WIXLIB

106THE REVIEW OF ECONOMICS AND STATISTICSfor future research is whether more can be donewith the current data to tly to adjust for the smallNI problem. It is the case, for example, that Nkis likely to be a decreasing function of k and thatcc is a decreasing function of Nk. Therefore, E is likely to be an increasing function of k. Theapproach taken in this study in dealing with thisproblem is simply to truncate the sample at thepoint where the size of the effect of k on e*appears to become large. An alternative approach would be to parameterixe the functionrelating k to cI (say ll 7, yZk y,k2 for kgreater than some value k), add this to (3). aEdtry to estimate the new parameters (7,, y2, y3, k)along with the others. The data may not be goodenough to allow anything sensible to come out ofthis, but it is a possible area for further research.Another possible approach is the following.Denote the density function in figure 1 for agiven age k as f(q*.0,),where qk is the log ofthe time in the event of an individual of agek and 8, is a vector of parameters. Let qpl”denote the minimum value of qk in a sample ofis an order statistic, and letsixe N*. qyg(q,n’“, 8 t , Nkl denote its density function. Thefunctional form of g depends, of course, on thefunctional form of f. The data used in this studyare observations on qp for k 35 and over.Gii (1) observations on qp”, (2) an assumptionabout the functional form of f, (3) a parameterization of the elements of 8, as functions of k,and (4) values for Nk or a parameterization of Nkas a function of k, one could estimate the parameters by maximum likelihood. Again, the datamay not be good enough to allow sensible estimates to be obtained using this approach, but it isanother possible area for further research.Until further work is done, the present resultsshould be interpreted with caution. If the sameestimation is done ten or twenty years hence, it islily that the estimated slowdown rates will besmaller than the currently estimated rates.Whether they will be only slightly smaller or a lotsmaller is an important open question.Note finally that if all ages are getting betterover time (say because of better nutrition, bettertraining methods, or better equipment), this willnot affect the estimated slowdown rates as long asall ages are getting better at the same rate.Progress like this will affect the estimated slow-down rates only if it differentially affects thevarious ages.The EconometricsLet d, 1 if k 5 k, and d, 0 if k k,.Using this notation, substituting (1) into (31, andusing the restrictions in (2) yields the equation tobe estimated:r, u, n,k ‘ (1- d,)x(k:- 2k,k k’) Ed,k 35,.,K.(4)There are four parameters to estimate, al. ,u5, and k,, where it should he remembered thatd, is a function of k,. K is the oldest age in thesample period. There are age gaps in the sampleperiod because of the exclusion of observationswith dominated times.Let i* be the predicted value of r, from equation (4) for a particular set of coefficient estimates. The main interest in this study is in thederivative of ?* with respect to k. This derivativeisai,/ak & 26,(1 - d,)(k- i3),(5)where a hat over a coefficient denotes its estimate. This derivative is not a function of theestimate of the constant term ul in (4), and sothe size of the mnstant term is not of directconcern here.Equation (4) pertains to a particular event. Ifone is willing to assume that 1 .as, and k, arethe same across events, then the data on thedifferent events can be pooled and more efficientestimates obtained. It does not seem unreasonable that the derivatives are the same at least forevents close to each other in distance. When thedata are pooled, different constant terms areneeded for each event, since these obviously varywith distance. When the data were pooled for theresults below, the following equation was estimated (n is the number of events pooled):rrr BrD,;, . . . B.Rit 4 a,(1 - di,)(k;- 2k,k k*) lirt,i-l I., n;k-35,., Ki, (6)where rik is the log of the observed record forevent i and age k, Djit is a dummy variable that

HOW FAST DO OLD MEN SLOW DOWN?is equal to one when event i is equal to j andzero otherwise (j 1,. . , n), di, 1 if k s k,and di, 0 if k k,, cik is the measurementerror for event i and age k, and K, is the oldestage used for event i. Again, there arc age gaps inthe sample period for a given event because ofthe exclusion of dominated observations. The njSi coefficients in equation (6) are the n differentconstant terms.Return now to the estimation of equation 14).Since positive measurement error for rk is moreIikely than negative measurement error, the meanof elr is Iikely to be positive. If there is nonegative measurement error at all, then lL 2 0for all k. A positive mean for ex poses no @robIem in the estimation of equation (4) because thepositive mean is merely absorbed in the estimateof the constant term. If the mean of lL is g,define e; Ed - Z, where e; has mean zero.Equation (4) can then be rewritten with e: replacing eL and the constant term changed fromo, to a, Z. In this case a, is not identified, butthis is of no concern here because the derivativesdo not depend on (I,. One can thus estimate (41by nonlinear least squares in the usual way. Thisestimation procedure will be called the NLS procedure.There is, however, another estimation methodthat is of interest to consider. Under the assumption that c* 2 0 for all k, equation (4) can beestimated under the restriction that all estimatedresiduals are non-negative. This procedure iscommon in the estimation of frontier productionfimctions-see,for example, Aigner and Chu@6&?)and Schmidt (1976). The one added complication here is that equation (4) is nonlinear incoefficients. For Iinear equations the estimationproblem can be set up as a quadratic programming problem and solved by standard methods,but for nonlinear equations some other procedure must be found.The procedure used for the results below is thefollowing. In the standard case the coefficients inequation (4) are estimated by minimizing the sumof squared residuals X s&. Instead, one can. . .mmmuze a wetghted sum Z:,“ ,A& where A, isequal to 1 if & r 0 and is equal to a numbergreater than one if ZI 0. This penahis negative errors more than non-negative ones. For thework below a value of 1OQwas used for A, when107tx was less than zero. This was large enough tomake nearly all the estimated residuals non-negative at the optimum.5 This estimation procedurewill be called the “frontier” procedure.it turns out, as will be seen below, that theuse of the frontier procedure instead of theNLS procedure has only a small effect on theestimated slope coefficients and thus on the estimated derivatives. The use of the frontier procedure primarily affects the estimate of the constantterm, which is not of concern here.An attempt was also made to estimate theparameters of (4) under the assumption that etfollows a gamma distribution, as discussed inGreene (1980). The use of this distribution hasthe advantage of allowing the statistical properties of the maximum likelihood estimator to bereadily obtained, which the procedure discussedabove does not. It also accommodates quite flexible shapes of the error distributions. Unfortunately, sensible results could not be obtainedfollowing this approach. The estimates of the hvodistribution parameters (P and A in Greene’s(1980) notation) were usually not sensible, andconvergence was hard to obtain. It would beinteresting to see in future work if this approachcould be made to work, but the effort so far(which was considerable) was not successful.III.ResultsNLS EstimatesThe results of estimating the equation for eachevent by itself are presented first in table 1 (linesl-17). The estimates of or, k,, and rrs and theirestimated standard errors are presented alongwith the implied values of the derivatives at agesSO, 60, 75, and 95. (The implied value of thederivative for ages below is is &.I The estimation technique for these results is NLS.Set aside for the moment the 100 meter, 200meter, and marathon events. 01 the remainingfourteen events, two stand out as being cortsiderably different from the rest in table 1: 1OJKlO

THE REVIEW OF ECONOMICS AND STATISTICS108L-THET*BLEESTIMATIONRESULTSFOR ME TRACK AND ROAD RXING EVEN-IXDerivative at AgeLineSE@,)o 087.w92Road 2.000ii.ocQ12:182212::82::ai81ai787879& 0meter8 and 5K.s For 10,fKlOmeters there is asmall estimate of qs, which means that thederivatives grow vary siowiy with age. For 5K theoppositeis true. Note in particuiar that 10,000meters is quite different from 1OKeven though itis the same distance, and Iihawise for SK and5,000 meters. It may be that the 10,005 meter and5K results refiect considerable measurement erMI, given that they are so different from the rest,The other two events that have somewhat diiferent results are 30K and 25 miles. These bothhave slightly larger estimates of (1s than the eventsbetween 400 meters and the half marathon escept for 3005 meters. Two things could be soing-:;256on here. First, it may be that at roughly the 30Kdistance, the siowdown rate at a given age be@to increase, and this is what the estimates arepithing up. Second, the results may be unreliable.The 30K and 20 mile avents are not as popular asthe others, and so there is more of a potentialsmall Nk pmblem here. The potential small Nkproblem aiso reveais itself in the fact that thesamples are small for these hvo awns (12 and 11obsetvations, respectively). The samples are smallbecause many of the records were dominated byrecords at older ages and so were discarded. Thehigh number of dominated racords probably indicates a small Nk problem. It is thus an openquestion as to whether the 3OK and 20 mileresults are capturing an increase in the siowdownrate at a given age across distances or are simplydue to a small sample problem.

HOW FAST DO OLD MEN SLOW DOWN?The remaining five track events (400 metersthrough 5,ooOmeters) and five road racing events(10K through the half marathon) give similarresults. There is no evidence of anything varyingin a systematic way across distances. The impliedderivatives at age 60 across the fen distances arein remarkably close agreement; the range is only0.0100 at 10 miles and 20K to 0.0114 at 3000meters. There is more variation in the estimatesof a:2, where the range is 0.0042 at the halfmarathon to 0.0087 at SO00meters. The range atage 7.5 is 0.0134 at 10 miles to 0.0186 at 3000meters, and the range at age 95 is 0.0180 at 10miles to 0.0281 at 3000 m rs. The estimatedstandard errors for GE,and k, are fairly large forsome events.Given that no systematic variation across distances is evident in the ten events, it seems sensible to pool them to obtain more efficient estimates. The results of doing this are reported inlime 18 in table 1. The estimate of n2 is 0.0069,with an estimated standard error of 0.0006, andthe estimate of k, is 47.7, with an estimatedstandard error of 3.0. The derivatives are 0.0076at age 50, 0.0109 at age 60, 0.0157 at age 75, and0.0221 at age 95.7These pooling results are not sensitive to theexclusion of the 10,000 meters, 5K, 3OK, and 20mile events. When the observations from theseevents are included in the pooling, the estimatesof o2 and k, are 0.0069 and 48.3, respectively,and the derivatives at ages JO, 60, 75, and 95 are0.0075, 0.0109, 0.0159, and 0.0227, respectively.‘Under the assumption that lt is normally distributed.which cannc t be quite right because of the truncation issues,an F-test can be used to test the hypothesis that CC*.q, andk, are the same across the ten wents. There arc 27 restrictions, and the number of observations in the pa&d regression is 256. The Fnlucwas 2.17. which compares with thecritical value at the 1% level of 1.82, and so the hypothesis isrejected. Similar results were obtained when other se*i ofevents were used. The hypothesis that the mflidents are thesame across the specified events was usually rejccte& although the computed F-values were usudiy “01 too muchabove the critical values. Cfl e hmthesis that the coefficientsare the mmc for MK and 20 miles was, however, not rejected81 the 5% level.)I am not inclined m take these rejections as strong evidenceagzdnst pooling. Tbe Eomputcd F-values were never taa faxfrom not rejecting the null hypothesis; the sample size is smallrelative to the number of restrictiow and there scans to beno compelling reason for believing that the coefficients changeacmss the particular events, especially since no systematicpatterns across the ten events were evident when the aqua(ions were estimated individually.109These estimates are very close to the estimatespresented in table 1 when the four events areexcluded.Consider now the 100 meter, 200 meter, 3OK,20 mile, and marathon events. For 100 meters theresults indicate that the rate of slowdown issmaller than it is for the other events. The estimated age at which the quadratic takes over issimilar for 100 meters versus the pried sample(46.5 versus 47.71, but the sizes of the derivativesare smaller. For example, at age 60 the slowdownrate is 0.0083 compared to 0.0109 for the pooledsample. At age 95 the rate is 0.017s compared to0.0221 for the pooled sample.The results for 200 meters are quite differentfrom the rest. Tbe estimated age at which thequadratic takes over is 65.8, which is much higherthan the other estimates. Also, the estimate of n,is much larger, which means that once thequadratic takes over, the estimated increase inthe slowdown rate is larger than it is for the otherevents. The derivatives at age 60 are similar for100 and 200 meters, but the derivative is noticeably larger at age 75 for 200 meters and considcrably larger at age 95 (0.0403 versus 0.0175). Because the 200 meter results stand out as beingmuch different from the rest-both from the 100meter results and from the results for 400 metersand above-they should be interpreted with considerable caution. It seems likely, for example,that the increase in the slowdown rate after age64 has been overestimated.Given that the results for 30K and 20 miles aresimilar to each other and differ somewhat fromthe rest, it is of interest to pool the two events.The results of thii pooling are presented in line19 in table 1. Comparing lines 18 and 19, it canbe seen that the estimated slowdown rate for 30Kand 20 miles is lower at the younger ages andhigher at the older ages. Although nof shown inthe table, the age at which the slowdown ratebecomes greater for 3OKand 20 miles is about 59.By age 95 the estimated slowdown rate is 0.0299for 30K and 20 miles versus 0.0221 for rhe others.The results for the marathon in line 17 continue the pattern of the estimated slowdown ratebeing lower at the younger ages and higher at theolder ages. Although not shown in the table, theage at which the slowdown rate becomes greaterfor the marathon compared to the pooled eventsin line 18 is about 63. The estimated age at which

110THE REVIEW OF ECONOMICS AND STATISTICSthe quadratic takes over is 58.2, which is higherthan all the other estimates except the one for200 meters. The estimate of a2 is 0.0063, whichmeans that until age 58 the estimated slowdownrate is constant at 0.63% per year. After age 58the estimated slowdown rate picks up fairlyrapidly (the estimate of crs is large), and by age95 the derivative is by far the largest of any eventat 0.0515. This derivative is even much largerthan the derivative for the 30K and 20 mileevents.The differences between the marathon derivatives and the other derivatives at the older agesare large enough to make one question whetherthe marathon results should be trusted. Theremay be, however, more to the marathon than amere 6.2 miles beyond 20 miles. Anyone who hasnm the last 6.2 miles in a marathon can appreciate this. If there is an important nonlinearity ingoing from 20 miles to the marathon, one mightexpect there to be a more rapid increase in therate of slowing down at older ages for themarathon. This is what the current results show,although the estimated size of the effects shouldbe taken with considerable caution.Frontier EstimatesThe final estimates in table 1 were obtainedusing the frontier procedure. Results are presented for 100 meters, 200 meters, pooled 400meters through the half marathon, pooled 30Kand 20 miles, and the marathon. The resultsusing the frontier procedure are quite similar tothe other results. None of the comparisons andmnclusions discussed above are changed by thefrontier results, although the results for 200 meters (line 21) are somewhat less utreme using thefrontier method than they are using NL,S.Figure 3 presents plots of actual and prediaedvalues for four events-100 meters, 200 meters,5000 meters, and the marathon. The actual valuesare the values used in the estimation, and so theydo not include the dominated values (which wereexcluded) and the values excluded at the highages. The predicted values are from the fmtttierestimates. For 100 meters, 200 meters, and themarathon, the frontier estimates are presented inlines 20, 21, and 24, respectively, in table 1. Thefrontier estimates for 5lXMlmeters were obtained,but they are not presented in table 1.The plots for 100 meters and 5000 meters showthat the curvature for the quadratic is quite modest once the quadratic takes over. The plo

HOW FAST DO OLD MEN SLOW DOWN? 10s and k, and one between k, and, say, k,. The two linear segments were restricted to touch at k,, and both k, and k, were estimated. This speci- fication did not work for the individual events because the estimates were too collinear,