# Aggregation Estimation Issues

I'm analyzing the effect of a law enforcement measure over time on reported violence for districts in a city. If I consider the city as a whole I have access to a lot of possible control variables (e.g., unemployment, poverty, police expenditures) that I don’t have at the more dis-aggregate district level.

I'm wondering what I lose by eschewing a panel analysis (with the possibility of fixed effects to control for unobserved heterogeneity and cross sectional variation) compared to the time series city analysis.

• Welcome to CV. Do you only have data on one city? How many districts/precincts do you have embedded within your city? Does your intervention/initiative affect all precincts/districts? – Thomas Bilach Jul 24 '20 at 3:46
• Thanks! Yes, one city, about 11 districts in the city, and the law was applied city wide. Although of course there are differences across districts in terms of police resources and deprivation levels. Some covariates I have at both city and district level but most are at city level. So the trade off – as I see it – is a lot of cross sectional variation at the district level plus the possibility of apply fixed effects to control for unobserved heterogeneity -vs- richer city-wide data but where the treatment effect detection might be flattened out at the more aggregate level. – cel Jul 24 '20 at 15:34
• Does the initiative go into effect at the same time for all districts? – Thomas Bilach Jul 24 '20 at 16:14
• Yes, Thomas, the law change goes into effect at the same time across all districts. – cel Jul 24 '20 at 16:33

It appears the 'effective date' of your intervention is uniform across all districts in the city.

A fixed effects approach is not likely to perform better than simply pooling your data. For instance, you might wish to estimate something like this:

$$\text{Crime}_{dt} = \beta_{0} + \beta_{1}\text{Policy}_{t} + \alpha_{d} + \epsilon_{dt}$$

where you observe districts $$d$$ across time periods $$t$$. The variable $$\text{Crime}_{dt}$$ is some arbitrary crime outcome (e.g., robbery rate) observed across your 'district-time' periods. $$\text{Policy}_{t}$$ is a treatment dummy 'turning on' at period $$t$$. Here, $$\alpha_{d}$$ is your district effect.

Note, the dummy will have, or should have, the same "within-district" pattern. Estimation of "district" fixed effects, which could be achieved via the inclusion of dummies for all districts, is not likely to yield anything useful in this scenario, even with a limited number of control variables. In fact, your estimate of $$\beta_{1}$$ should be similar to the pooled estimate that ignores the panel structure. Moreover, estimation of discrete time effects for all $$t$$ periods is likely to absorb a discrete treatment variable that uniformly affects all districts.

I don't presume that you only seek to model the effects of the initiative using a discrete treatment indicator. But without more information, it is difficult to advise you how to proceed. I suppose some districts received a greater dose of the initiative/intervention, in which case you could attempt estimation of the above equation with a continuous treatment variable. Or, maybe you could investigate heterogeneity of treatment effects by city section.

You will lose all cross-sectional variation by aggregating your data up to the city level. Is there another neighboring city/county similar to yours which could act as a control in this setting? Your options are somewhat limited unless you can find other jurisdictions, either at the district or city level, where the intervention was absent.

If you acquired a sufficient time series then I suppose you could look into interrupted time series modeling. Also, there is a relatively new package in R for estimating causal effects.

I hope this helps!