 Both their estimate and $I_{SB}$ show a strong rise before the crisis in 2007, a drop in 2008 (much more severe for $I_{SB}$), and a growth after 2008, with $I_{SB}$ increasing at a faster pace, especially in 2011.

Ref. \cite{Gravelle2013} argues that an entity-based approach to SB, such as that of the FSB, "[...] may omit SB activities undertaken by banks that may contribute to systemic risk." Furthermore, as observed by Adrian {\em et al.} \cite{Adrianetal2013} "[...] the shadow banking system comprises many different entities and activities. In addition, the types of entities and activities which are of particular concern will change in the future, in response to new regulations". Along similar lines, Pozsar {\em et al} \cite{PozsarEtAl2013} conclude: "[...] the reform effort has done little to address the tendency of large institutional cash pools to form outside the banking system. Thus, we expect shadow banking to be a significant part of the financial system, although almost certainly in a different form, for the foreseeable future". 
These arguments suggest that the FSB estimate is likely to provide a lower bound to the size of the SB system. $I_{SB}$ may instead be considered as a \textit{theoretical upper bound}, as it measures the amount of assets that are missing from a hypothetical economy in which PRG holds across all scales of asset sizes.  


A few comments are in order about $I_{SB}$:
\begin{itemize}
  \item $I_{SB}$ is a genuine systemic indicator, as it depends on a collective property of the economy. It is hard to manipulate and simple to compute, as it requires only data publicly available.
   \item $I_{SB}$ does not rely on a detailed list of entities and/or activities which contribute to the SB system; it is therefore robust to change in regulation and fiscal policy.
  \item $I_{SB}$ implicitly attributes SB activities to the largest financial firms which populate the top tail of assets distribution. It is well documented that the main financial firms originated most of the SB activities before the crisis \cite{Fein2013}. Yet, $I_{SB}$ also crucially depends on the exponent $\gamma$, whose estimate depends on the shape of the distribution in the intermediate range. In particular, \textit{ceteris paribus}, $I_{SB}$ is expected to increase if the exponent $\gamma$ decreases and {\em vice-versa}. 
\end{itemize}

A comparison between Table \ref{tab:estimatedParetoCoeff} and Fig. \ref{fig:SBIndex_comparison} shows how $I_{SB}$ is (anti)correlated with $\hat\gamma$ and $\hat{\gamma}_{\rm fin}$: when the assets distribution gets broader (i.e. $\gamma$ and $\gamma_{\rm fin}$ decrease), $I_{SB}$ increases and {\em vice-versa}. After the 2007-08 crisis, the correlation of $I_{SB}$ with $\hat{\gamma}_{\rm fin}$ is much stronger than with $\hat\gamma$. This is a further indication that the behavior of financial firms is at the core of the dynamics of $I_{SB}$.





\subsubsection*{A Simple Proportional Random Growth Model with Shadow Banking}
\label{sec:model}

Malevergne {\em et al.} \cite{Sornette2013} discuss a simple PRG model which displays a Paretian distribution. In their notation \cite{Sornette2013}, firms grow according to a log-normal stochastic process with drift $\mu$ and variance $\sigma$, they disappear from the market according to a Poisson point process at rate $h$, and new firms enter the market at rate $\nu$. In the following we neglect the possibility of an exogenous growth of economy (i.e. we take $c_0=d=0$ in the notation of \cite{Sornette2013}), and we assume that all new firms have initial size equal to one. Malevergne {\em et al.} \cite{Sornette2013} show that the top tail of the equilibrium size distribution has a power law shape with a Pareto exponent given by Eq. (7) in Ref. \cite{Sornette2013}:
\begin{equation}
\gamma = \frac{1}{2} \left[ \left(1 -2 \frac{\mu}{\sigma^2}\right) + \sqrt{\left( 1 -2 \frac{\mu}{\sigma^2} \right)^2 + 8 \frac{h}{\sigma^2}} \, \right].
\label{eq:index}
\end{equation}
We introduce a modification of the original model, in order to reproduce the cut-off displayed by the Forbes data, i.e. in the presence of SB. According to a Poisson point process at rate $\lambda$, the largest firm $i^*$ in the economy (with size $S_{i^*}=\max_i S_i$) moves a fraction $\epsilon$ of its assets outside the regular banking system to the SB system, reducing its \textit{observed} size to $(1-\epsilon) S_{i^*}$. The parameter $\lambda$ is therefore a proxy for the \textit{intensity} of the activity feeding the SB system.

Table \ref{tab:lambda} reports a calibration of the parameters of the modified PRG model for the period 2005-2012 based on the FG2000 list of firms (see caption of the table). We set $\epsilon=0.1$ and we located the value of $\lambda$ that yields the best match between the simulated and the observed firm size distributions (see caption of Table \ref{tab:lambda}). Figure \ref{fig:simulationVsObserved} shows the quality of our calibration of the model for 2012.
The same fitting procedure was performed for different values of $\epsilon$; for $\epsilon \in \left(0,0.1\right)$ the ``flux'' $\epsilon\lambda$ of capital flow into the SB system results independent of $\epsilon$. This is reasonable, because when $\epsilon$ is very small and $\lambda$ very large, wealth is repeatedly drawn into the SB system from the same firm (the largest one). 

\begin{figure}[htbp]
   \centering
   \includegraphics[width=4in]{Fig5.pdf}
   \caption{Comparison between the observed (cross) and the simulated (bold line) distributions for 2012. 
   Inset: the estimate of $\lambda$ in the period 2005-2012.}
   \label{fig:simulationVsObserved}
\label{fig:model}
\end{figure}

According to the estimate of $\lambda$ reported in Table \ref{tab:lambda}, the \textit{intensity} of SB activity peaked in 2007 before the financial crisis, when the {\em originate-to-distribute} activities implemented by asset-backed securities and other credit derivatives probably reached their zenith \cite{PozsarEtAl2013}. In 2008 and 2009 the SB activity showed a dramatic fall in agreement with the sharp decline in all economic activities and the supposed breakdown of the SB system; but from 2010 to 2012 we observe a renewed increase, even though not at pre-crisis rates. This dynamic is fully consistent with the evolution of the size of SB system reported in Fig. \ref{fig:SBIndex_comparison}, being $\lambda$ a proxy for the \textit{intensity} of the activity feeding the SB system. Considering that the largest firm is of the order of \$ 3 trillion in 2013, our result $\epsilon\lambda\approx 1.2$ suggests a flow of capital into the SB system that is currently progressing at approximately \$ 3.5 trillions a year.


\section*{Conclusions and Outlook}
\label{sec:conclusionsOutlook}
  
This paper takes a non-standard approach to studying the properties of an economy. 
Based on solid evidence in the literature \cite{Gabaix2009}, we consider the Pareto distribution for asset sizes as an empirical law of an economy. The observation of power law distributions in economics is a remarkably solid piece of empirical evidence, dating back to the work of Pareto \cite{Pareto}. This empirical law arises from a generic mechanism --  proportional random growth -- that is expected to work in particular for financial firms. The actual distribution of firm sizes, at the global scale, closely follows this empirical law in the middle range, but deviates markedly from it in the upper tail, which is populated entirely by financial firms. 
We invoke SB as the element that would reconcile observations with the expected law. This allows us to derive an index that identifies the size of SB with the missing mass in the top tail of the asset size distribution. This approach resembles the one leading astrophysicists to invoke {\em dark matter} and {\em dark energy} in order to reconcile empirical observations with the law of gravitation (current estimates suggest that dark matter and dark energy account for approximately 95\% of the total mass in the universe). Likewise, the observation of a truncated power law in the distribution of asset sizes, points to the existence of {\em dark assets} that account for the missing mass in the top tail of the distribution.

Our estimate of the SB size is silent about the precise nature of SB activities and entities, as well as about the mechanisms that generate the observed departure from the theoretical power law behavior. The missing mass from the top tail of the distribution does not necessarily correspond to hidden assets