ICES Journal of Marine Science: Journal du Conseil Advance Access originally published online on July 19, 2008
ICES Journal of Marine Science: Journal du Conseil 2008 65(7):1227-1234; doi:10.1093/icesjms/fsn115
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Sensitivity of predicted cohort size and catches to errors in estimates of fishing mortality in the terminal year
Sea Fisheries Institute, Ko
taja 1, 81-332 Gdynia, Poland
tel: +48 58 73 56 267; fax: +48 58 73 56 110; e-mail: horbowy{at}mir.gdynia.pl
Horbowy, J. 2008. Sensitivity of predicted cohort size and catches to errors in estimates of fishing mortality in the terminal year. – ICES Journal of Marine Science, 65: 1227–1234.Formulae for the sensitivity of projected cohort size and catches to errors (bias) in estimates of fishing mortality in the terminal year were developed. Assessment models allowing for random errors in the observed catches as well as models in which catches are treated as exact were considered. The formulae were applied to a Gulf of Riga (Baltic Sea) herring assessment to show how well they estimated prediction errors and to evaluate the effect of assessment errors on predictions. The errors propagate quickly with time, and the higher the fishing mortality, the bigger the projection error. The errors in predicted catches are somewhat lower than the errors in predicted cohort sizes. The formulae developed show that with moderate error in estimated fishing mortality (20%), the errors in predicted cohort size can reach 100%, and the errors in predicted catches may be 50% for fishing mortality estimated at 1.0 in the terminal year and the status quo prediction. As the Gulf of Riga herring case demonstrates, the overall error in predicted stock size and catches may be lower when terminal fishing mortality is underestimated at some ages and overestimated at others (cancelling effect).
Keywords: error, prediction, sensitivity, stock assessment
Received 5 December 2007; accepted 8 June 2008; advance access publication 19 July 2008.
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Introduction |
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The main instruments of fishery management are regulations of catch volume or fishing effort. Management advice in terms of total allowable catch (TAC) or fishing effort is often given based on projections of catches and stock size under a range of fishing mortality (fishing effort) options starting from the results of a stock assessment. Deterministic catch and stock projections are fairly simple tasks, but estimating their precision is complicated, because several unknowns and uncertainties have to be considered. Estimates of stock size and fishing mortality for the most recent year, which represent the starting point for a projection, are only available with error. They are influenced by a number of error sources, such as noisy input data (catch volume, discards, age distribution, weight-at-age, survey, and cpue indices) and uncertainty in the assessment model structure. Moreover, recruitment abundance, growth rate (weight-at-age), exploitation pattern, natural mortality, and maturity in the years being projected must be assumed or predicted. The uncertainty inherent in all these elements contributes to the uncertainty of a projection. For many stocks, great uncertainty comes from recent estimates of stock size and fishing mortality.
ICES, the International Council for the Exploration of the Sea, is the main advisory body on fishery management for stocks exploited in most European waters. The standard ICES procedure for estimating stock size and fishing mortality generally involves the use of eXtended Survivors Analysis (XSA; Shepherd, 1999), in which catches are treated as exact. Another method used relatively frequently is Integrated Catch Analysis (ICA; Patterson, 1998), in which random errors in catches are allowed for the period for which fishing mortality is assumed separable into year and age effects. In the USA and Canada, ADAPT VPA (Gavaris, 1988) and statistical catch-at-age models are used, e.g. CAGEAN (Deriso et al., 1985) and stock synthesis (Methot, 1990). In statistical catch-at-age models, observed catches are assumed to be measured with random error.
Much literature has been devoted to analysing prediction error. Rivard (1981) used the delta method to estimate the variance of catch projections for cod (Gadus morhua) in the Gulf of St Lawrence. Pope (1983) developed ANOVA TAC, an approach based on the separability of an age- and year-effect in fishing mortality (F) and a Taylor’s series approximation to estimate prediction variance. The method mirrored standard projection practice within ICES and allowed approximate analysis of variance components. Pope and Gray (1983) used Monte Carlo simulations to evaluate the precision of projection of some North Sea stocks. Prediction variance was lower when status quo projections were performed. Errors in the catch projections of several North Atlantic pelagic and demersal stocks were evaluated by Rivard and Foy (1987), who calculated the variance of catch prediction from the variance of the input data, and concluded that estimates of stock size and the abundance of prerecruit fish are key factors in the projections. The ICES prediction method was evaluated by Brander (1987) and van Beek and Pastoors (1999). Brander (1987) compared observed catches with ICES-predicted catches, adjusted to account for changing fishing mortality. Van Beek and Pastoors (1999) compared the observed (estimated using XSA) fishing mortality with that predicted to produce the landings. They found no relationship between the series of fishing mortality. Recently, there has been increasing use of computer-intensive methods, and Monte Carlo simulations and Bayesian approaches have been applied to investigate prediction variance (e.g. Gavaris et al., 2000; Restrepo et al., 2000; Patterson et al., 2001; Simmonds, 2003; Kell et al., 2005). The results generally show that errors are not additive. Maunder et al. (2006) demonstrate that computer-intensive methods may still be impractical for complex models owing to their enormous call on computer power.
In most of the works cited above, different methods of estimation and analysis of prediction variance were presented and applied. However, there was no simple way to answer questions such as: how sensitive are predictions of catch and stock size to relative errors in terminal fishing mortality of, for example, 10%? Here, I have developed methods for estimating errors in predicted cohort size and catches resulting from erroneous estimation of fishing mortality in the terminal year. I believe that they are applicable to the results of assessment models in which observed catches are assumed exact, as well as to models allowing random errors in catches.
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Methods |
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The stock and catch projections performed in a given calendar year typically rely on the results of the most recent stock assessment which provides estimates of F and stock size for the previous year. They provide projections of stock size and catches for the next year using a range of assumed levels of fishing mortality. In ICES nomenclature, the previous year, i.e. the most recent year for which estimates of stock size and fishing mortality are available, is referred to as the terminal year, the given year as the current year, and the next year as the management year (ICES, 2004). This nomenclature is used here.
Derivation of error propagation formulae
Let Nt and Ft denote the true values of stock size at the beginning of year t and the fishing mortality in year t, respectively. Then, from the standard theory, the catch, Ct, at a given age during year t is
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| (1) |
t denotes the average Nt during year t:
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| (2) |
Let us assume that the ratio of estimated to true fishing mortality, Ftest/Ft = a, and the ratio of estimated to true catches, Ctest/Ct = b. Then, Ctest can be expressed as
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| (3) |
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| (4a) |
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| (4b) |
Inserting Approximation (4b) into Equation (3), the ratio of estimated stock size to true stock size at the beginning of year t can be derived:
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| (5) |
Fstatus quo is usually defined as that estimated for the terminal year in the assessment, or the average F for the recent 2–3 years (ICES, 2004). In the present analysis, I use the first option. Assuming that fishing mortality in the current year and the management year differ by factors p1 and p2 from Fstatus quo, respectively (i.e. Ft+1 = p1Ft and Ft+2 = p2Ft), then the estimated (at the beginning of the current year) and projected cohort sizes (at the beginning of and the year following the management year) are
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| (6a) |
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| (6b) |
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| (6c) |
Therefore, from Ftest = aFt and Equation (5), the ratios of the estimated and projected stock size to the true stock size at the start of the current year is
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| (7a) |
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| (7b) |
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| (7c) |
The true catch in the current year for true F and N values is
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| (8a) |
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| (8b) |
The ratio of projected to true catches in the current year for p1 
0 [see Appendix for the development of Equations (9a) and (9b)] is
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| (9a) |
0:
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| (9b) |
The above formulae refer to projections with estimated Fstatus quo or fractions of that value. If another value of fishing mortality, Fref, is used for the management year (e.g. F0.1 or Fmax), Equation (6c) becomes
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| (6c') |
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| (7c') |
Similarly, Equations (9b) will take the form
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| (9b') |
For the assessment methods in which catches are treated as exact (e.g. VPA and XSA), b = 1 and the formulae simplify. For example, at Fstatus quo, the ratio of predicted to true stock size in the following year [the equivalent of Equation (7c)] is
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| (7c'') |
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| (9b'') |
Simulation study
It seems reasonable to assume that estimation errors a and b are correlated, at least to some extent. Greater precision in the estimates of F (a closer to 1) should generally lead to better estimation of catch size (b closer to 1). Simulations were performed to examine possible correlations between a and b and to determine how well the equations approximate errors in the projected stock size and catches. An artificial stock was generated for a period of 15 years, with ages 2–8. F in the first 5 years was 0.5, in the next 5 years 0.8, and in the final 5 years 1. The selectivity was constant and flat for ages 4 and older, and M was assumed to be 0.2. The mortality parameters for the stock resembled those of the eastern Baltic cod stock, and initial numbers and recruitment were taken from the XSA assessment (ICES, 2006). The catches and several sets of survey indices of stock size were generated from the stock, assuming lognormally distributed random errors with a survey CV ranging from 0.05 to 0.5 with steps of 0.05, and a catch CV of 0.25. Finally, the stock was assessed using the model of Deriso et al. (1985) for each of the generated sets of survey indices, and a and b from these assessments and generated values were calculated.
Next, standard ICES projections were performed for the generated stocks for both true and estimated (biased) F in the terminal year (year 15), and a range of p1 and p2 values. The results of these projections were used to calculate the ratios of projected to true stock size and projected to true catches. Finally, the ratios were compared with the ratios calculated using the analytical formulae developed above.
Gulf of Riga herring case study
The formulae were applied to a Gulf of Riga herring (Clupea harengus) assessment and its predictions, to evaluate the effects of assessment error on predictions. The predictions depend not only on assessment estimates in the terminal year, but also on recruitment estimates and recruitment assumptions. In addition, weight-at-age in the catch and in the stock has to be assumed or predicted.
The Gulf of Riga herring is a relatively well-defined stock, and only a few fish migrate outside the Gulf. The most recent assessment of the stock using XSA (ICES, 2006) was assumed to provide "true" estimates of stock size and F in earlier years (before 2002) based on the well-known virtue of convergence of VPA. Assessment and prediction of the stock and catches was taken from ICES (2002), and a new prediction based on the 2002 assessment was performed. XSA estimates of survivors aged 3–8 at the beginning of 2002 were used for the prediction. For ages 1–2, the numbers at the beginning of 2002 were taken from recruitment estimates, and recruitment levels for age 1 in 2003 and 2004 were assumed to be the long-term average. The p1 and p2 values were taken as ratios of the true F values (i.e. estimated in the 2006 assessment) of consecutive year classes in 2002 and 2003 to the true F values of those year classes in 2001. Consequently, the fishing mortalities for 2002 and 2003 were taken as the F in 2001 from the 2002 assessment multiplied by the realized p1 and p2 values. The values of F at age 1 in the assumed or predicted year classes (i.e. year classes 2000–2002) were calculated such that they corresponded to realized catches from assumed (predicted) recruitment. To calculate spawning-stock biomass (SSB) and yield, observed weights-at-age were used, so the difference between predicted and observed stock size could be attributed to errors in the assessment of stock size in the terminal year and the recruitment assumptions. Prediction was performed for the years 2002–2004 (stock size) and 2002/2003 (catches). SSB was calculated as that at 1 January. Finally, ratios of predicted to true stock and catch numbers-at-age for 2002–2004 and 2002/2003, respectively, were calculated and compared with their respective ratios, derived using the formulae above for b = 1 (in XSA, catches are treated as exact).
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Results |
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Simulation study
In the simulation study, the ratio of estimated to true stock size and the ratios of projected to true stock sizes and catches obtained in the projections from the generated stock were similar to the respective ratios calculated from the proposed analytical formulae, for exact catches (b = 1) and catches with random errors. The differences between simulated ratios and those estimated from the analytical equations were usually <1–2%. These differences increased with fishing mortality, as expected, because Approximation (4b) worsens when Z is high.
Figure 1 shows the ratios of estimated to true stock sizes in the terminal year for a range of values of a and F and b = 1, and for comparison, the curve showing the inverse proportionality to a. The relative bias in terminal N increases with bias in F and decreases with increasing F. When F is small, the bias is close to being inversely proportional to the error in F (measured as the ratio of estimated to true values). The inverse of F error is the upper limit of the error in stock numbers in the terminal year. At 20% underestimation of terminal F and true F within a range of 0.6–1.0 (the range typical for many demersal stocks), the stock size is overestimated by 14–18%. Pelagic stocks are often exploited with a value of F of 0.2–0.4, so a 20% underestimation of F in the terminal year leads to a 20–23% overestimation of the terminal stock size.
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The ratios of estimated to true abundances and catches depend on fishing mortality in the terminal year, assessment error (represented here by a and b), and the F multiplication factors p1 and p2. Errors resulting from erroneous assessments increase with projection time. In Figure 2a, I show the ratios of estimated to true stock sizes at the beginning of the terminal and current years, along with the ratios of projected to true stock sizes at the beginning of and in the year following the management year, for a stock assessment which assumes catches to be exact. The status quo fishing mortality was assumed for the projections (p1 = p2 = 1), and three options for assessment error (ratio of estimated to true F) were included: 10, 20, and 50% underestimation of terminal F, i.e. a = 0.9, 0.8, and 0.5, respectively. Calculations were performed for three levels of F: low (0.2), medium (0.6), and high (1.0). The propagation of errors in stock size is relatively low for low F. With 20% underestimation in a low terminal year F, the error in estimated stock numbers is 23% in that year, and increases to 38% in the year following the management year. However, the same underestimation of terminal year F assumed at 1.0 leads to a stock number error of 14% in the terminal year, increasing to almost 110% at the beginning of the year following the management year. With an assessment error of 50%, the stock projection errors in the third year are enormous and may reach a few hundred per cent. When F in the terminal year is overestimated, the ratios between projected and true stock sizes are of the order of the inverse to the respective ratios when F is underestimated.
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The ratios of projected to true catches (catch errors) in the current and management years are smaller than the errors in stock numbers (Figure 2b). This is because estimates of stock size and fishing mortality are negatively correlated, so their errors (underestimated F and overestimated N, or vice versa) cancel each other out to some extent. With 20% underestimation in a low terminal year F, the error in the status quo catch projected for the current year is 4%, and it increases to 8% in the management year. The error is much larger for the same proportional underestimation of a large terminal F, increasing from 22% in the current year to 49% in the management year. With a terminal F error of 50%, the errors in the catch projected for the management year are much larger, varying from 22% for low F to >170% for high F.
Gulf of Riga herring case study
The ratios for the Gulf of Riga herring stock numbers-at-age, estimated and projected based on the assessment conducted in 2002, to those based on stock numbers estimated in the 2006 assessment were similar to the errors estimated by the analytical formulae; the results were similar for the ratios of projected to realized catches (Table 1). Always, the difference was <0.01. The errors weighted by stock biomass or yield-at-age were the same as the ratios of projected biomass to "true" biomass (i.e. determined in the 2006 assessment) and projected to realized catches. At most ages, the stock size was overestimated in the 2002 assessment, but at age 2 in 2002 (assessment) and at age 1 in 2003 (recruitment assumption), stock numbers were underestimated by 60–70%. As these two year classes appear to have been relatively strong, they contributed largely to both stock biomass and yield, resulting in some compensation for the errors. Biomass errors weighted by stock biomass-at-age and yield errors weighted by catch weight-at-age produced average errors of biomass and yield equal to the ratios of predicted biomass and yield to the true biomass and yield (Table 1).
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When the assessment assumes errors in the catches (b
1), the dependence of errors in stock numbers and catches on the error in the estimate of terminal F is more complicated than for precise catches. As already indicated above, it can be expected that a and b (i.e. errors in F and in estimated catches) are correlated. This expectation was supported by the simulation study, which showed that errors in estimated catches depend significantly on errors in terminal F (Figure 3).
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Discussion |
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Brander (1987) analysed historical predictions of nine Northeast Atlantic stocks and showed that predictions for the current year were more precise than those for the subsequent year. The relationship between estimated prediction errors for the current and following years was similar to that resulting from Formulae (9a) and (9b), i.e. errors for the subsequent year (on a relative scale) were close to being the square of those from the current year.
The errors increase with F, p1, and p2, i.e. with increasing cumulative fishing mortality during the projection period. F options lower than status quo lead to smaller errors than those higher than status quo. In contrast, Pope (1972) showed that the convergence of VPA assessment is quicker for higher cumulative F, so higher estimates of terminal F may result in lower errors in that parameter. The formulae for errors are of the form b exp[A(1–a)]/a [Equations (5), (7a)–(7c), (7c'), and (9b')] or b exp[A(1–a)] [Equations (9a) and (9b)], where A depends on F, p1, and p2. The value of A will most often be <3 (for F and p1 
1 or smaller, and p2 between 0 and 1.5). The examples of functions b exp[A(1–a)]/a and b exp[A(1–a)], as dependent on a for A from 0.5 to 3 and b = 1, are presented in Figure 4. With status quo fishing mortality of 0.5 or 1.0, A is 1 and 2, respectively [Equation (9b)]. Then, when estimated F differs from true F by ±20%, the errors of projected catches range from 20 to 50% when F is underestimated and from –20 to –30% when it is overestimated. For that range of F and its error, the errors in projected stock sizes ranged from 60 to 100% when F was underestimated and from –35 to –50% when it was overestimated. In light of the size of the projection errors presented with a moderate bias in terminal F, it is no surprise that van Beek and Pastoors (1999) failed to find a relationship between the observed F and the predicted F to produce the realized landings.
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To apply the formulae developed, some proxies for errors in F are needed. The lower limit of the error may be obtained from the retrospective estimates of F, estimates that may be related to the estimates from the most recent assessments, which were treated as most reliable, at least for the historical part of the F time-series. Obviously, true errors may be higher. Even when retrospective estimates of F do not vary much between years, it is not certain whether or not they are biased. For eastern Baltic cod, the average retrospective deviation in the estimates of terminal F is 0.06 when the same assessment method and its parameterization is used (ICES, 2006). However, the method, its parameterization, and its tuning fleets are changed by the working groups from time to time, so the comparison of assessments made in consecutive years gives much greater differences between historical estimates of terminal F and the F obtained in most recent assessments. For Baltic cod, the deviations in terminal F (obtained in historical assessments) from fishing mortalities derived in the most recent assessment range from 10% to 50%, with an average of 30% (Figure 5). Therefore, the possible error in predicted catches and stock sizes attributable to erroneous specification of terminal F may vary from 20% to well over 100%.
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For assessment methods assuming errors in catches, the equations developed are useful when approximate errors in both terminal F and catches are known. Otherwise, the relationship between a and b should be specified and used for error calculations. A rough estimate of b may be obtained from the assessment model as the ratio of observed to estimated (treated as true) catches. Such an estimate of b should, however, be considered as its lower bound. The ratio between the right sides of Equations (7c) and (7c') is exp[p2(1–a)Ft], <1 for overestimation of terminal F (a > 1) and >1 if terminal F is underestimated (a < 1). Consequently, if catches are treated as exact (b = 1), the reference F strategy produces less error in cohort size than the strategy with status quo F. However, for some combinations of a, F, and p2 values, the status quo F strategy leads to lower catch errors [Equation (9b)] than the reference F strategy [Equation (9b')], although it is the opposite for other combinations.
It should be stressed that the formulae developed here do not refer directly to errors in total stock and catch, but rather show sensitivity of cohort size to error in the estimates of terminal F. The errors in overall stock size and catch may be as high as estimated using the equations above when terminal F is consistently under- or overestimated at all ages. However, as demonstrated for Gulf of Riga herring, F may be underestimated at some ages and overestimated at others, so some cohorts are overestimated and others underestimated, and the errors partially cancel each other out, leading to lower overall errors in stock sizes and catches. The equations can be used too at a cohort level, e.g. to investigate the propagation of errors in assumed or predicted recruitment to predicted stock sizes and catches.
Age was omitted from the equations developed for simplicity of presentation, and the formulae presented refer to errors in the assessment and projection of a given cohort abundance and errors in catches realized from that cohort. Obviously, errors in total stock sizes and catch numbers will result from errors in the assessment and the projection of cohorts comprising the stock.
In the analysis presented here, only errors in assessment and predictions resulting from the erroneous estimation of terminal F were considered. There are other sources of error in the predictions of catch and stock size, such as weight-at-age and maturity for current and management years, which have to be assumed or predicted. These elements are not considered here, but for many stocks, they will have a lesser impact on the overall prediction error than the error in stock assessment. Weights-at-age generally do not change much from year to year, and maturities are often kept constant owing to a lack of adequate data. A detailed discussion of these and other factors creating uncertainty in stock assessment and predictions is presented by Patterson et al. (2001).
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Appendix |
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TopIntroductionMethodsResultsDiscussionAppendixReferences |
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Derivation of the ratio of projected to exact catches in the current year [Equation (9a)]
Applying Equations (8a), (8b), and (5) and Approximation (4b) to terms
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Derivation of the ratio of projected to exact catches in the management year [Equation (9b)]
Ct+2 and Cestt+2 have functional forms of Ct+1 and Cestt+1 (with p1 replaced by p2) multiplied by exp(–M–p1Ft) and exp(–M–p1aFt), respectively. Therefore, the ratio Cestt+2/Ct+2 equals the right side of Equation (9a), with p1 replaced by p2, multiplied by exp[p1(1–a)Ft]:
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Acknowledgements |
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I thank Carl O’Brien, Beatriz Roel, and anonymous reviewers for helpful comments on earlier drafts of this manuscript.
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References |
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