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Here is the situation:

  • There was an intervention $T$.
  • I am interested in how this intervention changed the minutes spent using a computer in 5 counties: $C_{a,t}, C_{b,t}, C_{c,t}, C_{d,t}, C_{e,t}$
  • I have daily pre/post data on the minutes/spent using a computer on 200 individuals from these 5 counties
  • In addition, I found 'control' groups for each of the 5 treatment counties who showed similar trends to a specific county: $C_{a,c}, C_{b,c}, C_{c,c}, C_{d,c}, C_{e,c}$
  • I have daily pre/post data on the minutes/spent using a computer on 200 individuals from these 5 counties, as well

My question is - do I need to account for the fact that there are different counties other than using fixed effects? In other words, I am expecting the treatment effect to be the SAME across all 5 counties.

Here is what I was thinking. This is estimated at the individual (i) and day (t) level:

$Y_{it} = \alpha_{it} + \beta T_{it} + \gamma t_{it} + \delta (T_{it} · t_{it}) + C_{it} + \epsilon_{it}$

Where

  • $\alpha$ = constant term
  • $\beta$ = treatment group specific effect
  • $\gamma$ = time trend common to control and treatment groups
  • $\delta$ = true effect of treatment
  • $C$ = fixed effect for county

Is this an accurate estimate of the treatment effect?

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  • $\begingroup$ It seems about right. I have some follow-up questions. First, does treatment begin at the same time $t$ for all individuals within your counties? Second, does treatment affect all 200 persons across the different counties? I wonder if there is an extra layer to exploit. Third, do the individuals comprising your control group come from five different counties, or do they come from the same counties as those in the treatment group? $\endgroup$ – Thomas Bilach Nov 8 '20 at 17:44
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The confusing part about your equation, at least for me, is your inclusion of a county fixed effect and a treatment dummy at the individual level. Let me elaborate. Here is your difference-in-differences (DiD) equation (Note: I swapped the variable $t_{it}$ for $P_{t}$ as it is easier on the eyes and it usually denotes a post-treatment indicator that is unit-invariant):

$$ y_{ict} = α + \beta T_{i} + \gamma P_{t} + \delta(T_{i} \times P_{t}) + C_{c} + \epsilon_{ict}, $$

where you observe outcome $y_{ict}$ for individual $i$ within county $c$ across days $t$. As per your post, $\alpha$ is a constant (i.e., a global intercept) and should not be subscripted; it does not vary over time or across individuals. $T_{i}$ is a treatment dummy for treated individuals; it should equal 1 for the 200 treated individuals across the five different counties, 0 otherwise. $P_{t}$ is a post-treatment indicator equal to 1 for all days after treatment commences in both treatment and control groups, 0 otherwise. Unless I misunderstood your equation, you also want to estimate county fixed effects. If so, it should be $c$-subscripted. It shouldn't change anything with respect to your point estimates. Running the foregoing equation in software, as is, will return an estimate for $\delta$. Your county effect, however, will be dropped as it is collinear with the treatment dummy. Your estimate of $\delta$ will remain unchanged.

But let's see if we can improve your approach. Since you observe the same individuals before and after treatment, you can estimate a DiD equation using individual fixed effects. Your treatment appears well-defined at the level of the individual. Here is what I think you should do:

$$ y_{it} = α_{i} + \beta T_{i} + \gamma P_{t} + \delta(T_{i} \times P_{t}) + \theta X_{it} + \epsilon_{it}, $$

where

  • $\alpha_{i}$ denotes individual fixed effects
  • $T_{i}$ is your treatment dummy for treated individuals (i.e., it varies across persons but not over time and therefore has no $t$-subscript)
  • $P_{t}$ is your post-treatment indicator (i.e., it varies over time but exhibits the same pattern across all individuals and therefore has no $i$-subscript)
  • $X_{it}$ denotes a vector of time-varying, individual-level control variables

The treatment dummy $T_{i}$ will be absorbed by the individual fixed effects. Again, don't worry. Because you observe the same individuals before and after treatment, you can estimate individual fixed effects and your point estimates will remain unchanged. It is worth noting, however, that these fixed effects might soak up some of the residual variance, which in turn, might reduce the standard error associated with $\delta$. Review this post for more information.

If you're a purist who hates to see warning messages and/or NA values in your regression output, you could also estimate the following equation:

$$ y_{it} = α_{i} + \gamma_{t} + \delta D_{it} + \theta X_{it} + \epsilon_{it}, $$

where $\alpha_{i}$ and $\gamma_{t}$ represent individual and day fixed effects, respectively. The variable $D_{it}$ is a treatment dummy. It is your interaction term from earlier, just represented in a different way. It equals 1 if a person is treated and in the post-treatment period. You can instantiate this variable manually if you so desire (i.e., $D_{it} = T_{i} \times P_{t}$). Your point estimates should be similar across the two specifications (equivalence is assumed in the absence of covariates). This equation is also useful in settings where treatment timing is not standardized across your $i$ units, and thus it can be used in a wider variety of circumstances.

In sum, DiD methods are usually applied to aggregate level data (e.g., cities, counties, states, etc.), but they can also be used at a lower level if we observe the same $i$ units (e.g., individuals) over time as well. If applied at the level of the individual, the individual fixed effects should absorb more variation and likely reduce the size of your standard errors. Again, these recommendations assume you observe the same individuals across time.

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  • $\begingroup$ Do you think I need to estimate fixed effects for all counties? It will use up a lot of DF. As I said, I matched treatment and control counties. So can't I just use an index for the matched pair? $\endgroup$ – Jamal Mantburg Nov 18 '20 at 17:03
  • $\begingroup$ Do you mean you will eat up more degrees of freedom with individual fixed effects? $\endgroup$ – Thomas Bilach Nov 18 '20 at 17:10
  • $\begingroup$ Yes. That's what I meant. $\endgroup$ – Jamal Mantburg Nov 18 '20 at 17:13
  • $\begingroup$ How many time periods (days) do you have? $\endgroup$ – Thomas Bilach Nov 18 '20 at 17:15
  • $\begingroup$ There's like 50 days in the BEFORE and 50 days in the AFTER $\endgroup$ – Jamal Mantburg Nov 18 '20 at 17:17

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