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basic_empirics/BasicEmpirMethods.html

@@ -940,7 +940,7 @@ <h2> Contents </h2>
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_2720/3993614049.py:4: SettingWithCopyWarning: 
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_2762/3993614049.py:4: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
@@ -1600,7 +1600,7 @@ <h2> Contents </h2>
 Model:                            OLS   Adj. R-squared:                  0.608
 Method:                 Least Squares   F-statistic:                     171.4
 Date:                Fri, 10 Nov 2023   Prob (F-statistic):           4.16e-24
-Time:                        08:07:03   Log-Likelihood:                -119.71
+Time:                        08:14:56   Log-Likelihood:                -119.71
 No. Observations:                 111   AIC:                             243.4
 Df Residuals:                     109   BIC:                             248.8
 Df Model:                           1                                         

basic_empirics/LogisticReg.html

@@ -1141,29 +1141,29 @@ <h2> Contents </h2>
 <section id="interpreting-coefficients-log-odds-ratio">
 <span id="sec-loglogitinterpret"></span><h4><span class="section-number">13.2.2.4. </span>Interpreting coefficients (log odds ratio)<a class="headerlink" href="#interpreting-coefficients-log-odds-ratio" title="Permalink to this heading">#</a></h4>
 <p>The odds ratio in the logistic model is provides a nice way to interpret logit model coefficients. Let <span class="math notranslate nohighlight">\(z\equiv X^T\beta = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}\)</span>. The logistic model is stated by the probability that the binary categorical dependent variable equals one <span class="math notranslate nohighlight">\(y_i=1\)</span>.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-ee860652-3193-48f7-a1bd-372ccd95f10b">
-<span class="eqno">(13.11)<a class="headerlink" href="#equation-ee860652-3193-48f7-a1bd-372ccd95f10b" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-f6e74078-e905-42f7-a486-7317612d994b">
+<span class="eqno">(13.11)<a class="headerlink" href="#equation-f6e74078-e905-42f7-a486-7317612d994b" title="Permalink to this equation">#</a></span>\[\begin{equation}
   P(y_i=1|X,\theta) = \frac{e^z}{1 + e^z}
 \end{equation}\]</div>
 <p>Given this equation, we know that the probability of the dependent variable being zero <span class="math notranslate nohighlight">\(y_i=0\)</span> is just one minus the probability above.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-dc5c7f4a-5b93-4c95-97c6-1b1dbbe9cb58">
-<span class="eqno">(13.12)<a class="headerlink" href="#equation-dc5c7f4a-5b93-4c95-97c6-1b1dbbe9cb58" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-63df07e2-a99c-4dfe-a268-ebd0bf4935c9">
+<span class="eqno">(13.12)<a class="headerlink" href="#equation-63df07e2-a99c-4dfe-a268-ebd0bf4935c9" title="Permalink to this equation">#</a></span>\[\begin{equation}
   P(y_i=0|X,\theta) = 1 - P(y_i=1|X,\theta) = 1 - \frac{e^z}{1 + e^z} = \frac{1}{1 + e^z}
 \end{equation}\]</div>
 <p>The odds ratio is a common way of expressing the probability of an event versus all other events. For example, if the probability of your favorite team winning a game is <span class="math notranslate nohighlight">\(P(win)=0.8\)</span>, then we know that the probability of your favorite team losing that game is <span class="math notranslate nohighlight">\(P(lose)=1-P(win)=0.2\)</span>. The odds ratio is the ratio of these two probabilities.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-6d572d03-9c4e-49e6-95f1-42e3f372fea1">
-<span class="eqno">(13.13)<a class="headerlink" href="#equation-6d572d03-9c4e-49e6-95f1-42e3f372fea1" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-4daf6d79-8a28-43f0-a7ea-ab45f64d7cdf">
+<span class="eqno">(13.13)<a class="headerlink" href="#equation-4daf6d79-8a28-43f0-a7ea-ab45f64d7cdf" title="Permalink to this equation">#</a></span>\[\begin{equation}
   \frac{P(win)}{P(lose)} = \frac{P(win)}{1 - P(win)} = \frac{0.8}{0.2} = \frac{4}{1} \quad\text{or}\quad 4
 \end{equation}\]</div>
 <p>The odds ratio tells you that the probability of your team winning is four times as likely as your team losing. A gambler would say that your odds are 4-to-1. Another way of saying it is that your team will win four out of five times and will lose 1 out of five times.</p>
 <p>In the logistic model, the odds ratio reduces the problem nicely.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-08682f55-fc6d-4e3c-bdb6-daf92ae6d8dd">
-<span class="eqno">(13.14)<a class="headerlink" href="#equation-08682f55-fc6d-4e3c-bdb6-daf92ae6d8dd" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-18cc280d-8b86-40c7-a87f-44fd0f334972">
+<span class="eqno">(13.14)<a class="headerlink" href="#equation-18cc280d-8b86-40c7-a87f-44fd0f334972" title="Permalink to this equation">#</a></span>\[\begin{equation}
   \frac{P(y_i=1|X,\theta)}{1 - P(y_i=1|X,\theta)} = \frac{\frac{e^z}{1 + e^z}}{\frac{1}{1 + e^z}} = e^z
 \end{equation}\]</div>
 <p>If we take the log of both sides, we see that the log odds ratio is equal to the linear predictor <span class="math notranslate nohighlight">\(z\equiv X^T\beta = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}\)</span>.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-a6785f3d-6ae0-46d3-abb0-543b6ffde234">
-<span class="eqno">(13.15)<a class="headerlink" href="#equation-a6785f3d-6ae0-46d3-abb0-543b6ffde234" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-a3e73411-6b51-4185-afaf-33357aee1137">
+<span class="eqno">(13.15)<a class="headerlink" href="#equation-a3e73411-6b51-4185-afaf-33357aee1137" title="Permalink to this equation">#</a></span>\[\begin{equation}
   \ln\left(\frac{P(y_i=1|X,\theta)}{1 - P(y_i=1|X,\theta)}\right) = z = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}
 \end{equation}\]</div>
 <p>So the interpretation of the coeficients <span class="math notranslate nohighlight">\(\beta_k\)</span> is that a one-unit increase of the variable <span class="math notranslate nohighlight">\(x_{k,i}\)</span> increases the odds ratio or the odds of <span class="math notranslate nohighlight">\(y_i=1\)</span> by <span class="math notranslate nohighlight">\(\beta_{k,i}\)</span> percent.</p>
@@ -1175,18 +1175,18 @@ <h2> Contents </h2>
 <p>The multinomial logit model is a natural extension of the logit model. In contrast to the logit model in which the dependent variable has only two categories, the multinomial logit model accomodates <span class="math notranslate nohighlight">\(J\geq2\)</span> categories in the dependent variable. Let <span class="math notranslate nohighlight">\(\eta_j\)</span> be the linear predictor for the <span class="math notranslate nohighlight">\(j\)</span>th category.
 $<span class="math notranslate nohighlight">\( \eta_j\equiv \beta_{j,0} + \beta_{j,1}x_{1,i} + ...\beta_{j,K}x_{K,i} \quad\forall y_i = j \)</span>$</p>
 <p>The multinomial logit model gives the probability of <span class="math notranslate nohighlight">\(y_i=j\)</span> relative to some reference category <span class="math notranslate nohighlight">\(J\)</span> that is left out.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-3aa96cd6-eac7-4892-a370-66b758debda0">
-<span class="eqno">(13.16)<a class="headerlink" href="#equation-3aa96cd6-eac7-4892-a370-66b758debda0" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-35d8f320-74c1-4f3d-b5e4-38a73d908198">
+<span class="eqno">(13.16)<a class="headerlink" href="#equation-35d8f320-74c1-4f3d-b5e4-38a73d908198" title="Permalink to this equation">#</a></span>\[\begin{equation}
   Pr(y_i=j|X,\theta) = \frac{e^{\eta_j}}{1 + \sum_v^{J-1}e^{\eta_v}} \quad\text{for}\quad 1\leq j\leq J-1
 \end{equation}\]</div>
 <p>Once the <span class="math notranslate nohighlight">\(J-1\)</span> sets of coefficients are estimated, the final <span class="math notranslate nohighlight">\(J\)</span>th set of coefficients are a residual based on the following expression.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-aaff249d-6d36-4f61-8fc4-c9121c77f20d">
-<span class="eqno">(13.17)<a class="headerlink" href="#equation-aaff249d-6d36-4f61-8fc4-c9121c77f20d" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-28b4d2d6-f5bd-41c9-8ff7-5c1bc19a56e6">
+<span class="eqno">(13.17)<a class="headerlink" href="#equation-28b4d2d6-f5bd-41c9-8ff7-5c1bc19a56e6" title="Permalink to this equation">#</a></span>\[\begin{equation}
   Pr(y_i=J|X,\theta) = \frac{1}{1 + \sum_v^{J-1}e^{\eta_v}}
 \end{equation}\]</div>
 <p>The analogous log odds ratio interpretation applies to the multinomial logit model.</p>
-<div class="amsmath math notranslate nohighlight" id="equation-e49ab23e-2478-4263-bd3d-c5b0d6b42fe5">
-<span class="eqno">(13.18)<a class="headerlink" href="#equation-e49ab23e-2478-4263-bd3d-c5b0d6b42fe5" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-fb2b90c3-be20-4e3e-8c5e-3004b522ceec">
+<span class="eqno">(13.18)<a class="headerlink" href="#equation-fb2b90c3-be20-4e3e-8c5e-3004b522ceec" title="Permalink to this equation">#</a></span>\[\begin{equation}
   \ln\left(\frac{Pr(y_i=j|X,\theta)}{Pr(y_i=J|X,\theta)}\right) = \eta_j = \beta_{j,0} + \beta_{j,1}x_{1,i} + ...\beta_{j,K}x_{K,i} \quad\text{for}\quad 1\leq j \leq J-1
 \end{equation}\]</div>
 <p>This is the odds ratio of <span class="math notranslate nohighlight">\(y_i=j\)</span> relative to <span class="math notranslate nohighlight">\(y_i=J\)</span>. The interpretation of the <span class="math notranslate nohighlight">\(\beta_{j,k}\)</span> coefficient is the predicted percentage change in the log odds ratio of <span class="math notranslate nohighlight">\(y_i=j\)</span> to <span class="math notranslate nohighlight">\(y_i=J\)</span> from a one-unit increase in variable <span class="math notranslate nohighlight">\(x_{k,i}\)</span>.</p>

python/SciPy.html

@@ -678,7 +678,7 @@ <h2> Contents </h2>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> message: The solution converged.
  success: True
   status: 1
-     fun: [-1.235e-13 -1.017e-13]
+     fun: [-1.239e-13 -1.013e-13]
        x: [ 8.463e-01  2.331e+00]
     nfev: 10
     fjac: [[-8.332e-01  5.529e-01]
@@ -688,7 +688,7 @@ <h2> Contents </h2>
 
 The solution for (x, y) is: [0.84630378 2.33101497]
 
-The error values for eq1 and eq2 at the solution are: [-1.23456800e-13 -1.01696429e-13]
+The error values for eq1 and eq2 at the solution are: [-1.2390089e-13 -1.0125234e-13]
 </pre></div>
 </div>
 </div>

struct_est/SMM.html

@@ -1289,7 +1289,7 @@ <h2> Contents </h2>
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-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM1_1= 612.3371352249138  sig_SMM1_1= 197.26434895262162
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM1_1= 612.3319048024534  sig_SMM1_1= 197.26303566149244
 </pre></div>
 </div>
 </div>
@@ -1317,18 +1317,18 @@ <h2> Contents </h2>
 <div class="cell_output docutils container">
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Data mean of scores = 341.90869565217395 , Data variance of scores = 7827.997292398056
 
-Model mean 1 = 341.6692110494425 , Model variance 1 = 7827.864496338213
+Model mean 1 = 341.6690130387879 , Model variance 1 = 7827.866466512297
 
-Error vector 1 = [-7.00434373e-04 -1.69642445e-05]
+Error vector 1 = [-7.01013506e-04 -1.67125614e-05]
 
 Results from scipy.opmtimize.minimize:
   message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_&lt;=_PGTOL
   success: True
    status: 0
-      fun: 4.908960959342433e-07
+      fun: 4.916992449560748e-07
         x: [ 6.123e+02  1.973e+02]
       nit: 17
-      jac: [-7.436e-07  2.350e-06]
+      jac: [-7.454e-07  2.357e-06]
      nfev: 72
      njev: 24
  hess_inv: &lt;2x2 LbfgsInvHessProduct with dtype=float64&gt;
@@ -1453,19 +1453,19 @@ <h2> Contents </h2>
 </div>
 <div class="cell_output docutils container">
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Jacobian matrix of derivatives of moment error functions is:
-[[ 0.00089749 -0.00290433]
- [-0.00114132  0.00445698]]
+[[ 0.0008975  -0.00290434]
+ [-0.00114133  0.004457  ]]
 
 Weighting matrix W is:
 [[1. 0.]
  [0. 1.]]
 
 Variance-covariance matrix of estimated parameter vector is:
-[[602535.18442996 163802.17330123]
- [163802.17330123  44883.79092131]]
+[[602499.98744184 163793.83380005]
+ [163793.83380005  44881.85524372]]
 
-Std. err. mu_hat= 776.23139876583
-Std. err. sig_hat= 211.85794986573154
+Std. err. mu_hat= 776.2087267235777
+Std. err. sig_hat= 211.8533814781294
 </pre></div>
 </div>
 </div>
@@ -1554,12 +1554,12 @@ <h2> Contents </h2>
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 <div class="cell_output docutils container">
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2nd stage est. of var-cov matrix of moment error vec across sims:
-[[ 0.00033411 -0.00142289]
- [-0.00142289  0.01592879]]
+[[ 0.00033411 -0.00142288]
+ [-0.00142288  0.01592874]]
 
 2nd state est. of optimal weighting matrix:
-[[4830.88530228  431.53378728]
- [ 431.53378728  101.32749623]]
+[[4830.85282784  431.53075653]
+ [ 431.53075653  101.32740541]]
 </pre></div>
 </div>
 </div>
@@ -1577,7 +1577,7 @@ <h2> Contents </h2>
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 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM2_1= 619.4303074248937  sig_SMM2_1= 199.0747813692372
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM2_1= 619.4295828414915  sig_SMM2_1= 199.0745499003818
 </pre></div>
 </div>
 </div>
@@ -1607,15 +1607,15 @@ <h2> Contents </h2>
  [-0.0011259   0.00443426]]
 
 Weighting matrix W is:
-[[4830.88530228  431.53378728]
- [ 431.53378728  101.32749623]]
+[[4830.85282784  431.53075653]
+ [ 431.53075653  101.32740541]]
 
 Variance-covariance matrix of estimated parameter vector is:
-[[2397.38054356  745.29670501]
- [ 745.29670501  232.01757158]]
+[[2397.35566572  745.28979427]
+ [ 745.28979427  232.01568064]]
 
-Std. err. mu_hat= 48.963052841479445
-Std. err. sig_hat= 15.232123016118733
+Std. err. mu_hat= 48.96279879380848
+Std. err. sig_hat= 15.232060945338231
 </pre></div>
 </div>
 </div>
@@ -1865,7 +1865,7 @@ <h2> Contents </h2>
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 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM4_1= 362.560593472098  sig_SMM4_1 46.5751519565219
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM4_1= 362.560593472098  sig_SMM4_1 46.57515195652185
   message: CONVERGENCE: REL_REDUCTION_OF_F_&lt;=_FACTR*EPSMCH
   success: True
    status: 0
@@ -2016,8 +2016,8 @@ <h2> Contents </h2>
 [[33.48770545 23.53170341]
  [23.53170341 18.13459776]]
 
-Std. err. mu_hat= 5.786856266717126
-Std. err. sig_hat= 4.258473642422245
+Std. err. mu_hat= 5.786856266717139
+Std. err. sig_hat= 4.258473642422251
 </pre></div>
 </div>
 </div>
@@ -2139,11 +2139,11 @@ <h2> Contents </h2>
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM4_2= 362.5605400454758  sig_SMM4_2 46.57507128065564
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mu_SMM4_2= 362.5605400454758  sig_SMM4_2 46.57507128065559
  message: Optimization terminated successfully
  success: True
   status: 0
-     fun: 0.9984266286568926
+     fun: 0.9984266286568925
        x: [ 3.626e+02  4.658e+01]
      nit: 1
      jac: [ 5.467e-02  8.255e-02]
@@ -2188,7 +2188,7 @@ <h2> Contents </h2>
 
 Error vector (pct. dev.) = [-0.98        0.04678571  0.11720721 -0.075     ]
 
-Criterion func val = 0.9984266286568926
+Criterion func val = 0.9984266286568925
 </pre></div>
 </div>
 </div>
@@ -2273,8 +2273,8 @@ <h2> Contents </h2>
 [[3.53697411 2.47391182]
  [2.47391182 1.77184156]]
 
-Std. err. mu_hat= 1.8806844791000217
-Std. err. sig_hat= 1.3311053920876885
+Std. err. mu_hat= 1.8806844791000032
+Std. err. sig_hat= 1.3311053920876743
 </pre></div>
 </div>
 </div>