Chapter 5 Transformations

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5.1 Elasticity

An elasticity coefficient is the ratio of the percentage change in the forecast variable (\(y\)) to the percentage change in the predictor variable (\(x\)). Mathematically, the elasticity is defined as \((dy/dx)\times(x/y)\). Consider the log-log model, \[ \log y=\beta_0+\beta_1 \log x + \varepsilon. \] Express \(y\) as a function of \(x\) and show that the coefficient \(\beta_1\) is the elasticity coefficient.

\[ \log(y) = \beta_0 + \beta_1 \log(x) + \varepsilon \qquad \mid \mathbb{E}(\cdot \mid x)\] Taking the conditional expectation on the left and right hand side of the quation

\[\mathbb{E}(\log(y) \mid x) = \mathbb{E}(\beta_0 + \beta_1 \log(x) + \varepsilon \mid x)\] leads to the following as \(\mathbb{E}(\varepsilon\mid x) = 0\).

\[ \log(y) = \beta_0 + \beta_1 \log(x) \qquad \mid \frac{\partial}{\partial x}\] After taking derivatives of the left and right side with respect to \(x\) we receive. The left hand side follows due to the implicit differentiation.

\[ \frac{\partial}{\partial x}\log(y) = \frac{\partial}{\partial x}(\beta_0 + \beta_1 \log(x))\]

\[\frac{\partial\log(y)}{\partial x} = 0 + \beta_1 \frac{1}{x}\] \[\frac{1}{y} \frac{\partial y}{\partial x} = \beta_1 \frac{1}{x} \qquad \mid \cdot x\]

After rearranging we get the definition of the the elasticity.

\[\beta_1 = \frac{x}{y} \frac{\partial y}{\partial x}\]