Recitation 7: Examining Entry and Exit Data

Introducing the Data

This recitation will provide a guide to the data you will use for your last homework assignment, where you will estimate the dynamic duopoly model.

We will use data from Dearing and Blevins (2024). Let’s see what it looks like:

using DataFrames, DataFramesMeta, CSV

data = CSV.read("../data/clubstore_county.csv",DataFrame)

19320×9 DataFrame

19295 rows omitted

Row	market	year	active1	active2	active3	lactive1	lactive2	lactive3	pop
	Int64	Int64	Int64	Int64	Int64	Int64	Int64	Int64	Int64
1	1	2010	0	0	0	0	0	0	2
2	1	2011	0	0	0	0	0	0	2
3	1	2012	0	0	0	0	0	0	2
4	1	2013	0	0	0	0	0	0	2
5	1	2014	0	0	0	0	0	0	2
6	1	2015	0	0	0	0	0	0	2
7	1	2016	0	0	0	0	0	0	2
8	1	2017	0	0	0	0	0	0	2
9	1	2018	0	0	0	0	0	0	2
10	1	2019	0	0	0	0	0	0	2
11	1	2020	0	0	0	0	0	0	2
12	1	2021	0	0	0	0	0	0	2
13	2	2010	1	0	0	1	0	0	4
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
19309	1610	2010	0	0	0	0	0	0	2
19310	1610	2011	0	0	0	0	0	0	2
19311	1610	2012	0	0	0	0	0	0	2
19312	1610	2013	0	0	0	0	0	0	2
19313	1610	2014	0	0	0	0	0	0	2
19314	1610	2015	0	0	0	0	0	0	2
19315	1610	2016	0	0	0	0	0	0	2
19316	1610	2017	0	0	0	0	0	0	2
19317	1610	2018	0	0	0	0	0	0	2
19318	1610	2019	0	0	0	0	0	0	2
19319	1610	2020	0	0	0	0	0	0	2
19320	1610	2021	0	0	0	0	0	0	2

This is market level panel data that indicates the presence of three club stores in each market and each year. The variable active1 indicates the presence of firm 1 and so on. Similarly, the variable lactive1 indicates if firm was active in the previous year. As you will recall, these lagged values are state variables for the Markov Perfect Equilibrium.

The variable pop is a categorical variable that ranks the size of each market by population. This variable is the observable \(x\) that we introduced in our description of the model.

By plotting activity rates by this variable, we can get a sense of how it matters.

using StatsPlots, Statistics

d = @chain data begin
    groupby([:year,:pop])
    @combine :number = mean(:active1 .+ :active2 .+ :active3) 
    @transform :pop = string.(:pop)
end
@df d plot(:year,:number,group=:pop,title = "Average number of active firms by year and market size")

Depicting entry and exit

We can derive entry and exit indicators by comparing activity dummies with their lagged values. For example, for firm 1 entry patterns look like this:

@chain data begin
    @subset :lactive1.==0
    groupby(:pop)
    @combine :entry = mean(:active1)
    @df plot(:pop,:entry)
end

And exit patterns look like this:

@chain data begin
    @subset :lactive1.==1
    groupby(:pop)
    @combine :exit = mean(:active1.==0)
    @df plot(:pop,:exit)
end

Mapping to the model

An obvious problem is that there are three firms in the data and our model has only two firms. It would not be too difficult to extend the model to three firms but we have already jumped away from reality by assuming symmetry. For the purposes of the assignment let’s ignore firm 3 and consider only firm 2.

Recall that equillibrium is characterized by a symmetric policy function \(p(x,a,a')\), the probability of being active given the market size \(x\), the previous activity of the firm (\(a\)) and the previous activity of their competitor:

policies = @chain data begin
    groupby([:pop,:lactive1,:lactive2])
    @combine :p1 = mean(:active1) :p2 = mean(:active2) :num_obs = length(:active1)
    @orderby :pop :lactive1 :lactive2
end

18×6 DataFrame

Row	pop	lactive1	lactive2	p1	p2	num_obs
	Int64	Int64	Int64	Float64	Float64	Int64
1	1	0	0	0.000471032	0.0	6369
2	1	0	1	0.0	1.0	19
3	1	1	0	0.826087	0.0	23
4	2	0	0	0.00169651	0.00113101	5305
5	2	0	1	0.0	0.96875	128
6	2	1	0	0.974545	0.0	275
7	3	0	0	0.0122468	0.00471032	2123
8	3	0	1	0.00930233	0.990698	215
9	3	1	0	0.984489	0.00729927	1096
10	3	1	1	0.95	1.0	20
11	4	0	0	0.0380349	0.0126783	631
12	4	0	1	0.0128617	0.987138	311
13	4	1	0	0.986077	0.019656	1221
14	4	1	1	0.968504	0.984252	254
15	5	0	0	0.0454545	0.0568182	88
16	5	0	1	0.0117647	0.996078	255
17	5	1	0	0.989035	0.0504386	456
18	5	1	1	0.988701	0.990584	531

In theory, this gives two separate estimates of the equilibrium choice probabilities (due to the assumed symmetry).

How do you want to estimate the model?

There are five parameters in the model (\(\phi_{0},\phi_{1},\phi_{2},\phi_{3},\phi_{4}\)). There are so many ways to estimate these parameters. You could

1. Target particular moments (i.e. mean rates of activity, entry, or exit by market size) by simulating decisions using the distribution of initial market states that is in the data.
2. Many sensible moments could be calculated in the model directly using the choice probabilities \(p\) that come from the model solution, so you could do minimum distance without needing to use simulation.
3. One example of the above would be to simply do minimum distance on the estimated choice probabilities using choice probabilities at the model solution.
4. You could also just do maximum likelihood.

It will be completely up to you but you can use any remaining time in recitation to write down your plan for estimating the model and getting standard errors.

References

Dearing, Adam, and Jason R. Blevins. 2024. “Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games.” Review of Economic Studies.