<p>By including both regressors, you are effectively trying to find the <em>partial</em> 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?</p>
<p>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).</p>

CC BY-SA 4.0
2020-12-27T19:45:04.027Z
You say that: "The coefficients can be biased if the two variables are correlated with each other" are you sure this is correct?
Jesper Hybel
CC BY-SA 4.0
2020-12-27T20:55:26.200Z
I would think that correlation between the two would not the issue at hand here. Do countries with higher temperatures tend to have higher or lower variance in those temperatures? I imagine not. That would just affect the precision of those estimates. It was more so just a thought experiment of an approach to some research I am doing.
Brennan
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2020-12-28T06:21:40.677Z
@JesperHybel I don't see why there would be confusion about that. Independence of the regressors is a prerequisite for the Classical Linear Model assumptions.
ahorn
CC BY-SA 4.0
2020-12-28T06:23:09.520Z
@Brennan, great. In that case, the model holds.
ahorn
CC BY-SA 4.0
2020-12-28T09:15:39.090Z
@ahorn "Independence of the regressors is a prerequisite for the Classical Linear Model assumptions". Independence of regressors from what? Independence is a relation not a property. Are you saying regressors should be independent from each other?
Jesper Hybel
CC BY-SA 4.0
2020-12-28T11:10:57.550Z
Yes. The regressors need to be statistically [independent](https://en.wikipedia.org/wiki/Independence_(probability_theory)). The coefficients will not show the partial effects of the respective variables if they covary. The data matrix must be invertible (nonsingular) in a CLM, and this only happens when there is full column rank - i.e. no two regressors are linear combinations of the other. Or, at least, little correlation between any two regressors is desired. [Wikipedia describes how there should be little multicollinearity.](https://en.wikipedia.org/wiki/Linear_regression#Assumptions)
ahorn
CC BY-SA 4.0
2020-12-28T12:36:45.670Z
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.
Jesper Hybel
CC BY-SA 4.0
2020-12-28T15:24:37.297Z
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. I said that it is the other way round—an absence of multicollinearity implies unbiasedness. There is no use personally targeting me, because I didn't develop the Ordinary Least Squares formula. I really don't like the tone with which you are addressing me.
ahorn
CC BY-SA 4.0
2020-12-28T16:30:38.697Z
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.
Jesper Hybel
CC BY-SA 4.0
2020-12-28T16:35:23.573Z
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.
Jesper Hybel
CC BY-SA 4.0
2020-12-28T16:42:32.413Z
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).
Jesper Hybel
CC BY-SA 4.0
2020-12-30T12:25:16.340Z
@JesperHybel yes, you do mention another criterion now, that of the correlation between regressors and the error term (more intuitively thought of as correlation between regressors and the dependent variable, or endogeneity). Re: your second-last comment—now we're finally getting to the meat of the matter. I claim that when two regressors are highly correlated, then the OLS coefficients cannot accurately identify the individual effect of each regressor. There would be a lack of statistical power to do so. Regarding your last comment, I need more space to explain, in my next comment.
ahorn
CC BY-SA 4.0
2020-12-30T12:27:51.533Z
@JesperHybel: "Only if", i.e. $\impliedby$, means a reverse implication. On the other hand, "if", i.e. $\implies$, means a forward implication. That explains why the two statements are opposites. :)
ahorn
CC BY-SA 4.0
2020-12-27T18:59:13.097Z
2020-12-27T18:59:13.097Z
ahorn
<p>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:</p>

<p><span class="math-container">$$y_{it} = \alpha_0 + \alpha_{1}T_{it} + \alpha_{2}\sigma_{it}^T + u_{it}$$</span></p>

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

<p>i.e. testing if countries with greater temperature volatility <span class="math-container">$\implies$</span> more severe weather <span class="math-container">$\implies$</span> negatively impacts yields</p>

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

CC BY-SA 4.0
2020-06-01T13:25:36.257Z
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.
mark leeds
CC BY-SA 4.0
2020-06-01T16:22:44.747Z
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)
Brennan
CC BY-SA 4.0
2020-06-01T18:33:51.120Z
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)
brunosalcedo
CC BY-SA 4.0
2020-06-02T01:10:18.980Z
@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.
mark leeds
<econometrics>
Can one use the standard deviation of a variable as a regressor?
Brennan
# Can one use the standard deviation of a variable as a regressor?

(Asked by: username_0 on 2020-05-31)

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.

**Tags:** econometrics
> **Comments:**
> **username_1**: 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.  
> **username_0**: 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)  
> **username_2**: 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)  
> **username_1**: @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.  

---

## 1 answers

### Answer by username_3 on 2020-12-27. Score: -1

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).

> **Comments:**
> **username_4**: You say that: "The coefficients can be biased if the two variables are correlated with each other" are you sure this is correct?  
> **username_0**: I would think that correlation between the two would not the issue at hand here. Do countries with higher temperatures tend to have higher or lower variance in those temperatures? I imagine not. That would just affect the precision of those estimates. It was more so just a thought experiment of an approach to some research I am doing.  
> **username_3**: @JesperHybel I don't see why there would be confusion about that. Independence of the regressors is a prerequisite for the Classical Linear Model assumptions.  
> **username_3**: @Brennan, great. In that case, the model holds.  
> **username_4**: @ahorn "Independence of the regressors is a prerequisite for the Classical Linear Model assumptions". Independence of regressors from what? Independence is a relation not a property. Are you saying regressors should be independent from each other?  
> **username_3**: Yes. The regressors need to be statistically [independent](https://en.wikipedia.org/wiki/Independence_(probability_theory)). The coefficients will not show the partial effects of the respective variables if they covary. The data matrix must be invertible (nonsingular) in a CLM, and this only happens when there is full column rank - i.e. no two regressors are linear combinations of the other. Or, at least, little correlation between any two regressors is desired. [Wikipedia describes how there should be little multicollinearity.](https://en.wikipedia.org/wiki/Linear_regression#Assumptions)  
> **username_4**: 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.  
> **username_3**: 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. I said that it is the other way round—an absence of multicollinearity implies unbiasedness. There is no use personally targeting me, because I didn't develop the Ordinary Least Squares formula. I really don't like the tone with which you are addressing me.  
> **username_4**: 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.  
> **username_4**: 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.  
> **username_4**: 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).  
> **username_3**: @JesperHybel yes, you do mention another criterion now, that of the correlation between regressors and the error term (more intuitively thought of as correlation between regressors and the dependent variable, or endogeneity). Re: your second-last comment—now we're finally getting to the meat of the matter. I claim that when two regressors are highly correlated, then the OLS coefficients cannot accurately identify the individual effect of each regressor. There would be a lack of statistical power to do so. Regarding your last comment, I need more space to explain, in my next comment.  
> **username_3**: @JesperHybel: "Only if", i.e. $\impliedby$, means a reverse implication. On the other hand, "if", i.e. $\implies$, means a forward implication. That explains why the two statements are opposites. :)  

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