Dummy variables in regression with log-transformed continous variables I have a question dealing with logarithmic variables, log-sum and dummy variables in a multiple regression.
First of all I’ve never worked with log-sums before.
I have a model with y that is log-transformed. The independent variables are: one log-transformed, two dummy variables and one variables that is a log-sum. I want to use the model to predict some y-values. What is the correct way to do the “un-logging”? I’ve read that for example if I had this model:
$\ln Y=\alpha + \beta_1 \times \ln X_1 \to Y=e^{\alpha+\beta_1\times \ln X_1}$
But how should I deal with the dummy variables?
I have a lot of observations where the values of the independent variables are known (x-values), but I'm not sure how I should transform the variables. I have run the model with the known x-values (including the dummies) and after that exp. I have some knowledge about the topic and I know that the predicted y-values are not trustworthy. That gives me the idea that I am doing something wrong with the transformation of the dummy variables.
Would really appreciate if someone could help me with this.
 A: You can express your DGP as:
$$y=\alpha \cdot x^\beta\cdot \gamma^d \cdot\varepsilon$$
Here $x$ is a positive continuous variable, and $d$ is binary, either 0 or 1. Everything Greek is an unknown parameter, and the error term $\varepsilon$ is mean one, not zero. Note that $\gamma^1=\gamma$ and $\gamma^0=1$, so d toggles a multiplicative factor.
Now we take natural logs to linearize:
$$\ln y= \ln\alpha +  \beta \cdot \ln x + d \cdot \ln \gamma + \ln \varepsilon =a + b \cdot \ln x + c \cdot d + e $$
The point is that you don't need to transform your dummy variables when you fit the model.
Unfortunately, things are not that easy if you need to get back to $y$ from $\ln y$. You can use the Duan smearing approach:
$$\mathbf{E}[ y \vert x,d] \approx \exp\{\hat a + \hat b \cdot \ln x + \hat c \cdot d \} \cdot \frac{1}{N} \sum_{i=1}^N \exp\{\hat e_i\}$$

For the log sum, you probably have something like a coefficient restriction in mind:
$$y=\alpha \cdot x^\beta\cdot \gamma^d \cdot (z_1 \cdot z_2)^\eta \cdot \varepsilon,$$ which gets you
$$\ln y= \ln\alpha +  \beta \cdot \ln x + d \cdot \ln \gamma + \eta \cdot (\ln z_1 + \ln z_2) + \ln \varepsilon $$
