Identification in Dynamic Discrete Models

UBC ECON 567 Assignment

Author

Paul Schrimpf

Published

March 28, 2024

Please write your answers in a literate programming format, such as a Pluto, Jupyter, or Quarto notebook. Turn in both the notebook file and an html or pdf.

Simulation and Estimation

Problem 1

Adapt the equilibrium calculation, simulation, and estimation code in dyanmicgame.jl to compute an equlibrium, simulate, and estimate with a single agent firm (N=1). Most of the code will just work, except the transition function, but it is not needed here. Check your code by simulating some data with N=1 and Nexternal=2, and then estimating the model with the simulated data. When simulating, set T to 20_000 so that it easier to distinguish estimation noise from some other problem. Make a table similar to one in dynamicgame.jl at the end of the “Estimation” section comparing the true payoffs and estimated payoffs.

Fitted and True Choice Probabilities

The function DG.equlibrium returns a tuple consisting the output of NLsolve.solve and the equilibrium choice probabilities.

res, choicep = DG.equilibrium(g)

The equilibrium choice probabilities are in a 3-dimensional array of size number of players by number of actions by number of states. choicep[i,a,s] is the probability player i chooses action a in state s.

Problem 2

Compare the true choice probabilities with the estimated choice probabilities from the model. To calculate the estimated choice probabilities, create a new DG.DynamicGame with the payoff function given by the estimated payoffs from problem 1. You may use the code below to get started. Create a table and/or figure that compares the estimated and true choice probabilities.

function createufunction(Eu::AbstractArray)
    N = size(Eu,1)
    Nstates = size(Eu,3)
    Nchoices = size(Eu,2)
    Nexternal=Int(log2(Nstates))-N
    states = BitVector.(digits.(0:(2^(N+Nexternal)-1), base=2, pad=N+Nexternal))
    @show states
    statedict = Dict(statevec(x)=>x for x in 1:length(states))
    u(i,a,x::Integer) = Eu[i,a[1],x]
    u(i,a,s::AbstractVector) = u(i,a,statedict[s])
    return(u)
end

û = createufunction(Eu) # assuming you used Eu as the estimated payoffs in Problem 1
ĝ = DG.DynamicGame(N,û, 0.9, Ex, 1:2, 1:ns)
res, choicep̂ = DG.equilibrium(ĝ)

# create table and/or figure comparing choicep̂ and choicep

Counterfactual Choice Probabilities

Problem 3

Suppose the payoff of action 2 in states 2, 4, 6, and 8 is decreased by 0.25. Compute the true and estimated change in choice probabilities. Compare the true and estimated change in choice probabilities in a figure or table.

You can create an appropriate shifted payoff function and new choice probabilities with the following code.

u2(i,a,s) = u(i,a,s) + (s % 2 == 0)*(-0.25)
g2 = DG.DynamicGame(N, u2, 0.9, Ex, 1:2, 1:ns)
res2, choicep2 = DG.equilibrium(g2)

Incorrect Payoff Normalization

Problem 4

The estimation code assumes the payoff of action 1 is 0 in all states. What if this assumption is incorrect? To explore what happens, simulate data where the payoff of action 1 is -(s-3.5)/5*(s % 2==1) in state s, and the payoff of action 2 is the same as in problems 1-3. Then estimate the model assuming the payoff of action 1 is 0. Finally, calculate the change in conditional choice probabilities from decreasing the payoff of action 2 in states 2, 4, 6, and 8 by 0.25 as in problem 3. Does an incorrect normalization affect the estimated change in choice probabilities?

Shift in Transitions

Problem 5

Repeat the analysis in problem 4, but instead of a shift in payoffs, suppose the transition probability of the exogenous state changes. Consider a change of Ex with pstay=0.7, to pstay=0.9. Comment on your findings.

Implications

Problem 6

Read Kalouptsidi, Scott, and Souza-Rodrigues (2021). What findings of theirs do the above simulations illustrate?

For further reading, consider looking at Kalouptsidi, Scott, and Souza-Rodrigues (2017) and Kalouptsidi et al. (2024).

References

Kalouptsidi, Myrto, Yuichi Kitamura, Lucas Lima, and Eduardo Souza-Rodrigues. 2024. “Counterfactual Analysis for Structural Dynamic Discrete Choice Models.” https://kitamura.sites.yale.edu/sites/default/files/files/Dynamic-Partial-ID-20240318.pdf.

Kalouptsidi, Myrto, Paul T. Scott, and Eduardo Souza-Rodrigues. 2017. “On the Non-Identification of Counterfactuals in Dynamic Discrete Games.” International Journal of Industrial Organization 50: 362–71. https://doi.org/https://doi.org/10.1016/j.ijindorg.2016.02.003.

———. 2021. “Identification of Counterfactuals in Dynamic Discrete Choice Models.” Quantitative Economics 12 (2): 351–403. https://doi.org/https://doi.org/10.3982/QE1253.