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I'm working a location based home pricing model and I have a regression which looks like this:

$$Price=\beta_0+\beta_1x_1+\beta_2x_1^2+\beta_3x_2+\beta_4x_2^2$$

where $x_1$ is longitude and $x_2$ is latitude.

I'm finding statistical significance in longitude, longitude squared and latitude squared but not in regular latitude itself. I know that in economic theory quadratic terms are used to capture the effect of diminishing returns, but in this case I find it odd that the quadratic is significant but the regular variable isn't.

How do I interpret these results?

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When both the level and the square affect the dependent variable (and if the coefficient on the square is negative), we have a case where the variable in question initially affects positively the dependent variable, but after a point it affects it negatively.

When only the square is deemed to affect the dependent variable, then the effect is monotonic and accelerating, while its direction is indicated by the sign of the coefficient.

Real world interpretation: can you rationalize why latitude could have a monotonic effect, while longitude has not? Can you match this to the geographical areas you use data on? Is there anything special along latitude as regards the weather, socioeconomic conditions etc? And also, why longitude has an up-and-down effect? What could be the possible reason here?

These are the cases when empirical studies become thrilling: when unexpected econometric results provide new insights into the real world.

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That can happen and is not odd.

For your model the partial effect of a change in $x_1$ on the average price equals $\beta_1 + 2\beta_2 x_1$ (use calculus for simple derivation). Thus, $\beta_1$ captures the effect of an increase in $x_1$ from zero. The coefficient of $x_1$ being insignificant just means this effect of changing from zero is insignificant. In other words, $x_1=0$ happens to be close to the turning point, which can happen in some applications (why not?). We wouldn't worry if the turning point is $age=43$ for a smoking regression. There can be other applications where the turning point is close to zero.

In many applications, $x_1=0$ is not very meaningful (e.g., $x_1$ is years of schooling), though I don't know it for your data. Often we subtract the sample mean (or any particular value) from $x_1$ so that the coefficient of the transformed variable ($x_1-c$) indicates the effect of a change of $x_1$ from $x_1=c$. This way, you can also compare easily the results from the linear model and the quadratic model.

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