Jean-Louis Arcand

Marcel Dagenais

Errors in Variables and the Empirics of Economic Growth

Keywords: errors in variables, economic growth JEL: C52, C21, O41

, Sachs and Warner (1997a), Easterly and Levine (1997), Levine and Zervos (1998)-and …nd that many of their results are "fragile". We argue that the application of our estimator to cross-sectional empirical studies of the determinants of growth yields important insights which may qualify previous …ndings in the literature, especially given the errors in variables problems which are known to plague commonly used cross-sectional datasets.

Introduction 1.Motivation

In two celebrated articles, [START_REF] Barro | Economic Growth in a Cross Section of Countries[END_REF] (henceforth, Barro) and [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF] (henceforth, MRW) provided in ‡uential empirical contributions that largely shaped the stylized facts accepted by most economists as to the determinants of economic growth. In another contribution, Sachs and Warner (1997a) and Sachs and Warner (1997b) (henceforth, SW) provided widely cited evidence on the fundamental factors that determine growth as well as the sources of slow growth in the economies of sub-saharan Africa. This paper considers whether the results reported by Barro, MRW and SW, as well as two other in ‡uential papers - [START_REF] Easterly | Africa's Growth Tragedy: Policies and Ethnic Divisions[END_REF] (henceforth, EL), and [START_REF] Levine | Stock Markets, Banks, and Economic Growth[END_REF] (henceforth, LZ)-We acknowledge the …nancial support of the PARADI project, funded by a grant from the Canadian International Development Agency (CIDA). We also wish to thank numerous seminar participants and an anonymous referee. The usual disclaimer applies.

y CERDI-CNRS, Université d'Auvergne and European Development Network (EUDN). 65 boulevard François Mitterrand, 63000, Clermont Ferrand, France. Email: arcandjl@alum.mit.edu.

z Marcel Dagenais died on 14 February, 2001. This paper is dedicated to his memory. He was professor of economics at the Université de Montréal. He is sorely missed by all those who knew him and had the good fortune to be touched by his rare combination of joie de vivre and analytical rigour.

are signi…cantly a¤ected by an error in variables problem, and whether correcting for this problem changes our perception of the forces driving economic growth.

These papers touch on or explicitly test many of the fundamental questions associated with the empirics of long-run economic growth, including: conditional convergence (all of the papers), the role of government (Barro), the human capital augmented Solow model (MRW), geography (SW), ethnolinguistic fragmentation as an explanation for low growth in sub-saharan Africa (EL and SW), and stock markets and banking (LZ). While this list of growth topics is far from being exhaustive, we would argue that it constitutes a fairly representative sample of the issues that have occupied growth empiricists over the past decade and a half.

The problem and a potential solution

Despite the care which was put into the construction of the data used in these papers, it is widely believed that cross-sectional international data are plagued by errors in variables (EV). [START_REF] Srinivasan | Data Base for Development Analysis: An Overview[END_REF], for example, does not mince his words: "The disturbing conclusion emerging... is that the situation with respect to the quality, coverage, intertemporal and international comparability of published data on vital aspects of the development process is still abysmal in spite of decades of e¤orts at improvements." 1 If the regressors included in commonly estimated growth rate or per capita GDP equations di¤er from the "true" regressors because of EV, then the regression error structure does not satisfy the usual Gauss-Markov conditions and the estimated parameters will be inconsistent. 2 In this paper we reconsider widely-publicized cross-sectional results in the economic growth literature in the light of the "higher moments"instrumental variables (IV) estimator, developed by [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF], which is robust, under quite reasonable assumptions, to EV. We test the null-hypothesis of the absence of EV using a Hausman-type test, and assess the validity of our proposed IVs using both the standard Sargan test of the overidentifying restrictions and a recent instrument validity test proposed by Hahn and Hausman (2002a).

Is it worth the trouble?

While properly accounting for EV in empirical growth regressions is interesting from the econometric perspective, the empirically-minded reader will be wondering out loud "so what?"After all, the proof of the pudding is in the eating: does the estimator we propose yield results which di¤er substantially enough from the OLS results to warrant its use in practice? Do our results change one's understanding of the empirics of economic growth? If the answer is "no,"then we have merely uncovered an empirical "fact"(the presence of EV in less developed country (LDC) national income accounting data) which was already well-known, and our econometric artillery 1 p. 23-24. In the context of his review of empirical studies of the e¤ects of trade policy orientation on growth, [START_REF] Edwards | Openness, Trade Liberalization, and Growth in Developing Countries[END_REF]Edwards ( ) (p. 1390) ) notes that "in order to gain further insights into these issues, it is fundamental to adopt econometric methodologies that deal speci…cally with errors in variables, that investigate formally the robustness of speci…c results, and that rely systematically on sensitivity analysis... from an econometric perspective, one of the most serious shortcomings of the cross-section papers discussed [in his survey]... is the lack of e¤orts to implement in a systematic way a battery of tests that deal with the degree of robustness (or fragility) of the results."

2 [START_REF] Nelson | Vector Attenuation Bias in the Classical Errors-in-Variables Model[END_REF] consider the classic case of attenuation bias of OLS estimates when several variables are a¤ected by measurement error. is merely overkill. If the answer is "yes", then the problem of EV is su¢ ciently important to merit its careful consideration in formulating and testing hypotheses regarding the determinants of economic growth.

Unsurprisingly, our answer to the above questions is "yes." We argue that many of our results using the higher moments estimator are su¢ ciently di¤erent from those in the papers we consider to warrant (i) more careful scrutiny of the cross-sectional data and (ii) the application of our estimator in preference to OLS. Our results suggest that several of the conclusions regarding the empirics of economic growth drawn by the papers considered here may not be as strong as they may appear. Table 1 concisely summarizes our main empirical …ndings.

Errors in variables versus speci…cation error

We are not alone in subjecting a sample of empirical results from the large cross-sectional growth literature to some form of test for their robustness. For example, [START_REF] Levine | A Sensitivity Analysis of Cross Country Growth Regressions[END_REF] use Leamer's extreme bounds analysis to show that almost all variables included in a plethora of commonly estimated cross-country growth rate regressions are not robust to variations in the set of explanatory variables. Their analysis can be interpreted as a test of the underlying structural model used to justify the inclusion or exclusion of certain variables from the estimated relationship. Another example is furnished by [START_REF] Temple | Robustness Tests of the Augmented Solow Model[END_REF] who considers the robustness of the MRW results to classical EV using the [START_REF] Klepper | Consistent Sets of Estimates for Regressions with Errors in All Variables[END_REF] reverse regression technique as well as classical method of moments estimators.

As in Temple, the analysis in this paper does not explicitly question the validity of the underlying model but focuses on the properties of the regressors included in the estimated equation. Of course, since growth equations may su¤er from speci…cation error, and since the test procedure we use can be interpreted as a variant on the well-known Hausman instrumental variables (IV) speci…cation test, it is likely that a portion of what we ascribe to EV can be attributed to errors in speci…cation. The empirical researcher in search of a means of testing the robustness of her results is therefore left with the following choice : (i) the use of Leamer's extreme bounds analysis if EV may be considered relatively unimportant and the major concern is the speci…cation of the estimated equation, (ii) the use of our estimator if the validity of the speci…cation is less in doubt than is the "purity" of the underlying data. Keeping this last caveat in mind, it would seem to be of considerable interest to subject empirical results such as those considered here to a test for the presence of EV.

2 The Estimator and the test for errors in variables 2.1 The econometric problem and previous solutions using higher moments

The data employed in most growth regressions is almost certainly subject to EV. Since even aggregate time series for industrialized countries are, according to [START_REF] Morgenstern | On the Accuracy of Economic Observations[END_REF], [START_REF] Langaskens | A New Method to Estimate Measurement Errors in National Income Account Statistics: The Belgian Case[END_REF] and Lipsey and Tice (1989), subject to important EV, a fortiori one would expect this problem to plague a broad cross-section of data from 100 di¤erent countries, many of them LDCs where national income accounting practice is sketchy at best. The standard response of the econometrician to an EV problem is of course to resort to IV techniques in order to obtain consistent parameter estimates.

The problems with this approach in the context of cross-sectional growth regressions are that (i) some of the potential excluded instruments may in fact be correlated with the regression errors and should probably be included themselves as endogenous variables in a more complete simultaneous structural form, leading to the need for additional (unavailable) exogenous instruments, (ii) eligible instruments may simply not be available for a broad enough cross section of countries to permit the use of IV techniques, and (iii) it may not be feasible to verify that the proposed instruments satisfy the desired orthogonality assumptions because of the absence of overidenti…cation.

Given the three points raised above, this paper proposes an alternative to standard IV techniques that has received no attention in the empirical growth literature: the use of consistent estimators based on sample moments of order higher than two. There are a number of such estimators available. Estimators based on third-order sample moments have been proposed by [START_REF] Geary | Inherent Relations Between Random Variables[END_REF], [START_REF] Drion | Estimation of the Parameters of a Straight Line and of the Variance of the Variables, If They Are Both Subject to Error[END_REF] and [START_REF] Pal | Consistent Moment Estimators of Regression Coe¢ cients in the Presence of Errors in Variables[END_REF], while [START_REF] Geary | Inherent Relations Between Random Variables[END_REF] and [START_REF] Pal | Consistent Moment Estimators of Regression Coe¢ cients in the Presence of Errors in Variables[END_REF] also propose estimators based on fourth-order cumulants.

The problem with these estimators, however, is that their behavior is substantially more erratic than the corresponding least squares estimators (see, e.g., [START_REF] Kendall | The Advanced Theory of Statistics[END_REF], [START_REF] Malinvaud | Méthodes Statistiques de l'Econométrie[END_REF]). One possible solution to this endemic instability, which is adopted in this paper, is to use a higher moments estimator suggested by [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF] which is essentially a linear matrix-weighted combination of third and fourth moment estimators. As pointed out by [START_REF] Pal | Consistent Moment Estimators of Regression Coe¢ cients in the Presence of Errors in Variables[END_REF], all higher moments estimators can be considered as a special type of IV estimators where the instruments are given by functions of the original variables raised to some power.

The estimator

Basic notation

The typical growth regression can be written in matrix notation as

y 1 = e y 2 + u; (1)
where e y 2 is an N r matrix of exogenous explanatory variables measured without error, with empirical distribution such that p lim e y 2 0 e y 2 N = Q, where Q is a …nite non-singular matrix, N is sample size, and where y 1 is the N 1 vector of observations of the dependent variable. For notational convenience, equation (1) corresponds to the growth regression in which all variables known ex ante to be measured without error (such as the intercept term or continent dummies) have been "partialled out" of the speci…cation. The N 1 vector u is assumed to be distributed N (0; 2 u I N ). The r 1 vector and 2 u are unknown parameters. The problem of EV is posed econometrically by assuming that e y 2 is unobservable and that instead one observes the matrix y 2 , where

y 2 = e y 2 + V (2)
and V is an N r matrix of normally distributed errors in the variables. Furthermore, we assume that V ar [V ec(V )] = I N where V ar [:] stands for the covariance matrix and where is an r r symmetric positive de…nite matrix. This assumption implies that (i) the errors in the variables are independent between observations, but (ii) not between variables.

Discussion of the underlying assumptions

In the context of a set of regressors based on a heterogeneous group of approximately 100 di¤erent countries, both these assumptions appear to be reasonable as a …rst approximation.

Indeed, there is no reason to believe that errors in the statistical procedures used to measure, say, investment, are signi…cantly correlated across countries (this is particularly true given the great heterogeneity in the standard of accounting practices among LDCs and DCs), although problems might arise within subsets of countries which follow similar accounting procedures.

Without studying the national income accounting procedures of all countries in the sample, however, this assumption appears reasonable, prima facie.3

Second, there is good reason to believe that an LDC where the measurement of one macroeconomic aggregate is subject to error will also display similar lacunae in the measurement of other series. Finally, the assumption also implies that, for a given variable, the errors in measurement are homoskedastic. This assumption may not be so reasonable if one believes either that the accuracy of the data is an increasing function of the level of development, or that the accuracy of the data is correlated with an unknown set of variables (which may include some of the regressors). In the …rst case, a correction based speci…cally on the hypothesized relationship between the magnitude of the EV and the explanatory variable in question is called for. In the second case a correction for heteroskedasticity of unknown form (e.g., [START_REF] White | A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity[END_REF]) might seem to be appropriate.4

The proposed instrument set

Given the above considerations, it seems most appropriate to use a regression estimator which remains consistent in the presence of EV. The estimator used here is one of the estimators suggested by [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF], where the matrix of feasible instruments, denoted by Z = (z 1 ; z 2 ; z 3 ; z 4 ; z 5 ; z 6 ; z 7 ), is given by:

z 1 = y 2 y 2 ; z 2 = y 2 y 1 ; z 3 = y 1 y 1 ; z 4 = y 2 y 2 y 2 3y 2 y 0 2 y 2 N I r ; z 5 = y 2 y 2 y 1 2y 2 y 0 2 y 1 N I r y 1 0 r y 0 2 y 1 N I r ; z 6 = y 2 y 1 y 1 y 2 y 0 1 y 1 N y 1 y 0 1 y 2 N ; z 7 = y 1 y 1 y 1 3y 1 y 0 1 y 1 N ;
and where the symbol designates the Hadamard element-by-element matrix multiplication operator, I r is an r-dimensional identity matrix, and r is an r 1 vector of ones. Detailed proofs of the orthogonality of these instruments with respect to the disturbance term are provided in [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF].

The proposed estimator is a [START_REF] Fuller | Some Properties of a Modi…cation of the Limited Information Estimator[END_REF] modi…ed IV estimator, with the "Fuller constant" set equal to 1. This estimator possesses …nite moments for all values of the "concentration parameter" associated with the reduced forms, as well as good small sample properties. Moreover, [START_REF] Hahn | Estimation with Weak Instruments: Accuracy of Higher Order Bias and MSE Approximations[END_REF] have provided extensive Montecarlo experiment results that show that this estimator performs well when compared to other prominent IV estimators, under weak instruments, an issue we shall address below.

The resulting "higher moments" estimator, which we shall denote by H , is consistent when there are EV and is also much less erratic than other estimators based on sample moments of order higher than two heretofore suggested in the literature.5 Details concerning the construction of the Fuller estimator based on these instruments, as well as the variance-covariance matrix, are provided in [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF].

Testing for errors in variables

The test for the presence of EV that we apply is a [START_REF] Hausman | Speci…cation Tests in Econometrics[END_REF] test. This asymptotic test is most easily performed by the following procedure. First, run the augmented regression by OLS:

y 1 = y 2 + ŵ H + " (3)
where ŵ = y 2 P Z y 2 , P Z = Z(Z 0 Z) 1 Z 0 ; H is a vector of parameters and " is the vector of the regression errors. Second, test H = 0 using the usual F -test.

If there are no errors in the variables, y 2 = e y 2 and y 1 = y 2 + u. Therefore under the null hypothesis of no errors in the variables, " = u and H = 0.

In what follows, we will also apply the equivalent t-test to each individual right-hand-side (RHS) variable so as to attempt to isolate the source of any existing EV-induced bias. From equation (3), it is readily seen that the power of the test that any given element of H is equal to zero and hence that the associated variable has no EV, will depend in part on the collinearity between the columns of the ! matrix. As for any other statistical test, a low p-value associated with an individual element of H may lead one to reject the null of the absence of EV in the corresponding variable with a certain degree of con…dence; but a high p-value may not indicate its absence. It may stem simply from a lack of power of the test, due for example to a problem of collinearity. Furthermore, as for any other test, the validity of our test is conditional on the fact that the model is correctly speci…ed.

Montecarlo evidence on the performance of the estimator

Since, in [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF] We began by running a regression of the level of GDP per capita in 1985 on a constant, the population growth rate, the investment ratio and the enrollment rate (all variables are in logs), using the 98 observations in the MRW dataset. Variables were then scaled so that the intercept and all coe¢ cients were equal to 1. We then used these scaled variables to generate levels of the dependent variable according to:

y 1 = 1 + e y 2 +e y 3 +e y 4 + u;
where u is the normal regression error term. Normal random errors were then added to e y 3 so as to yield y 3 =e y 3 + V , and the variance of u was set so as to ensure an R 2 of 0.778, which corresponds to the empirical value in the baseline MRW regression. As in the original Montecarlo experiments presented in [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF], the ratio of the variance of measurement errors V to the variance of e y 3 was initially set equal to 0.3. In experiment 1, we instrumented all variables using Z, whereas in experiment 2, only y 3 was assumed to be subject to EV, with e y 2 and e y 4 serving as their own instruments.

The main results that emerge from Montecarlo experiments 1 and 2, reported in Table 2, when one instruments using Z = (z 1 ; z 4 ), are the following. First, bias is substantially smaller for H than for OLS . Second, Root Mean-Squared Error (RMSE) is generally higher for H than for OLS , though the RMSE can be reduced for the H estimator if one limits the number of variables assumed to be measured with error. Third, the size of type I errors are extremely high for OLS , with respect to the theoretical value of 5%, whereas the corresponding …gures are quite close to the true value for H . 6 Fourth, the power of the joint EV test is quite low with the sample size considered here (this is in contrast to the larger sample sizes considered in [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF]). From the empirical perspective, this implies, if one rejects the null of the absence of EV, that there is a strong presumption that they are present.

As with the Montecarlo experiments reported in [START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF], the estimator based on the full set of instruments (z 1 through z 7 ) performs less well than the estimator based on z 1 and z 4 alone. This is true in terms of bias, RMSE and the size of type I errors.

The power of the joint EV tests is the same for both estimators.7 For all of these reasons, we shall present empirical results based on instrumenting with z 1 and z 4 alone.

In experiment 3, we set the coe¢ cient on e y 4 equal to zero. The correlation between e y 3 and e y 4 in the results presented in Table 3 is equal to 0.633, and we increase the relative magnitude of the variance of V , from 0.3 to 0.8. Despite the fact that e y 4 was not a¤ected by an EV problem, and that, in a simple regression asymptotic bias disappears when the coe¢ cient in question is equal to zero, the bias of the OLS estimate of 4 was substantial (0.204). Most importantly, the size of the type I error associated with the test that 4 = 0 was large (0.398),

indicating that a t-test, based on a 95% con…dence level of the null that 4 = 0, would have been incorrectly rejected almost 40% of the time. Tu put it another way: in the Montecarlo experiment corresponding to Table 3, the OLS estimate of 4 was equal to 0.204, with an associated t-statistic of 1.747 (p-value = 0.081), whereas the higher moments estimate of 4 was 0.073, with a t-statistic of 0.266 (p-value = 0.790).

In the context of the cross-sectional data used in the papers considered here, where the explanatory variables are often highly correlated and the magnitude of EV on at least some variables is likely to be large, this experiment reveals that it is quite easy for one to conclude that a given variable is statistically signi…cant using OLS, when in fact its statistical signi…cance is spurious and stems from an EV problem a¤ecting another variable.

Instrument admissibility and instrument choice

A potential concern with the use of higher moments of the explanatory variables themselves as excluded instruments is that they may su¤er from a "weak instruments" problem, an issue that has come to the forefront of the econometrics literature in recent years. As is by now well known, weak instruments can lead to bias in IV estimation, and this bias does not vanish even with large sample sizes.8 Note that a …rst test of the underlying orthogonality conditions will be provided by the usual Sargan test of the overidentifying restrictions, although a more recent and robust diagnostic test of instrument validity will also be used, given that the Sargan test is known to possess poor size properties.

The Hahn-Hausman test

A recent procedure proposed by Hahn and Hausman (2002a) provides a joint test of instrument orthogonality and instrument relevance (i.e. their "strength"). In terms of instrument relevance, and in contrast to the by-now standard Shea (1997) partial R 2 and F statistics diagnostics on the reduced forms, which are based on the null hypothesis of weak instruments,

Hahn and Hausman base their procedure on the null of strong instruments.

Consider the Bias-adjusted 2SLS estimator (B2SLS), which is an example of a k-class estimator, of which conventional 2SLS, Limited Information Maximum Likelihood (LIML) and the Fuller estimator are special cases. As an illustration, consider the particularly simple situation in which y 2 is a scalar (i.e. r = 1). Then the k-class instrumental variables estimator for is de…ned by

b B2SLS = y 0 2 P Z y 1 y 0 2 M Z y 1 y 0 2 P Z y 2 y 0 2 M Z y 2 ;
where

M Z = I K P Z is the orthogonal complement to P Z . For = K 2 N 1 K 2 N
, where K denotes the number of excluded IVs, we obtain the B2SLS estimator proposed by [START_REF] Donald | Choosing the Number of Instruments[END_REF], whereas = 0 corresponds to conventional 2SLS.9

The Hahn and Hausman (2002a) test for the validity of the IVs is constructed by running the B2SLS regression in its usual "forward" form, and comparing the result to that obtained by running the "reverse" regression, in which the jointly endogenous RHS variable y 2 is moved to the left-hand-side (LHS), and the dependent variable y 1 is entered on the RHS. The reverse B2SLS estimator is given by

b RB2SLS = y 0 1 P Z y 1 y 0 1 M Z y 1 y 0 1 P Z y 2 y 0 1 M Z y 2 :
The basis for the Hahn-Hausman test is that, if the speci…cation is correct and the instruments are "strong", standard …rst-order asymptotics imply that there will be very little di¤erence between the results one obtains using the forward (b B2SLS ) or reverse regressions (b RB2SLS ).

The test, referred to as the m 2 test statistic, is standardized by using a second-order expression for the variance of the di¤erence between the forward and reverse estimators, and can be read as a simple t-statistic. 10 More formally,

m 2 = b d 2 = p b w 2 where b d 2 = p N (b B2SLS b RB2SLS ); and b w 2 = 2(K 1)(N 1) 2 4 ";LIM L (N 1)b 2 LIM L h y 0 2 P Z y 2 ( K 1 N K )y 0 2 M Z y 2 i 2 ;
where b LIM L and 2 ";LIM L are the LIML estimates of and 2 " . 11 In most of the situations considered in this paper, there will be more than one RHS variable potentially measured with error, and the m 2 test statistic is therefore not applicable. In this case, Hahn and Hausman (2002a) show that there are r 1 di¤erent reverse regressions that can be run but that no gains in e¢ ciency are achieved by stacking the various parameter estimates obtained through these di¤erent normalizations: the best that one can do is to use one di¤erence between forward and reverse results. If we arbitrarily consider the reverse regression in which y 2 is put on the LHS and y 1 on the RHS, with y j ; j > 3 denoting the other RHS variables subject to EV, then the m 3 test statistic is given by m

3 = b d 3 = p b w 3 where b d 3 = p N (b 2;B2SLS b 2;RB2SLS ),
and:

12 b w 3 = 2 K 1 N K 2 4 i=N X i=1 0 @ y 1i j=r X j=2 j;LIM L y ji 1 A 2 3 5 2 2 2;LIM L 2 4 y 0 2 P Z y 2 ( K 1 N K )y 0 2 M Z y 2 j=r X j=3 (y 0 2 P Z y j ( K 1 N K ))y 0 2 M Z y j ) 2 y 0 j P Z y j ( K 1 N K ))y 0 j M Z y j 3 5 2 :

The Andrews IV selection procedure

In the empirical applications that follow, we will sometimes be confronted with situations in which a speci…cation in which all variables are assumed to be a¤ected by EV, and other speci…cations in which only a subset of variables are assumed to be a¤ected by EV are "equivalent" in the sense that both are not rejected by the Sargan test of the overidentifying restrictions or the Hahn-Hausman test. In such cases, there is a need for a statistical basis upon which to choose amongst these various speci…cations, although the low power of the joint EV test sometimes renders this exercise less straightforward than one might wish.

The solution we adopt is the Andrews (1999) instrument selection procedure, which is based on the Bayesian (BIC), Akaike (AIC) and Hannan-Quinn (HQ) model selection information criteria. These tests are based on the J test statistic for the over-identifying restrictions, from which one subtracts a "bonus term" that rewards instrument sets that use more exclusion restrictions. 13 Though the Andrews test is in principle geared towards weeding out invalid instruments, the author suggests that one should limit its application to small sets of potential instruments. The instrument set which minimizes the instrument selection criteria (IV-BIC, IV-AIC and IV-HQIC) is then the preferred choice.

Empirical Results

For the empirical results presented below, our approach was as follows. First, after reproducing the OLS results reported in the original paper, we applied the higher moments estimator based on Z = (z 1 ; z 4 ). We also systematically computed the partial R 2 and F -statistics for the "partialled out" reduced forms so as to provide a rough indication, based on the usual rules of thumb, of those RHS variables in the regressions for which our proposed instrument set was "weak". 14 Second, if the joint EV test rejected the absence of EV with a relatively low p-value, or if the same was true for the t-test on any individual variable (keeping in mind the relatively low power of the joint test as revealed by our Montecarlo experiments), we allowed those variables that appeared less a¤ected by EV (as indicated by a particularly high p-value on the individual t-test for EV) to act as their own instruments. This process was carried out subject to the condition that the Sargan test did not reject at the 10% level, using the Andrews IV selection procedure in order to rank di¤erent potential instrument sets.

Ideally, we also attempted to ensure that the Hahn-Hausman test did not reject at the 10% level, though there were several instances where the Sargan test did not reject while the Hahn-

Hausman test did: a priori, this indicates that the instruments contained in Z are likely to be orthogonal to the disturbance term, but that they are weak. Finite sample bias for our IV estimates is, in this case, a possibility, though its e¤ects should be limited by our use of the Fuller estimator.

3.1 Mankiw, Romer and Weil

Level regressions

Tables 4 and5 present our results for the MRW level regressions, while Tables 6 and7 present our results for the growth regressions. 15 Five aspects of the results presented in Table 4 are worth emphasizing. First, there is a strong indication of EV given the low p-value associated with the joint EV test, especially in light of the relatively low power of this test as revealed by our Montecarlo results. Second, it would appear that the coe¢ cient associated with the population growth plus technical change plus depreciation (n + g + ) variable is grossly underestimated (in absolute value) by OLS, and that this bias stems from EV on this variable. Third, our

proposed instrument set appears to be reasonably orthogonal to the error term, as shown by the p-value associated with the Sargan test, and there is little indication of a weak instruments problem on the basis of the m 3 test statistic. Fourth, the parameter restriction implied by the human capital augmented Solow model is soundly rejected at the 1% level. Thus, although the H estimate is in some sense less precise than the corresponding OLS estimate, the magnitude of the EV in the MRW level equation is su¢ cient for us to be able to reject one of the key restrictions implied by the human capital augmented Solow model. This suggests, contrary to what is claimed by MRW, that the human capital augmented Solow model does not provide a good explanation for observed cross-country di¤erences in per capita GDP, once EV are taken into account.

Finally, the preceding results are con…rmed when we allow the log investment ratio and log schooling to act as their own instruments (RHS of the Table ), as would seem reasonable on the basis of the individual t-tests for the presence of EV. The m 3 test statistic rises slightly with respect to the speci…cation in which all variables are allowed to be subject to EV, indicating that there may be a slight weak instruments problem in this speci…cation. On the other hand, all three Andrews IV selection criteria come down in favor of the speci…cation in which all variables are allowed to be subject to EV.

In Table 5, we impose the Solow restriction, despite its being rejected in Table 4. In this case, the joint EV test no longer rejects the null of the absence of EV. Note also that the estimated standard errors using H are much larger, and that both the Sargan and the m 3 test statistics reject: this is a pattern that sometimes emerges when EV concerns are not present and one can rely on the OLS coe¢ cient estimates. 16

Growth regressions

Table 6 presents the unrestricted growth regression results. When one allows all variables to be subject to EV, the joint EV test for the absence of EV does not reject. The standard errors of the H estimates are large when compared with their OLS counterparts and the point estimates of the coe¢ cients often fall substantially. However, both the Sargan and m 3 test statistics do not reject at the 10% level. In a second set of estimates, basing oneself on the individual t-tests for the presence of EV, all variables except the initial level of GDP per capita were allowed to act as their own instruments. In this case, the absence of EV on initial GDP per capita is strongly rejected (p-value = 0:029), while the Sargan and m 3 test statistics continue not to reject instrument validity. In comparison with the OLS results, these estimates based on the higher moments estimator yield a much lower annual rate of convergence, which is indistinguishable from zero at the usual levels of con…dence, as well as point estimates of the marginal impacts of the population growth rate and schooling that are 50% lower and also statistically indistinguishable from zero.

Given that the Solow restriction, in contrast to the level regressions, is not rejected in the results presented in Table 6, we present results for the restricted speci…cation in Table 7. As with the unrestricted speci…cation, it is only when initial GDP per capita is the only variable assumed to be a¤ected by EV that the EV test rejects. The gain in e¢ ciency furnished by the imposition of the Solow restriction allows one to obtain structural parameter estimates (of and ) that are measured relatively precisely and are of the same order of magnitude as those obtained using OLS. As with the unrestricted speci…cation, the main impact of EV is to bias the implied annual rate of convergence upward. Using OLS, one obtains OLS = 0:010 (tstatistic = 5:552) which is cut in half, to H = 0:006 (t-statistic = 1:725) once EV are controlled for. Given our Montecarlo results concerning the relatively low levels of bias yielded by the higher moments estimator, we believe that the 0.006 …gure is likely to be closer to the actual annual rate of convergence than is the 0.010 …gure. Note also that our proposed instrument set is not rejected, either by the Sargan or the m 3 test statistics.

1 6 While these di¤erences between the results of our tests for the unrestricted versus the restricted regressions may appear at …rst sight to be incoherent, they are readily explainable. Let e y3 (here, the n + g + variable) be measured with error: y3 = e y3 + V , and suppose that the investment ratio variable, denoted by e y2, is measured without signi…cant error (this is what is suggested by the corresponding p-values in Table 4). Assuming, for simplicity, that there is no correlation between e y2 and y3 (the actual correlation is not zero, but it is very small) the asymptotic bias of the OLS estimate of the coe¢ cient of y3 would depend on the magnitude of 2 V = 2 y 3 whereas the asymptotic bias of the coe¢ cient estimate in the restricted regression model -in this case the explanatory variable is (e y2 y3)-is given by 2

V =( 2 e y 2 + 2 y 3 ): It follows, therefore, that 2 V =( 2 e y 2 + 2 y 3 ) < 2 V = 2 y 3 :
Hence, one would expect the OLS coe¢ cient on the restricted model to be less biased asymptotically. This explains our failure to reject the null hypothesis of no EV in the restricted regression, while our test detected EV in the unrestricted regressions.

Barro

Tables 8 presents results for Barro, for the sample restricted to countries with initial GDP per capita (in 1960) greater than $1,000, which a priori should be less a¤ected by EV. The joint EV test rejects the null of the absence of EV, the Sargan test of the overidentifying restrictions does not reject, while the m 3 test, taken in conjunction with the Sargan result, indicates that our proposed instruments are weak. 17 The H coe¢ cients associated with initial GDP per capita and government consumption expenditures fall substantially, in absolute value, with respect to the OLS results, with the t-statistic associated with g c =y going from t OLS = 3:736 to t H = 0:421. 18 Barro's explanation for the deleterious impact of government consumption expenditures on economic growth was that "government consumption [as opposed to government investment] has no direct e¤ect on private productivity (or private property rights), but lowered saving and growth through the distorting e¤ects from taxation or government-expenditure programs." (p. 430) Our H results cast doubt on this …nding. Conversely, the absolute value of the coe¢ cient associated with PPI60DEV increases, as does the associated t-statistic.

In the RHS of the Table, initial GDP per capita and both enrollment rates are allowed to act as their own instruments, given the relatively high p-values on the associated individual tests for the absence of EV on these variables. The point estimates change very little, the coe¢ cient associated with government consumption expenditures continues to remain statistically indistinguishable from zero, the Sargan test still does not reject, while the m 3 test statistic falls substantially (though it remains statistically signi…cant at the usual levels of con…dence), indicating that a portion of the weak instruments problem has been eliminated, though there are still good reasons to continue implementing our IVs using the Fuller estimator.

Sachs and Warner

Results for SW are presented in Tables 9 and10. In Table 9, the joint EV test rejects the absence of EV, the Sargan test does not reject our proposed instrument set, whereas the m 3 test statistic indicates that our instruments are, however, weak. The H estimates of the coe¢ cients associated with initial GDP per economically active member of the population and the growth rate of the economically active population (minus the total growth rate of the population) fall substantially (in absolute value) and become statistically indistinguishable from zero. The H coe¢ cients associated with the landlocked and tropical climate dummies fall somewhat (both variables act as their own instruments) and are measured much less precisely than in the OLS case. All other coe¢ cients change relatively little and remain statistically signi…cant at the 1 7 Note also that the partial F -tests on the reduced forms for primary enrollment, government consumption expenditure, and PPI60DEV are all extremely low.

1 8 Note that we have found that correcting for EV using our higher moments estimator decreases the coe¢ cient on the initial level of GDP per capita, which would appear to contradict the received wisdom that EV biases coe¢ cients downwards. In reality, this nugget of received wisdom is, simply put, wrong. In a simple regression with a single explanatory variable, when there is measurement error on the said variable, the OLS estimator of the coe¢ cient associated with this variable is indeed asymptotically biased toward zero and this bias disappears when the true value of the coe¢ cient is zero. This is no longer the case, however, when there are several explanatory variables a¤ected by measurement errors, unless they are perfectly uncorrelated. The biases associated with errors of measurement can go in either direction for any given coe¢ cient, and this bias does not reduce to zero even if the true coe¢ cient is zero. See the illustration in [START_REF] Dagenais | Parameter Estimation in Regression Models with Errors in the Variables and Autocorrelated Disturbances[END_REF], as well as the bias results from the Montecarlo experiments reported in Tables 2 and3 of the present paper. usual levels of con…dence.

In Table 10, we allow the institutional quality index and the growth rate of the economically active population to act as their own instruments, on the basis of the individual t-statistics for the presence of EV reported in the previous table. We continued to instrument central government saving and natural resource exports, despite the relatively high p-values of the individual test for EV associated with these variables presented in Table 10, because failure to do so resulted in a sound rejection of instrument validity by the Hahn-Hausman m 3 test statistic.

Two di¤erences with respect to the results presented in Table 9 are apparent. First, the dummy for a tropical climate returns (as in the OLS results) to being negative and statistically signi…cant. Second, and most importantly, instrument validity is not rejected by the m 3 statistic.

The upshot is that, contrary to what is reported by SW, (i) there is no statistically signi…cant conditional convergence result, and (ii) being landlocked does not penalize countries in terms of their growth rate of GDP per capita, ceteris paribus. This last …nding casts doubt on one of the main empirical results advanced by proponents of the "geographical" (as opposed to "institutional") view of the determinants of economic growth. 19

Easterly and Levine

Results for EL are presented in Tables 11 and12. In Table 11 we present results in which only the decade and continent dummies are assumed to be measured without error. Point estimates change very little with respect to the OLS results, with the notable exception of the coe¢ cient associated with ethnolinguistic fragmentation, which becomes statistically insigni…cant. The joint test for the presence of EV, though it does not reject at the 10% level, suggests the presence of EV, especially when considered along with the individual t-tests for initial GDP per capita, the same variable squared, and ethnolinguistic fragmentation. On the other hand, the Sargan test rejects the validity of the overidentifying restrictions, whereas the m 3 statistic suggests a severe weak instruments problem.

In Table 12, we allow all variables, apart from initial GDP per capita and ethnolinguistic fragmentation, to act as their own instruments, on the basis of the individual t-tests for the presence of EV reported in Table 11. The di¤erences between the OLS and the H results, in terms both of the coe¢ cient estimates and the associated t-statistics, are extremely small, apart from ethnolinguistic fragmentation, for which the H coe¢ cient is half the value of its OLS counterpart and is statistically indistinguishable from zero. The joint EV test rejects the null of the absence of EV, the Sargan test does not reject (if one takes a 10% critical p-value) and, most importantly, the m 3 test statistic does not reject. Our results based on the higher moments estimator therefore suggest, contrary to the main argument given in EL, that ethnolinguistic fragmentation is not one of the main reasons behind the poor growth performance of sub-saharan Africa.

1 9 Note that while all three Andrews IV selection criteria come down in favor of the estimates presented in Table 9, the rejection of IV validity by the Hahn-Hausman test in Table 9 leads us to prefer the results presented in Table 10.

Levine and Zervos

Results for LZ are presented in Table 13. In the LHS of the table, we allow all variables to be a¤ected by EV. The point estimates change very little with respect to the OLS results, with the exception of government investment expenditures ("government", in the Table ), which was statistically insigni…cant, at the usual levels of con…dence, in the OLS results, and becomes even less signi…cant once the H estimator is applied. The joint EV test does not reject, and the same is true of the test of the overidentifying restrictions and the Hahn-Hausman test.

On the basis of the individual t-tests for the presence of EV presented in the LHS of the Table, the RHS presents results in which only initial GDP per capita and government are allowed to be a¤ected by EV, with all other variables serving as their own instruments. Both the Sargan and the Hahn-Hausman tests fail to reject the null of instrument validity, while the p-value of the joint EV test falls substantially, with the individual test for EV on government becoming statistically signi…cant at the 10% level of con…dence. On the other hand, all other coe¢ cients are not signi…cantly di¤erent from their OLS counterparts, and it would therefore appear that the LZ results are not su¢ ciently a¤ected by an EV problem for it to be appropriate to prefer the H results to the original estimates based on OLS.

Concluding Remarks

In this paper we have subjected several well-known cross-sectional studies which propose a set of "stylized facts" regarding the determinants of the growth process to tests for the presence of EV in the explanatory variables. The implication of the results presented here is that EV can matter, and that several well-known "stylized facts" in the cross-sectional growth literature are not robust to controlling for EV. In particular, results such as the validity of the human capital-augmented Solow model (MRW), the deleterious impact of government consumption expenditures (Barro), the negative impact of being landlocked (SW), and the explanation for the low growth rate of sub-saharan Africa based on high levels of ethnolinguistic fragmentation (EL), are overturned once errors in variables are controlled for using our higher moments estimator.

While, as Robert Solow puts it, "few econometricians have ever been forced by the facts to abandon a …rmly held belief," we believe that our paper indicates that further work on testing the robustness of results in the empirical growth literature using econometric methods which are robust to errors in variables is certainly warranted. This is particularly true in the context of panel data, since it is well-known that the usual covariance transformations, such as the "within" procedure or …rst-di¤erencing, often exacerbate problems of errors in variables, and that GMM procedures applied to …rst-di¤erenced data may not solve the problem when serial correlation is present.

Paper

Joint

"

  True" relationship: y 1 = 1 + e y 2 +e y 3 +e y 4 + u; variable a¤ected by EV: y 3 =e y 3 + V; experiment carried out with 98 observations, R 2 = 0:778; ratio of variance of V to variance of e y 3 equal to 0:3, "true" size of type I errors equal to 0:05, and 5000 replications.

Table 1 :

 1 A summary of our results concerning measurement error in cross-sectional growth regressionsCoe¢ cientOLS Hz 1 and z 4 z 1 through z 7 all vars. only y 3 all vars. only y 3

		null of	Some point Basic result
		the absence of estimates	of paper
		measurement	signi…cantly that is
		error	changed	overturned
	Barro	rejected	yes	government
	1991			consumption
				expenditures no
				longer signi…cant
	MRW (levels)	rejected	yes	human capital
	1992			augmented Solow
				model rejected
	MRW (growth)	not rejected	yes	annual rate of
	1992			convergence
				halved
	Sachs-Warner	rejected	yes	landlocked dummy
	1997			and initial GDP
				per capita no
				longer signi…cant
	Easterly-Levine	rejected	yes	ethnolinguitic
	1997			diversity no
				longer signi…cant
	Levine-Zervos	not rejected	no	none
	1998

Table 2 :

 2 Results of Montecarlo experiments 1 and 2

The assumption may be not be quite so ino¤ensive when using Heston-Summers data since the authors extend their data to all "non-benchmark" countries using the same method. This may result in some degree of correlation between countries. Note however that our "higher moment estimator" does remain consistent when measurement errors are correlated between observations. Its asymptotic covariance matrix, however, would be di¤erent.

The second problem was addressed in the empirical work underlying this paper through the use of Huber-White standard errors. The changes in statistical inference that resulted were not su¢ ciently important to warrant their inclusion in the results presented below.

Note that other implementations of the proposed instrument set are possible, apart from the Fuller-estimator chosen here. These include GMM (the road taken in an earlier paper by[START_REF] Dagenais | Higher Moment Estimators for Linear Regression Models with Errors in the Variables[END_REF]), Nagar (or bias-adjusted 2SLS, see[START_REF] Donald | Choosing the Number of Instruments[END_REF]), or general k-class estimation. TheHahn and Hausman (2002a) tests used in this paper are based on using the proposed instrument set in the context of the Nagar estimator.

The sizes of the type I errors were computed as the percentage of replications for which the true value of the parameter was not included within the 95% con…dence interval.

Interestingly, the bias that emerges using z1 through z7, when there is only one variable assumed to be measured with error, takes on almost exactly the same value as that for OLS, suggesting that a situation of weak instruments is present (it is well-known that IV is biased towards OLS when the concentration parameter is near zero).

The standard discussion is provided by[START_REF] Bound | Problems with Instrumental Variables Estimation When the Correlation Between the Instruments and the Endogenous Explanatory Variable is Weak[END_REF]. See also the excellent surveys by[START_REF] Stock | A Survey of Weak Instruments and Weak Identi…cation in Generalized Methods of Moments[END_REF] andHahn and Hausman (2003), and a recent very short primer on the ensuing biases byHahn and Hausman (2002b).

Note that B2SLS only becomes a meaningful alternative to 2SLS once the degree of overidenti…cation is strictly greater than 1 since B2SLS is identical to 2SLS when K = 2 and r = 1.1 0 Asymptotic properties of the test are presented in[START_REF] Hausman | Asymptotic Properties of the Hahn-Hausman Test for Weak Instruments[END_REF], and the Montecarlo evidence shows "that the weak-instrument asymptotic distributions provide good approximations to the …nite sample distributions for samples of size

100." We thank Professor Hausman for informing us that his recent Montecarlo experiments suggest that the B2SLS-based version of the test is to be preferred to the 2SLS-based version.1 1 Note that one can replace the LIML estimates of the nuisance parameters by their Nagar or Fuller counterparts. This did not change the empirical results presented below appreciably.

2 This formula is a simple generalization of that given inHahn and Hausman (2002a), equation 9.4. 1 3 The version of the test used here is therefore based on the standard IV implementation of our proposed instrument set, since the J statistic does not exist for the Fuller estimator.1 4 It is worth noting that these diagnostics can be extremely misleading, as pointed out by[START_REF] Cruz | On the Validity of Econometric Techniques with Weak Instruments. Inference on Returns to Education Using Compulsory School Attendance Laws[END_REF]. This is why we do not place undue emphasis on their values.

5 All data used in this paper are available publicly and the TSP code used in all computations is, of course, available upon request.

Coe¢ cient

"True" relationship: y 1 = 1 + e y 2 +e y 3 + (0 e y 4 ) + u; variable a¤ected by EV: y 3 =e y 3 + V; experiment carried out with 98 observations, R 2 = 0:592; ratio of variance of V to variance of e y 3 equal to 0:8, "true" size of type I errors equal to 0:05, and 5000 replications. Implied value of (annual rate of conv.) 0:010

(5:552) 0:009

(2:640) 0:006

Implied value of (physical capital coef.) 0:288