<p>Let $X = h(Y)$. A first-order Taylor expansion around $E(Y) = \mu_y$ gives</p>

<p>$$X \approx h[\mu_y] + h'[\mu_y]\cdot [Y-\mu_y]$$</p>

<p>This easily leads to</p>

<p>$$\sigma^2_x \approx \big(h'[\mu_y]\big)^2\cdot \sigma^2_y$$</p>

<p>Since </p>

<p>$$\eta_{x,y} \equiv h'\cdot \frac Yh \implies \eta_{x,y}(\mu_y) = h'(\mu_y)\frac {\mu_y}{h(\mu_y)} \implies h'(\mu_y) = \eta_{x,y}(\mu_y) \cdot \frac {h(\mu_y)}{\mu_y}$$</p>

<p>Substituting,</p>

<p>$$\sigma^2_x \approx \left(\eta_{x,y}(\mu_y) \cdot \frac {h(\mu_y)}{\mu_y}\right)^2\cdot \sigma^2_y$$</p>

<p>Taking the square root we are led to </p>

<p>$$\frac {\sigma_x}{h(\mu_y)}  \approx \eta_{x,y}(\mu_y) \cdot \frac {\sigma_y}{\mu_y} $$</p>

<p>The difference of the above from the expression in the question is that</p>

<p>a) The elasticity must be evaluated at the center of the Taylor expansion used
and<br>
b) $h(\mu_y)$ is usually equal to $E(X)=E[h(Y)]=\mu_x$ only to a first approximation, due to Jensen's inequality.  </p>

<p>Of course if the relation between $X$ and $Y$ is linear, and/or the elasticity is constant, things specialize.</p>

<p>In any case, it is valid as an approximation to write</p>

<p>$${\rm cv}_x \approx \eta_{x,y}(\mu_y) \cdot {\rm cv}_y $$</p>

<p>where "cv" stands for "<a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" rel="nofollow">coefficient of variation</a>".</p>

CC BY-SA 3.0
2015-03-11T20:30:08.590Z
2015-03-11T20:30:08.590Z
Alecos Papadopoulos
<p>I am interested in two variables $x, y$. Their (observed empirical) standard deviations are $\sigma_x$, $\sigma_y$. I know the elasticity of $x$ w.r.t. $y$ is $\eta_{x,y}$.</p>

<p>Let $x$ be fully determined through $y$, which is the only source of randomness. Is it true that</p>

<p>$$ \frac{\sigma_x}{mean(X)} = \eta_{x,y} \cdot \frac{\sigma_y}{E[y]}$$</p>

<p>?</p>

<p>I can't find anything on this relationship, but it appears that appropriately scaled  standard deviations of two variables should be connected through their elasticity.</p>

CC BY-SA 3.0
2015-03-11T19:25:12.973Z
It seems like the end of your question got cut-off.
Ubiquitous
<econometrics>
Standard Deviations and Elasticities
FooBar
# Standard Deviations and Elasticities

(Asked by: username_0 on 2015-03-11)

I am interested in two variables $x, y$. Their (observed empirical) standard deviations are $\sigma\_x$, $\sigma\_y$. I know the elasticity of $x$ w.r.t. $y$ is $\eta\_{x,y}$.

Let $x$ be fully determined through $y$, which is the only source of randomness. Is it true that

$$ \frac{\sigma\_x}{mean(X)} = \eta\_{x,y} \cdot \frac{\sigma\_y}{E[y]}$$

?

I can't find anything on this relationship, but it appears that appropriately scaled standard deviations of two variables should be connected through their elasticity.

**Tags:** econometrics

> **Comments:**
> **username_1**: It seems like the end of your question got cut-off.  

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## 1 answers

### Answer by username_2 on 2015-03-11. Score: 1 | Accepted answer

Let $X = h(Y)$. A first-order Taylor expansion around $E(Y) = \mu\_y$ gives

$$X \approx h[\mu\_y] + h'[\mu\_y]\cdot [Y-\mu\_y]$$

This easily leads to

$$\sigma^2\_x \approx \big(h'[\mu\_y]\big)^2\cdot \sigma^2\_y$$

Since

$$\eta\_{x,y} \equiv h'\cdot \frac Yh \implies \eta\_{x,y}(\mu\_y) = h'(\mu\_y)\frac {\mu\_y}{h(\mu\_y)} \implies h'(\mu\_y) = \eta\_{x,y}(\mu\_y) \cdot \frac {h(\mu\_y)}{\mu\_y}$$

Substituting,

$$\sigma^2\_x \approx \left(\eta\_{x,y}(\mu\_y) \cdot \frac {h(\mu\_y)}{\mu\_y}\right)^2\cdot \sigma^2\_y$$

Taking the square root we are led to

$$\frac {\sigma\_x}{h(\mu\_y)} \approx \eta\_{x,y}(\mu\_y) \cdot \frac {\sigma\_y}{\mu\_y} $$

The difference of the above from the expression in the question is that

a) The elasticity must be evaluated at the center of the Taylor expansion used
and  
b) $h(\mu\_y)$ is usually equal to $E(X)=E[h(Y)]=\mu\_x$ only to a first approximation, due to Jensen's inequality.

Of course if the relation between $X$ and $Y$ is linear, and/or the elasticity is constant, things specialize.

In any case, it is valid as an approximation to write

$${\rm cv}\_x \approx \eta\_{x,y}(\mu\_y) \cdot {\rm cv}\_y $$

where "cv" stands for "[coefficient of variation](http://en.wikipedia.org/wiki/Coefficient_of_variation)".


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