<p>Let <span class="math-container">$Y_{ijst}$</span> be some outcome variable of firm <span class="math-container">$i$</span> of industry <span class="math-container">$j$</span>, located in state <span class="math-container">$s$</span>, at time <span class="math-container">$t$</span>. Normally, if we want to run any differences-in-differences specification where treatment varies across firms <span class="math-container">$i$</span> and time <span class="math-container">$t$</span>, I would throw in firm fixed effects <span class="math-container">$\lambda_i$</span> and time fixed effects <span class="math-container">$\tau_t$</span> in my regression specification, and then, the coefficient of my treatment variable <span class="math-container">$x_{it}$</span> will give me causal effect of <span class="math-container">$x_{it}$</span> on outcome <span class="math-container">$Y_{ijst}$</span>.</p>
<p>Suppose instead of <span class="math-container">$x_{it}$</span>, my regressor of interest is the following interaction term: <span class="math-container">$z_{st}*w_j$</span>: something which varies across state x time (<span class="math-container">$s$</span> and <span class="math-container">$t$</span>) and the other which varies across industries (<span class="math-container">$j$</span>). Then, in this specification, does the ad-hoc addition of fixed effects - firm fixed effects (<span class="math-container">$\lambda_i$</span>, which subsumes industry and state fixed effects) and time fixed effects (<span class="math-container">$\tau_t$</span>) work for identification of causal effect? Or shall I also include other fixed effects - state-time, industry-time, state-industry? Mainly asking because my coefficient turns insignificant once I add the last two.</p>

CC BY-SA 4.0
2022-05-13T00:40:20.033Z
Just to clarify, the $y$ variable in $x_{it}=z_{st}\cdot y_{j}$ is not the outcome variable $Y$, correct?
Evangelos Con
CC BY-SA 4.0
2022-05-13T05:41:58.377Z
Outcome variable is still $Y_{ijst}$. Just changing the independent variable from $x_{it}$ to $z_{st}*w_j$. I might have confused you with $y_j$ being the interaction. Sorry that was an error. I have made the right edit now. Thanks.
virtue_signal
CC BY-SA 4.0
2022-05-14T00:10:08.790Z
Is $w_j$ a categorical variable? If so, you could just create a series of interactions $z_{st} \cdot 1(w_j=1)$ ... $z_{st} \cdot 1(w_j=W)$, where $1(w_j = a)$ is an indicator variable for when $w_j=a$.
Evangelos Con
CC BY-SA 4.0
2022-05-15T09:12:33.237Z
$w_j$ is a continuous variables. Do I need to add the additional fixed effects mentioned in the question?
virtue_signal
<applied-econometrics><fixed-effects>
On adding fixed effects
virtue_signal
# On adding fixed effects

(Asked by: username_0 on 2022-05-12)

Let $Y\_{ijst}$ be some outcome variable of firm $i$ of industry $j$, located in state $s$, at time $t$. Normally, if we want to run any differences-in-differences specification where treatment varies across firms $i$ and time $t$, I would throw in firm fixed effects $\lambda\_i$ and time fixed effects $\tau\_t$ in my regression specification, and then, the coefficient of my treatment variable $x\_{it}$ will give me causal effect of $x\_{it}$ on outcome $Y\_{ijst}$.

Suppose instead of $x\_{it}$, my regressor of interest is the following interaction term: $z\_{st}\*w\_j$: something which varies across state x time ($s$ and $t$) and the other which varies across industries ($j$). Then, in this specification, does the ad-hoc addition of fixed effects - firm fixed effects ($\lambda\_i$, which subsumes industry and state fixed effects) and time fixed effects ($\tau\_t$) work for identification of causal effect? Or shall I also include other fixed effects - state-time, industry-time, state-industry? Mainly asking because my coefficient turns insignificant once I add the last two.

**Tags:** applied-econometrics, fixed-effects

> **Comments:**
> **username_1**: Just to clarify, the $y$ variable in $x_{it}=z_{st}\cdot y_{j}$ is not the outcome variable $Y$, correct?  
> **username_0**: Outcome variable is still $Y_{ijst}$. Just changing the independent variable from $x_{it}$ to $z_{st}*w_j$. I might have confused you with $y_j$ being the interaction. Sorry that was an error. I have made the right edit now. Thanks.  
> **username_1**: Is $w_j$ a categorical variable? If so, you could just create a series of interactions $z_{st} \cdot 1(w_j=1)$ ... $z_{st} \cdot 1(w_j=W)$, where $1(w_j = a)$ is an indicator variable for when $w_j=a$.  
> **username_0**: $w_j$ is a continuous variables. Do I need to add the additional fixed effects mentioned in the question?  

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## 0 answers

