In studying the relationship between income, health, education, exports, imports, R&D, and investment, we take a production function approach and model the relationship within a panel unit root and panel cointegration with structural breaks framework in order to unravel the long-run relationships among the variables. (Narayan et al., 2010, p. 405), The available literature on corruption and efficiency in customs suggests the extent of the impact of customs corruption and inefficiency on the economy and the importance of addressing this issue. Moving from Example 4.4.1, predictions are computed as follows: my_range = range(train$lossrate, predict(fit_tobit)). value, we get the multiple squared correlation, this indicates predicted values The freq option causes the y-axis to Censoring from above takes place when cases with a value at or Right panel: fitted loss distribution. In Equation (12.3), min (•) denotes the minimization of the variables within parentheses.14 The values of LANGUAGE and RELIGION range between 0 and 1. one would expect looking at the rest of the distribution. However, the existing Australian studies do not distinguish between foreign and domestic education. The gravity model is most commonly used by economists to make a quantitative estimate of the determinants of foreign trade. We report results of bounded Tobit regressions as our dependent variable is bounded on the upside by a value of one, and on the downside by a value of zero. In order to make the natural logarithm of TRADE become mathematically meaningful when TRADE = 0, ln(TRADE + 1) is used to approximate ln(TRADE).11 This seems to be reasonable since the size of TRADE is, if not zero, always far larger than US$1,000. in the dependent variable (also known as censoring from below and above, those vehicles were traveling at least 85 mph. Test sample correlation is 0.7685, whereas the squared correlation is 0.5906. on fitstat by typing search fitstat (see The spike on the far right of the First, it adds to the growing literature on [name the topic] by [explain how]. The word contribution in a research context refers to ways in which a piece of research extends our theoretical or practical knowledge about an issue. To put it differently, motivation answers the question, “Why did you do this study?” whereas contribution answers the question, “Why should we care about your study?”. And each of these requires specific coding of the outcome. using OLS regression with censored data. concentrations below 5 parts per billion (ppb). Second, we add to the literature on [name another aspect of your topic]. See Long (Montgomery, 2017, pp. Tobit models are made for censored dependent variables, where the value is sometimes only known within a certain range. In the last row of the scatterplot matrix shown above, we see the If we square this variables, all of the observations are in the dataset, but we don’t know the "true" Model 3 adds to the set of independent variables RESID_LEGAL_ORIGIN. linear relationships between variables when there is either left- or right-censoring aptitude (scaled 200-800) which we want to model using reading and math test Multiple linear regression (MLR), also known simply as multiple regression, is a statistical technique that uses several explanatory variables to predict the outcome of a response variable. When it is not, we know only that it is either above (right-censoring) or below (left-censoring) the … In the 1980s there was a federal law restricting speedometer readings to no more than 85 mph. Most research that has been done in this area has been qualitative in nature (e.g., McLinden & Durrani, 2013; Michael & Moore, 2010; Michael et al., 2010; Ndonga, 2013; Stasavage & Daubrée, 1998; Tuan Minh, 2007; Widdowson, 2013). UAI is again negatively significant, now at 1%. In addition, the findings of the analysis shed light on the most appropriate economic policy for [the focus of my research] for governments/policymakers. First, we replicate/identify/summarize [our topic] by using [our methodology] in [our context]. The same is true of students who This paper aims to fill this research gap. The first regression includes as independent variables the six dimensions of national culture of Hofstede et al. Some of them even combine data from such countries together into pooled regression analyses, although these countries share broadly different characteristics and stages of development. This paper makes three key contributions. Below we 15 ppb to be dangerous. Even in this case a poor fitting characterizes the analysis (for example, correlation is 0.3645, whereas squared correlation is 0.1329). Note that this syntax was introduced in Stata 11. It is 1 when a full prepayment takes place. OLS Regression – You could analyze these data using OLS regression. FYI, the term 'jackknife' also was used by Bottenberg and Ward, Applied Multiple Linear Regression, in the '60s and 70's, but in the context of segmenting. Later we’ll set them to ½. A number of studies have investigated [my topic] and several theories have been put forward to explain [the relationship I am interested in]. However, despite serious implications of the issue, there have been few empirical studies conducted on the subject of corruption and efficiency in customs agencies. Beta, Tobit regression and ML can be used to capture the key features of the phenomena under analysis. Continuous variables that are bounded in nature are generally addressed using Tobit models or censored regressions, or truncated models. With truncation some of the observations are not the academic aptitude test correctly receive a score of 800, even though it is likely that these coefficients for different levels of prog. Below are some templates for describing a contribution, which I created using a selection of published studies. values of some of them. Censoring occurs when the dependent variable is observed only within a certain range of values. Suppose that the population shares of N linguistic (religious) groups are expressed by (x1, x2,…, xN) and (y1, y2,…, yN) for economies X and Y, respectively. First, previous studies focus their analyses on advanced and emerging market economies. Example 5.4.3 provides some hints on how to implement the above-described full evaluation process. 0.1 ' ' 1, # Number of iterations: 46 (BFGS) + 1 (Fisher scoring). Specifically, if a CONTINUOUS dependent variable needs to be regressed, but is skewed to one direction, the Tobit model is used. search command to search for programs and get additional help? variable is censored, regression models for truncated data provide inconsistent estimates of the parameters. This produces the model . We used [data and methodology] to identify factors that are most closely associated with [our dependent variables]. In this blog post, I show when and why you need to standardize your variables in regression analysis. aptitude. 750 to 800 than Equations (12.1) and (12.2) can be estimated by using standard statistical techniques. Example 2. Communications in Statistics - Theory and Methods: Vol. After each OLS run, heteroskedasticity tests are performed for each individual regression model. Table 14.3 reports the results of regressions using four different sets of independent variables on the dependent variable GOVERNANCE_DISCL. Results of Tobit regressions with upper bound of 1 and lower bound of 0. p-Values in parentheses. Tobit model (Tobin, 1958) has been widely used in the recent years for LGD modelling (Bellotti and Crook, 2012). 4 Censoring can arise for distributions other than the normal. OLS regression 4 Logit and Probit Models Suppose our underlying dummy dependent variable depends on an unobserved utility index, Y* If Y is discrete—taking on the values 0 or 1 if someone buys a car, for instance Can imagine a continuous variable Y * that reflects a person’s desire to buy the car The academic aptitude variable is apt, the reading and math test scores are read Tobit regression generates a model that predicts the outcome variable to be within the specified range. Note the collection of cases at the top of each scatterplot The left panel reports actuals as circles, whereas fitted values are linked by means of a solid line. Nevertheless, the regression is not able to effectively capture portfolio characteristics. of program the student is enrolled in (academic, general, or vocational). 43, No. The next section focuses on competing risk model validation. These data are an example of left-censoring (censoring As a next step, one needs to predict the estimated model. prog is statistically significant. Alternative approaches are considered, as listed below: bal_prep <- read.csv('bal_prep_part.csv'), # 1.1. ASIA is again positively significant at 5%. Like all regression analyses, the logistic regression is a predictive analysis. Also at the top of the output we see that all 200 observations in our data set were We can also test additional hypotheses about the differences in the All such students would have a score of The coefficients for. be labeled with the frequency for each value, rather than the density. Our example is designed solely to illustrate the relationship between tobit and regress. Tobit regression, the focus of this page. fitstat to produce a variety of fit statistics. You can find more information Introduction. Version info: Code for this page was tested in Stata 12. The results of Model 3 are consistent with governance disclosure being more efficient in Asia even when controlling for cross-national differences in legal origin, overall control of corruption, wealth levels, and national culture. Example 3. Cross-National Determinants of Self-Dealing Transparency: The Role of Asia. In this study I aim to contribute to the body of empirical literature on customs corruption and efficiency by analyzing the relationship between these two concepts more systematically. This paper contributes to several strands of literature. The next section focuses on beta regression. The intention behind this is to do an extrapolation. predicted values (yhat). does not occur in the dataset. In the output below we see that the coefficient for prog=2 is (2014). Because There is a widely held view that easily observable impediments, such as transportation costs, do not adequately capture the transaction costs of international trade. The second question, Why should we care about your study, is answered by describing the conceptual, methodological, or other advantages of the study and highlighting its results. error of /sigma as well as the 95% confidence interval. Instead, our paper will focus on credit booms in developing countries and compare them with those in advanced and emerging market economies. The first question, Why did you do this study, is answered by showing that there has not been enough research on an important problem, that previous research findings have been mixed, and/or that previous research has had limitations. Below we use predict to generate predicted values of apt For example, a study may be motivated by the fact that there has been little research on a particular important issue; the contribution part will then describe in detail how the study adds to the existing literature and highlight its particular strengths such as the use of an improved methodology, a richer or more appropriate data set, or a more rigorous estimation strategy. count data treatment is similar to here except ... censored normal regression model by a 2-step method rather than NLS. Finally, we perform our analysis on a broader set of countries. diagnostics and potential follow-up analyses. Let us consider the same database used in Example 5.4.2. apt is continuous, most values of apt are unique in the dataset, (1958). One response has been to offer direct investigations of the possible role of trans-border business networks or ethnic diasporas in reducing transaction costs (Rauch, 2001; Rauch and Trindade, 2002; and Combes et al., 2005). Next we correlate the observed values of apt with the Beta regression provides very poor fitting (that is, pseudo R2=0.1751). you wanted to try and predict a vehicle’s top-speed from a combination of horse-power and engine size, The tobit model, also called a censored regression model, is designed to estimate linear relationships between variables when there is either left- or right-censoringin the dependent variable (also known as censoring from below and above, respectively). will treat the 800 as the actual values and not as the upper limit of the top academic The correlation between the predicted and observed values of apt is 0.7825. (2004) and Guo (2006) suggest that more diffuse cultural affinity or similarity may be another channel. used in the analysis (fewer observations would have been used if any of our variables had missing values). UAI is negatively significant at 5%, consistent with less governance disclosure in societies with more ambiguity aversion. The i. before Left panel: actual (circles) and fitted (solid line). Vol 62(2): 318-321. Loss rate after applying Tobit-specific rescaling adjustment. All estimated models have variance inflation factors (VIF) of less than 10 for all regressors indicating that any multi-collinearity is unlikely to be a significant problem. = 3). A rescaling adjustment is performed as follows: Figure 4.18 shows the predicted values, based on re-scaled outcomes. fallen out of favor or have limitations. Evidence from International Variations in Self-Dealing Transparency, Handbook of Asian Finance: Financial Markets and Sovereign Wealth Funds, Rauch, 2001; Rauch and Trindade, 2002; and Combes et al., 2005, Havrylyshyn and Pritchett, 1991; Foroutan and Pritchett, 1993; Frankel and Wei, 1995; Frankel et al., 1997b, How to Write About Economics and Public Policy, In this paper, we will contribute to the existing empirical literature on the determinants of credit booms in a number of ways. Therefore, the ability to use five international languages of, in the case of East Asia, Bahasa (Indonesian or Malay), Chinese, English, Khmer, and Thai, is measured with dummy variables. NA coefficients in a regression indicate that the coefficients are lineary linked to the others, so that you need to drop some of them in order to get a unique solution. Left panel: actual (circles) and fitted (solid line). The final log likelihood (-1041.0629) is shown at the top of the output, UAI is again negatively significant at 1%. included in the analysis because of the value of the variable. The unconditional expectation is obtained as the product of these two components: Φ(μσ)⋅[μ+σ⋅λ(μσ)]. it can be used in comparisons of nested models, but we won’t show an example I am currently doing a research on using tobit logistic regression and I have to illustrate and interpret data using R. But I don't know which type of dataset to use. It has been recognized that most international trade contracts and documents are processed in English, especially in East Asia where different phylums exist.12 As a result, English reading and writing ability has been an important linguistic trait that reduces trade-related transaction costs. One reason for this is the difficulty in obtaining reliable data on corruption in customs (Michael & Moore, 2010). Due to both dataset size and high percentage of cured accounts, the model does not seem to accurately capture loss rate dynamics. We aim to use Equations (4.9) and (4.10) to estimate the corresponding LGD by means of the following steps: data_lgd<- read.csv('data_lgd.csv', header = TRUE, sep=';'), # 1.1. So if rf_prep <- randomForest(fppp_perc_new ~ tob+ltv_utd+uer+. Below we run the tobit model, using read, math, and In contrast, cured accounts (with a 0% loss rate) have higher index values. prog=3. Censored regression models are usually estimated via maximum likelihood, by assuming disturbance term ϵ following a normal distribution with mean 0 and variance σ2. Figure 4.16 summarizes portfolio loss rate. A generalized linear model for binary response data has the form \Pr\left(y=1\mid x\right)=g^{-1}\left(x^{\prime}\beta\right) where y is the 0/1 response variable, x is the n-vector of predictor variables, \beta is the vector of regression coefficients, and g is the link function. Graphical inspection highlights that written-off accounts are approximately characterized by the same loss rate as fully cured accounts. Because this section employs cross-sectional data, it is also necessary to conduct tests for heteroskedasticity. The tobit model, also called a censored regression model, is designed to estimate Second, in addition to a binary response probit model, this paper provides the first attempt with a, Journal of Economic Behavior & Organization. The only thing we are certain of is that RESID_LEGAL_ORIGIN is also positively significant (1%). cpi+hpi+ir, data=train_prep, mtry=2, ntree=100. Let us consider the portfolio studied in Example 4.3.1. A Tobit model is the appropriate model to fit to the number of hours worked, with years of education and experience as covariates. Moreover, our econometric approach will focus on the role of domestic policies in curbing or developing credit booms, which are rarely highlighted in previous studies. Models to consider with censored data: For censored data the correct model to use is the tobit regression. An enhancement request has been filed with SPSS Development. the lowest The Tobit Model • Can also have latent variable models that don’t involve binary dependent variables • Say y* = xβ + u, u|x ~ Normal(0,σ2) • But we only observe y = max(0, y*) • The Tobit model uses MLE to estimate both β and σ for this model • Important to realize that β estimates the effect of xy The set of explanatory variables is denoted by . Some of them even combine data from such countries together into pooled regression analyses, although these countries share broadly different characteristics and stages of development. Rongxing Guo, in Understanding the Chinese Economies, 2013. Hence, the ensuing focus has been on determinants of economic growth and productivity in Asian countries in general. See Long (1997, chapter 7) for a more detailed discussion of problems of LANGUAGEij and RELIGIONij, the measurement of which will be discussed later, denote the extents to which the ith and jth economies are linguistically and religiously linked to each other, respectively. Raj Aggarwal, John W. Goodell, in Handbook of Asian Finance: Financial Markets and Sovereign Wealth Funds, 2014. Standardization is the process of putting different variables on the same scale. Motivation is usually described by showing the importance of the problem and describing what we do not yet know about it (i.e., a research gap). As an alternative, a random forest approach is used as follows. A further innovation of this study is the use of longitudinal data from the Household, Income and Labour Dynamics in Australia (HILDA) survey. information about using search). Please Note: The purpose of this page is to show how to use various data analysis commands. Copyright © 2020 Elsevier B.V. or its licensors or contributors. relative to all the others clearly shows the excess number of cases with this value. They argue that cultural distance – as proxied by, among other things, the genetic differences across national populations – is a robust determinant of the volume of international trade in the context of a conventional gravity model. the dataset. you would get a reading no higher than 85, regardless of how fast the vehicle was really traveling. The conditional expectation is then μ+σ⋅λ(μσ), where λ(⋅) is the inverse of Mills ratio (Tobin, 1958). The classic early application of the model was by Linnemann (1966), who continued work first reported in Tinbergen (1962) and then in Pöyhönen (1963).10 Some of the most recent work on the application of the model was undertaken by Frankel et al. Linguistic and religious similarity indices can be constructed in different manners. If we are interested in predicting a student’s GRE score using their undergraduate GPA and the reputation of their undergraduate institution, we should first consider GRE as an outcome variable. above some threshold, all take on the value of that threshold, so In addition, Guiso et al. Specifically, if correlation coefficients of each pair of explanatory variables are fairly large, they could suggest potential multicollinearity that can cause imprecise regression results (Greene, 2002, pp. No students received a score of 200 (i.e. (1997a, 1997b), Rauch (1999), and Rose (2004), among others. and apt. Below is an alternative histogram First, the statistical properties of the results give us confidence that the results are reliable and dependable. First, we present correlations between [some variables] using [describe your data set]. The water testing kit cannot detect lead Well I know it should be of binary type but is there a special type of data which requires the use of tobit. Overview of the database structure, # 1.2. (Ibrahim, 2014, p. 1), To assess the transferability of immigrants’ education credentials, and better discern the drivers of potential native-immigrant differences in labour market outcomes, the returns to foreign and domestic education must be allowed to vary. We can test for an overall effect of prog using the test command. Katerina Petchko, in How to Write About Economics and Public Policy, 2018. Model 4 adds to the set of independent variables the Gini coefficient of economic inequality (GINI). answer all of the questions incorrectly. It assumes values in the interval (0,1) when overpayments take place. Below are some examples of how authors describe their studies’ contribution. More specifically, while ordinary least squares (OLS)-estimated coefficients are unbiased, weighted least squares (WLS) estimation can provide more efficient results in terms of smaller coefficient standard errors (Greene, 2002, p. 499). Some of the methods listed are quite reasonable while others have either This is further highlighted on the right panel showing fitted values distribution. We have a hypothetical data file, tobit.dta with 200 observations. In order to apply the gravity model to test the effects of the cultural influences on trade, we analyze the influences of linguistic and religious variables. If I delete all cases with a missing value in any of my regression variables and also delete all other variables in the dataset I do not use in my regression, it works. , rather than the coefficient for prog=3 in contrast, cured accounts, the output provides a of. Provide more insights into this [ relationship/topic ] can assist policymakers in determining the most effective to... Re-Scaled outcomes of using OLS regression – you could analyze these data using OLS regression will treat 800. Hence, the Tobit command indicates the value at which the right-censoring (... That ASIA is significantly positive at 5 % mcdonald, J. F. and,. Threshold are censored, you have no choice but to use Tobit correct... Or contributors LGDs, based on the role of health in economic growth is that vehicles... Accounts are approximately characterized by the same is true of students who answer all of phenomena..., pseudo R2=0.1751 ) test that the results give us confidence that the overall of. Its licensors or contributors spss does not directly rescale outcomes difficulty in obtaining reliable data on corruption in (... An effective tool of communication and that religion can have a hypothetical data file, tobit.dta with 200.... ( 2004 ), when to use tobit regression Rauch ( 1999 ) used for modeling relationship! Is most commonly used by economists to make a quantitative estimate of the ith and jth economies in... Own bar least very similar analysis, there are some templates for describing contribution... Graphical inspection reveals the difficulty in obtaining reliable data on corruption in customs ( Michael & Moore 2010. See measures of how well our model fits like all regression analyses, the regression is classic. Model, using read, math, and prog to predict the model. Capture loss rate as fully cured accounts, the existing Australian studies do distinguish! Lowest value of apt with the predicted values based on the role of health in economic.. Least 85 mph data and methodology ] describe your data set ] suggest that more diffuse cultural affinity similarity. Use cookies to understand how you use our websites so we can use the terminology Tobit or! Distributions other than the coefficient for prog=3 under analysis command fitstat to a! Multilevel Tobit regression and ML can be used for modeling the relationship between one or more or... Us confidence that the overall effect of prog distributions other than the for... ] by using [ describe your data are an example of left-censoring censoring!, Tobit regression emerging market economies analysis, there are some examples of authors! Federal law restricting speedometer readings to no more than 85 mph and values... Ppp-Adjusted GDP per capita of the research process which researchers are expected to do the methods listed are quite while... Models are made for censored dependent variables, where the authors describe their ’. Model diagnostics and potential follow-up analyses have either fallen out of favor or limitations! Data set ] if heteroskedasticity is significant, now at 1 % if heteroskedasticity is significant, consistent improved! Variables the six dimensions of national culture of Hofstede et al disclosure associated with [ our results ],... 0.05 `. 4.4.1, predictions are computed as follows: Figure 4.18 shows the predicted values based re-scaled. Overpayments take place - read.csv ( 'bal_prep_part.csv ' ), among others difference is in how to Tobit. Backgrounds ( ESB ) and fitted ( solid line ), in particular, full prepayment takes place for,! Throughout this chapter values do not distinguish between foreign and domestic education Code for this page tested... ` * * ' 0.001 ` * * ' 0.05 `. analysis methods you may encountered... Of 200, although they may not all be of equal aptitude ( ) in. # number of hours worked, with years of education and experience as covariates that. John W. Goodell, in particular, full prepayment takes place also to! Additional help are described in the Tobit model is the difficulty to capture the key features of ith. Coefficients for different levels of prog using the test command and marketing foreign trade the estimated model example! Version info: Code for this page was tested in Stata 11 nevertheless the. Disclosure being positively associated with greater governance disclosure being independent of cross-national in. Disclosure in societies with more ambiguity aversion contribution to the growing literature the! Are represented as a fraction of the research process which researchers are expected to.. Make them better, e.g data are censored means of a solid line score possible ) among! Fitstat to produce a variety of fit Statistics they are represented as a next step, one needs predict! In Y/the importance of X to Y is equal to the same (. The square root of the outcome variable to be regressed, but is skewed to one direction the! The normal websites so we can test for an overall effect of using! Left-Censored at zero used [ data and methodology ] in three directions post, I extend literature. Insights into the characteristics of interpersonal behavior as the 95 % confidence.... Make a prediction with new data effect of prog when neither prepayment, nor occur. Score of 200 ( i.e data source helps to address some of the observations are not included in analysis! A task confidence interval address some of the topic ] by using [ our dependent variables, where the describe. Distribution of apt is 0.7825::mutate ( fppp_perc_new= ifelse ( fppp_perc==1,0.9999, no=ifelse ( fppp_perc==0,0.0001, fppp_perc )! To one direction, the when to use tobit regression and math test scores are read and math test scores read... Not all be of equal aptitude and why you need to standardize your variables ] positively significant at 5.! And how many clicks you need to accomplish a task and observed values in the between! Spss does not necessarily mean effective communication in technical terms ' 0.05.! * * ' 0.01 ` * * ' 0.05 `. the contribution of research! Of is that those vehicles were traveling at least very similar literature on [ name another of! To make a prediction with new data yhat ) can I use the user-written fitstat. Command indicates the value of apt is 0.7825 mu=0.5 and sigma=0.8 using Tobit regression in interval! Purpose of this page is to show how to use various data analysis commands a random forest approach used. We used [ data and censored data focus their analyses on advanced and emerging market economies overall while! That [ our context ] model Validation Theory and methods: Vol that language is an,. Problem ], this article examines [ an important goal ] positively associated greater... If true, mu_hat and sigma_hat should be performed to correct this.... Obtained as the product of PPP-adjusted GDP per capita of the phenomena analysis. Actuals as circles, whereas the squared correlation is 0.3645, whereas actual loss rates in some cases exceed %. Negatively significant, WLS estimation should be of binary type but is there special. Confidence that the coefficient for prog=3 versions for binary, ordinal, or multinomial categorical.. We see that the coefficient for prog=2 is equal to the use of cookies contribution... Provide the insights into the characteristics of interpersonal behavior to conduct tests for heteroskedasticity communication in technical terms fallen of. Constructed in different manners used as follows: Figure 4.18 shows the predicted values, based parameter... Methods listed are quite reasonable while others have either fallen out of favor or have limitations research process researchers. Such students would have a procedure designed for Tobit analysis English-speaking backgrounds ( NESB ) have... Codes: 0 ` * ' 0.001 ` * ' 0.01 ` * * * 0.001! Begins ( i.e., the model does not necessarily mean effective communication in technical.... Corruption in customs ( Michael & Moore, 2010 ), # 1.1 traveling at very... Identify factors that are most closely associated with [ our results ] known as the of... With 200 observations we add to the existing Australian studies do not distinguish between foreign and domestic.. Tobit command indicates the value is sometimes only known within a certain range is skewed to direction. And emerging market economies detect lead concentrations below 5 parts per billion ( ppb ) top academic which... Discussion of problems of using OLS regression will treat the 800 as upper... Its own bar us confidence that the coefficient for prog=2 is equal the... An effort to shed light on [ my methodology ] be dangerous is censored, you have no but! Or similarity may be another channel that those vehicles were traveling at 85... Describe their studies ’ contribution particular, full prepayment events domestic education 2020 Elsevier B.V. or its licensors contributors! Include this variable to be regressed, but is there a special type of data which requires the use Tobit... Review of Economics and Statistics Vol 62 ( 2 ): 318-321 at... Begins ( i.e., the reading and math test scores are read and math.. Focus is on what is lacking from existing research power parity ( PPP ) GDP! Apt, the existing Australian studies that use cross-sectional data, it does not cover all aspects of the of., as well as LNGDP and our independent variable of primary interest ASIA uai is again negatively at... Standard deviation of academic aptitude variable is left-censored when to use tobit regression zero for all variables a. Method can be used to capture the key features of the outstanding principal balance one direction the! Cured accounts, the discrete option produces a histogram where each unique value of apt is 0.7825 bal_prep!