Modeling prices with the Hedonic regression

Question

I'm using the concept of Hedonic regression in order to model the prices for real estates. I'm having some trouble with my approach.

What I have and what I do

• my data consists out of real estates with following charcteristics: price | livingArea | propertyArea | condoFloorNumber | roomCount | elevator | garage | quiet | etc.
• I run a robust regression without intercept lmRob(price ~ . -1)

What I want

• a model with which I can predict the price of real estates, but which are not in the used data set
• also it would be nice to have some constraints on the coefficients

Problems

• very often I get bad values for the coefficients ex: bathroomCount = -80000. it's not possible that with a additive bathroom , the price of the house will sink with 80.000€

• also I tried to use the function pcls in order to put some constraints on the coefficients, but this method gave very bad results. In the plot Y = price and X = livingArea. as you can see, the regression line isn't correct.

  • another thought was to transform the regression problem into a maximization or minimization problem, but didn't managed to do it
  • also I tried to use different regression methods lm, lmrob, ltsReg, MARS, but they also give me bad coefficients. (sometimes this bad coefficients make a good price estimation)
  • I think that the big number of dummy variables damages a little bit the regression

Is my approach false?

Does someone have some hints, tricks for me? (I'm not a statistician)

[UPDATE]

This is how the plotted data looks like. LivingArea is the only non-dummy variable.

[UPDATE 2]

y = bX 

     means

y = b_0*X_0 + b_1*X_1 + ... + b_k*X_k

     which is an equation system like this:

y[0] = b_0*X_0[0] + b_1*X_1[0] + ... + b_k*X_k[0]
.
.
.
y[n] = b_0*X_0[n] + b_1*X_1[n] + ... + b_k*X_k[n]

Did I got it right?

If so, isn't possible to add some inequality constraints equation to it. example:

b_0 >= 2000
b_2 <= b_0/2

[UPDATE 3]

I'm running the regression without intercept, because if all the characteristics of a real estate = 0, then of course it'S price = 0. Nobody would pay for an apartment with 0m². but it seems that the regression line where it was used an intercept (blue) looks far more better than the regression line without intercept (green). I can't understand why it is so. and why doesn't the regression line without intercept start at the point (0,0)?

Answer 1

This type of approach clearly can work (and has evidently been used by tax authorities to set property taxes on my house for many years), so there needs to be some investigation of the sources of this difficulty.

Understanding the nature of this data set is very important. If it is to be used for predicting prices of properties not in the data set you must be very certain that it is adequately representative of the population of properties of interest. It's possible there is some peculiarity in the way this particular sample was collected, so that some particular combinations of co-linear factors are leading to things like the negative coefficients for bathroom numbers. Re-evaluate the sample collection and the data coding, an oft-overlooked source of difficulty. Also, for your PCA-based approaches, the signs of coefficients for principal components depend on the directions of the associated eigenvectors, making it all too easy to create errors when you try to go back to the space of the original factors. Check that, too.

You didn't specify the standard errors of your coefficient estimates, so some of your apparently anomalous coefficients might not be significantly different from 0. For example, a -80K coefficient per bathroom with a standard error of +/- 100K would not really be an issue; that probably just means that the high co-linearity makes it difficult to determine a value per bathroom, given its high association with land area, numbers of bedrooms, and so forth. If that's the case you should retain the coefficient when making predictions, as the apparently anomalous coefficient for bathrooms is probably helping to correct for price over-estimates based on some of its co-linear factors alone.

You could try to figure out which combinations of factors are leading to these problems. Although stepwise selection of factors is not wise for building a final model, for troubleshooting you might consider starting with a simple model of price-bathroom relations and adding more factors to see which combinations of factors are leading to your problem.

You also should take advantage of information from structured re-sampling of your data set to evaluate these issues. You don't say whether or how you have approached this crucial aspect of model validation. If you have, then cross-validation or bootstrap resampling may have already provided insights into the sources of your difficulty. If you haven't, consult An Introduction to Statistical Learning or similar references to see how to proceed.

Answer 2

I know this is an old post; hope the message helps someone reading this thread who might approach the same problem. The logical premise that 0 rooms; and 0 living area = zero value is improper because what the model is ignoring is the underlying value of land. This also affects the "geographic" dispersion characteristics because exactly equal home sizes; baths; beds, living area etc will still show value differences correlated to location (and underlying land value) - but as house gets old, smaller, etc - the value will converge at land value; not at zero.

Answer 3

I think your last remark ("I think that the big number of dummy variables damages a little bit the regression") is spot on. The very anormal values you observe for some regession coefficients clearly points to multicollinearity. You might want to try ridge regression or principal components regression.

Answer 4

1. Use intercept in your model. This is very strong assumption, that when all variables would equal 0, then predicted price should be zero and it is not required, especially, that such a real estate would not appear in a train or test data set. Even if it is true, that real estate with all characteristics equal to zero should have price equal to zero, it require assumption, that your model control all factors of estate price, which is rather false. If you think it is so obvious that only real estate with all characteristics equal to zero should have price equal to zero, how much would you pay for real estate with 5m squared area? Because I would still be able to pay zero dollars for such an estate. Am I unreasonable or this assumption is unreasonable?

2. Use semi-logarithmic regression, that is predict logarithm of dependant variable. When you use normal linear regression, it is like assuming, that total value of estate is equal to sum of value of every single characteristic. This can be wrong assumption. I think, that for example, if estate is in quiet area it increases the value of estate by x%, let's say by 30%, no by x\$, not by certain fixed amount of dollars. That is however, what your model is assuming. It is unreasonable to assume, that values of two estates, that one of them have 100 m and second have 10000 m would be increased by the same amount of dollars, let's say by 10000$. It rather would be increased by certain percentage. Use semi-logarithmic model, by that your model will assume, that final value of estate can be reduced to multiplication of every of its properties. It usually works better in such a situations, and it is standard way of proceeding in real estate value model (look in literature, don't try to invent a wheel).

3. Try to add other terms to your model: for example squares of variables and their interactions.

4. Investigate outliers. At least two points looks like outliers, try to figure out what they are, and why they have strange variable values. Maybe you can find error in this data or you can add some feature or interactions to capture such cases.