notebooks/LinearModel.ipynb

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    "# Example MSM.jl\n",
    "\n",
    "This notebook shows how to use MSM. Our goal is to estimate the linear model using the Method of Simulated Moments (MSM). We observe a dependent variable $y_{t}$ and a set of independent variables $\\boldsymbol{x_{1}}$, ..., $\\boldsymbol{x_{t}}$, ... $\\boldsymbol{x_{T}}$ and we want to estimate the intercept $\\alpha$ and the slope coefficient $\\boldsymbol{\\beta}$ :\n",
    "\n",
    "$$ y_{t} = \\alpha + \\boldsymbol{x_{t}}\\boldsymbol{{\\beta}} + e_{t} $$\n",
    "\n",
    "where $e_{t}$ are error terms, potentially serially correlated. \n",
    "\n",
    "Here, it is clearly an overkill to use the MSM because we have a closed-form solution:\n",
    "\n",
    "$$ \\boldsymbol{\\beta_{OLS}} = (X'X)^{-1}X'y $$\n",
    "\n",
    "For the sake of this example, let's pretend we ignore the existence of this formula. By only simulating $y_{t}$, can we recover $\\boldsymbol{{\\beta}}$ and to inference on parameters?\n"
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    "## Step 0: Simulating data\n",
    "\n",
    "Let's fix the value of $\\boldsymbol{\\beta} = \\boldsymbol{\\beta_{0}}$ and simulate a sample according to the linear model described above. Our goal is to recover the true value of $\\boldsymbol{\\beta}$ using moment conditions and simulated data."
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     "text": [
      "True intercept = 0.5662374165061859\n",
      "True coefficient beta0 = [0.5908446386657102, 0.7667970365022592]\n",
      "Serial correlation coefficient theta0 = 0.0\n"
     ]
    }
   ],
   "source": [
    "using Distributions\n",
    "using Random\n",
    "using Plots\n",
    "using LaTeXStrings\n",
    "gr()\n",
    "Random.seed!(1234)  #for replicability reasons\n",
    "T = 100000          #number of periods\n",
    "P = 2               #number of dependent variables\n",
    "beta0 = rand(P)     #choose true coefficients by drawing from a uniform distribution on [0,1]\n",
    "alpha0 = rand(1)[]  #intercept\n",
    "theta0 = 0.0        #coefficient to create serial correlation in the error terms\n",
    "println(\"True intercept = $(alpha0)\")\n",
    "println(\"True coefficient beta0 = $(beta0)\")\n",
    "println(\"Serial correlation coefficient theta0 = $(theta0)\")\n",
    "\n",
    "# Generation of error terms\n",
    "# row = individual dimension\n",
    "# column = time dimension \n",
    "U = zeros(T)\n",
    "d = Normal()\n",
    "U[1] = rand(d, 1)[] #first error term\n",
    "# loop over time periods\n",
    "for t = 2:T\n",
    "    U[t] = rand(d, 1)[] + theta0*U[t-1]\n",
    "end"
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   "source": [
    "# Let's simulate the dependent variables x_t\n",
    "x = zeros(T, P)\n",
    "d = Uniform(0, 5)\n",
    "for p = 1:P  \n",
    "    x[:,p] = rand(d, T)\n",
    "end\n",
    "\n",
    "# Let's calculate the resulting y_t\n",
    "y = zeros(T)\n",
    "for t=1:T\n",
    "    y[t] = alpha0 + x[t,1]*beta0[1] + x[t,2]*beta0[2] + U[t]\n",
    "end"
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       "  350.776,252.407 581.274,95.9539 \n",
       "  \"/>\n",
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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = scatter(x[1:100,1], y[1:100], xlabel = \"x1\", ylabel = \"y\", legend=:none, smooth=true)\n",
    "p2 = scatter(x[1:100,2], y[1:100], xlabel = \"x2\", ylabel = \"y\", legend=:none, smooth=true)\n",
    "p = plot(p1, p2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "868f51ef",
   "metadata": {},
   "source": [
    "## Step 1: Initializing a MSMProblem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c3639331",
   "metadata": {},
   "outputs": [],
   "source": [
    "using MSM\n",
    "using DataStructures\n",
    "using OrderedCollections\n",
    "using Random\n",
    "using Distributions\n",
    "using Statistics\n",
    "using LinearAlgebra\n",
    "\n",
    "# Here we set the maximum number of function evaluations to be 10000.\n",
    "# We also select a global optimizer (see BlackBoxOptim.jl) and a local minimizer (see Optim.jl)\n",
    "# https://github.com/robertfeldt/BlackBoxOptim.jl\n",
    "# https://julianlsolvers.github.io/Optim.jl\n",
    "myProblem = MSMProblem(options = MSMOptions(maxFuncEvals=10000, globalOptimizer = :dxnes, localOptimizer = :LBFGS));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7474d727",
   "metadata": {},
   "source": [
    "## Step 2. Set empirical moments and weight matrix\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "44fc2cc5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       " 0.063999  0.0         0.0         0.0         0.0\n",
       " 0.0       0.00817344  0.0         0.0         0.0\n",
       " 0.0       0.0         0.00760982  0.0         0.0\n",
       " 0.0       0.0         0.0         2.41806e-5  0.0\n",
       " 0.0       0.0         0.0         0.0         2.0803e-5"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dictEmpiricalMoments = OrderedDict{String,Array{Float64,1}}()\n",
    "dictEmpiricalMoments[\"mean\"] = [mean(y)] #informative on the intercept\n",
    "dictEmpiricalMoments[\"mean_x1y\"] = [mean(x[:,1] .* y)] #informative on betas\n",
    "dictEmpiricalMoments[\"mean_x2y\"] = [mean(x[:,2] .* y)] #informative on betas\n",
    "dictEmpiricalMoments[\"mean_x1y^2\"] = [mean((x[:,1] .* y).^2)] #informative on betas\n",
    "dictEmpiricalMoments[\"mean_x2y^2\"] = [mean((x[:,2] .* y).^2)] #informative on betas\n",
    "set_empirical_moments!(myProblem, dictEmpiricalMoments)\n",
    "\n",
    "W = Matrix(1.0 .* I(length(dictEmpiricalMoments)))#initialization\n",
    "#Special case: diagonal matrix\n",
    "#Sum of square percentage deviations from empirical moments\n",
    "#(you may choose something else)\n",
    "for (indexMoment, k) in enumerate(keys(dictEmpiricalMoments))\n",
    "    W[indexMoment,indexMoment] = 1.0/(dictEmpiricalMoments[k][1])^2\n",
    "end\n",
    "\n",
    "set_weight_matrix!(myProblem, W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "e6c6411d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedDict{String, Vector{Float64}} with 5 entries:\n",
       "  \"mean\"       => [3.95288]\n",
       "  \"mean_x1y\"   => [11.0611]\n",
       "  \"mean_x2y\"   => [11.4634]\n",
       "  \"mean_x1y^2\" => [203.361]\n",
       "  \"mean_x2y^2\" => [219.249]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "myProblem.empiricalMoments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "5e4630ec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       " 0.063999  0.0         0.0         0.0         0.0\n",
       " 0.0       0.00817344  0.0         0.0         0.0\n",
       " 0.0       0.0         0.00760982  0.0         0.0\n",
       " 0.0       0.0         0.0         2.41806e-5  0.0\n",
       " 0.0       0.0         0.0         0.0         2.0803e-5"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "myProblem.W"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73f0a610",
   "metadata": {},
   "source": [
    "## Step 3. Set priors\n",
    "\n",
    "One has to attach priors values the `MSMProblem` we are going to use.\n",
    "This is achieved using the function  `set_priors!()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "532a3feb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedDict{String, Vector{Float64}} with 3 entries:\n",
       "  \"alpha\" => [0.5, 0.001, 1.0]\n",
       "  \"beta1\" => [0.5, 0.001, 1.0]\n",
       "  \"beta2\" => [0.5, 0.001, 1.0]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dictPriors = OrderedDict{String,Array{Float64,1}}()\n",
    "dictPriors[\"alpha\"] = [0.5, 0.001, 1.0]\n",
    "dictPriors[\"beta1\"] = [0.5, 0.001, 1.0]\n",
    "dictPriors[\"beta2\"] = [0.5, 0.001, 1.0]\n",
    "\n",
    "set_priors!(myProblem, dictPriors)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d68138d0",
   "metadata": {},
   "source": [
    "## Step 4: Specifying the function generating simulated moments using parameter values\n",
    "\n",
    "This is maybe the most crucial step. The objective function must generate an **ordered dictionary** containing the **keys of dictEmpiricalMoments**. Importantly, we \"freeze\" randomness during the minimization step. One way to do \n",
    "that is to generate draws from a Uniform([0,1]) outside of the objective function and to use [inverse transform sampling](https://en.wikipedia.org/wiki/Inverse_transform_sampling) to generate draws from a normal distribution. Otherwise the objective function would be \"noisy\" and the minimization algorithms would have a hard time finding \n",
    "the global minimum. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "bb1f4336",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::MSM.var\"#objective_function_MSM#6\"{MSMProblem}) (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# x[1] corresponds to the intercept\n",
    "# x[1] corresponds to beta1\n",
    "# x[3] corresponds to beta2\n",
    "function functionLinearModel(x; uniform_draws::Array{Float64,1}, simX::Array{Float64,2}, nbDraws::Int64 = length(uniform_draws), burnInPerc::Int64 = 0)\n",
    "    T = nbDraws\n",
    "    P = 2       #number of dependent variables\n",
    "\n",
    "    alpha = x[1]\n",
    "    beta = x[2:end]\n",
    "    theta = 0.0     #coefficient to create serial correlation in the error terms\n",
    "\n",
    "    # Creation of error terms\n",
    "    # row = individual dimension\n",
    "    # column = time dimension\n",
    "    U = zeros(T)\n",
    "    d = Normal()\n",
    "    # Inverse cdf (i.e. quantile)\n",
    "    gaussian_draws = quantile.(d, uniform_draws)\n",
    "    U[1] = gaussian_draws[1] #first error term\n",
    "\n",
    "    # loop over time periods\n",
    "    for t = 2:T\n",
    "        U[t] = gaussian_draws[t] + theta*U[t-1]\n",
    "    end\n",
    "\n",
    "    # Let's calculate the resulting y_t\n",
    "    y = zeros(T)\n",
    "\n",
    "    for t=1:T\n",
    "        y[t] = alpha + simX[t,1]*beta[1] + simX[t,2]*beta[2] + U[t]\n",
    "    end\n",
    "\n",
    "    # Get rid of the burn-in phase:\n",
    "    #------------------------------\n",
    "    startT = max(1, Int(nbDraws * (burnInPerc / 100)))\n",
    "\n",
    "    # Moments:\n",
    "    #---------\n",
    "    output = OrderedDict{String,Float64}()\n",
    "    output[\"mean\"] = mean(y[startT:nbDraws])\n",
    "    output[\"mean_x1y\"] = mean(simX[startT:nbDraws,1] .* y[startT:nbDraws])\n",
    "    output[\"mean_x2y\"] = mean(simX[startT:nbDraws,2] .* y[startT:nbDraws])\n",
    "    output[\"mean_x1y^2\"] = mean((simX[startT:nbDraws,1] .* y[startT:nbDraws]).^2)\n",
    "    output[\"mean_x2y^2\"] = mean((simX[startT:nbDraws,2] .* y[startT:nbDraws]).^2)\n",
    "\n",
    "    return output\n",
    "end\n",
    "\n",
    "# Let's freeze the randomness during the minimization\n",
    "d_Uni = Uniform(0,1)\n",
    "nbDraws = T #number of draws in the simulated data\n",
    "uniform_draws = rand(d_Uni, nbDraws)\n",
    "simX = zeros(length(uniform_draws), 2)\n",
    "d = Uniform(0, 5)\n",
    "for p = 1:2\n",
    "  simX[:,p] = rand(d, length(uniform_draws))\n",
    "end\n",
    "\n",
    "set_simulate_empirical_moments!(myProblem, x -> functionLinearModel(x, uniform_draws = uniform_draws, simX = simX))\n",
    "\n",
    "# Construct the objective function using:\n",
    "#* the function: parameter -> simulated moments\n",
    "#* emprical moments values\n",
    "#* emprical moments weights\n",
    "construct_objective_function!(myProblem)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6eb88ad3",
   "metadata": {},
   "source": [
    "## Step 5. Running the optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "42e954cb",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: 1 worker(s) detected\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hello\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Starting optimization in serial\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting optimization with optimizer BlackBoxOptim.DXNESOpt{Float64, BlackBoxOptim.RandomBound{BlackBoxOptim.ContinuousRectSearchSpace}}\n",
      "0.00 secs, 0 evals, 0 steps\n",
      "σ=1.0 η[x]=1.0 η[σ]=0.0 η[B]=0.0 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "0.51 secs, 40 evals, 5 steps, fitness=0.013412357\n",
      "σ=0.3231388748314356 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "1.01 secs, 88 evals, 11 steps, fitness=0.006538342\n",
      "σ=0.12696951926116984 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "1.51 secs, 224 evals, 28 steps, fitness=0.000254157\n",
      "σ=0.0019318620918446104 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-17 |path|=NaN speed=NaN\n",
      "2.02 secs, 368 evals, 46 steps, fitness=0.000021192\n",
      "σ=0.001273528080705865 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.163336342344337e-17 |path|=NaN speed=NaN\n",
      "2.53 secs, 504 evals, 63 steps, fitness=0.000015145\n",
      "σ=8.171795965231979e-5 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "3.03 secs, 640 evals, 80 steps, fitness=0.000014382\n",
      "σ=6.80857890420365e-5 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-17 |path|=NaN speed=NaN\n",
      "3.56 secs, 776 evals, 97 steps, fitness=0.000013270\n",
      "σ=9.164513252847426e-5 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.7755575615628914e-17 |path|=NaN speed=NaN\n",
      "4.06 secs, 904 evals, 113 steps, fitness=0.000011755\n",
      "σ=0.0003199043160393348 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "4.57 secs, 1048 evals, 131 steps, fitness=0.000008738\n",
      "σ=0.0001802633676075704 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "5.09 secs, 1184 evals, 148 steps, fitness=0.000005343\n",
      "σ=0.00030823085386238726 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=3.3306690738754696e-16 |path|=NaN speed=NaN\n",
      "5.61 secs, 1320 evals, 165 steps, fitness=0.000004574\n",
      "σ=4.7263330241336614e-5 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "6.12 secs, 1464 evals, 183 steps, fitness=0.000004571\n",
      "σ=3.112295174107389e-7 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "6.64 secs, 1600 evals, 200 steps, fitness=0.000004571\n",
      "σ=1.5798855538744493e-9 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=6.661338147750939e-16 |path|=NaN speed=NaN\n",
      "7.15 secs, 1736 evals, 217 steps, fitness=0.000004571\n",
      "σ=3.4458196599372467e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "7.66 secs, 1872 evals, 234 steps, fitness=0.000004571\n",
      "σ=3.3369958129119496e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n",
      "8.16 secs, 2000 evals, 250 steps, fitness=0.000004571\n",
      "σ=9.614505782985999e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "8.69 secs, 2136 evals, 267 steps, fitness=0.000004571\n",
      "σ=7.252628193687001e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "9.22 secs, 2272 evals, 284 steps, fitness=0.000004571\n",
      "σ=3.129231894254055e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "9.73 secs, 2408 evals, 301 steps, fitness=0.000004571\n",
      "σ=3.49112426695687e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "10.26 secs, 2544 evals, 318 steps, fitness=0.000004571\n",
      "σ=1.9979007763580364e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "10.78 secs, 2688 evals, 336 steps, fitness=0.000004571\n",
      "σ=8.096473676311934e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "11.29 secs, 2832 evals, 354 steps, fitness=0.000004571\n",
      "σ=1.5421946344377525e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "11.80 secs, 2976 evals, 372 steps, fitness=0.000004571\n",
      "σ=3.436688324656759e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "12.33 secs, 3112 evals, 389 steps, fitness=0.000004571\n",
      "σ=1.1547785819656451e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n",
      "12.84 secs, 3248 evals, 406 steps, fitness=0.000004571\n",
      "σ=4.175432124825909e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "13.36 secs, 3384 evals, 423 steps, fitness=0.000004571\n",
      "σ=7.77638241440553e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "13.89 secs, 3528 evals, 441 steps, fitness=0.000004571\n",
      "σ=5.676476940947333e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "14.41 secs, 3656 evals, 457 steps, fitness=0.000004571\n",
      "σ=1.9878736925040075e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n",
      "14.93 secs, 3800 evals, 475 steps, fitness=0.000004571\n",
      "σ=5.92975789006064e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "15.46 secs, 3944 evals, 493 steps, fitness=0.000004571\n",
      "σ=3.232881032467251e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n",
      "15.97 secs, 4072 evals, 509 steps, fitness=0.000004571\n",
      "σ=3.2814534406290677e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=8.881784197001252e-16 |path|=NaN speed=NaN\n",
      "16.49 secs, 4208 evals, 526 steps, fitness=0.000004571\n",
      "σ=1.6391356139176742e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=6.661338147750939e-16 |path|=NaN speed=NaN\n",
      "17.02 secs, 4352 evals, 544 steps, fitness=0.000004571\n",
      "σ=4.5162622386838553e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=6.661338147750939e-16 |path|=NaN speed=NaN\n",
      "17.53 secs, 4488 evals, 561 steps, fitness=0.000004571\n",
      "σ=2.661476838960059e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=6.661338147750939e-16 |path|=NaN speed=NaN\n",
      "18.05 secs, 4632 evals, 579 steps, fitness=0.000004571\n",
      "σ=3.0558172084219107e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "18.57 secs, 4768 evals, 596 steps, fitness=0.000004571\n",
      "σ=6.678326166122459e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "19.10 secs, 4912 evals, 614 steps, fitness=0.000004571\n",
      "σ=4.635135943410719e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "19.61 secs, 5048 evals, 631 steps, fitness=0.000004571\n",
      "σ=7.3546015424717e-14 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "20.14 secs, 5192 evals, 649 steps, fitness=0.000004571\n",
      "σ=1.0212017376143658e-14 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "20.66 secs, 5336 evals, 667 steps, fitness=0.000004571\n",
      "σ=8.418838385071591e-15 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "21.17 secs, 5464 evals, 683 steps, fitness=0.000004571\n",
      "σ=8.168378098788119e-15 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=3.3306690738754696e-16 |path|=NaN speed=NaN\n",
      "21.68 secs, 5600 evals, 700 steps, fitness=0.000004571\n",
      "σ=4.125887713689103e-15 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "22.19 secs, 5736 evals, 717 steps, fitness=0.000004571\n",
      "σ=4.859630732537271e-16 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "22.69 secs, 5864 evals, 733 steps, fitness=0.000004571\n",
      "σ=1.322584921197474e-16 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "23.19 secs, 5992 evals, 749 steps, fitness=0.000004571\n",
      "σ=1.249212805155748e-16 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "23.69 secs, 6128 evals, 766 steps, fitness=0.000004571\n",
      "σ=4.937267150625628e-17 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=3.3306690738754696e-16 |path|=NaN speed=NaN\n",
      "24.21 secs, 6272 evals, 784 steps, fitness=0.000004571\n",
      "σ=1.4944672955257812e-17 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.6653345369377348e-16 |path|=NaN speed=NaN\n",
      "24.73 secs, 6416 evals, 802 steps, fitness=0.000004571\n",
      "σ=4.7845281344492785e-18 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1657341758564144e-15 |path|=NaN speed=NaN\n",
      "25.24 secs, 6552 evals, 819 steps, fitness=0.000004571\n",
      "σ=6.7764207063611e-18 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1657341758564144e-15 |path|=NaN speed=NaN\n",
      "25.75 secs, 6688 evals, 836 steps, fitness=0.000004571\n",
      "σ=2.3363106662338516e-18 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.828670879282072e-16 |path|=NaN speed=NaN\n",
      "26.27 secs, 6824 evals, 853 steps, fitness=0.000004571\n",
      "σ=2.387286955841922e-18 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "26.79 secs, 6952 evals, 869 steps, fitness=0.000004571\n",
      "σ=1.3346861775159403e-18 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "27.31 secs, 7080 evals, 885 steps, fitness=0.000004571\n",
      "σ=6.173833083933908e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.7755575615628914e-16 |path|=NaN speed=NaN\n",
      "27.84 secs, 7208 evals, 901 steps, fitness=0.000004571\n",
      "σ=4.366952005821759e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.632783294297951e-16 |path|=NaN speed=NaN\n",
      "28.36 secs, 7344 evals, 918 steps, fitness=0.000004571\n",
      "σ=3.4434801875401033e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.580741590018647e-16 |path|=NaN speed=NaN\n",
      "28.89 secs, 7480 evals, 935 steps, fitness=0.000004571\n",
      "σ=9.144010467192238e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.494005416219807e-16 |path|=NaN speed=NaN\n",
      "29.41 secs, 7624 evals, 953 steps, fitness=0.000004571\n",
      "σ=3.5252961355930194e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.08995101411847e-16 |path|=NaN speed=NaN\n",
      "29.93 secs, 7744 evals, 968 steps, fitness=0.000004571\n",
      "σ=3.0221457535067885e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=3.0531133177191805e-16 |path|=NaN speed=NaN\n",
      "30.45 secs, 7888 evals, 986 steps, fitness=0.000004571\n",
      "σ=3.201608885066378e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.636779683484747e-16 |path|=NaN speed=NaN\n",
      "30.96 secs, 8024 evals, 1003 steps, fitness=0.000004571\n",
      "σ=1.6060788499097339e-19 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=8.98586760555986e-16 |path|=NaN speed=NaN\n",
      "31.48 secs, 8168 evals, 1021 steps, fitness=0.000004571\n",
      "σ=8.948607188425664e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.298117831235686e-16 |path|=NaN speed=NaN\n",
      "32.00 secs, 8312 evals, 1039 steps, fitness=0.000004571\n",
      "σ=7.385870049368578e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=8.534839501805891e-16 |path|=NaN speed=NaN\n",
      "32.51 secs, 8448 evals, 1056 steps, fitness=0.000004571\n",
      "σ=2.35069377441437e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.771561172376096e-16 |path|=NaN speed=NaN\n",
      "33.03 secs, 8592 evals, 1074 steps, fitness=0.000004571\n",
      "σ=3.3072769285306106e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.806255641895632e-18 |path|=NaN speed=NaN\n",
      "33.55 secs, 8736 evals, 1092 steps, fitness=0.000004571\n",
      "σ=9.281204582343359e-21 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=6.314393452555578e-16 |path|=NaN speed=NaN\n",
      "34.07 secs, 8880 evals, 1110 steps, fitness=0.000004571\n",
      "σ=1.1080682142101856e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.061650767297806e-15 |path|=NaN speed=NaN\n",
      "34.59 secs, 9000 evals, 1125 steps, fitness=0.000004571\n",
      "σ=7.414187615970699e-21 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1761425167122752e-15 |path|=NaN speed=NaN\n",
      "35.11 secs, 9128 evals, 1141 steps, fitness=0.000004571\n",
      "σ=4.574499956064197e-21 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.71445146547012e-16 |path|=NaN speed=NaN\n",
      "35.61 secs, 9256 evals, 1157 steps, fitness=0.000004571\n",
      "σ=3.002442682755566e-21 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.214306433183765e-15 |path|=NaN speed=NaN\n",
      "36.12 secs, 9384 evals, 1173 steps, fitness=0.000004571\n",
      "σ=1.6207642971852247e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.547373340571312e-15 |path|=NaN speed=NaN\n",
      "36.63 secs, 9520 evals, 1190 steps, fitness=0.000004571\n",
      "σ=1.7960739742213737e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.3877787807814457e-15 |path|=NaN speed=NaN\n",
      "37.14 secs, 9656 evals, 1207 steps, fitness=0.000004571\n",
      "σ=1.167110064229858e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.609823385706477e-15 |path|=NaN speed=NaN\n",
      "37.67 secs, 9800 evals, 1225 steps, fitness=0.000004571\n",
      "σ=1.2860423034033957e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.5543122344752192e-15 |path|=NaN speed=NaN\n",
      "38.18 secs, 9936 evals, 1242 steps, fitness=0.000004571\n",
      "σ=1.5244558670485814e-20 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.43689570931383e-16 |path|=NaN speed=NaN\n",
      "\n",
      "Optimization stopped after 1251 steps and 38.45 seconds\n",
      "Termination reason: Max number of function evaluations (10000) reached\n",
      "Steps per second = 32.54\n",
      "Function evals per second = 260.29\n",
      "Improvements/step = NaN\n",
      "Total function evaluations = 10008\n",
      "\n",
      "\n",
      "Best candidate found: [0.578509, 0.582018, 0.765104]\n",
      "\n",
      "Fitness: 0.000004571\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(Any[4.570643667309179e-6], Any[[0.5785088395951437, 0.5820179576998394, 0.7651041114910982]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Global optimization:\n",
    "msm_optimize!(myProblem, verbose = false)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46b5a287",
   "metadata": {},
   "source": [
    "## Step 6. Analysing Results\n",
    "\n",
    "### A. Point estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "ce116f90",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Minimum objective function = 4.570643667309179e-6\n",
      "Estimated value for alpha = 0.5785088395951437. True value for beta1 = 0.5662374165061859\n",
      "Estimated value for beta1 = 0.5820179576998394. True value for beta1 = 0.5908446386657102\n",
      "Estimated value for beta2 = 0.7651041114910982. True value for beta2 = 0.7667970365022592\n"
     ]
    }
   ],
   "source": [
    "minimizer = msm_minimizer(myProblem)\n",
    "minimum_val = msm_minimum(myProblem)\n",
    "\n",
    "println(\"Minimum objective function = $(minimum_val)\")\n",
    "println(\"Estimated value for alpha = $(minimizer[1]). True value for beta1 = $(alpha0[1])\")\n",
    "println(\"Estimated value for beta1 = $(minimizer[2]). True value for beta1 = $(beta0[1])\")\n",
    "println(\"Estimated value for beta2 = $(minimizer[3]). True value for beta2 = $(beta0[2])\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffdc93ac",
   "metadata": {},
   "source": [
    "### B. Inference\n",
    "\n",
    "### Estimation of the distance matrix $\\Sigma_0$\n",
    "\n",
    "Let's calculate the variance-covariance matrix of the **\"distance matrix\"** (using the terminolgy of [Duffie and Singleton (1993)](https://www.jstor.org/stable/2951768?seq=1)). Here we know that errors are not correlated (the serial correlation coefficient is set to 0 in the code above). in the presence of serial correlation, an HAC estimation would be needed (see the package [CovarianceMatrices](https://github.com/gragusa/CovarianceMatrices.jl) for instance)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "ad1876d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       "   2.942     12.1631    13.6279    346.754    392.581\n",
       "  12.1631    81.0139    47.795    2369.6     1503.16\n",
       "  13.6279    47.795     87.8405   1490.87    2622.29\n",
       " 346.754   2369.6     1490.87    77664.1    50175.4\n",
       " 392.581   1503.16    2622.29    50175.4    87292.2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Empirical Series\n",
    "#-----------------\n",
    "X = zeros(T, 5)\n",
    "\n",
    "X[:,1] = y\n",
    "X[:,2] = (x[:,1] .* y)\n",
    "X[:,3] = (x[:,2] .* y)\n",
    "X[:,4] = (x[:,1] .* y).^2\n",
    "X[:,5] = (x[:,2] .* y).^2\n",
    "\n",
    "Sigma0 = cov(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05cf8015",
   "metadata": {},
   "source": [
    "### Asymptotic variance\n",
    "\n",
    "#### Theory\n",
    "\n",
    "The asymptotic variance of the MSM estimate is calculated using the usual **GMM sandwich formula**, corrected to take into account simulation noise. \n",
    "\n",
    "$$\n",
    "AsymptoticVarianceMSM = (1 + \\tau)*AsymptoticVarianceGMM\n",
    "$$\n",
    "\n",
    "Here we are trying to match unconditional moments from time series. In this case, $\\tau = \\frac{tData}{tSimulation}$, where $tData$ is the number of periods in the empirical data and $tSimulation$ is the number of time periods in the simulated data. \n",
    "\n",
    "See [Duffie and Singleton (1993)](https://www.jstor.org/stable/2951768?seq=1) and [Gouriéroux and Montfort (1996)](https://www.jstor.org/stable/3533164?seq=1) for details on how to choose $\\tau$. \n",
    "\n",
    "#### Practice\n",
    "\n",
    "Calculating the asymptotic variance using MSM.jl is done in two steps:\n",
    "* setting the value of the **\"distance matrix\"** using the function `set_Sigma0!`\n",
    "* calculating the asymptotic variance using the function `calculate_Avar!`\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "f3efa137",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Symmetric{Float64, Matrix{Float64}}:\n",
       "  52.0436   -11.8728   -8.70917\n",
       " -11.8728     7.56591  -2.28261\n",
       "  -8.70917   -2.28261   6.63642"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set_Sigma0!(myProblem, Sigma0)\n",
    "calculate_Avar!(myProblem, minimizer, tau = T/nbDraws) # nbDraws = number of draws in the simulated data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2715e37",
   "metadata": {},
   "source": [
    "Once the asymptotic variance has been calculated, a summary table can be obtained using the\n",
    "function `summary_table`. This function has four inputs:\n",
    "* a MSMProblem\n",
    "* the minimizer of the objective function\n",
    "* the length of the empirical sample\n",
    "* the confidence level associated to the test H0: $\\theta_i = 0$, H1: $\\theta_i != 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "fe16cadf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>Estimate</th><th>StdError</th><th>tValue</th><th>pValue</th><th>ConfIntervalLower</th><th>ConfIntervalUpper</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>3 rows × 6 columns</p><tr><th>1</th><td>0.578509</td><td>0.0228131</td><td>25.3587</td><td>0.0</td><td>0.540985</td><td>0.616033</td></tr><tr><th>2</th><td>0.582018</td><td>0.00869823</td><td>66.9123</td><td>0.0</td><td>0.567711</td><td>0.596325</td></tr><tr><th>3</th><td>0.765104</td><td>0.00814642</td><td>93.9191</td><td>0.0</td><td>0.751704</td><td>0.778504</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccc}\n",
       "\t& Estimate & StdError & tValue & pValue & ConfIntervalLower & ConfIntervalUpper\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.578509 & 0.0228131 & 25.3587 & 0.0 & 0.540985 & 0.616033 \\\\\n",
       "\t2 & 0.582018 & 0.00869823 & 66.9123 & 0.0 & 0.567711 & 0.596325 \\\\\n",
       "\t3 & 0.765104 & 0.00814642 & 93.9191 & 0.0 & 0.751704 & 0.778504 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m3×6 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m Estimate \u001b[0m\u001b[1m StdError   \u001b[0m\u001b[1m tValue  \u001b[0m\u001b[1m pValue  \u001b[0m\u001b[1m ConfIntervalLower \u001b[0m\u001b[1m ConfInterval\u001b[0m ⋯\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Float64  \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64           \u001b[0m\u001b[90m Float64     \u001b[0m ⋯\n",
       "─────┼──────────────────────────────────────────────────────────────────────────\n",
       "   1 │ 0.578509  0.0228131   25.3587      0.0           0.540985           0.6 ⋯\n",
       "   2 │ 0.582018  0.00869823  66.9123      0.0           0.567711           0.5\n",
       "   3 │ 0.765104  0.00814642  93.9191      0.0           0.751704           0.7\n",
       "\u001b[31m                                                                1 column omitted\u001b[0m"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = summary_table(myProblem, minimizer, T, 0.05)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f577905",
   "metadata": {},
   "source": [
    "### Comparison with OLS\n",
    "\n",
    "Let's compare the MSM results to the one obtained using OLS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "c7f19abf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, CholeskyPivoted{Float64, Matrix{Float64}}}}, Matrix{Float64}}\n",
       "\n",
       "y ~ 1 + x1 + x2\n",
       "\n",
       "Coefficients:\n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "                Coef.  Std. Error       t  Pr(>|t|)  Lower 95%  Upper 95%\n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  0.571592  0.00836568   68.33    <1e-99   0.555196   0.587989\n",
       "x1           0.588239  0.00218787  268.86    <1e-99   0.583951   0.592528\n",
       "x2           0.767633  0.00219593  349.57    <1e-99   0.763329   0.771937\n",
       "─────────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DataFrames, GLM\n",
    "data = DataFrame(x1=x[:,1], x2=x[:,2], y= y[:]);\n",
    "ols = lm(@formula(y ~ x1 + x2), data)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82b5455d",
   "metadata": {},
   "source": [
    "In terms of point estimates, MSM and OLS estimates are rather close. Regarding standard errors, the OLS estimates are more precisely estimated. This is not surprising given that simulation noise impacts the accuracy of the MSM estimate and that OLS uses much more information than MSM. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7407a03",
   "metadata": {},
   "source": [
    "### C. Slices of the objective function\n",
    "\n",
    "One can \"visually check\" that the [local identification identification](https://en.wikipedia.org/wiki/Generalized_method_of_moments) is met by \"slicing\" the objective function. That is, we hold all the variables constant, and we calculate how the value of objective function changes when moving along one dimension. Hopefully, we will see nice U-shaped plots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "9c42764d",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: slicing along alpha\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
     ]
    },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: slicing along beta1\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n",
      "┌ Info: slicing along beta2\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
     ]
    }
   ],
   "source": [
    "# Slices\n",
    "vXGrid, vYGrid = msm_slices(myProblem, minimizer, nbPoints = 7);\n",
    "\n",
    "p1 = plot(vXGrid[:, 1],vYGrid[:, 1],title = L\"\\alpha\", label = \"\",linewidth = 3, xrotation = 45)\n",
    "plot!(p1, [minimizer[1]], seriestype = :vline, label = \"\",linewidth = 1)\n",
    "p2 = plot(vXGrid[:, 2],vYGrid[:, 2],title = L\"\\beta_1\", label = \"\",linewidth = 3, xrotation = 45)\n",
    "plot!(p2, [minimizer[2]], seriestype = :vline, label = \"\",linewidth = 1)\n",
    "p3 = plot(vXGrid[:, 3],vYGrid[:, 3],title = L\"\\beta_2\", label = \"\",linewidth = 3, xrotation = 45)\n",
    "plot!(p3, [minimizer[3]], seriestype = :vline, label = \"\",linewidth = 1)\n",
    "plot_combined = plot(p1, p2, p3)\n",
    "display(plot_combined)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44e174a8",
   "metadata": {},
   "source": [
    "## Appendix"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b464a7b",
   "metadata": {},
   "source": [
    "### Efficient MSM\n",
    "\n",
    "Set the weight matrix `W` to be the inverse of the distance matrix $\\Sigma_0$ (See [Duffie and Singleton (1993)](https://www.jstor.org/stable/2951768?seq=1))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "57fe3f12",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::MSM.var\"#objective_function_MSM#6\"{MSMProblem}) (generic function with 1 method)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "efficient_W = inv(Sigma0)\n",
    "set_weight_matrix!(myProblem, efficient_W)\n",
    "construct_objective_function!(myProblem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "83e0c167",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hello\n",
      "Starting optimization with optimizer BlackBoxOptim.DXNESOpt{Float64, BlackBoxOptim.RandomBound{BlackBoxOptim.ContinuousRectSearchSpace}}\n",
      "0.00 secs, 0 evals, 0 steps\n",
      "σ=1.0 η[x]=1.0 η[σ]=0.0 η[B]=0.0 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: 1 worker(s) detected\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n",
      "┌ Info: Starting optimization in serial\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.50 secs, 144 evals, 18 steps, fitness=0.012017301\n",
      "σ=4.530713998845915 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.734723475976807e-17 |path|=NaN speed=NaN\n",
      "1.01 secs, 288 evals, 36 steps, fitness=0.012017301\n",
      "σ=8.197938709640567 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=3.2959746043559335e-17 |path|=NaN speed=NaN\n",
      "1.52 secs, 432 evals, 54 steps, fitness=0.012017301\n",
      "σ=7.647776122450567 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.0816681711721685e-17 |path|=NaN speed=NaN\n",
      "2.03 secs, 592 evals, 74 steps, fitness=0.005261989\n",
      "σ=3.673846161069867 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.632783294297951e-17 |path|=NaN speed=NaN\n",
      "2.55 secs, 752 evals, 94 steps, fitness=0.005261989\n",
      "σ=0.3226758783270161 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.5265566588595902e-16 |path|=NaN speed=NaN\n",
      "3.05 secs, 912 evals, 114 steps, fitness=0.000296044\n",
      "σ=0.0019921748954311613 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.6653345369377348e-16 |path|=NaN speed=NaN\n",
      "3.55 secs, 1072 evals, 134 steps, fitness=0.000181642\n",
      "σ=0.000951314089091043 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.498001805406602e-16 |path|=NaN speed=NaN\n",
      "4.06 secs, 1232 evals, 154 steps, fitness=0.000077365\n",
      "σ=0.0006936683420398699 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "4.56 secs, 1392 evals, 174 steps, fitness=0.000077224\n",
      "σ=2.459373186254941e-6 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "5.07 secs, 1552 evals, 194 steps, fitness=0.000077224\n",
      "σ=3.3555118438810167e-7 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.6653345369377348e-16 |path|=NaN speed=NaN\n",
      "5.58 secs, 1712 evals, 214 steps, fitness=0.000077224\n",
      "σ=1.5273594914117382e-9 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "6.09 secs, 1872 evals, 234 steps, fitness=0.000077224\n",
      "σ=8.05883550778327e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "6.59 secs, 2032 evals, 254 steps, fitness=0.000077224\n",
      "σ=1.584563548150276e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.7755575615628914e-16 |path|=NaN speed=NaN\n",
      "7.10 secs, 2176 evals, 272 steps, fitness=0.000077224\n",
      "σ=3.8645731211651276e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "7.62 secs, 2336 evals, 292 steps, fitness=0.000077224\n",
      "σ=4.1411835722249485e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-17 |path|=NaN speed=NaN\n",
      "8.14 secs, 2488 evals, 311 steps, fitness=0.000077224\n",
      "σ=1.4533589310471189e-9 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.7755575615628914e-16 |path|=NaN speed=NaN\n",
      "8.66 secs, 2648 evals, 331 steps, fitness=0.000077224\n",
      "σ=4.3047427975994366e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "9.19 secs, 2800 evals, 350 steps, fitness=0.000077224\n",
      "σ=2.3300064865265185e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.996003610813204e-16 |path|=NaN speed=NaN\n",
      "9.70 secs, 2944 evals, 368 steps, fitness=0.000077224\n",
      "σ=7.006517663472592e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.996003610813204e-16 |path|=NaN speed=NaN\n",
      "10.22 secs, 3080 evals, 385 steps, fitness=0.000077224\n",
      "σ=2.125827578756748e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.996003610813204e-16 |path|=NaN speed=NaN\n",
      "10.74 secs, 3200 evals, 400 steps, fitness=0.000077224\n",
      "σ=9.407760625172715e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-16 |path|=NaN speed=NaN\n",
      "11.27 secs, 3336 evals, 417 steps, fitness=0.000077224\n",
      "σ=1.1244003019752318e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.220446049250313e-16 |path|=NaN speed=NaN\n",
      "11.78 secs, 3488 evals, 436 steps, fitness=0.000077224\n",
      "σ=2.072342395126247e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-17 |path|=NaN speed=NaN\n",
      "12.30 secs, 3640 evals, 455 steps, fitness=0.000077224\n",
      "σ=8.626344707807327e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.6653345369377348e-16 |path|=NaN speed=NaN\n",
      "12.80 secs, 3768 evals, 471 steps, fitness=0.000077224\n",
      "σ=1.9590040761767694e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1102230246251565e-16 |path|=NaN speed=NaN\n",
      "13.31 secs, 3912 evals, 489 steps, fitness=0.000077224\n",
      "σ=6.9873709105612006e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.551115123125783e-17 |path|=NaN speed=NaN\n",
      "13.82 secs, 4064 evals, 508 steps, fitness=0.000077224\n",
      "σ=2.747784774080344e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=0.0 |path|=NaN speed=NaN\n",
      "14.32 secs, 4208 evals, 526 steps, fitness=0.000077224\n",
      "σ=1.888014843597344e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n",
      "14.85 secs, 4360 evals, 545 steps, fitness=0.000077224\n",
      "σ=9.684618626137892e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=8.326672684688674e-16 |path|=NaN speed=NaN\n",
      "15.36 secs, 4496 evals, 562 steps, fitness=0.000077224\n",
      "σ=1.0144181951328111e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=6.661338147750939e-16 |path|=NaN speed=NaN\n",
      "15.88 secs, 4648 evals, 581 steps, fitness=0.000077224\n",
      "σ=6.956608602154481e-13 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=3.885780586188048e-16 |path|=NaN speed=NaN\n",
      "16.39 secs, 4800 evals, 600 steps, fitness=0.000077224\n",
      "σ=1.1012799810608936e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.0547118733938987e-15 |path|=NaN speed=NaN\n",
      "16.90 secs, 4944 evals, 618 steps, fitness=0.000077224\n",
      "σ=2.7263591530254676e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.7486012637846216e-15 |path|=NaN speed=NaN\n",
      "17.41 secs, 5096 evals, 637 steps, fitness=0.000077224\n",
      "σ=7.360094020890408e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.1094237467877974e-15 |path|=NaN speed=NaN\n",
      "17.93 secs, 5248 evals, 656 steps, fitness=0.000077224\n",
      "σ=1.0126961638406069e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.8033131371785203e-15 |path|=NaN speed=NaN\n",
      "18.45 secs, 5392 evals, 674 steps, fitness=0.000077224\n",
      "σ=6.541845073954297e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.915134717478395e-15 |path|=NaN speed=NaN\n",
      "18.97 secs, 5528 evals, 691 steps, fitness=0.000077224\n",
      "σ=1.5220117598614036e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.1649348980190553e-15 |path|=NaN speed=NaN\n",
      "19.48 secs, 5656 evals, 707 steps, fitness=0.000077224\n",
      "σ=1.1587026500289601e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.0261570199409107e-15 |path|=NaN speed=NaN\n",
      "20.00 secs, 5800 evals, 725 steps, fitness=0.000077224\n",
      "σ=5.861785686913875e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.275957200481571e-15 |path|=NaN speed=NaN\n",
      "20.52 secs, 5944 evals, 743 steps, fitness=0.000077224\n",
      "σ=2.9703975852667397e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.8908485888147197e-15 |path|=NaN speed=NaN\n",
      "21.03 secs, 6080 evals, 760 steps, fitness=0.000077224\n",
      "σ=1.39455584041613e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.123301534595612e-15 |path|=NaN speed=NaN\n",
      "21.55 secs, 6232 evals, 779 steps, fitness=0.000077224\n",
      "σ=1.8747614665124982e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.1822821327788233e-15 |path|=NaN speed=NaN\n",
      "22.06 secs, 6368 evals, 796 steps, fitness=0.000077224\n",
      "σ=4.0022300821724463e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.581268532253489e-15 |path|=NaN speed=NaN\n",
      "22.59 secs, 6520 evals, 815 steps, fitness=0.000077224\n",
      "σ=6.67268755170261e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.7755575615628914e-15 |path|=NaN speed=NaN\n"
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     "text": [
      "23.10 secs, 6664 evals, 833 steps, fitness=0.000077224\n",
      "σ=5.437534508789445e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.3869795029440866e-15 |path|=NaN speed=NaN\n",
      "23.60 secs, 6792 evals, 849 steps, fitness=0.000077224\n",
      "σ=3.8965650682692575e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.6930901125533637e-15 |path|=NaN speed=NaN\n",
      "24.11 secs, 6936 evals, 867 steps, fitness=0.000077224\n",
      "σ=3.8466857690147836e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.159339953157541e-16 |path|=NaN speed=NaN\n",
      "24.63 secs, 7088 evals, 886 steps, fitness=0.000077224\n",
      "σ=5.12248278524265e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.771561172376096e-16 |path|=NaN speed=NaN\n",
      "25.15 secs, 7240 evals, 905 steps, fitness=0.000077224\n",
      "σ=7.647130503176857e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.771561172376096e-16 |path|=NaN speed=NaN\n",
      "25.66 secs, 7392 evals, 924 steps, fitness=0.000077224\n",
      "σ=7.433571141132404e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.4155343563970746e-15 |path|=NaN speed=NaN\n",
      "26.17 secs, 7536 evals, 942 steps, fitness=0.000077224\n",
      "σ=2.6796495840525843e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.8041124150158794e-15 |path|=NaN speed=NaN\n",
      "26.68 secs, 7688 evals, 961 steps, fitness=0.000077224\n",
      "σ=1.4803724208577696e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.4988010832439613e-15 |path|=NaN speed=NaN\n",
      "27.20 secs, 7840 evals, 980 steps, fitness=0.000077224\n",
      "σ=5.374934498612881e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.6653345369377348e-15 |path|=NaN speed=NaN\n",
      "27.71 secs, 7992 evals, 999 steps, fitness=0.000077224\n",
      "σ=3.103852126916872e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.762479051592436e-15 |path|=NaN speed=NaN\n",
      "28.21 secs, 8136 evals, 1017 steps, fitness=0.000077224\n",
      "σ=2.1142633396685713e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.0547118733938987e-15 |path|=NaN speed=NaN\n",
      "28.72 secs, 8288 evals, 1036 steps, fitness=0.000077224\n",
      "σ=6.19807816355574e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=1.1379786002407855e-15 |path|=NaN speed=NaN\n",
      "29.24 secs, 8440 evals, 1055 steps, fitness=0.000077224\n",
      "σ=1.3333944622905892e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.147060721024445e-16 |path|=NaN speed=NaN\n",
      "29.75 secs, 8592 evals, 1074 steps, fitness=0.000077224\n",
      "σ=5.370929864169454e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.494005416219807e-16 |path|=NaN speed=NaN\n",
      "30.26 secs, 8744 evals, 1093 steps, fitness=0.000077224\n",
      "σ=3.208027390999012e-12 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=5.152128723651117e-16 |path|=NaN speed=NaN\n",
      "30.78 secs, 8896 evals, 1112 steps, fitness=0.000077224\n",
      "σ=1.9338623301406916e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.440892098500626e-16 |path|=NaN speed=NaN\n",
      "31.30 secs, 9048 evals, 1131 steps, fitness=0.000077224\n",
      "σ=1.0097458084023759e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.8102520310824275e-16 |path|=NaN speed=NaN\n",
      "31.81 secs, 9200 evals, 1150 steps, fitness=0.000077224\n",
      "σ=3.948570471413722e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=7.494005416219807e-16 |path|=NaN speed=NaN\n",
      "32.33 secs, 9352 evals, 1169 steps, fitness=0.000077224\n",
      "σ=3.8203197003356594e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.718447854656915e-16 |path|=NaN speed=NaN\n",
      "32.84 secs, 9504 evals, 1188 steps, fitness=0.000077224\n",
      "σ=5.09809610220223e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=2.7755575615628914e-17 |path|=NaN speed=NaN\n",
      "33.36 secs, 9656 evals, 1207 steps, fitness=0.000077224\n",
      "σ=1.4981251428525263e-10 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=4.475586568020162e-16 |path|=NaN speed=NaN\n",
      "33.88 secs, 9808 evals, 1226 steps, fitness=0.000077224\n",
      "σ=6.907331736381087e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.922618282587337e-16 |path|=NaN speed=NaN\n",
      "34.39 secs, 9960 evals, 1245 steps, fitness=0.000077224\n",
      "σ=1.9667133626277604e-11 η[x]=1.0 η[σ]=1.5714285714285714 η[B]=0.06349206349206349 |tr(ln_B)|=9.992007221626409e-16 |path|=NaN speed=NaN\n",
      "\n",
      "Optimization stopped after 1251 steps and 34.56 seconds\n",
      "Termination reason: Max number of function evaluations (10000) reached\n",
      "Steps per second = 36.20\n",
      "Function evals per second = 289.62\n",
      "Improvements/step = NaN\n",
      "Total function evaluations = 10008\n",
      "\n",
      "\n",
      "Best candidate found: [0.57546, 0.589797, 0.762882]\n",
      "\n",
      "Fitness: 0.000077224\n",
      "\n"
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     "data": {
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       "(Any[7.72242594803831e-5], Any[[0.5754602018145192, 0.5897970480427388, 0.762882204121741]])"
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     "execution_count": 18,
     "metadata": {},
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   ],
   "source": [
    "# Global optimization:\n",
    "msm_optimize!(myProblem, verbose = false)"
   ]
  },
  {
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   "execution_count": 19,
   "id": "80579084",
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     "output_type": "stream",
     "text": [
      "Minimum objective function = 7.72242594803831e-5\n",
      "Estimated value for alpha = 0.5754602018145192. True value for beta1 = 0.5662374165061859 \n",
      "\n",
      "Estimated value for beta1 = 0.5897970480427388. True value for beta1 = 0.5908446386657102 \n",
      "\n",
      "Estimated value for beta2 = 0.762882204121741. True value for beta2 = 0.7667970365022592 \n",
      "\n"
     ]
    },
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       "\\begin{tabular}{r|cccccc}\n",
       "\t& Estimate & StdError & tValue & pValue & ConfIntervalLower & ConfIntervalUpper\\\\\n",
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       "\u001b[1m3×6 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m Estimate \u001b[0m\u001b[1m StdError   \u001b[0m\u001b[1m tValue   \u001b[0m\u001b[1m pValue  \u001b[0m\u001b[1m ConfIntervalLower \u001b[0m\u001b[1m ConfInterva\u001b[0m ⋯\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Float64  \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Float64  \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64           \u001b[0m\u001b[90m Float64    \u001b[0m ⋯\n",
       "─────┼──────────────────────────────────────────────────────────────────────────\n",
       "   1 │ 0.57546   0.0226358    25.4226      0.0           0.538228           0. ⋯\n",
       "   2 │ 0.589797  0.00751405   78.4925      0.0           0.577438           0.\n",
       "   3 │ 0.762882  0.00710704  107.342       0.0           0.751192           0.\n",
       "\u001b[31m                                                                1 column omitted\u001b[0m"
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   ],
   "source": [
    "minimizer = msm_minimizer(myProblem)\n",
    "minimum_val = msm_minimum(myProblem)\n",
    "\n",
    "println(\"Minimum objective function = $(minimum_val)\")\n",
    "println(\"Estimated value for alpha = $(minimizer[1]). True value for beta1 = $(alpha0[1]) \\n\")\n",
    "println(\"Estimated value for beta1 = $(minimizer[2]). True value for beta1 = $(beta0[1]) \\n\")\n",
    "println(\"Estimated value for beta2 = $(minimizer[3]). True value for beta2 = $(beta0[2]) \\n\")\n",
    "\n",
    "set_Sigma0!(myProblem, Sigma0)\n",
    "calculate_Avar!(myProblem, minimizer, tau = T/nbDraws) # nbDraws = number of draws in the simulated data\n",
    "df = summary_table(myProblem, minimizer, T, 0.05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "48129f73",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: slicing along alpha\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: slicing along beta1\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n",
      "┌ Info: slicing along beta2\n",
      "└ @ MSM /home/julien/.julia/dev/MSM/src/generic.jl:350\n"
     ]
    }
   ],
   "source": [
    "# Slices\n",
    "vXGrid, vYGrid = msm_slices(myProblem, minimizer, nbPoints = 7);\n",
    "\n",
    "p1 = plot(vXGrid[:, 1],vYGrid[:, 1],title = L\"\\alpha\", label = \"\",linewidth = 3, xrotation = 45)\n",
    "plot!(p1, [minimizer[1]], seriestype = :vline, label = \"\",linewidth = 1)\n",
    "p2 = plot(vXGrid[:, 2],vYGrid[:, 2],title = L\"\\beta_1\", label = \"\",linewidth = 3, xrotation = 45)\n",
    "plot!(p2, [minimizer[2]], seriestype = :vline, label = \"\",linewidth = 1)\n",
    "p3 = plot(vXGrid[:, 3],vYGrid[:, 3],title = L\"\\beta_2\", label = \"\",linewidth = 3, xrotation = 45)\n",
    "plot!(p3, [minimizer[3]], seriestype = :vline, label = \"\",linewidth = 1)\n",
    "plot_combined = plot(p1, p2, p3)\n",
    "display(plot_combined)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acd7f8b2",
   "metadata": {},
   "source": [
    "### Version"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "23d981bc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.6.1\n",
      "Commit 6aaedecc44 (2021-04-23 05:59 UTC)\n",
      "Platform Info:\n",
      "  OS: Linux (x86_64-pc-linux-gnu)\n",
      "  CPU: Intel(R) Core(TM) i7-8850H CPU @ 2.60GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-11.0.1 (ORCJIT, skylake)\n"
     ]
    }
   ],
   "source": [
    "versioninfo()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.6.1",
   "language": "julia",
   "name": "julia-1.6"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
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