metadata: authors: 0: author: Canavire Bacarreza,Gustavo Javier
1: author: Rios Avila,Fernando
2: author: Sacco Capurro,Flavia Giannina
count: World
docna: 0: docna: Recovering Income Distribution in the
            Presence of Interval-Censored Data
docty: Policy Research Working Paper
abstracts: cdata!: This paper proposes a method to analyze
            interval-censored data, using multiple imputation based on a
            heteroskedastic interval regression approach. The proposed
            model aims to obtain a synthetic data set that can be used
            for standard analysis, including standard linear regression,
            quantile regression, or poverty and inequality estimation.
            The paper presents two applications to show the performance
            of the method. First, it runs a Monte Carlo simulation to
            show the method's performance under the assumption of
            multiplicative heteroskedasticity, with and without
            conditional normality. Second, it uses the proposed
            methodology to analyze labor income data in Grenada for
            2013&#x2013;20, where the salary data are
            interval-censored according to the salary intervals
            prespecified in the survey questionnaire. The results
            obtained are consistent across both exercises.
display_title: Recovering Income Distribution in
            the Presence of Interval-Censored Data
listing_relative_url: /research/2022/08/33886176/recovering-income-distribution-presence-interval-censored-data
topicv3: Inequality and Shared Prosperity m1327655 700,Employment Tools and Methods m1504070 2760,Macroeconomic Modelling and Statistics
            m1504530 3108
publishtoextweb_dt: 2022-08-22T00:00:00Z
dois: 10.1596/1813-9450-10147
author_keys: 0: author_key: 715971
1: author_key: 1974997
2: author_key: 1579490
model_extractions: Policy Research Working Paper 10147 Recovering Income Distribution in the Presence of Interval-Censored Data _Gustavo Canavire-Bacarreza Fernando Rios-Avila Flavia Sacco-Capurro_ Poverty and Equity Global Practice August 2022
Policy Research Working Paper 10147

# **Abstract**

This paper proposes a method to analyze interval-censored data, using multiple imputation based on a heteroskedastic interval regression approach. The proposed model aims to obtain a synthetic data set that can be used for standard analysis, including standard linear regression, quantile regression, or poverty and inequality estimation. The paper presents two applications to show the performance of the method. First, it runs a Monte Carlo simulation to show the method’s performance under the assumption of multiplicative heteroskedasticity, with and without conditional normality. Second, it uses the proposed methodology to analyze labor income data in Grenada for 2013–20, where the salary data are interval-censored according to the salary intervals prespecified in the survey questionnaire. The results obtained are consistent across both exercises.

This paper is a product of the Poverty and Equity Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at gcanavire@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
**Recovering Income Distribution in the Presence of Interval-Censored Data**<sup>*****</sup>

Gustavo Canavire-Bacarreza<sup>†</sup> Fernando Rios-Avila<sup>‡</sup>

# Flavia Sacco-Capurro<sup>§</sup>

**Keywords:** interval-censored data, Monte Carlo simulation, heteroskedastic interval regression,

wages JEL Codes: C150, C340, J3

> * The findings, interpretations, and conclusions expressed in this paper do not necessarily reflect the views of the World Bank, the Executive Directors of the World Bank or the governments they represent. The World Bank does not guarantee the accuracy of the data included in this work. The authors would like to thank Ximena del Carpio, Leonardo Luchetti, Carlos Ospino, Daniel Mahler and the participants at the 5th IZA Labor Statistics Workshop: The Measurement of Incomes, Living Costs and Standards of Living and the 2022 Stata Conference for helpful comments and suggestions.

> † The World Bank, gcanavire@worldbank.org

> ‡ Levy Institute at Bard College, f.rios.a@gmail.com

> § The World Bank, fsaccocapurro@worldbank.org
# 1. Introduction

Labor force surveys are a useful data source to understand employment dynamics in both developing and developed countries. These surveys provide vast information on the labor market status at higher frequency levels than living conditions surveys. In some cases, they are the only source of information to describe and examine the structure of the labor markets. In the Latin American and the Caribbean region, countries like Bolivia, Costa Rica, Ecuador, Jamaica, Mexico, Peru, and Uruguay collect their labor force surveys quarterly as opposed to a yearly basis, which is the case of most household and living standard surveys.

One of the key features of these labor surveys is that they provide information on the wages and salaries of workers. This allows to estimate job market trends and obtain inequality measures of labor income among workers. However, the full income distribution in many countries cannot be retrieved because labor income is reported in brackets. Because of this, the estimation of inequality or poverty measures, as well as regression type analysis, is difficult. This is the case of the labor force survey for all countries in the Organization of Eastern Caribbean States (OECS).

This is not unique to the Caribbean region. Countries like Colombia, Germany, Australia, New Zealand, Bosnia and Herzegovina, North Macedonia and Serbia, among others, have similar data collection protocols for their micro census (Walter and Weimer 2018). In the U.S, the current population survey (CPS) collects detailed family income only once a year, in the March supplement, but collects family income in brackets on a monthly basis.

One argument in favor of using interval-censored questions to collect information on income is the higher response rate compared to questions asking to report exact amounts (Wang et al., 2013). This happens because income information is considered “sensitive”, and people are reluctant to report actual earnings, and may choose not to respond to those questions at all (Moore et al., 2000; Hagenaars and Vos, 1988). Field tests conducted in the past have shown that asking follow-up income questions in a series of unfolding brackets achieves superior results in terms of response rates for income amounts, as was the case of the National Health Interview Survey (NHIS) and the Behavioral Risk Factor Surveillance System Survey (BRFSS), both administered by the Center for Disease Control and Prevention of the United States (Angelov and Ekstrom 2018, Yan et al. 2018).
However, even though this form of data collection solves the problem of underreporting or misreporting, it raises a problem for recovering the full wage (income) distribution, which is key to understanding and analyzing inequality.

To better use the information from these types of surveys, we propose an imputation approach to simulate the distribution of the data that is only available in brackets. The method is an extension on the imputation approach described in Royston (2007), which considers heteroskedastic errors to model the conditional distribution of the censored data. The estimated conditional distribution is then used impute the data using draws from the estimated conditional distribution. Once the imputed data is obtained, standard aggregation methods (Rubin, 1987) can be used to analyze the censored data as if it were fully observed. For example, it can be used calculate poverty or inequality measures, as well as perform regression analysis. To demonstrate the flexibility of this approach, we use a Monte Carlo simulation to analyze the sensitivity of our method. As an empirical example, we use the approach to analyze wage inequality in Grenada utilizing the country’s Labor Force Survey.

Other approaches exist in the literature, and have been used for analyzing this kind of data. To measure income inequality with right-censored (top-coded) data, Jenkins et al. (2011) propose multiple-imputation methods for estimation and inference where censored observations are imputed using draws from a flexible parametric model fitted to the censored distribution, such as Generalized Beta of the second kind (GB2), Sigh-Maddala or Dagum distributions. Chen (2017) provides a generalized approach for the estimation of parametric income distributions using grouped data, showing its consistency through complementary simulation results. More recently, Walter and Weimer (2018) propose an iterative kernel density algorithm that generates pseudo samples from the interval-censored income variable to estimate poverty and inequality indicators. While the interval regression approach we propose fits with the models described in Chen (2017), Jenkins et al (2011), and Walter and Weimer (2018), these papers focus on recovering the unconditional distribution of income, without considering the relationship with explanatory variables.

Zhou et al. (2017) and Chih-Yuan et al. (2021) propose methodologies for the estimation of conditional quantile regressions using interval censored data, under different distributional
assumptions. While this approach can be used for analyzing interval censored data, it only focuses on estimating conditional quantile regressions, requiring specialized software that are not readily available. In contrast, the method we propose can be applied not only for the estimation of conditional quantile regressions, but also for the estimation of unconditional distribution statistics.

Other studies, like the one proposed by Han et al. (2020), construct new measures of the income distribution and estimate poverty in the U.S. using data from the monthly Current Population Survey (CPS). They address the problem of censored income data using draws from the empirical income distribution observed in the last March supplement. A similar method is proposed by Parolin and Wimer (2020), who produce monthly updates of the Supplemental Poverty Measure (SPM) rates with demographic data from the CPS and poverty data from the previous March supplement of the CPS. However, these studies seek to obtain income estimates using the uncensored distribution of previous years, which is not always available with other data sources, like the ones analyzed in this paper.

Buutner and Rassler (2008) propose a multiple imputation approach, similar to ours, to analyze wages from the German Institute of Employment Research (IAB) employment survey. While their method focuses on the analysis top coded data, we expand the approach to analyze data with a more generalized censoring structure.

The paper is organized as follows. Section 2 introduces the model and the econometric issues associated with the imputation method; Section 3 provides a Monte Carlo simulation exercise to analyze the performance of the methodology; Section 4 uses the methodology to analyze labor income distribution changes in Grenada using the 2013-2020 series of the Labor Force Survey. Section 5 concludes.

# 2. Methodology

To address the problem of interval-censored data, we propose a multiple imputation approach based on a heteroskedastic interval-regression model. An interval-regression model is a generalization of the Tobit model that allows using a mixture of censored and completely observed data, even if the censoring thresholds are unique to each individual. The goal of the model is to
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# **3. Simulation studies**

# **3.1. Setup**

We examine the performance of our proposed estimator under several simulation scenarios, using data structures with explicit multiplicative heteroskedasticity, similar to the ones proposed in Machado and Santos-Silva (2019), and with a varying coefficient model structure, as in Hsu, Wen and Chen (2021). In both cases, the goal is to simulate data that would show heterogeneity when using conditional quantile regressions for the estimation. This structure is flexible enough to also allow the estimation of other distribution based regressions such as unconditional quantile regressions (Firpo, Fortin and Lemieux, 2009) and Recentered Influence function regressions in general (Rios-Avila, 2020).

The first set of simulations is designed to study the performance of the estimator under the assumption of multiplicative heteroskedasticity assuming the following functional form: 0 1 1 2 2 1 2) (15) Where 1~b yy= ββ and + ββ2 xx52/5 + ββ . Following Machado and Santos-Silva (2019), xx + vvσσ(xx , xx we use two different functional forms for 1 2): xx rrUUuul ii(0.5) xx ~χχ σσ(xx , xx 1 1 2 0 1 1 2 2 σσ (xx , xx ) = γγ + γγ xx + γγ xx (16aa) 2 1 2 σσ (xx , xx ) = bb<sup>γγ0+γγ1xx1+γγ2xx2</sup> (16bb) In both cases, we require that 1 2) to be strictly positive. The first case, equation (16a), imposes the assumption of linear hetoreskedasticity and provides a closed form solution for the σσ(xx , xx corresponding quantile coefficients. The second option, equation (16b), guarantees standard deviation to be strictly positive, but does not have a closed form solution for the corresponding conditional quantile regression coefficients. As described in Machado and Santos-Silva (2019), this data generating process also guarantees that quantiles will not cross, and thus the
corresponding coefficients can be estimated directly using standard conditional quantile regression estimators.

Using this data structure, we consider four different distributions for the error : Normal distribution, logistic distribution, chi square distribution with 5 degrees of freedom, and uniform vv distribution. All of them were adjusted to have a mean 0 and standard deviation 1. Whereas the first two distributions are meant to show how sensitive is the estimator to the normality assumption, the third and fourth aim to show how sensitive the results are to cases where the error has a skew distribution, or a distribution with limited range. With this considerations, the data generating process is defined as: 1 2 1 2) 1 2 yy= xx + xx + vv∗(1 −0.5xx + 0.2xx (17aa) yy= xx + xx + vv∗bb<sup>0.6−0.5+0.2xx2</sup> (17bb) Equation (19a) imposes a structure that is similar to the multiplicative normality under linear heteroskedasticity (equation 17a), whereas the second equation imposes a skew conditional distribution of the outcome.

In all scenarios, we assume that data is subject to interval censoring, such that l = ⌋ & u ii respectively. In addition, we also assume if ii ii<sup>= ⌈yy</sup> ii<sup>⌉, where ⌊. ⌋ and ⌈. ⌉ represent the nearest integers that is lower or higher than</sup> < −1 or > 10, the lower and upper thresholds,<sup>yy</sup> ii ⌊yy respectively, will be undefined. ii ii yy yy
For the implementation and analysis, we use 2500 replications, with a sample size of 1,000 observations. Replications using sample sizes of 500 and 2,000 are provided in the appendix, with results that are qualitatively similar. We focus on the comparison of conditional quantile regressions for the 10<sup>th</sup> , 50<sup>th</sup> and 90<sup>th</sup> quantiles, as well as for the 10<sup>th</sup> , 50<sup>th</sup> and 90<sup>th</sup> unconditional quantiles. Quantile regressions were estimated using the fast algorithm developed in Chernozhukov et al (2022) and implemented via the Stata command -qrprocess-, whereas the unconditional quantile regressions were estimated following Firpo, Fortin and Lemieux (2009) and implanted via the Stata command -rifhdreg- (Rios-Avila, 2020). Finally, the simulation was implemented using -parallel- (Vega Yon and Quistorff, 2019). Finally, our imputation method is implanted with a new user-written program -intreg_mi-, which is available upon request.

While population parameters for conditional quantile regressions for some of the data generating exists, there are no close form solutions for the population parameters corresponding to the RIF regressions. Because of this, we assume that average estimates using fully observed data represent the population parameters, for the calculation of the relevant statistics. Thus, by construction, the bias of the model estimations using fully observed data is zero.

# **3.2. Results**

Tables 1 to 3 provide a summary of the results for the Monte Carlo simulations using the different data generating processes. In each table, we present the bias of our imputation procedure compared to the average parameters using fully observed data as if they were the asymptotic population parameters. For both conditional and unconditional quantile regressions, the bias observed using the multiple imputation data is small when the homoscedastic error is assumed to follow a symmetric bell curve distribution, regardless of the type of heteroskedasticity implied by the dgp.

When the errors follow a chi2 distribution or uniform distribution, we observe some bias, especially for the lower quantile coefficients. The bias, however, is considerably smaller if the data generating process assumes an functional form with exponential heteroskedasticity. Finally, the results using the varying coefficient structure reveal low bias in both cases. In all simulations, the bias magnitude did not depend on the sample size (see appendix).
In terms of the mean absolute error (MAE), we present the ratio between the MAE for the imputed data and the MAE for the fully observed data. Except for cases when the bias is large, the MAE for the imputed data is somewhat smaller than the one using fully observed data, by almost 10%. It is possible that this gain in the precision of the point estimate may be simulation specific, since we are indirectly using parametric structures for the estimation of the quantile regressions. In terms of standard errors ratio, which compares the average standard errors of the imputed data to fully observed data, we observe that the standard errors for imputed data are about 15% larger on average, than the standard errors based on fully observed data. This is expected given the information loss due to the nature of the interval censored data.
# Table 1. Monte Carlo Simulation: N=1000, Linear Heteroskedasticity

||||u~n|ormal|||u~l|ogistic|||u~|Chi2|||u~u|niform||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|y= xβ+|u∗γx|||MAE|StErr|||MAE|StErr|||MAE|StErr|||MAE|StErr|
|||TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|
||<br>x1|2.011|0.008|-0.014|0.235|1.955|-0.030|-0.094|0.156|1.848|0.321|3.886|0.915|2.094|0.258|2.073|0.690|
|CQR-Q10|x2|0.798|0.002|-0.030|0.155|0.803|-0.006|-0.063|0.134|0.831|0.123|1.490|0.377|0.786|0.052|0.293|0.296|
||cons|-2.381|-0.008|0.006|0.292|-2.247|0.022|-0.084|0.207|-1.989|-0.471|5.425|1.035|-2.572|-0.264|1.946|0.749|
||x1|1.000|0.001|-0.055|0.070|1.003|-0.001|-0.056|0.086|1.168|0.020|-0.029|0.094|0.996|0.004|-0.053|0.043|
|CQR-Q50|x2|1.001|-0.002|-0.055|0.071|0.997|-0.001|-0.025|0.088|0.969|0.000|-0.048|0.084|0.999|0.001|-0.074|0.041|
||cons|-0.001|0.002|-0.042|0.072|-0.002|0.002|-0.060|0.088|-0.385|0.023|-0.032|0.096|0.005|-0.005|-0.042|0.045|
||x1|-0.009|0.000|-0.049|0.103|0.041|0.005|-0.066|0.096|-0.051|-0.011|-0.051|0.048|-0.097|-0.008|-0.019|0.146|
|CQR-Q90|x2|1.199|0.001|-0.041|0.110|1.191|0.000|-0.067|0.109|1.215|-0.004|-0.054|0.047|1.216|-0.001|0.024|0.145|
||cons|2.383|-0.001|-0.059|0.099|2.252|0.007|-0.061|0.095|2.486|-0.003|-0.048|0.046|2.573|-0.042|0.057|0.141|
||x1|2.097|0.008|-0.005|0.159|1.915|-0.021|-0.065|0.114|1.840|0.188|0.903|0.287|2.539|0.236|0.760|0.459|
|UQR-Q10|x2|0.611|0.001|-0.021|0.060|0.602|-0.008|-0.067|0.047|0.684|0.079|0.312|0.252|0.651|0.027|0.075|0.191|
||cons|-2.537|-0.006|0.006|0.130|-2.342|0.016|-0.063|0.108|-2.370|-0.174|0.557|0.332|-3.046|-0.163|0.352|0.401|
||x1|1.006|0.000|-0.083|0.130|1.026|-0.007|-0.059|0.159|1.165|0.039|-0.011|0.161|0.945|0.012|-0.070|0.066|
|UQR-Q50|x2|0.929|0.000|-0.058|0.110|0.919|-0.002|-0.061|0.128|0.945|0.000|-0.067|0.144|0.921|0.007|-0.096|0.062|
||cons|0.131|0.001|-0.069|0.122|0.120|0.004|-0.072|0.140|-0.199|0.020|0.001|0.137|0.190|-0.004|-0.076|0.073|
||x1|0.052|-0.001|-0.076|0.103|0.106|0.004|-0.061|0.108|0.014|-0.004|-0.055|0.034|-0.003|-0.004|-0.045|0.166|
|UQR-Q90|x2|1.466|0.001|-0.053|0.192|1.492|-0.004|-0.089|0.213|1.455|-0.004|-0.087|0.111|1.484|-0.015|0.052|0.231|
||cons|2.263|-0.001|-0.075|0.129|2.134|0.010|-0.063|0.134|2.369|0.000|-0.060|0.068|2.314|0.010|0.003|0.176|
# Table 2 Monte Carlo Simulation: N=1000, exponential Heteroskedasticity

||||u~n|ormal|||u~lo|gistic|||u~|Chi2|||u~u|niform||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|y= xβ+ u|∗b<sup>γx</sup>|||MAE|StErr|||MAE|StErr|||MAE|StErr|||MAE|StErr|
|||TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|
||<br>x1|1.639|-0.001|-0.032|0.168|1.603|-0.009|-0.066|0.143|1.536|-0.030|0.301|0.484|1.691|-0.012|0.228|0.320|
|CQR-Q10|x2|0.743|0.004|-0.040|0.173|0.758|0.003|-0.078|0.162|0.788|0.018|0.258|0.465|0.726|0.046|0.501|0.268|
||cons|-1.280|-0.003|-0.026|0.160|-1.209|-0.013|-0.081|0.129|-1.072|-0.050|0.434|0.456|-1.384|-0.008|0.352|0.344|
||x1|1.000|0.000|-0.076|0.115|0.999|0.000|-0.045|0.158|1.102|0.025|0.009|0.152|1.001|-0.001|-0.123|0.045|
|CQR-Q50|x2|0.999|0.000|-0.060|0.114|0.998|-0.003|-0.027|0.158|0.959|-0.009|-0.012|0.154|1.002|0.007|-0.104|0.043|
||cons|0.000|0.001|-0.078|0.119|0.004|0.003|-0.032|0.162|-0.204|0.053|0.195|0.154|0.000|-0.008|-0.105|0.045|
||x1|0.364|-0.002|-0.066|0.170|0.392|0.009|-0.062|0.154|0.331|-0.003|-0.075|0.053|0.306|0.014|0.148|0.270|
|CQR-Q90|x2|1.255|-0.001|-0.065|0.162|1.239|-0.002|-0.040|0.157|1.269|-0.007|-0.078|0.035|1.274|-0.012|0.152|0.253|
||cons|1.279|0.001|-0.066|0.164|1.216|0.012|-0.061|0.147|1.340|-0.010|-0.083|0.047|1.386|-0.040|0.268|0.266|
||x1|1.613|-0.002|-0.184|0.256|1.478|0.018|-0.097|0.258|1.273|0.071|0.059|0.265|1.734|-0.125|0.171|0.089|
|UQR-Q10|x2|0.582|0.001|-0.097|0.107|0.565|0.006|-0.052|0.100|0.648|-0.003|-0.106|0.180|0.670|-0.084|0.219|0.042|
||cons|-1.533|-0.001|-0.135|0.274|-1.390|-0.019|-0.043|0.256|-1.417|-0.042|-0.138|0.348|-1.843|0.200|0.496|0.167|
||x1|1.003|0.002|-0.072|0.246|1.025|-0.015|-0.053|0.273|1.099|0.037|0.045|0.284|0.922|0.072|0.152|0.181|
|UQR-Q50|x2|0.850|0.001|-0.079|0.183|0.853|-0.009|-0.074|0.200|0.860|-0.031|0.037|0.219|0.839|0.042|0.077|0.139|
||cons|0.170|0.000|-0.079|0.207|0.152|0.016|-0.047|0.221|0.023|0.059|0.191|0.233|0.245|-0.077|0.144|0.182|
||x1|0.430|-0.002|-0.047|0.077|0.446|-0.006|-0.068|0.050|0.383|-0.020|-0.075|0.000|0.429|0.009|-0.010|0.129|
|UQR-Q90|x2|1.624|-0.001|-0.124|0.484|1.612|-0.024|-0.105|0.433|1.585|-0.064|-0.055|0.295|1.630|0.041|-0.083|0.558|
||cons|1.212|0.002|-0.134|0.323|1.190|0.027|-0.127|0.290|1.305|0.066|-0.073|0.163|1.217|-0.043|-0.078|0.377|
Table 3 Monte Carlo Simulation: N=1000, Varying coefficient structure

||||Typ|e1|||Typ|e2||
|---|---|---|---|---|---|---|---|---|---|
|||||MAE|StErr|||MAE|StErr|
|y=xβ(|t)|TRUE|Bias|Ratio|Ratio|TRUE|Bias|Ratio|Ratio|
||x1|-1.140|0.000|-0.030|0.103|-0.092|0.011|0.033|0.154|
|<br>CQR-Q10|x2|-0.035|0.010|-0.038|0.120|-0.086|0.010|0.016|0.156|
||cons|0.356|-0.010|-0.046|0.182|-0.086|-0.043|0.178|0.170|
||x1|0.404|-0.002|-0.052|0.067|0.845|-0.009|-0.021|0.056|
|CQR-Q50|x2|0.601|-0.001|-0.050|0.065|0.841|-0.004|-0.016|0.045|
||cons|0.998|0.003|-0.031|0.085|0.853|0.029|0.011|0.059|
||x1|1.938|0.001|-0.048|0.098|2.282|-0.001|-0.045|0.045|
|CQR-Q90|x2|1.236|-0.005|-0.038|0.098|2.280|-0.010|-0.067|0.054|
||cons|1.644|0.004|-0.046|0.139|2.309|-0.002|-0.066|0.073|
||x1|-1.211|0.002|-0.046|0.170|-0.097|0.018|-0.027|0.174|
|UQR-Q10|x2|-0.044|0.002|-0.023|0.072|-0.078|0.012|-0.031|0.157|
||cons|0.482|-0.004|-0.025|0.066|-0.074|-0.056|0.078|0.162|
||x1|0.418|-0.003|-0.059|0.119|0.900|-0.008|-0.023|0.031|
|UQR-Q50|x2|0.535|-0.003|-0.071|0.112|0.737|-0.005|-0.015|0.026|
||cons|0.899|0.007|-0.061|0.124|0.757|0.017|-0.031|0.028|
||x1|1.982|0.001|-0.079|0.150|2.214|-0.002|-0.039|0.051|
|UQR-Q90|x2|1.296|0.002|-0.060|0.136|2.321|0.001|-0.022|0.077|
||cons|1.778|-0.003|-0.093|0.141|2.499|-0.006|-0.031|0.057|

# **4. Wage inequality in Grenada**

This illustration focuses on an empirical application of our proposed method for the case of Grenada, focusing on the description of wage inequality trends in the country between 2013 and 2020 using the annual Labor Force Survey (LFS). This survey provides the only source of information that can be used to describe the status of the labor market and the distribution of labor income in the country.

One major limitation of this survey, however, is the collection of labor income data. Compared to standard household surveys or labor force surveys in most developed countries, labor income recorded in the LFS in Grenada is only available in brackets. Furthermore, there is a large proportion of the employed population who do not declare their labor income. Table 4 provides an overview of the labor income distribution across time.
_Table 4 Labor Income distribution by year_

|**Year**|**2013**|**2014**|**2015**|**2016**|**2017**|**2018**|**2019**|**2020**|
|---|---|---|---|---|---|---|---|---|
|>200|3.0|1.2|3.7|3.5|1.4|0.2|0.0|0.4|
|200-399|6.9|5.8|6.3|5.3|4.1|1.6|1.2|1.1|
|400-799|15.4|15.9|12.3|14.2|13.7|9.0|8.3|10.3|
|800-1199|19.1|20.0|18.3|18.7|21.1|20.4|23.8|24.6|
|1200-1999|17.7|17.4|13.9|13.1|18.4|14.7|14.9|15.9|
|2000-3999|15.6|11.3|11.2|11.5|10.5|9.7|12.8|11.8|
|4000-5999|2.6|2.4|2.4|2.2|2.2|1.6|1.2|2.1|
|6000+|2.0|1.2|0.6|0.6|0.7|1.0|1.0|0.5|
|Not stated|17.7|24.8|31.3|30.9|27.9|41.8|36.7|33.2|

In this case, we face two types of problems. On the one hand, we only have access to intervalcensored data, which is insufficient to analyze changes in the distribution of earnings in the country, and, on the other hand, we have an increasing proportion of individuals who do not declare income. We apply the imputation procedure previously described to address both problems, estimating the interval-censored regression for each year, with a set of household-level characteristics and job type characteristics. The sample of interest includes all adults who declared to be employed, even if they did not state their income.

We make the simplifying assumption that people who did not state income are randomly distributed conditional on observed characteristics. To account for the fact that characteristics may differ across those who did or did not state their incomes, an inverse probability weighting strategy is used to estimate the interval regression model. Finally, the imputation procedure is implemented as discussed in section 3 but assuming no lower and upper bounds for the imputed wages. Nevertheless, the maximum imputed wage for those who do not state their income is capped at the maximum predicted among those who declare their income. In all cases, imputed earnings are adjusted by inflation.
_Figure 1 Average Monthly Earnings by Year and Gender_ <!-- Start of picture text --> 1600<br>1500<br>All<br>Men<br>1400<br>Women<br>1300<br>1200<br>2012 2014 2016 2018 2020<br>year<br>Average Monthly earning<br><!-- End of picture text --> The results suggest that after a small decline in average real monthly earnings from 2013 to 2016, there was a slight improvement in the following two years, with a small decline in 2019, with average wages remaining at stable levels in 2020, despite the Covid-19 pandemic.<sup>7</sup> The results also suggest that the gender earnings gap has shown a somewhat increasing trend between 2013 and 2019, although it is predicted to decline a little in 2020.

7 This estimate does not take into account the decline in labor force participation observed during the pandemic.
_Figure 2 Selected Quantiles and Gini coefficient across Years_ <!-- Start of picture text --> 46<br>2500<br>44<br>1500<br>42<br>1000<br>40 Q10<br>Q50<br>Q90<br>600 38 Gini<br>36<br>300<br>34<br>32<br>2012 2014 2016 2018 2020<br>year<br>Gini points<br>Monthly earnings (log scale)<br><!-- End of picture text --> In terms of inequality, the estimates suggest that it has declined substantially across the years. The estimated Gini coefficient fell from 44.2 Gini points in 2015 to 34.1 in 2019, with a significant increase in 2020. This decline in inequality seems to have been driven by faster growth in the lower and middle sections of the wage distribution and a small decline in the upper section of the distribution.

# **5. Conclusion**

In this paper, we present an imputation strategy that can be used to analyze interval-censored data. Our method proposes that a flexible enough interval regression model can be used to impute interval-censored data, which allows to recover the full distribution of data, and can be further analyzed using standard statistical methods.

The main limitation of our strategy is the assumption of conditional normality, which is required for the estimation of the interval regression model using standard software. The principles of the imputation approach, however, could be extended to allow for more flexible moment specifications, as well as error distributions.
Nevertheless, the Monte Carlo simulation suggests that as long as the latent error has a symmetric bell shaped distribution, regression analysis using the imputed data show small bias, with performance that is comparable to analyzing the uncensored data. Furthermore, when the heteroskedasticity structure is given by an exponential function, biases are small even when the latent error follows a skew or a limited distribution.

For the specific case of Grenada, we only had access to interval-censored data, which is insufficient to analyze changes in the distribution of earnings in the country, and, on the other hand, we have an increasing proportion of individuals who do not declare income. We apply the imputation procedure to address both problems, estimating the interval-censored regression for each year, with a set of household-level characteristics and job type characteristics. The results suggest that earned income inequality in this country has declined, which coincides with other economic performance indicators in the country.