Inference on transformed variables
Inference for Linear Regression in R
Jo Hardin
Professor, Pomona College
Interpreting coefficients - linear
$Y = \beta_0 + \beta_1 \cdot X + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$
$E[Y_X] = \beta_0 + \beta_1 \cdot X$
$E[Y_{X+1}] = \beta_0 + \beta_1 \cdot (X+1)$
$\beta_1 = E[Y_{X+1}] - E[Y_X]$
Interpreting coefficients - nonlinear X
$Y = \beta_0 + \beta_1 \cdot \ln(X) + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$
$E[Y_{\ln(X)}] = \beta_0 + \beta_1 \cdot \ln(X)$
$E[Y_{\ln(X)+1}] = \beta_0 + \beta_1 \cdot (\ln(X)+1)$
$\beta_1 = E[Y_{\ln(X)+1}] - E[Y_{\ln(X)}]$
Interpreting coefficients - nonlinear Y
$\ln(Y) = \beta_0 + \beta_1 \cdot X + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$
$E[\ln(Y)_X] = \beta_0 + \beta_1 \cdot X$
$E[\ln(Y)_{X+1}] = \beta_0 + \beta_1 \cdot (X+1)$
$\beta_1 = E[\ln(Y)_{X+1}] - E[\ln(Y)_X]$
Interpreting coefficients - both nonlinear
$\ln(Y) = \beta_0 + \beta_1 \cdot \ln(X) + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$
$E[\ln(Y)_{\ln(X)}] = \beta_0 + \beta_1 \cdot \ln(X)$
$E[\ln(Y)_{\ln(X)+1}] = \beta_0 + \beta_1 \cdot (\ln(X)+1)$
$\beta_1 = E[\ln(Y)_{\ln(X)+1}] - E[\ln(Y)_{\ln{X}}]$
Interpreting coefficients - both natural log (special case)
$\ln(Y) = \beta_0 + \beta_1 \cdot \ln(X) + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$
$E[\ln(Y)_{\ln(X)}] = \beta_0 + \beta_1 \cdot \ln(X)$
$E[\ln(Y)_{\ln(X)+1}] = \beta_0 + \beta_1 \cdot (\ln(X)+1)$
$\beta_1 = E[\ln(Y)_{\ln(X)+1}] - E[\ln(Y)_{\ln{X}}]$
OR (when $X$ and $Y$ are both transformed using natural log):
$\beta_1 = $ percent change in $Y$ for each 1% change in $X$
Let's practice!
Inference for Linear Regression in R
