Mean and Variance of Transformed Random Variables

Mean and Variance of Transformed Random Variables

To estimate the moments (expected value and variance) of the logarithm of aggregate sales, we can utilize a Taylor expansion for the moments of the function $f(x) := \log(x)$, where $x$ is a random variable $X$. Below are the key derivatives:

• First Derivative: $f'(x) = \frac{1}{\ln(10) x}$
• Second Derivative: $f''(x) = -\frac{1}{\ln(10) x^2}$

Using these, the expected value and variance of $\log(X)$ can be approximated as follows:

Expected Value:

$$\operatorname{E}\left[\log(X)\right] \approx \log(\operatorname{E}\left[X\right]) - \frac{1}{2 \operatorname{E}\left[X\right]^2 \ln(10)} \operatorname{var}\left[X\right]$$

Variance:

$$\operatorname{var}\left[\log(X)\right] \approx \frac{1}{\left(\ln(10) \operatorname{E}[X]\right)^2} \operatorname{var}\left[X\right]$$

The approximation's order depends on the magnitude of fluctuations in $X$. For large national economies' gross exports or imports, linear terms suffice, allowing us to express $\operatorname{E}\left[\log(X)\right] \approx \log(\operatorname{E}\left[X\right])$ and $\operatorname{var}\left[\log(X)\right]$ as shown above.

In the context of the variance equation, $\operatorname{var}\left[\log(X)\right]$ and $\operatorname{var}\left[X\right]$ are proportional. Notably:

• $\sigma^2(\log(X)) \sim 1$
• $\sigma^2(X) \sim 10^{20}$ if $\bar X \sim 10^{11}$ in EUR.

Additionally, the variance of the ratio $X/\bar X$ is:

$$\operatorname{var}\left[\frac{X}{\bar X}\right] = \frac{\operatorname{var}[X]}{\bar X^2} \approx \ln^2(10) \operatorname{var}[\log(X)]$$

The variance of log levels is closer in magnitude to the variance of $(X_t/X)$, differing by a factor of $\ln^2(10) \approx 5.3$.

Note: Alternative derivations can be made using approximate linear relations or first-order approximations, leading to similar conclusions about the proportionality between the variances of $X$ and $\log(X)$.