estimate of governance for each country. We can also use these expressions to form confidence intervals for governance, that we sometimes also informally refer to as "margins of error". In particular, given our assumptions there is a 90 percent probability that the true level of governance falls in a range of +/- 1.64 x SD(gj|yj) around the estimate of governance.<sup>20</sup>

# _Estimating the Parameters of the UCM_

In order to implement (2) and (3), we need to first estimate the unknown parameters αk, βk and σk for every indicator k. This in turn requires us to distinguish between "representative" and "non-representative" indicators, which we treat differently in the estimation process. Representative indicators are indicators that cover a set of countries in which the distribution of governance is likely to be similar to that in the world as a whole. Practically these include all of our indicators with large cross-country coverage of developed and developing indicators. In contrast non-representative indicators cover either specific regions (for example the BEEPS survey of transition economies or the Latinobarometer survey of Latin American countries), or particular income levels (for example the World Bank CPIA ratings that cover only developing countries). Our classification of "representative" and "non-representative" indicators is given in Table 1 of this paper.

For the set of representative indicators, we use the assumption of normality of gj and εjk to write down the likelihood function of the observed data. The assumption of representativeness is crucial here because it justifies our assumption of a common distribution of governance across these different sources. As useful notation, let 2 2 2 α = (α1<sup>,...,</sup> αK j )<sup>′</sup> , β = (β1<sup>,...,</sup> βK j )<sup>′</sup> , σ = (σ1 ,..., σK j )<sup>′</sup> and let B and Σ be diagonal matrices with β and σ<sup>2</sup> on the diagonal. Using this notation, the mean of yj is α and the variance ' ' of yj is Ω ≡ ββ + BΣB . The contribution to the log-likelihood of country j therefore given by: Summing these over all countries j and then maximizing over the unknown parameters delivers our maximum-likelihood estimates of αk, βk and σk for every representative indicator k. Identification requires that we have a minimum of three representative indicators. Note that the number of data sources available for each country varies, and so the dimension of yj and α is Kjx1, and conformably Ω is KjxKj. This way we are able to compile the likelihood function even though there potentially are missing observations for each country even among the representative indicators.

We cannot apply this method to non-representative indicators. To see why, consider the maximum-likelihood estimate of αk, which unsurprisingly is the mean score across countries on indicator k. It is straightforward to see from Equation (1) that the expected value of the sample mean of scores on indicator k is αk +βk ⋅ ~~g~~ k , where<sup>~~g~~</sup> k denotes the average level of governance in the sample of countries covered by indicator

> 20 See Kaufmann, Kraay and Zoido-Lobatón (1999a) for the Bayesian interpretation of these expressions.

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