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.  
  
  

Running a Regression (Using R Statistics Software)
Step-by-step example of how to do a regression using R statistics software (including the models below). I'll walk through the code for running a multivariate regression - plus we'll run a number of slightly more complicated examples to ensure it's all clear.
Video 16:30 - www.youtube.com/watch?v=Ktks5K95uQM 

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

Level-level Regression

y

x

Δy=β1Δx 

“If you change x by one, 
we’d expect y  to change by β1"

Interpreting Level-Level Regression Coefficient Estimate Results
We run a level-level regression and interpret the regression coefficient estimate results. Simple example of regression analysis with a level-level model.

Video 5:00 - www.youtube.com/watch?v=TJACbJspao0

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:  

but the quoted interpretation is approximately true for values -0.1 < β1 < 0.1 (and it's much easier to remember.)

Log-Level Regression Coefficient Estimate Interpretation
We run a log-level regression (using R) and interpret the regression coefficient estimate results. A nice simple example of regression analysis with a log-level model.

Video 6:40 - www.youtube.com/watch?v=wXC2kViEGz8 

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. 

Level-Log Regression Coefficient Estimates
We run a level-log regression and help understand the regression coefficient estimates. A nice simple example of regression analysis. 

Video 6:50 www.youtube.com/watch?v=L9ZL6_DB4fQ

Log-Log Regression

%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. 

Log-Log Regression Coefficient Estimate Results
We do a log-log regression and explain the regression coefficient estimate results. Simple example of regression analysis with a log-log model.

Video 5:30 www.youtube.com/watch?v=NZCSt9Wkpkl