Consider the log-linear regression model \(\ln (y)=\beta_{1}+\beta_{2} x+\delta_{1} D+\delta_{2}(x \times D)+e\). If the regression errors are normally distributed \(N\left(0, \sigma^{2}\right)\), then

This is the marginal effect, the percentage change, in \(E(y \mid x, D)\) given a unit change in \(x\) in the log-linear model.

d. A fitted log-linear model for house price, where \(S Q F T(x)\) is the house's living area (100s of square feet) and \(U T O W N(D)\) is an indicator variable with \(U T O W N=1\) for houses near a university, and zero otherwise, is

Use equation (XR7.11.4) to calculate the marginal effect of \(S Q F T\) on house price, for a house with \(U T O W N=1\) and for a house with \(U T O W N=0\).

e. Let \(b_{2}\) and \(d_{2}\) be the least squares estimators of \(\beta_{2}\) and \(\delta_{2}\) in equation (XR7.11.4). Write down the formula for the standard error of the estimated value \(100\left(b_{2}+d_{2} D\right)\), for a given \(D\).

f. Multiply both sides in (XR7.11.3) by \(x\), and by \(100 / 100\), and rearrange to obtain

Interpreting \(100 \partial x / x\) as the percentage change in \(x\), we find that the elasticity of expected price with respect to a percentage change in \(x\) is \(\left(\beta_{2}+\delta_{2} D\right) x\).
g. Apply the result in equation (XR7.11.5) to calculate the elasticities of expected house price with respect to a change in price for a house of 2500 square feet, when \(U T O W N=1\) and when \(U T O W N=0\).
h. Let \(b_{2}\) and \(d_{2}\) be the least squares estimators of \(\beta_{2}\) and \(\delta_{2}\) in equation (XR7.11.5). Write down the formula for the standard error of the estimated value \(\left(b_{2}+d_{2} D\right) x\), given \(D\) and \(x\).

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E(yx, D) = exp(B +2x + D+ 2(xD)) exp(o/2) (XR7.11.1)