probit model econometrics

Explain how you estimate the coefficient parameters in the probit model. \[\begin{align} The dataset provides the firms information. This process is applied until all features in the dataset are exhausted. Just in the glm() command we need to specify the family argument to be family = binomial(link="logit"). Recall: If there is a firm which defaulted present in the test set and our Probit model can identify it 84% of the time. Poirier and Rudd (1988) discussed the Probit model with dependence in time-series 1 Anselin, Florax and Rey (2004) wrote a comprehensive review about econometrics for spatial models. For example, under body_mass_g', the 0.006644753 suggests that for one unit increase in body_mass_g' weight, the logit coefficient for Chinstrap' relative to Adelie' will go up by that amount, 0.006644753. Love podcasts or audiobooks? The ordered probit model can be used to model a discrete dependent variable that takes ordered multinomial outcomes, e.g., y = 1, 2, , m. A common example is self-assessed health, with categorical outcomes such as excellent, good, fair, poor. Learn on the go with our new app. You can refer to the Econometrics Learning Material for the results of the Probit model. I would be pleased to receive feedback or questions on any of the above. The associated likelihood functions and derivation of marginal effects are available there . My experience with ordered probit is limited, but generally I would get results that indicate coefficients moving from category 1 to category 2, category 2 to category 3, etc. The Probit model corrects the distortion created in the linear probability model and limits the probability of default between 0 and 1. Burnett (1997) proposed the following bivariate probit model for the presence of a gender economics course in the curriculum of a liberal arts college: Prob [yi = 1, y2 = 11 xi, x2] = $2 (x'i0i + y y., P). Augmenting this model with $Y_i^*$, we can have the likelihood contribution from observation $i$, $p(y_i|y_i^*)=1_{y_i=0}1_{y_i^*\leq 0}+1_{y_i=1}1_{y_i^*> 0}$, where $1_A$ is an indicator function that takes the value of 1 when condition $A$ is satisfied. Probit model has been used to analyze the socioeconomic factors affecting milk consumption of households. where ${\bf{B}}_n = ({\bf{B}}_0^{-1} + {\bf{X}}^{\top}{\bf{X}})^{-1}$, and $\beta_n= {\bf{B}}_n({\bf{B}}_0^{-1}\beta_0 + {\bf{X}}^{\top}{\bf{Y}}^*)$. There are two features that we do not need, such as Firm ID and year, so, we will drop them. Probit model Probit models are pretty much similiar to logit models (see above). SPSS and AMOS, EVIEWS Smart PLS, STATA The default variable takes the value of 1 if the firm defaulted, and the value of 0 otherwise. The fact that we have a probit, a logit, and the LPM is just a statement to the fact that we don't know what the "right" model is. We can see that coefficient for logit and probit models could be quite different, but the average marginal effects are on contrary quite similliar. Logit and Probit models In document Time Series Econometrics Using Microfit 5.0(Page 132-135) The Logit and Probit options are appropriate when the dependent variable, yi, i = 1; 2; :::; n takes the value of 1 or 0. agents are faced with a choice between two alternatives. P[Y_i=1]&=P[Y_i^*\geq 0]\\ Privacy Policy. The Probit model corrects the distortion created in the linear probability model and limits the probability of default between 0 and 1. Question: Dave Giles, in his econometrics blog, has spent a few blog entries attacking the linear probability model. The probit model also has as dependent variable a binary outcome. Application: Determinants of hospitalization in Medelln. Press J to jump to the feed. where $Y_i^*={\bf{x}}_i^{\top}\beta+\mu_i$, $\mu_i\stackrel{iid} {\sim}N(0,1)$. Here, we will present the results of the Logit model only. 2 data, and developed generalized conditional moment (GCM) estimators which are computational attractive and relatively more ecient. The precision is the ratio tp / (tp + fp) where tp is the number of true positives and fp the number of false positives. For more details please refere to AER package documentation, page 140. link Cross-section data about resume, call-back and employer information for 4,870 fictitious resumes sent in response to employment advertisements in Chicago and Boston in 2001, in a randomized controlled experiment conducted by Bertrand and Mullainathan (2004). Four estimators (household size, income, milk preferences reason, and milk price) in the probit model were found statistically significant. The five ratios are those from the widely known Z-score developed by Altman (1968). We can not interpret magnitude from the regression table for probit model, only we can interpret the direction of the effect i.e. The dotted line represents the ROC curve of a purely random classifier; a good classifier stays as far away from that line as possible (toward the top-left corner). In this course, you will discover models and approaches that are designed to deal with challenges raised by the empirical econometric modelling and particular types of data. In the book Multilevel and Longitudinal Modeling using Stata , Rabe-Hesketh and Skrondal have a lot of exercises and over the years I've been trying to write Stata and R code to demonstrate. The model is specified with Random Parameters to accout for unobserved heterogeneity in data. The dependent variables in the model are y1 = presence of a gender economics course, y2 = presence of a women's studies program on the campus. B. the statistical inferences about causal effects are valid for the population studied. Masters in Economics (Econometrics & Statistics) who has a high proficiency in research, data analysis, data visualization, interpretation of obtained results, academic and business writing. Can I say that an increase in income reduces the probability of being in a poor health (5)? S/TA further proxies for the competitive situation of the company and ME/TL is a market-based measure of leverage. 16.4 The Logit Model for Binary Choice. Flashcards. For example, whether you defaulted on your credit or not. Albert, James H, and Siddhartha Chib. \end{align}\]. Why don't economic researchers use a regularized probit model? Match. In the existing code, the model only has an observed correlation term between the count model and the ordered model. The p-values for all of the variables are smaller than 0.05, so we will keep all of them. Test. In the process, the model attempts to explain the relative effect of differing explanatory variables on the different outcomes. Econometrics Academy - Bivariate Probit and Logit Models Bivariate Probit and Logit Models Bivariate probit and logit models, like the binary probit and logit models, use binary. By accepting all cookies, you agree to our use of cookies to deliver and maintain our services and site, improve the quality of Reddit, personalize Reddit content and advertising, and measure the effectiveness of advertising. The explanatory variables can be any risk metrics that reflect the firms financial strength, such as the financial leverage ratios, liquidity ratios or profitability ratios. A GLiM has three parts, a structural component, a link function, and a response distribution. Relative risk ratios allow an easier interpretation of the logit coefficients. Then, \[\begin{align} The resumes contained information concerning the ethnicity of the applicant. The polynomial regression adds polynomial or quadratic terms to the regression equation as follow: Quantile regression is an extension of linear regression used when the conditions of linear regression are not met. Learn. There are two ways that bidding occurs on eBay. How regression is used to find answers for questions, 12. Competently use regression, logit and probit analysis to quantify economic relationships using standard regression programmes (Stata and EViews) in simple applications. This is overall correct. Fits a smooth curve with a series of polynomial segments. In this model we runnig a linear regression in which the explained variable, Z, can have a value of 1, in the case of default, or a value of 0, when the firm is paying its debts. Most of the firms in this dataset have a WC/TA ratio in the range of 0.060.37. We'll use Boston data set. Reference: Learning Predictive Analytics with Python book, Chief Data Scientist at Prediction Consultants Advanced Analysis and Model Development. Results of Logit Model. Training an XGBoost model for Pricing Analysis using AWS SageMaker, Build your own machine learning model to predict the presence of heart disease, df = pd.read_csv(USCorporateDefault.csv), df.drop([Firm ID,Year], axis=1, inplace=True), sns.countplot(x=Default, data=df, palette=hls), count_no_default = len(df[df[Default]==0]), from sklearn.feature_selection import RFE, cols=[WC/TA, RE/TA, EBIT/TA, ME/TL, S/TA], X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42, stratify=y), params = pd.DataFrame(probit.fit().params,columns={'coef'},), result1['y_pred'] = result1['WC/TA'] * params['coef'][0] + result1['RE/TA'] * params['coef'][1] + result1['EBIT/TA'] * params['coef'][2] + result1['ME/TL'] * params['coef'][3] + result1['S/TA'] * params['coef'][4], result1[y_pred_Probit] = normsdist(result1[y_pred]), d = {'y_pred_proba': result1['y_pred_Probit']}, from sklearn.metrics import accuracy_score, print('Accuracy of Probit Model on test set: {:.2f}'.format(accuracy_score(y_test, y_pred))), from sklearn.metrics import confusion_matrix, confusion_matrix = confusion_matrix(y_test, y_pred), from sklearn.metrics import classification_report, print(classification_report(y_test, y_pred)), y_pred_proba = np.array(df23['y_pred_proba']), from sklearn.metrics import roc_auc_score, probit_roc_auc = roc_auc_score(y_test, y_pred), The receiver operating characteristic (ROC), Learning Predictive Analytics with Python book, https://polanitz8.wixsite.com/prediction/english. With a probit or logit function, the conditional probabilities are nonlinearly related to the independent variable (s). These are the logit coefficients relative to the reference category. Works by creating synthetic samples from the minor class (default) instead of creating copies. a specific case that we want to comment on (it might be a made up average family or whatever allows you to paint the picture). A large collection of fictitious resumes were created and the presupposed ethnicity (based on the sound of the name) was randomly assigned to each resume. It add polynomial terms or quadratic terms (square, cubes, etc) to a regression. Econometrics Theory and application of econometric models. The support is the number of occurrences of each class in y_test. Marginal effects would need to be computed to determine the likelihood with which one leaves a given category. &=P[\mathbf{x}_i^{\top}\beta+\mu_i\geq 0]\\ b Of the 13557 seen at PROBIT IV, 12072 were seen at both PROBIT II & III, 274 were not seen at either PROBIT II & III, 449 were seen at PROBIT II but not seen at III, and 762 were seen at PROBIT III but not seen at II. In this article, we present the user-written commands probitfe and logitfe, which fit probit and logit panel-data models with individual and time unobserved effects.Fixed-effects panel-data methods that estimate the unobserved effects can be severely biased because of the incidental parameter problem (Neyman and Scott, 1948, Econometrica 16: 1-32). In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. 1. In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution.It has applications in data analysis and machine learning, in particular exploratory statistical graphics and specialized regression modeling of binary response variables.. 5. 2013. the-probit-logit-models-uc3m 1/13 Downloaded from classifieds.independent.com on November 7, 2022 by guest . First, bidders can manually insert their bid into the proxy bidding system. For private sector credit, it has a positive relationship with reserves, tourism earnings, remittances and domestic exports. Probit and Logit Modelshttps://sites.google.com/site/econometricsacademy/masters-econometrics/probit-and-logit-modelsLecture: Probit and Logit Models.pdfhttp. Generally coefficients in probit models are not interpreted directly due to underlying distribution of the likelyhood function. Economics Econometrics Econometrics Final Exam: Multiple Choice 5.0 (1 review) Term 1 / 27 A statistical analysis is internally valid if: A. the regression R > 0.05. Probit Probit regression models the probability that Y = 1 Using the cumulative standard normal distribution function ( Z) evaluated at Z = 0 + 1 X 1i k ki since ( z) = Pr Z ) we have that the predicted probabilities of the probit model are between 0 and 1 Example Suppose we have only 1 regressor and Z = 2 + 3X 1 Interpreting regression with logarithm, 5. \beta|{\bf{Y}}^*, {\bf{X}} & \sim N(\beta_n,\bf{B}_n), Then we plug the variables into the formula to get a value of a latent variable (lets call it z). There is three type of penguins: Adelie, Gentoo and Chinstrap. A bivariate probit model is a 2-equation system in which each equation is a probit model. In table 5 of the paper (see Screenshot) the dependent variable is a categorical variable that ranges from 1 to 5, 1 being an excellent health status and 5 poor health status. (Albert and Chib 1993) implemented data augmentation (Tanner and Wong 1987) to apply a Gibbs sampling algorithm in this model. Create an account to follow your favorite communities and start taking part in conversations. Practical issues when running regression. We look at conventional methods for removing endogeneity bias in regression models, including the linear model and the probit model. We use a recursive bivariate probit strategy to address this concern. Either you can compute them customly, or you can use package stargazer, that computes p-values for you. https://polanitz8.wixsite.com/prediction/english. Under the general Women have a higher probability of being hospitalized than do men, and people with bad self perception of health condition also have a higher probability of being hospitalized. They are all artistically enhanced with visually stunning color, shadow and lighting effects. Brookings Papers on Economic Activity 2001 William C. Brainard 2002-01-01 For almost thirty years, Brookings Papers on Economic Activity (BPEA) has provided academic and business The precision is intuitively the ability of the classifier to not label a sample as positive if it is negative. The equation for the outcome (1) remains the same, but we add another equation. more likely or less likely get called back. A probit model (also called probit regression), is a way to perform regression for binary outcome variables. Because ethnicity is not typically included on a resume, resumes were differentiated on the basis of so-called "Caucasian sounding names" (such as Emily Walsh or Gregory Baker) and "African American sounding names" (such as Lakisha Washington or Jamal Jones). The explained variable receives only two values: value 1 which represents a firm that has reached default and value 0 which represents a stable firm. F1-Score: The harmonic average score of the Probit model on class #1 (i.e., the default class), which weights the precision and the recall together, is 81%. Goodness-of-fit 6. System of Linear Equations: Matrices and Economic-Business Application, 2. Data is on penguins and their characterstics. Upon receipt of the coefficients from the regression run one can multiply them by the firms explanatory variables in order to get the firms probability of default. Perhaps the authors just assume that the distance between categories is the same and the regressors have a linear effect to dependent variable. The we need to use multinomial logit model. Extensive experience in performing data analysis/visualization by using: Power-BI Excel. Spline regression. WC/TA captures the short-term liquidity of a firm, RE/TA and EBIT/TA measure historic and current profitability, respectively. the bivariate probit model is typically used where a dichotomous indicator is the outcome of interest and the determinants of the probable outcome includes qualitative information in the form of a dummy variable where, even after controlling for a set of covariates, the possibility that the dummy explanatory variable is endogenous cannot be ruled kaylaekerr. 11.3 Estimation and Inference in the Logit and Probit Models So far nothing has been said about how Logit and Probit models are estimated by statistical software. As in Shijaku (2013) and Salisu (2017) the estimated probit models fit . \end{align}\], # Prior precision (inverse of covariance), Bayesian Analysis of Binary and Polychotomous Response Data., The Impact of Subsidized Health Insurance on the Poor in, The Calculation of Posterior Distributions by Data Augmentation., Introduction to Bayesian Econometrics: A GUIded tour using R, Ramrez Hassan, Cardona Jimnez, and Cadavid Montoya 2013. \end{align}\]. However, by multiplying the results of the logistic distribution by an appropriate coefficient the distribution of the Probit model can be obtained. Reddit and its partners use cookies and similar technologies to provide you with a better experience. (For example, whether to use public We can then compare f(z1) and f(z2) values with z1 having lower income as variable and f() being the probit distribution function. If the outcome variable is categorical variable without inherent oreder(regular categorical), such as car manufacturers. Spatial probit and logit models Model specification In the spatial econometric literature the classical probit model has been adapted to account for spatial dependence in its versions as spatial lag or spatial error which we have reviewed in the case of linear models in Chapter 3. . I might be mistaken, so take my reply with a grain of salt. Part 2 of 5. The classification goal is to predict whether a firm will default (1) or not (0). The Logit and Probit models are estimated using the Maximum-Likelihood technique. &=1-P[\mu_i < -\mathbf{x}_i^{\top}\beta]\\ Would love for someone with more knowledge on this to correct me if Im wrong. However, the Z value, which measures the firms probability of default, may deviate from the range between zero and one, thus the main disadvantage of the model. A normal distribution can be described by two parameters. For more information, please see our But then, the same is true for the "wrong" nonlinear model! \end{Bmatrix}, 1 2 3 Justin L. Tobias (Purdue) The Tobit 2 / 1 TN_{(-\infty,0)}({\bf{x}}_i^{\top}\beta,1), & y_i= 0 \\ The results are virtually identical for logit and probit models run on the same data. Van de Ven and Van Pragg (1981) introduced the probit model with sample selection to allow for consistent estimation of in samples that suffer from selection on unobservables. For example, if it is believed that the decisions of sending at least one child to public school and that of voting in favor of a school budget are correlated (both decisions are binary), then the multivariate probit model would be . More precisely, my concern is that I don't know hot to interpret the coefficients in a paper I'm currently reading (Case et al. Data checking during PROBIT IV found one of these children had been incorrectly reported as deceased and data were amended. Variable lstat (percentage of lower status of the population). Data were collected and made available by Dr. Kristen Gorman and the Palmer Station, Antarctica LTER, a member of the Long Term Ecological Research Network. &+\beta_8\text{Fair}_i+\beta_9\text{Good}_i+\beta_{10}\text{Excellent}_i, If we look at the first row of the regression table, we can interpret it as following. 4. We have collected default information and five variables for default prediction: Working Capital (WC), Retained Earnings (RE), Earnings before interest and taxes (EBIT) and Sales (S), each divided by Total Assets (TA); and Market Value of Equity (ME) divided by Total Liabilities (TL). Probit Analysis and Economic Education. To have meaningful interpretation of effects we can calculate average marginal effects. Created by. In this paper, we use an autoregressive panel probit model where the autocorrelation in the discrete variable is driven by the autocorrelation in the latent variable. and our The standard deviation - the measure of the spread. Randomly choosing one of the k-nearest-neighbors and using it to create a similar, but randomly tweaked, new observations. The Probit model can be represented using the following formula: Pr (Y = 1|X) = (Z) = Z = (b0 + b1X1 + b2X2 + .. + bnXn) Where, Y is the dependent variable and represents the probability that the event will occur (hence, Y = 1) given the variables X. is the cumulative standard normal distribution function. The posterior distribution is $\pi(\beta,{\bf{Y^*}}|{\bf{y}},{\bf{X}})\propto\prod_{i=1}^n\left[\mathbf{1}_{y_i=0}1_{y_i^*< 0}+1_{y_i=1}1_{y_i^*\geq 0}\right] \times N_N({\bf{Y}}^*|{\bf{X}\beta},{\bf{I}}_N)\times N_K(\beta|\beta_0,{\bf{B}}_0)$ when taking a normal distribution as prior, $\beta\sim N(\beta_0,{\bf{B}}_0)$. The selection process for the outcome is modeled as. (b) [5]. Application 4. It seems from our results that female and health status are relevant variables for hospitalization, as their 95% credible intervals do not cross 0. The model is estimated for many firms using a linear regression from the form: Xij The explanatory variables (financial ratios) of firm i; j A coefficient that measures the importance of a variable in explaining default. Probit and logit models are among the most popular models. more likely or less likely get called back. where $TN_A$ denotes a truncated normal density in the interval $A$. We ensure identifiability by taking utility differences and fixing one error-term variance. You need to be really careful and specific with interpretation of models like these. Probit Model - Econometrics. Probit models use Maximum Likelihood Estimation "MLE" for estimates of the Betas. Coefficients and marginal effects Course outline 2 5. A Thorough Dive into the Ames Iowa Housing Dataset. It is known that the usual Heckman two-step procedure should not be used in the probit model: from a theoretical perspective, it is unsatisfactory, and likelihood methods are superior. Enroll for Free. A probit model (also called probit regression), is a way to perform regression for binary outcome variables.Binary outcome variables are dependent variables with two possibilities, like yes/no, positive test result/negative test result or single/not single. Since our dataset is balanced (i.e., both classes have exactly the same size), we will use a threshold of Z = 0.50 for the value of y_pred_Probit so that: Accuracy of Probit Model on test set: 0.80. In table 5 of the paper (see Screenshot) the dependent . The average WC/TA ratio (i.e., Working Capital divided by Total Assets) for the firms which defaulted is almost equal to that of the firms which didnt. I have a basic understanding of econometrics and I'd be happy about every input I can get from you guys. Interpretation of marginal effect for variable ethnicityafam is -0.324139363996093, which means if the person has african american sounding name, then he is 32% less likely to get call back from potential employer. At a high level, SMOTE: We are going to implement SMOTE in Python. It includes 4,000 records and 8 fields. Additionally, both functions have the characteristic of approaching 0 and 1 gradually (asymptotically), so the predicted probabilities are always sensible. Our new CrystalGraphics Chart and Diagram Slides for PowerPoint is a collection of over 1000 impressively designed data-driven chart and editable diagram s guaranteed to impress any audience. The recall is intuitively the ability of the classifier to find all the positive samples. The dependent variable is a binary response, commonly coded as a 0 or 1 variable. Our classes are imbalanced, and the ratio of no-default to default instances is 89:11. From the regression table we can see coefficient for ethnicityafam is -0.21685 (which is little bit different compared to logit model) that means that if the applicant have african american sounding name then he is less likely to recieve call back. Instead one relies on maximum likelihood estimation (MLE). The mean 2. Before we go ahead to balance the classes, lets do some more exploration. Of . Flashcards. By rejecting non-essential cookies, Reddit may still use certain cookies to ensure the proper functionality of our platform. We also see that there are posterior convergence issues (see Exercise 2). &=P[\mu_i < \mathbf{x}_i^{\top}\beta], The result is telling us that we have 599+661 correct predictions and 124+186 incorrect predictions. 2002 "Economic status and health in childhood"). You may have noticed that I over-sampled only on the training data, because by oversampling only on the training data, none of the information in the test data is being used to create synthetic observations, therefore, no information will bleed from test data into the model training. In statistics, a probit model (binary dependent variable case) is a type of regression in which the dependent variable can take only two values (0/1), for example, married or not married. Thank you The name comes from probability and unit.The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific category. The model can be expressed as (18) where and 0 = , j j+1, m = . Is probit the same as logistic regression? Now we have a perfect balanced data! Precision: Precision is about being precise, i.e., how precise our model is. Y_i^*|\beta,{\bf{y}},{\bf{X}}&\sim\begin{Bmatrix} The decision/choice is whether or not to have, do,. Cookie Notice Robust Standard Errors and OLS Standard Errors; Information Criteria (AIC/SIC) and Model Selection; Goodness-of-fit for Logit and Probit Models; VAR-VECM Goodness of fit; Panel Data. Problem statement. Except for the market value, all of these items are found in the balance sheet and income statement of the company. Probit models are pretty much similiar to logit models(see above). Interpretation of marginal effect for variable ethnicityafam is -0.0321574328415445, which means if the person has african american sounding name, then he is 32% less likely to get call back from potential employer. More precisely, my concern is that I don't know hot to interpret the coefficients in a paper I'm currently reading (Case et al. The median house value (mdev), in Boston Suburbs. beta = 1.0 means recall and precision are equally important. Probit model with sample selection. Whether you are eligible for the program or not, etc etc. Press question mark to learn the rest of the keyboard shortcuts In other words, if your body_mass_g' weight increases one unit, the chances of the penguin to be identified as Chinstrap' compared to the chances of being identified as Adelie' are higher. According to Key Concept 8.1, the expected change in the probability that Y = 1 Y = 1 due to a change in P /I ratio P / I r a t i o can be computed as follows: Compute the predicted probability that Y = 1 Y = 1 for the original value of X X. Compute the predicted probability that Y = 1 Y = 1 for X+X X + X. The receiver operating characteristic (ROC) curve is another common tool used with binary classifiers. Regression model for quantitative easing/tightening? This is the simple approach to model non-linear relationships. One advantage of quantile regression relative to ordinary least squares regression is that the quantile regression estimates are more robust against outliers in the response measurements. I'm using the program STATA to do so, and have the output of the regression, and of average marginal effects, but am not sure how to calculate average partial effect from there. . Modeling and estimating persistent discrete data can be challenging. Mathematically, the probit is the inverse of the cumulative distribution function of the . Recursive Feature Elimination (RFE) is based on the idea to repeatedly construct a model and choose either the best or worst performing feature, setting the feature aside and then repeating the process with the rest of the features. Cheers! Study Resources. 1 2 2 t 0 1 1 ' ^ ^ 1. y Gujarati . There is a latent (unobserved) random variable, Y i Y i , that defines the structure of the estimation problem Y i = {1, Y i 0 0, Y i < 0}, Y i = { 1, Y i 0 0, Y i < 0 }, The generalized linear model (GLiM) was developed to address such cases, and logit and probit models are special cases of GLiMs that are appropriate for binary variables (or multi-category response variables with some adaptations to the process). The RFE has helped us to understand that all the following features are relevant for the modeling: WC/TA, RE/TA, EBIT/TA, ME/TL, S/TA. Hello everyone, as the title already revealed my question is about the ordered probit model. How to interpret standard deviation vs coefficient. This model uses financial and other variables to predict the firms probability of default, and assumes that this probability has a cumulative standard-normal distribution, which is limited, by definition, to a range between 0 and 1: F(Zi) The firms cumulative probability of default, Zi The value obtained from estimating the Probit model, (Zi) The cumulative standard-normal distribution function from minus infinity (-) to the point Zi (i.e., the number of standard deviations).