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Mathematical Framework: Aggregate Volatility in Log Scale

Mathematical Framework: Aggregate Volatility in Log Scale​

In this section, we delve into the technical aspects of aggregate volatility, focusing on a population of agents contributing to total exports or imports. The aim is to express the variance of the time series of the total, var[XT]var[X_T], in terms of parameters describing the population of agents.

Understanding Aggregate Variance​

The aggregate variance var[Xt]var[X_t] measures the width of fluctuations in aggregate sales. By taking its square root, we arrive at the standard deviation of XtX_t: var1/2[Xt]=std[Xt]{var^{1/2}[X_t]} = std[X_t] However, for practical purposes, we focus on var[X]var[X] as a measure of volatility, leaving the square root for the final step.

In previous sections, we discussed how agent sales and their fluctuations are best described on a log scale. Here, we review how fluctuations expressed in log levels should be integrated into variance accounting.

Easy Facts About Volatility​

Consider a time series XtX_t of length TT. An estimator of its mean value is given by: [ \bar X \equiv \sum_t X_t / T ]

To account for deviations of elements of XtX_t, we define: [ (\Delta X)_t \equiv \Delta X \equiv X_t - \bar X ]

The unbiased sample variance of XtX_t is:

var(Xt)=∑t(ΔX)2T−1var(X_t) = \frac{\sum_t (\Delta X)^2}{T - 1}

This serves as an estimator of the population variance of the time series. The biased sample variance is: var^(Xt)=∑t(ΔX)2/T\hat {var} (X_t) = \sum_t (\Delta X)^2 / T

In a normal iid distribution, the variance computed on a finite sample of length TT relates to the variance at the large TT limit by: lim⁡T→+∞var(Xt)=var(Xt)(T−1)/T=var^(Xt)\lim_{T \to +\infty} var(X_t) = var(X_t) (T - 1)/T = \hat {var} (X_t)

For our purposes, using sample variances computed on time series of length T=17T = 17, we have: var(Xt)=1.0625 var^(Xt){var} (X_t) = 1.0625\ \hat {var} (X_t) This factor is crucial when comparing volatilities measured in different studies.

Aggregate Variance and Covariance​

The expression of total sales as a linear combination (X≡∑kSkX \equiv \sum_k S_{k}) is quite general. Components SkS_{k} can represent individual agents or groups of agents (sectors). The aggregate variance is the sum of cross covariances among these parts:

var[∑kSk]=cov(∑kSk,∑kSk)=∑k1,k2cov(Sk1,Sk2)var\left[\sum_k S_{k}\right] = cov \left( \sum\limits_{k} S_{k}, \sum\limits_{k} S_{k} \right) = \sum\limits_{k_1, k_2} cov(S_{k_1}, S_{k_2})

This property is essential as it provides a general expression of aggregate sales variance in terms of covariances among its parts. It remains valid regardless of the details of parts' fluctuations and their cross correlations. Here, we introduce the sample covariance operator:

cov(Sk1,Sk2)=E[ΔSk1,t⋅ΔSk2,t]T/(T−1)≡∑t((Sk1,t−Sk1ˉ)(Sk2,t−Sk2ˉ))/(T−1)cov(S_{k_1}, S_{k_2}) = E\left[ \Delta S_{k_1, t} \cdot \Delta S_{k_2, t} \right] T/(T - 1) \equiv \sum_t \left( \left(S_{k_1, t} - \bar {S_{k_1}}\right) \left(S_{k_2, t} - \bar {S_{k_2}}\right) \right) / (T - 1)

Aggregate variance encompasses all these elements, similar to how total sales must include all agents' sales for an exact match.

  1. In our case, T=2014−1997=17T = 2014 - 1997 = 17. See Section: Data Details for more information.
  2. This factor would be necessary when comparing volatilities measured in different studies.