Demand Estimation

Paul Schrimpf

Creative Commons Attribution-ShareAlike 4.0 International License

20.7 μs

This notebook will demonstrate some demand estimation methods. It focuses on the random coefficients logit model of demand for differentiated products of Berry, Levinsohn, & Pakes (1995).

9.6 μs

Simulation

Preferences

There are $J$ products with $K$ characteristics. Consumer $i$ 's utility from product $j$ in market $t$ is:

$u_{i j t} = x_{j t}^{'} β + x_{j t}^{'} Σ ν_{i t} + ξ_{j t} + ϵ_{i j t}$

where $x_{j t}$ is a vector product characteristics, $ν_{i t}$ is a vector of unobserved idiosyncratic tastes for characteristics, $ξ_{j t}$ is an unobserved product demand shock, and $ϵ_{i j t}$ is an independent Type-I extreme value shock.

Market Shares

The market share of product $j$ in market $t$ is given by an integral over $ν$ of logit probabilities:

$σ_{j} (x_{t}, ξ_{t}) = \int \frac{e^{x_{j t}^{'} β + x_{j t}^{'} Σ ν + ξ_{j t}}}{1 + \sum_{k = 1}^{J} e^{x_{k t}^{'} β + x_{k t}^{'} Σ ν_{i t} + ξ_{k t}}} d F_{ν} (ν)$

8.8 μs

Profit Maximization

We will assume that firm's choose price, which will be the first element of $x$ , ( $x_{j t} [1]$ ), while other product characteristics are fixed. Firms know the characteristics of all products and the whole vector $ξ_{t}$ when choosing $x_{j t} [1]$ . Firms' profit maximization problem is:

$max_{p} p M_{t} σ_{j} (x_{t} (p), ξ_{t}) - C_{j} (M_{t} σ_{j} (x_{t} (p), ξ_{t}))$

where $x_{t} (p)$ denotes $x_{t}$ with $x_{j t} [1]$ set to $p$ and all other elements held fixed, $C_{j}$ is the cost function of the firm, and $M_{t}$ is the market size.

The firm's first order condition is

$σ_{j} (x_{t} (p), ξ_{t}) + p \frac{\partial σ}{\partial p} (x_{t} (p), ξ_{t}) = C_{j}^{'} (M_{t} σ_{j} (x_{t} (p), ξ_{t})) \frac{\partial σ}{\partial p} (x_{t} (p), ξ_{t})$

Note that rearranging this equation gives the familiar price equals marginal cost plus markup formula:

$p_{j} - C_{j}^{'} = - \frac{σ_{j}}{\partial σ_{j} / \partial p_{j}}$

This is a markup since $\partial σ_{j} / \partial p_{j}$ is generally negative.

Marginal Costs

We will assume that marginal costs are log linear:

$C_{j}^{'} (M s) = \exp (w_{j t}^{'} γ + ω_{j t})$

where $w_{j t}$ are observed firm or product characteristics and $ω_{j t}$ is an unobserved cost shock.

Equilibrium

In equilibrium, each firm's price much satisfy the above first order condition given the other $J - 1$ prices. Thus, equilibrium prices satisfy as system of $J$ nonlinear equations.

15.3 μs

Simulating Data

We will use the BLPDemand.jl package for computing and simulating the equilibrium. See the docs for that package for more information.

In the simulation, exogenous product characteristics and cost shifters are $U (0, 1)$ . $ξ$ and $ω$ are normally distributed.

To ensure that finite equilibrium prices exist, it is important that all consumers dislike higher prices, i.e. $β [1] + ν [i, 1] * σ [1] < 0$ for all $i$ . Otherwise, the firm could charge an infinite price and just sell to the price loving consumers. Therefore, the first component of $ν$ (the random coefficient on price) is $- U (0, 1)$ , and the other components of $ν$ are normally distributed.

10.9 μs

To see the source for functions in BLPDemand.jl or any other package, you can use the CodeTracking package.

3.5 μs

xxxxxxxxxx
 
using Revise, CodeTracking, BLPDemand, PlutoUI

2.4 s

Enter cell code...xxxxxxxxxx
 
​

---

function simulateBLP(J, T, β::AbstractVector, σ::AbstractVector, γ::AbstractVector, S;
                     varξ=1, varω=1, randj=1.0, firmid=1:J, costf=:log)
  
  K = length(β)
  L = length(γ)
  dat = Array{MarketData,1}(undef, T)
  Ξ = Array{Vector,1}(undef,T)
  Ω = Array{Vector,1}(undef,T)
  for t in 1:T
    Jt, fid = if randj < 1.0
      inc = falses(J)
      while sum(inc)==0
        inc .= rand(J) .< randj
      end
      sum(inc), firmid[inc]
    else
      J, firmid
    end
    x = rand(K, Jt)
    ξ = randn(Jt)*varξ
    w = rand(L, Jt)
    ν = randn(K, S)
    ν[1,:] .= -rand(S) # make sure individuals' price coefficients are negative
    ω = randn(Jt)*varω

    c = costf == :log ? exp.(w'*γ + ω) : w'*γ + ω
    p = eqprices(c, β, σ, ξ, x[2:end,:], ν, firmid=fid)

    x[1,:] .= p
    s = share(x'*β+ξ, σ, x, ν)
    z = makeivblp(cat(x[2:end,:],w, dims=1), firmid=fid,
                  forceown=(length(firmid)!=length(unique(firmid))))
    dat[t] = MarketData(s, x, w, fid, z, z, ν)
    Ξ[t] = ξ
    Ω[t] = ω
  end

  return(dat=dat, ξ=Ξ, ω=Ω)
  
end

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let
    # get the function definition
    str = @code_string simulateBLP(J, T, β, σ, γ, S)
​
    # for nicer display in Pluto
    str = "```julia\n"*str*"\n```\n"
    Markdown.parse(str)
end

13.5 s

The hardest part of simulating the model is solving for equilibrium prices. This is done by eqprices function, which uses the approach of Morrow and Skerlos (2011).

4.8 μs

function eqprices(mc::AbstractVector,
                  β::AbstractVector, σ::AbstractVector,
                  ξ::AbstractVector,
                  x::AbstractMatrix, ν::AbstractMatrix;
                  firmid= 1:length(mc),
                  tol=sqrt(eps(eltype(mc))),
                  maxiter=10000, verbose=false)

  iter = 0
  dp = 10*tol
  focnorm = 10*tol
  p = mc*1.1
  pold = copy(p)
  samefirm = firmid.==firmid'  
  while (iter < maxiter) && ((dp > tol) || (focnorm > tol))
    s, ds, Λ, Γ = dsharedp(β, σ, p, x, ν, ξ)    
    ζ = inv(Λ)*(samefirm.*Γ)*(p - mc) - inv(Λ)*s
    focnorm = norm(Λ*(p-mc - ζ))
    pold, p = p, pold
    p .= mc .+ ζ
    dp = norm(p-pold)
    if verbose && (iter % 100 == 0)
      @show iter, p, focnorm
    end
    iter += 1    
  end
  if verbose
    @show iter, p, focnorm
  end
  return(p)  
end

29.0 ms

We use the following parameters for simulating the data.

3.6 μs

xxxxxxxxxx
 
begin
    J = 5 # number of products
    T = 100 # number of markets
    β = [-0.1, # price coefficients
        ones(2)...] # other product coefficients
    σ = [0.5, ones(length(β)-1)...] # Σ = Diagonal(σ)
    γ = ones(1) # marginal cost coefficients
    S = 10 # number of Monte-Carlo draws per market and product to integrate out ν
    sξ = 0.2 # standard deviation of demand shocks
    sω = 0.2 # standard deviation of cost shocks
end;

4.0 μs

xxxxxxxxxx
 
sim = simulateBLP(J, T, β, σ, γ, S, varξ=sξ, varω=sω);

2.6 s

NamedTuple{(:dat, :ξ, :ω),Tuple{Array{MarketData,1},Array{Array{T,1} where T,1},Array{Array{T,1} where T,1}}}

xxxxxxxxxx
 
typeof(sim)

69.0 ns

We can see that simulateBLP returns a named tuple. The first element of the tuple is an array of MarketData which represents data that we would observe. The next two are the unobserved demand and cost shocks.

MarketData is a struct defined in BLPDemand.jl.

Let's look at some summary statistics and figures describing the simulated data.

6.0 μs

xxxxxxxxxx
 
using Statistics, DataFrames

954 ms

		min	max	mean	std dev
	String	Real	Real	Float64	Float64
1	"Number of Products"	5	5	5.0	0.0
2	"Shares"	0.0274241	0.312975	0.132071	0.0505714
3	"Prices"	3.39449	9.52479	5.4538	0.88458
4	"Other Characteristics"	0.00183849	0.998827	0.509104	0.282729
5	"Cost Shifters"	0.0096932	0.999206	0.490112	0.281802
6	"Demand Instruments"	0.00183849	3.9834	1.99203	1.27632
7	"Supply Instruments"	0.00183849	3.9834	1.99203	1.27632

xxxxxxxxxx
 
begin
    function describe(d::Array{MarketData})
        T = length(d)
        funcs = [minimum, maximum, mean, std]
        cnames = ["min","max","mean","std dev"]
        rnames = ["Number of Products", "Shares", "Prices",
                "Other Characteristics",
                "Cost Shifters",
                "Demand Instruments",
                "Supply Instruments"]
        trans = [x->length(x.s),
                x->x.s,
                x->x.x[1,:],
                x->x.x[2:end,:],
                x->x.w,
                x->x.zd,
                x->x.zs]
        df = DataFrame([[f(vcat(t.(d)...)) for t in trans] for f in funcs])
        rename!(df, cnames)
        insertcols!(df, 1,Symbol(" ") => rnames)
        df
    end
    describe(sim.dat)
end

2.4 s

xxxxxxxxxx
 
using Plots

10.9 s

xxxxxxxxxx
 
let
    Plots.gr(fmt="png")
    j = 1
    k =1
    fig=scatter(vcat( (x->x.x[1,j]).(sim.dat)...),
        vcat( (x->x.s[k]).(sim.dat)...),
         ylabel="Share of Product $k", xaxis=:level)
    plot!(fig, xlabel="Price of Product $j")
    fig
end

8.2 s

xxxxxxxxxx
 
let
    j = 1
    k =1
    scatter(vcat( (x->x[j]).(sim.ξ)...),
        vcat( (x->x.s[k]).(sim.dat)...),
        xlabel="ξ[$j]",
        ylabel="Share of Product $k")
end

105 ms

Estimation

It is conventional to group together all product and market specific terms entering consumers' utility as

$δ_{j t} = x_{j t}^{'} β + ξ_{j t}$

Let $δ_{t}$ denote the vector $J$ values of $δ_{j t}$ in market $t$ .

Define the share as a function of $δ_{t}$ as

$σ_{j t} (δ_{t}; θ) = \int \frac{e^{δ_{j} t + x_{j t}^{'} Σ ν}}{1 + \sum_{k = 1}^{J} e^{δ_{k t} + x_{k t}^{'} Σ ν_{i t}}} d F_{ν} (ν)$

where $θ$ denotes all the parameters to be estimated.

Let $σ_{t} (δ_{t}; θ)$ denote the vector of $J$ shares.

Define the inverse of this function as $Δ_{t} (s_{t}; θ) = d$ where $d$ satisfies $σ_{t} (d; θ) = s_{t}$ .

The demand-side moment condition for this model can then be written as

$E [(\underset{ξ_{j} t}{\underset{⏟}{Δ_{j t} (s_{t}; θ) - x_{j t} β}}) z_{j t}^{D}] = 0$

Using the log-linear marginal specification, we can rearrange the first order condition above to isolate the cost shock:

$ω_{j t} = \log (p_{j t} + \frac{σ_{j t} (Δ_{t} (s_{t}; θ); θ)}{\frac{\partial σ_{j t}}{\partial p_{j}} (Δ_{t} (s_{t}; θ); θ)}) - w_{j t}^{'} γ$

and the supply side moments can be written as:

$E [\underset{ω_{j t}}{\underset{⏟}{(\log (p_{j t} + \frac{σ_{j t} (Δ_{t} (s_{t}; θ); θ)}{\frac{\partial σ_{j t}}{\partial p_{j}} (Δ_{t} (s_{t}; θ); θ)}) - w_{j t}^{'} γ)}} z_{j t}^{S}] = 0$

To more succintly denote the moments, define:

$g_{j t} (δ, θ) = (\begin{matrix} (δ_{j t} - x_{j t} β) z_{j t}^{D} \\ (\log (p_{j t} + \frac{σ_{j t} (δ_{t}; θ)}{\frac{\partial σ_{j t}}{\partial p_{j}} (δ_{t}; θ)}) - w_{j t}^{'} γ) z_{j t}^{S} \end{matrix})$

Then the combined moments conditions can be written as

$E [g_{j t} (Δ (s_{t}; θ), θ)] = 0$

To esitmate $θ$ we minimize a quadratic form in the empirical moments:

$\hat{θ} \in a r g min_{θ} {(\frac{1}{J T} \sum_{j, t} g_{j t} (Δ (s_{t}; θ), θ))}^{'} W (\frac{1}{J T} \sum_{j, t} g_{j t} (Δ (s_{t}; θ), θ))$

11.8 μs

82.0 ns

Nested Fixed Point

The original BLP paper estimated $θ$ using a "nested fixed point" (NFXP) approach. The function $Δ (s; θ)$ has no closed form expression. $Δ (s; θ)$ is the solution to $σ (Δ (s; θ); θ) = s$ . This equation can only be solved numerically with some sort of iterative algorithm. BLP rearrange the equation to write $δ$ as fixed point the point of a contraction mapping, $Δ (s; θ) = F (Δ (s; θ), θ, s)$ and then compute $Δ$ iteratively:

Choose $d_{0}$ arbitrarily
Define $d_{i} = F (d_{i + 1}, θ, s)$ for $i = 1, 2, . . .$
Continue until $∥ d_{i} - d_{i - 1} ∥$ is small enough
Use the final $d_{i}$ as $Δ (s, θ)$

This simple repeated application of a contraction mapping is robust, but not neccessarily the most efficient way to compute $Δ$ . Other methods can compute $Δ$ more quickly, but can be less stable. All methods involve some of iteration until convergence. The function delta in BLPDemand.jl computes $Δ$ .

The "nested" part of nested fixed point refers to the fact that minimizing the GMM objective function also involves some sort of iterative minimization algorithm. Each iteration of the minimization algorithm, we redo all the iterations needed to compute $Δ (s, θ)$ .

2.6 ms

Automatic Differentiation

We need to compute deriviatives of many of the functions above. $\frac{\partial σ}{\partial p}$ is part of the moment conditions. Minimization algorithms can be much more efficient if we provide the gradient and/or hessian of the objective function. Finally, the asymptotic variance of $θ$ involves the Jacobian of the moment conditions (i.e. $D_{θ} g_{j t} (Δ (s_{t}; θ), θ)$ ). While we could calculate these derivatives by hand and then write code to compute them, there is a less laborious and error-prone way. Automatic differentiation refers to taking code that computes some function, $f (x)$ and algorithmically applying the chain rule to compute $\frac{d f}{d x}$ . One of Julia's nicest features is how well it supports automatic differentiation. The two leading Julia packages for automatic differentiation are ForwardDiff.jl and Zygote.jl. ForwardDiff.jl is usually a good choice for function from $R^{n} \to R^{k}$ with $n$ not too much larger than $k$ . Zygote.jl is somewhat more restrictive in what code it is compatible with, but is more efficient if $n$ is much much larger than $k$ . You can read more in the Quantecon notes on automatic differentiation.

2.9 μs

Iter     Function value   Gradient norm 
     0     1.158102e+04     8.015509e+05
 * time: 0.009341955184936523
     1     1.924357e+02     1.126686e+05
 * time: 1.9586098194122314
     2     7.571640e+00     3.806780e+02
 * time: 1.974524974822998
     3     7.569459e+00     1.405256e+01
 * time: 1.998633861541748
     4     4.394585e+00     1.448938e+03
 * time: 2.0837879180908203
     5     4.364341e+00     3.530266e+01
 * time: 2.0992259979248047
     6     4.364101e+00     3.047149e+00
 * time: 2.1224489212036133
     7     4.363403e+00     2.152880e+02
 * time: 2.1677727699279785
     8     4.169330e+00     6.884843e+02
 * time: 2.213343858718872
     9     4.108017e+00     1.350131e+03
 * time: 2.2287039756774902
    10     4.038903e+00     4.962200e+02
 * time: 2.2513558864593506
    11     4.032905e+00     5.101109e+01
 * time: 2.2665657997131348
    12     4.031522e+00     3.735515e+01
 * time: 2.2816929817199707
    13     4.031358e+00     4.919446e+00
 * time: 2.3042078018188477
    14     4.031356e+00     6.030557e+00
 * time: 2.3707997798919678
    15     4.031171e+00     7.907360e+00
 * time: 2.4144129753112793
    16     3.944447e+00     8.525036e+02
 * time: 2.458922863006592
    17     3.911783e+00     1.384805e+03
 * time: 2.489116907119751
    18     3.855410e+00     2.188737e+03
 * time: 2.504194974899292
    19     3.597449e+00     4.505988e+03
 * time: 2.5271029472351074
    20     3.345880e+00     3.317219e+03
 * time: 2.588615894317627
    21     3.304821e+00     2.178381e+03
 * time: 2.603888988494873
    22     3.235112e+00     6.233731e+00
 * time: 3.4163057804107666
    23     3.235005e+00     8.420931e+01
 * time: 3.4542877674102783
    24     1.595482e+00     1.829725e+03
 * time: 3.515472888946533
    25     1.487061e+00     5.183353e+02
 * time: 3.5388197898864746
    26     1.481471e+00     5.661871e+00
 * time: 3.554438829421997
    27     1.481462e+00     5.655291e+00
 * time: 3.5779268741607666
    28     6.369042e-01     2.907135e+02
 * time: 3.639993906021118
    29     4.882034e-01     9.423588e+02
 * time: 3.6556828022003174
    30     4.722599e-01     2.181082e+02
 * time: 3.6712148189544678
    31     4.692708e-01     1.569599e+01
 * time: 3.68691086769104
    32     4.692615e-01     1.957743e+00
 * time: 3.7025837898254395
    33     4.690967e-01     8.172367e+01
 * time: 3.795501947402954
    34     4.170150e-01     4.707579e+02
 * time: 3.842383861541748
    35     3.874091e-01     8.745371e+02
 * time: 3.858219861984253
    36     3.660467e-01     1.680291e+02
 * time: 3.8745369911193848
    37     3.565735e-01     5.990861e+02
 * time: 3.8904759883880615
    38     3.393281e-01     6.071597e+02
 * time: 3.914761781692505
    39     3.361840e-01     5.630346e+02
 * time: 3.930722951889038
    40     3.315947e-01     3.883355e+01
 * time: 3.9465367794036865
    41     3.270270e-01     1.243660e+02
 * time: 3.9702279567718506
    42     3.260627e-01     7.921322e+01
 * time: 3.993623971939087
    43     3.259680e-01     9.265922e-01
 * time: 4.009253978729248
    44     3.259491e-01     6.831564e+00
 * time: 4.032655954360962
    45     3.257824e-01     1.187813e+01
 * time: 4.071793794631958
    46     3.239176e-01     5.372098e+01
 * time: 4.103302001953125
    47     3.233085e-01     9.303561e+01
 * time: 4.134528875350952
    48     3.218731e-01     1.392104e+02
 * time: 4.150173902511597
    49     3.202316e-01     2.240442e+02
 * time: 4.221021890640259
    50     3.199122e-01     9.829724e+01
 * time: 4.236719846725464
    51     3.189274e-01     1.641617e+02
 * time: 4.259667873382568
    52     3.178248e-01     2.557791e+02
 * time: 4.274986982345581
    53     3.153498e-01     1.790588e+02
 * time: 4.298233985900879
    54     3.137514e-01     7.463333e+01
 * time: 4.321682929992676
    55     3.132983e-01     1.441394e+02
 * time: 4.337328910827637
    56     3.117996e-01     1.723170e+02
 * time: 4.353243827819824
    57     3.100110e-01     1.656003e+02
 * time: 4.376643896102905
    58     3.097470e-01     2.063646e+02
 * time: 4.392545938491821
    59     3.087138e-01     3.113353e+01
 * time: 4.408326864242554
    60     3.077516e-01     5.728675e+01
 * time: 4.423904895782471
    61     3.074488e-01     8.128606e+01
 * time: 4.439549922943115
    62     3.065080e-01     1.423380e+02
 * time: 4.455223798751831
    63     3.055215e-01     2.248325e+02
 * time: 4.470853805541992
    64     3.051409e-01     9.599818e+01
 * time: 4.486663818359375
    65     3.042253e-01     1.544814e+02
 * time: 4.509979963302612
    66     3.031998e-01     2.342677e+02
 * time: 4.5255959033966064
    67     3.011996e-01     6.464122e+01
 * time: 4.548874855041504
    68     3.007170e-01     1.480817e+02
 * time: 4.564445972442627
    69     2.992811e-01     2.081977e+02
 * time: 4.579939842224121
    70     2.986292e-01     1.651863e+01
 * time: 4.595477819442749
    71     2.975897e-01     1.756935e+02
 * time: 4.654794931411743
    72     2.966141e-01     1.860799e+02
 * time: 4.678025960922241
    73     2.961896e-01     2.527389e+01
 * time: 4.693364858627319
    74     2.953896e-01     1.213568e+02
 * time: 4.716164827346802
    75     2.949674e-01     1.823298e+02
 * time: 4.7317869663238525
    76     2.942106e-01     5.112320e+01
 * time: 4.755225896835327
    77     2.938091e-01     3.383827e+01
 * time: 4.770978927612305
    78     2.933637e-01     1.121973e+02
 * time: 4.78657078742981
    79     2.930515e-01     5.772480e+01
 * time: 4.802248001098633
    80     2.927370e-01     1.379276e+02
 * time: 4.825741767883301
    81     2.919313e-01     1.604749e+02
 * time: 4.849093914031982
    82     2.909408e-01     8.290502e+01
 * time: 4.872331857681274
    83     2.906202e-01     2.313698e+01
 * time: 4.8879358768463135
    84     2.904082e-01     2.540116e+01
 * time: 4.903480768203735
    85     2.902697e-01     7.055657e+01
 * time: 4.918900966644287
    86     2.902208e-01     2.897104e+01
 * time: 4.942051887512207
    87     2.899351e-01     7.757287e+01
 * time: 4.972864866256714
    88     2.891741e-01     1.095429e+02
 * time: 4.99612283706665
    89     2.873303e-01     1.093396e+02
 * time: 5.019243001937866
    90     2.871501e-01     8.660222e+01
 * time: 5.561618804931641
    91     2.832469e-01     1.004631e+02
 * time: 5.633150815963745
    92     2.828676e-01     1.803296e+02
 * time: 5.655808925628662
    93     2.801536e-01     9.969537e+01
 * time: 5.678155899047852
    94     2.793153e-01     1.709469e+02
 * time: 5.701012849807739
    95     2.774047e-01     3.129373e+01
 * time: 5.724222898483276
    96     2.772003e-01     3.137401e+01
 * time: 5.739959001541138
    97     2.766189e-01     4.000743e+01
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   333     2.413364e-01     7.422739e-05
 * time: 11.388206958770752
   334     2.413364e-01     2.016650e-05
 * time: 11.403850793838501
   335     2.413364e-01     3.898515e-05
 * time: 11.465949773788452
   336     2.413364e-01     5.813111e-05
 * time: 11.496924877166748
   337     2.413364e-01     5.737829e-05
 * time: 11.512472867965698
   338     2.413364e-01     2.258345e-05
 * time: 11.527966976165771
   339     2.413364e-01     6.517463e-05
 * time: 11.558870792388916
   340     2.413364e-01     1.174185e-05
 * time: 11.582137823104858
   341     2.413364e-01     8.400931e-06
 * time: 11.60533094406128
   342     2.413364e-01     7.217852e-06
 * time: 11.62084698677063
   343     2.413364e-01     1.826584e-05
 * time: 12.1454598903656
   344     2.413364e-01     1.444287e-05
 * time: 12.168283939361572
   345     2.413364e-01     1.275383e-05
 * time: 12.183957815170288
   346     2.413364e-01     1.463366e-03
 * time: 12.67297077178955
   347     2.413364e-01     6.977897e-04
 * time: 12.696410894393921
   348     2.413364e-01     2.884721e-05
 * time: 12.720035791397095
   349     2.413364e-01     5.512581e-04
 * time: 12.758677959442139
   350     2.413364e-01     8.555417e-07
 * time: 12.781937837600708
   351     2.413364e-01     7.524316e-05
 * time: 12.828342914581299
   352     2.413364e-01     5.337661e-05
 * time: 12.843871831893921
   353     2.413364e-01     2.534011e-05
 * time: 12.905558824539185
   354     2.413364e-01     4.205067e-06
 * time: 12.97214388847351
   355     2.413364e-01     3.500206e-07
 * time: 12.987884998321533

xxxxxxxxxx
 
begin
    using Optim, LineSearches
    PlutoUI.with_terminal() do
        nfxp = estimateBLP(sim.dat, method=:NFXP, verbose=true,
                           optimizer = LBFGS(linesearch=LineSearches.HagerZhang()))
    end
end

13.9 s

MPEC

MPEC is an alternative approach to computing estimates promoted by Judd and Su and applied to this model by Dube, Fox, and Su (2012). MPEC stands for "Mathematical Programming with Equilibrium Constraints." It is based on the observation that the estimation problem can be expressed as a constrained minimization problem

$(\hat{θ}, \hat{δ}) = a r g min_{θ, δ} {(\frac{1}{J T} \sum_{j, t} g_{j t} (δ_{t}, θ))}^{'} W (\frac{1}{J T} \sum_{j, t} g_{j t} (δ_{t}, θ)) s.t. s_{t} = σ_{t} (δ_{t}; θ) \forall t$

Note that this problem no longer requires compute $Δ (s, θ)$ , so there is no more nesting of iterative computations. However, there are some tradeoffs. We now have a constrained minimization problem, which are slightly more complicated to solve than unconstrained problems. More importantly, the minimization problem is now over a much higher dimensional space ( $θ$ and $δ$ instead of just $θ$ ). Nonetheless, the MPEC formulation of the problem can sometimes lead to more efficient and/or more robust computation. The relative performance of MPEC and NFXP is very dependent on the implementation details (the minization algorithms, the way derivatives are calculated, and whether any sparsity patterns are taken advantage of).

6.2 μs

This is Ipopt version 3.13.2, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:   155509
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:    59009

Total number of variables............................:    17516
                     variables with only lower bounds:     5003
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:    17509
Total number of inequality constraints...............:        0
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        0

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  0.0000000e+00 5.38e+00 2.59e-03  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  2.4902150e+01 9.87e+00 1.94e+02  -1.0 3.70e+00   0.0 5.96e-01 1.00e+00h  1
   2  7.0778252e+00 3.72e+00 2.65e+02  -1.0 2.65e+00    -  8.01e-01 1.00e+00f  1
   3  5.7238522e+00 1.80e+00 8.38e+01  -1.0 2.45e+00    -  1.00e+00 6.89e-01h  1
   4  4.9730437e+00 1.65e+00 8.03e+01  -1.0 4.78e+00    -  1.00e+00 8.36e-02h  1
   5  4.4563632e+00 2.71e-01 3.80e+01  -1.0 1.01e+00    -  1.00e+00 1.00e+00f  1
   6  4.1119628e+00 4.64e-02 8.11e+01  -1.0 3.97e-01    -  4.59e-01 1.00e+00h  1
   7  4.0797294e+00 2.28e-03 5.14e-02  -1.0 1.03e-01  -0.5 1.00e+00 1.00e+00h  1
   8  4.0300757e+00 7.91e-04 6.33e-01  -2.5 5.11e-02  -1.0 8.51e-01 1.00e+00h  1
   9  3.3007868e+00 2.49e-01 2.86e-01  -2.5 8.45e-01    -  7.22e-01 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  2.0409823e+00 8.57e-01 7.93e-01  -2.5 2.12e+01    -  9.03e-02 8.05e-02f  1
  11  1.3077361e+00 8.14e-01 2.40e+00  -2.5 5.54e+00    -  6.76e-01 3.03e-01h  1
  12  2.7037103e-01 4.10e+00 7.95e-01  -2.5 6.87e+00    -  7.13e-01 8.33e-01h  1
  13  1.0600200e+00 4.68e+00 5.89e+01  -2.5 1.33e+01    -  8.28e-02 1.00e+00h  1
  14  7.7575740e-01 2.59e-01 1.47e+00  -2.5 7.21e-01  -1.4 5.19e-01 1.00e+00h  1
  15  6.9478453e-01 2.72e-02 6.32e-02  -2.5 9.22e-01  -1.9 1.00e+00 1.00e+00h  1
  16  5.2893994e-01 1.23e-01 2.86e-02  -2.5 2.44e+00  -2.4 1.00e+00 1.00e+00h  1
  17  3.3163234e-01 3.80e-01 8.92e-02  -2.5 4.07e+00  -2.9 1.00e+00 1.00e+00h  1
  18  2.7053490e-01 3.12e-01 1.44e-01  -2.5 3.07e+00  -3.3 4.63e-01 1.00e+00h  1
  19  2.5994722e-01 5.40e-02 1.27e-02  -2.5 1.80e+00  -2.9 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  20  2.3970933e-01 4.86e-01 1.44e-01  -2.5 8.91e+00  -3.4 6.86e-01 1.00e+00H  1
  21  2.3101273e-01 5.22e-02 1.70e-01  -2.5 1.29e+02  -3.9 1.93e-01 4.11e-02h  3
  22  2.1548498e-01 8.06e-01 5.38e-01  -2.5 2.21e+01  -3.4 1.00e+00 1.00e+00H  1
  23  2.8727140e-01 6.87e-01 7.96e-01  -2.5 9.06e+01  -3.9 1.88e-01 1.13e-01h  1
  24  1.3111627e-01 1.98e+01 4.67e+00  -2.5 4.99e+01    -  1.00e+00 9.95e-01h  1
  25  3.8830274e-01 1.38e+01 3.16e+01  -2.5 5.07e+01    -  6.15e-02 3.22e-01h  2
  26  6.5921831e-01 1.06e+01 1.22e+02  -2.5 4.72e+01  -4.4 1.53e-01 1.00e+00H  1
  27  1.3625812e+00 3.47e+00 8.93e+01  -2.5 1.35e+01    -  5.42e-01 1.00e+00h  1
  28  1.6239477e+00 1.66e-01 7.13e+00  -2.5 1.56e+00  -1.3 1.00e+00 1.00e+00h  1
  29  1.5611908e+00 2.64e-02 2.90e+00  -2.5 2.16e+00  -1.7 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  30  1.0878528e+00 1.78e+00 2.26e+01  -2.5 2.64e+01    -  1.00e+00 5.00e-01f  2
  31  5.9471167e-01 5.47e+00 8.52e+00  -2.5 4.95e+01    -  1.00e+00 4.93e-01h  1
  32  3.9199789e-01 1.99e+00 4.89e+00  -2.5 2.44e+01    -  1.00e+00 1.00e+00h  1
  33  1.9504844e-01 4.58e+00 3.63e+00  -2.5 6.01e+01    -  8.57e-01 4.76e-01h  1
  34  9.9082472e-02 6.70e+00 3.99e+00  -2.5 4.70e+01    -  1.00e+00 1.00e+00h  1
  35  1.1646569e-01 4.58e-02 1.27e+01  -2.5 2.43e+00    -  8.25e-02 1.00e+00h  1
  36  1.1728633e-01 5.55e-01 2.58e-01  -2.5 1.48e+01    -  1.00e+00 1.00e+00h  1
  37  1.3111244e-01 1.85e-01 1.29e-01  -2.5 7.65e+00    -  1.00e+00 1.00e+00h  1
  38  1.3322783e-01 7.96e-04 1.03e-03  -2.5 3.36e-01    -  1.00e+00 1.00e+00h  1
  39  6.3105600e-02 8.04e-01 1.22e+00  -3.8 8.68e+00    -  8.45e-01 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  40  5.8024469e-02 3.67e-01 2.18e-01  -3.8 1.08e+01    -  1.00e+00 1.00e+00h  1
  41  5.8878589e-02 6.15e-03 7.51e-03  -3.8 1.00e+00    -  1.00e+00 1.00e+00h  1
  42  5.8895170e-02 3.77e-07 4.33e-07  -3.8 1.68e-02    -  1.00e+00 1.00e+00h  1
  43  5.7983183e-02 2.97e-02 3.38e-02  -5.7 3.00e+00    -  9.77e-01 1.00e+00h  1
  44  5.8172630e-02 6.32e-04 1.01e-04  -5.7 4.02e-01    -  1.00e+00 1.00e+00h  1
  45  5.8173082e-02 1.17e-08 4.42e-09  -5.7 1.22e-03    -  1.00e+00 1.00e+00h  1
  46  5.8172911e-02 7.84e-06 6.78e-06  -8.6 4.80e-02    -  1.00e+00 1.00e+00h  1
  47  5.8172953e-02 2.90e-11 4.83e-12  -8.6 8.44e-05    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 47

                                   (scaled)                 (unscaled)
Objective...............:   5.8172953363477609e-02    5.8172953363477609e-02
Dual infeasibility......:   4.8316906031686813e-12    4.8316906031686813e-12
Constraint violation....:   2.8961721909581684e-11    2.8961721909581684e-11
Complementarity.........:   2.5059049512274208e-09    2.5059049512274208e-09
Overall NLP error.......:   2.5059049512274208e-09    2.5059049512274208e-09


Number of objective function evaluations             = 62
Number of objective gradient evaluations             = 48
Number of equality constraint evaluations            = 62
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 48
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 47
Total CPU secs in IPOPT (w/o function evaluations)   =     20.889
Total CPU secs in NLP function evaluations           =      4.698

EXIT: Optimal Solution Found.

x
 
PlutoUI.with_terminal() do
    mpec = estimateBLP(sim.dat, method=:MPEC, verbose=true)
end

6.8 s

83.0 ns