From 9e68a7f50bddd3ffaed7f148924437ece560593c Mon Sep 17 00:00:00 2001 From: rickecon Date: Fri, 10 Nov 2023 09:13:25 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20OpenSour?= =?UTF-8?q?ceEcon/CompMethods@2aa69f999e70d45716d6724a92054f639da6807c=20?= =?UTF-8?q?=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- basic_empirics/BasicEmpirMethods.html | 4 +- basic_empirics/LogisticReg.html | 32 +++++++------- python/SciPy.html | 4 +- searchindex.js | 2 +- struct_est/SMM.html | 60 +++++++++++++-------------- 5 files changed, 51 insertions(+), 51 deletions(-) diff --git a/basic_empirics/BasicEmpirMethods.html b/basic_empirics/BasicEmpirMethods.html index e5b6d2c..2ee8e0a 100644 --- a/basic_empirics/BasicEmpirMethods.html +++ b/basic_empirics/BasicEmpirMethods.html @@ -940,7 +940,7 @@

Contents

-
/tmp/ipykernel_2788/3993614049.py:4: SettingWithCopyWarning: 
+
/tmp/ipykernel_2739/3993614049.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 @@ 
 Contents 

 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:                        08:34:48   Log-Likelihood:                -119.71
+Time:                        09:12:57   Log-Likelihood:                -119.71
 No. Observations:                 111   AIC:                             243.4
 Df Residuals:                     109   BIC:                             248.8
 Df Model:                           1                                         
diff --git a/basic_empirics/LogisticReg.html b/basic_empirics/LogisticReg.html
index fcf2153..13b6508 100644
--- a/basic_empirics/LogisticReg.html
+++ b/basic_empirics/LogisticReg.html
@@ -1141,29 +1141,29 @@ 
 Contents 

 

 
13.2.2.4. Interpreting coefficients (log odds ratio)#

 
The odds ratio in the logistic model is provides a nice way to interpret logit model coefficients. Let \(z\equiv X^T\beta = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}\). The logistic model is stated by the probability that the binary categorical dependent variable equals one \(y_i=1\).

-

-(13.11)#\[\begin{equation}
+

+(13.11)#\[\begin{equation}
   P(y_i=1|X,\theta) = \frac{e^z}{1 + e^z}
 \end{equation}\]

 
Given this equation, we know that the probability of the dependent variable being zero \(y_i=0\) is just one minus the probability above.

-

-(13.12)#\[\begin{equation}
+

+(13.12)#\[\begin{equation}
   P(y_i=0|X,\theta) = 1 - P(y_i=1|X,\theta) = 1 - \frac{e^z}{1 + e^z} = \frac{1}{1 + e^z}
 \end{equation}\]

 
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 \(P(win)=0.8\), then we know that the probability of your favorite team losing that game is \(P(lose)=1-P(win)=0.2\). The odds ratio is the ratio of these two probabilities.

-

-(13.13)#\[\begin{equation}
+

+(13.13)#\[\begin{equation}
   \frac{P(win)}{P(lose)} = \frac{P(win)}{1 - P(win)} = \frac{0.8}{0.2} = \frac{4}{1} \quad\text{or}\quad 4
 \end{equation}\]

 
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.

 
In the logistic model, the odds ratio reduces the problem nicely.

-

-(13.14)#\[\begin{equation}
+

+(13.14)#\[\begin{equation}
   \frac{P(y_i=1|X,\theta)}{1 - P(y_i=1|X,\theta)} = \frac{\frac{e^z}{1 + e^z}}{\frac{1}{1 + e^z}} = e^z
 \end{equation}\]

 
If we take the log of both sides, we see that the log odds ratio is equal to the linear predictor \(z\equiv X^T\beta = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}\).

-

-(13.15)#\[\begin{equation}
+

+(13.15)#\[\begin{equation}
   \ln\left(\frac{P(y_i=1|X,\theta)}{1 - P(y_i=1|X,\theta)}\right) = z = \beta_0 + \beta_1 x_{1,i} + ...\beta_K x_{K,i}
 \end{equation}\]

 
So the interpretation of the coeficients \(\beta_k\) is that a one-unit increase of the variable \(x_{k,i}\) increases the odds ratio or the odds of \(y_i=1\) by \(\beta_{k,i}\) percent.

@@ -1175,18 +1175,18 @@ 
 Contents 

 
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 \(J\geq2\) categories in the dependent variable. Let \(\eta_j\) be the linear predictor for the \(j\)th category.
 $\( \eta_j\equiv \beta_{j,0} + \beta_{j,1}x_{1,i} + ...\beta_{j,K}x_{K,i} \quad\forall y_i = j \)$

 
The multinomial logit model gives the probability of \(y_i=j\) relative to some reference category \(J\) that is left out.

-

-(13.16)#\[\begin{equation}
+

+(13.16)#\[\begin{equation}
   Pr(y_i=j|X,\theta) = \frac{e^{\eta_j}}{1 + \sum_v^{J-1}e^{\eta_v}} \quad\text{for}\quad 1\leq j\leq J-1
 \end{equation}\]

 
Once the \(J-1\) sets of coefficients are estimated, the final \(J\)th set of coefficients are a residual based on the following expression.

-

-(13.17)#\[\begin{equation}
+

+(13.17)#\[\begin{equation}
   Pr(y_i=J|X,\theta) = \frac{1}{1 + \sum_v^{J-1}e^{\eta_v}}
 \end{equation}\]

 
The analogous log odds ratio interpretation applies to the multinomial logit model.

-

-(13.18)#\[\begin{equation}
+

+(13.18)#\[\begin{equation}
   \ln\left(\frac{Pr(y_i=j|X,\theta)}{Pr(y_i=J|X,\theta)}\right) = \eta_j = \beta_{j,0} + \beta_{j,1}x_{1,i} + ...\beta_{j,K}x_{K,i} \quad\text{for}\quad 1\leq j \leq J-1
 \end{equation}\]

 
This is the odds ratio of \(y_i=j\) relative to \(y_i=J\). The interpretation of the \(\beta_{j,k}\) coefficient is the predicted percentage change in the log odds ratio of \(y_i=j\) to \(y_i=J\) from a one-unit increase in variable \(x_{k,i}\).

diff --git a/python/SciPy.html b/python/SciPy.html
index 2c172d9..e5dfbd4 100644
--- a/python/SciPy.html
+++ b/python/SciPy.html
@@ -678,7 +678,7 @@ 
 Contents 

 
 message: The solution converged.
  success: True
   status: 1
-     fun: [-1.239e-13 -1.013e-13]
+     fun: [-1.235e-13 -1.017e-13]
        x: [ 8.463e-01  2.331e+00]
     nfev: 10
     fjac: [[-8.332e-01  5.529e-01]
@@ -688,7 +688,7 @@ 
 Contents 

 
 The solution for (x, y) is: [0.84630378 2.33101497]
 
-The error values for eq1 and eq2 at the solution are: [-1.2390089e-13 -1.0125234e-13]
+The error values for eq1 and eq2 at the solution are: [-1.23456800e-13 -1.01696429e-13]
 

 

 

diff --git a/searchindex.js b/searchindex.js
index 57fcde2..8517082 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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\ No newline at end of file
diff --git a/struct_est/SMM.html b/struct_est/SMM.html
index 3903a25..ee60edc 100644
--- a/struct_est/SMM.html
+++ b/struct_est/SMM.html
@@ -1289,7 +1289,7 @@ 
 Contents 

 

 

 

-
mu_SMM1_1= 612.3319048024534  sig_SMM1_1= 197.26303566149244
+
mu_SMM1_1= 612.3371352249138  sig_SMM1_1= 197.26434895262162
 

 

 

@@ -1317,18 +1317,18 @@ 
 Contents 

 

 
Data mean of scores = 341.90869565217395 , Data variance of scores = 7827.997292398056
 
-Model mean 1 = 341.6690130387879 , Model variance 1 = 7827.866466512297
+Model mean 1 = 341.6692110494425 , Model variance 1 = 7827.864496338213
 
-Error vector 1 = [-7.01013506e-04 -1.67125614e-05]
+Error vector 1 = [-7.00434373e-04 -1.69642445e-05]
 
 Results from scipy.opmtimize.minimize:
   message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
   success: True
    status: 0
-      fun: 4.916992449560748e-07
+      fun: 4.908960959342433e-07
         x: [ 6.123e+02  1.973e+02]
       nit: 17
-      jac: [-7.454e-07  2.357e-06]
+      jac: [-7.436e-07  2.350e-06]
      nfev: 72
      njev: 24
  hess_inv: <2x2 LbfgsInvHessProduct with dtype=float64>
@@ -1453,19 +1453,19 @@ 
 Contents 

 

 

 
Jacobian matrix of derivatives of moment error functions is:
-[[ 0.0008975  -0.00290434]
- [-0.00114133  0.004457  ]]
+[[ 0.00089749 -0.00290433]
+ [-0.00114132  0.00445698]]
 
 Weighting matrix W is:
 [[1. 0.]
  [0. 1.]]
 
 Variance-covariance matrix of estimated parameter vector is:
-[[602499.98744184 163793.83380005]
- [163793.83380005  44881.85524372]]
+[[602535.18442996 163802.17330123]
+ [163802.17330123  44883.79092131]]
 
-Std. err. mu_hat= 776.2087267235777
-Std. err. sig_hat= 211.8533814781294
+Std. err. mu_hat= 776.23139876583
+Std. err. sig_hat= 211.85794986573154
 

 

 

@@ -1554,12 +1554,12 @@ 
 Contents 

 

 

 
2nd stage est. of var-cov matrix of moment error vec across sims:
-[[ 0.00033411 -0.00142288]
- [-0.00142288  0.01592874]]
+[[ 0.00033411 -0.00142289]
+ [-0.00142289  0.01592879]]
 
 2nd state est. of optimal weighting matrix:
-[[4830.85282784  431.53075653]
- [ 431.53075653  101.32740541]]
+[[4830.88530228  431.53378728]
+ [ 431.53378728  101.32749623]]
 

 

 

@@ -1577,7 +1577,7 @@ 
 Contents 

 

 

 

-
mu_SMM2_1= 619.4295828414915  sig_SMM2_1= 199.0745499003818
+
mu_SMM2_1= 619.4303074248937  sig_SMM2_1= 199.0747813692372
 

 

 

@@ -1607,15 +1607,15 @@ 
 Contents 

  [-0.0011259   0.00443426]]
 
 Weighting matrix W is:
-[[4830.85282784  431.53075653]
- [ 431.53075653  101.32740541]]
+[[4830.88530228  431.53378728]
+ [ 431.53378728  101.32749623]]
 
 Variance-covariance matrix of estimated parameter vector is:
-[[2397.35566572  745.28979427]
- [ 745.28979427  232.01568064]]
+[[2397.38054356  745.29670501]
+ [ 745.29670501  232.01757158]]
 
-Std. err. mu_hat= 48.96279879380848
-Std. err. sig_hat= 15.232060945338231
+Std. err. mu_hat= 48.963052841479445
+Std. err. sig_hat= 15.232123016118733
@@ -1865,7 +1865,7 @@

Contents

-
mu_SMM4_1= 362.560593472098  sig_SMM4_1 46.57515195652185
+
mu_SMM4_1= 362.560593472098  sig_SMM4_1 46.5751519565219
   message: CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
   success: True
    status: 0
@@ -2016,8 +2016,8 @@ 
 Contents 

 [[33.48770545 23.53170341]
  [23.53170341 18.13459776]]
 
-Std. err. mu_hat= 5.786856266717139
-Std. err. sig_hat= 4.258473642422251
+Std. err. mu_hat= 5.786856266717126
+Std. err. sig_hat= 4.258473642422245
@@ -2139,11 +2139,11 @@

Contents

-
mu_SMM4_2= 362.5605400454758  sig_SMM4_2 46.57507128065559
+
mu_SMM4_2= 362.5605400454758  sig_SMM4_2 46.57507128065564
  message: Optimization terminated successfully
  success: True
   status: 0
-     fun: 0.9984266286568925
+     fun: 0.9984266286568926
        x: [ 3.626e+02  4.658e+01]
      nit: 1
      jac: [ 5.467e-02  8.255e-02]
@@ -2188,7 +2188,7 @@ 
 Contents 

 
 Error vector (pct. dev.) = [-0.98        0.04678571  0.11720721 -0.075     ]
 
-Criterion func val = 0.9984266286568925
+Criterion func val = 0.9984266286568926
@@ -2273,8 +2273,8 @@

Contents

[[3.53697411 2.47391182] [2.47391182 1.77184156]] -Std. err. mu_hat= 1.8806844791000032 -Std. err. sig_hat= 1.3311053920876743 +Std. err. mu_hat= 1.8806844791000217 +Std. err. sig_hat= 1.3311053920876885