### Interpret Regression Coefficient Estimates - {level-level, log-level, level-log & log-log regression}

 Interpreting Beta: how to interpret your estimate of your regression coefficients (given a level-level, log-level, level-log, and log-log regression)? Assumptions before we may interpret our results:  The Gauss–Markov assumptions* hold (in a lot of situations these assumptions may be relaxed - particularly if you are only interested in an approximation - but for now assume they strictly hold).   * If you're interested in more details, read the discussion here, or check out your textbook. Our coefficient estimates (our estimates of  below) are statistically significant and practically significant.  With a multivariate model, we assume that other independent variable(s) (x_2, x_3, ... x_n) are held constant.

 Model Dependent or Response  Variable (y) Independent or  Explanatory Variable (x) Interpretation of β Given a change in x,  how much do we expect y to change by? Video Review Given reader requests, I created short video explanations of how to interpret regression estimates y x Δy=β1Δx  “If you change x by one,  we’d expect y  to change by β1" Video 5:00 - www.youtube.com/watch?v=TJACbJspao0 Log-Level Regression x %Δy=100⋅β1⋅Δx “if we change x by 1 (unit), we’d expect our y variable to change by 100⋅β1 percent” Technically, the interpretation is the following:         $\dpi{100} \%\Delta y = 100 \cdot \left ( e^{\beta_1}-1 \right )$ but the quoted interpretation is approximately true for values -0.1 < β1 < 0.1 (and it's much easier to remember.) Video 6:40 - www.youtube.com/watch?v=wXC2kViEGz8 Level-Log Regression y Δy=(β1/100)%Δx   "If we increase x by one percent,  we expect y to increase by (β1/100) units of y." Note, you cannot include obs. for which x<=0 if x is then logged. You either can't calculate the regression coefficients, or may introduce bias. Video 6:50 www.youtube.com/watch?v=L9ZL6_DB4fQ %Δy=β1%Δx  “if we change x by one percent,  we’d expect y to change by β1 percent” Note, you cannot include obs. for which x<=0 if x is logged. You either can't calculate the regression coefficients, or may introduce bias.

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Curtis Kephart,
Jun 12, 2013, 4:47 PM