By how many units should NOx decrease to increase the mean housing Price by $10,000?

Question

. reg price nox dist rooms stratio radial

Source | SS df MS Number of obs = 506
-------------+---------------------------------- F(5, 500) = 163.14
Model | 2.6551e+10 5 5.3101e+09 Prob > F = 0.0000
Residual | 1.6275e+10 500 32549997.3 R-squared = 0.6200
-------------+---------------------------------- Adj R-squared = 0.6162
Total | 4.2826e+10 505 84803032 Root MSE = 5705.3

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price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
nox | -2960.624 399.0899 -7.42 0.000 -3744.724 -2176.524
dist | -963.2088 191.7126 -5.02 0.000 -1339.87 -586.5472
rooms | 6833.164 405.2006 16.86 0.000 6037.058 7629.269
stratio | -1238.796 143.7365 -8.62 0.000 -1521.198 -956.3944
radial | -19.0124 41.583 -0.46 0.648 -100.7113 62.68654
_cons | 22707.34 5579.43 4.07 0.000 11745.32 33669.35
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. sum nox

Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
nox | 506 5.549783 1.158395 3.85 8.71

. sum price

Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
price | 506 22511.51 9208.856 5000 50001

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Answer 1

The model estimated here is:

price = β₀ + β₁nox + β₂dist + β₃rooms + β₄stratio + β₅radial + u

Based on the attached regression output, the estimated model can be characterized by this equation:

price = 22707 - 2961nox - 963dist + 6833rooms - 1239stratio - 19radial

Because the question is focused on nox, we can ignore the other variables. From here, it's simple algebra to solve for the number of nox units required to increase the mean price by $10,000:

10,000 = -2961nox

nox = -3.38

If nox is a continous variable, we would say that a reduction of 3.38 units of nox would lead to an increase in the mean price of $10,000. If nox is a discrete variable, we would say that to increase price by $10,000, a minimum reduction of 4 units is required.