Per capita income, human capital, and inequality convergence: A latent-variable approach

Journal of Agricultural and Applied Economics, 2003 by Deepak, Sri Devi, Seale, James L Jr, Moss, Charles B

and [delta]^sub i^ is a random variable relating the observable variable to the unobservable latent variable. In the current case, [delta]^sub 5^ = [delta]^sub 6^ = [delta]^sub 7^ = 0.

Assuming that the residuals across equations (1) and (2) are uncorrelated, the implied covariance matrix can be derived as

where [beta] denotes the vector of unknown parameters,

This (unrestricted) specification yields 23 parameters to be estimated before normalization: the structural coefficients ([lambda]^sub 1^, [lambda]^sub 2^, [lambda]^sub 3^, [lambda]^sub 4^, [gamma]^sub 1^, [gamma]^sub 2^, [gamma]^sub 3^, and [gamma]^sub 4^); the variance parameter in the income equation ([psi]); the noncommunal portion of the indicator variances ([theta]^sub 11^, [theta]^sub 22^, [theta]^sub 33^, and [theta]^sub 44^); and the elements of the [Phi] matrix ([phi]^sub 11^, [phi]^sub 12^, [phi]^sub 13^, [phi]^sub 14^, [phi]^sub 22^, [phi]^sub 23^, [phi]^sub 24^, [phi]^sub 33^, [phi]^sub 34^, and [phi]^sub 44^). However, for estimation purposes, one must normalize either one of the [lambda]s or the variance of the latent variable, [phi]^sub 11^. We chose to set [phi]^sub 11^ to one making the latent variable an N(0, 1) variable and therefore making its statistical significance easily interpretable.

The parameters are estimated by maximum likelihood by searching over the parameter space to maximize:

where tr[ ] is the matrix trace operator, and S the sample variance matrix. Given the structure of the current problem, this search can be further simplified. Note that

By setting [Phi]^sub 22^ = S^sub 22^ from the sample variance matrix, the estimation can be considerably simplified. The intuition is that openness, government spending, and investment expenditures are exogenous to the model so their variance parameters are fixed in the same way the x' x matrix in a regression analysis is fixed.

After estimating the parameters in the model, we estimate the latent variable, human capital, by minimizing the weighted squared errors as proposed by Bartlett,

Results

The data of the 22 OECD countries are pooled over the 1955-1990 period for a total of 22 countries x 36 years = 792 observations. Equations (1) and (2) are estimated simultaneously by the maximum-likelihood estimator (MLE) with and without the restriction that [Phi]^sub 12^ = [Phi]^sub 21^ = 0. The likelihood-ratio test failed to reject this hypothesis at the .05 level-of-significance, so only the results of the restricted model are reported (Tables 1 and 2). All the estimated coefficients are statistically significant at any conventional level of significance.

The results from the latent-variable model (Table 1) show that the four indicators-public expenditure, secondary and higher levels of education, and consumption of newsprint-all load positively on human capital. Public expenditure on education (PE) has the largest effect on human-capital accumulation, probably because this investment results in higher levels of schooling for the population and in the improvement of skill levels and the level of technology. The factor with the next largest effect on human-capital accumulation is the percentage of the population with a secondary education (ES). The effects of the percentage of population with a university or equivalent degree (ET) and of the per capita consumption of newsprint (CN) are significantly positive and approximately the same. That the population percentage with a secondary education has a greater effect on human-capital accumulation than the percentage with a university education may be because secondary education is an input into university education, but it may also have implications for the industrial composition of the economy. In summary, increases in these four indicators increase the level of human capital for the 22 OECD countries.