# Can one use the standard deviation of a variable as a regressor?

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.

• 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. Jun 1 '20 at 13:25
• 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) Jun 1 '20 at 16:22
• 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) Jun 1 '20 at 18:33
• @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. Jun 2 '20 at 1:10

• 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. Dec 28 '20 at 12:36
• 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. Dec 28 '20 at 16:35