class: center, middle, inverse, title-slide .title[ # ScPoEconometrics: Advanced ] .subtitle[ ## Binary Response Models ] .author[ ### Bluebery Planterose ] .date[ ### SciencesPo Paris </br> 2023-04-04 ] --- layout: true <div class="my-footer"><img src="data:image/png;base64,#../../img/logo/ScPo-shield.png" style="height: 60px;"/></div> --- # Where Are We At? .pull-left[ **Last Time** * Panel Data Estimation * The *fixed effects estimator* * `fixest` package ] -- .pull-right[ **Today** 1. Binary Response Models! 2. Another cool app! 😎 ] --- class: separator, middle # Binary Response Models --- # Binary Response Models .pull-left[ So far, our models looked like this: `$$\begin{align} y &= b_0 + b_1 x + e \\ e &\sim N\left(0,\sigma^2\right) \end{align}$$` * The distributional assumption on `\(e\)`: * In priniciple implies that `\(y \in \mathbb{R}\)`. * test scores, earnings, crime rates, etc. are all continuous outcomes. ✅ ] -- .pull-right[ But some outcomes are clearly binary (i.e., either `TRUE` or `FALSE`): * You either work or you don't, * You either have children or you don't, * You either bought a product or you didn't, * You flipped a coin and it came up either heads or tails. ] --- # Binary Outcomes * Outcomes restricted to `FALSE` vs `TRUE`, or `0` vs `1`. * We'd have `\(y \in \{0,1\}\)`. * In those situations we are primarily interested in estimating the **response probability** or the **probability of success**: `$$p(x) = \Pr(y=1 | x)$$` * how does `\(p(x)\)` change as we change `\(x\)`? * we ask >If we increase `\(x\)` by one unit, how would the probability of `\(y=1\)` change? --- # Remembering Bernoulli Fun .pull-left[ Remember the [Bernoulli Distribution?](https://en.wikipedia.org/wiki/Bernoulli_distribution): We call a random variable `\(y \in \{0,1\}\)` such that `$$\begin{align} \Pr(y = 1) &= p \\ \Pr(y = 0) &= 1-p \\ p &\in[0,1] \end{align}$$` a *Bernoulli* random variable. ] -- .pull-right[ For us: *condition* those probabilities on a covariate `\(x\)` `$$\begin{align} \Pr(y = 1 | X = x) &= p(x) \\ \Pr(y = 0 | X = x) &= 1-p(x) \\ p(x) &\in[0,1] \end{align}$$` * Partcularly: *expected value* (i.e. the average) of `\(Y\)` given `\(x\)` $$ E[y | x] = p(x) \times 1 + (1-p(x)) \times 0 = p(x) $$ * We often model **conditional expectations** 😉 ] --- # The Linear Probability Model (LPM) * The simplest option. Model the response probability as $$ \Pr(y = 1 | x) = p(x) = \beta_0 + \beta_1 x_1 + \dots + \beta_K x_K $$ * Interpretation: *a 1 unit change in `\(x_1\)`, say, results in a change of `\(p(x)\)` of `\(\beta_1\)`.* ## Example: Mroz (1987) * Female labor market participation * How does `inlf` (*in labor force*) status depend on non-wife household income, her education, age and number of small children? --- # Mroz 1987 ```r data(mroz, package = "wooldridge") plot(factor(inlf) ~ age, data = mroz, ylevels = 2:1, ylab = "in labor force?") ``` <img src="data:image/png;base64,#07-probit_files/figure-html/unnamed-chunk-1-1.svg" style="display: block; margin: auto;" /> --- # Running the LPM .pull-left[ ```r LPM = lm(inlf ~ nwifeinc + educ + exper + I(exper^2) + age +I(age^2) + kidslt6, mroz) broom::tidy(LPM) ``` ``` ## # A tibble: 8 × 5 ## term estimate std.error statistic p.value ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 (Intercept) 0.322 0.486 0.662 5.08e- 1 ## 2 nwifeinc -0.00343 0.00145 -2.36 1.86e- 2 ## 3 educ 0.0375 0.00735 5.10 4.33e- 7 ## 4 exper 0.0383 0.00577 6.63 6.44e-11 ## 5 I(exper^2) -0.000565 0.000189 -2.98 2.96e- 3 ## 6 age -0.00112 0.0225 -0.0497 9.60e- 1 ## 7 I(age^2) -0.000182 0.000258 -0.706 4.80e- 1 ## 8 kidslt6 -0.260 0.0341 -7.64 6.72e-14 ``` ] .pull-right[ * **identical** to our previous linear regression models * Just `inlf` takes on only two values, 0 or 1. * Results: non-wife income increases by 10 (i.e 10,000 USD), `\(p(x)\)` falls by 0.034 (that's a small effect!), * an additional small child would reduce the probability of work by 0.26 (that's large). * So far, so simple. ✌️ ] --- # LPM: Predicting negative probabilities?! .pull-left[ <img src="data:image/png;base64,#07-probit_files/figure-html/unnamed-chunk-3-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ <br> <br> * LPM predictions of `\(p(x)\)` are not guaranteed to lie in unit interval `\([0,1]\)`. * Remember: `\(e \sim N\left(0,\sigma^2\right)\)` * here, some probs smaller than zero! * Particularly annoying if you want *predictions*: What is a probability of -0.3? 🤔 ] --- # LPM in Saturated Model: No Problem! .left-wide[ ```r library(dplyr) mroz %<>% # classify age into 3 and huswage into 2 classes mutate(age_fct = cut(age,breaks = 3,labels = FALSE), huswage_fct = cut(huswage, breaks = 2,labels = FALSE)) %>% mutate(classes = paste0("age_",age_fct,"_hus_",huswage_fct)) LPM_saturated = mroz %>% lm(inlf ~ classes, data = .) broom::tidy(LPM_saturated) ``` ``` ## # A tibble: 6 × 5 ## term estimate std.error statistic p.value ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 (Intercept) 0.611 0.0277 22.0 2.98e-83 ## 2 classesage_1_hus_2 -0.611 0.350 -1.75 8.11e- 2 ## 3 classesage_2_hus_1 -0.0257 0.0404 -0.635 5.25e- 1 ## 4 classesage_2_hus_2 -0.277 0.203 -1.37 1.72e- 1 ## 5 classesage_3_hus_1 -0.149 0.0494 -3.01 2.72e- 3 ## 6 classesage_3_hus_2 -0.111 0.350 -0.317 7.51e- 1 ``` ] .right-thin[ * *saturated model* : only have dummy explanatory variables * Each class: `\(p(x)\)` *within that cell*. ] --- # LPM in Saturated Model: No Problem! .left-wide[ <img src="data:image/png;base64,#07-probit_files/figure-html/saturated-1.svg" style="display: block; margin: auto;" /> ] .right-thin[ * Each line segment: `\(p(x)\)` *within that cell*. * E.g. women from the youngest age category and lowest husband income (class `age_1_hus_1`) have the highest probability of working (0.611). ] --- class: inverse # Task 1 (10 Minutes): Saturated LPM Define a *saturated* LPM as before $$ \Pr(y = 1 | x) = p(x) = \beta_0 + \beta_1 x_1 + \dots + \beta_K x_K $$ but restrict all `\(x_j \in \{0,1\}\)`. 1. Create a binary indicator `age_lt_50 = 1` for age smaller than 50 and `0` else and same for `husage_lt_50`. 1. Run a full interactions model (use the `*` syntax in your formula) of `age_lt_50 = 1` interacted with `husage_lt_50`. I.e. run the following LPM: $$ \Pr(y = 1 | x) =\beta_0 + \beta_1 \text{age_lt_50} + \beta_2 \text{husage_lt_50} + \beta_3 \times \text{age_lt_50} \times \text{husage_lt_50} $$ 1. `predict` `\(\Pr(y = 1 | x)\)` for each observation using your LPM. 1. What's the probability for a woman younger than 50 with a husband younger than 50? 1. make a plot similar to the one on the previous slide. --- # Nonlinear Binary Response Models In this class of models we change the way we model the response probability `\(p(x)\)`. Instead of the simple linear structure from above, we write $$ \Pr(y = 1 | x) = p(x) = G \left(\beta_0 + \beta_1 x_1 + \dots + \beta_K x_K \right) $$ * *almost* identical to LPM! * except the *linear index* `\(\beta_0 + \beta_1 x_1 + \dots + \beta_K x_K\)` is now inside some function `\(G(\cdot)\)`. * Main property of `\(G\)`: transforms any `\(z\in \mathbb{R}\)` into a number in the interval `\((0,1)\)`. * This immediately solves our problem of getting weird predictions for probabilities. --- # `\(G\)`: **probit** and **logit** .left-wide[ <img src="data:image/png;base64,#07-probit_files/figure-html/cdfs-1.svg" style="display: block; margin: auto;" /> ] .right-thin[ <br> For both **probit** and **logit** we see that: 1. any value `\(x\)` results in a value `\(p(x)\)` between 0 and 1 1. the higher `\(x\)`, the higher the resulting `\(p(x)\)`. 1. Logit has *fatter tails* than Probit. ] --- # Running probit and logit in `R`: the `glm` function * We use the `glm` function to run a **generalized linear model** * This *generalizes* our standard linear model. We have to specify a `family` and a `link`: ```r probit <- glm(inlf ~ age, data = mroz, family = binomial(link = "probit")) logit <- glm(inlf ~ age, data = mroz, family = binomial(link = "logit")) ``` --- # Interpretation .pull-left[ ```r modelsummary::modelsummary(list("probit" = probit,"logit" = logit)) ``` <table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:center;"> probit </th> <th style="text-align:center;"> logit </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> (Intercept) </td> <td style="text-align:center;"> 0.707 </td> <td style="text-align:center;"> 1.136 </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.248) </td> <td style="text-align:center;"> (0.398) </td> </tr> <tr> <td style="text-align:left;"> age </td> <td style="text-align:center;"> −0.013 </td> <td style="text-align:center;"> −0.020 </td> </tr> <tr> <td style="text-align:left;box-shadow: 0px 1px"> </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.006) </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.009) </td> </tr> <tr> <td style="text-align:left;"> Num.Obs. </td> <td style="text-align:center;"> 753 </td> <td style="text-align:center;"> 753 </td> </tr> <tr> <td style="text-align:left;"> AIC </td> <td style="text-align:center;"> 1028.9 </td> <td style="text-align:center;"> 1028.9 </td> </tr> <tr> <td style="text-align:left;"> BIC </td> <td style="text-align:center;"> 1038.1 </td> <td style="text-align:center;"> 1038.1 </td> </tr> <tr> <td style="text-align:left;"> Log.Lik. </td> <td style="text-align:center;"> −512.442 </td> <td style="text-align:center;"> −512.431 </td> </tr> <tr> <td style="text-align:left;"> F </td> <td style="text-align:center;"> 4.828 </td> <td style="text-align:center;"> 4.858 </td> </tr> <tr> <td style="text-align:left;"> RMSE </td> <td style="text-align:center;"> 0.49 </td> <td style="text-align:center;"> 0.49 </td> </tr> </tbody> </table> ] .pull-right[ * probit coefficient for `age` is -0.013 * logit: -0.02 for logit, * impact of age on the prob of working is **negative** * However, **how** negative? We can't tell! ] --- # Interpretation The model is $$ \Pr(y = 1 | \text{age})= G \left(x \beta\right) = G \left(\beta_0 + \beta_1 \text{age} \right) $$ and the *marginal effect* of `age` on the response probability is `$$\frac{\partial{\Pr(y = 1 | \text{age})}}{ \partial{\text{age}}} = g \left(\beta_0 + \beta_1 \text{age} \right) \beta_1$$` * function `\(g\)` is defined as `\(g(z) = \frac{dG}{dz}(z)\)` - the first derivative function of `\(G\)` (i.e. the *slope* of `\(G\)`). * given `\(G\)` that is nonlinear, this means that `\(g\)` will be non-constant. You are able to try this out yourself using this [app here](https://floswald.shinyapps.io/marginal_effects_of_logit_probit/): ```r ScPoApps::launchApp("marginal_effects_of_logit_probit") ``` or online --- # Interpretation So you can see that there is not one single *marginal effect* in those models, as that depends on *where we evaluate* the previous expression. In practice, there are two common approaches: 1. report effect at the average values of `\(x\)`: `$$g(\bar{x} \beta) \beta_j$$` 1. report the sample average of all marginal effects: `$$\frac{1}{n} \sum_{i=1}^N g(x_i \beta) \beta_j$$` Thankfully there are packages available that help us to compute those marginal effects fairly easily. One of them is called [`mfx`](https://cran.r-project.org/web/packages/mfx/), and we would use it as follows: --- # Interpretation ```r f <- "inlf ~ age + kidslt6 + nwifeinc" # setup a formula glms <- list() glms$probit <- glm(formula = f, data = mroz, family = binomial(link = "probit")) glms$logit <- glm(formula = f, data = mroz, family = binomial(link = "logit")) # now the marginal effects versions glms$probitMean <- mfx::probitmfx(formula = f, data = mroz, atmean = TRUE) glms$probitAvg <- mfx::probitmfx(formula = f, data = mroz, atmean = FALSE) glms$logitMean <- mfx::logitmfx(formula = f, data = mroz, atmean = TRUE) glms$logitAvg <- mfx::logitmfx(formula = f, data = mroz, atmean = FALSE) ``` --- # Interpretation <table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table"> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:center;"> probit </th> <th style="text-align:center;"> logit </th> <th style="text-align:center;"> probitMean </th> <th style="text-align:center;"> probitAvg </th> <th style="text-align:center;"> logitMean </th> <th style="text-align:center;"> logitAvg </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> (Intercept) </td> <td style="text-align:center;"> 2.080*** </td> <td style="text-align:center;"> 3.394*** </td> <td style="text-align:center;"> 2.080*** </td> <td style="text-align:center;"> 2.080*** </td> <td style="text-align:center;"> 3.394*** </td> <td style="text-align:center;"> 3.394*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.309) </td> <td style="text-align:center;"> (0.516) </td> <td style="text-align:center;"> (0.309) </td> <td style="text-align:center;"> (0.309) </td> <td style="text-align:center;"> (0.516) </td> <td style="text-align:center;"> (0.516) </td> </tr> <tr> <td style="text-align:left;"> age </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.013*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.014*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.013*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.035*** </td> <td style="text-align:center;"> −0.057*** </td> <td style="text-align:center;"> −0.057*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.003) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.003) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.011) </td> <td style="text-align:center;"> (0.011) </td> </tr> <tr> <td style="text-align:left;"> kidslt6 </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.314*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.290*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.322*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.292*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −0.800*** </td> <td style="text-align:center;"> −1.313*** </td> <td style="text-align:center;"> −1.313*** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.044) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.036) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.046) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.047) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.111) </td> <td style="text-align:center;"> (0.188) </td> <td style="text-align:center;"> (0.188) </td> </tr> <tr> <td style="text-align:left;"> nwifeinc </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.004** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.005** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.004** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.011** </td> <td style="text-align:center;"> −0.019** </td> <td style="text-align:center;"> −0.019** </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.001) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.002) </td> <td style="text-align:center;"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.004) </td> <td style="text-align:center;"> (0.007) </td> <td style="text-align:center;"> (0.002) </td> </tr> <tr> <td style="text-align:left;box-shadow: 0px 1px"> </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.004) </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.007) </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.004) </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.004) </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.007) </td> <td style="text-align:center;box-shadow: 0px 1px"> (0.007) </td> </tr> <tr> <td style="text-align:left;"> Num.Obs. </td> <td style="text-align:center;"> 753 </td> <td style="text-align:center;"> 753 </td> <td style="text-align:center;"> 753 </td> <td style="text-align:center;"> 753 </td> <td style="text-align:center;"> 753 </td> <td style="text-align:center;"> 753 </td> </tr> <tr> <td style="text-align:left;"> Log.Lik. </td> <td style="text-align:center;"> −478.395 </td> <td style="text-align:center;"> −478.377 </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:left;"> F </td> <td style="text-align:center;"> 21.784 </td> <td style="text-align:center;"> 20.280 </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:left;"> RMSE </td> <td style="text-align:center;"> 0.47 </td> <td style="text-align:center;"> 0.47 </td> <td style="text-align:center;"> 0.47 </td> <td style="text-align:center;"> 0.47 </td> <td style="text-align:center;"> 0.47 </td> <td style="text-align:center;"> 0.47 </td> </tr> </tbody> <tfoot><tr><td style="padding: 0; " colspan="100%"> <sup></sup> + p </td></tr></tfoot> </table> --- class: separator, middle # Goodness of Fit in Binary Models --- # GOF in Binary Models * There is no universally accepted `\(R^2\)` for binary models. * We can think of a *pseudo* `\(R^2\)` which compares our model to one without any regressors: ```r glms$probit0 <- update(glms$probit, formula = . ~ 1) # intercept model only 1 - as.vector(logLik(glms$probit)/logLik(glms$probit0)) ``` ``` ## [1] 0.07084972 ``` -- * But that's not super informative (unlike the standard `\(R^2\)`). Changes in likelihood value are highly non-linear, so that's not great. * Let's check **accuracy** - what's the proportion correctly predicted! `round(fitted(x))` assigns `1` if the predicted prob `\(> 0.5\)`. ```r prop.table(table(true = mroz$inlf, pred = round(fitted(glms$probit)))) ``` ``` ## pred ## true 0 1 ## 0 0.1699867 0.2616202 ## 1 0.1221780 0.4462151 ``` --- # GOF in Binary Models: ROC Curves * The 0.5 cutoff is arbitrary. What if all predicted probs are `\(> 0.5\)` but in the data there are about 50% of zeros? * Let's choose an *arbitrary cutoff* `\(c \in (0,1)\)` and check accuracy for each value. This gives a better overview. -- * Also, we can confront the **true positives rate** (TPR) with the **false positives rate** (FPR). 1. TPR: number of women correctly predicted to work divided by num of working women. 2. FPR: number of women incorrectly predicted to work divided by num of non-working women. -- * Plotting FPR vs TPR for each `\(c\)` defines the **ROC** (Receiver Operating Characteristics) Curve. * A good model has a ROC curve in the upper left corner: FPR = 0, TPR = 1. --- # GOF in Binary Models: ROC Curves .left-wide[ ```r library(ROCR) pred <- prediction(fitted(glms$probit), mroz$inlf) par(mfrow = c(1,2), mar = lowtop) plot(performance(pred,"acc")) plot(performance(pred,"tpr","fpr")) abline(0,1,lty = 2, col = "red") ``` <img src="data:image/png;base64,#07-probit_files/figure-html/unnamed-chunk-11-1.svg" style="display: block; margin: auto;" /> ] .right-thin[ <br> <br> * Best accuracy at around `\(c=0.6\)` * ROC always above 45 deg line. Better than random assignment (flipping a coin)! Yeah! ] --- class: inverse # Task 2 (10 Minutes): `SwissLabor` 1. Load the `SwissLabor` Dataset from the `AER` package with `data(SwissLabor, package = "AER")` 1. `skim` the data to get a quick overview. How many foreigners are in the data? 1. Run a probit model of `participation` on all other variables plus age squared. Which age has the largest impact on participation? 1. What is the marginal effect at the mean of all `\(x\)` of being a foreigner on participation? 1. Produce a ROC curve of this probit model and discuss it! --- class: title-slide-final, middle background-image: url(data:image/png;base64,#../../img/logo/ScPo-econ.png) background-size: 250px background-position: 9% 19% # END | | | | :--------------------------------------------------------------------------------------------------------- | :-------------------------------- | | <a href="mailto:bluebery.planterose@sciencespo.fr">.ScPored[<i class="fa fa-paper-plane fa-fw"></i>] | bluebery.planterose@sciencespo.fr | | <a href="https://github.com/ScPoEcon/ScPoEconometrics-Slides">.ScPored[<i class="fa fa-link fa-fw"></i>] | Original Slides from Florian Oswald | | <a href="https://scpoecon.github.io/ScPoEconometrics">.ScPored[<i class="fa fa-link fa-fw"></i>] | Book | | <a href="http://twitter.com/ScPoEcon">.ScPored[<i class="fa fa-twitter fa-fw"></i>] | @ScPoEcon | | <a href="http://github.com/ScPoEcon">.ScPored[<i class="fa fa-github fa-fw"></i>] | @ScPoEcon |