How does the hedonic regression price index work?

I'm interested in building a price index for car values. I've been reading a lot about the use of hedonic regression for creating these kinds of indices. However, I don't fully understand how it works.

Suppose I have data with the car prices, along with three characteristics ($$z_1, z_2$$ and $$z_3$$) which for instance show mileage, age and car model.

It looks like initially, the car prices at (for instance) each month are estimated using a regression model $$p_n^t = \beta_0^t + \sum_{k=1}^3 \beta_k^t z_{nk}^t + \epsilon_n^t$$. But then how is the index created? And why is this called hedonic regression as it looks like is just uses regular OLS regression?

• You need to explain what t is. The index could be the beta-0-t constant term with t defining some grouping of the observations n=1,...,N. Commented May 12, 2020 at 20:23

We first introduce a minimum of economic theory. Goods and services are sold on market where every sale results in an observable price. A market basket consist of several goods and each of these have a unit price. When you fill your basket in the supermarket the total expenditure $$P_b$$ for the basket $$b$$ is given as

$$P_b = \lambda_1 z_{1b }+ \lambda_2 z_{2b} + e_b$$

where $$z_{1b}$$ is the number of bottles of milk in the basket $$b$$ and $$\lambda_1$$ the price per bottle, $$z_{2b}$$ is the boxes of serial in the basket $$b$$ and $$\lambda_2$$ the price per box. Finally the error terms $$e_b$$ is how much the person buying basket $$b$$ decided to tip.

The thing to notice is offcourse that the coefficients $$\lambda_1$$ and $$\lambda_2$$ are prices.

Next we need the concept of a composite good, which we simple define as some good that has a vector of characteristics $$\mathbf w$$ that are deemed relevant for the price of the good but are not themselves traded at a market where they have observable unit prices. To make this connection apparent we assume that the price is a function of the charachteristics $$p(\mathbf w)$$. We then assume that some of the features $$\mathbf x$$ are observable and some not $$e$$. Hence it follows that $$\mathbf w =(\mathbf x,e)$$ and that $$p(\mathbf w) = p(\mathbf x,e)$$.

Impose now a functional form assumption of lineraity justified purely by its simplicity and you have

$$p(\mathbf x,e) = \mathbf x^\top \beta + e,$$

which in your particular case becomes

$$p_i = p(\mathbf x_i,e_i) = \mathbf x_i^\top \beta + e_i= \beta_0 + \sum_{k=1}^3 x_{k}\beta_{ki} + e_i,$$

where by analogy to the supermarket basket example we see $$\beta_k$$ as the unit price of the feature $$x_k$$. Since the prices are not observed but only inferred we call them "implicit prices".

How then do we get these to be called "hedonic"? The term "hedonic" has Greek roots and are associated with ancient greek philosophers for whom the pursuit of "happiness" was key (see for example wiki on hedonism). These philosphers are seen as the historical intellectual roots of utilitarianism where man seeks to achieve higher utility. The idea of economic actors as utility maximizer is today foundational for modern economics (not to deny existence of alternative schools, but ...).

Consider now an actor who has to buy a car. He maximize utiliy choosing the car with characteristics $$(x,e)$$ that bring him the most utility for the dollar $$p(x,e)$$ and use the rest of his money on a package other consumer goods $$c$$, hence he solves

$$max_{c,x} \ \ U(c,x,e)$$

$$s.t. \ \ I = c + p(x,e)$$

where $$c$$ is the standard Hicksian good with unit price standardize to 1. This maximization problem was studied by Rosen (1974) Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition and while he to my knowledge is not the first to use the term "hedonic prices" his article is the standard reference for giving the using of hedonic regressions an economic theoretical justification.

The key point to take away is simply that the characteristics are utility bearing so to speak because they enter the actors utility function.

Since we have already connected the term "hedonic" to "utility" and have seen that the coefficients are kind of prices we get the idea to call them: Hedonic prices.

If you want a historical explanation stating the first time use in economics by some author with major influence you have to do your own research.

How then is this used to create an index? You run the regression

$$p_i^t = \beta^t_0 + \sum_{k=1}^3 x_{k}\beta_{ki} + e^t_i$$

and for each type $$t$$ you get what remains $$\beta^t_0$$ of the price ones you have controlled for quality differences $$\sum_{k=1}^3 x_{k}\beta_{ki}$$ and thrown away any unsystematic varition $$e^t_i$$. Since $$\beta^t_0$$ is what remains of the price for good of type t ones you have controlled for quality differences, $$\beta^t_0$$ can be used to compare the different types $$t=1,...,T$$ on price and hence serves as an index.

Offcourse you need multiple observations for each type $$t$$. And you can complicate the model to let $$\beta_{ki}$$ be type dependent $$\beta^t_{ki}$$ and you can in general allow for non-linear function forms.

For a paper on the best functional form used in hedonic regressions on the housing market see Kuminoff et. al. (2010) Which hedonic models can we trust to recover the marginal willingness to pay for environmental amenities?

• So for instance, if $t$ is time, and $i \in \{1, \ldots n_t\}$ represents the number of cars at time $t$, then you run the regression over all $n_t$ cars at time $t$. This gives you estimates of $\beta_{kt}$ at time $t$, and if you want to calculate the index $P_{st}$ at time $t$ with respect to time $s$, then you use $\frac{\hat{p}_t(\bar{x_t})}{\hat{p}_s(\bar{x_t})}$ with $\bar{x}_t$ the average of the characteristics of all cars at time $t$ and $\hat{p}^t$ the estimated regression at time $t$? Then this is the Paasche-type index? Commented May 13, 2020 at 12:26
• No. Based on what I have written how do you get that conclusion? Commented May 13, 2020 at 13:00
• Not based on what you've written, but based on an OECD Statistics Working Papers 2011/01, called 'Hedonic Price Indexes for Housing', page 30. But I guess this is then called the 'characteristics' method. Commented May 13, 2020 at 13:22
• I do not see how that is related to your initial question. I actually state in my answer what you use as index. If you found the answer I provided helpful pls. consider accepting and upvoting. If you have further question - for example related to the paper you posted - I think it would be better to ask them in new post (and would be happy to help), comments are however not intended for that. Commented May 13, 2020 at 14:33
• Nevertheless looking at the paper the answer I have suggested would amount to what they refer to as the "time dummy method" Commented May 13, 2020 at 14:42