Abstract This series presents research findings based either directly on data from the German SocioEconomic Panel study (SOEP) or using SOEP data as part of an internationally comparable data set (e.g. CNEF, ECHP, LIS, LWS, CHER/PACO). SOEP is a truly multidisciplinary household panel study covering a wide range of social and behavioral sciences: economics, sociology, psychology, survey methodology, econometrics and applied statistics, educational science, political science, public health, behavioral genetics, demography, geography, and sport science. Abstract The decision to publish a submission in SOEPpapers is made by a board of editors chosen by the DIW Berlin to represent the wide range of disciplines covered by SOEP. There is no external referee process and papers are either accepted or rejected without revision. Papers appear in this series as works in progress and may also appear elsewhere. They often represent preliminary studies and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be requested from the author directly. Abstract Any opinions expressed in this series are those of the author(s) and not those of DIW Berlin. Research disseminated by DIW Berlin may include views on public policy issues, but the institute itself takes no institutional policy positions. Abstract The outcome of pursuing an upper or post-secondary education degree is uncertain. A student might not complete a chosen degree for a number of reasons, such as insufficient academic preparation or financial constraints. Thus, when considering whether to invest in post-secondary education, students must factor their probability of completing the degree into their decision. We study the role of this uncertainty in education choices using representative survey data from Germany. Students' subjective beliefs about the probability of completing a post-secondary education were elicited prior to them finishing their secondary education. We relate these subjective completion probabilities to students' subsequent education choices and outcomes. We find that these early beliefs are predictive of intentions to invest in education, actual subsequent investments in education, and degree completion. A structural choice model of sequential investment further reveals that the association between completion beliefs and investment choices is strongest for students with low academic skills and low preferences for post-secondary education. Subjective completion beliefs and the demand for post-secondary education Introduction After finishing compulsory education, students have to decide whether to continue investing in education. This choice is one of the most important career decisions young adults have to make, and is made under partial information; beliefs held at this early stage are thus central to post-secondary education choices. In this paper, we focus on one major source of uncertainty: the probability of dropping out of a chosen education track. We incorporate a measure of the perceived or subjective probability of being able to complete a post-secondary degree into simple models of education choice, using data from a large representative survey from Germany. This data offers at least two characteristics that contribute to our understanding of young adults' education decisions. First, subjective completion beliefs are elicited shortly before students finish their compulsory secondary education; the data track students over the next 14 years, which allows us to observe their subsequent education outcomes. In contrast, a large part of the literature focuses on beliefs of students who are already enrolled in college. Introduction At that point in time, students' beliefs are likely to have been updated by new information, and are thus likely to be different from the original beliefs upon which investment decisions were based. Second, the representative population sample allows us to extend the previous literature, which has been confined to college education, to more vocational forms of post-secondary education, such as different types of apprenticeships. Since students from less advantaged economic backgrounds are more likely to invest in vocational education, it is necessary to take these education tracks into account when designing informational policy interventions. Introduction We have three key objectives. First, we study the determinants of these subjective completion beliefs. Introduction Specifically, we assess the relative importance that students assign to factors such as past academic performance, personality measures, risk attitudes, family background, and labor market characteristics. Second, we quantify the importance of subjective completion beliefs for three key outcomes: Introduction the intention to invest in education, the actual subsequent investment, and the actual completion of the chosen education track. In addition to estimating the persistence of these early beliefs as predictors of education outcomes, we can compare the actual influence of academic, personality, risk, family, and local labor and education market variables on degree completion vis-à-vis belief formation. Introduction Third, we delve deeper into the relationship between beliefs and education by incorporating subjective probabilities into a sequential model of human capital investment. The model combines the different post-secondary options of the German system with forward-looking behavior and unobserved tastes or preferences for education that are correlated across choices. Introduction Our findings suggest that the most important determinants in the formation of completion beliefs are students' academic ability and personality. In contrast, subjective risk attitudes, family background variables (such as household income or parents' education), and local labor market variables (such as youth unemployment or various indicators of educational demand and supply) are given comparatively less weight by students when they are about to finish secondary education. These determinants have similar effects on intentions to invest in education. However, actual investment and completion depend to a larger extent on family characteristics, the state of the local labor market, and regional supply and demand in the post-secondary education market. When we consider different post-secondary options separately, we find that academic ability is the main driver of subjective beliefs for those choosing a university education. In contrast, the subjective beliefs of those choosing a vocational education seem to be driven more by non-academic factors. We also find that beliefs are most decisive for youths with low academic ability and weak preferences for education, a group that has largely been ignored in the present literature. Introduction Our results point to a strong persistence of early completion beliefs in post-secondary education choices and outcomes; beliefs are statistically significant predictors of intentions to invest, investments, and completion, even after accounting for a large set of background variables. The effects are also substantial in an economic sense. For instance, our models predict that an increase in completion beliefs by one standard deviation would be associated with a reduction in the number of youths who fail to invest in post-secondary education by about 20 to 30 percent, depending on the model specification. Although these estimates are correlational in nature, we show that they are robust in a bounding analysis which takes potential selection on unobservables into account. Introduction Understanding the role of uncertainty in individuals' post-secondary education choices is essential for designing effective education policies. For instance, if students' expectations are misaligned, providing additional information can be a cost-effective measure to enhance education choices and eventual career success (e.g., Bettinger et al., 2012; Jensen, 2010) . Related literature The incorporation of completion probabilities into theories of education choice has a long standing (e.g., Altonji 1993; Comay, Melnik and Pollatschek 1973; Manski 1989 ; see also Bound and Turner 2011 ). This literature emphasizes that " [d] ifferences in dropout probabilities may be more important than differences in ex post payoffs in determining the ex ante return to attending a particular school" (Altonji, 1993, p74) . Empirical approaches based on structural assumptions that distinguish ex ante from ex post returns to education include Carneiro et al. (2003) , Cunha, Heckman and Navarro (2005) , Cunha and Heckman (2007) , and Foley, Gallipoli and Green (2014) . 1 However, there is relatively little empirical work using completion probabilities directly; notable exceptions using predicted college completion probabilities include Hussey and Swinton (2011) , Fossen and Glocker (2014), and Castex (2015) . In general, when studying choice under uncertainty, researchers have to make assumptions about how expectations are formed (Manski, 2004) , and, most commonly, such work relies on rational expectations, e.g., that individuals' predictions are unbiased. To avoid imposing rational expectations, the most widely used alternative, which we follow in this paper, is to apply direct measures of elicited 1 Completion uncertainty also has important consequences beyond aggravating wage uncertainty. For example, various non-pecuniary aspects have been shown to be relevant to educational choice (see Oreopoulos and Salvanes, 2011 , for a recent summary). In order to benefit from them, staying in the chosen educational path and/or completing the degree might be crucial. Studies using elicited subjective beliefs about labor market prospects consistently find the (non-financial) consumption value of education or major-specific unobserved tastes to be the main drivers of educational choices (i.e., Huntington-Klein, 2015a; Wiswall and Zafar, 2015a) . Similar results are found in structural approaches that do not use subjective beliefs. For instance, D 'Haultfoeuille and Maurel (2013) use a sophisticated Roy model and find non-pecuniary aspects to be predominant in educational choice. Such preference-related factors are not affected by pure labor market uncertainty, but they can be affected by completion uncertainty. subjective beliefs (Manski, 2004) . Several studies have shown subjective beliefs to be meaningful in education choice models, and often superior to those constructed by rational expectations models (e.g., Attanasio and Kaufmann, 2014; Huntington-Klein, 2015b; Stinebrickner and Stinebrickner, 2012; Zafar, 2011a ) and a growing literature on college major choice uses such subjective beliefs (e.g., Arcidiacono et al., 2014; Arcidiacono, Hotz and Kang, 2013; Hastings et al., 2016; Huntington-Klein, 2016; Stinebrickner, 2012, 2014a,b; Wiswall and Zafar, 2015a) . In contrast to our paper, most of these studies assess beliefs about returns to or costs of educational choices, rather than about the likelihood of completion. In this respect, our analysis is most closely related to Wiswall and Zafar (2015a) , who also use students' subjective completion beliefs. We contribute to this literature in various dimensions: Our paper is the first to study subjective completion beliefs formed before the end of secondary education in a population survey; we follow students over time until they complete their post-secondary education, and we integrate elicited subjective completion probabilities into a sequential model of educational choice, as motivated by the theoretical literature. Related literature Most of the existing studies on the demand for post-secondary education focus solely on investment, rather than on intentions or completion; 2 we add to the literature by providing evidence on the relationship between subjective completion beliefs and each of these three outcomes in a unifying framework. Related literature Besides beliefs about returns to education, beliefs about one's own academic ability have recently been highlighted as a key factor in college choice (Arcidiacono, Hotz and Kang, 2013; Bond et al., 2016; Bulman, 2015; Stange, 2012; Stinebrickner, 2012, 2014b; Zafar, 2011b) : students learn about their own ability by observing their grade point averages [GPA] , and this updated information on their own ability is a main determinant of college enrollment, college-major choice, and dropout decisions. 3 While belief updating is essential for staying in school, results in this literature also point towards preexisting beliefs as key determinants of later investment and success in education. Related literature For example, Bond et al. (2016, p2) find that belief updating in response to SAT scores is too modest to explain the variation in college application choices, and conclude that there is a substantial amount of "inertia" in college choices, in the sense that they are "predetermined by non-academic factors and preexisting beliefs". In a similar vein, Stinebrickner and Stinebrickner (2014a, p468) stress the importance of timing interventions to inform students about their own ability before college entrance. Related literature We add to the literature by studying the persistence in education choice due to the ex ante perceived probability of completing a post-secondary education, using a representative population survey. 4 Our evidence supports and extends Zafar's presumption that "prior belief[s] [at the start of college] continue[s] to be important. In attempting to understand the choice of college majors, it might be useful to focus on students at earlier stages of their schooling (for example, in high school) and analyze their subjective beliefs" (Zafar, 2011b, p339f) . Related literature 2 Notable exceptions are Turner (2004) , Venti and Wise (1983) and Light and Strayer (2000) for completion. Similarly, the literature on intentions is still comparatively small, although it has been growing recently (e.g., Christofides et al., 2015; Wiswall and Zafar, 2015b; Zachary and Zafar, 2015) . Related literature 3 For evidence on students applying for college, see Bond et al. (2016); Bulman (2015) , and for students enrolled in college, Stinebrickner (2012, 2014a,b); Zafar (2011b) . Related literature 4 Thus far, studies have focused mainly on single institutions rather than representative samples. Exceptions include Milla (2014) and Bond et al. (2016) . Both assess students enrolled in college or who applied for college in representative samples. As opposed to us, they do not use an elicited measure of subjective beliefs, and focus on the selected population applying to, or enrolled in, college. Related literature Our study is also closely related to the literature on non-cognitive determinants of education success (see, Almlund et al., 2011; Borghans et al., 2006) . In particular, we show how subjective completion beliefs relate to the Big Five personality measures, risk attitudes, and locus of control, all of which are now ubiquitous in economic applications (see, for example, Borghans et al., 2006; Caliendo, CobbClark and Uhlendorff, 2015; Dohmen et al., 2010) . Moreover, we use estimates from our structural model to decompose the effects of academic ability, locus of control, and parental background into their direct impact on investment and their indirect impact via subjective completion beliefs. Of special interest to our design is the locus of control. Coleman and DeLeire (2003) hypothesize that students with a more internal locus of control (i.e., students who believe their actions affect their outcomes) have higher subjective beliefs about their own returns to education, which leads them to exert more effort and to invest in their human capital (for a recent review on the locus of control, see CobbClark, 2015) . Our results support the hypothesis that one's locus of control affects education choices via subjective beliefs: We estimate that virtually the entire effect of locus of control acts through subjective beliefs (for other variables, subjective beliefs play only a partial role). Thus, this paper also adds to recent contributions assessing the role of subjective beliefs as a mediator and a potential explanation of educational differentials in parental unemployment (Pinger, 2015) , family background (Keller and Neidhöfer, 2014) , or gender and migration (Tolsma, Need and De Jong, 2010) . Our framework might prove useful in studying the mediating role of subjective beliefs, since it integrates investment in upper and post-secondary education jointly in both reduced-form and structural models. Outline of the paper The remainder of this study proceeds as follows: In Section 2, we describe the institutional features of the education system in Germany and present the data. In Section 3, we assess determinants of subjective completion beliefs, relate these beliefs to education outcomes, and explain how the impact of subjective beliefs varies with selection on observables and unobservables. We develop and estimate a structural model of sequential human capital investment in Section 4. Section 5 concludes our paper by discussing and reviewing our key findings. Outline of the paper 2 Institutional setting, data, and descriptive statistics Institutional setting A simplified version of Germany's education system is depicted in Figure 1 , in which we briefly summarize the system's key features that are relevant to our analysis (more information can be found in Wölfel and Heineck, 2012) . The German education system is characterized by early tracking, which takes place after grade four (elementary school), at age nine to eleven years. 5 Based on grades and teachers' recommendations, the children are tracked into three streams according to their academic 5 With the exceptions of Berlin and Brandenburg, which track after grade six (ages 11 to 13 years). Source: Adaptation and extension of the overview provided by Wölfel and Heineck (2012) . Institutional setting ability. 6 The statistical agency in Germany (Statistisches Bundesamt, 2014, p27) reported that in 2012, ten percent of children were assigned to lower-track schools, 19 percent to intermediate-track schools, and 40 percent to upper-track schools (high schools). The remaining children enrolled in other, so-called comprehensive schools; in these, children are not separated, but follow the same tracking into upper, intermediate and lower streams at a later stage. Institutional setting At the time of entering the survey population, youths-ages 16 to 17 years-are in the midst of deciding whether to enroll in post-secondary education. Students completing lower or intermediary tracks have the opportunity to apply for and start a profession-specific apprenticeship or a vocational education. 7 Institutional setting Although investing directly in an apprenticeship is the most common path, they could also enroll in a consecutive school-track that leads to the university entrance qualification (German: Abitur ), the equivalent of a high school degree. 8 This high school degree is valuable even if students do not plan to attend university; students with high school degrees typically have better chances, compared to their peers who completed a lower track, when applying for highly competitive apprenticeship positions. Institutional setting In fact, some apprenticeship positions are exclusively available to high school graduates. In 2010, 20.9 percent of the newly signed apprenticeship contracts went to students with high school degrees (Statistisches Bundesamt, 2011 . Thus, we model this path separately and refer to it as tertiary apprenticeship. Institutional setting The decision is different for students already enrolled in high school (the upper track). In principle, they could also start an apprenticeship straight away. Alternatively, they could continue with their high school education and, upon completion of high school, either enroll in university or start a tertiary 6 The binding nature of these recommendations varies across states. 7 Students who started an apprenticeship before entering the survey population are excluded from our analyses. However, in 2011, only 10.6 percent started an apprenticeship before the age of 17 years (Statistisches Bundesamt, 2013, p17) . Institutional setting 8 Due to the limited time horizon of our sample we focus on early investment. The possibility of visiting complementary courses that allow students to go to university after apprenticeship completion is not modeled separately. Institutional setting apprenticeship. 9 Yet, their default choice is to continue in high school, as they are already enrolled and do not have to make an active choice to enroll. 10 For instance, of the class of 2011, four percent dropped out without a secondary degree, 17 percent completed the lower track, 36 percent the intermediary track, and 43 percent obtained high school degrees (Statistisches Bundesamt, 2013, p7) . It is important to realize that in Germany, an apprenticeship degree has a high standing and can have a reputation similar to a university degree-especially if the apprenticeship degree is acquired after completing high school. Data sources Our primary data source is the German Socio-Economic Panel [SOEP] , a large-scale representative household panel data set (Wagner et al., 2007 (Wagner et al., , 2008 . We focus on youths, ages 16 to 17 years, who have newly entered the survey population by answering the youth questionnaire between 2000 and 2013. Data sources The SOEP is a yearly household panel that provides a rich set of parental background information, and follows the young adults over time up to 14 years. Additionally, we combine the individual-level data with regional labor market information and educational supply and demand measures based on 96 geographic regions, which we will refer to as Ror (for their German name Raumordnungsregionen). 11 Data sources All regional information is matched according to the individual's residency when answering the youth questionnaire, and lagged by one year to avoid endogeneity or reverse causality. We only use variables assessed in the youth questionnaire as covariates to avoid any biases from conditioning on (future) outcomes. 12 Outcome variables We are interested in a student's intention to invest in post-secondary education, actual investment, and actual completion. We distinguish between not investing, and investing in one of the three most commonly taken educational tracks in Germany: At the time of answering the youth questionnaire, youths can choose not to invest (j = 0), to invest in an apprenticeship directly after completing either the lower or intermediate school track (j = 1), or to continue high school. After completing high school, students can decide whether to invest in an apprenticeship (j = 2) or in a university degree (j = 3). Outcome variables To assess their post-secondary investment intention, we use a self-reported measure elicited in the youth questionnaire: Students are asked to indicate which further educational degree, if any, they 9 Additionally, students could also drop out after completing high school, but this rarely occurs in practice (see also Fossen and Glocker, 2014) . Note that here university subsumes universities of applied sciences. While it would be interesting to consider those separately, we have to leave this to future research due to our current sample size. Outcome variables 10 All regressions include the indicator variable "In high school with 17". We also estimated the regressions of our main Table 3 separately for the two groups and present the results in Appendix, Table D4 . Outcome variables 11 A map of Ror is provided in Appendix, Figure B1 . The data source is INKAR 2013 provided by the German Federal Institute for Research on Building, Urban Affairs and Spatial Development (BBSR, 2013) . For more information, see Pinger (2015) who also uses this additional data source. Moreover, we add the number of universities (higher learning institutions) as a proxy for distance to university provided by the statistical agency of Germany (Statistisches Bundesamt). Outcome variables 12 One of the motivations for the structural model presented in Section 4 is to account for sequential decision-making and to avoid associated biases. Outcome variables plan to complete. The answers correspond to j = 0, 1, 2, 3. 13 To assess actual post-secondary educational investment, we use longitudinal information from the SOEP two years after the individual has answered the youth questionnaire. If, after this period, the individual is observed to have started an apprenticeship, a tertiary apprenticeship, or a university degree, this investment is recorded accordingly; otherwise, we record the individual as not having invested in post-secondary education. Outcome variables Actual post-secondary completion is assessed five years after the individual has answered the youth questionnaire, and constructed analogously to investment intention and actual investment. Subjective completion beliefs Our main variable of interest is subjective completion belief, p i , that is assessed by the following question in the youth questionnaire: Subjective completion beliefs Think about your future in your job and private life: how probable is it, in your opinion, that the following events will occur? Subjective completion beliefs [Please check off a probability on the scale from 0 percent to 100 percent.] You successfully finish your vocational training or university studies? 14 When we incorporate this measure into models of education choice in the next sections, we interpret this belief as the subjective probability of graduation from a post-secondary education track conditional on attendance, i.e., p i ≡ P r i (complete = 1|invest = 1) for student i. You successfully finish your vocational training or university studies? 14 Another interpretation would be that p i elicits the unconditional completion probability entailing also the belief of investing: P r i (complete = 1) = P r i (invest = 1)P r i (complete = 1|invest = 1). However, this is unlikely for at least two reasons. First, the question immediately preceding the subjective completion belief question in the survey directly assesses students' probability of investment in their preferred track, P r i (invest = 1), which raises our confidence that students were thinking about p i as the conditional probability. Second, the two subjective beliefs questions provide bounds on each other that can be used to check their coherency. Using the self-reported P r i (invest = 1), if p i referred to P r i (complete = 1), it could not be larger than P r i (invest = 1). However, this interpretation of p i as the unconditional probability is strongly rejected by the data. 15 In the appendix, we present results where we interpret p i as the unconditional probability, in which case our models need to be modified to control for intentions and the subjective investment probability; our findings are robust to this interpretation (cf. Table D2 ). You successfully finish your vocational training or university studies? 14 13 That means that, for consistency with the other outcome variables, students who want to enroll in an apprenticeship first and then continue with supplementary courses that prepare for university are subsumed into the apprenticeship category. You successfully finish your vocational training or university studies? 14 14 Students could answer on an eleven point scale: 0%, 10%, 20%,. . . , 100%. 15 We acknowledge that this could also be interpreted as students having difficulties to respond consistently to subjective beliefs questions: their responses would simply include some error. However, the magnitude of the inconsistency is so large that it cannot reasonably be interpreted as errors, and clearly points to pi being understood as the conditional probability. Estimation sample As stated before, we exclude from the estimation sample all individuals who have already started an apprenticeship at the time of responding to the youth questionnaire. Moreover, we exclude students with missing information in the core variables: subjective belief, education outcomes, and grade point average [GPA] . All other missing information are included along with corresponding indicator variables for missing information. This selection results in a sample size of 3,610 individual observations. In the longitudinal analysis, we additionally require at least two years of information to assess the end of secondary education and the start of a post-secondary education, which reduces the observations to 2,116. To assess educational completion, we restrict the sample to students who responded for at least five years of data collection, resulting in 1,372 observations. 16 Descriptive statistics In Figure 2 , we plot histograms of subjective beliefs by students' educational intentions. Overall, German students appear to be confident about finishing a post-secondary education, as most report a subjective probability above 50%. The distributions of students intending to invest in paths that first require a high school degree (tertiary apprenticeship and university, Panels C and D) are very similar in shape. Yet, the subjective beliefs of youths who intend to complete a university degree are more concentrated and slightly shifted to the right compared to their high school counterparts who aim for a tertiary apprenticeship position. The mode of the distribution of students who intend to start an apprenticeship without finishing high school lies at 100%. Finally, students with no intention to invest in further education display a much larger variance in their beliefs. The fact that positive beliefs are found among all intended investment levels, including the intention not to invest in further training or education, can be rationalized in a simple expected utility framework where students weight their utility from education by their beliefs about their completion probabilities, and report their intentions based on their highest expected utility. This is also the interpretation we pursue in the following analysis. Descriptive statistics Analogously, Table 1 presents descriptive statistics by intended investment level for our baseline sample. 17 At the bottom of the table, we present the sample shares of the intentions to invest: The modal category is the intended completion of an apprenticeship (36 percent), followed by university studies (32 percent). A substantial share wants to pursue a high school education and an apprenticeship (tertiary apprenticeship, 23 percent), and roughly nine percent do not intend to invest in any professional education. It is reassuring that these sample statistics are consistent with the population statistics presented above (Statistisches Bundesamt, 2013) . Descriptive statistics Individuals who intend to invest in a university education have the highest average confidence level about successfully completing their post-secondary education-80.5 percent. However, all young adults who report an intention to invest in some education exhibit a comparable level of completion beliefs, as opposed to those without educational intentions whose average belief at 69 percent is visibly lower. Descriptive statistics Educational investment intentions are positively related to our measures of academic ability. Average GPA clearly increases with the academic rigour of the intended post-secondary education, increasing from apprenticeship to tertiary apprenticeship to university. 18 Prior track recommendations at the age of ten years seem to be a good indicator for the intentions up to seven years later, which could either be a reflection of effective ability streaming, or a manifestation of students' expectations as a result of early tracking. Interestingly, having no educational intentions occurs in all tracks, and the largest share of students without intentions is found in high school. This could be explained by a default effect, as the survey elicits these intentions at a time when students not enrolled in high school have to make an active decision, as opposed to high school students, who can follow their track and decide after obtaining a high school degree. Descriptive statistics We assess youths' personalities by their locus of control, risk attitudes, and the Big Five personality inventory. 19 Educational intentions are positively associated with the locus of control, which measures the degree to which a person believes her life is under her own control. Among the standard Big Five inventory, intentions increase with openness, agreeableness, and extraversion but are less monotonically related to conscientiousness or neuroticism. Intending to go to university is positively associated with 18 The GPA refers to the student's average of the German and Math grade, which is standardized over the sample population we present in our main results from Table 3 . We also standardize the GPA within school track in Appendix, Table D5 to show that the choice of standardization does not drive the results. We further standardize GPA within federal states to show that different grading levels do not affect the results, see Appendix, Table D6 . 19 We standardize all the principal components of the personality variables (all but risk attitudes, which are assessed by one question only). The locus of control has been developed by Rotter (1966) , the Big Five inventory by Costa and McCrae (1992) and validated in the SOEP version by Hahn, Gottschling and Spinath (2012) . Risk attitudes have been introduced and extensively studied by Dohmen et al. (2011) . Notes: Cells contain sample means and standard deviations in parentheses. Intended investment levels in post-secondary education: no intention (0), apprenticeship (1), tertiary apprenticeship (2), or university (3). Individual characteristics are assessed at the time of answering the youth questionnaire (at age 17); p denotes the subjective completion belief; definitions of other variables are given in Appendix, Table B1 . more risk-loving attitudes, but the differentials across categories are small. Descriptive statistics Individual and family backgrounds are captured by individual's gender, number of siblings, whether they are second-generation immigrants (youths whose parents were both born in a foreign country), whether at least one parent has a college education, whether at least one parent is currently unemployed, and the logarithm of net household income. Intentions tend to be higher among males, children from smaller families, natives, youths with employed and college-educated parents, and youths with a higher household income. Descriptive statistics The regional labor and education market (Ror) characteristics relevant for the students' choice sets are a mix of (exogenous) educational supply and demand shifters. We use the cyclical component of the youth unemployment rate, as well as the number of apprenticeship positions, students, high school graduates, and universities in the region. Intentions to invest in more academically ambitious tracks are related to a more favorable local labor and education market. However, youths with no investment intentions are not faced with clearly worse or better labor/education market conditions. Descriptive statistics In the following analysis, we will also account for region and year of first questioning (which is almost identical to students' age) fixed effects. 20 Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs To analyze how the variables we discussed in the previous section relate to subjective completion beliefs, we estimate OLS regressions of the model Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs where i indexes individuals, p i is subjective completion belief, x i are varying sets of explanatory variables with corresponding vector of coefficients β p , and υ i is an unobserved error term. The estimates are presented in Table 2 . 21 In Column (1), the beliefs are explained solely by academic ability, year, and region fixed effects. In Column (2), we add the personality measures; in Column (3), individual and family background characteristics; and, finally, in Column (4), regional labor market measures. Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs The explained variation, as measured by the adjusted R 2 , increases substantially only when academic ability and personality measures are included in the regressions, but is relatively unaffected when 20 For some of the regressions, the number of students in a state is too small. To obtain consistent samples, we use a broader grouping by dividing Germany into the following five regions (and an indicator for missing values). Southern Germany: Baden-Wuerttemberg, Bavaria; Eastern Germany: Berlin, Brandenburg, Saxony, Saxony-Anhalt, Mecklenburg-Western Pomerania; Central Germany: Hesse, Thuringia; Western Germany: North Rhine-Westphalia, Rhineland-Palatinate, Saarland; Northern Germany: Bremen, Hamburg, Lower Saxony, Schleswig-Holstein. We present analogous results of our main Table 3 in Appendix, Table D5 were we use federal states fixed effects, as the jurisdiction over educational policies is at the federal state level. The results are very similar. Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs 21 Note that our dependent variable is a fraction. In the Appendix, Table D1 , we present fractional response regressions (as in Wooldridge, 1996, 2008) . The results are virtually indistinguishable from the OLS estimates. adding individual and family characteristics or regional youth labor market variables. 22 The joint significance tests for subsets of variables reported at the bottom panel of the table paint a similar picture: Academic ability and personality characteristics are highly significant across all regressions, and their associated F statistics in Column (4) are 16.5 and 26.6, respectively. Individual and family characteristics are jointly significant, but at the five percent level with an F statistic of 2.5. The local youth labor market characteristics are insignificant and their F statistic is just 0.3. Since these coefficient estimates are neither jointly nor individually significant, we omitted them from the table. Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs Thus, Table 2 indicates that youths' subjective completion probabilities reflect mainly their past academic record and personality traits. Youths' socio-economic family backgrounds are only mildly related to their completion beliefs, and the state of the local youth labor and education market seems not to affect their beliefs at this stage. Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs Looking at the determinants individually, all academic ability variables are consistently positive and significant. Somewhat surprisingly, already being enrolled in high school does not alter students' subjective completion beliefs, which might be a result of conditioning on prior track recommendations. Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs As hypothesized by Coleman and DeLeire (2003) , the locus of control is a very important determinant of subjective completion beliefs throughout the regressions, both in magnitude and significance. 23 Subjective completion beliefs and educational outcomes 3.1 Determinants of subjective completion beliefs Risk attitudes do not matter once family characteristics are accounted for. Our regressions indicate that among the Big Five measures of personality, conscientiousness is the most influential in shaping subjective beliefs. This finding highlights the importance of conscientiousness for education outcomes (see, inter alia, Borghans, Meijers and Ter Weel, 2006). While we find little evidence that openness or neuroticism influence completion beliefs, the effect of extraversion is about half as large as conscientiousness, and the effect of agreeableness, in turn, is about half as large as extraversion. On average, females seem to have lower subjective completion beliefs. This estimate is, however, only marginally significant (at least conditional on personality and academic ability). Household income is positively and significantly related to subjective completion beliefs. The remaining estimated coefficients are insignificant and mostly very small in magnitude. Education choice with subjective completion beliefs In this section, we turn to our central question of how subjective completion beliefs measured at age 17 years relate to educational intentions, investments, and, finally, degree attainment. To fix ideas, let individual i's utility u ij from choosing an uncertain post-secondary educational track (j ≥ 1) be Education choice with subjective completion beliefs where p i is subjective completion belief, µ ij andμ ij are the utilities from completing and dropping out, respectively, and ν ij is a utility component unaffected by completion. 24 The associated expected utility is Education choice with subjective completion beliefs Hence, adolescents get a baseline utility from attending a particular education trackμ ij . The subjective completion belief p i weights the utility differential between completing and not completing up or down. Education choice with subjective completion beliefs Since not investing in education does not involve completion uncertainty, the utility is simply Education choice with subjective completion beliefs In this section, we assess the investment in any post-secondary education U ij = U i for j ≥ 1, against not investing U i0 , which directly relates to the survey question of p i . A student prefers to invest in Education choice with subjective completion beliefs where, by standard normalization, µ i0 = 0. Taking averages across individuals, adding covariates that measure observed preferences and skills (i.e., 25 and assuming that ϵ i = ε i − ε i0 follows a standard normal distribution, we estimate probit models of the form Education choice with subjective completion beliefs We consider three different binary outcomes d i : First, whether a student intends to complete any post-secondary education, which is measured concurrently with subjective beliefs at age 17 years. Education choice with subjective completion beliefs Then, the expectation of (5) gives P (U i > 0), so that α = µ −μ. A similar interpretation is possible when d i represents the second outcome -a student actually invests in any post-secondary education, Education choice with subjective completion beliefs i.e., whether the student started an apprenticeship, a tertiary apprenticeship, or a university degree. Education choice with subjective completion beliefs This event can be a few months or a few years away. It then corresponds to the revealed preferences of actual investment in post-secondary education. Finally, the interpretation is somewhat different when d i stands for the third outcome-the completion of an apprenticeship or a university degree, an event that is at least a couple of years away. In this case, α gives an indication of the student's ability to incorporate information beyond that in x i into his or her forecast of d i = 1, an interpretation of subjective beliefs along the lines of Finkelstein and McGarry (2006) . The students process all their available information in forming their beliefs, meaning that relevant information over and above their subjective beliefs are either not used, not used efficiently, or influence the decision through a different channel from subjective completion uncertainty. Table 3 contains the estimation results. Panels (A) to (C) present the probit regressions of subjective completion beliefs on different education outcomes. In each panel, we report the estimated coefficients, robust standard errors (in parentheses), average marginal effects (in squared brackets), pseudo R 2 n for models estimated without and with p i , and sample statistics for the respective subsamples. Columns Table entries are coefficients (robust standard errors in parentheses; average marginal effects in squared brackets), from probit regressions (1)- (4) and probit regressions with endogenous explanatory variable (5)- (7). Variable p denotes the subjective completion beliefs. Other covariates: in column (1) in high school, region and time fixed effects, (2) adds academic, (3) adds personality, (4) to (7) family background, individual, and local labor market variables. In probit eev regressions the correlation between errors, ρ, is restricted as indicated in column headers: (5)ρ o , the estimated selection-on-observables; (6) 0.10; (7) (1) to (4) contain simple probit estimates of education outcomes on subjective beliefs and varying sets of covariates: The specification in Column (1) contains, apart from p i , only an indicator of whether the student is currently in high school, as well as region and year fixed effects. Thus, in this specification, any other variable acts on the intention to invest in education through its effect on p i . The next Columns progressively control for sets of academic (Column 2), personality (Column 3), and family and labor market variables (Column 4). We turn to the results in Columns (5)- (7) in the next section. Education choice with subjective completion beliefs Panel (A) contains results corresponding to the intention to invest in any post-secondary education. Education choice with subjective completion beliefs The uncertainty measured by p i is an important predictor for investment intentions: The coefficients on subjective beliefs are large and highly significant throughout the probit regressions. The average marginal effects are economically relevant. In the most parsimonious specification, increasing subjective beliefs in the population by one standard deviation increases intentions to invest in post-secondary education by 2.8 percentage points (0.140 × 0.2), which is quite large relative to the nine percent of students who do not intend to invest in a post-secondary education. These figures change little if we condition on increasing sets of background characteristics commonly considered in the literature. A one standard deviation change results in an increase of 2.1 percentage points using all background characteristics. Moreover, the increase in the R 2 when including subjective beliefs in the regression in Column (1) (from 2.9 to 4.9 precent) is similar to the increase when adding the full sets of both academic and personality variables (from 2.9 to 4.8 precent). Thus, the predictive power of subjective beliefs is equal to the one of academic and personality variables combined. In sum, subjective beliefs are strongly related to intended behavior (a result consistent with, for example, Huntington-Klein, 2015b). Education choice with subjective completion beliefs Several explanations can account for this contemporaneous correlation between beliefs and intentions. Education choice with subjective completion beliefs Therefore, we examine whether the link from beliefs to intention carries over to revealed preferences in actual investments at least two years later. In Panel (B), our dependent variable is now an indicator that equals one if the student started any post-secondary education. Compared to the sample used in (A), the sample in (B) only excludes students who are still in school and students who have not completed any subsequent questionnaires two years after the youth questionnaire at age 17. The average marginal effects are somewhat smaller in absolute value than for the intentions, ranging from 1.4 to 0.9 percentage points for a one standard deviation increase in subjective beliefs. The effects are of similar magnitude as before when considered relative to the group who fails to invest in postsecondary education, which comprises about four percent of youths: A one standard deviation increase in p i is associated with a predicted reduction of this group by between 34 percent to 22 percent. 26 This result suggests that subjective completion beliefs not only drive hypothetical, intended investment, but have real behavioral consequences. Compared to the previous results on students' intentions, the set of family background and labor market variables explain a somewhat larger fraction of the completion belief effect and exhibit more explanatory power. Education choice with subjective completion beliefs Finally, we compare how subjective completion beliefs relate to actual completion at least five years later. This correlation can be interpreted as the degree to which students can predict their future outcomes. 27 The estimation results are given in Panels (C). The average completion rate is roughly 54 percent. Unconditionally, a one standard deviation increase in subjective beliefs increases completion rates in the population by 3.4 percentage points. This effect shrinks to 3.0 percentage points when including the full set of individual, family, and regional characteristics. Thus, a one standard deviation increase in p i is predicted to reduce the group of students dropping out of post-secondary education by about 6.5 percent. For completing a post-secondary education, the explanatory power of family background and labor market characteristics is substantial relative to that of the other covariates. Education choice with subjective completion beliefs For instance, relative to a baseline of only fixed effects, the inclusion of p i , academic, and personality variables adds successively about three, three, and eight percent to the R 2 , while family and labor market variables add another 18 percent. This result stands in contrast to the results from the previous section, and suggests that students do not optimally take thier family and local labor market information into account when forming their beliefs. Robustness and Altonji-Elder-Taber bounds Taken together, the results show that subjective completion beliefs formed during secondary education are predictive over a long time horizon, which is consistent with substantial inertia due to preexisting beliefs in these choices, as was hypothesized in the studies cited above. Moreover, these early subjective completion beliefs are predictive even after accounting for a comprehensive set of previously identified, important characteristics. In the appendix, we present further results showing the robustness of these findings across a number of alternative specifications. Robustness and Altonji-Elder-Taber bounds First, we show that the results change little when using only students who expressed positive intentions, and additionally control for the subjective investment probability (cf. Table D2 ). We are therefore confident that the measure corresponds well to the conditional subjective probability of completing an educational degree. Next, we show that the results are robust to dichotomizing the subjective belief to a dummy variable (cf . Table D3 ), thus accounting for potential non-linearity, as discussed in Pinger (2015) . Robustness and Altonji-Elder-Taber bounds A key determinant is GPA, which the literature has found to be the main driver of learning about own ability among college students. To show that our results do not hinge on the measure of GPA we use, we assess various alternative standardizations and specifications. First, we standardize GPA within high schools, since grades might be based on different standards across school tracks (Table D5) , and within federal states, which accounts for potentially different grading standards across state education systems (Table D6) . Further, we use a fifth-order polynomial in GPA to demonstrate that the effects of beliefs are not spuriously picking up non-linearities in academic ability (Table D6) . Additionally, we use federal state dummies (Table D6) instead of the region dummies used in the main specification 27 Since completing a program and graduating takes some time, we consider only students whom we see at least five years after they have taken the youth questionnaire. Students who were interviewed in earlier years are more likely to have completed their degrees simply by virtue of having been in the sample for a longer period of time. However, this mechanism is captured by the year fixed effects, and is therefore unlikely to bias our results. A second concern is that some of the observations are censored: As of the time we observe them, some students have not yet completed their degree, but they might do so in the future. In this sense, our results should be interpreted as representing the average effect of completion beliefs on completion within a given time frame. Robustness and Altonji-Elder-Taber bounds (cf. footnote 20). Finally, we present separate estimations for students enrolled in high school when answering the youth questionnaire to account in a completely flexible way for the different default choices discussed above (Table D4) . Robustness and Altonji-Elder-Taber bounds Although the results in Columns (1)-(4) in Table 3 show that the coefficient on p i does not change much even after including a very large set of potential confounders, a remaining concern might be that the effect of subjective beliefs, α, is confounded with further unobserved variables, i.e., in terms of our empirical model, that there are unobserved variables in ϵ i that are correlated with p i . In order to show that our results are robust to this selection on unobservables, we go one step further and use a bounding strategy introduced by Altonji, Elder and Taber (2005a,b, 2008, hereafter, AET) for the coefficients by taking into account potential selection on unobservable tastes and preferences for education when estimating (5). We show how α varies if there is a correlation ρ between unobservable components of d i and p i , and we present a suggestive upper bound under the assumption that the correlation between observables is informative about the correlation between unobservables. In contrast to AET, our main variable is a fraction rather than an indicator. Instead of estimating a bivariate probit, we therefore estimate
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