Does Mental Productivity Decline with Age? Evidence from Chess Players* by Marco Bertoni (Padova and LSE) Giorgio Brunello (Padova, CESifo, IZA and ROA) Lorenzo Rocco (Padova) ** PRELIMINARY DRAFT ‐ DO NOT QUOTE WITHOUT PERMISSION Abstract: We use data on international chess tournaments to study the relationship between age and mental productivity in a brain‐intensive profession. Using chess has the advantage that both individual productivity and (relative) ability can be measured. Furthermore, chess players carry out an individual task, reducing concerns due to aggregation or team effects. We show that selective attrition is a relevant phenomenon for chess professionals, as less able players tend to leave chess in the earliest phases of their career. This imparts a positive bias on the age – productivity profile in the first 10 years in the profession. Selection is much less of a problem later on. We propose two alternative methods to correct for the effects of selection on productivity: re‐population of the selected sample by imputation, and using a measure of ability to control for un‐observables driving selection. Both methods produce similar results. Productivity peaks in the early to mid twenties and smoothly declines thereafter. We estimate that professional chess players aged 50 are less productive than players in their mid 20s by 7 to 8 percent. Keywords: aging, productivity, mental ability. JEL codes: D83, J14, J24. *
The authors are grateful to the audience at a seminar in Padova for comments and suggestions. We also thank Michele Bertoni for technical help with data collection. Financial support from the University of Padova (Grant CPDA093857 – Percorsi Lavorativi e Invecchiamento Attivo) is gratefully acknowledged. All errors are our own. ** Corresponding author. Address: Department of Economics and Management “M. Fanno”. Via del Santo, 33 – 35123 Padova ‐IT. E‐mail: [email protected] 1 Introduction There is a broad perception that mental ability declines with age, and not just for humans1. Unless experience and knowledge can fully compensate the decline in ability, productivity is also bound to decline. In many developed countries, population is ageing. If individual productivity declines with age, overall productivity will also decline, with important macroeconomic implications. In spite of the important implications for modern economies, surprisingly little is known about the relationship between age and productivity, and the little we know is not pointing unambiguously in the same direction. In his review of the empirical literature, Skirbekk, 2003, concludes that the evidence suggests that productivity follows an inverted U‐shaped profile, with significant decreases taking place from around age 502. Van Ours, 2009, on the other hand, finds that while physical productivity does decline after age 40, mental productivity – measured by publishing in economics journals – does not decline even after age 50. Borsch‐Supan, 2007, uses data on production workers of a large German car manufacturer and concludes that productivity does not decline at least up to age 60. Pekkarinen and Uusitalo, 2012, look at the population of finnish blue‐collar employees and use piece‐rate wages as proxies for output: their findings confirm that labour productivity stays roughly constant after age 40. Measuring the effects of age on productivity is difficult for a number of reasons. First, it is difficult to find reliable measures of individual productivity. Second, in many jobs individual productivity should include also the effects on the productivity of others, either because of knowledge spillovers or because some jobs involve a relevant team component. Third, the relationship between age and productivity in observed samples tends to be affected by selection. If more productive workers are more likely to stay longer in their jobs, selection may induce a spurious positive correlation between age and productivity. In this paper, we address these difficulties by studying the relationship between age and mental productivity for professional chess players. Focusing on chess players has a number of important advantages. First, we can compute a quality adjusted measure of individual productivity by looking at wins and losses in international tournaments, weighting each result with the measured talent of the opponent. Second, since chess players do not work in teams but compete against each other, the spillovers on the productivity of other players is likely to be minimal. Third, the chess profession has developed over time an accurate measure of individual (relative) ability in the game of chess, the ELO score, that can be used to control for selection. 1
See The Economist, 2004. Recent contributions in this area that use individual productivity data include Weinberg and Galenson, 2005 and Castellucci, Pica and Padula, 2010. 2
2 We have access to a large dataset which contains information both on all games played in international tournaments from 2008 to 2011 and on the evolution of individual ELO scores from 2001 to 2011. We use these data to document that selection into and out of professional chess varies with age and ability, and is particularly pronounced in the early part of the individual career for less able players. When selection on the basis of ability is age variant, both ordinary least squares and standard fixed effects methods are inadequate to estimate the underlying relationship between age and productivity. To deal with this problem, we propose two alternative estimation strategies: 1) we use the available measure of ability in our data to measure unobserved ability and control for endogenous selection; 2) we use imputation techniques to re‐populate the selected sample and estimate the relationship between age and productivity in the re‐populated sample using median regressions, using the property that median productivity depends on the relative position and not on the absolute value of individual productivity. We show that these two alternative methods produce similar results. In either case, controlling for selection reduces the substantial growth of relative productivity between age 15 and age 25 by about 10 percent. This suggests that observed productivity growth during the first period of professional chess is amplified by the fact that less talented players leave the game relatively early. Controlling for selection has instead minor effects on the second and longer part of professional life, from age 25 to age 60. Independently of whether we control or not for selection, estimated productivity declines rather smoothly. We estimate that productivity at age 50 is between 7 and 9 percent lower than productivity at age 25 when we control for selection, and between 6 and 10 percent lower when we do not take endogenous selection into account. Several studies (see Skirbekk, 2003) have shown that the decline of mental abilities from early adulthood is a universal phenomenon. Unless the acquisition of skills and experience on the job outweighs this decline, productivity at cognitive tasks is likely to fall with age. Our evidence from a brain – intensive activity – professional chess – shows such compensation is unlikely to completely offset the decline in numerical and reasoning abilities. The paper is organized as follows. In Section 1 we consider measures of ability and productivity for chess players. Section 2 introduces the data and Section 3 presents some evidence on selection patterns. Section 4 presents out proposed empirical approaches to deal with endogenous selection and Section 5 presents our results. Conclusions follow. 1. Measures of Ability and Productivity for Professional Chess Players Ranking players has been a critical issue in chess until the 1960s, when the ELO rating system was introduced by FIDE, the International Chess Federation. This system was developed by the Hungarian mathematician Arpad Elo and is based on a Thurstonian model for paired comparisons (see 3 Thurstone, 1927). In this section, we argue that ELO is not a measure of individual productivity but rather an indicator of individual (relative) ability in the game of chess. In the ELO system, the latent ability of player i,  i , is assumed to be normally distributed with mean si and standard deviation arbitrarily set at 2003. Let the outcome of a match between players i and j be the random variable zij  i   j . Player i wins if zij  0 . With independent abilities, the  si  s j 
 , where Φ is the cumulative distribution function of a   
probability of winning is pij  
standard normal random variable and σ=200√24. The expected ability si of player i is estimated by using the outcomes of the games she plays. Players are initially classified as unrated5. Starting from their first official ELO score, si 0 , the score after game g is obtained as follows sig 1  sig  K ( wij  pij ) [1] where wij is equal to one if player i wins, to 0.5 if she draws and to zero if she loses the match, pij is the expected winning probability of player i against player j, and K is a scale factor which weights the importance of a single game with respect to her entire previous career. This weight declines with the number of games played and with the ELO score6. Equation (1) is an updating rule which adjusts the ELO score when actual performance in the game differs from expected performance. When players’ current ELO perfectly predicts pij , no further update occurs. Thus, by construction, players’ ELO tends to converge to a stationary state, a feature that can be empirically observed, for instance, in the rating progress charts reported by FIDE for the first 100 top players. After an initial increase, the ELO profile flattens out. Only at its stationary state, ELO represents an accurate measure of players relative ability. Since only unexpected wins and losses matter in the updating mechanism, ELO cannot be considered a measure of productivity in the game of chess. The latter needs to depend on total rather 3
The normality assumption is based on observational data collected by Arpad Elo on the distribution of individual chess performance (see Gransmark and Gërdes, 2010). Currently, FIDE prefers to use a logistic distribution. 4
For example, consider two players with si‐sj=200. In this case, the likelihoods that players i and j win are Ф(200/200√2)=0.76 and 0.24 respectively. 5
The results of their first games and the ELO score of their opponents determine a provisional rating. The following conditions are required to obtain such rating: (see FIDE, 2012): 1) having played in at least one official FIDE tournament; 2) having completed a minimum of nine games against rated players and having scored at least one point against them (i.e., having won a match or having drawn two); 3) the initial score ought to be above a minimum rating floor – equivalent to 1400 ELO points for players in our sample, who obtained their first rating before 2009. 6
In practice, K = 30 for a player who has completed less than 30 games, K = 15 for players with a score lower than 2400 and K = 10 once a player's rating reaches 2400 and she has completed at least 30 games (see Glickman, 1995, for details). Using the example in footnote 2, if player i wins, her ELO score increases by 0.24*K, while if she loses her ELO decreases by ‐0.76*K. 4 than unexpected wins and losses. To illustrate, a player can be very productive in terms of winning several games per year and yet experience no change in ELO if these wins are expected7. We therefore distinguish between ELO and productivity Y: the former is an estimate of relative ability, which is refined whenever the player performs better or worse than expected and it is more accurate when it has reached its stationary state; the latter can be defined as the weighted outcome per game defined as Git
Yit 
 [ I ( win
j 1
it
) * ELO j 
1
I ( drawij ) * ELO j ]
2
Git
[2] where Git is the number of games played in international tournaments by player i in year t and I ( winit ) and I (drawit ) are dummies equal to 1 when either a win or a loss occurs. Each win has weight equal to 1, each draw is weighted 0.5 and each loss has zero weight. This measure of productivity is quality adjusted because each win or draw is weighted with the relative quality of the opponent. Since the weighted sum of wins and losses is divided by the number of games played, Yit is productivity per match8. 2. The Data We use data on all official FIDE (World Chess Federation) tournaments played worldwide between 2008 and 2011. These data are downloaded from the FIDE online archive9. Each tournament record reports the results of all games played by every participant (wins, losses or draws), their ELO score at the beginning of the tournament and at the date of the tournament. We merge these data with the official FIDE lists of rated players, which include quarterly information on the ELO scores of active players, their national federation, date of birth and gender. These lists are available since the early 2000. 7
Furthermore, two players with the same initial ELO but different K factors (i.e. different experience) have different ELO adjustments even if their game results are the same, making the use of ELO as a measure of productivity even more problematic. 8
Our weighting system implies that playing two games against players of a given strength and winning both is equivalent to playing two games against opponents twice as strong and winning only one game. It also implies that winning one game against a player with ELO score x yields more in terms of productivity than drawing one game against a player with ELO equal to 2*(x‐ε). 9
As of December 2012, the web address of this archive is http://ratings.fide.com. 5 Our initial sample consists of all male FIDE rated players born between 1948 and 1993 who were listed by 2008 and have played in at least one FIDE tournament between 2008 and 2011. From this sample, we drop players who obtained an official rating for the first time in 2008 and have played only in 2008, as we want to avoid considering “casual” players. For the remaining players, we only consider the outcomes of games played against rated players, both because we do not have a measure of ability for unrated opponents and because games against these opponents do not count for rating10. We also drop those players belonging to national federations with less than 30 affiliates. Our final sample consists of 40,545 players aged between 15 and 60 who are listed in 2008 and remain in the FIDE list from a minimum of 1 to a maximum of 3 years11, and of 140,074 observations. Table 1 presents descriptive statistics on productivity, ELO in 2006, age and number of games played. Age ranges from 15 to 60 and has an average of 38.09. Mean ELO score in 2006 ranges between 1,402 and 2,806 and averages at 2,07212. The annual number of games of active players range from 1 to 289 and averages at 17.45, and measured annual productivity ranges from 0 (no wins or draws in a year) to 2,551, with an average of 972. Figure 1 shows the distribution of annual productivity: there is a peak at zero (3.6% of observations), due to players who have never won or drawn a game in a single year, and an upper tail with few players having very high productivity. Our dataset also includes two variables at the federation‐by‐year level: the GDP per capita in real PPP 2005 (thousand) dollars and the number of internet users per 100 inhabitants. Both variables are drawn from the World Bank World Development Indicators. 3. Some evidence on the selection process in our data Since professional players enter and exit the FIDE lists every year, our raw data are affected by endogenous selection. In this section, we provide some evidence on selection, using the longitudinal information contained in the FIDE lists. We consider the pool of active players who were present in the lists in 2001 and track their activity and ELO until 2011. By so doing, we are able to follow each cohort for a maximum of ten years and to document selection over this relatively long time span. We distinguish between “stayers”, who were included in the FIDE list in 2001 and were active players between 2009 and 2011, and “dropouts”, who were not in the list between 2009 and 2011. For each group, we normalize ELO in 2001 with the age‐specific median value in the same year. Figure 2 plots for each age group the ability distribution of stayers and dropouts and shows two stylized facts: a) the distribution of ELO for stayers exhibits first order stochastic dominance with respect to the 10
In the few cases where annual productivity is missing in either 2009 or 2010 but not in 2008 and 2011, we estimate missing values by interpolation. 11
The number of players enrolled in the lists in 2008 is 40,545,, 37,396 of whom are still present in 2009, 33,475 in 2010 and 28,658 in 2011. 12
Chess grandmasters typically have ELO scores above 2500. 6 distribution for dropouts; b) stochastic dominance is clear for the younger age groups (15‐17 and 25‐
27) and tends to disappear for the older groups. This suggests that selection is not independent of age and ability, with weak players leaving chess at the early stages of their career. Regression analysis confirms these findings. Table 2 reports the estimates of a linear probability model where the dependent variable is a dummy equal to one if the individual dropouts of professional chess and to zero otherwise, and the controls include talent in 2001, measured with the log of the 2001 ELO score, age in 2001 and the interaction between age and talent. We also control for country fixed effects and use robust standard errors. The estimates confirm our expectations: less talented and younger players are more likely to drop out. The interaction term is positive, meaning that selection on ability weakens with age. When we split the sample between players aged up to 25 and more than 25 years in 2001 we see that the negative effect of age on the probability of dropping out is driven mainly by younger players, and that selection on ability is stronger for younger players. This reinforces the view that positive selection into the pool of active players is stronger for younger players. 4. Selection and the Age Productivity Gradient Let the relationship between productivity Y and age A in the population of professional chess players be given by D
ln yit   0    d Ait   i Ait   x X it   i   it (3) d
d 1
where y=1+Y to take into account that Y can be equal to zero, X is a vector of exogenous controls, α is individual innate ability and ε is a random error. Productivity depends on an age polynomial of order d. Moreover the effect of age on productivity is heterogeneous across players and depends on ability. Innate ability is normalized to have zero mean and it is assumed to be orthogonal to age in the population. Our assumptions on the distribution of ability ensure that the conditional mean of (3) in the population is given by D
E[ln yit | Ait , X it ]   0    d Ait   x X it (5) d
d 1
7 and that we can estimate the relationship between age and productivity using ordinary least squares. The conditional mean in the population and in the observed sample do not coincide, however, when individual players select in and out of the sample in a non‐random way. In the case of professional chess players, we expect that the decision to stay or leave the FIDE lists depends on individual ability and age. This implies that D
E (ln yit | Ait , X it )   0    d Ait   x X it  E ( i | Ait , X it )(1  Ait ) (6) d
d 1
and the conditional expectation of ability depends on players’ age. Since ability and age are correlated, failure to control for ability imparts a bias in the ordinary least squares estimates of the relationship between productivity and age. This bias cannot be removed using a fixed effects estimator – as in the case of Castellucci et al (2010) – because the within‐
transformation would only remove the innate component of ability. To fully remove ability in the selected sample, one would need to apply within‐player transformations to the growth rate of productivity rather than to its level, as done for instance by Pischke (2001), at the price of increasing the noise to signal ratio in the data. In this paper, we propose two empirical strategies to estimate [3] in the selected sample. In the first strategy, we measure individual ability using the information contained in the ELO score. In the second strategy, we use an imputation procedure which re‐populates the selected sample and by so doing re‐introduces the orthogonality between ability and age. We describe each strategy in turn. 4.1. Measuring Ability We have argued in Section 1 of the paper that the ELO score is a measure of individual (relative) ability, which changes whenever players attain unexpected wins and losses in their international games. Moreover, the updating process described in equation (1) is faster at the beginning of players career, i.e. typically, at early age. Thus ELO is just an estimate of players’ ability, less precise at early age. In our sample of players, we measure productivity from 2008 to 2011. We use the pre‐sample ~
information on the ELO score in 2006 to extract an estimate of ability ,  , by taking the residuals of the regression of ELO on age dummies and additional controls. We then estimate Eq. (3) by replacing α ~
with  13 and compute the correct standard errors by bootstrapping. 4.2 Imputation 13
We believe that this strategy is superior to the one that proxies ability with the ELO score in 2006. This measure varies only with age in the cross section, and ignores that ability can also vary for each individual over time. It is also superior, we believe, to using lagged measures of the ELO score, because reverse causality problems are less relevant. 8 The purpose of imputation is to reconstruct the population of chess players that would have been observed at time t=0 in the absence of endogenous attrition, and by so doing to restore the independence between age and ability required to obtain consistent estimates of equation (3). Our imputation method assumes that the self‐selection process depends on age but does not vary with the cohort of birth. In other words, different cohorts of chess players share the same pattern of selection, which varies with age. By virtue of this assumption, we reconstruct the population of older cohorts by using the patterns of self‐selection observed in the younger cohorts. For our imputation strategy we use the property that median productivity is invariant with respect to the absolute value of individual productivity of players above or below the median one. Intuitively, as long as we can establish whether the productivity of players who dropped out of chess lies below or above the median for their age group, this property permits us to impute to these players, whom we add back to the sample, an arbitrary productivity level and nonetheless obtain consistent estimates for median productivity. By so doing, we closely follow the imputation method described by Olivetti and Petrongolo (2008). Since the size of our sample is relatively small, we define cohorts as three contiguous years of birth and group players accordingly. Using the year 2008 as our reference, our youngest cohort ‐ c15 ‐ is aged 15 to 17 and our oldest cohort ‐ c60 ‐ is aged 60 to 62. The imputation strategy requires the following two assumptions: Assumption A1 (anchoring). Cohort c15 at time t=0 is not self‐selected. Therefore, E[ i | c15 , t  0]  Med[ i | c15 , t  0]  0 . Assumption A2 (stationarity). Selection depends on age and not on cohort. Let (log) productivity net of the effect of the exogenous controls in vector X be y ict , where c is the cohort. Then Med ( y itc | c  k , t  3)  Med ( y itc | c  k  3, t  0) for all values of k. Since each cohort consists of three contiguous years of birth, the implementation of Assumption 2 requires that we only retain the initial and final year in the sample (2008 and 2001). The imputation procedure consists of five steps, which are described below: 1)
We perform median regressions of yitc on age, cohort, country dummies and country by time effects and from the estimates we derived y itc . 9 2)
We consider the first cohort c15 at time t=0 and identify the players belonging to this cohort who drop out from the sample at any time between t=0 and t=314. 3)
We impute to each dropout identified in step 2) a productivity value that is arbitrarily larger (smaller) than median productivity computed in t=3 if y i 0c15  Med( y i 0c15 ) (
y i 0c15  Med( y i 0c15 ) . 4)
We compute median productivity at the end of the observation period (t=3) – or m3,15  Med( y i 3c15 ) ‐ on the repopulated sample. 5)
We compute m0,18  Med( y i 0c18 ) . If m0,18  m3,15 , we add to the cohort as many players (bots) with an arbitrary low productivity as required to re‐establish m0,18  m3,15 , which holds true under assumption A2. We add bots with arbitrarily high productivity if m0,18  m3,15 .This step repopulates cohort c18 at time t=0 and corrects for the self‐
selection affecting this cohort prior to t=0. The repopulated cohort c18 is used as the basis for the next iteration. 6)
We repeat steps 2‐5 for each cohort until the last, c60 . Under our assumptions, the outcome of the imputation procedure is a repopulated sample where Med ( i )  0 for all cohorts. To further illustrate this, notice that in step 3 the within‐cohort imputation preserves exactly the same composition of the cohort as it was at time t=0, because the players who dropped out between time t=0 and t=3 have been re‐introduced into the sample. Therefore, at the end of step 3 the cohort c15 at time t=3 is not self‐selected, because of Assumption A1. The median productivity of this re‐populated cohort is the counterfactual median that would have been observed had cohort c15 not suffered attrition between time t=0 and t=3. This median is used as the counterfactual median productivity of cohort c18 at time t=0 had this cohort not suffered any attrition before time t=0. 14
We allocate the dropouts with a productivity at time t=0 close to the median either above or below the median at time t=3 according to the following scheme: we compute the distribution of D 
y i 0c15  Med( y i 0c15 ) for all the players with D>0 and the distribution of D  Med(y i 0 c )  y i 0 c for the players with D '  0 . We impute 15
15
'
to dropouts a productivity that is arbitrarily large (small) if D  pct10 ( D ) ( D '  pct10 ( D ' ) ), where pct(10) is the 10th percentile. We impute an arbitrarily large value of productivity with probability P (1‐p) and an arbitrarily low value of productivity with probability 1‐P (p) if D  pct10 ( D ) ( D '  pct10 ( D ' ) ), where P
D'
D
and P ' 
. max(D )
max( D ' )
10 The actual cohort c18 at time t=0 is re‐populated with bots until its median productivity meets the counterfactual (step 5). Importantly, the bots added at time t=0 are assigned a arbitrarily large or small productivity depending on whether they ought to be located above or below the median player of cohort c18 at time t=0. The bots are considered as dropouts between t=0 and t=3 and treated accordingly. Finally, all dropouts (actual dropouts and bots) of cohort c18 are used to compute the median productivity that cohort c18 would have displayed at time t=3 if no attrition had occurred before time t=3. This value is the counterfactual productivity for the next cohort, and the previous process is iterated until the last cohort has been adjusted. The key assumption in this imputation method is that median productivity at a given age is invariant across cohorts. As suggested by equation (3), this assumption also implies that, conditional on age, median ability does not vary across cohorts. To check whether this assumption holds in our data, we use the longitudinal sample running from 2001 to 2011 to plot for each age group the annual changes in median ability. Since the youngest cohort in our data is aged 25 in 2011 and we can only follow players for ten years, the youngest cohort group for which this exercise can be performed is the one aged 25‐30 in each year of the sample. Figure 3 compares six age groups and shows that the percentage changes in median ability for each group has been very moderate, with the largest change being below 3 percent with respect to the 2001 baseline. This suggests that the effects of cohort of birth on selection are tiny, and that age is the main factor affecting the decision to leave or stay in the sample of professional chess players. In this method, the assumed absence of selection for the youngest cohort – our anchor – trickles down to the older cohorts because of our assumption of stationarity. In appendix A we show the results of a Monte Carlo experiment designed to evaluate how our imputation performs in a controlled setting. Results indicate that the procedure is capable of reproducing the medians of the original population rather well. Once the sample has been repopulated, equation (3) can be estimated using median regression. 5. Results and discussion The two approaches discussed in Section 4 are applied to two different datasets: the full dataset of professional players who played in international tournaments between 2008 and 2011 when we proxy talent with information extracted from ELO in 2006, and the dataset of professional players who played in 2008 and 2011 when we use imputation. Equation (3) suggests that log productivity depends on a polynomial in age. For each approach, we allow the data to establish the degree of the polynomial, starting from degree five. It turns out that a fifth order polynomial is adequate for the former approach, and a fourth order for the latter approach. 11 In the empirical specification of Eq. (3) we also control for country (chess federation) dummies and capture country specific time effects with real GDP per capita and the percentage of individuals with an internet connection. GDP per capita captures the economic conditions in the federation of the player, which are likely to affect participation to international tournaments and access to resources to improve game specific skills. Since chess training is often done on the internet, access to the web can affect training and performance. We also group adjacent cohorts into 6 groups and add to the regression cohort dummies. Since the distribution of productivity has a probability mass point at its lower bound (zero), we use median regressions, which are less sensitive to the presence of outliers. Table 3 reports the estimates of Equation (3) for each method. In columns (1) and (2) we report the estimates of the age polynomial when individual talent is measured using the information extracted from ELO in 2006 and when we use the imputation method. Both methods allow for clustering of standard errors at the level of the individual player, however, since in the former method ability is measured with a generated regressor, in that case we compute the correct standard errors by a clustered bootstrap procedure (400 replications). Figures 4 and 5 plot the predicted age‐
productivity profiles obtained with either method to correct for selectivity. In both figures we report the estimated profile when endogenous attrition is ignored (blue line) and when it is explicitly addressed (red line). Consider first Figure 4. Both the blue and the red line show productivity at age j relative to productivity at age 15. When we ignore selectivity issues, relative productivity peaks at around age 25 and declines rather smoothly until age 60. The estimated decline in productivity from the peak to the age 50 is about 10 percent (
1.125  1.25
). When we take selectivity into account, productivity still 1.25
peaks at around age 25, but the increase with respect to age 15 is much smaller (15 rather than 25 percent). As in the case when selectivity is ignored, estimated productivity declines rather smoothly after the peak is about 8.6 percent lower than the peak at age 50. The estimated gap between the blue and red line increases rapidly in the early twenties and remains more or less constant thereafter, suggesting that selection is particularly intense in the early period of the professional life of a chess player. Failure to control for selection leads to exaggerating the productivity gains early on, as part of these gains is due to less talented players selecting out of chess. Our results suggest that productivity in chess peaks relatively early and falls thereafter. Figure 5 presents a similar picture. Since the imputation method only uses the first and last year in the sample (2008 and 2011), the blue line in this figure is different from the blue line in Figure 4. Using the selected sample, we find that productivity peaks around age 30, when it is 20% higher than at age 15, and declines smoothly until age 60. The percent decline from peak to the age 50 is about 6%. In the re‐populated sample, productivity peaks earlier at around age 25 and declines until age 60, with an overall decrease from peak to age 50 of 7.3%. 12 We summarize our key results as follows: 1) mental productivity peaks in the mid twenties and smoothly declines afterwards; 2) endogenous sample selection concentrates in the first part of chess professional life. Since selection weeds out the less talented at chess, failure to control for selection exaggerates the productivity gains early on by about 10 percent; 3) productivity at age 50 is 7 to 8 percent lower than at age 25. Early selection is consistent with a model where individuals have imperfect information about their ability at chess and learn by playing against relatively good players in international tournaments. Learning takes place early on, and those realizing that they are low ability drop out from professional chess. Little selection occurs after this initial learning period is completed. Early selection patterns are likely to be amplified if learning involves also outside opportunities, and these opportunities decline with age. Conclusions [to come] 13 References Borsch‐Supan Axel and Matthias Weiss, 2007, Productivity and Age: Evidence from Work Teams at the Assembly Line, MEA Working Paper 148 Castellucci Fabrizio, Padula Mario and Giovanni Pica, 2011, The Age‐Productivity Gradient: Evidence from a Sample of F1 Drivers, Labour Economics, 18, 464‐473 Glickman, Mark, 1995, A Comprehensive Guide to Chess Ratings, American Chess Journal, 3, 59‐102 Gransmark, Patrik and Christer Gërdes, 2010, Strategic Behaviour across Gender: A Comparison of Female and Male Expert Chess Players, Labour Economics, 17, 766‐775 Pischke, Jörn‐Steffen, 2001, Continuous Training in Germany, Journal of Population Economics, 14, 523‐
48 Olivetti Claudia and Barbara Petrongolo, 2008, Unequal Pay or Unequal Employment? A Cross‐Country Analysis of Gender Gaps, Journal of Labor Economics, University of Chicago Press, vol. 26(4), pages 621‐654 Pekkarinen, Tuomas, and Roope Uusitalo, 2012. Aging and Productivity: Evidence from Piece Rates. IZA Discussion Paper 6909, Skirbekk, Vegard, 2003, Age and Individual Productivity: A Literature Survey, Max Planck Institute for Demographic Research Working Paper Thurstone, Louis Leon, 1927, A Law of Comparative Judgement, Psychological Review, 34, 273‐286 The Economist, 2004, Over 30 and Over the Hill, June 24th Van Ours, Jan, 2009, Will You Still Need Me – When I’m 64?, IZA Discussion Paper 4246 14 Appendix We perform a Monte Carlo experiment to evaluate how well our imputation strategy reproduces the initial population starting from a selected sample. Suppose that the data generating process is given by A 2 A 3
A 4
ln yitc  5  5 Aict  3 ict  ict  0.3 ict  20i  i Aict  60k21,24  90k24,27
(A1) 10 2
103
10 4
 30k54,57  10 P2  50 P5  60 P10  2GDPt  10 itc
where individual talent is distributed as i  N (0,1) and the noise term eictf is distributed as  i  N (0,1) . Cohort dummies are denoted by k and country dummies by P. Country by time effects are captured by real GDP, which varies by country and time. There are 16 cohorts, {15,18,21,...,60}, each composed of 3,000 individuals observed at time t=0 and at time t=3. We assume that cohort effects are common across pairs of cohorts {15,18} , {21,24}, {27,30} and different from zero only for cohorts {21,24}, {27,30} and {51,54}. There are 10 countries. Players are ranked on the basis of their name and allocated to countries (the first player is allocated to the first country, the second player to the second country, the eleventh player to the first country again and so on). We assume that country fixed effects are all zero except for countries 2, 5 and 10. The initial sample is composed of 96,000 players. Given the features of the data generating process, ability is orthogonal to age in the population. Players self‐select into chess on a year‐by‐year basis according to a selection rule which depends only on age and is defined as follows: Z itc  1 if Aitc  15 Z itc  1 if itc  0.035 Aitc  2.5 and Aitc  15 (A2) Z itc  0 otherwise where Z is an indicator that takes value 1 when player i keeps playing and 0 if he drops out. After the selection, the correlation between age and talent is assumed to be equal to 0.187, and average ability varies across age. The main feature of this selection process is that less talented players drop out as they get older, coherently with our estimates. We draw 1,000 times the initial population and for each draw we allow the selection rule to work. We then apply to the selected sample the imputation procedure described in the text and estimate a median regression of productivity on an age polynomial Med ( y | A)   0  1B1   2 B2   3 B3   4 B4   15 (A3) Due to the very strong multicollinearity among the powers of age that causes a marked instability in the estimates of each term of the polynomial, we orthogonalize the matrix [A,A2,A3,A4] into [B,B2,B3,B4]. The latter is a linear transformation of the former obtained by means of a Gram‐Schmidt procedure. Therefore, B=P*A, where P is the transforming matrix. Applying P to the vector of coefficients of the polynomial of age in model (A1), the equivalent data generating process is ln y itc  109.124  7.716 Bict  31.491Bict 2  7.115 Bict 3  0.714 Bict 4  20  i   i Aict
 60k21,24  90k24,27  30k54,57  10 P2  50 P5  60 P10  2GDPt  10 itc
(A4) Equations (A1) and (A4) are equivalent and represent the same function. Model (A3) is the benchmark against which we compare the results of our procedure, i.e. the estimates obtained from (A2) and step 1 in the imputation process. The estimates obtained from (A2) and step 1 are shown in Figure A1, where the simulated age‐productivity profile and its 95% confidence interval is plotted against the true profile. The two overlap almost perfectly and the confidence interval is remarkably narrow. 16 Tables and Figures Table 1. Descriptive Statistics Variable Mean
Std. Dev.
Min
Max
Age 38.09
12.31
15
60
2072.27
193.22
1402
2806
Games 17.45
19.20
1
289
Productivity 972.39
396.66
0
2551
GDP per capita (in thousand $ at constant prices) Internet users (per 100 inhabitants) 24.29
11.39
1.35
73.34
61.29
19.78
2.50
96.62
Mean ELO Score in 2006 Table 2. The effects of age and talent on the probability of dropping out from FIDE lists (1)
(2)
(3)
All players
Age<=25
Age>25
log(ELO) ‐2.555***
‐6.198***
‐1.525***
(0.197)
(0.831)
(0.315)
Age ‐0.266***
‐1.506***
‐0.0816
(0.0424)
(0.309)
(0.0612)
log(ELO)* Age 0.0343***
0.196***
0.0104
(0.00551)
(0.0401)
(0.00794)
Observations 14,063
3,690
10,373
R‐squared 0.119
0.162
0.096
Country dummies Yes
Yes
Yes
Note: Robust standard errors within parentheses. ELO stands for players’ ELO score in 2001 and age for age in 2001. Three, two and one star for statistically significant coefficients at the 1, 5 and 10% level of confidence. 17 Table 3 – Estimates of Eq. (3) using ELO residuals to measure individual talent (column (1)) and imputation (column (2)). Median regression. Dependent variable: log productivity. Age With a measure of ability 0.291*** (0.058)
With imputation
0.153*** (0.032)
Age 2 ‐1.493*** (0.352)
‐0.636*** (0.157)
Age 3 0.372*** (0.101)
0.109*** (0.031)
Age 4 ‐0.045*** (0.014)
‐0.007*** (0.002)
Age 5 0.002*** (0.0007)
‐
N. obs 140,074
70,358
N. clusters 40,545
40,545
The regressions include country and cohort dummies, real GDP per capita and the percentage of internet users per 100 inhabitants. The regression in the first column includes also a measure of talent and its interaction with age. Three, two and one star for statistically significant coefficients at the 1, 5 and 10% level of confidence. Bootstrapped standard errors with 400 replications clustered at the individual level in the first column, robust standard errors clustered at the individual level in the second column. 18 Figure 1. The distribution of productivity in the sample 0
5.0e-04
Density
.001
.0015
Productivity
0
500
1000
1500
2000
2500
Age in 2001: 15-17
Cumulative Probability
Cumulative Probability
Figure 2. Cumulative Distribution of Ability: Stayers versus Dropouts 1
.8
.6
.4
.2
0
.9
1
1.1
Age in 2001: 25-27
1
.8
.6
.4
.2
0
1.2
.9
1
1.1
Ability
Ability
Dropouts
Age in 2001: 35-37
1
.8
.6
.4
.2
0
.9
1
1.1
Ability
Stayers
1.2
Stayers
Cumulative Probability
Cumulative Probability
Stayers
1.3
1.2
1.3
Dropouts
Age in 2001: 45-47
1
.8
.6
.4
.2
0
.9
1
1.1
1.2
Ability
Dropouts
Stayers
Dropouts
19 Figure 3. Changes in median ability by age group. Changes in Ability by Age Group
Baseline: 2001 median ELO
31-36
37-42
02 03 04 05 06 07 08 09 10 11
02 03 04 05 06 07 08 09 10 11
02 03 04 05 06 07 08 09 10 11
43-48
49-54
55-60
02 03 04 05 06 07 08 09 10 11
02 03 04 05 06 07 08 09 10 11
02 03 04 05 06 07 08 09 10 11
-2
3
-2
-1
0
1
2
% change
-1
0
1
2
3
25-30
Productivity (normalized at 1 at age 15)
1
1.05
1.1
1.15
1.2
1.25
Figure 4. Estimated age productivity profiles, with and without controlling for ability. Normalized at 1 at age 15. 10
20
30
40
50
60
Age
Controlling for ability
without controlling for ability
Notes: the graph plots the profiles obtained from estimation of model (3) with and without controls for ability and the interaction between age and ability. Coherently with equation (3), we plot exp(Ŷ), where Ŷ=Σd=0…5 αd Aged, i.e., the models are evaluated at μ=0. To ease interpretation we rescaled productivity using as baseline the value for age=15. 20 .9
1
Productivity
1.1
1.2
1.3
Figure 5 – Estimated age productivity profiles, using the imputation method to repopulate the sample. Normalized at 1 at age 15. 10
20
30
40
50
60
Age
Self-selected sample
Repopulated sample
Notes: see Figure 4. In this case, d=0,…,4 21 Figure A1. Simulated and true productivity profiles. 22