ESTIMATING SAVING FUNCTIONS WITH A ZERO-INFLATED BIVARIATE TOBIT MODEL * Alessandra Guariglia University of Kent at Canterbury.

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1 ESTIMATING SAVING FUNCTIONS WITH A ZERO-INFLATED BIVARIATE TOBIT MODEL * Alessandra Guariglia University of Kent at Canterbury and Atsushi Yoshida Osaka Prefecture University Abstract A zero-inflated bivariate Tobit model is developed to jointly estimate the determinants of saving in the form of bank deposits and voluntary contributions to Personal Pension Plans (PPP) in Britain. This model allows us to deal with the under-reporting of saving. We show that the estimates obtained with a standard bivariate Tobit model are seriously biased compared with those obtained using the zero-inflated model. Key words: Saving functions; Bivariate Tobit model; Misreporting; Zero-inflated model; selection. JEL classification: C3; C34; D. (Number of words: 3,763) * Corresponding author: Alessandra Guariglia, Department of Economics, Keynes College, University of Kent at Canterbury, Canterbury, Kent CT 7NP, United Kingdom. Tel: Fax: a.guariglia@ukc.ac.uk.

2 . INTRODUCTION Misreporting is one of the major problems that researchers encounter when using micro-data sets. Some respondents may deliberately not answer survey questions correctly, in particular when these questions are related to their income, saving and assets, or more in general to their economic status. In other cases, since the questionnaires are not always well designed, respondents may misunderstand the questions. Deaton and Irish (984) and Maki and Nishiyama (996) found problems of this type when examining the demand for durable goods; Erard (997) when dealing with noncompliance of audit-based tax reports; and Klein and Sherman (997) when analyzing the demand for new products. In many cases, the data contaminated by misreporting have higher frequency at zero than standard censored data. When applying the standard Tobit model to these data sets, the estimates of the parameters are likely to be biased, and in some cases, the numerical convergence may fail. Erard (997) and Maki and Nishiyama (996) proposed models to deal with the excessive-zeros problem. They introduced a censoring random variable that takes value if the answer to the relevant question is correct and 0 otherwise, and modelled it as a Probit. Their models are essentially equivalent to the double-hurdle model proposed by Cragg (97). In the context of count data models, Johnson, Kotz and Kemp (99, pp.86-87); Lambert (988); and Mullahy (986) suggested the zeroinflated model to deal with the excessive-zeros problem. Though the statistical aspects of the zero-inflated and double-hurdle models are similar, the likelihood function of the former model is much simpler. In particular, the multivariate double-hurdle model has a complex likelihood function at zero, so that the time of computation for the maximization is significantly longer compared to that required by the zero-inflated model.

3 3 In this paper, we apply the idea of the zero-inflated model to the bivariate Tobit case. Our model can thus be regarded as a finite mixture between a standard bivariate Tobit model and a degenerate distribution with mass point at zero. Using the British Household Panel Survey (BHPS thereafter), we estimate two saving functions: saving in the form of bank, postal and building society deposits (abbreviated as bank deposit thereafter), and saving in the form of voluntary contributions to Personal Pension Plans (PPP thereafter). The former saving is liquid, while the latter, which is tax-exempt, can generally be accessed only at retirement. The two types of saving have very different purposes (see Guariglia and Markose, 000). Among other reasons, people save in the form of bank deposits for a precautionary motive, i.e. to insure against income or health risk. On the other hand, people contribute to PPPs to ensure a comfortable life after retirement. We are concerned with the extent to which the sensitivities of these two types of saving with respect to respondents permanent (lifetime) income differ. We are also interested in calculating the threshold levels of permanent income, at which respondents start to save in the two above-mentioned forms. The rest of the paper is laid out as follows. In Section, we describe our data set; in Section 3, we illustrate our econometric model; and in Section 4, we present some descriptive statistics and our empirical results. Section 5 concludes.. THE DATA We use wave seven of the BHPS, which covers the year 997. In the BHPS, a representative sample of 0,000 individuals living in Great Britain was interviewed in 99. These individuals, together with their co-residents were interviewed again each year thereafter. The survey provides information on respondents' demographic,

4 4 occupational, educational, income and saving characteristics. We limit our analysis to all employees, excluding the self-employed, aged between 5 and 64. Our sample is thus made of,954 observations. In each wave, except wave, individuals are asked the following question on their participation to Personal Pension Plans: In the past year, [...], have you paid any contributions or premiums for a private personal pension, or had such contributions paid on your behalf by the Department of Social Security (DSS)? Those individuals who answer YES are then asked for the amount of their last contribution and the period it covered. Let us denote with S an individual's average weekly voluntary contribution to a PPP 3. Note that this form of saving is tax-exempt, and very illiquid since it cannot be used before retirement. In each wave, individuals are also asked the following questions on their saving behaviour in more traditional vehicles: Do you save any amount of your income, for example by putting something away now and then in a bank, building society, or Post Office account other than to meet regular bills? Please include share purchase schemes and Personal Equity Plan schemes. If the answer to this question is YES, then respondents (the savers) are asked the following: About how much on average do you personally manage to save a month? For more details on the BHPS, see Taylor (994) and Taylor (996). As we will show below, we model saving as a function of respondents permanent (lifetime) income, which we proxy with the fitted values obtained from a regression of their earnings on six lags of these same earnings and age dummies. This procedure constrains us to use the panel dimension of the BHPS only to construct our proxy for permanent income, but not in estimation. 3 Note that individuals with PPPs are divided according to whether their pension plans were taken before or after July 988. The former plans are the so-called Retirement Annuities, and all the contributions made to these plans are voluntary. The post-988 pension plans are such that the DSS itself makes a minimum contribution, which individuals can top-up with additional voluntary

5 5 We denote with S the average weekly amount saved by individuals in this liquid form. The variable is calculated as the answer to the latter question divided by four to obtain a weekly figure. All the relevant income and saving variables are expressed in 995 pounds THE ECONOMETRIC FRAMEWORK Let us define two latent variables, S * and S *, which are respectively the optimal saving as voluntary PPP contributions and the optimal saving as bank deposits. These two types of saving are determined by maximisation of the individual's lifetime utility function subject to a period-to-period budget constraint. In theory, negative values of either or both types of saving in some periods cannot be ruled out. However, the observed saving in the form of voluntary PPP contributions and bank saving (S and S ) are constrained to be nonnegative. Let us briefly consider the three cases in which S and/or S can be expected to take a value of 0. First, when an individual faces severe liquidity constraints, both S and S are likely to be 0. Second, when income risk is large, and income itself is relatively low, one can expect S to be positive and S to be 0. Third, in the presence of a moderately low income, accompanied by low income risk, S can be expected to take a positive value, while S is likely to be 0. These three cases can be regarded as the corner solutions of the above mentioned dynamic utility maximization problem. Alternatively, in the presence of a sufficiently large income, both S and S should be positive. contributions. Our measure of S only takes into account the voluntary contributions to PPPs, and not those contributions made by the DSS. 4 The variables are deflated using the Retail Price Index.

6 6 Both S and S are likely to suffer from an "excessive-zeros problem". Two types of respondents are likely to fall in the zero-saving region. The first type understood the question correctly and answered it truly. The second type, includes respondents (i) who regarded themselves excluded from the category of people who saved 'now and then'; (ii) who felt it troublesome to fill in the questions on the amounts saved, so that they incorrectly answered 'NO' to either questions on saving, just to avoid the subsequent questions on the amounts saved; (iii) who answered 'NO' because of ambiguous memory on whether they saved or not. In order to distinguish between these two types of "zero-saving respondents", we introduce a random variable, Z, which takes value for those who belong to the first type and to 0 otherwise. Our sample selection problem can then be represented as follows: Case : S = 0, S = 0 if {S * 0, S * 0 and Z= } or Z = 0 Case : S = 0, S = S * if S * 0, S * > 0 and Z= Case 3: S = S *, S = 0 if S * > 0, S * 0 and Z= Case 4: S = S *, S = S * if S * > 0, S * > 0 and Z= We assume that S* and S* are determined by the following equations: S * = α + Yβ + Y δ + Xγ + U () S * = α + Yβ + Y δ + Xγ + U, () where Y is a proxy for the respondent's permanent income and X is a row vector of his/her socio-economic characteristics 5. We allow the constant terms and the 5 Permanent income is proxied with the fitted values obtained from a regression of the respondent's usual real net monthly earnings on six lags of this same variable and age dummies (see Dynan et al.,

7 7 coefficients on permanent income and permanent income squared to differ for the two types of saving. This allows us to assess the extent to which both the direct effects of changes in permanent income on saving, and the threshold levels of permanent income at which people start saving differ in the two cases. On the other hand, in order to keep the number of parameters in the model within a reasonable range, the socio-economic parameters are assumed to be the same for the two types of saving. It is also assumed that U=(U,U ) is normally distributed with mean 0, variances σ and σ, and a correlation coefficient ρ. It follows that (S *,S *) is also normally distributed with mean (µ, µ )=( α + Yβ + Y δ + Xγ, α + Yβ + Y δ + Xγ), and the same variance-covariance matrix as U. Z is modelled as a Logit such that: Pr[Z=0] = exp(wη) / [+ exp(wη)], (3) where W is a row vector of socio-economic characteristics of respondents, which is related to their tendency to answer 'NO' incorrectly to either saving questions. Let us denote with f(s *, s *) and g(s, s ) the joint density functions of S * and S *, and of S and S, respectively. For case, the density function g(s, s ) can be obtained as follows: g(0,0) = Pr[Z = 0] + Pr[ S * 0, S * 0] Pr[Z =], (4) where Pr[ S * 0, S * 0] = 0 0 f(s *, s *) ds *ds *. The first term of the right-hand side of Equation (4) is the probability of the respondents answering 'NO' to either the saving questions because of the motives (i), 999 for a similar approach). We model saving as a function of permanent income rather than current income because the relationship between saving and the latter would reflect not only the true relationship between lifetime income and saving, but also the effects of transitory income and measurement error. Moreover, current income could be affected by endogeneity problems.

8 8 (ii) or (iii) explained above. The second term is the probability of answering 'NO' correctly to either saving questions. For cases and 3, we have respectively: g(0,s ) = [ 0 f(s *, s )ds *] Pr[Z = ] and (5) g(s,0) = [ 0 f(s, s *)ds *] Pr[Z = ] (6) Since S * and S * are normally distributed, the integrals of the right-hand sides of the equations above can be obtained as follows: 0 f(s *, s )ds * = + Φ σ µ ρ σ µ ρ σ µ φ σ S S (7) 0 f(s, s *)ds * = + Φ σ µ ρ σ µ ρ σ µ φ σ S S (8) where ρ = σ /(σ σ ) and φ(.) and Φ(.) are respectively the pdf and the cdf of the standard normal distribution. Finally, for case 4 we have: g(s,s ) = f(s,s ) Pr[Z =]. (9) The log likelihood function for our problem can thus be expressed as follows: L = {i: case } ln g(0,0) + {i: case } ln g(0,s i ) + {i: case 3} ln g(s i,0) + + {i: case 4} ln g(s i,s i ) (0) 4. DESCRIPTIVE STATISTICS AND EMPIRICAL RESULTS Table shows the distribution of permanent income for the entire sample, as well as for the cases in which respondents have no saving, bank deposits only, voluntary PPP contributions only, and both types of saving. The medians of weekly

9 9 permanent income in these cases are respectively 0, 00, 30, 3, and 34. In the 'no saving' case, the sample is about /3 of the total sample and the median of income is slightly smaller than in the other cases. It is interesting to note that 43 out of 95 respondents have a weekly permanent income higher than 388 (which represents the 90 th percentile of the sample) but state that they do not save in either form. It is likely that at least some of these individuals are giving an incorrect answer to the questions relative to whether they save or not 6. Table shows the distributions of the various types of saving by weekly permanent income bands. Within each income band, the amounts saved are generally smaller for those people who save only in the form of voluntary PPP contributions. Those respondents in the lower income bands tend to save mainly in the form of bank deposits, and those in the higher bands, in the form of both bank deposits and voluntary PPP contributions. The average amounts saved in bank deposits and/or voluntary PPP contributions tend to gradually increase with permanent income, with a generally more pronounced increase for the former. Table 3 shows the estimated parameters of Equations () and () obtained both with the zero-inflated model and with a standard bivariate Tobit model. Using the estimated coefficients on the constant terms, permanent income and permanent income squared, we calculated the threshold levels of weekly permanent income, at which female respondents aged between 5 and 9, with no educational qualifications start saving in the form of bank deposits or voluntary PPP contributions 7. In the zero-inflated model, these thresholds were respectively and 56.7, while they were 67.6 and in the standard bivariate Tobit model. 6 These individuals are likely to have answered NO to either question on saving to avoid subsequent questions on the amounts saved (see motive (ii) above).

10 0 The latter results are clearly upward biased. For 'males' holding a 'university degree or higher', who are expected to save more than the other categories, the threshold levels in the zero-inflated model were 3.8 and 433.3, while they were 69.0 and in the bivariate Tobit model. The upward bias characterizing the latter two numbers is even stronger than in the base case. The estimated parameters on permanent income and permanent income squared show that as their permanent income becomes larger, respondents tend to increase their bank deposits more than their PPP contributions. This is consistent with the descriptive results presented in Table. Compared to the 5-9 age band, saving appear to be higher in the band, and lower for those respondents aged 45 or over. The number of dependent children in the household has a negative effect on saving. Education is also a crucial determinant of saving. Respondents with a university degree or post graduate qualifications save slightly less than twice as much as those with O levels or vocational qualifications. Comparing the estimated coefficients on the educational variables obtained using the zero-inflated model with those obtained using the standard bivariate Tobit model, we can see that the variable 'university degree or higher' appears with opposite signs in the two cases. Moreover, the coefficients on the dummies for 'A levels' and 'O levels' are poorly determined in the latter case, while they are positive and statistically significant in the former. In particular, according to the standard bivariate Tobit model, having a college degree or a higher qualification has a negative effect on saving, while having A/O levels does not play any effect on saving. These results are hard to rationalize and can be seen as 7 Females, respondents aged between 5 and 9, and respondents with no educational qualifications represent the omitted categories in Table 3 (base case).

11 an indication of the relatively worse performance of the standard bivariate Tobit model compared to the zero-inflated model. In order to compare the general performance of the two models considered, using a more rigorous criterion, we calculated the Akaike's (973) Information Criterion (AIC) for both cases. The values were 6,638 for the zero-inflated model, and 6,779 for the standard bivariate Tobit model, showing the superiority of the former model. In the zero-inflation part of Table 3, coefficients that are negative and large in absolute value can be associated with a higher probability of the respondents answering correctly the question on whether they save in either form. We can see that the probability of answering correctly differs across educational levels 8. In particular, respondents with a higher educational level are more likely to answer incorrectly. This can be explained as follows. Individuals with higher education are generally those who have a higher income. As discussed while analyzing Table, a non negligible proportion of respondents with high income state that they do not save at all. Their answers, and consequently the answers of the highly educated respondents as well, are likely to be incorrect and thus to cause biases in the estimates obtained using the standard bivariate Tobit model. 5. CONCLUSIONS In this paper, we have developed a zero-inflated bivariate Tobit model to jointly estimate saving functions, while dealing with the under-reporting of saving. We have 8 We have assumed that the educational dummies are the main determinants of whether those respondents who state that they do not save are likely to tell the truth or lie.

12 compared the estimates obtained using the zero-inflated model with those obtained using a standard bivariate Tobit model, and shown that when the saving are underreported, the latter estimates are seriously biased. Moreover, we have shown that the zero-inflated bivariate Tobit model is superior to the standard bivariate Tobit model also according to the Akaike's (973) Information Criterion. The zero-inflated bivariate Tobit model is generally easier to handle than the double-hurdle model, and can be easily extended to the multivariate case. REFERENCES Akaike, H., 973. Information theory and an extension of the maximum likelihood principle. In B. Petrov and F. Csake, eds., Second International Symposium on Information Theory. Budapest: Akademiai Kiado. Cragg, J., 97. Some statistical models for limited dependent variables with application to the demand for durable goods. Econometrica, 39, Deaton, A. and M. Irish, 984. Statistical models for zero expenditures in household budgets. Journal of Public Economics, 3, Dynan, K, J. Skinner, and S. Zeldes, 999. Do the rich save more? Mimeo, Graduate School of Business, Columbia University. Erard, B., 997. Self-selection with measurement errors: A microeconometric analysis of the decision to seek tax assistance and its implications for tax compliance. Journal of Econometrics, 8, Guariglia, A. and S. Markose, 000. Voluntary contributions to Personal Pension Plans: evidence from the BHPS. Fiscal Studies,.

13 3 Johnson, N. L., Kotz, S. and A. W. Kemp, 99. Univariate Discrete Distributions. nd edition, John Wiley and Sons, New York. Klein, R. and R. Sherman, 997. Estimating new product demand from biased survey data. Journal of Econometrics, 76, Lambert, D., 988. Zero-Inflated Poisson regression with an application to defects in manufacturing. Technometrics, 34, -4. Maki, A. and S, Nishiyama., 996. An analysis of under-reporting for micro-data sets: The misreporting or double-hurdle model. Economics Letters, 5, -0. Mullahey, J., 986. Specification and testing of some modified count data models. Journal of Econometrics, 33, Taylor, A., 994. Appendix: characteristics, attrition and weighting. In (eds. N. Buck, J. Gershuny, D. Rose, and J. Scott) Changing Households: The British Household Panel Survey Colchester: ESRC Research Center on Micro-Social Change, University of Essex. Taylor, M. (ed.), 996. British Household Panel Survey user manual. Introduction, Technical Reports, and Appendices. Colchester: ESRC Research Center on Micro-Social Change, University of Essex.

14 4 Table : Distribution of Permanent Income by Respondents Saving Status TOTAL NO SAVING BANK DEPOSITS ONLY VOLUNTARY PPP CONTRIBUTIONS ONLY BOTH TYPES OF SAVING Permanent income percentile Perm. income Perm. income Perm. income Perm. income Perm. income.5% % % % % % % % Note: The unit of the permanent income cells is pounds per week. Source: BHPS, Wave 7. Table : Saving by Permanent Income Classes BANK DEPOSIT ONLY VOLUNTARY PPP CONTRIBUTIONS ONLY BOTH TYPES OF SAVING Permanent Income Share (%) Mean of B. D. Share (%) Mean of PPP contr. Share (%) Mean of B.D. Mean of PPP contr. Under Over Note: B.D. is an abbreviation for bank deposits. The unit of all permanent income, bank deposits, and PPP contributions cells is pounds per week. Source: BHPS, Wave 7.

15 5 Table 3: Estimation Results Zero-inflated bivariate Tobit model Standard bivariate Tobit model α (PPP contributions) α (Bank deposits) ß (PPP contributions) ß (Bank deposits) δ (PPP contributions) δ (Bank deposits) AGE Demographic variables Male Number of adults in household Number of dependent children in household Married/Cohabiting Education University degree or higher A levels / professional qualifications O levels / vocational qualifications Zero-inflation part University degree or higher A levels / professional qualifications O levels / vocational qualifications No qualifications ln(σ ) ln(σ ) tan(ρ) Others Akaike's Information Criterion 6,638 6,779 Notes: α and α represent the constant terms for the two types of saving; β and β are the coefficients on permanent income; and δ and δ, the coefficients on permanent income squared. The figures reported in italics are the ratios of the coefficient estimates to the estimated asymptotic standard errors. The categories: female, aged 5 to 9, with no educational qualifications are omitted. Source: BHPS, Wave 7.