metadata: authors: 0: author: Dollar, David
1: author: Kraay, Aart
docna: 0: docna: Institutions, trade, and growth : revisiting
            the evidence
docty: Policy Research Working Paper
subtopic: Trade Policy,Health Service Management and Delivery,International Trade and Trade Rules,Cholera,Malaria,Communicable Diseases,Leprosy,Law and Justice Institutions
teratopic: Poverty Reduction
abstracts: cdata!: Several recent papers have attempted to
            identify the partial effects of trade integration and
            institutional quality on long-run growth using the
            geographical determinants of trade and the historical
            determinants of institutions as instruments. The authors
            show that many of the specifications in these papers are
            weakly identified despite the apparently good performance of
            the instruments in first-stage regressions. Consequently,
            they argue that the cross-country variation in institutions,
            trade, and their geographical and historical determinants is
            not very informative about the partial effects of these
            variables on long-run growth.
repnme: repnme: Institutions, trade, and growth :
            revisiting the evidence
display_title: Institutions, trade, and growth :
            revisiting the evidence
listing_relative_url: /research/2003/03/2191891/institutions-trade-growth-revisiting-evidence
topicv3: Trade and Growth m1326018 466,Trade m1326924 698,Malaria m1327042 1420,Mortality m1504365 3661,Diagnostics m1504698 3625
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Poverty Reduction
origu: Development Research Group (DECRG)
no_of_pages: 36
model_extractions: POLICY RESEARCH WORKING PAPER 3004

# Abstract

Several recent papers have attempted to identify the partial effects of trade integration and institutional quality on long-run growth using the geographical determinants of trade and the historical determinants of institutions as instruments. Dollar and Kraay show that many of the specifications in these papers are weakly identified despite the apparently good performance of the instruments in first-stage regressions. Consequently, they argue that the cross-country variation in institutions, trade, and their geographical and historical determinants is not very informative about the partial effects of these variables on long-run growth.

This paper-a product of Investment Climate, Development Research Group-is part of a larger effort **in** the group to study institutions and development. Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Anna Bonfield, room MC3-354, telephone 202-473-1248, fax 202-522-3518, email address abonfield@worldbank.org. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at ddollar@worldbank.org or akraay@worldbank.org. March 2003. (29 pages) _The Policy Research Working Paper Senes disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An obhective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the azuthors. They do not necessanly represent the view of the World Bank, its Executive Directors, or the countries they represent._ Produced bv the Research Advisorv Staff
_Preliminary and incomplete Comments welcome_ Institutions, Trade, and Growth:<sup>Revisiting theEvidence</sup> David Dollar and Aart Kraay The World Bank 1818 H Street NW, Washington,<sup>DC20433,</sup><sup>**ddollar(fworldbank.ora.**</sup> **akraavAworldbank.org.** We would like to thank Romain Wacziarg<sup>and Jeff Sachs for</sup> helpful comments
# 1. **Introduction**

Two recent papers have proposed novel instruments and have used them<sup>to</sup> separately identify strong causal effects of trade and institutions on economic development. Frankel and Romer (1999, FR) show that the geographically determined component of trade as a fraction of GDP exerts a strong positive effect on growth<sup>in</sup> the very long run, while Acemoglu, Johnson and Robinson (2001, AJR) show that the historically determined component of current institutional quality exerts strong effects<sup>on</sup> development in a similar framework.

An open question, however, concerns the partial effects of institutions<sup>andtrade</sup> on growth in the long run. Several very recent papers have investigated this question, but come to strikingly different conclusions. Alcala and Ciccone (2002, AC) estimate<sup>a</sup> variety of regressions of GDP per worker on trade as a share of GDP and measures<sup>of</sup> institutional quality. They instrument for these two variables using the FR instrument as well as linguistic origins, and find that trade is quite significant, while institutional quality is occasionally but not consistently significant. Dollar and Kraay (2002, DK) estimate<sup>a</sup> specification very similar to that of AC. In contrast with AC, DK argue informally that<sup>the</sup> cross-country variation in the data is uninformative about the relative contribution<sup>of</sup> institutions and trade Finally, Rodrik, Subramanian and Trebbi (2002, RST) estimate<sup>a</sup> variety specifications similar to those of AC and DK in an effort to sort out<sup>the</sup> geographical, institutional, and trade-related determinants of development. In contrast with AC and OK, RST conclude that institutions matter while trade does not.' The objective of this paper is to show that empirical specifications which seek<sup>to</sup> isolate the partial effects of institutions and trade can suffer from serious identification problems. The issue, which we discussed informally in our previous paper, is straightforward. In many of the specifications in these papers, the historical and geographical instruments proposed in the previous literature tend to have<sup>good</sup>

> **1** In a related paper, Easterly and Levine (2002) attempt to disentangle the geographical<sup>and</sup> institutional determinants of development. In a number of specifications they control<sup>for various</sup> measures of policy, including trade policy. They generally find that institutions<sup>dominate</sup> geography and trade. Since their paper considers trade policy rather than trade itself,<sup>we do not</sup> consider it further here.
explanatory power for _both_ endogenous variables, institutions and trade. Consequently, the fitted values of institutions and trade from first-stage regressions are highly correlated with each other, and these specifications are at best weakly identified, even though first-stage regressions have apparently good explanatory power. In short, although the historical and geographical instruments for institutions and trade proposed by AJR and FR perform well in their respective univariate applications, they are not strong enough to identify the partial effects of these variables.

We document the identification problem using the partial R-squared diagnostic measures suggested by Shea (1997), and the minimum eigenvalue test of the null hypothesis of weak instruments suggested by Stock and Yogo (2001). In most cases, both diagnostic statistics as well as the formal tests verify the informal observation in DK that these models are not very well identified. Consequently, inferences about the structural parameters of interest, i.e. the causal effects of institutions and trade on longrun growth, cannot reliably be based on conventional t-statistics.<sup>2</sup> We therefore compute confidence intervals for these structural coefficients of interest using three methods suggested in the literature on inference with weak instruments: the AR statistic proposed by Anderson and Rubin (1949), the conditional likelihood ratio statistic proposed by Moreira (2002), and the S-statistic proposed by Startz, Zivot and Nelson (2001). These methods which take into account the low strength of the instruments produce substantially larger confidence intervals than those based on standard asymptotic theory in the specifications we consider. In many cases, confidence intervals for the parameters of interest cover (nearly) the entire real line. These results serve to formalize the informal intuition in our previous paper that the cross-country variation in trade, institutions, and their historical and geographical determinants is not very informative about their relative importance for growth in the long run.

Finally, in our previous paper we also investigated the relative contributions of institutions and trade over shorter horizons by looking at changes in decadal growth rates over the past four decades. We apply the same weak instrument diagnostics to these specifications. Although we find that dynamic regressions controlling only for trade and initial income are well-identified, the regressions which add institutional quality

**2** See Staiger and Stock (1997) for a seminal paper, and Stock, Wright, and Yogo (2002) for a recent survey of weak instruments.

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are poorly identified when institutional quality is taken as endogenous,<sup>but are</sup> reasonably well-identified when institutional quality is treated as exogenous.

The empirical examples that we discuss in this paper are<sup>specific to the recent</sup> debate on the relative importance of institutions and trade<sup>for long-rungrowth.We think</sup> however that our results have broader implications for the cross-country empirical growth literature, which has only in recent years begun to take seriously<sup>the endogeneity</sup> of many of the key determinants of growth. Our results here highlight<sup>the moregeneral</sup> difficulty in finding exogenous and independent<sup>sourcesof variationrequiredtoidentify</sup> the partial effects of such determinants of growth.

The remainder of this paper proceeds as follows. In Section<sup>2 we review some of</sup> the main empirical specifications that have been advanced<sup>to showthe effectsof</sup> institutions and trade on long-run growth. In Section 3 we describe and<sup>implement</sup> diagnostic tests for the presence of weak instruments, and show that<sup>most of the</sup> empirical specifications we consider in Section 2 are in fact weakly<sup>identified.InSection</sup> 4 we describe and implement three approaches to inference with<sup>weak instruments,and</sup> show that existing estimates of the partial effects of institutions<sup>and tradeare much less</sup> precisely estimated than conventional t-statistics would suggest.<sup>In</sup> Section<sup>5we</sup> examine the strength of the identification of the decadal regressions<sup>fromour previous</sup> paper. Section 6 concludes.

# **2. Empirical Background**

All of the papers discussed above estimate variants on<sup>the followinglinear</sup> instrumental variables regression:

*(1) In(GDP Per Capita), = 0 +P, Institutions, **j** +P 2 .Trade, + 3'Controls1 +ui

Cross-country differences in GDP per capita today primarily reflect<sup>differencesingrowth</sup> over the very long run since a date in the distant past when initial<sup>incomes werenot very</sup> different across counties. Consequently, this.specification can be thought of<sup>as</sup> capturing the determinants of growth in the very long run. Institutions<sup>and trade areboth</sup> treated as endogenous, and are instrumented using their deep historical<sup>and</sup> 3
geographical determinants. Each of the papers discussed above differs somewhat in the measures of institutions and trade used, the control variables included in the regression, and the choice of instruments. Table 1 summarizes the key differences across the main specifications of interest in each of these papers.

The top part of Tables 2 and 3 report some of the main results of these papers. To conserve on space, we report only the coefficients on the main endogenous variables of interest: institutions, trade, and their instruments in the first-stage regressions. Table 2 reports selected results from DK, while Table 3 reports selected results from RST. We do not report results from AC as we were unable to obtain their dataset at the time of writing. As a benchmark, the first two columns of Table 2 present regressions from DK in the spirit of the original AJR and FR papers, estimating the effects of institutions and trade separately, without controlling for the other variable. Specification 2.1 shows the effects of institutions only, analogous to the basic specification of AJR. The main differences with AJR is that DK use linguistic origins rather than settler mortality to instrument for institutional quality, and use a different measure of institutional quality. Nevertheless, the results are broadly similar to that of AJR. As the OLS regressions show, institutional quality is strongly positively correlated with per capita incomes. The IV regressions show a statistically and economically significant effect of institutions on per capita income, with a slope coefficient larger than the corresponding OLS estimate. Specification 2.2 shows the analog of the main FR specification. Here there are again only two differences with the original paper: the trade/GDP ratio is measured at PPP, and the only included exogenous variable is the logarithm of population. Again the results are very similar to those of the original paper, with a large and significant effect of trade on growth. In both specifications 2.1 and 2.2, the first-stage regressions of institutions and trade on their respective instruments perform quite well, with comfortably large F-statistics of 13.3 and 28.8.

Specification 2.3 reports the main DK result on the partial effects of institutions and trade. Both institutions and trade enter positively in the OLS regression, with t- statistics greater than 2. However, in the IV regression only institutional quality has a t- statistic greater than 2, while the coefficient on trade falls to near zero and the t-statistic falls to 0.3. In our other paper, we showed however that this apparent result on the importance of institutions relative to trade was quite fragile, and that both variables were 4
insignificant once four influential observations were discarded. Here however we focus on this main specification, and show more formally below that the<sup>apparentsignificance</sup> of institutional quality vanishes once we take weak identification into account.

Table 3 presents the baseline RST specification, and three variants.<sup>RST</sup> regress log per capita GDP on distance from the equator, the same institutional<sup>quality</sup> variable as in DK, and trade as a fraction of GDP in local currency units.<sup>3Intheir</sup> preferred 80-country specification, they use settler mortality from AJR and distance<sup>from</sup> FR as instruments. The first column of Table 3 replicates their basic specification.<sup>To</sup> conserve on space, we again suppress the coefficient on distance<sup>from the equator (and</sup> all other included exogenous variables). The IV estimate of<sup>the effect of institutionsis</sup> positive, large, and highly significant using a conventional t-test. In contrast,<sup>trade</sup> enters negatively and insignificantly.

RST go on to provide a number of robustness checks on this basic<sup>variant.A</sup> subset of these are reported in the remaining columns of Table 3. In specification<sup>3.2,</sup> they replace the local currency trade share with the PPP trade share<sup>in order to make</sup> their results more comparable with DK and AC. In specification 3.3 they return to local currency trade shares, but add measures of domestic market size<sup>(thelogarithmsof</sup> area and population) in order to make their results more comparable<sup>with the originalFR</sup> paper which included these variables. In specification 3.4 we report<sup>one of several</sup> specifications produced by RST including a variety of other variables<sup>capturing different</sup> channels through which geography might matter for development. We focus on<sup>acluster</sup> of three variables they identify as jointly significant, consisting of land area<sup>in thetropics,</sup> number of frost-days per year, and the incidence of malaria. In each of these robustness checks, they find that institutions enter the IV specification positively and with 3In their paper, RST argue that this is preferable to using PPP-adjusted<sup>trade shares as do AC</sup> and DK. The sketch a simple model in which an increase in productivity<sup>which raises therelative</sup> price of non-tradeables relative to the benchmark country will also raise the PPP-adjusted trade ratio. This is because the PPP-adjusted GDP will fall to reflect the higher price<sup>of non-tradeables.</sup> Since the increase in productivity raises income, this can introduce a spurious correlation between income and the PPP-trade share. While the choice of denominator for the trade<sup>ratio is</sup> not the focus of this paper, we note in passing that we are not entirely convinced<sup>by this</sup> argument. First, it as long as these other sources of productivity differences<sup>are uncorrelatedwith</sup> the FR instrument, this potential bias will not matter for the IV coefficient estimates. Second, at a more intuitive level, if we want to capture the size of the traded sector relative<sup>to overalleconomic</sup> activity, it seems natural to use an internationally-comparable measure of overall economic activity in constructing trade shares.

_5_
t-statistics in excess of two. In contrast, trade enters negatively in three of the four specifications, and in all cases the t-statistics are well below 2. Based on these findings, as well as the basic DK finding in specification 2.3, they conclude in the title of their paper that "institutions rule". Below we aregue that these results could be interpreted more cautiously once we take the strength of the identification into account.

# **3.** Are the Partial Effects of Institutions and Trade Identified?

The key question in this paper is the extent to which the empirical strategy of the papers discussed in the previous section is a good one for capturing the partial effects of institutions and trade on growth in the very long run. Our focus here is on one shortcoming of this approach - that existing instruments at best only weakly identify the structural parameters of interest. In order to review the tools on which we base this conclusion, some further notation is useful. The structural model in Equation (1) can be written in the usual matrix notation as

- (2) y = _Xp_ + u

where y is an nxl vector of observations on the dependent variable, X is an nxk matrix of endogenous regressors, 1 is a kxl vector of structural coefficients, and u is an nxl vector of disturbances.<sup>4</sup> Let Z represent an nxq matrix of instruments. Then the reduced form consists of the following k+1 regressions of y and the columns of X on Z:

- (3) y=Ze+v X=ZX+e

where 0 is an qxl vector and r is a qxk matrix of reduced form coefficients, and v and e are nxl and nxk matrices of reduced-form disturbances. Let a and E denote the variance of vj and the covariance matrix of ej respectively, and let Q denote the covariance matrix of (v, e,')'.

> 4 In what follows we suppress the additional exogenous control variables in the regression, by interpreting y, X, and the instruments as the residuals from a regression of these variables on the included exogenous control variables.

6
In order for the historical and geographical variables<sup>in Z to be valid instruments,</sup> they need to be independent of u, which we assume<sup>to be thecase, i.e.E[Z'u]=O.In</sup> order for the structural parameters p to be identified,<sup>the twotextbook order andrank</sup> conditions must also be satisfied, i.e. the number of instruments<sup>is greater than the</sup> number of endogenous variables (q>k), and the<sup>matrix</sup> r<sup>mustbe fullrank(rank(r)=q).</sup> Using this notation, the instrumental variables (IV)<sup>estimator of 3 is</sup> = (X'Pz X)-' X'Pz y where Pz - Z(Z'Z)-'Z<sup>for any matrix Z.</sup> A rapidly-growing recent literature on weak instruments<sup>haspointedout that the</sup> validity of the usual approximation that ,IV is normally<sup>distributedin finite samples</sup> depends crucially on the strength of the instruments.<sup>Inthe case of only one</sup> endogenous variable, the instruments are weak<sup>if they jointly have poor explanatory</sup> power for the endogenous variable in the reduced-form<sup>equationfor X.Nelson and</sup> Startz (1990), Staiger and Stock (1997) show that<sup>when the instrumentsare weakin this</sup> sense, the IV estimator is biased towards the mean<sup>of theOLS estimator, and isnot</sup> normally distributed. Thus inferences about the structural<sup>parametersbasedonusual t-</sup> tests can be highly misleading.

When there are multiple endogenous variables, and<sup>so more thanone first-stage</sup> regression, it is less obvious how to gauge the strength<sup>of the instrumentsand the</sup> quality of the normal approximation. An important<sup>point,however,is that it is not</sup> sufficient for the F-statistics in the individual first-stage<sup>regressions to be large.5</sup> Shea (1997) has proposed a simple diagnostic statistic<sup>for determiningthe</sup> strength of instruments when there are multiple<sup>endogenous variables.If X, and X</sup> 2<sup>are</sup> two endogenous variables, then the instruments are<sup>strongif the componentof the fitted</sup> value of **X,** from the first-stage regressions that is<sup>orthogonalto the fittedvalue of X</sup> 2<sup>is</sup> highly correlated with the component of X, that<sup>is orthogonalto X</sup> 2<sup>.In the caseof a</sup> single endogenous variable, this criterion is simply<sup>the R-squaredfromthe first-stage</sup>

> **5** Shea (1997) gives a simple example. In the case of two<sup>instruments Z, andZ</sup> 2<sup>for the two</sup> endogenous variables, X, and X2, if ZI is is a good predictor<sup>of X, andof X</sup> 2<sup>, butZ</sup> 2<sup>iS</sup> uncorrelated with both endogenous variables, then<sup>the first-stageF-statistics will be largebut the</sup> model will not be identified since there is in effect only<sup>oneinstrument for the two endogenous</sup> variables.

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regression. While this statistic is intuitive and easily calculated, results on its distribution are not provided in Shea (1997) and so this statistic does not lend itself well to formal tests of the null hypothesis of weak instruments.

More generally, Stock, Wright and Yogo (2002) point out that the validity of the 1 1 normal approximation depends on the concentration matrix ^1 = y 2']F'Z'Zr **2** being large in the sense of the smallest eigenvalue of this matrix being large. Note that the concentration matrix can be small for a variety of reasons. All instruments could be nearly uncorrelated with the endogenous variables, i.e. all the elements of F are small relative to the variances of the errorterms in the first-stage regressions. The instruments could also be very highly correlated with each other so that Z'Z is not full rank, i.e. there are essentially fewer instruments than endogenous variables. Finally concentration matrix could also be small if the rank condition for identification fails, i.e. F. is not full rank. This can occur if, as is the case in some of the applications considered here, each of the instruments has strong explanatory power for both endogenous variables.

Stock and Yogo (2001) develop a test of the null hypothesis that the instruments are weak based on this idea. In particular, they provide critical values for a test of the null hypothesis of weak instruments based on the minimum eigenvalue of _F_ = 7<sup>1/2</sup> vXvPz,71/2, where i = X'(l - Pz )X/(n - q) is a consistent estimate of the covariance matrix of errors from the first-stage regressions.<sup>6</sup> In the version of the test we implement below, the researcher specifies a null hypothesis that the instruments are weak if the maximum size distortion of Wald tests about parameters of interest under the null of weak instruments is equal to some pre-specified value. Stock and Yogo (2001) then tabulate the critical values of the minimum eigenvalue of G for which the null of weak instruments should be rejected.

**6** When the instruments are weak, they show that E[G] = 2 / q _+_ I. One drawback of this test in our current application is that it is derived under the assumption of iid homoskedastic error terms in the structural equation. The same is true for the approaches to inference discussed below. We are unaware of tests for weak instruments which relax this assumption, and it is difficult to say how important deviations from this assumption are for the results we obtain.

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With these tools in hand we can investigate the strength of the instruments<sup>in the</sup> specifications in Tables 2 and 3. Consider first the core<sup>DK specificationinregression</sup> 2.3. In the first-stage regressions, we see that the linguistic<sup>origin variables and the FR</sup> instrument are highly significant predictors of both endogenous<sup>variables.In our</sup> previous paper we noted that this implied a very high correlation<sup>betweenthe fitted</sup> values of institutions and trade in the second-stage regression,<sup>whichmade theIV</sup> estimates highly sensitive to a few influential observations.<sup>Table 2reports that this</sup> correlation of fitted values from the first-stage regression<sup>is 0.74,indicating strong</sup> collinearity in the second-stage regression. From this we concluded<sup>informallythat the</sup> cross-sectional variation in institutions, trade and incomes was<sup>not very informative</sup> about the partial effects of the former two on the latter.

The weak instrument diagnostics discussed in the previous subjection<sup>help us to</sup> make this point more precisely. Consider first the Shea<sup>(1997) partial R-squared</sup> statistics. We find in our specification that these statistics are<sup>very small at only 0.02</sup> and 0.04 respectively. In contrast, the R-squareds from the first-stage<sup>regressions in the</sup> first two univariate specifications in Table 2 were 0.14<sup>and0.29.Thisprovides a first</sup> indication of the idea that while linguistic origin and distance<sup>are reasonableinstruments</sup> when the effects of institutions and trade are estimated separately,<sup>they are quite poor</sup> instruments when the effects of the two are considered jointly.

The middle panel of Table 2 also reports the eigenvalues of<sup>the matrix of</sup> estimated coefficients in the first-stage regression, F, as<sup>wellas the eigenvalues of G.</sup> The first observation here is that the ratio of the largest to the smallest<sup>eigenvalues ofr,</sup> i.e. the condition number of r, is nearly 100<sup>and thesmallest eigenvalueis small at</sup> 0.025. This indicates that the matrix of estimated coefficients<sup>in</sup> the<sup>first-stage</sup> regression is very close to not being of full rank, contributing<sup>to weakidentification of the</sup> parameters of interest. The smallest eigenvalue of G<sup>is1.36.To interpret this value,</sup> suppose we were to define the set of weak instruments very<sup>narrowly as only those</sup> instruments that would lead to size distortions of 25% or more<sup>(i.e.the size of a nominal</sup> 5% test about a parameter of interest were actually<sup>25%),we wouldnot reject the null</sup> hypothesis that the instruments in this application are weak<sup>according to this stringent</sup> criterion, since the critical value provided by Stock<sup>andYogo is5.75.</sup> 9
The middle panel of Table 3 also reports the same diagnostic statistics for selected specifications from RST. In all of these specifications, the first-stage regressions perform well: seven of the eight first-stage F-statistics are larger than 10, and even the smallest is still respectable at 8. However, these F-statistics are not sufficient to diagnose the strength of the instruments. Closer inspection of the first-stage regressions shows the same tell-tale signs of potential problems as in the DK specification in Table 2. In seven of the eight first-stage regressions in Table 3, the instrument for institutions (settler mortality) has strong explanatory power for both institutions and trade, while the instrument for trade (the FR constructed trade share) is a significant predictor of both institutions and trade at the 5% level in seven of the eight specifications as well.

Table 3 also presents the same diagnostic statistics on instrument quality as in Table 2. The Shea partial R-squared statistics are small in most cases, although not as extremely so as in the basic DK specification. In four of the eight first-stage regressions, the partial R-squareds are less than 0.1, and in seven out of eight they are less than 0.2. More formally, in each of the four specifications, we cannot reject the null hypothesis that the instruments are weak. However, the size of the set of weak instruments under the null is different in the four specifications. In specifications 2 and 4, the minimum eigenvalues of G are 1.95 and 3.24, and we cannot reject even the most conservative set of weak instruments tabulated in Stock and Yogo (2001). In specifications 1 and 3, the minimum eigenvalues of G are 6.2 and 6.8, and we cannot reject the null that the instruments fall in the larger set of weak instruments that would lead to size distortions of at least 5 percent, i.e. a test with nominal size of 5% would in fact have a true size of 10%.

These results suggest that there are reasons to question inferences about the importance of trade and institutions based on conventional t-statistics in the main specifications considered by RST as well. In the next section of the paper we discuss approaches to inference that are valid even when the instruments are weak, and show that the confidence intervals for the parameters of interest are much larger than simple t- statistics would suggest.

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# **4 How Large are the Partial Effects of Institutions and Trade?**

In this section we briefly describe three approaches to constructing valid confidence intervals when instruments are weak. The first is based on the AndersonRubin statistic: which was first proposed by Anderson and Rubin (1949). Under very general conditions, including for any arbitrary concentration matrix ,u, Staiger and Stock (1997) show that q times the AR statistic converges in distribution to a %<sup>2</sup> (q) random variable. Since the asymptotic distribution of the AR statistic does not depend on any nuisance parameters even when the instruments are arbitrarily weak, it is pivotal and can be used for inference. Wang and Zivot (1998) suggest the use of this result to form valid (1-a).100% confidence sets for the elements of **3** by 'inverting" this statistic, i.e. by finding the set of values of Po such that AR(po3) • s ) where x2 (q) denotes the (1-ca).100 percentile of **qa** a x<sup>2</sup> (q) distribution. In the case of two endogenous variables, the joint confidence sets for the two elements of ,B can take a variety of shapes. Confidence intervals for the individual elements of 1 can be obtained by projecting these joint confidence sets on the appropriate axes.<sup>7</sup> Figure 1 shows an example of two such 95% confidence sets, and the associated confidence intervals. The top panel shows the case where the 95% confidence set is the area inside an ellipse, corresponding to the coefficients on institutions and trade in RST specification 3.1. The confidence intervals for the coefficients on trade and institutions are obtained by projecting the height and the width of the ellipse on the corresponding axes. The bottom panel shows a case where the 95% confidence set is the area inside a hyperbola, corresponding to the DK specification 7Kleibergen (2002) develops an altemative test which corrects the deficiency of the AR statistic which is that it has low power when the number of instruments is large. In our applications with at most three instruments this is not a concern, and inferences based on Kleibergen's K-statistic are almost identical to those based on the AR statistic.

**11**
2.3. Projecting this on the two axes shows that the 95% confidence intervals-for the coefficients on institutions and trade are the entire real line. That is, the data is so uninformative due to weak identification that it is consistent with any value of these parameters. We will discuss the specific cases to which these confidence sets apply below when we present the empirical results in more detail.

Moreira (2002) has proposed a conditional likelihood ratio statistic as the basis for inference when instruments are weak. Moreira observes that it is possible to write the usual likelihood ratio statistic for the linear IV model as a function of two orthogonal sufficient statistics: S = Z'Ybo and T = Z'YQ-'Ao where Y -(y,X), bo (1, -Po')', and AO = (Po, **Ik).** His key observation is that since S is a function only of data and the parameters of interest, the likelihood ratio statistic conditional on the value of T observed in the data can be used to make inferences about the parameters of interest. In particular, Moreira shows that the likelihood ratio statistic can be written as: (5) LR = S'S Xmin where S = (Z'Z)-'S(bo'Qbo)-<sup>112</sup> and T (Z'ZY1T(AO'Q-1Aoy)'<sup>2</sup> are the standardized versions of S and T, and Xmin denotes the smallest eigenvalue of (S,T)' (s, T). Confidence intervals for ,B can be constructed by inverting this statistic in the same way as the AR statistic. The LR statistic does not have a known distribution on which to base critical values. However, Moreira (2002) suggests a strategy for simulating this distribution empirically to obtain the necessary critical values, which we adopt here.<sup>8</sup> The final approach to inference with weak instruments is based on the S-statistic proposed by Startz, Nelson and Zivot (2001). This statistic is defined as:

**8** In particular, since S follows a standard normal distribution, the distribution of LR conditional on the value of T observed in the data can be simulated by repeatedly taking draws from the distribution of S and evaluating LR at the observed value of T.

12
where X is a vector containing all of the reduced-form<sup>coefficient estimates,</sup>

**_,f()_** /V,IV pi Iv is the IV estimate of the ij element of 13; and p1 o is<sup>its value</sup>

under the null hypothesis. The S-statistic<sup>is</sup> the<sup>square of the standardized value of</sup> T,(X), and rejects the null when it is large. To<sup>provideanintuition for this statistic,</sup> Startz, Nelson and Zivot (2001 ) note that the numerator<sup>of,(x) is the distance between</sup> the IV estimate and its hypothesized value. The<sup>denominator is the middlepart of the</sup> concentration matrix, and is small when identification<sup>is weak.Thus the S-statistic will</sup> reject the null if the coefficient estimate is far from<sup>the null, or ifidentification isweak.</sup> Startz, Nelson and Zivot (2001) show that the<sup>S statistic convergesto aX2(1)</sup> random variable when the instruments are good,<sup>and is boundedaboveby a</sup> x<sup>2</sup> (q - k + 1) random variable when the instruments are<sup>weak.To construct confidence</sup> intervals for the ith element of 13, we need only<sup>find the set of hypothesizednullvalues</sup> for which the S-statistic is less than the appropriate<sup>critical value.Figure2 illustrates</sup> one such confidence set, for the coefficient<sup>oninstitutionsin the RST specification 2.1.</sup> Note that the confidence set need not be the usual<sup>closedinterval,but can consist of two</sup> disconnected intervals. A further convenient property<sup>of this approach to inference is</sup> that it is possible to construct confidence intervals<sup>for a single parameter directly, and-</sup> that projection methods needed for the AR<sup>andLR statistic arenot requiredin this case.</sup> The bottom panels of Tables 2 and 3 present confidence<sup>intervalsconstructed</sup> using these three approaches to inference that<sup>are robust to weak instruments.For</sup> reference purposes, we first report 95% confidence<sup>intervals for thecoefficients on</sup> institutions and trade based on the usual Wald<sup>statistics, i.e.the estimated coefficient</sup> plus or minus two estimated standard errors. In the<sup>remainingrows we reportthe 95%</sup> confidence intervals based on the AR,<sup>LR, and S-statistics,respectively.</sup> Consider first the main DK specification in the third<sup>columnof Table1.</sup> Conventional confidence intervals for the coefficients<sup>on institutions andtradeare (0.57,</sup> 1.95) and (-1.00, 1.35) respectively, suggesting<sup>a positive andsignificant effect of</sup> institutions and an insignificant effect of trade.<sup>Confidenceintervals that arerobustto</sup> 13
weak instruments lead to quite different conclusions. All three statistics give confidence intervals that consist of the entire real line for both the coefficients on institutions and on trade. This finding formalizes the informal observation in our previous paper that the cross-country variation in institutions, trade, and their geographical and historical determinants is not sufficiently informative to pin down the partial effects on growth of either of these endogenous variables.

The RST results are less stark, but still show that inferences based on the usual t-statistics can be misleading. In the basic RST specification 3.1, the 95% confidence intervals for institutions based on the AR and LR statistics are much wider than those based on the usual Wald statistic, but still contain only positive values.The confidence interval for institutions based on the S-statistic consists of two disjoint intervals, (-0, - 1.77) and (1.67, **o).** The former two tests suggest that the partial effect of institutions is positive, but much less tightly estimated than a conventional t-test would suggest. The third statistic indicates that the coefficient on institutions is in fact very imprecisely estimated, in that the data are consistent with a wide range of both positive and negative values for this coefficient. The intervals for the coefficient on trade based on the AR, LR and S-statistics are also much larger that those based on conventional t-statistics, but mostly tend to include negative values. Overall, the substantially-larger confidence intervals based on weak instrument asymptotics suggest that the partial effects of institutions and trade are not very tightly estimated.

The remaining specifications consider some of the robustness checks in RST. First, RST argue that their finding on the relative importance of trade is robust to using PPP trade shares to measure trade. The results in specification 3.2 suggest that this equation is also weakly identified. In the bottom panel of Table 3 we see that the AR, LR and S-statistics all produce very large confidence intervals in the form of two disjoint sets which exclude only what might be thought of as a priori reasonable values for these parameters in the general vicinity of the OLS estimates. The results of this specification are therefore more similar to those in the basic DK finding, in which the partial effects of institutions and trade are essentially unidentified. The same is true in specification 3.4 with additional geographical control variables. The correct 95% confidence intervals span nearly the entire real line, indicating that the data are consistent with virtually any partial effect of institutions and trade. In specification 2.3 the AR and LR statistics give 14
confidence intervals that are similar (but substantially wider) to conventional ones, while those based on the S-statistic encompass a very large range of parameter values.

To sum up, we have seen that the basic DK conclusion that the cross-sectional data in their specification is uninformative about the partial effects of institutions and trade is supported by the more formal weak instrument asymptotic approximations used here. The same techniques also suggest that the RST findings on the relative importance of institutions may be less robust than meets the eye. In their specifications using PPP trade shares and adding additional geographical controls, the data appears consistent with virtually any values of the parameters of interest, due to the weak identification of these equations. In their basic specification, the different approaches to inference lead to different precise conclusions. Qualitatively however they all lead to much larger confidence intervals than those based on t-statistics, especially in the case of the S-statistic.

# **5. Decadal Regression Results**

How might we proceed from the negative findings above that the partial effects of institutions and trade are not identified in cross-country regressions of levels of per capita GDP on instrumented institutions and trade? One possibility which we explored in our previous paper was to instead look at decadal changes in growth, trade and institutional quality in the hopes that the partial effects of the latter two variables would be better identified in this framework relying on internal instruments. Of course, a drawback of this approach is that by construction it cannot pick up the very long-run effects of institutions and trade emphasized in the levels regressions above. Moreover, while we showed that there is non-trivial variation in measures of trade and institutional quality even at decadal frequencies, we are less sure how informative this variation is about true unobserved changes in these variables. In our previous paper we found that changes in trade volumes were a significant predictor of changes in decadal growth rates. However, the effects of changes in five measures of institutional quality on changes in growth were very imprecisely estimated and rarely statistically significant across specifications.

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In this section we deploy the tools descriped above to verify the strength of the instruments in the decadal regressions of our previous paper. We began with the following standard cross-country growth regression:

**(7) YCt =Po** + P1 **YC,t-k +** P2'Xct +C + Yt + **VCt**

where **Yct** is log-level of per capita GDP in country c at time t, YC,t-k iS its lag k years ago (k=1 0 years in our application using decadal data) and Xc, is a set of control variables which are measured as averages over the decade between t-k and t. We will consider trade volumes and measures of institutional quality among the variables in X.

Subtracting lagged income from both sides of the equation gives the more conventional formulation in which the dependent variable is growth, regressed on initial income and a set of control variables. The disturbance term in the regression consists of an unobserved country effect that is constant over time, **jc,** an unobserved period effect that is common across countries, **-yt,** and a component that varies across both countries and years which we assume to be uncorrelated over time, vct.

We adopted the approach of Caselli, Esquivel and Lefort (1996), which is to estimate equation (7) in differences, using appropriate lags of the right-hand side variables as instruments. In particular, we estimate the following regression: (8) **yct yc,t = (Yc,t-k Yc,t-2k)+p2(Xct -Xc,t-k)+(Yt** Yt-*)+(Vct **VC,t-k)** This is nothing more than a regression of growth on lagged growth, and on changes in the set of explanatory variables. Or, subtracting lagged growth from both sides of the equation, we have changes in growth from one decade to the next as a function of initial growth and changes in the explanatory variables. Our identifying assumption is that while trade volumes and institutional quality may be correlated with the contemporaneous and lagged shocks to GDP growth (E[Xdtvc,..t-]#0 for s2O), they are uncorrelated with future shocks to GDP growth, (E[(XYdv,,t+]=O for s>0). We therefore instrument for lagged growth using the log-level of per capita income at the beginning of decade t-2k, and we instrument for the changes in trade and institutional quality using their levels at the beginning of decade t-k.

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The first column of Table 4 reports the<sup>resultsof this specification when X</sup> contains only the share of trade<sup>in GDP.TheOLS andIV estimates replicate those in</sup> our previous paper. We find a positive<sup>effectof trade on growthat thisdecadal</sup> frequency, with a coefficient that is<sup>economicallysensible andhas a t-statistic greater</sup> than 2. In the first-stage regressions,<sup>we findthat while only initial incomeisa significant</sup> predictor of growth, both initial income<sup>andinitial trade aresignificant predictors of trade.</sup> This suggests some concerns about<sup>the strength of the identification.The diagnostic</sup> statistics for the presence of weak instruments<sup>paint a somewhatmixedpicture as well.</sup> The Shea partial R-squared for growth<sup>is very small,while the partialR-squaredfor trade</sup> is a respectable 0.25. On the other<sup>hand,theStock-Yogo test statistic for the nullof</sup> weak instruments is large enough<sup>that wecanreject thenull.The bottom panel reports</sup> conventional Wald-based confidence<sup>intervalsas well asintervals based on the S-</sup> statistic.<sup>9</sup> While the confidence interval<sup>for the coefficient oninitial incomeis very large,</sup> the confidence interval for the coefficient<sup>on tradeisessentially thesameas that based</sup> on the usual Wald statistic. This gives<sup>us someconfidence in the robustnessof this</sup> basic finding on the effects of trade<sup>on growthat decadal frequencies,even thoughthe</sup> coefficient on initial income appears<sup>to be less well identified.</sup> The middle panel of Table 4 reports<sup>results addingone of the five time-varying</sup> measures of institutional quality we considered<sup>in our previous paper.Following Clague,</sup> Keefer, Knack and Olson (1999) we<sup>proxy for the strength of property rights protection</sup> using what they refer to as "contract-intensive<sup>money",i.e.the share of currencyin</sup> circulation held inside the banking system.<sup>In themiddlepanel of Table 4, we enter this</sup> variable and treat it as endogenous.<sup>The top part of the table again replicatesthe results</sup> of our previous paper. We find that<sup>the coefficienton tradedoes notchange very much</sup> relative to the previous case, while<sup>the measureof institutional quality enters negatively</sup> with a very large standard error. However,<sup>the diagnostic statistics for the presenceof</sup> weak instruments strongly point to identification<sup>problemsin this specification.The Shea</sup> partial R-squared statistics are tiny,<sup>as is theminimumeigenvalueof the concentration</sup> matrix. Not surprisingly, in this case<sup>confidence intervals based on the S-statistic cover</sup> 9 In this section we report only the<sup>confidence intervalsbased onthe S-statisticbecause of its</sup> convenient property that projection<sup>methods are not required.This is most useful in the middle</sup> specification of Table 4 where we have<sup>threeendogenous variables,andensuresthat the results</sup> are comparable across these three specifications.

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the entire real line, suggesting that this specification is entirely uninformative about all of the parameters of interest.

The reason for the poor performance of the IV estimator is however somewhat different from the levels regressions discussed before. In the levels regressions the problem was that the instruments for trade also explained institutional quality while the instruments for institutional quality also predicted trade well. This is not the case here. In our previous paper we noted that initial levels of trade predict subsequent changes in trade but not changes in institutions. Conversely initial levels of institutions predict changes in institutions but not changes in trade. However, the problem here is that initial institutions also predicts the endogenous change in initial income, while initial levels of per capita income that are supposed to instrument for the change in initial income also explain changes in trade and changes in institutions. This is why the identification of this specification breaks down.'° We obtain rather more encouraging results when we treat institutions as exogenous. In our previous paper we noted that the large estimated standard errors on the coefficient on institutional quality in the dynamic regressions might be due to weak instruments. We then also discussed results in which we treated institutions as exogenous, which if anything were likely to exaggerate the effects of institutions on growth. In these specifications we continued to fail to find a significant impact of institutions on growth, while trade continued to enter positively and significantly. In the final panel of Table 4 we revisit this altemative specification. The results are quite similar to those in the first panel where institutional quality was omitted entirely. The main difference now however is that the Stock-Yogo test favours the null hypothesis of weak instruments. Closer inspection of the first-stage regressions suggests that this may be because initial levels of income are predicting both lagged growth and changes in trade. When we compute appropriate confidence intervals using the S-statistic, we

**1'** RST argue that the significance of trade in the decadal regressions in DK is not robust to the inclusion of selected regional dummies and their interactions with decadal dummies. For example, for the specification using contract-intensive money as a proxy for institutional quality, when these variables are estimated, nothing is significant in the IV regression. In light of the results above, this is perhaps not very surprising. Since these additional exogenous variables are likely to be good predictors of all three endogenous variables, this is likely to further weaken the performance of the instruments. Moreover, given that these additional dummy variables are not significant in this specification, we are not sure that it is a good strategy to include them and undermine any hopes of identification.

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find once.again that the coefficient on initial income is not very precisely estimated. Nevertheless we do find that the coefficient on trade is fairly precisely estimated, with a confidence interval very similar to that based on a conventional Wald test.

Overall these results suggest that it is also difficult to identify the partial causal effects of institutions on growth in this shorter-run framework. We do not appear to have sufficiently informative variation in the decadal data to identify the partial effects of institutions and trade treating both as endogenous. However, the specification which includes institutions but treats this variable as exogenous still serves a useful purpose, since it helps to control for a spurious correlation of changes in trade with changes in institutional quality, even if we cannot properly identify the partial effects of the latter.

# 6. **Conclusions**

In this paper we have argued that existing attempts to isolate the partial effects of institutions and trade on growth in the long run suffer from serious identification problems. The reason for this is simple -- existing historical and geographical instruments in the literature tend to have strong predictive power for _both_ institutions and trade. As a result, these specifications are only weakly identified, despite the apparent good performance of first-stage regressions.

How do we move forward from such a negative result? One possibility is to rely more on the time-series variation in institutions, trade and growth, in the hopes that internal instruments will be less weak than in the cross-sectional applications we have described. This is the approach we followed in our previous paper. However, this approach appears to yield mixed results regarding the strength of the instruments. If we treat institutions as exogenous, we are at least able to uncover a significant partial association between trade and growth which survives the inclusion of a variety of proxies for institutional quality. However, if we allow institutions to be endogenous, we find that the model is too weakly identified to be able to sharply estimate any of the parameters of interest.

Clearly simple cross-country linear instrumental variables regressions, either in levels or in decadal differences, cannot provide definitive answers about questions as 19
complex as the interacting roles of institutions and trade for growth. At best<sup>this</sup> approach can summarize interesting partial correlations in the data while at<sup>least</sup> partially controlling for obvious endogeneity problems. Combining this approach<sup>with</sup> more microeconometric evidence, case studies, and descriptions of<sup>historical episodes</sup> is likely to constitute fruifful research agenda. In the meantime, the empirical<sup>examples</sup> we provide here illustrate a broader problem in the cross-country empirical growth literature, which only in recent years has begun to take serious the problem<sup>of</sup> endogeneity of many of the key determinants of growth. Finding compelling<sup>instruments</sup> for these endogenous variables is difficult to begin with. If in addition we<sup>are interested</sup> in sorting out the partial effects of multiple endogenous determinants of growth,<sup>the</sup> search for instruments is further complicated by the need to find exogenous _and_ independent sources of variation in order to identify partial effects of interest.

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