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A few months ago I interned at this organization; and, as a going away present, I decided to spend my last week, with whatever off time I had, to investigate the factors that affect teacher salaries. One problem that I ran into with teacher salaries was that the distribution for the given state was skewed. I had a lot of observations that clung to the lower end of the wage spectrum. I tried resolving this by incorporating a Comparable Wage Index into my dependent variable (teacher wages), but the results I found were completely out of date for the scope of my project. I instead decided to log my dependent variable. This was nice because now my wages had a normal distribution and it just looked perfect in the histogram. When I started testing down, I got to the point where I was left with one last independent variable, property tax returns. The problem with my normative wages was also apparent in my property tax return observations. I had a huge skew of property tax return numbers towards the lower end of the spectrum. So, I logged this variable as well and it still passed the null hypothesis test just fine.

I am not sure if this is precisely correct, but by comparing the change of one logged variable to another logged variable gave me the elasticity. Assuming that this is correct, my regression equation (something like LogWages = B0 + B1(LogPropertyTaxReturns)) shows the elasticity between the two variables. Is this meaningful though? If my goal was to see which variable most affected teacher salaries in any given county of my state, then is showing the elasticity between the two variables helpful? We want to raise the counties with the lowest teacher salaries up higher to increase their living standards, but I fear that I've extrapolated so far away from the real observations that my concluding regression equation is meaningless.

Edit: One of my bigger fears is that I should have used a non-linear model to show the relationship. I feel that forcing both the dependent and independent variable to cooperate in a this linear regression is misleading in some way.

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It's absolutely meaningful. Look at the definition of the elasticity. You basically have information on the direction of the relationship between Wages and PropertyTaxReturns. Moreover, you have an estimated measure of that relationship. Since it is Log-Log, the wages will change by B1 percents per one percent change of the property tax return. You could do a time series analysis of that to confirm it. Actually, just graphing wages and property tax return over time would be enough to see what is the relationship. Thats a basic way that does not account for hidden variables and etc. –  Koba Nov 18 at 22:41
    
@Koba Thanks for commenting so quickly. Isn't the problem though that elasticity changes along the curve? My biggest regret is that I may have forced the model to be linear, forcing the elasticity to be pretty stable. Thinking back, it might have actually been better to have a non-linear model to reflect this skew that I was talking about. –  rosenjcb Nov 18 at 22:46
    
There is nothing wrong in transforming the variables using log, square root, reciprocals, or other methods. You are not forcing anything. You use the transformations to find the linear relationship between the variables. Sometimes it easy like you just use y=b0+b1*x. Other times variables are linearly related in more complicated way like for example log(y)=b0+b1*(1/x). The last function might give you a good linear relationship, but it is harder to interpret, so the less transformation you can use the better. –  Koba Nov 18 at 22:57
    
The log-log function is pretty straightforward log(y)=b0+b1*log(x). B1 is precisely the percent change in y per one percent change in x in your cross-section analysis. Again, if you have this data for a certain period of time you can just graph it to see the relationship. –  Koba Nov 18 at 22:58
    
I've logged variables before and done other transformations for my regression models. I was just worried that elasticity gave a pretense of meeting. Though, thinking back, the model was linear, it just had the problem of having dependent and independent variables with skewed distributions. –  rosenjcb Nov 18 at 23:00

5 Answers 5

up vote 4 down vote accepted

The answer to the question is yes, it is indeed meaningful (at least mathematically speaking). If you have the linear equation:

$$ W = \beta_0 + \beta_1 PTR $$

Then $\beta_1=\frac{\partial W }{\partial PTR}$, meaning that $\beta_1$ represents the marginal change of $PTR$ over $W$. Now, if you have

$$ log(W) = \beta_0 + \beta_1 log(PTR) $$

Then $\beta_1=\frac{\partial W}{\partial PTR}\cdot\frac{PTR}{W}$, which is the very definition of elasticity.

Generally speaking, linear transformations only affect the interpretation given to the coefficients, but the validity of the regression itself (in broad economic terms) is given by the model's assumptions and the economic phenomena being analyzed.

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Like people have said in the comments, log-log is commonly used. It amounts to estimating a constant elasticity model $Y = \alpha X^\beta$, which is a commonly used functional form within economics. Once you take logs, this becomes $\ln Y = \ln \alpha + \beta \ln X$. You can read more about this here.

I guess your question is whether or not using this functional form makes sense in your particular model. It's hard to say. As with any ordinary linear regression, you're making an assumption about functional form. You can at least just think about it as a linear approximation that makes more sense after the log-log transformation.

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Alright, the other respondents have covered the logic behind a log-log regression pretty well, so I'm just going to add some practical tips. If you want to check whether your specification is reasonable, and your problem is the assumption of a constant elasticity, try splitting the sample into groups based on percentiles of $x$ and recalculating $\alpha$ and $\beta$. Then see how much they differ. You can even do this by using dummies and interaction terms for each of the percentiles, and then use an $F$ test to determine joint significance of the interaction terms. In other words: $$\log y_i = \alpha + \beta \log x_i +\sum_{j=2}^S \gamma_j\chi_j +\lambda_j\chi_j\log x_i $$ where $\chi_j$ is your percentile dummy. Then test whether the $\gamma$'s and $\lambda$'s are jointly significant. This is by no means formal, but it may give you a rough idea how reasonable having a constant elasticity is.

Note that as a representation of "true" underlying decision making all transformations that result in a linear regression are wrong. In fact, all models are going to be wrong. The question is really: is the statistic you've gotten from this model useful to your problem? If your study is focused on determining an underlying model, is this a moment that tells you something interesting about that deeper model? If you're more policy oriented, will an approximation with constant elasticity get you close enough to the truth that further improvements are irrelevant? Either are extremely difficult questions to answer as an outside observer. But if the only alternative you're worried about is variable elasticity, the kind of test I outlined above may give you some peace of mind.

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The other answers covered the main issues, I would like to respond to the "Edit" made by the OP in the question:

Edit: One of my bigger fears is that I should have used a non-linear model to show the relationship. I feel that forcing both the dependent and independent variable to cooperate in a this linear regression is misleading in some way.

We tend to forget that "transforming a variable" leads to a new variable, whose behavior may be totally different than the "original one". The easiest example is to compare the graphs of a variable and its square.

So by considering the natural logarithms of your variables, you no longer examine the relation between them, but a relation between some function of them.
It is fortunate that the mathematical concept of "logarithm" can be linked to the concept of "elasticity", which describes a relation between percentage changes, which is something we understand from an economic point of view and we can meaningfully interpret and use.

If the variables can be reasonably said to exhibit a "linear relationship in logarithms", it means that their levels (i.e. the actual variables) have a non-linear relationship:

$$\ln y \approx a+b\ln x \Rightarrow y \approx e^a + x^b$$

So why not estimate a non-linear model?
In (mathematical) principle, there is no reason why not. Some practical issues are:

1) There are too many forms of non-linear relationships, there is only one linear relationship (structurally speaking). It is a matter of "search costs" for the most suitable specification.

2) The non-linear relationship obtained may not have a clear economic explanation. Why this is a problem? Because, we are not uncovering "laws of nature" here, unchanged through time and space. We are approximating a social phenomenon. Having an approximation which, moreover, can only be presented as a mathematical formula, without an economic reasoning that validates and supports it, makes the result very thin.

3) Non-linear estimation is less stable, as regards the mechanics of the estimation algorithm.

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I would say that your model in this case doesn't seem meaningful if your "goal was to see which variable most affected teacher salaries in any given county of my state". You have just shown what the correlation between (the logs of) wages and property tax returns is. You should at least use a multiple regression.

Of course, you could keep going and develop a fully fledged, proper, identification strategy with the appropriate methodological tools in order to estimate the intensity of each causal effect and find the biggest... In reality, you most likely won't be able to do it given the complexities of such a task. It's just a continuum of refinements and you're near the crudest possible model used to explain wages, very far from what I would consider the acceptable approximations of an answer to the question implicit in your goal. You should try to enlist the help of an econometrician.

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