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 $\epsilon \sim N(0, \sigma_\epsilon)$

$E[Y_X] = \beta_0 + \beta_1 \cdot X$

$E[Y_{X+1}] = \beta_0 + \beta_1 \cdot (X+1)$

$\beta_1 = E[Y_{X+1}] - E[Y_X]$

Inference for Linear Regression in R

Interpreting coefficients - nonlinear X

$Y = \beta_0 + \beta_1 \cdot \ln(X) + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$

$E[Y_{\ln(X)}] = \beta_0 + \beta_1 \cdot \ln(X)$

$E[Y_{\ln(X)+1}] = \beta_0 + \beta_1 \cdot (\ln(X)+1)$

$\beta_1 = E[Y_{\ln(X)+1}] - E[Y_{\ln(X)}]$

Inference for Linear Regression in R

Interpreting coefficients - nonlinear Y

$\ln(Y) = \beta_0 + \beta_1 \cdot X + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$

$E[\ln(Y)_X] = \beta_0 + \beta_1 \cdot X$

$E[\ln(Y)_{X+1}] = \beta_0 + \beta_1 \cdot (X+1)$

$\beta_1 = E[\ln(Y)_{X+1}] - E[\ln(Y)_X]$

Inference for Linear Regression in R

Interpreting coefficients - both nonlinear

$\ln(Y) = \beta_0 + \beta_1 \cdot \ln(X) + \epsilon$, where $\epsilon \sim N(0, \sigma_\epsilon)$

$E[\ln(Y)_{\ln(X)}] = \beta_0 + \beta_1 \cdot \ln(X)$

$E[\ln(Y)_{\ln(X)+1}] = \beta_0 + \beta_1 \cdot (\ln(X)+1)$

$\beta_1 = E[\ln(Y)_{\ln(X)+1}] - E[\ln(Y)_{\ln{X}}]$

Inference for Linear Regression in R

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 $X$

Inference for Linear Regression in R

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Inference for Linear Regression in R

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