name: xaringan-title class: left, middle # Econometr铆a I <br> ## Variables Cualitativas <br> <br> <img src="images/lognig.png" width="280" /> ### Carlos Yanes | Departamento de Econom铆a | 2024-05-05 --- class: middle, inverse .left-column[ # 馃槄 ] .right-column[ # Preguntas de la sesion anterior? ] --- background-size: 100% background-image: url(https://media.giphy.com/media/IkmyynhYBEqme9IMFH/giphy.gif) ??? Image test. Taken from gyfty. --- # Modelos de Regresi贸n M煤ltiple: (k) Par谩metros -- - Recordemos que: -- `$$y_{i}=\beta_{0}+\beta_{1}x_{1i}+\beta_{2}x_{2i}+\beta_{3}x_{3i}+ \cdots+\beta_{k-1}x_{(k-1)i}+ \mu_{i}$$` -- - Se tiene que: --
Un n煤mero de `\(k\)` par谩metros desconocidos que depende del n煤mero de variables de control. --
`\(k-1\)` regresores. --- # Modelos de Regresi贸n M煤ltiple: K Par谩metros -- Recuerde que de lo que tenemos estad铆sticamente, podemos expresarlo de forma **matricial**: -- - Una medida de variabilidad de la variable _dependiente_ es: -- `$$y'y = \sum \limits_{i=1}^{n} y_{i}^{2}$$` -- - De la descomposici贸n **ortogonal** de (Y) podemos tener: -- `$$\begin{aligned} y'y=& (\widehat{y}+u)'(\widehat{y}+u)\\ =& \widehat{y}'\widehat{y}+ 2\widehat{y}'\widehat{u}+\widehat{u}'\widehat{u}\\ =& \widehat{y}'\widehat{y}+\widehat{u}'\widehat{u} \end{aligned}$$` -- - Lo que tambi茅n puede ser establecido como: -- `$$\sum \limits_{i=1}^{n} y_{i}^{2}= \sum \limits_{i=1}^{n} \widehat{y}_{i}^{2} + \sum \limits_{i=1}^{n} \widehat{u}_{i}^{2}$$` --- # Modelos de Regresi贸n M煤ltiple: K Par谩metros Si de la expresi贸n anterior restamos su media con un vector de unos (1) llamado **f** de tama帽o `\(n \times 1\)` entonces encontramos: -- `$$(y-\textbf{f}\bar{y})'(y-\textbf{f}\bar{y})=(\widehat{y}-\textbf{f}\bar{y})'(\widehat{y}-\textbf{f}\bar{y})+\widehat{u}'\widehat{u}$$` -- - De tal forma que es lo mismo que: `$$\underbrace{\sum \limits_{i=1}^{n} (y_{i}-\bar{y})^{2}}_{SST}= \underbrace{\sum \limits_{i=1}^{n} (\widehat{y}_{i}-\bar{y})^{2}}_{SSE} + \underbrace{\sum \limits_{i=1}^{n} \widehat{u}_{i}^{2}}_{SSR}$$` -- Los t茅rminos de la expresi贸n **SST** hacen referencia a la suma total al cuadrado (fuente de variaci贸n principal), **SSE** es la suma al cuadrado del modelo o _suma explicada_ y por 煤ltimo, **SSR** la suma de los residuos al cuadrado de nuestro modelo. --- # Modelos de Regresi贸n M煤ltiple: K Par谩metros --
De lo anterior, para tener al **coeficiente de determinaci贸n**, la formula mas usada es: `$$R^{2}=1-\frac{SSR}{SST}$$` -- Note que `\(SSR= \sum_i \left( y_i - \hat{y}_i \right)^2 = \sum_i e_i^2\)` y que `\(SST= \sum_i \left( y_i - \bar{y}_i \right)^2\)` -- Para lo cual se debe **pensar** que: -- `$$\text{Si} \quad SSR \downarrow \; \Rightarrow R^{2} \; \uparrow$$` -- > A medida que se _adicionan_ nuevas variables al modelo de regresi贸n, automaticamente el .blue[R2] aumenta. -- - Al tener ese problema entonces, hay que entrar a solucionarlo. --- class: title-slide-section-grey, middle # Nuevamente.. El `\(R^2-Ajustado\)` <br> <img src="images/lognig.png" width="380" /> --- # Modelos de Regresi贸n M煤ltiple: `\(R^2\)` ajustado --
Una **soluci贸n** es .blue[penalizar] el n煤mero de variables, _p.e_ con el `\(R^2\)` **ajustado**. -- $$ \overline{R}^{2} = 1 - \dfrac{\sum_i \left( y_i - \hat{y}_i \right)^2/(n-k-1)}{\sum_i \left( y_i - \overline{y} \right)^2/(n-1)}$$ -- *Nota:* `\(R^2\)` Ajustado necesariamente no esta entre 0 y 1. -- ### Mucho cuidado!! -- Hay que tener en cuenta las ventajas y desventajas de a帽adir o quitar variables: -- **Menos variables** -- - Generalmente explican menos variaci贸n en `\(y\)` - Proporcionan interpretaciones y visualizaciones sencillas (*parsimoniosas*) - Puede ser necesario preocuparse por el sesgo de las variables omitidas --- # Modelos de Regresi贸n M煤ltiple: `\(R^2\)` ajustado -- **M谩s variables** -- - Es m谩s probable que se encuentren relaciones *esp煤reas* (estad铆sticamente significativas debido a la casualidad; no reflejan una verdadera relaci贸n a nivel de la poblaci贸n) -- - Es m谩s dif铆cil interpretar el modelo -- - Es posible que se pasen por alto variables importantes: sigue habiendo un sesgo de variables omitidas --- # Modelos de Regresi贸n M煤ltiple: `\(R^2\)` ajustado Tomemos los datos de _autos_ : <div class="tabwid"><style>.cl-725f0b14{}.cl-725b9ae2{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-725b9af6{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:italic;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-725ce6e0{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-725ce6ea{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 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0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-725cf15a{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-725cf15b{width:1.05in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-725cf162{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-725cf163{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-725cf164{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-725cf16c{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-725f0b14'><thead><tr style="overflow-wrap:break-word;"><th colspan="6"class="cl-725cf108"><p class="cl-725ce6e0"><span class="cl-725b9ae2">Tabla #1: Regresi贸n Simple</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-725cf108"><p class="cl-725ce6e0"><span class="cl-725b9ae2"></span></p></th><th class="cl-725cf112"><p class="cl-725ce6ea"><span class="cl-725b9ae2">Estimate</span></p></th><th class="cl-725cf113"><p class="cl-725ce6ea"><span class="cl-725b9ae2">Standard Error</span></p></th><th class="cl-725cf11c"><p class="cl-725ce6ea"><span class="cl-725b9ae2">t value</span></p></th><th class="cl-725cf11d"><p class="cl-725ce6ea"><span class="cl-725b9ae2">Pr(>|t|)</span></p></th><th class="cl-725cf11e"><p class="cl-725ce6e0"><span class="cl-725b9ae2"></span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-725cf11f"><p class="cl-725ce6e0"><span class="cl-725b9ae2">(Intercept)</span></p></td><td class="cl-725cf126"><p class="cl-725ce6ea"><span class="cl-725b9ae2">11,253.061</span></p></td><td class="cl-725cf127"><p class="cl-725ce6ea"><span class="cl-725b9ae2">1,170.813</span></p></td><td class="cl-725cf128"><p class="cl-725ce6ea"><span class="cl-725b9ae2">9.611</span></p></td><td class="cl-725cf130"><p class="cl-725ce6ea"><span class="cl-725b9ae2">0.0000</span></p></td><td class="cl-725cf131"><p class="cl-725ce6e0"><span class="cl-725b9ae2">***</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-725cf132"><p class="cl-725ce6e0"><span class="cl-725b9ae2">mpg</span></p></td><td class="cl-725cf13a"><p class="cl-725ce6ea"><span class="cl-725b9ae2">-238.894</span></p></td><td class="cl-725cf13b"><p class="cl-725ce6ea"><span class="cl-725b9ae2">53.077</span></p></td><td class="cl-725cf13c"><p class="cl-725ce6ea"><span class="cl-725b9ae2">-4.501</span></p></td><td class="cl-725cf144"><p class="cl-725ce6ea"><span class="cl-725b9ae2">0.0000</span></p></td><td class="cl-725cf145"><p class="cl-725ce6e0"><span class="cl-725b9ae2">***</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-725cf14e"><p class="cl-725ce6ea"><span class="cl-725b9af6">Signif. codes: 0 <= '***' < 0.001 < '**' < 0.01 < '*' < 0.05</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-725cf15a"><p class="cl-725ce6e0"><span class="cl-725b9ae2"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-725cf15a"><p class="cl-725ce6e0"><span class="cl-725b9ae2">Residual standard error: 2624 on 72 degrees of freedom</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-725cf15a"><p class="cl-725ce6e0"><span class="cl-725b9ae2">Multiple R-squared: 0.2196, Adjusted R-squared: 0.2087</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-725cf15a"><p class="cl-725ce6e0"><span class="cl-725b9ae2">F-statistic: 20.26 on 72 and 1 DF, p-value: 0.0000</span></p></td></tr></tfoot></table></div> --- # Modelos de Regresi贸n M煤ltiple: `\(R^2\)` ajustado Con dos variables: <div class="tabwid"><style>.cl-727aedd4{}.cl-7277dc0c{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-7277dc0d{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:italic;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-7279184c{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-7279184d{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-727920d0{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920da{width:0.965in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920db{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920dc{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920e4{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920e5{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920e6{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920ee{width:0.965in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920ef{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920f0{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920f1{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920f8{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920f9{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-727920fa{width:0.965in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792102{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792103{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792104{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279210c{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279210d{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279210e{width:0.965in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792116{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792117{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792118{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792120{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792121{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792122{width:0.965in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279212a{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279212b{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792134{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792135{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792136{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792137{width:0.965in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279213e{width:1.287in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-7279213f{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792140{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-72792148{width:0.4in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-727aedd4'><thead><tr style="overflow-wrap:break-word;"><th colspan="6"class="cl-727920d0"><p class="cl-7279184c"><span class="cl-7277dc0c">Tabla #2: Regresi贸n M煤ltiple</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-727920d0"><p class="cl-7279184c"><span class="cl-7277dc0c"></span></p></th><th class="cl-727920da"><p class="cl-7279184d"><span class="cl-7277dc0c">Estimate</span></p></th><th class="cl-727920db"><p class="cl-7279184d"><span class="cl-7277dc0c">Standard Error</span></p></th><th class="cl-727920dc"><p class="cl-7279184d"><span class="cl-7277dc0c">t value</span></p></th><th class="cl-727920e4"><p class="cl-7279184d"><span class="cl-7277dc0c">Pr(>|t|)</span></p></th><th class="cl-727920e5"><p class="cl-7279184c"><span class="cl-7277dc0c"></span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-727920e6"><p class="cl-7279184c"><span class="cl-7277dc0c">(Intercept)</span></p></td><td class="cl-727920ee"><p class="cl-7279184d"><span class="cl-7277dc0c">5,864.305</span></p></td><td class="cl-727920ef"><p class="cl-7279184d"><span class="cl-7277dc0c">5,888.103</span></p></td><td class="cl-727920f0"><p class="cl-7279184d"><span class="cl-7277dc0c">0.996</span></p></td><td class="cl-727920f1"><p class="cl-7279184d"><span class="cl-7277dc0c">0.3227</span></p></td><td class="cl-727920f8"><p class="cl-7279184c"><span class="cl-7277dc0c"> </span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-727920f9"><p class="cl-7279184c"><span class="cl-7277dc0c">mpg</span></p></td><td class="cl-727920fa"><p class="cl-7279184d"><span class="cl-7277dc0c">-173.702</span></p></td><td class="cl-72792102"><p class="cl-7279184d"><span class="cl-7277dc0c">87.723</span></p></td><td class="cl-72792103"><p class="cl-7279184d"><span class="cl-7277dc0c">-1.980</span></p></td><td class="cl-72792104"><p class="cl-7279184d"><span class="cl-7277dc0c">0.0516</span></p></td><td class="cl-7279210c"><p class="cl-7279184c"><span class="cl-7277dc0c"> .</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-7279210d"><p class="cl-7279184c"><span class="cl-7277dc0c">length</span></p></td><td class="cl-7279210e"><p class="cl-7279184d"><span class="cl-7277dc0c">21.286</span></p></td><td class="cl-72792116"><p class="cl-7279184d"><span class="cl-7277dc0c">22.793</span></p></td><td class="cl-72792117"><p class="cl-7279184d"><span class="cl-7277dc0c">0.934</span></p></td><td class="cl-72792118"><p class="cl-7279184d"><span class="cl-7277dc0c">0.3535</span></p></td><td class="cl-72792120"><p class="cl-7279184c"><span class="cl-7277dc0c"> </span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-72792121"><p class="cl-7279184d"><span class="cl-7277dc0d">Signif. codes: 0 <= '***' < 0.001 < '**' < 0.01 < '*' < 0.05</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-72792136"><p class="cl-7279184c"><span class="cl-7277dc0c"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-72792136"><p class="cl-7279184c"><span class="cl-7277dc0c">Residual standard error: 2626 on 71 degrees of freedom</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-72792136"><p class="cl-7279184c"><span class="cl-7277dc0c">Multiple R-squared: 0.2291, Adjusted R-squared: 0.2073</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-72792136"><p class="cl-7279184c"><span class="cl-7277dc0c">F-statistic: 10.55 on 71 and 2 DF, p-value: 0.0001</span></p></td></tr></tfoot></table></div> --- # Modelos de Regresi贸n M煤ltiple: `\(R^2\)` ajustado Con tres variables: <div class="tabwid"><style>.cl-7296e9ee{}.cl-72935810{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-7293581a{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:italic;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-72949824{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-7294982e{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 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0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-7296e9ee'><thead><tr style="overflow-wrap:break-word;"><th colspan="6"class="cl-7294a198"><p class="cl-72949824"><span class="cl-72935810">Tabla #3: Regresi贸n M煤ltiple con mas k</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-7294a198"><p class="cl-72949824"><span class="cl-72935810"></span></p></th><th class="cl-7294a1a2"><p class="cl-7294982e"><span class="cl-72935810">Estimate</span></p></th><th class="cl-7294a1a3"><p class="cl-7294982e"><span class="cl-72935810">Standard Error</span></p></th><th class="cl-7294a1ac"><p class="cl-7294982e"><span class="cl-72935810">t value</span></p></th><th class="cl-7294a1ad"><p class="cl-7294982e"><span class="cl-72935810">Pr(>|t|)</span></p></th><th class="cl-7294a1ae"><p class="cl-72949824"><span class="cl-72935810"></span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-7294a1b6"><p class="cl-72949824"><span class="cl-72935810">(Intercept)</span></p></td><td class="cl-7294a1b7"><p class="cl-7294982e"><span class="cl-72935810">14,542.434</span></p></td><td class="cl-7294a1b8"><p class="cl-7294982e"><span class="cl-72935810">5,890.632</span></p></td><td class="cl-7294a1c0"><p class="cl-7294982e"><span class="cl-72935810">2.469</span></p></td><td class="cl-7294a1c1"><p class="cl-7294982e"><span class="cl-72935810">0.0160</span></p></td><td class="cl-7294a1c2"><p class="cl-72949824"><span class="cl-72935810"> *</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-7294a1ca"><p class="cl-72949824"><span class="cl-72935810">mpg</span></p></td><td class="cl-7294a1cb"><p class="cl-7294982e"><span class="cl-72935810">-86.789</span></p></td><td class="cl-7294a1d4"><p class="cl-7294982e"><span class="cl-72935810">83.943</span></p></td><td class="cl-7294a1d5"><p class="cl-7294982e"><span class="cl-72935810">-1.034</span></p></td><td class="cl-7294a1d6"><p class="cl-7294982e"><span class="cl-72935810">0.3047</span></p></td><td class="cl-7294a1d7"><p class="cl-72949824"><span class="cl-72935810"> </span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-7294a1de"><p class="cl-72949824"><span class="cl-72935810">length</span></p></td><td class="cl-7294a1df"><p class="cl-7294982e"><span class="cl-72935810">-104.868</span></p></td><td class="cl-7294a1e0"><p class="cl-7294982e"><span class="cl-72935810">39.722</span></p></td><td class="cl-7294a1e8"><p class="cl-7294982e"><span class="cl-72935810">-2.640</span></p></td><td class="cl-7294a1e9"><p class="cl-7294982e"><span class="cl-72935810">0.0102</span></p></td><td class="cl-7294a1ea"><p class="cl-72949824"><span class="cl-72935810"> *</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-7294a1f2"><p class="cl-72949824"><span class="cl-72935810">weight</span></p></td><td class="cl-7294a1f3"><p class="cl-7294982e"><span class="cl-72935810">4.365</span></p></td><td class="cl-7294a1fc"><p class="cl-7294982e"><span class="cl-72935810">1.167</span></p></td><td class="cl-7294a1fd"><p class="cl-7294982e"><span class="cl-72935810">3.739</span></p></td><td class="cl-7294a1fe"><p class="cl-7294982e"><span class="cl-72935810">0.0004</span></p></td><td class="cl-7294a206"><p class="cl-72949824"><span class="cl-72935810">***</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-7294a207"><p class="cl-7294982e"><span class="cl-7293581a">Signif. codes: 0 <= '***' < 0.001 < '**' < 0.01 < '*' < 0.05</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-7294a21a"><p class="cl-72949824"><span class="cl-72935810"></span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-7294a21a"><p class="cl-72949824"><span class="cl-72935810">Residual standard error: 2415 on 70 degrees of freedom</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-7294a21a"><p class="cl-72949824"><span class="cl-72935810">Multiple R-squared: 0.3574, Adjusted R-squared: 0.3298</span></p></td></tr><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-7294a21a"><p class="cl-72949824"><span class="cl-72935810">F-statistic: 12.98 on 70 and 3 DF, p-value: 0.0000</span></p></td></tr></tfoot></table></div> --- # Modelos de Regresi贸n M煤ltiple: `\(R^2\)` ajustado -- - Nuestro modelo en la medida que hemos incorporado nuevas variables ha cambiado la m茅trica del `\(R^2\)`. -- | **Variables** | **Coeficiente R2** | **Variaci贸n R** | **R-Ajustado** | **Variaci贸n Ajustado** | |---------------|--------------------|-----------------|------------------------|------------------------| | 1 | 0.2196 | | 0.2087 | | | 2 | 0.2291 | 4% | 0.2073 | -0.6% | | 3 | 0.3574 | 56% | 0.3298 | 59% | -- <ru-blockquote> **Proporci贸n** de la _variaci贸n_ muestral de `\(Y\)` que es explicada por las `\(X_{i}\)`, o tambi茅n se puede definir como la variaci贸n en (Y) que es explicada por las variables **independientes** pero con _castigo_ en los grados de libertad por incluir esas nuevas variables.</ru-blockquote> -- - Otra forma de establecer la .blue[formula] del `\(R^2\)` ajustado es: -- `$$\overline{R}^{2} = 1 - \dfrac{SSR/(n-k-1)}{SST/(n-1)}$$` --- class: title-slide-section-red, middle # Sesgo por variable omitida <br> <img src="images/lognig.png" width="380" /> --- # Sesgo por variable omitida -- **El sesgo de la variable omitida** (SVO) surge cuando omitimos una variable que -- 1. Afecta a nuestra variable de resultado `\(y\)` -- 2. Se correlaciona con una variable explicativa `\(x_j\)`. -- Como su nombre indica, esta situaci贸n provoca un sesgo en nuestra estimaci贸n de `\(\beta_j\)`. -- **Ejemplo** -- Imagine un modelo de regresi贸n con varios individuos `\(i\)` de su nivel de ingresos -- $$ \text{Pago}_i = \beta_0 + \beta_1 \text{Escolaridad}_i + \beta_2 \text{Genero}_i + u_i $$ -- Donde -- - `\(\text{Escolaridad}_i\)` son los a帽os aprobados y cursados por el individuo `\(i\)`. -- - `\(\text{Genero}_i\)` una variable _indicador_ (dummy) del genero de `\(i\)` haciendo referencia a si este es masculino. --- # Sesgo por variable omitida Entonces -- - `\(\beta_1\)`: Es el retorno econ贸mico por cada a帽o de educaci贸n que tiene el individuo (*ceteris paribus*) -- - `\(\beta_2\)`: la prima por ser hombre (*ceteris paribus*) <br>Si `\(\beta_2 > 0\)` o `\(\beta_2 < 0\)`, entonces existe una **discriminaci贸n** contra las mujeres (hombres): ya que alguno(a)s reciben menos salario por raz贸n de su g茅nero. -- Para nuestro modelo poblacional -- `$$\text{Pago}_i = \beta_0 + \beta_1 \text{Escolaridad}_i + \beta_2 \text{Genero}_i + u_i$$` -- Si nos concentramos en la estimaci贸n solo de **Escolaridad**, es decir, omitimos la variable de genero, el modelo ahora ser谩 _p.e._, -- `$$\text{Pago}_i = \beta_0 + \beta_1 \text{Escolaridad}_i + \left(\beta_2 \text{Genero}_i + u_i\right)$$` -- Ahora vamos a tener -- `$$\text{Pago}_i = \beta_0 + \beta_1 \text{Escolaridad}_i + \varepsilon_i$$` -- Donde `\(\varepsilon_i = \beta_2\; \text{Genero}_i + u_i\)`. --- # Sesgo por variable omitida -- La condici贸n de **exogeneidad** ya no se cumple. pero incluso si -- `\(\mathop{\boldsymbol{E}}\left[ u | X \right] = 0\)`, no es cierto que `\(\mathop{\boldsymbol{E}}\left[ \varepsilon | X \right] = 0\)` de tal forma que `\(\beta_2 \neq 0\)`. -- Esto es, `\(\mathop{\boldsymbol{E}}\left[ \varepsilon | \text{genero} = 1 \right] = \beta_2 + \mathop{\boldsymbol{E}}\left[ u | \text{genero} = 1 \right] \neq 0\)`. -- **Ahora entonces MCO es Sesgado.** --- # Sesgo por variable omitida Veamos un gr谩fico: <img src="Class06_files/figure-html/plot1-1.svg" style="display: block; margin: auto;" /> --- # Sesgo por variable omitida Tenemos un modelo: `\(\text{Pago}_i = 20.5 + 10.4 \times \text{Escolaridad}_i\)` <img src="Class06_files/figure-html/plot2-1.svg" style="display: block; margin: auto;" /> --- # Sesgo por variable omitida Variable omitida: Genero (**<font color="#e64173">Masculino</font>** y **<font color="#314f4f">Femenino</font>**) <img src="Class06_files/figure-html/plot3-1.svg" style="display: block; margin: auto;" /> --- # Sesgo por variable omitida Variable omitida: Genero (**<font color="#e64173">Masculino</font>** y **<font color="#314f4f">Femenino</font>**) <img src="Class06_files/figure-html/plot4-1.svg" style="display: block; margin: auto;" /> --- # Sesgo por variable omitida Regresi贸n insesgada: `\(\text{Pago}_i = 20.4 + 10.4 \times \text{Escolaridad}_i + 0.1 \times \text{Genero}_i\)` <img src="Class06_files/figure-html/plot5-1.svg" style="display: block; margin: auto;" /> --- <img src="Class06_files/figure-html/venng2-1.svg" style="display: block; margin: auto;" /> --- class: title-slide-section-grey, middle # Sesgo por variable omitida <br> <img src="images/lognig.png" width="380" /> --- # Sesgo por variable omitida -- ## Como corregirlo -- 1. No **omita variables** -- 2. Instrumentalizando o por MCO (dos etapas) --- # Modelos de Regresi贸n M煤ltiple: variables irrelevantes -- Se tiene: `$$y= \beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\mu$$` -- - `\(x_{3}\)` no tiene efecto sobre `\(y\)` una vez se control贸 por `\(x_{1}\)` y `\(x_{2}\)`, es decir, `\(\beta_{3}=0\)`. -- - Puede o no estar **correlacionado** `\(x_{3}\)` con `\(x_{1}\)` o `\(x_{2}\)`. -- - `\(E[y|x_{1},x_{2},x_{3}]=E[y|x_{1},x_{2}]=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}\)` -- - No sabemos que `\(\beta_{3}=0\)`, por lo que estimamos: `$$\widehat{y}= \widehat{\beta}_{0}+\widehat{\beta}_{1}x_{1}+\widehat{\beta}_{2}x_{2}+\widehat{\beta}_{3}x_{3}+\mu$$` -- - .blue[Sobre-especificar] el modelo, no afecta el **insesgamiento** de los estimadores de MCO. --- class: title-slide-section-grey, middle # Variables Cualitativas <br> <img src="images/lognig.png" width="380" /> --- # Variables Cualitativas --
**Dummy** Son variables que son catalogadas como _ficticias_ y su manera de abordar se hace a partir de una variable _binaria_. --
Generalmente son variables de **escala nominal** o caracter铆stica. -- `$$D=\left\{\begin{matrix} 1 & \text{si la caracter铆stica est谩 presente}\\ 0 & \text{si la caracter铆stica no est谩 presente} \end{matrix}\right.$$` --
Debe escogerse un grupo **base** con la cual se hacen _comparaciones_ (elecci贸n puede ser arbitraria). --- # Variables Cualitativas | **Obs** | **Ingreso** | **Educaci贸n** | **Experiencia** | **Masculino** | |--------------|------------------|--------------------|----------------------|-------------------| | 1 | 3.15 | 11 | 12 | 0 | | 2 | 2.92 | 12 | 3 | 1 | | 3 | 5.4 | 16 | 5 | 0 | | `\(\vdots\)` | 6.00 | 14 | 7 | 0 | | 324 | 11.2 | 15 | 6 | 1 | | 325 | 15.3 | 16 | 12 | 0 | --- # Variables Cualitativas <ru-blockquote>Son de tipo **categ贸rico** p.e: (sexo, estrato, sector, religi贸n, raza, etc). En los modelos debemos _transformarlas_ en variables *binarias* o de tipo (0,1)</ru-blockquote> -- - Considere por ejemplo la siguiente estimaci贸n: -- `$$Ingreso_i=\beta_0+ \beta_1 \; \text{Masculino}_i+ \mu_i$$` -- _Donde_
`\(Ingreso_i\)` es una variable _continua_ que mide el nivel de ingreso (pago recibido) de un individuo cualquiera. --
`\(Masculino_i\)` es una variable _cualitativa_ que mide define el sexo o genero de un individuo cualquiera. -- > La interpretaci贸n de `\(\beta_1\)` es la diferencia esperada entre hombres y mujeres en el ingreso. El par谩metro `\(\beta_0\)` es el ingreso promedio de las mujeres `\((Masculino_{i}=0)\)` y la parte de `\(\beta_0+\beta_1\)` es el ingreso promedio de las **hombres** --- # Variables Cualitativas Derivaci贸n -- $$ `\begin{aligned} \mathop{\boldsymbol{E}}\left[ \text{Ingreso} | \text{Femenino} \right] &= \mathop{\boldsymbol{E}}\left[ \beta_0 + \beta_1\times 0 + u_i \right] \\ &= \mathop{\boldsymbol{E}}\left[ \beta_0 + 0 + u_i \right] \\ &= \beta_0 \end{aligned}` $$ -- $$ `\begin{aligned} \mathop{\boldsymbol{E}}\left[ \text{Ingreso} | \text{Masculino} \right] &= \mathop{\boldsymbol{E}}\left[ \beta_0 + \beta_1\times \color{#e64173}{1} + u_i \right] \\ &= \mathop{\boldsymbol{E}}\left[ \beta_0 + \beta_1 + u_i \right] \\ &= \beta_0 + \beta_1 \end{aligned}` $$ -- **Nota:** Si no hay mas _variables_ explicativas o controles, entonces `\(\hat{\beta}_1\)` es igual a la diferencia de medias, _p.e._, `\(\overline{x}_\text{Masculino} - \overline{x}_\text{Femenino}\)`. -- **Nota<sub>2</sub>:** El supuesto de *mantener todo lo demas constante* se aplica de igual manera para los modelos de regresi贸n m煤ltiple. --- # Variables Cualitativas `\(y_i = \beta_0 + \beta_1 x_i + u_i\)` para variable binaria `\(Masculino_i\)` o `\(x_i = \{\color{#314f4f}{0}, \, \color{#e64173}{1}\}\)` <img src="Class06_files/figure-html/ploty 1-1.svg" style="display: block; margin: auto;" /> --- # Variables Cualitativas `\(y_i = \beta_0 + \beta_1 x_i + u_i\)` para variable binaria `\(Masculino_i\)` o `\(x_i = \{\color{#314f4f}{0}, \, \color{#e64173}{1}\}\)` <img src="Class06_files/figure-html/ploty 2-1.svg" style="display: block; margin: auto;" /> --- # Variables Cualitativas : mas categor铆as --
Si tuvi茅ramos una variable _ordinal_ con 3 **categor铆as** -- - El grupo _base_ es el primero por "default" _se omite por multicolinealidad_ . -- `$$D_{2}=\left\{\begin{matrix} 1 & \text{para grupo 2}\\ 0 & \text{en otro caso} \end{matrix}\right. \quad \quad D_{3}=\left\{\begin{matrix} 1 & \text{para grupo 3}\\ 0 & \text{en otro caso} \end{matrix}\right.$$` -- - Lo que tendr铆amos a modo de ecuaciones: -- `$$\begin{aligned} y &= \beta_{0} + \left( \beta_{2}-\beta_{0} \right) D_{2} +\left( \beta_{3}-\beta_{0} \right) D_{3} +\beta_i x_i + \mu \\ &= \beta_{0} + \alpha_{2} D_{2} + \alpha_{3} D_{3} +\beta_i x_i + \mu \\ \end{aligned}$$` --
Hay que mirar que todas son _diferencias_ `\(\alpha_i\)` _p.e_ `\((\alpha_2, \alpha_3)\)` son los mismos par谩metros de la regresi贸n, solo que son diferencias con respecto al grupo base. --- # Variables Cualitativas en
.pull-left[ ``` #> #> Call: #> lm(formula = y ~ x, data = base_1) #> #> Residuals: #> Min 1Q Median 3Q Max #> -606.10 -249.92 -11.71 296.38 704.54 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 4916.9 161.7 30.401 2.93e-16 *** #> xB 233.9 220.4 1.061 0.303 #> xC -119.9 220.4 -0.544 0.593 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 396.2 on 17 degrees of freedom #> Multiple R-squared: 0.1448, Adjusted R-squared: 0.04422 #> F-statistic: 1.44 on 2 and 17 DF, p-value: 0.2645 ``` ] .pull-right[ - **R** y muchos softwares autom谩ticamente generan las dummies m煤ltiple - La variable `\(x\)` hace referencia a los tipos de A, B y C respectivamente. - Note que **A** es el grupo base o de referencia, es decir, `\(\beta_0\)` es el _promedio_ de esa variable. - Los par谩metros `\(xB\)` y `\(xC\)` son en efecto `\(\beta_2\)` y `\(\beta_3\)` que son las diferencias con respecto al grupo de **referencia** o base. - La significancia es ut铆l para decir si existe o no diferencias entre grupos. ] --- # Variables Cualitativas en
.pull-left[ ``` #> #> Call: #> lm(formula = y ~ 0 + x, data = base_1) #> #> Residuals: #> Min 1Q Median 3Q Max #> -606.10 -249.92 -11.71 296.38 704.54 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> xA 4916.9 161.7 30.40 2.93e-16 *** #> xB 5150.8 149.7 34.40 < 2e-16 *** #> xC 4797.0 149.7 32.04 < 2e-16 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 396.2 on 17 degrees of freedom #> Multiple R-squared: 0.9946, Adjusted R-squared: 0.9937 #> F-statistic: 1045 on 3 and 17 DF, p-value: < 2.2e-16 ``` ] .pull-right[ - Cuando se da la opci贸n sin intercepto, todas las caracter铆sticas de la variable **cualitativa** muestran sus respectivos promedios. - Esta opci贸n solo se usa mas como informaci贸n que como objetivo final de la estimaci贸n de la regresi贸n. - El _asunto_ de _omitir_ una de las caracter铆sticas de la variable **categ贸rica** es para evitar caer en la trampa de _dummies_ y entonces tener el problema de multicolinealidad. ] --- # Variables Cualitativas en
.pull-left[ ``` #> #> Call: #> lm(formula = y ~ x, data = base_1) #> #> Residuals: #> Min 1Q Median 3Q Max #> -606.10 -249.92 -11.71 296.38 704.54 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 5150.8 149.7 34.399 <2e-16 *** #> xA -233.9 220.4 -1.061 0.303 #> xC -353.9 211.8 -1.671 0.113 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 396.2 on 17 degrees of freedom #> Multiple R-squared: 0.1448, Adjusted R-squared: 0.04422 #> F-statistic: 1.44 on 2 and 17 DF, p-value: 0.2645 ``` ] .pull_right[ - En **R** para cambiar el grupo de referencia solo es usar la opci贸n `relevel`, _p.e_ : `datos$x<- relevel(datos$x,"B")`. - Recuerde que la variable categ贸rica o cualitativa debe ser clasificada _primero_ como `factor` para poder interactuar con ella _p.e_ : `datos$x<-as.factor(datos$x)`. Si ya su variable de entrada es clasificada como factor, se puede omitir esta parte. - Las diferencias ahora son con base a la opci贸n **B**. ] --- class: middle, inverse, center # 馃數 Y de las interacciones? --- class: inverse # Variables Cualitativas con interacciones -- Las interacciones permiten que el efecto de una variable cambie en funci贸n del nivel de otra variable. -- **Preguntas** -- 1. 驴Cambia el efecto de la escolarizaci贸n sobre el salario en funci贸n del genero (sexo)? -- 1. 驴Cambia el efecto del g茅nero en el salario seg煤n la raza? -- 1. 驴Cambia el efecto de la escolaridad en el salario seg煤n la experiencia? --- # Variables Cualitativas con interacciones --
Dos _variables_ cualitativas: sexo, estado civil. --
Grupo <span style="color:blue">**base**</span>: hombre soltero. -- `$$Mujer=\left\{\begin{matrix} 1 & \text{si es mujer}\\ 0 & \text{si es hombre} \end{matrix}\right. \quad \quad Casado=\left\{\begin{matrix} 1 & \text{si est谩 casado}\\ 0 & \text{si est谩 soltero} \end{matrix}\right.$$` -- - La base de datos puede ser: | **Obs** | **Ingreso** | **Genero (Fem=1)** | **E. Civil (Cas=1)** | **Interacci贸n** | |--------------|------------------|--------------------|----------------------|-------------------| | 1 | 3.15 | 0 | 1 | 0 | | 2 | 2.92 | 1 | 0 | 0 | | 3 | 5.4 | 0 | 1 | 0 | | `\(\vdots\)` | 6.00 | 0 | 0 | 0 | | 324 | 11.2 | 1 | 1 | 1 | | 325 | 15.3 | 0 | 0 | 0 | --- # Variables Cualitativas con interacciones --
Suponga que tiene el siguiente modelo: `$$y= \beta_0+\beta_1 \; femenino + \beta_2 casado + \beta_3 \; femenino \times casado +\beta_i x_i + \mu_i$$` `$$\begin{aligned} E(y_{i}|x_{i}, mujer=0, \: casado=0)&= \beta_0+\beta x_{i}\\ E(y_{i}|x_{i}, mujer=0, \: casado=1)&= \beta_0+\beta_2 +\beta x_{i}\\ E(y_{i}|x_{i}, mujer=1, \: casado=0)&= \beta_0+ \beta_1 +\beta x_{i}\\ E(y_{i}|x_{i}, mujer=1, \: casado=1)&= \beta_0+ \beta_1 + \beta_2 + \beta_3 + \beta x_{i} \end{aligned}$$` --
Donde `\(\beta_1\)` es el _efecto diferencial de ser mujer_; `\(\beta_2\)` es el <span style="color:blue">efecto diferencial de ser casado</span> y `\(\beta_3\)` es el <span style="color:black">**efecto diferencial de ser mujer casada**</span>. Puede probarse si _diferencial_ en sexo (estado civil) depende del estado civil (sexo). `\(H_{0}:\beta_3=0\)`. --- # Variables Cualitativas con interacciones --
Tomemos ahora otro ejemplo pero haciendo **interacci贸n** con una variable _cuantitativa_. --
Mediremos ahora en un nuevo modelo: el salario, el genero y a帽adiremos la escolaridad o n煤mero de a帽os de estudio de la persona. `$$Salario_i= \beta_0+\beta_1 \; Femenino_i + \beta_2 \; Escolaridad_i +\mu_i$$` _En el anterior se mira la escolaridad de igual forma o manera para todos._ -- - Al a帽adir un t茅rmino de **interacci贸n**, se hace con el objeto de ver como varia la escolaridad por genero o grupo. El modelo entonces es: -- `\(Salario_i= \beta_0+\beta_1 \; Femenino_i + \beta_2 \; Escolaridad_i +\color{#2b59c3}{\beta_3} \; Femenino_i \times \color{#e64173}{Escolaridad_i}+\mu_i\)` --- # Variables Cualitativas con interacciones La _escolaridad_ tiene el mismo efecto para todos (**<font color="#e64173">F</font>**) y para (**<font color="#314f4f">M</font>**) <img src="Class06_files/figure-html/it graph 1-1.svg" style="display: block; margin: auto;" /> --- # Variables Cualitativas con interacciones La _escolaridad_ tiene distinto efecto para los grupos de (**<font color="#e64173">F</font>**) y (**<font color="#314f4f">M</font>**) <img src="Class06_files/figure-html/it graph 2-1.svg" style="display: block; margin: auto;" /> --- # Variables Cualitativas con interacciones -- > La interpretaci贸n del _efecto de la interacci贸n_ puede ser un poco complejo, pero la clave<sup>.pink[*]</sup> es entender la parte matematica. .footnote[.pink[*] Como suele ocurrir con la econometr铆a.] `$$\text{Ingreso}_i = \beta_0 + \beta_1 \, \text{Femenino}_i + \beta_2 \, \text{Escolaridad}_i + \beta_3 \, \text{Femenino}_i\times\text{Escolaridad}_i + u_i$$` -- Rendimiento esperado de un a帽o adicional de escolarizaci贸n para las mujeres: `$$\begin{aligned} \mathop{\boldsymbol{E}}\left[ \text{Ingreso}_i | \text{Femenino} \land \text{Escolaridad} = \phi + 1 \right] - \mathop{\boldsymbol{E}}\left[ \text{Ingreso}_i | \text{Femenino} \land \text{Escolaridad} = \phi \right] &= \\ \mathop{\boldsymbol{E}}\left[ \beta_0 + \beta_1 (\phi+1) + \beta_2 + \beta_3 (\phi + 1) + u_i \right] - \mathop{\boldsymbol{E}}\left[ \beta_0 + \beta_1 \phi + \beta_2 + \beta_3 \phi + u_i \right] &= \\ \beta_1 + \beta_3 \end{aligned}$$` -- Del mismo modo, `\(\beta_1\)` da el rendimiento esperado de un a帽o adicional de escolarizaci贸n para los hombres. As铆, `\(\beta_3\)` da la **diferencia en los rendimientos de la escolarizaci贸n** para mujeres y hombres. --- class: title-slide-section-grey # Bibliograf铆a
Gujarati, D. N., & Porter, D. C. (2011). *Econometria B谩sica*. Ed. Porto Alegre: AMGH..
Stock, J. H., Watson, M. W., & Larri贸n, R. S. (2012). *Introducci贸n a la Econometr铆a*.
Wooldridge, J. M. (2015). *Introductory econometrics: A modern approach*. Cengage learning. --- class: title-slide-final, middle # Gracias por su atenci贸n! ## Alguna pregunta adicional? ### Carlos Andres Yanes Guerra
cayanes@uninorte.edu.co
keynes37