### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    #! format: off
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
    #! format: on
end

# ╔═╡ 38aa4818-b037-44a9-a27a-3b27118d159e
using Pkg

# ╔═╡ a0829c7b-3843-436b-b52a-3db8b46d3cea
# Let's load all required packages
begin
	using CSV
	using PlutoUI
    using LinearAlgebra  # for norm()
    using Plots
    using DataFrames
    using StatsBase  # counts
    using Optim
	using JuMP
	using Ipopt
	using StatsPlots
end

# ╔═╡ 47af6108-4804-4f94-8795-c7832a4a396b


# ╔═╡ 79dee14b-5c6c-4358-ba09-1490d23575bc
html"<button onclick='present()'>present</button>"

# ╔═╡ 43e3f61c-9bb0-11eb-35bc-ade31ecafbe8
md"

# Estimating the Rust Bus Model

*Compuational Economics 2025*

>Florian Oswald

"

# ╔═╡ 9825a678-af24-42c9-a3ad-c22715946d92
PlutoUI.TableOfContents()

# ╔═╡ a40f7be1-9af9-421e-a616-2353fc1a1dc5
md"""
* in this notebook we continue with the Bus Engine Replacement model.
* We heard in class last week about the NFXP and MPEC methods.
* Both require us to formulate the likelihood function in this model class. (other models could be estimated with a GMM setup, all depends on what the model outputs).

"""

# ╔═╡ 7572c64b-1f86-45dc-8e41-a1f378622d72
md"""
## Credits
"""

# ╔═╡ c0f2b80c-7e73-4c3a-836d-ee0b976c3424
Markdown.MD(Markdown.Admonition("tip","Info",[md"""The code in this notebook is partially based on what I learned from looking at [Bertel Schjerning's](https://bschjerning.com/) MATLAB [code](https://github.com/dseconf/DSE2021/tree/master/02_NFXP_Schjerning/code/zurcher_matlab) for the Dynamic Structural Econometrics Workshop 2021. In particular, I copied their data setup almost one for one. My contribution, if any, is to show how one can achieve reasonably good performance with automatic differentiation via JuMP. Notice, however, that the original method of Rust (and the one used in Bertel's Matlab code) takes fully advantage of the mathematical properties of the model, deriving analytic gradients for the likelihood function *as well as* the expected value function (!), via a so-called *Newton-Kantorovich* value function iteration technique and thus achieves very good performance indeed. More info on this is in their [Econometrica comment](https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA12605), and (maybe more suited as a starting point) the [slides of Bertel for that workshop](https://github.com/dseconf/DSE2021/tree/master/02_NFXP_Schjerning/slides). My presentation of the model is very close to what [Fedor Iskhakov](https://fedor.iskh.me) shows in [his great course (lession 28)](https://fedor.iskh.me/compecon)"""]))

# ╔═╡ 48b5b838-c5b7-4992-b020-e653f0a8451b
md"

# Model Recap

$$\begin{eqnarray}
T^*(EV)(x,d) &\equiv& \sum_{x' \in X} \log \big( \exp[v(x',0)] + \exp[v(x',1)] \big) \pi(x'|x,d) \\
v(x,d) &=& u(x,d) + \beta EV(x,d)
\end{eqnarray}$$


$$P(d|x) = \frac{\exp[v(x,d)]}{\sum_{d'\in \{0,1\}} \exp[v(x,d')]}$$


So, object $P$ (the ccp) depends on the solution of the structural model, since $v$ depends on $EV$. To make this clear, we will index $P(d|x, EV_\theta)$ from here on.
"


# ╔═╡ 4698df16-21ee-4b40-a36e-899532caa0c8
Markdown.MD(Markdown.Admonition("info","Info",[md"""Notice that $\theta$ will stand for all parameters of the utility function, and that the components of $\pi$ are a subset of those."""]))

# ╔═╡ 10a47b0e-bea3-43f6-9235-ca5034ef3bf3
md"""

## Likelihood Function

This model outputs a conditional probability distribution - the $P$s. *Estimating* such a model means to use Maximum Likelihood to see how *likely* the observed data could have been generated from this particular model at hand.

The likelihood function in this problem looks like this:

$$\mathcal{L}_n(\theta, EV_\theta) = \prod_{i=1}^{162}\prod_{t=2}^{T_i} P(d_{i,t}|x_{i,t}, EV_\theta) \pi(x_{i,t}|x_{i,t-1},d_{i,t-1})$$

* We have unique 162 buses in our 4 groups
* Each bus has a different number of observations $T_i$
* The CCP (i.e. the result of the structural model) is $P(d|x, EV_\theta)$
* The *exogenous* state transition is$\pi(x_t|x_{t-1},d_{t-1})$ 

## Log Likelihood

* As in our homework, we work with the log of the likelihood function, i.e.

$$\ell_n(\theta, EV_\theta) = \sum_{i=1}^{162}\sum_{t=2}^{T_i} \log P(d_{i,t}|x_{i,t}, EV_\theta) + \log \pi(x_{i,t}|x_{i,t-1},d_{i,t-1})$$

**Estimation Problem**

In reality, this is an optimization problem. Given that the Bellman Operator $T$ has a unique fixed point for each $\theta$, we *just* have to choose $\theta$!

$$\max_{\theta} \ell_n(\theta, EV_\theta)$$

* This is an *unconstrained* problem.
* In practice, we again maximise the *average* log likelihood function, i.e. divide by $n$.

# NFXP: Double Loop


* We heard last time that the nested fixed point (NFXP) approach has an inner (find $EV = T(EV)$, given $\theta$) and an outer loop (find $\theta = \arg \max \ell_n$)
* We set up all the code to compute $EV = T(EV)$ and the CCP function. So now we can proceed to build the data-interface for the estimation!

## Components

1. log likelihood function: evaluates $\ell$ above.
1. *nested* log likelihood function: takes a candidate $\theta^{(k)}$, updates `Harold`, computes model solution, and evaluates log likelihood
1. `nfxp` a function that loads data, sets starting values, and calls a nonlinear optimizer on the nested likelihood function.



## Model Solving 

Remember that we developed the solution to the model [in a separate notebook](https://floswald.github.io/NumericalMethods/lecture10-zurcher/) last week in class.
* I copied the required functions at the bottom of this notebook into the **Function Library**

Remember the main components we needed were:

1. A Data type `Harold` to represent Mr Zurcher
1. A Bellman Operator
1. A Value function iterator
1. A function to compute the probability of Replacing


"""

# ╔═╡ 34e52272-5b28-4c9a-9a27-7b13df589870
md"""

## Likelihood Function

* We will set this up such that we can estimate the parameters for the milage transition in $\pi$ inside or outside the structural model. We will call those approaches *partial* or *full* Maximum likelihood.
* Let us start from the innermost component: the likelihood function.
* So, for a certain solution of the model, represented by $P$, or here, an array that tells us how the model predicts the replacement probability at each *observed* state (in `p_model`), and what the actual decision of Mr Zurcher was (in `replace_data`)
"""

# ╔═╡ 387d9de3-9dc1-4e3c-8b9b-cbb7cc9cb85c
md"
## Nested Likelihood Function

* For a given choice of $\theta$ parameters, solve the model, and evaluate the likelihood function, given that solution.
"

# ╔═╡ d3221610-1791-4d87-acdd-4a5bd9a2b37a
md"
## NXFP: Full Monty!

* Sets up the data
* gives `nested_likelihood` to a numerical solver and runs the `BFGS` algorithm
* returns solution
"

# ╔═╡ 62893e3f-195f-4581-aa07-7a8b9115560b
md"
# Quick Look at Data

* We run `nfxp` for a certain setup of model parameters. `n` for example is the number of grid points.
* Let's quickly look at the data do see what those choices imply.
"

# ╔═╡ 5168797e-01d0-4697-b796-c0e270550348
# a simple frequency count is sufficient to estimate how mileage bins increase
function partialMLE(dx1::Vector) 
	c = counts(dx1) ./ length(dx1)
	c[1:(end-1)]  # forget about smallest category
end

# ╔═╡ 57632a86-1479-4331-a8e8-94865193a4eb
@bind n Slider(50:200, show_value = true)

# ╔═╡ c6ed299c-b9a7-47e7-ab3a-959031aa7d0a
md"""

# Time to Roll!

* Time to run this now.
* We looked at the code to solve for the model last time, so that should be ok.
* But what is going to go a *success* for our estimation exercise? 
* Well, we want to get close to 👇 this! We use Groups 1,2,3,4.

$(LocalResource("./Rust-table9.png"))
"""

# ╔═╡ 2e555776-ce29-4791-9542-27093d132a0e
md"
#

* ok let's see!
"

# ╔═╡ 165022ec-6952-4844-bfc4-c6cac6a501b1
md"
🎉
(running with β=0.9999 as in Rust would get even closer but it takes too long!)
"

# ╔═╡ f83be35c-b0d9-430a-a7a9-e5a09fd392b2
md"""
# MPEC

In the MPEC formulation we try to solve this problem instead: 

$$\max_{\theta , EV} \ell_n(\theta, EV; X)\quad \text{s.t. }EV = T(EV)$$

* The first thing to note here is that we have 2 sets of choice variables: structural parameters (the ``\theta``s) *and* things that represent the model ($EV$)
* Suppose we have a `JuMP` model `m`
* depending on whether or not we want to estimate the mileage progression process via full MLE, we need to add variables (and a corresponding contraint for the probabilities to sum to one) for this or not. 
* We use the same dataframe `d` as before as built by `busdata`
* `N` is the number of mileage states, `J` is the number of bins that the state can at most move up
"""

# ╔═╡ 0f3c4e0d-6a60-44b0-99cf-6251e5ad7b9a
md"""
* next, we have to build some expressions that will be needed to define the constraints, hence all the components needed to compute `EV`
* Notice that we can reference objects stored on `m` with a dict-like accessor `m[:x]`
"""

# ╔═╡ 6f778c42-6beb-40e4-8956-65bf046c9489
md"""
* Then we set up the objective function. Looking at the above expression, that is the likelihood function at the parameter choices ``\theta``. 
* Notice that we need to add a time index to our data frame first.
* We group by `busid` and add a column for each observation (i.e. each *period*)
* Again, we setup the likelihood function for full MLE of transition probabilities or not
"""

# ╔═╡ e4209b2e-0a59-4162-8d93-23ff2f819af1
function add_obj!(m::JuMP.Model,doθ,d; θlast = nothing)

	# add time period index to each bus 
    gd = groupby(d, :id)
    dd = transform(gd, :id => (x -> 1:length(x)) => :it)
    icounter = combine(gd, nrow)  # now many rows for each bus: Tᵢ
    I = nrow(icounter)  # number of unique buses

	if isnothing(θlast)
		θlast = m[:θlast]
	end
	
	# objective function
    # This is the likelihood function 
    if doθ
        @NLobjective(m, Max, 
        sum( log( (gd[i][it,:d]==false) * m[:pkeep][ gd[i][it,:x] ] + (gd[i][it,:d]==true) * (1 - m[:pkeep][ gd[i][it,:x] ]) )
             + log(θlast[gd[i][it,:dx1] + 1]) 
        for i in 1:I, it in 1:icounter[i,:nrow] )  )

    else
        @NLobjective(m, Max, 
        sum( log( (gd[i][it,:d]==false) * m[:pkeep][ gd[i][it,:x] ] + (gd[i][it,:d]==true) * (1 - m[:pkeep][ gd[i][it,:x] ]) )
             for i in 1:I, it in 1:icounter[i,:nrow] )  )
    end
end


# ╔═╡ d6962b11-99b2-449c-bbd5-17cc096bf2ef
md"""
* Finally, the constraints on this optimization problem: the structure of the model.
* In our case, this is the requirement that the expected value function has the correct form. In particular, we want that at mileage state $x_i$,

$$EV(x_i) = \sum_{j=0}^J \log \big( \exp[v(x_{i+j},0)] + \exp[v(x_{i+j},1)] \big) \pi(x_{i+j}|x_{i})$$

* So, the optimizer *chooses* the value on the LHS of this equality, and the constraint is that this choice *needs to coincide with the RHS*, i.e. the definition of the Expected Value function as per our model.
* The only slight issue is that at the upper bound of th state space, the transitions in $\pi$ are no longer unrestricted, hence we need to take care of indexing a little bit. 
* Basically we add three different types of constraint, depending on how many slots $x_i$ can move up 

"""

# ╔═╡ 50269188-a202-410d-bd0e-53f35cc44b49
function add_constraints!(m::JuMP.Model,J,N; θlast = nothing)
	# if we did not give this argument, grab it from the model
	if isnothing(θlast)
		θlast = m[:θlast]
	end
	# 1. when state can move up any number of slots: easy
    @NLconstraint(m, evcon[i = 1:(N-J+1)], 
    m[:EV][i] == sum( log(exp(m[:VK][i + j]) + exp(m[:VR])) * θlast[j+1] for j in 0:(J-1)  ))

    # 2. not all state progressions are possible: need to sum last bins      
    @NLconstraint(m, evconJ[i = (N-J+2):(N-1)], 
    m[:EV][i] == sum(
		log(exp(m[:VK][i+j]) + exp(m[:VR])) * θlast[j+1] for j in 0:(N-i-1)
	    ) + 
		log(exp(m[:VK][N]) + exp(m[:VR])) * (1-sum( θlast[k+1] for k in 0:(N-i-1))) 
        )
    
    # 3. bellman equation at the final state
    @NLconstraint(m, evconN,
        m[:EV][N] == log(exp(m[:VK][N]) + exp(m[:VR]))
    )
		end

# ╔═╡ 651e2a7a-1ffd-4cab-aef8-c51fb4582533
md"""
### Running MPEC via `JuMP`

Ok, time to put it all together and run the JuMP model!
"""

# ╔═╡ a89f585a-165e-42a2-b426-574b4b57cc59
md"""
🎉
"""

# ╔═╡ 4b1c7427-ecf1-4275-a592-ce620be8969d
md"""
🎉🎉
"""

# ╔═╡ b95bea83-1ed8-412f-aecb-34b12bfcf1e9
md"""
* So, in the partial MLE case, we seem to get reasonably close results with either method, which is good. 
* In the full MLE case, we get exactly the same results in both methods as well (which is also good).
* The higher we make $\beta$, the closer we get to John Rust's table above. 
* There is an issue with convergence for MPEC however at very high values (it does not converge to a valid solution) - the NXFP solution takes forever.
* I suspect the slight difference in solutions (and possibly the failure to converge at very high $\beta$ ) in the full MLE case comes from the way I handle the residual probability to move up the last mileage bin: you can see that the NFXP method estimates only 2 probabilities (the third is given as their complement), while in MPEC I estimate all three. For some reason I was not able to fix that, so that's a loose end.
* More details, also timings [at the dedicated repo](https://github.com/floswald/Zurcher.jl)
"""

# ╔═╡ efd9b30b-8dd9-4656-b95b-13a999a58111
md"""
# Larger State Space

* Rust then increase `n` to 175.
* The estimates slightly change, particularly for 

$(LocalResource("./Rust-table10.png"))

"""

# ╔═╡ 0ea31a68-74ea-4cbd-a920-9ac384cfc67a
md"
#
"

# ╔═╡ 1848a939-7c2b-436f-ac45-03f5371e6ce5
md"
# Function Library
"

# ╔═╡ c5edc6f2-380c-47a8-8eaa-e44868f83eb6
function make_trans(θ, n)
	transition = zeros(n, n);
	p = [θ ; 1 - sum(θ)]
	if any(p .< 0)
		@warn "negative probability in make_trans" maxlog=1
	end
	np = length(p)

	# standard block
	for i = 0:n - np
		transition[i + 1,(i + 1):(i + np)] = p
	end

	for k in 1:(np-1)
		transition[n - k,(n - k):n] = [p[1:k]...; 1 - sum(p[1:k])]
	end
	transition[n,n] = 1.0
	return transition
end

# ╔═╡ 684ac370-bb0d-4f42-8e3c-65f4f68329a8
mutable struct Harold
	# parameters
	n::Int
	maxmiles::Int
	RC::Float64 
	c::Float64 
	θ::Vector{Float64}
	β::Float64 

	# numerical settings
	tol::Float64

	# state space
	mileage::Vector{Float64}
	transition::Matrix{Float64}

	function Harold(;n = 175,maxmiles = 450, RC = 11.7257,c = 2.45569,
					  θ = [0.0937, 0.4475 ,0.4459, 0.0127],
					  β = 0.999, tol =  1e-12)

		this = new()   # create new empty Harold instance
		this.n = n
		this.maxmiles = maxmiles
		this.RC = RC
		this.c = c
		this.θ = θ
		this.β = β
		this.tol = tol

		# build state space
		this.mileage = collect(0.0:n - 1)
		this.transition = make_trans(θ, n)
		return this
	end
end

# ╔═╡ daf361e3-9e68-4f64-8e33-a51ce443d3fa
function loglikelihood(h::Harold,p_model,
					   replace_data::BitArray,
					   miles_data::Vector{Int64},
					   miles_increase_data::Vector{Int64};do_θ = false)

	# check data arrays are consistent
	nt = length(replace_data)
	@assert(nt == length(miles_data))
	@assert(nt == length(miles_increase_data))

	# 1. get model-implied Probability of replace at miles_data
	prob_replace = p_model[miles_data]  
	prob_keep    = 1 .- prob_replace 

	# 2. for each observed discrete choice (replace or not)
	# compute the model implied probability of that happening
	logL = log.(prob_keep .* (.!(replace_data)) .+ (prob_replace .* replace_data))

	# 3. compute likelihood of mileage transition
	# adjust \theta for last element
	# this is tricky because implicitly need to enforce constraint that all p sum to 1
	# can get negative numbers here for the last element.
	if do_θ
		p_miles = clamp!([h.θ ; 1 .- sum(h.θ)], 0.0, 1.0)  # the clamp!() constrains the values to [0,1] mechanically
		logL = logL .+ log.(p_miles[miles_increase_data .+ 1])
	end

	mean((-1) * logL)
end

# ╔═╡ 2f7d6cd1-70e4-4d67-8285-9f6d513e6903
function add_vars(m::JuMP.Model,N, h::Harold, doθ,d)
	@variable(m, θc >= 0 , start = 0.0)
    @variable(m, RC >= 0 , start = 0.0)
	@variable(m, -50.0 <= EV[1:N] <= 50.0)  # need to help solver here!

    if doθ
        p_0 = partialMLE(d.dx1)  # get a starting value for mileage
        J = length(p_0) + 1 # the partialMLE function discards the smallest bin, but later on sums up all probs 
        @variable(m, θlast[1:(J)] >= 0)
        set_start_value.(θlast[1:(J-1)], p_0)
        set_start_value(θlast[J], 0.0)
        @constraint(m, sum(θlast) == 1)
		return J

    else
        J = length(h.θ) + 1
        θlast = [h.θ; 1 - sum(h.θ)]
		return J, θlast
    end
end

# ╔═╡ f265b96a-ac85-4785-b4c3-83f85ed2ceca
function add_expressions!(m,N,h::Harold)
    @NLexpression(m, opcost[i = 1:N], -0.001 * h.mileage[i] * m[:θc])
    @NLexpression(m, VK[i = 1:N], opcost[i] + h.β * m[:EV][i])  # value of keep
    @NLexpression(m, VR         , -m[:RC] + opcost[1] + h.β * m[:EV][1])  # replace
    @NLexpression(m, diffV[i = 1:N], VR - VK[i] )  # payoff difference
    @NLexpression(m, pkeep[i = 1:N], 1 / (1 + exp(diffV[i])))
end

# ╔═╡ 2bd5b2d5-7bc1-4986-89b0-88dea6d8bf74
function busdata(z::Harold; bustype = 4) 
	d = CSV.read(joinpath("buses.csv"), DataFrame, header = false)
	select!(d, 1,2,5,7)
	rename!(d, [:id, :bustype, :d1, :odometer])

	d = filter(x -> x.bustype .<= bustype, d)

	# discretize odometer
	transform!(d, :odometer => (x -> Int.(ceil.(x .* z.n ./ (z.maxmiles * 1000)))) => :x)

	# replacement indicator
	dd = [d.d1[2:end] ; 0]

	# mileage increases
	dx1 = d.x .- [0;d.x[1:(end-1)]]
	dx1 = dx1 .* (1 .- d.d1) .+ d.x .* d.d1

	# make new dataframe
	df = [select(d, :id, :x, :bustype) DataFrame(dx1 = dx1, d = BitArray(dd))]

	# get rid of first observation for each bus
	idx = df.id .== [0; df.id[1:end-1]]
	df = df[idx,:]
end

# ╔═╡ fe4345fd-3eb2-471a-9be6-0e7d39cb2d1a
bd = busdata(Harold())

# ╔═╡ d77e1c95-9ecb-4678-978f-436dfbde916b
function mpec(; β = 0.9, is_silent = false, doθ = false,n = 175, θ = [0.107, 0.5152 ,0.3622, 0.0143,0.0009])

    h = Harold(β = β, n = n, θ = θ)
    N = h.n # number of mileage states
	d = busdata(h)

    # create Jump model
    m = Model(Ipopt.Optimizer)
    set_optimizer_attribute(m, MOI.Silent(), is_silent)

	if doθ
		J = add_vars(m,N, h, doθ,d)  
	else
		J,θlast = add_vars(m,N, h, doθ,d)
	end
	add_expressions!(m,N,h)
	add_obj!(m,doθ,d, θlast = doθ ? nothing : θlast)
	add_constraints!(m,J,N, θlast = doθ ? nothing : θlast)

    JuMP.optimize!(m)

    if doθ
        (RC = value(m[:RC]), θc = value(m[:θc]), θp = value.(m[:θlast]))
    else
        (RC = value(m[:RC]), θc = value(m[:θc]))
    end

end

# ╔═╡ 5b7051fd-a0ac-49db-b2a6-e037e523e0ca
mpec()

# ╔═╡ 4eacd72d-0b94-41cf-8762-65b9e1b6120b
mpec(β = 0.99, n = 150)

# ╔═╡ 7889081f-a2db-4032-b8bd-a54ae78344f2
mpec(β = 0.99, n = 175, doθ = true)

# ╔═╡ 91b85c93-a2d4-45bc-b181-b51695bf6178
function dplots(n)
	dd = busdata(Harold(n = n))
	b2 = groupby(dd, [:x, :bustype])
	bh2 = combine(b2, :d => mean => :mean_replaced, nrow)

	p1 = @df bh2 bar(:x, :mean_replaced, group=:bustype, title="avg replacement indicator", bar_position=:dodge, ylab="share replaced", xlab="miles", alpha=0.9, legend = :topleft, ylims = (0,1))
	p2 = @df bh2 groupedbar(:x, :nrow, group=:bustype, bar_position=:stack, 	xlab="mileage state", ylab="number of buses", title="mileage states by bus groups", bar_width=1,ylims = (0,400))
	plot(p1,p2)
end

# ╔═╡ 1b0fb366-0557-4dc4-bf79-553f4d068737
dplots(n)

# ╔═╡ e0a0d3fc-b183-4b10-bead-a13170c61538
function bellman(h::Harold, ev0::Vector)
	maintainance = -0.001 .* h.mileage .* h.c  # StepRange of length n
	v0 = maintainance .+ h.β .* ev0  # vector of length n
	v1 = -h.RC + maintainance[1] + h.β * ev0[1]  #  a scalar. if you replace, you start at zero miles
	M  = maximum(vcat(v1, v0))  #  largest value in both vs
	logsum = M .+ log.(exp.(v0 .- M) .+ exp(v1 - M))
	ev1 = h.transition * logsum #  matrix multiplication
	ev1
end

# ╔═╡ bbc4c659-1577-4af0-a892-d980858800fc
function ccp(h::Harold, ev::Vector)
	maintainance = -0.001 .* h.mileage .* h.c  # StepRange of length n
	v0 = maintainance .+ h.β .* ev  # vector of length n
	v1 = -h.RC + maintainance[1] + h.β * ev[1] 
	1.0 ./ (exp.(v0 .- v1) .+ 1)
end

# ╔═╡ 8149b1eb-c401-45eb-8262-f0615336fcd9
function vfi(h::Harold)
	if h.β >= 1.0
		throw(ArgumentError("value function iteration will not converge with β >= $(h.β)"))
	end
	ev0 = zeros(h.n)  # starting value
	err = 100.0
	iters = 0

	while err > h.tol
		iters += 1
		ev1 = bellman(h, ev0)
		err = norm(abs.(ev0 .- ev1))
		ev0[:] .= ev1   #  [:] do not reallocate a new object
	end
	return (ev0, iters)
end

# ╔═╡ ec467235-26c1-45c7-886d-d3c5a572a784
function nested_likelihood(x::Vector{Float64}, h::Harold, d::DataFrame)
	# update Harold
	h.RC = x[1]
	h.c  = x[2]
	if length(x) > 2
		h.θ  = x[3:end]
		h.transition = make_trans(h.θ,h.n)
	end

	# compute structural model 
	sol, iters = vfi(h)
	pr  = ccp(h, sol)

	# evaluate likelihood function 
	loglikelihood(h, pr, d.d, d.x, d.dx1, do_θ = length(x) > 2)
end

# ╔═╡ 724b6d55-9757-4c58-94ef-9de0e8f64669
function nfxp(; n = 175, β = 0.9, is_silent = false, doθ = false)
	z = Harold(RC = 0.0, c = 0.0, β = β,n = n)

	# get data
	d = busdata(z)
	# get initial transition probabilities
	if doθ
		p_0 = partialMLE(d.dx1)
		x_0 = [0.0, 0.0, p_0...]  # starting value with thetas
	else
		x_0 = [0.0, 0.0]  # starting value without thetas
	end

	# optimize likelihood which calls the structural model

	r = Optim.optimize( x -> nested_likelihood(x, z, d), x_0 ,BFGS(),  Optim.Options(show_trace  = !is_silent ))
	o = Optim.minimizer(r)
	if doθ
		(RC = o[1], θc = o[2], θp = o[3:end])
	else
		(RC = o[1], θc = o[2])
	end

end

# ╔═╡ 06645c70-959c-490e-b3e5-c5eebe42268e
nfxp()

# ╔═╡ 31c0c8b6-3630-432a-97c6-f59842b7185c
nfxp(n = 90,β = 0.999)

# ╔═╡ ed23d7fd-80ac-4cff-a791-4535c9450db1
nfxp(n = 90,β = 0.999, doθ = true)

# ╔═╡ 107fbf54-be6b-4177-a821-6cb9697abe5f
nfxp()

# ╔═╡ 5997ef6d-a83d-4bd2-93cf-ab8ee79acae5
nfxp(β = 0.99, n = 150)

# ╔═╡ b75dc6ef-f0f6-49ba-866f-1dd8aeacb622
nfxp(β = 0.99, n = 175, doθ = true)

# ╔═╡ ca35cfb9-344a-4345-a560-76393364b863
nfxp(n = 175,β = 0.99, doθ = true)

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[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "555d1076590a6cc2fdee2ef1469451f872d8b41b"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "936081b536ae4aa65415d869287d43ef3cb576b2"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.53.0+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b70c870239dc3d7bc094eb2d6be9b73d27bef280"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.44+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.oneTBB_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "7d0ea0f4895ef2f5cb83645fa689e52cb55cf493"
uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
version = "2021.12.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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