\]
Bivariate Data
\[r_{xy}=\frac{s_{xy}}{s_x s_y}\quad\hat{y}=b_1+b_2x\quad b_1=\bar{y}-b_2\bar{x}\quad b_2=\frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2}=r_{xy}\frac{s_y}{s_x}\]
\[TSS=\sum(y_i-\bar{y})^2\quad ResSS=\sum(y_i-\hat{y}_i)^2\quad R^2=1-\frac{ResSS}{TSS}\]
\[b_2\pm t^*_{\alpha/2;n-2}\times s_{b_2}\quad t=\frac{b_2-\beta_2}{s_{b_2}}\quad s_{b_2}=\frac{s_e}{\sqrt{\sum(x_i-\bar{x})^2}}\quad s_e=\sqrt{\frac{ResSS}{n-2}}\]
Multivariate Data
\[\hat{y}=b_1+b_2x_2+\ldots+b_kx_k\quad b_j=\frac{\sum_{i=1}^n\tilde{x}_{ji}(y_i-\bar{y})}{\sum_{i=1}^n\tilde{x}_{ji}^2}\]
\[b_j\pm t^*_{\alpha/2,n-k}\times s_{b_j}\quad t=\frac{b_j-\beta_j}{s_{b_j}}\quad s_{b_j}=\frac{s_e}{\sqrt{\sum_{i=1}^n\tilde{x}_{ji}^2}}\quad s_e=\sqrt{\frac{ResSS}{n-k}}\]
\[\bar{R}^2=1-\frac{ResSS/(n-k)}{TSS/(n-1)}\quad F_{q,n-k}=\frac{(ResSS_r-ResSS_u)/q}{ResSS_u/(n-k)}\]
\[invFtail(q,n-k,p)\rightarrow F^*:P[F>F^*]=p\]