decompose.ipynb

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    "## Decomposing employment growth\n",
    "Employment of firm $i$ in age $a$ is $L_{ia}$. The object of interest is $L_{ia}/L_{i1}$.\n",
    "\n",
    "Even if the firm has more than one CEOs in a given year, we have a discrete number of events, in which the CEO team changes (at least one hiring or at least one firing). I have code that captures these years.\n",
    "\n",
    "The life of a firm can then be split into $n_{ia}$ non-overlapping intervals, each with a different CEO team. (When there is only one CEO per year, this is simply the number of CEOs over time.) Each interval can be characterized by an employment growth $L_{it_2}/L_{it_1}$, where $t_1$ and $t_1$ are the beginning and the end of the interval.\n",
    "\n",
    "Take logs,\n",
    "$$\n",
    "\\ln (L_{ia}/L_{i1}) = \\sum_{n} \\ln (L_{it_2}/L_{it_1}) \n",
    "= n_{ia} \\cdot \\bar{\\ln (L_{it_2}/L_{it_1})}.\n",
    "$$\n",
    "Total log growth is the product of the number of CEO spells and the average growth in each spell.\n",
    "\n",
    "The LHS is data, and $n_{ia}$ on the RHS is also data (I have it computed). We can compute the average per-CEO growth as the ratio. \n",
    "\n",
    "We can then shut down variation in either $n$ or per-CEO growth to show which explains, in an accounting sense, more variation in employment growth across firms. Given your figures, I strongly suspect it will be $n$."
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