Dynamics and heterogeneity of subjective stock market expectation updates

Transcription

1 Dynamics and heterogeneity of subjective stock market expectation updates Florian Heiss University of Dusseldorf Michael Hurd RAND, Santa Monica Maarten van Rooij De Nederlandsche Bank, Amsterdam Tobias Rossmann University of Munich Joachim Winter University of Munich 15th February 2018 Abstract: Between 2004 and 2016, we elicited individuals subjective expectations of stock market returns in a Dutch internet panel at bi-annual intervals. In this paper, we show that respondents update their expectations more strongly in the loss domain than in the gain domain, in particular with respect to the probabilities of extreme changes in the stock market. We develop a panel data model that allows us to study the importance of past stock market returns for individuals updating behavior. Our results reveal substantial heterogeneity in updating behavior. A large share of respondents expectations appear to track recent market movements, while only few responses are consistent with a random walk model of the stock market in that they report stable return expectations that match historic returns. We also study whether socio-economic characteristics predict respondents expectation types and switching between types over time.

2 1 Introduction Expectations are crucial in all decisions where outcomes only materialize in the future and are subject to uncertainty. These comprise, amongst others, decisions regarding education, health, insurance, and household finance. One might argue such intertemporal decisions are among the most important ones individuals make. Recent literature has therefore started to elicit expectations from individuals in large-scale surveys. 1 For example, data on subjective stock market expectations contributed to the understanding of the stock market participation puzzle (see, amongst others, Dominitz and Manski, 2007; Hurd, 2009; Hudomiet et al., 2011; Hurd et al., 2011). These papers also find evidence for substantial heterogeneity in the population. 2 However, while the cross-sectional distribution of subjective stock market expectations is well understood, less is known on how individuals form and update their expectations. This paper reports on findings from a study that collected data on subjective stock markets expectations over a twelve-year period in a large, representative internet panel in the Netherlands. Expectations are elicited using a probabilistic format and refer to the one-year ahead rate of return of the Amsterdam Stock Exchange index (AEX), with four questions on gains and losses, respectively. We thus have for each respondent and each interview date eight responses that correspond to well-defined points on the subjective distribution of oneyear rates of returns. Results from the first two surveys, conducted in 2004 and 2006, are reported in Hurd et al. (2011). In that paper, we documented substantial heterogeneity in 1 Surveys which include questions on subjective probabilities are, amongst others, the Health and Retirement Study (HRS), the Survey of Health, Ageing and Retirement in Europe (SHARE), the English Longitudinal Study of Ageing (ELSA), the Korean Longitudinal Study of Ageing (KLoSA), the Survey of Economic Expectations (SEE), the Survey of Consumer Expectations (SCE) and the Understanding America Study (UAS). For excellent overviews on subjective expectations, see Manski (2004), Hurd (2009) or Manski (2017). 2 In studies exploiting natural experiments, exposure to such dramatic events as wars, hunger episodes, recessions, and other crises as well as health shocks such as flu epidemics has been shown to predict a variety of adult outcomes. These include subjective expectations, trust attitudes, risk attitudes, and investment behavior, as well as various economic and health outcomes (inter alia, Malmendier and Nagel, 2011; Akbulut-Yuksel, 2014; Kesternich et al., 2014, 2015; Bauer et al., 2014; Black et al., 2015; Van den Berg et al., 2016; Malmendier and Shen, 2015). The data we analyze in this paper do not allow us to investigate whether such shocks predict heterogeneity in stock market expectations levels or changes. 1

3 financial market expectations. The present paper extends the analysis using more recent data from surveys conducted in 2008, 2009, 2010, 2012, 2014, and 2016 that is, throughout the financial market crisis. The long panel dimension in combination with the great level of detail on individuals subjective distributions allows us to analyze stock market expectations and more importantly updating behavior in greater detail than previous literature. We find that respondents update their expectations more strongly in the loss domain compared to the gain domain. This particularly applies to questions on extreme changes in the stock market, i.e. questions on the chance that the stock market will change by more than 20 or 30 percent. In the year of the financial crisis, for example, individuals adjusted their expectations by roughly five percentage points across the entire loss domain. In the gain domain, in contrast, respondents did on average not update their expectations on extreme changes. We also find that for a given respondent expectations on extreme changes in the gain domain are considerably less volatile over time than comparable expectations in the gain domain. Another contribution is the analysis of the importance of past stock market returns for the updating behavior of stock market expectations. In line with previous literature, we describe the population by three distinct updating types, who differ in how they use the recent stock market performance to update expectations. The Random Walk (RW) type does not take short-term stock market performance into account, as she only uses the long-run historical average return to predict future returns. The Persistence (P) type beliefs that recent fluctuations in the stock market persist into the near future, while the Mean Reversion (MR) type expects them to be reverted. We use two approaches to analyze respondents updating types. First, we estimate the type distribution in the population, which is consistent with the reported expectations, using an ordinal methodology introduced by Dominitz and Manski (2011). Similar to their results, we find that the fractions of respondents consistent with the (RW, MR, P) types are (0.30, 0.26, 0.44). In addition, our data allows us a within-respondent analysis of expectation type stability over time. We show that updating types are quite stable 2

4 over time with almost 50% of respondents being classified as having the same expectation type in every observed wave. We find mixed evidence for socio-economic variables predicting expectation types and switching behavior between types. Second, we develop a panel data model which combines the discrete types approach to heterogeneity of stock market expectations formation and the response model for subjective probabilities by Kleinjans and Van Soest (2014). In particular, our model is able to distinguish between the three different expectations types and at the same time take heterogeneity of rounding behavior into account. Our estimates for this model suggest that, compared to the Dominitz and Manski (2011) methodology, more individuals are classified as P type and less as MR, while the RW share is roughly the same. We also find that socio-demographic variables affect updating behavior. Interestingly, the model also suggests that next to the past one-year stock market return, returns on shorter horizons, such as the one-week return, have also a significant impact on individuals expectation updates. We conclude by noting that our proposed panel data model can also be applied to analyze probability updates in other domains, such as housing prices, subjective health, retirement and more. The remainder of this paper is organized as follows: We first describe our data (Section 2) and then focus on the dynamics of stock market expectations (Section 3). Expectation types and the role of past returns are analyzed in Section 4. While Section 4.1 focuses on the ordinal methodology introduced by Dominitz and Manski (2011), Section 4.2 will present our proposed panel data model. Section 5 concludes. 2 Data and descriptive statistics The study was conducted using the CentER Panel. About 2,000 Dutch households are interviewed online every spring in 2004, 2006, 2008, 2009, 2010, 2012, 2014 and 2016, making a total of eight waves (see Figure 1). While the majority of respondents participate right away, others who do not are contacted again three or four weeks later. The questionnaire contains variables on stock market experience, knowledge of average long- 3

5 AEX [weekly] year Figure 1: Amsterdam Stock Market Index (AEX), The vertical lines show the timing of the spring interviews. term returns for investment in risky and safe assets, and past trading history. In addition, it also includes probabilistic expectations questions on stock market returns over a one-year horizon. We ask for the chances that an investment in a broad investment fund will generate gains of more than 0%, 10%, 20%, and 30%, as well as losses of more than 0%, 10%, 20%, and 30% percent, for a total of eight questions. The four questions within each sequence (gains and losses) are always presented with increasing absolute threshold returns, but the gain and loss sequences are presented in random order (even though we did not find a significant order effect in a pre-test of our survey). The sequence of gain and loss questions starts with a short introduction explaining that the respondent has to imagine that he unexpectedly received 10,000 Euro from a rich relative and is thinking of putting the money into a mutual fund invested in blue chip stocks (like those in the Amsterdam AEX stock market index). The wording of the first question in the gain sequence reads as follows: Suppose you put the 10,000 Euro in the stock mutual fund and left it in for one year. What are the chances that you would make money where 0 means absolutely no chance and 100 means absolutely certain; that is what are the chances that in a year your investment would be worth more than 10,000 Euro? The other questions in this sequence use a very similar wording, with different numbers and 4

6 adjusted to reflect the gain and loss sequence where appropriate. In addition, households from the CentER Panels also participate in the DNB Household Survey (DHS), formerly known as the CentER Savings Survey. We match our data with the very detailed DHS data on variables such as education, income, employment, marital status, savings and portfolio choice. Overall, our (unbalanced) panel dataset contains 5,718 individuals, who are observed in up to eight waves between 2004 and 2016, making a total of 16,565 observations. Panel A of Table 1 displays standard summary statistics for the eight probabilistic expectation questions. Overall, people are quite pessimistic regarding the future stock market performance. The average subjective probability that the stock market will make any loss is 42.7% and therefore almost as high as the average subjective probability that the stock market will make any gain (40.9%). For the questions on more extreme changes in the stock market (gains and loss of more than 10, 20 and 30 percent), respondents assign on average more probability mass to negative events than to positive events. For example, the average subjective probability that the stock market will increase by more than 30% is roughly 7%, compared to a 15% chance that the stock market will decrease by more than 30%. The item non-response is similar across questions and around 18 percent. We also fit individual-specific cumulative distribution functions (c.d.f.) using all eight questions on subjective stock market expectations. We report both parametric and non-parametric estimates of the mean and variance of the subjective distributions in Panel B and C, respectively. 3 Both are, however, very close. The estimates for the first moment suggest that respondents on average expect the stock market index to decrease by roughly one percent in the next year. This is in stark contrast to the actual average one-year return of the AEX in this time period, which is around four percent. Again, respondents seem to be very pessim- 3 Following Hurd et al. (2011) we obtain the parametric estimates under the assumption of normal returns using non-linear least squares. For the non-parametric estimates, we average historic one-year stock market returns within each interval, which is defined by the eight different return thresholds, and use the respondent-specific probabilities as weights. For details see Hurd et al. (2011). 5

7 Table 1: Descriptive statistics for stock market expectations A: Stock market expectations [in %] Observations Mean SD Min Max Gain > 0% 13, Gain > 10% 13, Gain > 20% 13, Gain > 30% 13, Loss > 0% 13, Loss > 10% 13, Loss > 20% 13, Loss > 30% 13, B: Parametric estimates [in %] mean 9, SD 9, C: Non-parametric estimates [in %] mean 10, SD 10, D: Covariates Female 16, Age < 45 16, Age > 64 16, Low education 16, High education 16, Partner 16, Net income in 1000 Euro 16, Late 16, Notes: Summary statistics for different variables; N = 16, 565. Varying number of observations due to item-nonresponse (Panel A, B, C and D) and monotonicity assumptions (Panel B). istic regarding the future stock market performance. Panel D describes our sample regarding standard socio-economic characteristics. Overall, there are slightly less females than males in the sample. One third is younger than 45 years, while one quarter is aged 65 and above. One third completed no more than primary school 6

8 or prevocational training (low education), while another third completed higher vocational training or university education (high education). The average net income is around 2,700 Euro per month. 3 Dynamics of stock market expectations Respondents stock market expectations display considerable variation over time. Figure 2 displays average stock market expectations over time for each of the eight expectation questions. In general, expectations seem to follow the Business cycle. The financial crisis in 2009 and the following European sovereign debt crisis in 2012, for example, coincide with dips in expectations in the gain domain (left panel) and hills in the loss domain (right panel). Similarly, expectations were more optimistic during the boom of Expectations [%] Expectations [%] Year Year Gain > 0% Gain > 10% Gain > 20% Gain > 30% Loss > 0% Loss > 10% Loss > 20% Loss > 30% Figure 2: Dynamics of subjective stock market expectations Overall, the largest adjustments are made for the questions on any gain and any loss. If individuals are asked about more extreme events (gains and losses of more than 10, 20 or 30 percent), average expectations display considerably less variation over time. For example, 7

9 the average probability that the AEX will increase by more than 30 percent in the next year varies only between 5.5% (in 2012) and 9% in (2014). This pattern changes if we look at questions in the loss domain. In 2009, respondents shifted their entire distribution by roughly five percentage points upwards. Interestingly, the sovereign debt crisis in 2012 did not have such an effect. Overall, it seems as if due to the financial crisis respondents shifted probability mass to negative outcomes of the distribution. This pattern can also be shown more formally when looking at the within-respondent disagreement over time. In particular, we are interested in how strongly respondents update their expectations from wave to wave. We therefore estimate respondent-specific (sample) standard deviations of answers across waves for each of the eight questions separately. For clarification consider the following example. Respondent A (B) is observed in four (two) waves. The corresponding responses for the question on a positive stock market return (Gain > 0%) are given by (70, 80, 60, 60) and (80, 80), respectively. The within-respondent (sample) standard deviation across waves for the question on positive stock market returns would then be 9.57 for respondent A and zero for respondent B. For each respondent, we calculate the standard deviation across waves for all of the eight expectations questions. 4 Table 2 displays summary statistics for our measure of within-respondent disagreement. Again, the largest adjustments are made for questions on any gain or any loss. The more extreme the outcome of the question gets, the less volatile are the answers to that particular question. More interestingly, however, is the difference between the gain and loss domain. While there is almost no difference for the question on any gain or loss, the picture changes when we look at questions on larger gains and losses. Here, answers in the loss domain are considerably more volatile than in the gain domain. For the questions on gains and losses of more than 30 percent, the difference in average standard deviation amounts to roughly five percentage points (12.06% versus 6.73%). In line with evidence from Figure 2, it seems that respondents tend to adjust their expectations more in the loss domain than the gain domain. 4 Note that individuals have to be observed at least twice in order to calculates the (sample) standard deviation. 8

10 Table 2: Summary statistics for within-respondent disagreement (across years). Gains Mean p25 p50 p75 Min Max N Gain > 0% ,783 Gain > 10% ,732 Gain > 20% ,709 Gain > 30% ,701 Losses Loss > 0% ,778 Loss > 10% ,705 Loss > 20% ,686 Loss > 30% ,665 Notes: This table reports summary statistics for the within-respondent disagreement, i.e. sample standard deviation, across waves for each of the eight probabilistic questions on stock market returns. The across-wave standard deviation is only defined if the respondent answers the question in at least two waves. For details see text. Last, we also calculate the traditional measure of disagreement i.e. the cross-sectional standard deviation of expectations in a given year. This measure is shown to correlate with several other measures of economic uncertainty and is frequently used in the literature. 5 In total, we calculate eight disagreement measures, one for each expectation question. As shown in Figure 3, uncertainty measures which are based on dispersion of expectations in the loss domain behave somewhat better. First, they suggest a similar disagreement measure, both in levels and adjustments over time. Second, as suggested by previous literature, disagreement based on loss expectations behaves counter-cyclically, i.e. you can observe a clear increase in 2009 and for the Loss>0% questions also in The correlations between the four disagreement measures in the loss domain varies between 0.65 and In contrast, the disagreement measures in the gain domain perform poorly. They are rather uncorrelated and the four measures are quite different in levels. These findings might indicate that uncertainty 5 See for example, Zarnowitz and Lambros (1987), Bachmann et al. (2013) and XXX Bloom. 9

11 indexes which are based on loss framing might be more appropriate as a uncertainty index than measures in a gain frame Cross sectional disagreement Cross sectional disagreement Year Year Gain > 0% Gain > 10% Gain > 20% Gain > 30% Loss > 0% Loss > 10% Loss > 20% Loss > 30% Figure 3: Cross-sectional disagreement of expectations over time. Last, we provide evidence that both expectations themselves and their association with socioeconomic variables are stable over time. As shown by previous literature, expectations vary considerably across individuals (e.g. Manski, 2004; Hurd, 2009). Table 3 confirms this result and additionally provides evidence that these associations are stable over time. In particular, the sign, significance and even the magnitudes of the coefficients seem to be remarkably stable over time. Moreover, Figure 4 provides direct evidence for the intertemporal stability of expectations. In particular, we find positive correlations between the mean of the subjective distribution in one wave (normalized to the cross-sectional average) to the next wave. This indicates that respondents, who are relatively pessimistic in one wave, are also relatively pessimistic in the next wave. In fact, this finding emerges between all observed waves. Overall, Table 3 and Figure 4 provide strong evidence that both expectations per se and its associations with socio-economic variables are rather stable over time. 10

12 Table 3: Probability of positive stock market return: Cross-sectional heterogeneity (1) (2) (3) (4) (5) (6) (7) (8) (9) RE Female Age > Age < Low education High education Partner Net income in 1000 Euro Late Stock ownership Constant Observations Notes: Column 1 also includes year FE. Standard errors in brackets are clustered at the individual level (Column 1); *** p <0.01, ** p <0.05, * p <0.1. N = 691 N = 735 N = 697 N = N = 593 N = 644 N = X axis and y axis are normalized to the average alpha in the respective year. Outliers dropped Figure 4: Correlation alpha 4 Past stock market returns and updating types In the following we will argue that respondents use the recent stock market performance when updating their expectations. 6 Following the literature, we divide the population into three 6 Of course, our data does not allow us to infer whether or not individuals actually use the past AEX performance to revise their expectations, and if so whether they focus on the one-year return. Our analysis 11

13 different expectation types. The Random Walk (RW) type beliefs that stock market returns are i.i.d. over time and therefore uses the long-run historical average return to predict the future. The Persistence (P) type believes that short-term fluctuations in the stock market will persist into the near future, while the Mean Reversion (MR) type expects them to be reverted. 7 For clarification, consider the following example. A respondent is interviewed on her stock market expectations in 2004 and 2006 a period in which the AEX index increased almost monotonically (see Figure 1) and more importantly, the one-year return in 2006 was higher than the one-year return in If she were a RW type, she would hardly adjust their expectations in 2006, as the long-run historical average return will only marginally be affected by those two additional years. In contrast, a P type would positively update her 2004 expectations, because she beliefs the (positive) recent stock market performance to persist into the near future. Similarly, if she were a MR type, she would lower her expectations in The example illustrates how probabilistic survey answers from respondents who are observed at least twice can be used to identify different updating types. For our analysis we will both use an ordinal methodology introduced by Dominitz and Manski (2011) and develop a panel data model which combines the discrete types approach to heterogeneity of stock market expectations formation and the response model for subjective probabilities by Kleinjans and Van Soest (2014). In particular, our model will be able to distinguish between the three different expectations types (RW, MR, P) and at the same time take heterogeneity of rounding behavior into account. can only show whether or not observed behavior is consistent with on of our three types. See Dominitz and Manski (2011) for a broader discussion. 7 We follow Dominitz and Manski (2011) in the description of the three types. Note, however, that in Armona et al. (2016), for example, the Persistence type is referred to as Momentum type. 12

14 4.1 Ordinal methodology As noted earlier, respondents expectations are considered to be consistent with the RW type if expectations hardly change between waves. Applying the methodology introduced by Dominitz and Manski (2011), we define respondents answers to be consistent with the RW type if they differ by at most five percentage points. Similarly, if respondents update their expectations by more than five percentage points, they are classified as MR or P type, depending on the update s direction and the past short-term stock market performance. In particular, we calculate the expectation type for every individual if answers for both the current and previous wave is available. By definition we cannot get type distributions for the first wave in Figure 5 displays the cross-sectional distribution of expectations types over time. In particular, we focus on updating behavior of the question on a positive stock market return (Gain > 0%). The average type shares (RW, MR, P) in the population which are consistent with the reported probabilities are given by (0.30, 0.26, 0.44). 8 Even though there are small differences, the shares are remarkably stable over time. The majority of reported probabilities are consistent with type P, while the share of MR and RW types is roughly the same. It is, however, important to note that Figure 5 cannot say anything about the withinrespondent type stability. In fact, it could well be that individuals quite frequently switch their updating types and the stability of cross-sectional types shares is just the byproduct of some other mechanism. Using the panel structure of our data, we thus examine the within-respondent variation in expectation types directly. In particular, we define a variable indicating the degree to which respondents switch between updating types across waves. Respondents are classified as No switchers if their answers are consistent with the same expectations type in all observed waves. Stark switchers are people who switch from P to MR, 8 Note that we also calculated the type shares based on the seven other expectation questions. Comparing questions on identical absolute thresholds, such as on Gain > 10% or Loss > 10%, yields almost identical results. However, using questions with higher absolute thresholds typically assigns more respondents to the RW type, while type P is still more prevalent than MR. Similarly, increasing the range of updates which are considered to be consistent with type RW also increases the prevalence of type RW, while again more responses are consistent with type P than MR. 13

15 Persistence Random Walk Mean Reversion Figure 5: Type distributions in the population over time or vice versa. Mild switchers are considered to be respondents who switch between RW and P, RW and MR or vice versa. Note that this definition is quite strict in the following sense: Imagine we observe an individual in eight waves, and thus could identify seven expectation types. Even if six of those types were identical, say type P, but the seventh type was MR (RW), we would classify this respondent as Stark (Mild) switcher. Table 4 displays the switching type distribution among the population in relation to how often their expectation type (ET) is observed in the data. Note that in order to observe one expectation type (ET=1), respondents have to be observed at least twice. Observing two expectation types (ET=2), requires the respondent to be observed three times and so on. On average, almost half of the population displays stable expectation types, while one quarter displays mild and another quarter stark switching behavior. All respondents with only one observed expectation type are by definition No switchers. Naturally, as we observe more expectation types, switching becomes more prevalent. For example, for ET=3, the switching shares in the population (No, Mild, Stark switchers) are given by (0.22, 0.45, 0.33). We also calculated the switching shares using the seven other expectation questions. As earlier, we repeated the analysis with the other seven expectation questions and found little difference 14

16 between the gain and loss domain. However, using questions with higher absolute thresholds suggests that switching behavior becomes less prevalent. For example, the average switching type distribution in the population (No, Mild, Stark switchers) which is based on the question on Gain > 30% is given by (0.71, 0.21, 0.08), providing strong evidence for type stability over time. This difference might be explained by the fact that individuals report more stable expectations when they are asked about drastic changes in the stock market. For example, when asked about the likelihood that the stock market will increase by more than 30 percent, many individuals report a likelihood of 0% in all waves, explaining why less respondents are classified as expectations type switchers. Table 4: Stability of expectation types over waves Total ET=1 ET=2 ET=3 ET=4 ET=5 ET=6 ET=7 No switchers Mild switchers Stark switchers N 2, Notes: This table shows the distribution of individuals who switch or do not switch their expectation type (ET) in relation to how often their type is observed in the data. Results are based on responses to the question on a positive stock market return (Gain > 0%). Heterogeneity in expectation types and switching behavior Even though there is evidence for stability of expectation types, some respondents report probabilities which are in line with quite unstable updating types. In a next step we examine whether switching between expectation types can be predicted by socio-economic variables. We therefore estimate pooled ordered probit models, with our dependent variable taking a value of zero if we observe no switching, and one (two) if we observe mild (stark) switching. Every column in Table 5 corresponds the analysis of switching behavioral of one of the eight probability questions. Overall, there are almost no differences among subgroups. In fact, the only pattern that emerges is that education might play a significantly role in explaining switching behavior. In particular, our estimates suggest that respondents with higher vocational training or university education (high education) are less likely to switch their 15

17 Table 5: Heterogeneity in switching behavior Switching type (1) (2) (3) (4) (5) (6) (7) (8) Gain > 0% Gain > 10% Gain > 20% Gain > 30% Loss > 0% Loss > 10% Loss > 20% Loss > 30% main Female [0.55] [-1.48] [0.07] [1.30] [0.32] [0.80] [2.29] [2.90] Age < [-2.84] [0.86] [2.31] [2.26] [-1.22] [-1.62] [0.39] [-0.26] Age > [2.27] [-0.45] [-1.36] [-1.77] [2.34] [0.48] [-0.24] [-0.50] Low education [-1.09] [-1.27] [-0.54] [-0.07] [0.73] [-0.17] [1.10] [1.74] High education [-0.45] [-1.12] [-2.96] [-2.94] [-0.90] [-2.18] [-2.62] [-2.53] Partner [-0.67] [-0.27] [1.00] [0.14] [0.57] [1.02] [-1.09] [-0.93] Net income in 1000 Euro [0.71] [0.49] [0.54] [-0.38] [-1.08] [-1.78] [-1.60] [-1.07] Late [1.87] [1.31] [1.89] [1.18] [-0.35] [0.69] [0.72] [1.81] / cut [9.76] [11.30] [12.11] [11.70] [10.94] [9.54] [9.46] [10.40] cut [19.15] [21.55] [22.15] [19.61] [19.82] [19.39] [19.90] [19.49] Observations Notes: Pooled ordered probit models. Dependent variable is indicator variable for switching type: No switching (0), Mild switching (1) and Stark switching (2). For details see text. Each column uses a different probability questions to calculate the expectations types and switching types. Models also includes time fixed effects. Standard errors are clustered at the individual level; *** p <0.01, ** p <0.05, * p <0.1. expectation types. In a last step we use pooled multinational logit models to predict expectation types by a set of standard covariates. With RW being the baseline category, Table 6 displays the estimated coefficients. Again, no systematic pattern emerges with most coefficients being insignificantly different from zero. There is some evidence that older respondents are indeed less likely to be of type P and MR in comparison to RW. This finding can, however, not be found for all expectation questions. Overall, our findings are therefore in line with Dominitz and Manski (2011) who state that it appears that demographic groups differ systematically in the magnitudes of their expectations for equity return but not in the way they revise expectations (page 362). 16

18 Table 6: Heterogeneity in updating types Expectation type (1) (2) (3) (4) (5) (6) (7) (8) Gain > 0% Gain > 10% Gain > 20% Gain > 30% Loss > 0% Loss > 10% Loss > 20% Loss > 30% Persistence Female [-1.86] [-0.41] [1.67] [2.09] [3.21] [3.36] [5.07] [3.66] Age < [-1.59] [2.22] [5.39] [4.14] [-1.34] [0.81] [2.25] [3.45] Age > [0.52] [-2.32] [-2.93] [-1.95] [0.43] [-1.68] [-2.03] [-0.62] Low education [-0.88] [-0.47] [0.07] [0.87] [2.22] [0.55] [1.82] [2.66] High education [-0.73] [-0.08] [-2.04] [-1.79] [1.08] [-1.75] [-1.75] [-2.22] Partner [0.89] [1.63] [2.96] [1.50] [1.37] [2.33] [0.54] [0.27] Net income in 1000 Euro [0.66] [-0.06] [0.06] [0.26] [-1.87] [-0.63] [-1.07] [-2.22] Late [-0.01] [-0.30] [0.90] [0.53] [1.78] [2.95] [2.65] [3.87] Constant [4.63] [0.90] [-6.68] [-11.56] [-2.51] [-4.56] [-8.61] [-10.56] Mean reversion Female [-2.43] [-1.27] [-0.33] [-0.02] [1.08] [3.23] [4.42] [3.36] Age < [-0.70] [2.43] [4.37] [3.10] [-0.25] [-0.86] [0.77] [1.29] Age > [1.14] [-3.00] [-2.94] [-3.68] [-1.21] [-2.08] [-2.49] [-2.04] Low education [-0.90] [-1.69] [-0.73] [1.19] [0.25] [-0.82] [0.70] [2.95] High education [-0.62] [-0.46] [-2.10] [-0.81] [0.06] [-2.12] [-3.10] [-2.60] Partner [-0.67] [0.08] [0.21] [0.16] [0.86] [0.61] [-1.54] [-1.20] Net income in 1000 Euro [0.84] [0.19] [0.95] [0.31] [-1.38] [1.02] [1.27] [-0.13] Late [2.52] [1.32] [2.36] [1.21] [-1.39] [-0.06] [-0.48] [0.55] Constant [1.04] [0.74] [-6.16] [-11.35] [2.82] [2.06] [-3.27] [-8.12] Observations Notes: Pooled multinomial logit models. Omitted category is RW. Model also includes time fixed effects and an indicator for the number of observed rounding types of the individual. Each of the eight columns corresponds to one of the eight probability questions which are used to determine the expectations types and switching types. Standard errors are clustered at the individual level; *** p <0.01, ** p <0.05, * p <

19 4.2 Panel data model In the following section we abandon the ordinal methodology by Dominitz and Manski (2011) and propose a panel data model, which combines the discrete type approach to heterogeneity of stock market expectations formation and the response model for subjective probabilities by Kleinjans and Van Soest (2014). In their model subjective probabilities are determined by a linear index of observable, potentially time-varying covariates, a random effect, and an idiosyncratic error. We extend their model by making the role of past returns explicit and by allowing for discrete expectation types that differ in how they take past returns into account. In addition, we focus on probability updates between waves rather than on subjective probabilities in one particular wave. Similarly to Kleinjans and Van Soest (2014), we explicitly model heterogeneity in rounding behavior, a phenomenon which is ubiquitous in data on subjective expectations. One major advantage of our model compared to the approach by Dominitz and Manski (2011) is that our resulting expectation type distribution in the population does not depend on a somewhat ad-hoc cutoff of five percent. For tractability reasons, we only model updates for one probability question, such as the subjective probability of a positive return (Gain > 0%). An extension to multiple questions is left for future work. This could follow the ideas in Hurd et al. (2011) who estimate the mean and variance of the subjective return distributions under the assumption of a normal distribution, or Bellemare et al. (2012) who approximate the distribution non-parametrically using splines. Our outcome variable the probability update of a respondent, i.e. the difference in reported subjective expectations between two waves is modeled as a function of a latent variable. The latent variable is then mapped into the reported (and observed) probability; we therefore specify a separate reporting model Probability updates Suppose that the (latent) probability update of respondent i in wave t, i.e. the difference between respondents subjective probability in wave t and wave t 1 is given by 18

20 K yit = D ik f k (X t ) + x it β k + ɛ itk, (1) k=1 where D ik denotes a dummy variable indicating expectation type k. f k (X t ) is a function of the history of past stock returns at period t, X t. This function captures how individuals of type k process past stock market information. x it is a vector of potentially time-varying covariates (if any) and ɛ itk denotes the idiosyncratic error. For now, assume that X t contains only the difference between the past one-year AEX return in wave t and wave t 1, (r t ), and that the function f( ) takes the linear from f k (X t ) = f k (r t ) = γ k + δ k r t. The different expectations types can then be characterized in terms of δ k : k = 1 is the Random Walk (RW) type. For these individuals, δ 1 is equal to zero, as they do not use the past one-year stock market to predict future returns, but rather use the long-run historical average return. k = 2 is the Mean Reversion (MR) type. These individuals belief that recent stock market changes will be reversed in the near future, implying that δ 2 will be strictly negative. k = 3 is the Persistence (P) type. These individuals belief that recent stock market changes will persist into the near future, implying that δ 3 will be strictly positive. Expectation type probabilities Pr (D ik = 1 z i +... ), k = 1,..., K, are modeled as a function of (time-constant) covariates z i using a multinomial logit (MNL) specification. 19

21 4.2.2 Reporting model (rounding) The reporting model is a simplified version of Kleinjans and Van Soest (2014). 9 In particular, we assume that the population can be described by the following rounding types: R it = 1 : the subjective probability is rounded to a multiple of 1 R it = 2 : the subjective probability is rounded to a multiple of 5 R it = 3 : the subjective probability is rounded to a multiple of 10 R it = 4 : the subjective probability is rounded to a multiple of 25 R it = 5 : the subjective probability is rounded to a multiple of 50 This implies that the true subjective probabilities will lie within an interval of ±0.5m percentage points, where m denotes the multiple to which the respondent rounds. For example, consider an individual who reports a subjective probability of 20%. If she were rounding type 3, her true subjective probability would lie within the interval of [15%; 25%]. Note also that each reported probability partially reveals individual s rounding type. Sticking to our example, the reported probability of our respondent is only consistent with rounding types 1, 2 or 3, but not with rounding types 4 or 5. Similarly, a reported probability of say 18% would uniquely identify this respondent to be of type Our outcome variable the probability update between two waves is therefore the difference between two possibly rounded subjective probabilities. Assuming that the rounding type is constant across two waves, the true probability update then lies within the interval of ±2 0.5m = ±m: R it = 1 : R it = 2 : R it = 3 : the probability update lies within an interval of ±1 percentage points the probability update lies within an interval of ±5 percentage points the probability update lies within an interval of ±10 percentage points 9 For tractability reasons, we do not model item nonresponse and epistemic uncertainty expressed by 50% answers. But note that the results in Kleinjans and Van Soest (2014) indicate that the unobservables in their item-nonresponse equation are correlated with the unobservables in the other models, which implies that ignoring item nonresponse may lead to selection bias. 10 Reference to Tail and Center Rounding of Probabilistic Expectations in the Health and Retirement Study Giustinelli, Manski and Molinari (2018) 20

22 R it = 4 : R it = 5 : the probability update lies within an interval of ±25 percentage points the probability update lies within an interval of ±50 percentage points Again sticking to our example, suppose the respondents increases her reported probability in the next wave to 50%. The resulting probability update would thus be 30 percentage points. If she were of rounding type 3, her true probability update will lie within the interval of [20;40] percentage points. Rounding type probabilities are assumed to be the same for every individual. However, a simple extension of the model could also parametrize these probabilities, for example by using an ordered response model (ordered probit in the appendix) Distributional assumptions and estimation All the ɛ s are i.i.d. normal and independent of each other. The entire model is then estimated by Maximum Likelihood Results The estimates from our model are reported in Table 7. Panels A, B and C report the return coefficient and coefficients of additional covariates (Model 1 only) for three different latent classes, respectively. While in Class 1 the return coefficient is restricted to be zero, the return coefficient in Class 2 and Class 3 are unrestricted. Most importantly, Class 2 displays a significantly negative return coefficient, while Class 3 displays a significantly positive coefficient. According to our earlier classification regarding the sign of the return coefficient, we thus label Class 1, 2 and 3 as type RW, MR and P, respectively. Interestingly, no other variable seems to have an impact on the probability updates. In fact, all coefficients in the first three panels except the return coefficient are insignificant. Panels E and F report the coefficients from the multinomial logit specification for the class probabilities with Class 1 (RW) being the reference category. Similarly to our results in Table 6, we only find a weakly significant coefficient for female, suggesting that that fe- 21

23 males (in comparison to males) are less likely to be of type Class 2 (MR) than of Class 1 (RW). This result can, however, not be found when comparing Class 3 (P) and Class 1 (RW). Panel G provides strong evidence for heterogeneity in rounding behavior in the population. In fact, 52% of respondents are estimated to round to a multiple of 5 (rounding type 2), and 36% to a multiple of 10 (rounding type 3). 9% of people seem not to round at all (rounding type 1) and the share of people who round to multiples of 25 or 50 is estimated to be 1% each (rounding types 4 and 5). Overall, our estimates suggest that more than nine in 10 respondents report a rounded value. Lastly, our estimates suggest that the shares of responses which are consistent with the expectations types (RW, MR, P) are given by (0.17, 0.21, 0.62). Similarly to the Dominitz and Manski (2011) methodology from the previous section, most answers are consistent with a P type, even though our model suggest a slightly larger fraction. While the RW and MR shares were both around 26% in Dominitz and Manski (2011), our model would suggest that fewer people are in fact of type RW and more of type MR. However, these results are remarkably similar to our earlier findings and support the claim that a substantial part of respondents use indeed the past performance of the stock index to update their expectations Model extensions We next hypothesize that individuals might not only focus on the past one-year return of the stock market index, but also on other short-run returns. In particular, you could imagine that short-run events, such as television news, have a significant effect on your updating behavior. We thus extend our model to include also the difference between returns of three months, one month as well as one week. Estimates are reported in Table 8. Interestingly, all but one return coefficient in Class 2 are negative, while most coefficients in Class 3 are positive. This is again in line with our interpretation of the Persistence types (Class 3) and Mean Reversion types (Class 2). More importantly, we find evidence that most return differences have a significant impact on repor- 22

24 Table 7: Model estimates A: Class 1 Model 1 Model 2 Cl1 (Intercept) 0.95 (1.11) 0.58 (0.43) Cl1 nettohh 0.14 (0.28) Cl1 old 0.51 (0.99) Cl1 young 0.51 (0.94) Cl1 partner 1.04 (1.04) B: Class 2 Cl2 (Intercept) 3.02 (5.58) 1.08 (2.18) Cl2 returndiff52weeks 0.15 (0.05) 0.14 (0.05) Cl2 nettohh 0.82 (1.50) Cl2 old 0.13 (4.54) Cl2 young 1.25 (5.20) Cl2 partner 0.81 (5.53) C: Class 3 Cl3 (Intercept) 2.01 (2.60) 1.58 (0.99) Cl3 returndiff52weeks 0.26 (0.02) 0.27 (0.02) Cl3 nettohh 0.02 (0.53) Cl3 old 0.20 (1.92) Cl3 young 0.33 (2.26) Cl3 partner 0.59 (2.43) D: Standard deviations sigma Cl (0.23) 7.72 (0.23) sigma Cl (0.74) (0.72) sigma Cl (0.38) (0.38) E: Cl2 probabilities Class 2 (Intercept) 0.41 (0.26) 0.48 (0.26) Class 2 female 0.46 (0.25) 0.46 (0.24) F: Cl3 probabilities Class 3 (Intercept) 1.35 (0.14) 1.36 (0.14) Class 3 female 0.12 (0.16) 0.12 (0.16) G: Rounding type prob. Prob. rounding type (0.00) 0.09 (0.00) Prob. rounding type (0.01) 0.54 (0.01) Prob. rounding type (0.01) 0.35 (0.01) Prob. rounding type (0.00) 0.01 (0.00) Class 1 share Class 2 share Class 3 share N*t N loglik AIC Notes: Standard errors in parentheses; *** p <0.01, ** p <0.05, * p <

25 ted expectations. For example, the differences in the one-week return have a highly significant impact on expectations, which is negative for type MR and positive for type P. We again find evidence that socio-demographic characteristics seem not to play a major role. However, we do find that slightly less responses are now consistent with type Persistence. In fact, the (RW, MR, P) shares in the population are now given by (0.17, 0.33, 0.50) and are much closer to the values suggested by the ordinal measure by Dominitz and Manski (2011). 5 Conclusion This paper provides evidence that most people do not have rational expectations of equity returns. In the absence of private information, expectations across individuals should be identical. In contrast, we show that expectations vary considerably across different subgroups of the population, and that these associations are stable across years. Using the methodology by Dominitz and Manski (2011), we replicate their findings in a different sample and extend their analysis. In particular, we find that expectations types are rather stable concepts and provide some evidence that education might predict switching between expectation types. We propose a panel data model which combines the discrete expectation type approach and a response model for subjective probabilities by Kleinjans and Van Soest (2014). Our results suggest that the majority of people seems to belief that recent market performance will persist into the near future, while fewer people belief that it will be reversed in the near future. We also find that events in the very short run, such as stock market returns of the previous week, have a significant impact on reported expectations. Our model can easily be adapted for other expectations, such as for example house price expectations elicited in the SCE Housing Survey. Future research is, however, needed to understand the exact expectation formation process of individuals. 24