# Functional form for regression in a hedonic pricing model

I am doing a regression where I use the hedonic pricing model. Hedonic pricing treats a marketed good, usually a house, as a sum of individual goods (characteristics or attributes) that cannot be sold separately in the market. The main objective of a hedonic pricing model is to estimate the contribution of such characteristics or attributes to the price of the house.

In the next step I have written the semi-log functional form/equation which is the following

I am really bad with equations. Is there anyone that could comment if they see anything wrong with these equations, and if i have to change it? I would really appreciate any comments and feedback.

## 1 Answer

It is widely considered that economic theory does not suggest any particular functional form for a hedonic pricing model. See for example Cassel & Mendelsohn 1985 which references several sources for that view. So theory does not rule out a linear functional form, as in your first equation, or a semi-log functional form, as in your second equation, but other forms are also possible.

With so many dependent variables, the range of possible functional forms is very large, since different variables may appear in the model in different ways, using a combination of mathematical operations, eg addition, multiplication, logs, etc. I would recommend thinking carefully about how the different variables might be expected to influence price. For example, price might be expected (at least in some localities) to be roughly proportional to size of house, other things being equal. On the other hand, although price might be expected to fall with distance from amenities, it seems unlikely that doubling the distance would halve the price. That might suggest multiplication for the former and addition (with the coefficient expected to be negative) for the latter, so that the right-hand side of the regression equation might include an expression such as:

$$S_1(\beta_1^LL_1+\beta_2^LL_2+…)$$

where $$S_1$$ is size of house and the $$L_i$$ are distances from various amenities.

Looking at your specific equations, the first has no constant term and a regression would therefore force the intercept to pass through the origin. Even if you are not interested in the value of $$P$$ when all the independent variables are zero, this restricts the position of the whole regression line in a way which may not reflect the true relationship between the variables. It would be more common to include a constant term.

In the second equation as formulated, the term $$\beta_0$$ is redundant since, for any value $$k$$ it might take, the same result would be obtained if each of the $$\beta_i (i = 1,…,n)$$ were multiplied by $$k$$. It would be better to include a $$+$$ sign immediately following $$\beta_0$$ which would then be a constant term and no longer redundant.

• Thank you for your answer. But if I may ask anything further, how would you rewrite my first equation? If you could've rewritten it the way you would've done it would have been really nice to see. Thank you for your answer anyway! Commented Jun 3, 2022 at 22:12
• There isn't a single correct way to rewrite the equation since much depends on both the choice of variables and the region from which the dataset is drawn. One way to include the suggestions in my answer would be, assuming $S_1$ is size of property: $$P=\beta_0+\sum_{i=1}^l\beta_i^LL_i+\sum_{j=1}^n\beta_j^SS_j+\sum_{k=1}^m\beta_k^mE_k+\sum_{i=1}^l\beta_i^{INT}S_1L_i+\varepsilon$$ Here the $\beta_i^{INT}$ are coefficients of what are known as interaction terms, in this case the product of size and location variables such as distance from amenities. Commented Jun 4, 2022 at 20:38