BAYESIAN ANALYSIS OF TOBIT QUANTILE REGRESSION WITH ADAPTIVE LASSO PENALTY IN HOUSEHOLD EXPENDITURE FOR CIGARETTE CONSUMPTION

: Tobit Quantile Regression with Adaptive Lasso Penalty is a quantile regression model on censored data that adds Lasso's adaptive penalty to its parameter estimation. The estimation of the regression parameters is solved by Bayesian analysis. Parameters are assumed to follow a certain distribution called the prior distribution. Using the sample information along with the prior distribution, the conditional posterior distribution is searched using the Box-Tiao rule. Computational solutions are solved by the MCMC Gibbs Sampling algorithm. Gibbs Sampling can generate samples based on the conditional posterior distribution of each parameter in order to obtain a posterior joint distribution. Tobit Quantile Regression with Adaptive Lasso Penalty was applied to data on Household Expenditure for Cigarette Consumption in 2011. As a comparison for data analysis, Tobit Quantile Regression was used. The results of data analysis show that the Tobit Quantile Regression model with Adaptive Lasso Penalty is better than the Tobit Quantile Regression.


INTRODUCTION
Research, especially surveys in various fields, produces outputs with various data characteristics. One of them is data that is censored at a certain value. This is what inspired Tobin [1] introducing the Tobit method in the regression model. Tobit regression is able to illustrate the relationship between a censored respon variable and its predictor variables. Unfortunately, Tobit regression is not helpful enough for data analysis that does not meet the normality assumption. Furthermore, Powell [2] introduces Tobit Quantile Regression, where the conditional mean function on parameter estimation is replaced by a function of quantiles. Tobit Quantile Regression model is able to provide a more complete explanation of the relationship between response variables and predictor variables than simple Tobit Regression. In Tobit Quantile Regression, the estimation of the regression coefficients for each quantile can be obtained.. method to minimize errors that depend on the sum of the absolute values of the coefficients. Lasso regression is able to eliminate the variables in the regression model by reducing the coefficient to zero. However, Zou [4] states that Lasso Regression for optimal l results in variable selection that is not always consistent.. To solve this problem, Zou [4] introduces the Adaptive Lasso method in which an adaptive weight is applied on the l1 penalty.
This paper will discuss about Tobit Quantile Regression with Adaptive Lasso Penalty. Adaptive weights are added to the estimation of the Tobit Quantile Regression parameter, so that the parameter estimate is the solution of minimizing the Loss Function plus the absolute sum of the adaptive weights. The form of such an optimization function cannot be solved explicitly, so the parameter estimation of this study is determined using Bayesian approach.
Bayesian analysis uses knowledge of prior distributions to find a complete conditional posterior distribution. Furthermore the joint posterior distribution can be found. This characteristic relates the Bayesian approach to the Markov Chain Monte Carlo (MCMC) method using the Gibbs Sampling algorithm. MCMC comes from a combination of two things, namely: Markov Chain and Monte-Carlo. Sampling of this method uses the Markov Chain principle, which generates a new sample based on only one previous sample. The algorithm used is the Gibbs Sampling algorithm.
Cigarette consumers in Indonesia are spread across all economic levels. Based on data from the 2011 Indonesian Household Socio-Economic Survey (SUSETI) by the World Bank in Cintiani [5], household spending on cigarette consumption is quite varied. For households with no active smokers, the cigarette consumption expenditure is 0. Thus this data has censored data criteria with a lower limit of 0. Furthermore, in this paper, Tobit Quantile Regression with Adaptive Lasso Penalty will be applied in the case of Household Expenditure for Cigarette Consumption. Quantile regression was applied to determine the regression analysis for each selected quantile.

LITERATURE REVIEW 2.1. Loss Function
Parameter estimation in quantile regression was obtained by developing the LAD (Least Absolute Deviation) method, which is given different weights for different quantiles. In the τth quantile, for the underprediction case where the error is greater than or equal to zero, the weight used is τ. As for the overprediction case where the error is less than zero, the weight used is 1-τ.
According to Koenker [6], the loss function in Quantile Regression is defined: The form of the Loss Function in Alhamzawi and Yu [7] can also be expressed:

Tobit Regression
The censored data in Tobit regression according to Greene [8] is when the response variable values in a certain range are all changed to one value. The structure of the censored data is a discrete-scale response variable for the censored data, and a continuous scale for those that are not.

The Tobit Regression Equation is
: with: where:

Adaptive Lasso
LASSO stands for Least Absolute Shrinkage and Selection Operator, which is an estimation method that minimizes the sum of the squares of errors that depend on the sum of the absolute values of the coefficients. In other words, the lasso regression coefficient minimizes: James, et al [9] stated that Lasso Regression has the same formula as Ridge Regression, the only difference is that Ridge Regression uses the ∑ & + & constraint whereas Lasso Regression uses the ∑ | & | ≤ result, Lasso Regression can be used for variable selection, so that the model obtained using Lasso Regression will be easier to interpret than using Ridge Regression. To produce consistent variable selection, Zou [4] introduced the Adaptive Lasso. The adaptive weight is added to the l1 penalty so that the equation becomes: by defining lj=l , C, we get:

Tobit Quantile Regression with Adaptive Lasso Penalty
Mosteller and Tukey [10] stated that in order to get a more complete explanation of the data, several different regression curves can be calculated according to various percentage points of the data distribution. Quantile regression was first introduced by Koenker and Basset [11]. In this model, various quantile functions are estimated from a distribution of Y as a function in X. One of the developments of quantile regression is tobit quantile regression. Based on Powell [2], the Tobit Quantile Regression coefficient estimator can be estimated by finding a solution that minimizesfrom: where ! (. ) is the Loss Function.
Alhamzawi [12] added the adaptive lasso penalty to the tobit quantile regression in equation (9) so that the τ -th quantile regression is the solution to minimize β from: This equation contains a Loss Function which is not differentiable at point 0. Therefore, this minimization problem cannot be solved explicitly. The method used in this paper is the Bayesian approach.

Bayesian Analysis of Tobit Quantile Regression with Adaptive Lasso Penalty
Kozumi and Kobayashi [13] stated that minimizing the Loss Function is equivalent to finding the posterior maximum of the estimator with an error following the Asymmetric Laplace distribution. Furthermore " , i=1,2,...,n are assumed to have ALD distribution where The form of the Laplace asymmetric distribution in (11) can be expressed as a mixture of the normal and exponential distribution families. We use the equations of Andrews and Mallows [14], that is, for a, b > 0, applies: As a result, the likelihood of the data is stated as:  The prior distribution for other parameters follows Alhamzawi [12], is: where " = max ( % , " -).
The complete conditional posterior distribution of each parameter is then searched using the Box-Tiao method. The results obtained are: a. The conditional posterior distribution of " * b. The conditional posterior distribution ofis normal multivariat distribution with ). c. The conditional posterior distribution of . Based on the complete conditional posterior distribution obtained, then the Gibbs Sampling process is carried out to estimate the parameter.

MCMC Gibbs Sampling
MCMC comes from a combination of two things, namely: Markov Chain and Monte-Carlo. Ravenswaiij, et al. [15] stated that Monte-Carlo is a method of estimating the parameters of the distribution by taking a random sample from the distribution. Markov Chain in MCMC is the idea that random samples are generated from sequential processes with certain rules. Each random sample is used as a stepping stone to generate the next random sample (forming a chain). The rule of Markov Chain is that each new sample only depends on one previous sample. Walsh [16] states that Gibbs Sampling aims to build a Markov Chain that converges to the targeted distribution. The key of Gibbs Sampling is to use a univariate complete conditional distribution, where the random variables have a fixed value except for one variable.
The steps of the Gibbs Sampling process carried out in this study are as follows: a. Determine the τ -th quantile to be estimated. This study analyzed the regression model on three quantiles, τ = 0.25, τ = 0.5, and τ = 0.75. Each quantile will be modeled using Tobit Quantile Regression and Tobit Quantile Regression with Adaptive Lasso Penalty. Tobit Quantile Regression and Tobit Quantile Regression with Adaptive Lasso Penalty were analyzed using the Brq package in R software.

Estimation of Regression Coefficient Parameter
Tobit Quantile Regression (BTQR) and Tobit Quantile Regression with Adaptive Lasso Penalty (BALTQR) will be analyzed using Bayesian approach. After the complete conditional prior and posterior distributions are determined, then the solution is carried out using the MCMC Gibbs Sampling method. The following is a summary of the regression coefficient parameter values in both models.

. Comparison of Tobit Quantile Regression and Tobit Quantile Regression with Adaptive Lasso Penalty
Comparison of the goodness of the model in this study will be carried out by finding the Mean Absolute Deviation (MAD) in each quantile, which is formulated: The smaller the MAD value, the more accurate the model will be. MAD was calculated using R software. Summary of MAD for each quantile in both models is shown in Table 2 below. Obtained at the 0.25th and 0.75th quantiles, the Tobit Quantile Regression with Adaptive Lasso Penalty model has a smaller MAD than the Tobit Quantile Regression without using a penalty. As for the 0.50th quantile, both MAD values are the same.

CONCLUSION
Based on the value of Mean Absolute Deviation, in the case of Household Expenditure for Cigarette Consumption, Tobit Quantile Regression with Adaptive Lasso Penalty produces better parameter estimates than Tobit Quantile Regression at quantiles 0.25 and 0.75. Further research can develop the model by replacing the Adaptive Lasso penalty with another penalty such as Ridge or Elastic Net and using a larger data set.