Calculating income elasticity of demand

Question

My model is as follows: $$Y=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_3+\beta_4X_4.$$

My income variable is represented by $X_2$. When it comes to calculating the income elasticity of demand (demand being represented by $Y$) I am aware that the formula is $\beta \frac{\bar{I}}{\bar{Q}}$, where $I$ is income and $Q$ is the demand variable.

My question is, when running a least squares regression, using EViews, would I run a regression with my $Y$ dependent variable and all my other independent variables ($X_1, X_2, X_3, X_4$ and the intercept) or do I just run the regression with $Y$ and $X_2$? (In this case are we assuming that all other variables do not matter? Or that we are holding these other variables constant?).

Moreover, I am aware that one can log the values in a regression and then extract the coefficient value as the elasticity value without needing to conduct more calculations. In this case, would we run the regression with $\log{Y}$ dependent variable and all my other independent variables logged ($X_1, X_2, X_3, X_4$ and the intercept) or do I just run the regression with $\log{Y}$ and $\log{X_2}$, or do I include the other independent variables and only log the $Y$ variable and $X_2$ variable.

I hope my question makes some sense. Essentially, I want to understand how one would calculate the income elasticity of demand in EViews based on the above model.

Any help with be greatly appreciated. Thank you.

Answer 1

To get a better estimation on the income elasticity of demand you should include all the X-variables in the model. The estimated coefficient you will get for your income elasticity variable in the complete model will be considered "Income elasticity of demand while holding other X-variables constant". If you omit the other X-variables, you do indeed assume that they do not have an effect on demand and thus do not belong in the model.

If you omit the other X-variables from your model while they do actually have an effect on demand, you will be committing a specification error and your estimations will be biased, inconsistent and your standard error estimations will be invalid. This will basically make your estimations very misleading.

Whether to take the log on one or both sides - or only on individual variables depends on the model and should be based on the specific model and previous research on the topic. Taking the log of a variable is often used to stabilize the variance of the variable or to show some sort of relative change in the variable. Hope this helps!