regression-equations.tex

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\begin{document}
\section{How to write out regression equations?}

\begin{quote}
We estimate the effect of foreign managers on firm performance in an annual firm panel. Outcomes at firm $i$, in sector $s$, in year $t$ are modeled as
\begin{equation}
	y_{ist} = \beta F_{it} + \gamma \mathbf X_{it} + \alpha_i + \mu_{st} + \varepsilon_{ist},
\end{equation}
with $F_{it}$ denoting an indicator taking the value one when firm $i$, has a foreign manager in year $t$. The vector $\mathbf X_{it}$ denotes time varying firm controls. We also include firm fixed effects and sector-year fixed effects.
\end{quote}

Regression equations are mathematical equations, not notes to yourself. They should make sense as such. Use the rules below and always check whether the algebra in your equation is correct.

\begin{enumerate}
	\item  Write out all dimensions that vary, even if they are related. $y_{ist}$ for firm $i$ in sector $s$ in year $t$, even though $s$ is, strictly speaking, a function of $i$ and $t$.
	\item For right-hand side variables, take careful account of the dimensions that vary. $P_{st}$ for a sectoral price index, $F_{it}$ for a firm variable (I could not care less about industries at this stage, unless they directly affect $F_{it}$).
	\item Always write out the error term. 
	\item Variables are either single-letter variables or short words, but not both. Either $\gamma M_{it}$ or $\gamma \text{EXPAT}_{it}$, but never $\gamma M_{it}\text{EXPAT}_{it}$. 
	\item If you use longer words, use the \texttt{\textbackslash text\{\}} command so that the letters don't fall apart. $\text{EXPAT}_{it}$ is a variable, $EXPAT_{it}$ is the product of five variables. Again, this is math.
	\item Do not write out multiplication signs for interacted variables, we are used to reading mathematical formula like this. $M_{it}F_{jt}$, not $M_{it}*F_{jt}$. If you really must, to reduce complexity, use \texttt{\textbackslash cdot} or \texttt{\textbackslash times}, mot $*$.
	\item The proper notation for fixed effects is a greek parameter indexed by what varies. For example, firm fixed effects are $\alpha_i$, not $FIRM_i$ (Is there such a variable called FIRM? What values does it take?), nor $\sum_j \alpha_j D_j$ (Since $D_j$ is zero for all other firms but $i=j$, this is the same as $\alpha_i$, just much harder to read.)

	\item Interacted fixed effects are $\mu_{st}$, not $\mu_s\mu_t$. Think about what your regression is doing mathematically.

	\item Don't use the same greek letter for different parameters, even for fixed effects. For example, $\alpha_i + \alpha_t$ would algebraically mean that the time effect for year 1997 is the same as the firm effect for firm 1997. That is surely not what you meant to say.

	\item If you are controlling for a non-parametric function, feel free to write $f(\text{age}_{it})$, for example. This is ok for polynomials, lowess, kernel, spline, binscatter, anything that is a function of $\text{age}_{it}$.

	\item Use punctuation as you would in a sentence.
\end{enumerate}

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