The following equation describes the median housing price in...

The following equation describes the median housing price in a community in terms of amount of pollution (nox for nitrous oxide) and the average number of rooms in houses in the community (rooms): log(price) = ?? + ??log(nox) + ??rooms + u. (i) What are the probable signs of ?? and ??? What is the interpretation of ??? Explain. (ii) Why might nox [or more precisely, log(nox)] and rooms be negatively correlated? If this is the case, does the simple regression of log(price) on log(nox) produce an upward or a downward biased estimator of ??? (iii) Using the data in HPRICE2, the following equations were estimated: log(price) = 11.71 – 1.043 log(nox), n = 506, R² = .264. log(price) = 9.23 – .718 log(nox) + .306 rooms, n = 506, R² = .514. Is the relationship between the simple and multiple regression estimates of the elasticity of price with respect to nox what you would have predicted, given your answer in part (ii)? Does this mean that –.718 is definitely closer to the true elasticity than –1.043?

Transcript

00:01 So here we're given this regression model for the price of a house where the price of a house is described in terms of its exposure to pollution and the number of rooms, right? so if this was just sort of a naive regression, we would expect beta 1 to be negative because pollution is bad, right? that's all that is saying that higher levels of pollution would reduce the price.

00:25 And beta 1 is capturing this sort of elasticity, right? beta 1 is measuring the price elasticity to pollution right when pollution goes up by 10 % beta 1 tells you by how much prices fall right it is in particular the derivative of the log of price with respect to the log of pollution right and that is by definition the elasticity so beta 1 is the elasticity parameter for pollution beta 2 is a semi elasticity we would would expect it to be positive and we expect it to be positive because very simply rooms are good, right? we expect that more rooms, more space, bigger house means that the house is worth more, right? very simply.

01:14 Okay, now we're getting into real econometrics in two, right? why might knox or pollution in rooms be negatively correlated, right? well, there's, you could again talk about this, but in general, i would expect they are correlated because wealth, right? rich people are going to buy by bigger houses in better areas.

01:40 That's simply it, right? i would expect that as people get richer, they can afford more house.

01:47 They would choose to buy more house in better, less polluted areas.

01:55 So in some sense, we have this correlation between the coefficients caused by this sort of missing variable, right? we've got this wealth variable lurking in the background, and wealth is trying to drive both prices, sorry, both rooms and pollution exposure together.

02:19 So now we're asked, let's suppose we dropped.

02:25 If we dropped then rooms, right, we'd have a missing variable problem, a real missing barrier problem, right? there would be this rooms variable capturing wealth in the error term, right? and this would be negatively correlated with beta 1.

02:45 So think about what's trying to go on when the model fits the reduced form equation, right? if we estimate it just sort of log price equals to beta zero plus beta one log nox plus some error term, this error term now contains the missing variable, right? that is correlated.

03:14 So you've got what economists might call an endogeneity problem.

03:20 We can rewrite this missing variable.

03:22 Now, i, sorry, so, epsilon t is equal to beta 2 times rooms plus ut, right, or just u the error term u.

03:34 And i know that rooms is correlated with log nox.

03:41 So i could rewrite this as a function of log nox, right? this is going to be b2 times log nox times the correlation between rooms and log knocks, right? basically what i'm doing is i'm replacing rooms with a function of log knocks in a very simple way.

04:06 So when i estimate this, what i'm going to get, if i substitute this back in, i get beta 0 plus beta 1 plus beta 2.

04:20 Correlation log knox plus you right the model is going to say look we're trying to minimize the sum of squared errors we have this systematic correlation in the error term that's that's associated with log knocks the coefficient our beta one hat that we actually estimate is going to try to shove that correlation into its estimate of beta one hat um to try to minimize the sum of square errors, right? and because this is negative, right? remember, these correlations are negative.

04:54 This is positive.

04:56 This is the true beta one is negative, right? so we have a negative beta 1.

05:02 Then we're going to be subtracting another negative term...

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