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I am wondering if you could use the standard deviation of a variable as a regressor in an econometrics model? Consider the following hypothetical model:

$$y_{it} = \alpha_0 + \alpha_{1}T_{it} + \alpha_{2}\sigma_{it}^T + u_{it}$$

Where $y$ is the some outcome variable (e.g. crop yields) in country $i$ at time $t$, $T$ is temperature, and $\sigma^T$ is the standard deviation of temperature. Could one interpret $\alpha_2$ in such a way that would indicate how increased volatility in temperature would impact crop yields?

i.e. testing if countries with greater temperature volatility $\implies$ more severe weather $\implies$ negatively impacts yields

What comes to mind is the use of volatility indices as independent variables, and I wonder if this would lead to a similar interpretation.

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    $\begingroup$ I'm not familiar with that approach but if the volatility is closely related to the temperature level ( which is often the case with stock returns. don't know if its true with temperature ) then you'll have two highly correlated regressors which will be problematic. $\endgroup$
    – mark leeds
    Jun 1 '20 at 13:25
  • $\begingroup$ That is a great point that I was also considering. It is problematic to use both temperature and its standard deviation. I would likely run them separately to ask separate questions (e.g. marginal effects of temperature vs. volatility of temperature effects) $\endgroup$
    – Brennan
    Jun 1 '20 at 16:22
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    $\begingroup$ I am not sure how you plan to estimate the variance of temperature in a given country at a given date. Since presumably, you only have one observation for the temperature of a country at a given time, and you are already using that observation as an additional regression. (Unless by sigma you mean temperature dispersion across different regions of each country) $\endgroup$ Jun 1 '20 at 18:33
  • $\begingroup$ @Brennan: I'm not sure if running them seperately is valid either because, if the other variable is significant, then you kind of have a mis-specified model by leaving it out. Maybe someone else can comment on that approach ? It's an interesting idea but I'm not so sure it will be valid statistically. $\endgroup$
    – mark leeds
    Jun 2 '20 at 1:10
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By including both regressors, you are effectively trying to find the partial effect of either the temperature level, or variance, on crop yield. That is, the effect of the one, after holding the other constant. Does that approach make sense? Do you need to know the effect of a rise in temperature, for a constant variance in temperature? Or, do you need to know the effect of a change in the deviation in temperature, for a given level in temperature?

The multivariate approach finds those partial effects. The coefficients can be biased if the two variables are correlated with each other—you can test multicollinearity by regressing the one regressor on the other (or, for higher numbers of regressors, one would look at the Variance Inflation Factor).

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    $\begingroup$ You say that: "The coefficients can be biased if the two variables are correlated with each other" are you sure this is correct? $\endgroup$ Dec 27 '20 at 19:45
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    $\begingroup$ I'm not sure I understand you correctly, so I will introduce an example. Consider the model $y_i = \beta_0 + \beta_1 x_{i1} + \beta_{2} x_{i2} + u_i$ for $i=1,...,N$ where $\mathbb E[u_i \lvert \{x_{i1},x_{i2}\}_{i=1}^N] = 0$ from which it follows that $\beta_1$ is partial effect of $x_{i1}$ and from which it follows that OLS estimator defined as $\hat \beta := (X^\top X)^{-1}X^\top y$ is unbiased and consistent. However, this does not imply that $Cov(x_{i1},x_{i2}) = 0$ as implied by independence of regressors with regressors. So if that is what you are saying, I am afraid you are wrong. $\endgroup$ Dec 28 '20 at 12:36
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    $\begingroup$ Independence between regressors does not imply unbiasedness and unbiasedness does not imply independece between regressors. To claim otherwise is simply wrong. What is standardly assumed to get unbiasedness is strict mean independence between regressors and the error term. I am not targeting you but your statements. $\endgroup$ Dec 28 '20 at 16:30
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    $\begingroup$ Lack of perfect multicollinearity is requirement because otherwise $(X^\top X)^{-1}$ does not exist and then the OLS estimator does not exist (this is also known as the rank condition). Multicollinearity can be a problem but the problem is not about bias. $\endgroup$ Dec 28 '20 at 16:35
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    $\begingroup$ Also note that logically these two statements "It's only unbiased and consistent if there is independence in the regressors in the first place. I did not say that unbiasedness implies independence." are simply contradictory. If OLS is only unbiased if there is independence then surely OLS being unbiased implies independence (otherwise you must remove the word "only" in your first sentence). $\endgroup$ Dec 28 '20 at 16:42

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