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Generalizations that aren't? Evidence on education and growth
European Economic Review 45 (2001) 905}918
Heterogeneity and the Growth Process
Generalizations that aren't? Evidence on education and growth Jonathan R.W. Temple Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, UK and CEPR
Abstract Several papers have suggested that the relationship between changes in average schooling and growth is weak in the cross-country data. This might call into question the relevance of micro estimates of returns to schooling, at least for developing countries. This paper examines the reliability of some of the aggregate evidence, and presents an alternative framework for analysing these questions. 2001 Elsevier Science B.V. All rights reserved. JEL classixcation: O15 Keywords: Education; Growth; Human capital
1. Introduction When economists and politicians discuss the policies that could raise growth rates, education usually takes centre stage. Yet the empirical evidence that education matters for growth is surprisingly mixed. One of the most in#uential papers is titled &Where has all the education gone?', re#ecting a perception that enormous investments in education since the 1960s have yielded a surprisingly meagre growth payo!, at least for developing countries. In this article, I will propose changes to the conventional empirical approach, which might allow us to arrive at more precise estimates of the growth e!ect of education. E-mail address: [email protected] (J.R.W. Temple). 0014-2921/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 4 - 2 9 2 1 ( 0 1 ) 0 0 1 1 6 - 7
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The main aim is to examine the evidence for something that I will call the &Pritchett hypothesis.' Associated with Pritchett (1999), this is the claim that increases in measured educational attainment have done little to raise growth in less developed countries, perhaps due to the institutional environments in which those increases have taken place, or the low quality of schooling, or limited demand for skilled labour. The economics of the Pritchett hypothesis } in particular, the suggestion that the success of educational expansion is contingent upon getting other things right } is surely correct. Here the focus is on the statistics. Is it really the case that most developing countries have not bene"ted from increases in attainment? For this issue, like others in the growth "eld, it is useful to think in terms of possible generalizations and their reliability. It may be the case that education has done little for growth in one or two countries, yet been essential elsewhere. In Section 2, I will describe empirical methods that could permit more reliable generalizations to be drawn. The central ideas are those of the robust statistics literature. In my view, this is a "eld that should inform empirical work on growth more often than it does at present. The paper then turns to speci"cation issues particular to the study of education and growth (Section 3). One important point should be made at the outset. The paper does not revisit the much-discussed correlation between growth and the initial level of human capital. Instead, the focus is on the correlation between growth and changes in human capital } a correlation that one would expect to observe if the micro estimates of returns to schooling genuinely capture a productivity e!ect, rather than signalling or other aspects of labour market organisation. The two best-known papers to examine this relationship are Benhabib and Spiegel (1994) and Pritchett (1999), and Sections 4 and 5 revisit their "ndings. Section 6 presents an alternative empirical framework for these questions and some new results, before Section 7 concludes.
2. The statistics of reliable generalizations The motivation of most empirical work on growth is to draw conclusions about the growth process that are true for the majority of countries. A clear problem here is that parameters are likely to vary across countries, yet this heterogeneity immediately undermines claims about the validity of a particular generalization. To put this another way, in a "eld like empirical growth, every model that can be estimated is mis-speci"ed. As soon as we recognise this, we should also start to think carefully about estimation issues, because there are sometimes better ways to estimate a mis-speci"ed model than least squares } including ways that may inform us about the precise nature of the misspeci"cation.
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One estimator that seems particularly well suited to growth questions is least trimmed squares (LTS). The estimator minimizes the sum of squares, but over a fraction of the observations, say 75%. The fraction that is used to calculate the "nal estimates is not chosen by the researcher, but determined endogenously by the estimator. How should we think about this? LTS searches out the part of the sample for which the current model has greatest explanatory power. By comparing the LTS and OLS results, we can also start to assess the validity of a generalization. When the LTS and OLS estimates are close, this tends to suggest that small changes to the sample would leave our conclusions about parameters essentially unchanged. When the two results di!er, it is possible that the OLS results are driven by a small minority of observations. Most possible subsets of the sample would yield di!erent conclusions about the growth process, and that directly calls into question the validity of our attempted generalization. These arguments must be used very carefully, given our knowledge that least squares is unbiased and e$cient in the presence of normal disturbances, and that omitting some observations in a non-random way will tend to generate biases. One possibility (not unimportant when studying growth) is that the disturbances are non-normal, so that least squares is ine$cient. This in itself can justify the use of a robust estimator like LTS, but there is also a rather more general argument. In the cross-section context, a robust estimator may be useful for investigating the presence of parameter heterogeneity. Since least trimmed squares can be used to detect such heterogeneity, it can be used to o!er some insight into the validity of a particular regression speci"cation, and that is the perspective taken here. I should note that this contrasts with my previous work using the LTS estimator. In that work, I have used LTS to identify possible outliers. The procedure typically adopted is to note observations with large residuals in the LTS estimates, and then omit these from an otherwise straightforward leastsquares regression. This allows us to check that a particular result does not depend on, say, the diamond mines of Botswana. Yet to end the analysis at this stage is somewhat unsatisfactory. In principle, the "nding that a model explains some observations much less well than others is useful information. It may be an indication that the model is mis-speci"ed, and that a more general and #exible speci"cation should be sought. This suggests the use of least trimmed squares as an exploratory tool. A sizeable di!erence between the OLS and LTS estimates can be regarded as a warning sign that a model may be mis-speci"ed. Rather than simply omit Note that this estimator should not be confused with another, &trimmed least squares'. More details on the LTS estimator can be found in Temple (2000a) and the references therein. Unless the standard error of a parameter estimate is greatly a!ected by a leverage point. See Temple (2000a) for more discussion.
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observations } a potentially dangerous procedure } I will also use the LTS results to inform decisions about the regression speci"cation. The next two sections of the paper apply these ideas to education and growth, and examine whether existing results are sensitive to outliers and alternative choices of functional form.
3. Empirical frameworks for education and growth A common starting point for thinking about education and growth is to write down an aggregate production function that looks something like: >"AK?H@¸\?\@,
(1)
where > is output, A total factor productivity, K physical capital, H the aggregate stock of human capital, and ¸ the size of the labour force. I have always found this formulation slightly odd, because it seems to imply that there is a lump of human capital that can be substituted for labour, rather than being embodied in the skills and knowledge of individual members of the labour force. A simple rewriting clari"es the intuition behind this speci"cation. Let us identify H with total years of schooling, and de"ne average years of schooling S"H/¸. Then we have
> K ? "A S@. ¸ ¸ Output per worker is a function of the capital}labour ratio and an index of labour quality. We can also begin to see that this particular speci"cation is implausible: it implies that raising average schooling from 0.1 to 0.2 years will raise output per worker by the same factor (2@ ) as raising average schooling from 4 to 8 years. Hence, speci"cation (1) imposes much higher returns to schooling at low levels of schooling than at high ones. With this in mind, various papers (probably starting with the "rst draft of Bils and Klenow, 1998) have explored an alternative form, with a more direct connection to conventional assumptions:
> K ? "A eX1. ¸ ¸
(2)
The key feature of this speci"cation is that an extra year of average schooling will always raise output in the same proportion, independently of the current level of schooling (and conditional on the capital stock). This is an attractive feature partly because it is shared with the standard semi-logarithmic formulation for individual earnings, widely used by labour economists to estimate returns to schooling from microeconomic survey data. In these earnings
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regressions, an extra year of an individual's schooling is often assumed to have the same proportional e!ect on earnings, regardless of the level of schooling. Simple extensions allow the return to vary across schooling levels. Several researchers have carried out cross-country accounting exercises that impute productivity e!ects in this fashion, using estimates of the return to schooling from micro data (Klenow and Rodriguez-Clare, 1997; Hall and Jones, 1999; Woessmann, 2000). Although the accounting exercise is informative for some questions, often we would like to test the productivity impact of schooling directly. Until that can be done, it is hard to gauge whether micro estimates of the private returns to schooling also capture the social returns, or are in fact driven by signalling e!ects and other aspects of labour market organisation. This is not a new point: Arrow (1973, p. 215) suggested that macroeconomic data might be needed to test the relevance of signalling and "ltering, admittedly while expressing some scepticism about the potential usefulness of this approach. More recently, Griliches (1997, p. S333) suggested that direct tests of productivity e!ects of education are likely to rely on the estimation of production functions. This means that, for the speci"c task of examining the Pritchett hypothesis, we want to estimate the parameter z, rather than impute it. This parameter should certainly not be interpreted as the social return, not least because it does not incorporate the opportunity cost of the resources used in educational provision. Yet it remains of considerable interest, since it provides a way of either con"rming the importance of education suggested by micro studies, or calling into question } as in Pritchett (1999) } the relevance of estimated private returns in thinking about social returns and education policy. Although Eq. (2) is a useful starting point, for our purposes it may not be su$ciently general. In particular, when we examine the cross-country evidence, we may want the productivity e!ect of an extra year's schooling to vary with the level of schooling. This suggests the use of a more #exible speci"cation, familiar from the work of Hall and Jones (1999) among others,
> K ? e(1. "A ¸ ¸
(3)
The function (S) tells us how the productivity e!ect of schooling varies with the level of schooling. Accounting exercises typically take the function to be piecewise linear, and use estimates of returns to schooling for di!erent levels (primary, secondary, tertiary) to impute the parameters. For testing the productivity of schooling directly, we might be interested in seeing what the data can tell us about the form of (S), and that is one aim of what follows. These issues are discussed in much greater detail in Temple (2000b). External e!ects of schooling provide another reason for looking at the macro data.
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Table 1 Regression Estimation Observations
1 OLS 78
2 LTS 78
3 OLS 78
4 OLS 78
5 LTS 78
6 RWLS 73
Constant
0.269 (2.68) 0.457 (5.09) 0.209 (0.96) 0.063 (0.72)
0.140
0.275 (3.46) 0.432 (5.08) 0.266 (1.38)
0.255 (2.67) 0.438 (4.86) 0.168 (0.80) 0.182 (1.90)
0.119
0.168 (2.87) 0.463 (7.18) 0.250 (1.50) 0.183 (2.23)
log K !log K 2 log ¸ !log ¸ 2 log S !log S 2
0.595 0.010 0.157
S !S 2
0.030 0.221
0.015 (0.52)
1/S !1/S 2 R Joint signi"cance
0.577
0.52 n.a.
n.a. n.a.
0.51 n.a.
0.078 (1.72)
0.030
0.056 (1.27)
0.54 0.19
n.a. n.a.
0.65 0.07
Heteroscedasticity-consistent t-ratios in parentheses (not available for least trimmed squares). Regression 6 excludes Botswana, Lesotho, Iraq, Rwanda and Saudi Arabia. The entry in the row &Joint signi"cance' is the p-value from a Wald test of joint signi"cance for the two schooling terms.
4. Benhabib and Spiegel revisited The paper by Benhabib and Spiegel (1994) is among the "rst to note the weak sample correlation between education and growth in the cross-country data. They found that changes in average schooling have little explanatory power for changes in output per worker. In previous work, I have pointed out that the use of least trimmed squares on the same data set gives very di!erent results (Temple, 1999). Yet rather than simply eliminating outliers, we may want to explore alternative, more general speci"cations that provide a good "t for the whole sample. Table 1 presents some results. Regression 1 replicates the estimates of Benhabib and Spiegel, while regression 2 is the same speci"cation estimated by least trimmed squares; in both cases, the log di!erence in output is explained by that in physical capital, the labour force, and average years of schooling, for the period 1965}1985. The di!erence in point estimates across the two estimation methods is marked; Temple (1999) explores this in more detail. The table also shows new results, based on the speci"cation in (3) above, and that sometimes allow for
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curvature in the function (S). The growth regression is now log > !log > " # (log K !log K )# (log ¸ !log ¸ ) 2 2 2 # (S )! (S )#. 2 When the simple case (S)"zS is adopted (regression 3) the estimate of the e!ect of education lacks precision. Regression 4 allows the function (S) to be non-linear,
(S)" # log S# (1/S). This is one of several non-linear speci"cations that I have tried; it is the one that works best, and has another advantage in that it nests Benhabib and Spiegel's formulation. The two schooling variables are jointly signi"cant only at the 20% level, but one attractive feature of these results is that the least trimmed squares estimates of the schooling coe$cients are similar to the OLS estimates (compare regressions 4 and 5). The LTS estimates can also be used to identify a small number of outliers, so that regression 6 excludes the "ve observations with the highest residuals in the LTS estimates. Now the two schooling terms are jointly signi"cant at the 10% level; but the magnitude of the e!ect is small, as I will discuss in the next section.
5. Where has all the education gone? Versions of an in#uential paper by Pritchett (1999) have been circulating for at least 5 years. Pritchett's paper was probably the "rst to try to reconcile aggregate evidence on education and growth with estimates of the returns to schooling based on micro data. Like Benhabib and Spiegel, he found that variation in the change in average schooling plays little role in explaining the cross-country variation in growth rates. Pritchett suggests that this somewhat undermines the conventional interpretation of estimates of returns to schooling, at least for developing countries. His speci"cation relies on the explicit construction of a measure of the value of human capital. This leads to the following speci"cation for output per worker:
> K ? "A (eP1!1)@, ¸ ¸
(4)
where r should, under Pritchett's assumptions, correspond to the return to schooling. He sets this return to 0.10 in line with micro estimates. When Eq. (4) is used as the basis for a growth regression, Pritchett "nds that is not signi"cantly di!erent from zero. In other words, one cannot reject the hypothesis that education has no role in output determination, even when using a measure of human capital that takes the micro evidence at face value.
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We can test the functional form by using a more general function that nests Eq. (4), and that potentially allows more #exibility in the return to schooling:
> K ? "A (eP1)@ (1!e\P1 )@ , ¸ ¸
(5)
where the Pritchett speci"cation corresponds to the restriction " ". Using data largely drawn from Pritchett (1999), and the same sample of 91 countries, I have estimated a growth regression based on Eq. (5). is estimated to be 0.90 with a standard error of 0.31 and found to be !0.11 with a standard error of 0.05. Not surprisingly, the restriction that " is strongly rejected (the Wald test yields a p-value of 0.01). Eqs. (4) and (5) are both somewhat unconventional and (5) in particular is hard to interpret. Given the evidence that the Pritchett speci"cation (4) may be unsatisfactory, I have considered alternatives suggested by the earlier discussion. In Table 2, regressions 7}10 show the results from estimating a growth regression based on the (S)"zS speci"cation, using Pritchett's data for output and capital per worker, and very similar data for schooling. The dependent variable is the log di!erence in output per worker, 1960}1987. I follow Pritchett in imposing constant returns to scale, and in examining the sensitivity of the results to the inclusion of regional dummies. The table reports least-squares results and those from robust methods. Although the least-squares estimates provide a relatively precise estimate of z, the point estimate is much lower in the least trimmed squares results. It is certainly lower than the typical range of private returns to schooling, which would tend to imply a coe$cient in the range 0.05 to 0.15. Hence, the data continue to provide some support for the Pritchett hypothesis, at least when using this speci"cation. Does an alternative speci"cation for (S) a!ect the results? I adopt the same non-linear form as before. The least-squares estimates are imprecise, and very di!erent to the least trimmed squares results, which suggest a stronger e!ect. When the LTS estimates are used to identify possible outliers, least squares gives more precise "ndings, and the two terms in schooling are jointly signi"cant at the 1% level. One pleasing aspect of these results is that the parameter estimates for the two schooling terms are now similar to those reported in Table 1, based on slightly di!erent data on output, capital and schooling, and a slightly di!erent sample of countries.
For comparison with Pritchett's results, I use his data for growth rates in output and capital per worker. The main set of results in his paper is based on schooling data from Barro and Lee (1993); I follow this choice, but use data from Barro and Lee (2000) to obtain the relevant data for Mali and the Central African Republic. Using this data set it is possible to replicate Pritchett's results almost exactly.
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Table 2 Regression Estimation Observations Regional dummies
7 OLS 91 No
0.000 (0.13) log (K /¸ )!log(K /¸ ) 0.490 2 2 (8.18) S !S 0.080 2 (2.56) log S !log S 2
8 LTS 91 No
Constant
0.025
9 OLS 91 Yes
10 LTS 91 Yes
!0.126 (0.51) 0.462 (5.97) 0.062 0.013 (1.76)
1/S !1/S 2 R Joint signi"cance
0.67 n.a.
n.a. n.a.
0.71 n.a.
n.a. n.a.
11 OLS 91 Yes
12 LTS 91 Yes
0.107 (0.39) 0.483 (6.25)
13 RWLS 84 Yes !0.325 (1.24) 0.585 (14.0)
0.011 (0.09) 0.035 (0.56)
0.170
0.71 0.30
n.a. n.a.
0.041
0.181 (1.83) 0.066 (1.40) 0.85 0.01
Regression 13 excludes Hong Kong, Haiti, Indonesia, Jordan, Mozambique, the Democratic Republic of Congo (formerly Zaire) and Zimbabwe. See Table 1 for other notes.
This does not yet tell us anything interesting, in that I have not discussed the magnitude of the e!ect. As noted by Hall and Jones (1999), the proportional impact of an additional year of average schooling on labour productivity, expressed as a &return', can be assessed by evaluating the derivative of (S) at di!erent levels of average schooling. When this is done, the parameter estimates in Tables 1 and 2 suggest that the impact of schooling is close to micro estimates only at low levels of average schooling (0}3 years) and is below 5% for higher levels. It continues to be quite hard to reject the Pritchett hypothesis.
6. An alternative framework All the empirical work considered thus far is a little unsatisfactory in one respect or another. Some previous results are sensitive to the inclusion or exclusion of a handful of observations, nearly all are sensitive to decisions about functional form, and many non-linear speci"cations work rather badly. The standard frameworks are also unsatisfactory from a theoretical point of view, in Note that this calculation, like the growth regressions, holds physical capital constant. Allowing physical capital to be endogenously determined will tend to raise the overall impact of schooling. Under the assumptions of perfect international capital mobility and Cobb}Douglas technology, it is easy to calculate the overall impact, and show that it depends positively on the capital share.
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that the correspondence between the simple aggregate models and the micro evidence is unclear, and this makes the estimated parameters harder to interpret. For example, the case that parameters in an aggregate production function correspond to either private or social returns to schooling is not straightforward, and there are several strong assumptions involved. An alternative way forward is suggested by the work of Jenkins (1995). She examines the role of education in labour productivity growth in the UK, using a production function to estimate directly the productivity impact of changes in educational attainment. In its most general form, this work is based on a Cobb}Douglas production function with constant returns to scale, but with an index of labour quality that is a CES aggregator of uneducated labour (¸ ) and educated labour (¸ ): >"AK?[¸\M#(1#)¸\M]\\? M. If we denote the fraction that are educated by h then we can rewrite this as
> K ? "A [(1!h)\M#(1#)h\M]\\? M ¸ ¸
(6)
and so as before output per worker depends on the capital}labour ratio and an index of labour quality. In practice, it is di$cult to identify the elasticity of substitution between skill types using the available data. An attractive simpli"cation is to adopt the case where educated and uneducated labour are perfect substitutes at a ratio captured by (1#). Then output per worker is
> K ? "A [(1!h)#(1#)h]\?. ¸ ¸
(7)
We can interpret this in terms of homogeneous e$ciency units of labour. Education raises the e$ciency units of labour possessed by a particular individual, across all tasks they perform } but these e$ciency units do not themselves di!er in their nature across individuals, and in principle a su$ciently large group of university professors could substitute for an Olympic athlete. It is a nice idea but clearly unrealistic, and later I will brie#y discuss one means of relaxing this assumption. The perfect substitutes case retains some attractive features. There is an immediate connection to the simpler interpretations of earnings functions based on survey data, in that the production technology ties down the returns to di!erent levels of schooling, independently of the relative supplies of educated and uneducated workers. The ratio of marginal products of the two types of labour is (1#) and so the parameter is easy to interpret. In particular, we are interested in trying to reject the hypothesis that "0, in other words the hypothesis that educated and uneducated workers are equally productive.
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As a proxy for the fraction of educated workers in the labour force, I use the proportion of the population that attained (but not necessarily completed) secondary school, based on the most recent version of the Barro and Lee (2000) data set. Eq. (7) is used to specify a growth regression that can be estimated by non-linear least squares, for a sample of 90 countries. The sample corresponds to that used by Pritchett, except that Malta is excluded due to lack of the relevant schooling data. This exercise yields a point estimate for of 1.52, implying that the marginal product of educated workers is 2.52 times that of the uneducated. However, the 95% con"dence interval for is wide at (!0.45, 3.49) and the parameter is signi"cantly di!erent from zero only at the 15% level. Once again it is hard to reject the Pritchett hypothesis. That said, one advantage of this result is that it clari"es the degree of uncertainty: obviously it is also hard to rule out a large e!ect of education on productivity, and this again suggests that one should be very wary about drawing "rm conclusions using the aggregate data. A remaining objection to this exercise is that, by only using the proportion of the labour force that has attained secondary schooling, the greater degree of information in average years of schooling is disregarded. One response is to adopt a "ner disaggregation of labour types. I have explored models that allow for three types of worker (no schooling, only secondary level attained, higher education attained) and four types (male/female, secondary schooling attained/unattained) and that also use information on di!erential labour force participation rates for men and women. As with the simpler model, the usual result is imprecise estimates. Other extensions are possible. One could explore the use of alternative data for the physical capital stock, or adopt &secondary school completed' as a proxy for the proportion of educated workers. A particularly interesting extension would be to make use of the literacy rate. In principle, this could allow a direct estimate of the marginal product of the literate relative to that of the illiterate. One reason this approach is attractive is that the literacy rate is a widely available measure of competence, and is thus a measure of schooling &output', rather than an input measure like years of schooling or educational expenditure. In other words, one could assess the productivity impact of educational
Using the proportion with some secondary schooling seems to work better than using the proportion of the population with any schooling. Although I should also note a reservation, namely that Barro and Lee (1993, p. 367) "nd the literacy rate to be highly correlated with the proportion of the population recorded in census/survey data as without schooling (in fact, they use the literacy rate to impute this proportion for some countries). They also note that education aims to provide a much wider range of skills. See Romer (1990) for a growth paper that does use data on literacy.
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achievement with less concern that the aggregate results are chie#y explained by the low quality of schooling in some poorer countries.
7. Conclusions The principal message of this paper is that the aggregate evidence on education and growth, for large samples of countries, continues to be clouded with uncertainty. This is particularly so when one tries to con"rm the importance of schooling found in estimates of earnings regressions, by using the change in average educational attainment to explain growth at the aggregate level. The earlier empirical work on this topic is fragile in some respects, but the importance of education hardly leaps out of the cross-country data. With some patience and ingenuity, apparently precise estimates of schooling's e!ect can be extracted, but the degree of data mining involved here (such as the investigation of several non-linear speci"cations, and the non-random exclusion of observations) renders the "ndings about parameters tentative to say the least. But I should also draw attention to another danger in interpreting the evidence as a whole. Some results are occasionally used to argue that schooling does not matter for growth. This is a mistake, for at least three reasons. Such a statement ignores the evidence for the importance of the level of schooling (rather than the change in schooling) in explaining subsequent growth; it ignores the high standard errors typically associated with the relevant parameters; and it also fails to note that the evidence for the claim &schooling has no e!ect' is often just as unreliable as that for any other claim. How can the situation be improved? Time will help, as more and better data become available. There could also be high returns to revising existing data, as in Barro and Lee (2000), and the careful study of the data for the OECD countries that is included in the work of de la Fuente and Domenech (2000). At least until we have greater con"dence in the data, analysis of sensitivity to measurement error should also be a key element of work in this "eld, as in Krueger and Lindahl (1998). Another major improvement might lie in greater attention to empirical frameworks and their interpretation, for both the micro- and macro data. Estimation of the nth earnings function or growth regression may ultimately be less useful and informative than a new approach to exploring these questions, or a new interpretation of the existing body of work.
Another possible extension would be to make a more plausible assumption about the elasticity of substitution between di!erent skill types, by drawing on previous studies. However, it might be di$cult to establish an uncontroversial "gure for the elasticity at this level of aggregation. See Woessmann (2000) for a useful and detailed review of measurement issues.
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It is perhaps also becoming clear that the questions currently driving this research are not necessarily the most interesting in the "eld. Measuring the impact of education on productivity is important, perhaps essential; but it is not clear to what extent decisions about education spending will ever be informed by empirical evidence, especially given that policy on education has (or should have) many ends other than raising productivity. A perhaps more interesting task for future research is to explore the "ne detail of the institutional and incentive structure that best allocates a "xed amount of educational expenditure. These questions have an immediate connection to policy, and the need for further research in this area is apparent more than ever.
Acknowledgements Various people have made helpful contributions. I am particularly grateful to Antonio Ciccone, Damon Clark, Angel de la Fuente, Sir James Mirrlees, Stephen Nickell, Stephen Redding and Ludger Woessmann for helpful comments and discussion, and to Jess Benhabib and Lant Pritchett for supplying data.
References Arrow, K.J., 1973. Higher education as a "lter. Journal of Public Economics 2, 193}216. Barro, R.J., Lee, J.-W., 1993. International comparisons of educational attainment. Journal of Monetary Economics 32, 363}394. Barro, R.J., Lee, J.-W., 2000. International data on educational attainment: Updates and implications. Working paper no. 42, CID, Harvard. Oxford Economic Papers, July 2001, forthcoming. Benhabib, J., Spiegel, M.M., 1994. The role of human capital in economic development: Evidence from aggregate cross-country data. Journal of Monetary Economics 34, 143}173. Bils, M., Klenow, P.J., 1998. Does schooling cause growth or the other way around? Working paper no. 6393, NBER, Cambridge, MA. de la Fuente, A., Domenech, R., 2000. Human capital in growth regressions: How much di!erence does data quality make? Discussion paper no. 2466, CEPR, London. Griliches, Z., 1997. Education, human capital and growth: A personal perspective. Journal of Labor Economics 15, S330}344. Hall, R.E., Jones, C.I., 1999. Why do some countries produce so much more output per worker than others? Quarterly Journal of Economics 114, 83}116. Jenkins, H., 1995. Education and production in the United Kingdom. Discussion paper no. 101, Nu$eld College, Oxford. Klenow, P.J., Rodriguez-Clare, A., 1997. The neoclassical revival in growth economics: Has it gone too far? NBER macroeconomics annual, 73}103. Krueger, A.B., Lindahl, M., 1998. Education for growth: Why and for whom? Manuscript, Princeton University, Princeton, NJ. Pritchett, L., 1999. Where has all the education gone? Manuscript, World Bank, Washington, DC. Romer, P.M., 1990. Human capital and growth: Theory and evidence. Carnegie-Rochester Conference Series on Public Policy 32, 251}286.
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Temple, J.R.W., 1999. A positive e!ect of human capital on growth. Economics Letters 65, 131}134. Temple, J.R.W., 2000a. Growth regressions and what the textbooks don't tell you. Bulletin of Economic Research 52, 181}205. Temple, J.R.W., 2000b. Growth e!ects of education and social capital in the OECD countries. Manuscript, University of Oxford. Woessmann, L., 2000. Specifying human capital: A review, some extensions, and development e!ects. Manuscript, Kiel Institute of World Economics.
Heterogeneity and the Growth Process
Generalizations that aren't? Evidence on education and growth Jonathan R.W. Temple Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, UK and CEPR
Abstract Several papers have suggested that the relationship between changes in average schooling and growth is weak in the cross-country data. This might call into question the relevance of micro estimates of returns to schooling, at least for developing countries. This paper examines the reliability of some of the aggregate evidence, and presents an alternative framework for analysing these questions. 2001 Elsevier Science B.V. All rights reserved. JEL classixcation: O15 Keywords: Education; Growth; Human capital
1. Introduction When economists and politicians discuss the policies that could raise growth rates, education usually takes centre stage. Yet the empirical evidence that education matters for growth is surprisingly mixed. One of the most in#uential papers is titled &Where has all the education gone?', re#ecting a perception that enormous investments in education since the 1960s have yielded a surprisingly meagre growth payo!, at least for developing countries. In this article, I will propose changes to the conventional empirical approach, which might allow us to arrive at more precise estimates of the growth e!ect of education. E-mail address: [email protected] (J.R.W. Temple). 0014-2921/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 4 - 2 9 2 1 ( 0 1 ) 0 0 1 1 6 - 7
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The main aim is to examine the evidence for something that I will call the &Pritchett hypothesis.' Associated with Pritchett (1999), this is the claim that increases in measured educational attainment have done little to raise growth in less developed countries, perhaps due to the institutional environments in which those increases have taken place, or the low quality of schooling, or limited demand for skilled labour. The economics of the Pritchett hypothesis } in particular, the suggestion that the success of educational expansion is contingent upon getting other things right } is surely correct. Here the focus is on the statistics. Is it really the case that most developing countries have not bene"ted from increases in attainment? For this issue, like others in the growth "eld, it is useful to think in terms of possible generalizations and their reliability. It may be the case that education has done little for growth in one or two countries, yet been essential elsewhere. In Section 2, I will describe empirical methods that could permit more reliable generalizations to be drawn. The central ideas are those of the robust statistics literature. In my view, this is a "eld that should inform empirical work on growth more often than it does at present. The paper then turns to speci"cation issues particular to the study of education and growth (Section 3). One important point should be made at the outset. The paper does not revisit the much-discussed correlation between growth and the initial level of human capital. Instead, the focus is on the correlation between growth and changes in human capital } a correlation that one would expect to observe if the micro estimates of returns to schooling genuinely capture a productivity e!ect, rather than signalling or other aspects of labour market organisation. The two best-known papers to examine this relationship are Benhabib and Spiegel (1994) and Pritchett (1999), and Sections 4 and 5 revisit their "ndings. Section 6 presents an alternative empirical framework for these questions and some new results, before Section 7 concludes.
2. The statistics of reliable generalizations The motivation of most empirical work on growth is to draw conclusions about the growth process that are true for the majority of countries. A clear problem here is that parameters are likely to vary across countries, yet this heterogeneity immediately undermines claims about the validity of a particular generalization. To put this another way, in a "eld like empirical growth, every model that can be estimated is mis-speci"ed. As soon as we recognise this, we should also start to think carefully about estimation issues, because there are sometimes better ways to estimate a mis-speci"ed model than least squares } including ways that may inform us about the precise nature of the misspeci"cation.
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One estimator that seems particularly well suited to growth questions is least trimmed squares (LTS). The estimator minimizes the sum of squares, but over a fraction of the observations, say 75%. The fraction that is used to calculate the "nal estimates is not chosen by the researcher, but determined endogenously by the estimator. How should we think about this? LTS searches out the part of the sample for which the current model has greatest explanatory power. By comparing the LTS and OLS results, we can also start to assess the validity of a generalization. When the LTS and OLS estimates are close, this tends to suggest that small changes to the sample would leave our conclusions about parameters essentially unchanged. When the two results di!er, it is possible that the OLS results are driven by a small minority of observations. Most possible subsets of the sample would yield di!erent conclusions about the growth process, and that directly calls into question the validity of our attempted generalization. These arguments must be used very carefully, given our knowledge that least squares is unbiased and e$cient in the presence of normal disturbances, and that omitting some observations in a non-random way will tend to generate biases. One possibility (not unimportant when studying growth) is that the disturbances are non-normal, so that least squares is ine$cient. This in itself can justify the use of a robust estimator like LTS, but there is also a rather more general argument. In the cross-section context, a robust estimator may be useful for investigating the presence of parameter heterogeneity. Since least trimmed squares can be used to detect such heterogeneity, it can be used to o!er some insight into the validity of a particular regression speci"cation, and that is the perspective taken here. I should note that this contrasts with my previous work using the LTS estimator. In that work, I have used LTS to identify possible outliers. The procedure typically adopted is to note observations with large residuals in the LTS estimates, and then omit these from an otherwise straightforward leastsquares regression. This allows us to check that a particular result does not depend on, say, the diamond mines of Botswana. Yet to end the analysis at this stage is somewhat unsatisfactory. In principle, the "nding that a model explains some observations much less well than others is useful information. It may be an indication that the model is mis-speci"ed, and that a more general and #exible speci"cation should be sought. This suggests the use of least trimmed squares as an exploratory tool. A sizeable di!erence between the OLS and LTS estimates can be regarded as a warning sign that a model may be mis-speci"ed. Rather than simply omit Note that this estimator should not be confused with another, &trimmed least squares'. More details on the LTS estimator can be found in Temple (2000a) and the references therein. Unless the standard error of a parameter estimate is greatly a!ected by a leverage point. See Temple (2000a) for more discussion.
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observations } a potentially dangerous procedure } I will also use the LTS results to inform decisions about the regression speci"cation. The next two sections of the paper apply these ideas to education and growth, and examine whether existing results are sensitive to outliers and alternative choices of functional form.
3. Empirical frameworks for education and growth A common starting point for thinking about education and growth is to write down an aggregate production function that looks something like: >"AK?H@¸\?\@,
(1)
where > is output, A total factor productivity, K physical capital, H the aggregate stock of human capital, and ¸ the size of the labour force. I have always found this formulation slightly odd, because it seems to imply that there is a lump of human capital that can be substituted for labour, rather than being embodied in the skills and knowledge of individual members of the labour force. A simple rewriting clari"es the intuition behind this speci"cation. Let us identify H with total years of schooling, and de"ne average years of schooling S"H/¸. Then we have
> K ? "A S@. ¸ ¸ Output per worker is a function of the capital}labour ratio and an index of labour quality. We can also begin to see that this particular speci"cation is implausible: it implies that raising average schooling from 0.1 to 0.2 years will raise output per worker by the same factor (2@ ) as raising average schooling from 4 to 8 years. Hence, speci"cation (1) imposes much higher returns to schooling at low levels of schooling than at high ones. With this in mind, various papers (probably starting with the "rst draft of Bils and Klenow, 1998) have explored an alternative form, with a more direct connection to conventional assumptions:
> K ? "A eX1. ¸ ¸
(2)
The key feature of this speci"cation is that an extra year of average schooling will always raise output in the same proportion, independently of the current level of schooling (and conditional on the capital stock). This is an attractive feature partly because it is shared with the standard semi-logarithmic formulation for individual earnings, widely used by labour economists to estimate returns to schooling from microeconomic survey data. In these earnings
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regressions, an extra year of an individual's schooling is often assumed to have the same proportional e!ect on earnings, regardless of the level of schooling. Simple extensions allow the return to vary across schooling levels. Several researchers have carried out cross-country accounting exercises that impute productivity e!ects in this fashion, using estimates of the return to schooling from micro data (Klenow and Rodriguez-Clare, 1997; Hall and Jones, 1999; Woessmann, 2000). Although the accounting exercise is informative for some questions, often we would like to test the productivity impact of schooling directly. Until that can be done, it is hard to gauge whether micro estimates of the private returns to schooling also capture the social returns, or are in fact driven by signalling e!ects and other aspects of labour market organisation. This is not a new point: Arrow (1973, p. 215) suggested that macroeconomic data might be needed to test the relevance of signalling and "ltering, admittedly while expressing some scepticism about the potential usefulness of this approach. More recently, Griliches (1997, p. S333) suggested that direct tests of productivity e!ects of education are likely to rely on the estimation of production functions. This means that, for the speci"c task of examining the Pritchett hypothesis, we want to estimate the parameter z, rather than impute it. This parameter should certainly not be interpreted as the social return, not least because it does not incorporate the opportunity cost of the resources used in educational provision. Yet it remains of considerable interest, since it provides a way of either con"rming the importance of education suggested by micro studies, or calling into question } as in Pritchett (1999) } the relevance of estimated private returns in thinking about social returns and education policy. Although Eq. (2) is a useful starting point, for our purposes it may not be su$ciently general. In particular, when we examine the cross-country evidence, we may want the productivity e!ect of an extra year's schooling to vary with the level of schooling. This suggests the use of a more #exible speci"cation, familiar from the work of Hall and Jones (1999) among others,
> K ? e(1. "A ¸ ¸
(3)
The function (S) tells us how the productivity e!ect of schooling varies with the level of schooling. Accounting exercises typically take the function to be piecewise linear, and use estimates of returns to schooling for di!erent levels (primary, secondary, tertiary) to impute the parameters. For testing the productivity of schooling directly, we might be interested in seeing what the data can tell us about the form of (S), and that is one aim of what follows. These issues are discussed in much greater detail in Temple (2000b). External e!ects of schooling provide another reason for looking at the macro data.
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Table 1 Regression Estimation Observations
1 OLS 78
2 LTS 78
3 OLS 78
4 OLS 78
5 LTS 78
6 RWLS 73
Constant
0.269 (2.68) 0.457 (5.09) 0.209 (0.96) 0.063 (0.72)
0.140
0.275 (3.46) 0.432 (5.08) 0.266 (1.38)
0.255 (2.67) 0.438 (4.86) 0.168 (0.80) 0.182 (1.90)
0.119
0.168 (2.87) 0.463 (7.18) 0.250 (1.50) 0.183 (2.23)
log K !log K 2 log ¸ !log ¸ 2 log S !log S 2
0.595 0.010 0.157
S !S 2
0.030 0.221
0.015 (0.52)
1/S !1/S 2 R Joint signi"cance
0.577
0.52 n.a.
n.a. n.a.
0.51 n.a.
0.078 (1.72)
0.030
0.056 (1.27)
0.54 0.19
n.a. n.a.
0.65 0.07
Heteroscedasticity-consistent t-ratios in parentheses (not available for least trimmed squares). Regression 6 excludes Botswana, Lesotho, Iraq, Rwanda and Saudi Arabia. The entry in the row &Joint signi"cance' is the p-value from a Wald test of joint signi"cance for the two schooling terms.
4. Benhabib and Spiegel revisited The paper by Benhabib and Spiegel (1994) is among the "rst to note the weak sample correlation between education and growth in the cross-country data. They found that changes in average schooling have little explanatory power for changes in output per worker. In previous work, I have pointed out that the use of least trimmed squares on the same data set gives very di!erent results (Temple, 1999). Yet rather than simply eliminating outliers, we may want to explore alternative, more general speci"cations that provide a good "t for the whole sample. Table 1 presents some results. Regression 1 replicates the estimates of Benhabib and Spiegel, while regression 2 is the same speci"cation estimated by least trimmed squares; in both cases, the log di!erence in output is explained by that in physical capital, the labour force, and average years of schooling, for the period 1965}1985. The di!erence in point estimates across the two estimation methods is marked; Temple (1999) explores this in more detail. The table also shows new results, based on the speci"cation in (3) above, and that sometimes allow for
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curvature in the function (S). The growth regression is now log > !log > " # (log K !log K )# (log ¸ !log ¸ ) 2 2 2 # (S )! (S )#. 2 When the simple case (S)"zS is adopted (regression 3) the estimate of the e!ect of education lacks precision. Regression 4 allows the function (S) to be non-linear,
(S)" # log S# (1/S). This is one of several non-linear speci"cations that I have tried; it is the one that works best, and has another advantage in that it nests Benhabib and Spiegel's formulation. The two schooling variables are jointly signi"cant only at the 20% level, but one attractive feature of these results is that the least trimmed squares estimates of the schooling coe$cients are similar to the OLS estimates (compare regressions 4 and 5). The LTS estimates can also be used to identify a small number of outliers, so that regression 6 excludes the "ve observations with the highest residuals in the LTS estimates. Now the two schooling terms are jointly signi"cant at the 10% level; but the magnitude of the e!ect is small, as I will discuss in the next section.
5. Where has all the education gone? Versions of an in#uential paper by Pritchett (1999) have been circulating for at least 5 years. Pritchett's paper was probably the "rst to try to reconcile aggregate evidence on education and growth with estimates of the returns to schooling based on micro data. Like Benhabib and Spiegel, he found that variation in the change in average schooling plays little role in explaining the cross-country variation in growth rates. Pritchett suggests that this somewhat undermines the conventional interpretation of estimates of returns to schooling, at least for developing countries. His speci"cation relies on the explicit construction of a measure of the value of human capital. This leads to the following speci"cation for output per worker:
> K ? "A (eP1!1)@, ¸ ¸
(4)
where r should, under Pritchett's assumptions, correspond to the return to schooling. He sets this return to 0.10 in line with micro estimates. When Eq. (4) is used as the basis for a growth regression, Pritchett "nds that is not signi"cantly di!erent from zero. In other words, one cannot reject the hypothesis that education has no role in output determination, even when using a measure of human capital that takes the micro evidence at face value.
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We can test the functional form by using a more general function that nests Eq. (4), and that potentially allows more #exibility in the return to schooling:
> K ? "A (eP1)@ (1!e\P1 )@ , ¸ ¸
(5)
where the Pritchett speci"cation corresponds to the restriction " ". Using data largely drawn from Pritchett (1999), and the same sample of 91 countries, I have estimated a growth regression based on Eq. (5). is estimated to be 0.90 with a standard error of 0.31 and found to be !0.11 with a standard error of 0.05. Not surprisingly, the restriction that " is strongly rejected (the Wald test yields a p-value of 0.01). Eqs. (4) and (5) are both somewhat unconventional and (5) in particular is hard to interpret. Given the evidence that the Pritchett speci"cation (4) may be unsatisfactory, I have considered alternatives suggested by the earlier discussion. In Table 2, regressions 7}10 show the results from estimating a growth regression based on the (S)"zS speci"cation, using Pritchett's data for output and capital per worker, and very similar data for schooling. The dependent variable is the log di!erence in output per worker, 1960}1987. I follow Pritchett in imposing constant returns to scale, and in examining the sensitivity of the results to the inclusion of regional dummies. The table reports least-squares results and those from robust methods. Although the least-squares estimates provide a relatively precise estimate of z, the point estimate is much lower in the least trimmed squares results. It is certainly lower than the typical range of private returns to schooling, which would tend to imply a coe$cient in the range 0.05 to 0.15. Hence, the data continue to provide some support for the Pritchett hypothesis, at least when using this speci"cation. Does an alternative speci"cation for (S) a!ect the results? I adopt the same non-linear form as before. The least-squares estimates are imprecise, and very di!erent to the least trimmed squares results, which suggest a stronger e!ect. When the LTS estimates are used to identify possible outliers, least squares gives more precise "ndings, and the two terms in schooling are jointly signi"cant at the 1% level. One pleasing aspect of these results is that the parameter estimates for the two schooling terms are now similar to those reported in Table 1, based on slightly di!erent data on output, capital and schooling, and a slightly di!erent sample of countries.
For comparison with Pritchett's results, I use his data for growth rates in output and capital per worker. The main set of results in his paper is based on schooling data from Barro and Lee (1993); I follow this choice, but use data from Barro and Lee (2000) to obtain the relevant data for Mali and the Central African Republic. Using this data set it is possible to replicate Pritchett's results almost exactly.
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Table 2 Regression Estimation Observations Regional dummies
7 OLS 91 No
0.000 (0.13) log (K /¸ )!log(K /¸ ) 0.490 2 2 (8.18) S !S 0.080 2 (2.56) log S !log S 2
8 LTS 91 No
Constant
0.025
9 OLS 91 Yes
10 LTS 91 Yes
!0.126 (0.51) 0.462 (5.97) 0.062 0.013 (1.76)
1/S !1/S 2 R Joint signi"cance
0.67 n.a.
n.a. n.a.
0.71 n.a.
n.a. n.a.
11 OLS 91 Yes
12 LTS 91 Yes
0.107 (0.39) 0.483 (6.25)
13 RWLS 84 Yes !0.325 (1.24) 0.585 (14.0)
0.011 (0.09) 0.035 (0.56)
0.170
0.71 0.30
n.a. n.a.
0.041
0.181 (1.83) 0.066 (1.40) 0.85 0.01
Regression 13 excludes Hong Kong, Haiti, Indonesia, Jordan, Mozambique, the Democratic Republic of Congo (formerly Zaire) and Zimbabwe. See Table 1 for other notes.
This does not yet tell us anything interesting, in that I have not discussed the magnitude of the e!ect. As noted by Hall and Jones (1999), the proportional impact of an additional year of average schooling on labour productivity, expressed as a &return', can be assessed by evaluating the derivative of (S) at di!erent levels of average schooling. When this is done, the parameter estimates in Tables 1 and 2 suggest that the impact of schooling is close to micro estimates only at low levels of average schooling (0}3 years) and is below 5% for higher levels. It continues to be quite hard to reject the Pritchett hypothesis.
6. An alternative framework All the empirical work considered thus far is a little unsatisfactory in one respect or another. Some previous results are sensitive to the inclusion or exclusion of a handful of observations, nearly all are sensitive to decisions about functional form, and many non-linear speci"cations work rather badly. The standard frameworks are also unsatisfactory from a theoretical point of view, in Note that this calculation, like the growth regressions, holds physical capital constant. Allowing physical capital to be endogenously determined will tend to raise the overall impact of schooling. Under the assumptions of perfect international capital mobility and Cobb}Douglas technology, it is easy to calculate the overall impact, and show that it depends positively on the capital share.
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that the correspondence between the simple aggregate models and the micro evidence is unclear, and this makes the estimated parameters harder to interpret. For example, the case that parameters in an aggregate production function correspond to either private or social returns to schooling is not straightforward, and there are several strong assumptions involved. An alternative way forward is suggested by the work of Jenkins (1995). She examines the role of education in labour productivity growth in the UK, using a production function to estimate directly the productivity impact of changes in educational attainment. In its most general form, this work is based on a Cobb}Douglas production function with constant returns to scale, but with an index of labour quality that is a CES aggregator of uneducated labour (¸ ) and educated labour (¸ ): >"AK?[¸\M#(1#)¸\M]\\? M. If we denote the fraction that are educated by h then we can rewrite this as
> K ? "A [(1!h)\M#(1#)h\M]\\? M ¸ ¸
(6)
and so as before output per worker depends on the capital}labour ratio and an index of labour quality. In practice, it is di$cult to identify the elasticity of substitution between skill types using the available data. An attractive simpli"cation is to adopt the case where educated and uneducated labour are perfect substitutes at a ratio captured by (1#). Then output per worker is
> K ? "A [(1!h)#(1#)h]\?. ¸ ¸
(7)
We can interpret this in terms of homogeneous e$ciency units of labour. Education raises the e$ciency units of labour possessed by a particular individual, across all tasks they perform } but these e$ciency units do not themselves di!er in their nature across individuals, and in principle a su$ciently large group of university professors could substitute for an Olympic athlete. It is a nice idea but clearly unrealistic, and later I will brie#y discuss one means of relaxing this assumption. The perfect substitutes case retains some attractive features. There is an immediate connection to the simpler interpretations of earnings functions based on survey data, in that the production technology ties down the returns to di!erent levels of schooling, independently of the relative supplies of educated and uneducated workers. The ratio of marginal products of the two types of labour is (1#) and so the parameter is easy to interpret. In particular, we are interested in trying to reject the hypothesis that "0, in other words the hypothesis that educated and uneducated workers are equally productive.
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As a proxy for the fraction of educated workers in the labour force, I use the proportion of the population that attained (but not necessarily completed) secondary school, based on the most recent version of the Barro and Lee (2000) data set. Eq. (7) is used to specify a growth regression that can be estimated by non-linear least squares, for a sample of 90 countries. The sample corresponds to that used by Pritchett, except that Malta is excluded due to lack of the relevant schooling data. This exercise yields a point estimate for of 1.52, implying that the marginal product of educated workers is 2.52 times that of the uneducated. However, the 95% con"dence interval for is wide at (!0.45, 3.49) and the parameter is signi"cantly di!erent from zero only at the 15% level. Once again it is hard to reject the Pritchett hypothesis. That said, one advantage of this result is that it clari"es the degree of uncertainty: obviously it is also hard to rule out a large e!ect of education on productivity, and this again suggests that one should be very wary about drawing "rm conclusions using the aggregate data. A remaining objection to this exercise is that, by only using the proportion of the labour force that has attained secondary schooling, the greater degree of information in average years of schooling is disregarded. One response is to adopt a "ner disaggregation of labour types. I have explored models that allow for three types of worker (no schooling, only secondary level attained, higher education attained) and four types (male/female, secondary schooling attained/unattained) and that also use information on di!erential labour force participation rates for men and women. As with the simpler model, the usual result is imprecise estimates. Other extensions are possible. One could explore the use of alternative data for the physical capital stock, or adopt &secondary school completed' as a proxy for the proportion of educated workers. A particularly interesting extension would be to make use of the literacy rate. In principle, this could allow a direct estimate of the marginal product of the literate relative to that of the illiterate. One reason this approach is attractive is that the literacy rate is a widely available measure of competence, and is thus a measure of schooling &output', rather than an input measure like years of schooling or educational expenditure. In other words, one could assess the productivity impact of educational
Using the proportion with some secondary schooling seems to work better than using the proportion of the population with any schooling. Although I should also note a reservation, namely that Barro and Lee (1993, p. 367) "nd the literacy rate to be highly correlated with the proportion of the population recorded in census/survey data as without schooling (in fact, they use the literacy rate to impute this proportion for some countries). They also note that education aims to provide a much wider range of skills. See Romer (1990) for a growth paper that does use data on literacy.
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achievement with less concern that the aggregate results are chie#y explained by the low quality of schooling in some poorer countries.
7. Conclusions The principal message of this paper is that the aggregate evidence on education and growth, for large samples of countries, continues to be clouded with uncertainty. This is particularly so when one tries to con"rm the importance of schooling found in estimates of earnings regressions, by using the change in average educational attainment to explain growth at the aggregate level. The earlier empirical work on this topic is fragile in some respects, but the importance of education hardly leaps out of the cross-country data. With some patience and ingenuity, apparently precise estimates of schooling's e!ect can be extracted, but the degree of data mining involved here (such as the investigation of several non-linear speci"cations, and the non-random exclusion of observations) renders the "ndings about parameters tentative to say the least. But I should also draw attention to another danger in interpreting the evidence as a whole. Some results are occasionally used to argue that schooling does not matter for growth. This is a mistake, for at least three reasons. Such a statement ignores the evidence for the importance of the level of schooling (rather than the change in schooling) in explaining subsequent growth; it ignores the high standard errors typically associated with the relevant parameters; and it also fails to note that the evidence for the claim &schooling has no e!ect' is often just as unreliable as that for any other claim. How can the situation be improved? Time will help, as more and better data become available. There could also be high returns to revising existing data, as in Barro and Lee (2000), and the careful study of the data for the OECD countries that is included in the work of de la Fuente and Domenech (2000). At least until we have greater con"dence in the data, analysis of sensitivity to measurement error should also be a key element of work in this "eld, as in Krueger and Lindahl (1998). Another major improvement might lie in greater attention to empirical frameworks and their interpretation, for both the micro- and macro data. Estimation of the nth earnings function or growth regression may ultimately be less useful and informative than a new approach to exploring these questions, or a new interpretation of the existing body of work.
Another possible extension would be to make a more plausible assumption about the elasticity of substitution between di!erent skill types, by drawing on previous studies. However, it might be di$cult to establish an uncontroversial "gure for the elasticity at this level of aggregation. See Woessmann (2000) for a useful and detailed review of measurement issues.
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It is perhaps also becoming clear that the questions currently driving this research are not necessarily the most interesting in the "eld. Measuring the impact of education on productivity is important, perhaps essential; but it is not clear to what extent decisions about education spending will ever be informed by empirical evidence, especially given that policy on education has (or should have) many ends other than raising productivity. A perhaps more interesting task for future research is to explore the "ne detail of the institutional and incentive structure that best allocates a "xed amount of educational expenditure. These questions have an immediate connection to policy, and the need for further research in this area is apparent more than ever.
Acknowledgements Various people have made helpful contributions. I am particularly grateful to Antonio Ciccone, Damon Clark, Angel de la Fuente, Sir James Mirrlees, Stephen Nickell, Stephen Redding and Ludger Woessmann for helpful comments and discussion, and to Jess Benhabib and Lant Pritchett for supplying data.
References Arrow, K.J., 1973. Higher education as a "lter. Journal of Public Economics 2, 193}216. Barro, R.J., Lee, J.-W., 1993. International comparisons of educational attainment. Journal of Monetary Economics 32, 363}394. Barro, R.J., Lee, J.-W., 2000. International data on educational attainment: Updates and implications. Working paper no. 42, CID, Harvard. Oxford Economic Papers, July 2001, forthcoming. Benhabib, J., Spiegel, M.M., 1994. The role of human capital in economic development: Evidence from aggregate cross-country data. Journal of Monetary Economics 34, 143}173. Bils, M., Klenow, P.J., 1998. Does schooling cause growth or the other way around? Working paper no. 6393, NBER, Cambridge, MA. de la Fuente, A., Domenech, R., 2000. Human capital in growth regressions: How much di!erence does data quality make? Discussion paper no. 2466, CEPR, London. Griliches, Z., 1997. Education, human capital and growth: A personal perspective. Journal of Labor Economics 15, S330}344. Hall, R.E., Jones, C.I., 1999. Why do some countries produce so much more output per worker than others? Quarterly Journal of Economics 114, 83}116. Jenkins, H., 1995. Education and production in the United Kingdom. Discussion paper no. 101, Nu$eld College, Oxford. Klenow, P.J., Rodriguez-Clare, A., 1997. The neoclassical revival in growth economics: Has it gone too far? NBER macroeconomics annual, 73}103. Krueger, A.B., Lindahl, M., 1998. Education for growth: Why and for whom? Manuscript, Princeton University, Princeton, NJ. Pritchett, L., 1999. Where has all the education gone? Manuscript, World Bank, Washington, DC. Romer, P.M., 1990. Human capital and growth: Theory and evidence. Carnegie-Rochester Conference Series on Public Policy 32, 251}286.
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Temple, J.R.W., 1999. A positive e!ect of human capital on growth. Economics Letters 65, 131}134. Temple, J.R.W., 2000a. Growth regressions and what the textbooks don't tell you. Bulletin of Economic Research 52, 181}205. Temple, J.R.W., 2000b. Growth e!ects of education and social capital in the OECD countries. Manuscript, University of Oxford. Woessmann, L., 2000. Specifying human capital: A review, some extensions, and development e!ects. Manuscript, Kiel Institute of World Economics.