Journal of Probability and Statistics

Volume 2019, Article ID 2723069, 8 pages

https://doi.org/10.1155/2019/2723069

## Retirement Consumption Puzzle in Malaysia: Evidence from Bayesian Quantile Regression Model

^{1}School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, 06010 UUM Sintok, Kedah DA, Malaysia^{2}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia

Correspondence should be addressed to Noriszura Ismail; ym.ude.mku@in

Received 12 August 2018; Accepted 13 December 2018; Published 1 January 2019

Guest Editor: Rahim Alhamzawi

Copyright © 2019 Ros Idayuwati Alaudin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The objective of this study is to use the Bayesian quantile regression for studying the retirement consumption puzzle, which is defined as the drop in consumption upon retirement, using the cross-sectional data of the Malaysian Household Expenditure Survey (HES) 2009/2010. Three different measures of consumption, namely, total expenditure, work-related expenditure, and nonwork-related expenditure, are suggested for studying the retirement consumption puzzle. The results show that the drop in consumption upon retirement is significant and has a regressive distributional effect as indicated by larger drops at lower percentiles and smaller drops at higher percentiles. The smaller drops among higher consumption retirees (or higher income retirees) may imply that they have more savings and/or retirement benefits than the smaller consumption retirees (or lower income retirees). Comparison between the three types of consumption shows that the work-related expenditure has a uniform drop across the distribution. The drop under the nonwork-related expenditure varies across the distribution, implying that it is the source behind the variation of the consumption drop.

#### 1. Introduction

A life cycle theory is a major economic theory that relates consumption and saving behavior, which states that individuals desire to maintain their level of consumption in their entire lifetime [1]. The marginal utility consumption should remain smooth throughout retirement transition because the change in income during retirement should be predictable [2]. On the contrary, a number of previous studies found a one-time significant drop in consumption in the early years of retirement, a situation which is known as retirement consumption puzzle.

Over the past three decades, several studies focusing on the smoothing or stable path of consumption have been carried out. Hamermesh [3], who was among the first to study retirement consumption puzzle in the United States, showed that an individual was unable to sustain the level of real consumption prior to retirement due to inadequate retirement savings. Later, several more studies have been conducted in other countries such as Banks et al. [4], Hurd and Rohwedder [5], Schwerdt [6], Wakabayashi [7], and Battistin et al. [8]. A review on several studies related to the retirement consumption puzzle can be found in Attanasio and Weber [9]. There are a number of studies which found that the decline in consumption upon retirement is due to several reasons. As examples, Haider and Stephens [10], Smith [11], and Blau [12] found that the consumption drop is due to the unexpected retirement which resulted from illnesses, disabilities, or involuntarily unemployment. Blau [12] also developed a modified life cycle model which incorporated the uncertainty in the timing of retirement.

In terms of consumption measures, several studies utilized food expenditure as a prominent substitution for the actual consumption during retirement. However, Aguiar and Hurst [13] found that food expenditure is a poor proxy for the actual consumption because retirees consumed the same quantity of food, as well as its quality, even when food expenditure has declined. The main reason is that the retirees consumed home production food in their retirement since they have more time to prepare their meal and survey for cheaper food. Later, Hurst et al. [14] and Fisher et al. [15] proposed a broader measure of consumption which can be categorized according to several different types of expenditure and showed that a broader consumption measure can be used to eliminate retirement consumption puzzle.

Previous studies on the drop in retirement consumption were mostly concentrated at the mean distribution, which covered only certain parts of distribution, and may lead to poor estimation of parameters especially in long-tailed distributions. The traditional mean model estimates only the average effects of the whole data and do not allow for an understanding of any potential distributional impacts (or heterogeneity potential). Furthermore, the mean regression model is based on least squares estimation and thus has a significant sensitivity (or is not robust) to outliers. Several studies have been carried out to study the distributional aspects of retirement consumption puzzle, including Bernheim et al. [16] who estimated the retirement consumption drop using subgroups of wealth and income replacement rate and Aguila et al. [17] who used low and high consumption households to examine the drop in retirement expenditures. Recently, Fisher and Marchand [18] used the quantile regression model to investigate the drop in consumption upon retirement.

Quantile regression model is different from the traditional mean regression model as it uses the Least Absolute Deviation (LAD), instead of the least square error, which is able to rectify the weaknesses prevalent in the usual regression framework. Quantile regression model also allows the impact of each regression parameter to be analysed based on different selected quantiles. In short, quantile regression model has several advantages; it is a distribution-free model which does not adhere to any distributional assumptions, it is robust to outliers, it does not require independence assumption, and it allows the analysis of regression parameters to be extended beyond central locations.

Quantile regression model is based on the works of Koenker and Bassett [19] and Koenker and Hallock [20] and is gaining rapid interests by other researchers. The model has been developed for linear models with continuous responses and has been applied in various fields such as finance and economics [21], ecology [22], environmental epidemiology [23], criminology [24], and climate change [25]. Several extensions to the quantile regression model have been suggested and applied in other areas, such as Yu and Moyeed [26] who proposed Bayesian quantile regression model, Machado and Silva [27] who proposed quantile regression model for discrete data, Hewson and Yu [28] who suggested quantile regression model for binary data within the Bayesian framework, Reich et al. [25] who introduced Bayesian spatial quantile regression model, and Fuzi et al. [29] who applied Bayesian quantile regression model for claim count data in insurance area.

In this paper, we use the Bayesian quantile regression model to examine the drop in consumption upon retirement. The Bayesian quantile regression has the combined advantages of both the quantile regression and the Bayesian approach. The quantile regression is a distribution-free model and robust to data, while the Bayesian approach allows the complete univariate and joint posterior distribution of each parameter to be generated by the MCMC simulations. Several motivational examples are worth mentioning here. For instance, in the field of environmental study, the quantile regression model allows the investigation of whether the effects of environmental exposure change depending on the level of respiratory health of the population [23], while in the area of insurance pricing and ratemaking, the Bayesian quantile regression models handle the parameters of covariates (or the risk factors) as random variables [29]. We use a cross-sectional data of Household Expenditure Survey (HES) 2009/2010 in Malaysia to investigate the changes in consumption drop across the population. We also expand the consumption measure into three categories, namely, total expenditure (TE), work-related expenditure (WRE), and nonwork-related expenditure (NWRE), to identify the source behind the variation of consumption drop.

#### 2. Materials and Methods

##### 2.1. Mean Regression

In our study, the drop in mean consumption upon retirement is estimated using the ordinary least square (OLS):where is the log of response variable (consumption measures), is the binary variable which equals one for retirees and zero for working households, and is the vector of control variables consisting of demographic and socioeconomic variables (gender, marital status, ethnic, etc.). The regression parameter, , represents the mean consumption difference between working and retired households.

##### 2.2. Frequentist Quantile Regression

The frequentist quantile regression model from Koenker and Bassett [19] is used in our study for comparison purposes. The same equation shown in (1) is used, but the quantile regression is now fitted to the conditional differences in the log of consumption between working households (preretirement) and retirees (postretirement) at the th quantile.

Let be the vector of continuous response variables, which in our case can be represented by the three different consumption measures, and be the associated row-vector of covariates consisting of demographic and socioeconomic characteristics of households. The classical regression model focuses on the expectation of variable , conditional on the values of variables** X**, which can be summarized as . On the other hand, the quantile regression model extends this approach to , where the quantile are fixed values between 0 and 1, allowing us to study the conditional distribution of on** X **at different locations.

The regression parameter, , can be obtained usingwhere the loss function is a piecewise linear function and is the indicator function. Equivalently, the loss function can be written asEquation (3) can be minimized using linear programming, while the confidence interval can be obtained using bootstrap method. R statistical program with* quantreg* package [30] is used in this study to fit the frequentist quantile regression model.

##### 2.3. Bayesian Quantile Regression

The Bayesian approach for quantile regression model was introduced by Yu and Moyeed [26] who formed the likelihood function using the asymmetric Laplace distribution (ALD). A random variable follows the ALD when the density function is given bywhere and is the loss function in equation (3). Under the Bayesian approach, the MCMC can be used to obtain the posterior distributions of the unknown parameters. The posterior distribution of parameter is given bywhere is the prior distribution of and is the likelihood function which is formed by joining the independently distributed ALD. The joint likelihood function can be written asSince there is no specific conjugate prior distribution for generating the posterior distribution, we use the uniform prior distribution for all in our study. The prior distribution for the scale parameter is the inverse-gamma distribution, or ~inverse-gamma, which allows the Gibbs sampling algorithm in the MCMC to update and tune for obtaining good acceptance rates.

#### 3. Results and Discussion

##### 3.1. Sample Data

The sample data from Household Expenditure Survey (HES) 2009/2010 is used in our study. The data contains information on monthly expenditure, together with demographic and socioeconomic characteristics of each household. The selected sample is a cross-sectional data consisting of 6480 household heads.

Three different measures of consumption are used for the response variable, namely, total expenditure (TE), work-related expenditure (WRE), and nonwork-related expenditure (NWRE). TE consists of food at home, alcohol and cigarettes, home appliances and furniture, clothing, education, entertainment and recreation, health, insurance, outside food (restaurant and café), transportation (own vehicle and public transport), personal care, rental, utility, and other services (such as legal services, tax services, and government agency). WRE and NWRE are constructed using the definitions in Aguiar and Hurst [31] and Fisher and Marchand [18]. WRE consists of outside food, personal care, public transport, and clothing, whereas NWRE contains food at home, alcohol, utilities, and entertainment.

##### 3.2. Model Development

The MCMC simulations via Gibbs sampling algorithm are used to generate 5,000 posterior samples of each regression parameter*. *The first 1,000 runs of posterior samples are discarded as burn-ins to lessen the effect of initial simulations, and the process resulted in 4,000 final posterior samples for each regression parameter. The values of are initialized at zero, while the inverse-gamma parameters are set at* a *= 0.01 and* b *= 0.01.

##### 3.3. Summary Statistics

Table 1 provides the summary statistics for the sample. More than half of household heads (69%) are located in urban areas, a large majority (78%) are married, and more than half (64%) belongs to Bumiputera ethnic. In terms of educational attainment, 14% of household heads are university or college graduates, 35% are high school graduates, and the balances are below high school education (16%), and others (35%) (attend informal or religious education). For occupational groups, almost 13% of household heads are professionals, while 19% and 13% are administrative supports and technicians, respectively. In terms of employment type, almost half of household heads are private sector employees, followed by self-employed (19%) and government employees (13%). The proportions of age group are quite equally divided, with the exception of younger household heads (age less than 25). With regard to the status of living quarters, more than half (67%) are homeowners, followed by renters (24%).