سوالات احتمالاتی و پاسخ های نادرست: نرخ جایگزینی درآمد بازنشستگی

We study responses to subjective retirement income replacement rate expectations questions in a survey of Dutch employees. One out of three respondents is unable to provide probabilities satisfying the requirements of a cumulative distribution function. We show that using probabilistic survey questions yields an endogenous sample selection when these individuals are removed from the sample, biasing the results toward more pessimistic expectations and excess uncertainty in the replacement rate. These biases are most prevalent for less-educated individuals.

The standard life-cycle consumption model with uncertain (pension) income predicts that consumption during one's working life is positively related to what individuals expect to receive as income, and negatively related to income uncertainty (Caballero, 1990). In most empirical work in economics (see e.g., Feldstein, 1974, Attanasio and Rohwedder, 2003 and Kapteyn et al., 2005), individual-specific expectations and uncertainty are not available, leading authors to assume static, rational expectations. To avoid making such strict assumptions regarding the expectation formation process, several studies suggest using individuals’ subjective expectations of future income (Dominitz and Manski, 2006 and Guiso et al., 2009). Of particular interest, and the ones used in this study, are probabilistic questions of the type suggested by Dominitz and Manski (1997) and Manski (2004) which allow the researcher to elicit the subjective cumulative distribution function of an individual's pension income. Manski (2004) provides an overview of the use of probabilistic questions, which have become increasingly popular in recent years and have been used to assess the likelihood of general events (such as inflation and social security benefits) as well as person-specific events (such as mortality and one's economic situation). Manski suggests two reasons why eliciting expectations in a probabilistic way is better than eliciting expectations from vaguely defined answer categories (e.g., an event is “very likely” or “not too likely”). First, the numerical scale allows comparisons among individuals. Second, the consistency of a respondent's answers can be checked using the laws of probability. This article contributes to the empirical literature by investigating whether incorrect (or inconsistent) answers to probabilistic survey questions, and their subsequent removal from the sample, lead to endogenous sample selection. For example, less-educated and less financially literate persons may be less likely to answer such questions in a meaningful way—that is, their answers may not satisfy certain laws of probability—and may also have a different retirement income replacement rate. Simply excluding these observations when analyzing the determinants of the subjective replacement rate or subjective uncertainty, as is commonly done in other papers (e.g., Dominitz and Manski, 2006), can therefore result in endogenous sample selection and bias the parameter estimates. The quality of subjective expectations, elicited using probabilistic survey questions, is examined in other papers.1Hurd and McGarry, 1995 and Hurd and McGarry, 2002 investigate the validity of subjective survival probabilities and find that individuals are well able to predict their own mortality, underlining Manski's (2004) conclusion that probabilistic survey questions are informative. Dominitz (1998) shows that next-year income expectations are able to predict subsequent realizations reasonably well. In addition, data on expectations and subsequent realizations are used in Dominitz (2001), Das and van Soest (1997), Das and Donkers (1999), and Stephens (2004). The overviews of Hurd (2009) and Pesaran and Weale (2006) also emphasize the predictive power of subjective probabilities. However, Dominitz and Manski, 1996 and Dominitz and Manski, 1997 signal some evidence that not all respondents answer correctly. Dominitz and Manski (1996) use survey software that automatically signals mistakes in the probabilities entered by the respondent, after which the respondent must correct the mistake, but still allows the researcher to keep track of them. The authors find that 7 percent of respondents violate the monotonicity of answers, and 40 percent provide answers incompatible with the (previously elicited) median of the subjective distribution of future income. Dominitz and Manski (1997) report 21 percent item non-response, 8 percent providing constant probabilities over the thresholds, and 5 percent violating monotonicity. Kleinjans and van Soest (2010) show that two common fears associated with probabilistic questions, namely, non-response and focal points (e.g., answering 0 percent, 50 percent, or 100 percent), do not affect the determinants of retirement expectations, but that individuals round off probabilities instead. Manski and Molinari (2010) investigate the extent of rounding in more detail and find heterogeneity in answering patterns, with a small fraction (11 percent of the respondents) always rounding up to multiples of 50. We do not address the issues of rounding or focal points. More closely related to our study are Dominitz and Manski (2006), who compare the sample statistics of non-respondents to those in their final sample, using data from the Survey of Economic Expectations (SEE). Non-respondents, defined as providing either missing values, incomplete, non-valid, or unusable answers for estimating the subjective distribution, in the SEE are more likely to be female, less likely to be non-Hispanic whites, less likely to be labor force participants, less likely to be married, and less likely to be high school or college graduates. Furthermore, the probability of non-response is non-monotonic in age. The degree of non-response, Dominitz and Manski (2006) claim, compares favorably to that from the Health and Retirement Study, and hence the authors conclude that selection effects are not an issue. By contrast, our empirical findings provide strong evidence of endogenous sample selection effects when omitting incorrect answers from an analysis of expected retirement income replacement rates. We use responses to the Dutch Pension Barometer survey, described in detail in Section 2, which follows the Dominitz and Manski, 1997 and Dominitz and Manski, 2006 approach. By eliciting points on the (subjective) distribution function of future pension income, these questions allow the researcher to compute estimates of the expected replacement rate, that is, the ratio of expected pension income to current income, as well as its standard deviation, which can be interpreted as uncertainty regarding the replacement rate. We find that about one-third of the respondents are unable to answer correctly, and that the incidence of violations correlates with observable background variables, such as education, income, and gender. De Bresser and van Soest (2011) find similar results using the same data source, but use only the sample with correct responses, thus implicitly assuming exogenous sample selection. A new finding is that excluding those individuals for whom it is not possible to compute the expectation or standard deviation of future pension income results in endogenous sample selection. The resulting biases are quantified by predicting both the expected replacement rate and the standard deviation of the replacement rate using a linear model without correcting for selection effects and using a Heckman model that does correct for possible endogenous selection effects. This quantification shows that ignoring endogenous sample selection yields a downward bias in the predicted expected replacement rate and an upward bias in the predicted uncertainty (standard deviation) of the replacement rate. These biases are largest for less-educated individuals. The paper is organized as follows. Section 2 discusses the data and the exact wording of the survey questions. Section 3 examines the incidence of violations of the laws of probability and how it relates to individual characteristics. Section 4 discusses the computation of the expected value and standard deviation of the replacement rate and relates these to individual characteristics. It also quantifies the consequences for the parameter estimates of ignoring endogenous sample selection. Finally, Section 5 presents our conclusions.

This paper has been the first to quantify the selection bias due to incorrectly answering probabilistic survey questions concerning pension entitlements in a sample of Dutch employees. Given the mentioned selected samples reported in Dominitz and Manski, 1997 and Dominitz and Manski, 2006 and Delavande and Rohwedder (2008), a similar bias is likely to be found in other datasets, obtained from different respondents using different methods of interview and on different topics. Two related questions8 arise: do respondents have a correct estimation of the replacement rate, or should we keep assuming rational expectations in empirical work? More interesting, once the selection bias is controlled for, is the gap between subjective and objective replacement rates bigger or smaller? Knowing an individual's true replacement rate is notoriously difficult, and hence we resort to aggregate statistics, taken from administrative data collected by the Central Bureau of Statistics of The Netherlands. Table 8 reports replacement rates by gender and immigrant status, where the latter serves as a proxy for education. The first observation from this table is the fact that women have higher replacement rates than men. In Table 6, we have seen a strong negative effect of being female on the expected replacement rate, if we do not control for endogenous sample selection. Controlling for selection, the significance of this negative effect disappears, suggesting that the selection-correction pushes the replacement rates toward the objective replacement rate. The second observation we make is that non-western immigrants (as a proxy for the less educated) have a much higher replacement rate, on average 14 percent higher, compared to both non-immigrants and western immigrants. A similar difference is obtained in Table 7, where the difference between predicted replacement rates for the sample average and the elementary educated equals 8 percent for the earliest retirement age, and 10 percent for the latest retirement age. Using a linear model, the differences are 6 percent and 8 percent, respectively. This supports the hypothesis that the selection correction yields more plausible results compared to using a linear model without selection correction. Moreover, this comparison shows that the elicited expectations are, on average, plausible, and should provide better results in empirical analysis than using strong rationality assumptions.Future research should be devoted to prevent a reduction in sample size and a selected sample using alternative ways of eliciting expectations. A first step has been taken by Delavande and Rohwedder (2008), using a visual format for eliciting the distribution of social security benefits. Delavande and Rohwedder obtain prima facie evidence of an increase in sample size, as well as a smaller standard deviation of the distribution of social security benefits. However, the fact that we use replacement rates, not the level of pension benefits, makes it difficult to compare their findings to ours. The ultimate test would be to use both the probabilistic format and an alternative way of eliciting expectations (such as the visual format of Delavande and Rohwedder (2008)). For the probabilistic format, respondents should be forced to provide consistent answers, but the information on inconsistent responses should be stored. Only then can we make a comparison between the various formats, as well as give a judgment on the informative value of the subjective expectations. As an application, one can use the expected pension income and pension risk based on the (computed) replacement rates and, for instance, estimate a life-cycle model of consumption without the need to arbitrarily assume how expectations are formed. An important implication of our findings for such research is that one must account for the endogenous selection effects due to incorrect responses. An interesting extension of our study is to examine how persistent expectations are. That is, we have shown some dispersion in the expected replacement rate in a (pooled) cross-section, but to obtain further insights in expectation formation, it would be of interest to analyze at the individual level how expectations are updated when new information becomes available. Hence, the persistency of these expectations can tell us something about how expectations are formed. Such an extension may require a longer panel than we currently have and is left for future research.