Generalising the Generalised Method of Moments

The original research paper which presented the Generalised Method of Moments was set in reasonably abstract terms, using mathematical tools such as separable metric spaces and open sets. The abstraction is a good idea, as it puts the theory in a wider context than its immediate statistical application. Theorems in the paper describe the behaviour of two key parameters in a particular minimised function. These results are part of a much wider theory of parameter distribution in minimised functionals, including Noether's theorem which is important in the Risk Symmetries described previously in this blog.

Macromodelling as an exercise

Here's a small exercise in macromodelling which helps with understanding how the economy works. The exercise involves modelling mathematically a simple economy in full, capturing all essential aspects of the macroeconomy - the interaction between money, saving, and the goods market, as well as debt, price determination, international trade, and so on. The modelling should be microfounded, so that the decisions of individual people and companies in the economy should be considered in determining aggregate behaviour. The economy should have enough people, companies, and other actors in it to allow for all of the aspects of the economy, but no more.

The exercise offers benefits in that one comes away from it with better understanding of the macroeconomy, and more knowledge of the contents and limitations of other people's models. I set the exercise's time limit for myself of a couple of hours, so I always (so far) fail to model adequately, but the benefits still accrue.

More on geography's relation with the growth effects of technological gaps

I should clarify and expand my post on Monday on the relation of geography, growth, and technological gaps. When a country has a close trading or geographical relation with another country, then if the countries have similar levels of telephone ownership, the growth of the country with the lowest level is promoted to a greater extent than if the countries have a wider difference of ownership. If the countries are more separated in their relationship, then the lower level country's growth is promoted to a greater extent when the gap is greater. If we look at computer ownership rather than telephone ownership, it is usually only the first case which applies, so a narrow gap is usually better for growth.

I think there are at least three effects in operation:
1. Straightforward technology transfer between countries
2. Clustering effects
3. Terms of trade changes

The first accounts for the positive relations between the gap and growth, the last two account for the negative relations.

Technological transfer's growth effects split by size of gap

For the model in my last post, a gap from the United States in per capita telephone ownership tends to increase economic growth. This is an average result for all countries combined.

If we look at countries split according to whether they have a small, medium, or large gap from the United States, there is a fuller picture. Low gap countries see little positive growth impact from each extra telephone in the gap. Medium gap countries see a very large positive impact from each extra telephone. Large gap countries see a negative impact from each extra telephone in the gap, but the effects are not significant. Perhaps a country requires some technological development in order to take advantage of gaps, for example by transferring technology from abroad.

I am not sure whether the effects are capturing improvements in growth (if the telephones measure greater efficiency increasing growth) or shifts in growth (if the change in technology lowers the growth of countries which are not technologically connected). A bit of both, probably.

Geographical bias of technological transfers' effects on growth

I am studying equations of the form

growth in a country = a0 + a1*technological gap to a leader country + a2*other variables + error

The leader country varies. I've tried France, Japan, and the United States. If the data is split into different continents, we can see if there are any regional effects linking leader countries with greater growth effects in different continents.

For gaps in the number of telephones and cell lines, there is a link, and in the direction that may be expected. The biggest growth effect in Asian countries is from a gap with Japan, and in European countries from a gap with France. The effect of a gap is not significant or beneficial in the Americas or Africa.

Capitalism and the Last Man

The book "The End of History and the Last Man" attracted a great deal of media attention around a decade ago. It argues that history has ended in the sense that capitalism has been shown to be the best arrangement of the economy for the greatest happiness of the mass of the population. It has the appeal of a big-picture description of the driving themes of history, and only lasts a few hundred pages.

The economic content of the End of History looks like a claim of classical Pareto optimality at a macroeconomic level, with some empirical evidence. It doesn't seem appealing for a variety of reasons: much of the world is still in a pre-capitalist state, the empirical evidence deals with socialist states observed to date rather than future ones, capitalism may be challenged by non-socialist economic systems, capitalism suffers from many problems macroeconomically, and other criticisms.

The neglected part of the thesis - from a publicity viewpoint - is the Last Man bit of the title. It contends that under capitalism, although the maximum aggregate amount of happiness is created, individuals may be dissatisfied with their lot because they are comparatively constrained in their ability to gain power (this might not be the exact statement; it has been a while since I read the book). Thus, capitalism denies an important part of human nature, and the Last Man is the idealised embodiment of this happy but incomplete individual.

The idea draws from Nietzsche, via Kojeve. Nietzsche's original ideas, in Thus Spoke Zarathrustra, were far more developed, and capitalism may be viewed not as characterising the Last Man, but the people who lived before them. Whereas pre-capitalist morality was dictated by religions and societies tested by trial and error, morality in capitalism is substantially determined by the demands of the economic system. Capitalist morality is not the outcome of the recognition of the origins and operation of the pre-capitalist morality and retention of its moralities whilst rejecting its logic - which would characterise the Last Man - but rather the acceptance of capitalist morality and logic as a partial replacement for traditional morality and its logic. The social awareness without self-acceptance that characterises the Last Man is absent.

Capitalism unlike pre-capitalist systems has had limited historical testing and so its Last Man would adopt a morality which would not necessarily promote the continuation of humanity but solely of capitalism itself. A transition beyond the capitalist last man may be required for such promotion, either to follow Nietzsche's recommendation of asserting the individual's morality rather than the system's, or to become pre-capitalist morality's last man. This transition may be viewed as necessary in view of the pending disaster of global warming, or the millions of children who die needlessly every year through poverty, or (for a vegetarian like me) the enormous suffering of animals in factory farms.

SADC women's equality

The member states of the Southern African Development Community reportedly have agreed to introduce new legislation domestically to grant greater equality for women in their constitutions. The SADC website hasn't yet posted any information on the change, so the information is second hand which is never a good sign but oh well.

If equality is enforced in the workplace, then it should be good for economic growth. Empirical evidence suggests that increased women's equality is associated with increased economic growth, although there are questions to what extent causality applies since women's equality could be increased by the pressures of economic expansion or it could be acting as a proxy, at least initially, for pro-capital measures such as the suppression of union rights. On theoretical grounds, the link between greater equality and short term per capita growth is compelling.

The link above argues that female protection should be extended into the domestic environment as well. In a high HIV environment, another argument could be used to link economic robustness with such protection.

Consistency of instruments across specifications

When one has multiple specifications for testing an empirical model using instrumental variable or GMM estimation, then the best instruments for each specification may vary according to exogeneity and correlation criteria. Should a fixed instrument set be used across all specifications, or should the instruments be the best for each specification?

Changing the instrument set risks changing the estimation bias from it, unless the instrument sets all have the same correlation with the error term, which is uncommon in economic applications. Similarly, the uncertainty levels will probably not be directly comparable across specifications.

There are several reasons why correlation with an error term could arise. One is that the instrument set is correlated with some of the variables in the specification, while another is that some of the instruments should have been included in the specification. If the instrumental set is correlated with some of the non-instrumented variables, and these variables do not change across specifications, then the correlation across specifications will change if the set varies between them. In this case, it may be best to use the same set across all specifications. If the non-instrumented variables change across specifications, then the correlation is likely to change across specifications even if the same set is used, so it may be best to use the optimal set for each individual specification.

I think the above comments may vary slightly between instrumental and GMM system estimation due to the way instruments are allocated under the GMM system, but the idea is the same.

Weak instruments or endogeneity?

Sometimes in choosing instruments for an instrumental variable estimation one has a choice of two instrument sets. The first is highly correlated with the variables to be instrumented, but exogeneity tests indicate that it is correlated with the error term. The second is weakly correlated with the instrumented variables but exogeneity tests indicate that it is not correlated with the error term. Which to choose, if no better sets can be found?

The correlation with an error term means that the coefficient estimates produced by the instruments will be biased; the weak correlation with the instrumented variables will increase the uncertainty in the estimates. If the weakly correlated instruments produce coefficient estimates with an acceptable standard error, then it is probably best to go with them. Otherwise, using the error correlated instruments may be best, since the biased confidence interval will probably lie within the other confidence interval, if the error term correlation is not colossal.

Similar considerations apply when choosing GMM system instruments.

The benefits of openness and technology transfer

I ran two estimations at the weekend, augmenting a neoclassical growth model. The first estimation included openness (imports plus exports divided by GDP) as an explanatory variable, the second had electric meter imports as an explanatory variable. The dataset, preparation and variable manipulation were standard for growth empirics.

Openness was found to be associated with an average (over countries and years) growth increase of 0.35 percent per year, while electric meter imports were associated with a 0.1 percent increase. In the latter case, electric meter imports are acting as a proxy for a variety of influences on growth, not controlled by the classical variables of education and capital accumulation. In so far as electric meter imports capture the benefits of technology transfer more specifically than openness does, then the gap between the two numbers could very roughly measure the non-technology transfer benefits of openness, equalling 0.25 percent per year.

Selecting instruments in GMM

An instrument is a variable which is not correlated with the error term in an estimation model and is related in some way with the other variables in the model. This definition is mine and relates to generalised method of moments estimation, and is a bit broader than the classical definition described in a second. If one has enough instruments, it is possible to estimate the parameter values in an equation, such as b in y(t) = b.y(t-1) + error, without the estimation biases which would arise if they were estimated using the estimation variables correlated with the error term.

For example, if the error is positively correlated with y(t-1), then when y(t-1) is large, the error is large too and so b will tend to be estimated as being larger than it really is. It might not be y(t-1) which is increasing the size of the error (when one might say that the parameter estimate is sensible), but rather some missing variables which are correlated with y(t-1). So an incomplete model specification leads to a bias in b. Instruments overcome the problem since they only capture the effect of the y(t-1) which is not due to the missing variables.

Although the tendency in empirical economic growth analysis using GMM estimators is to select instruments from lagged variables within the estimation model, it is equally valid to choose variables which are not included among them. One important criterion in the case of GMM system and GMM difference estimators is that the instrumental variables should be at least moderately correlated with the variables in the estimation, to minimise the uncertainty of the resulting estimates of the estimation variables' effects on growth. In this respect, the two GMM estimators are the same as classical instrumental variable estimators, for whom instruments should preferably be highly correlated with the estimation variables.

The moment conditions which give rise to GMM estimation can also admit more complex conditions relating the instruments to the estimation variables. For example, an instrument may have to have its squared value highly correlated with the squared value of the estimation variables. The instrumental relationships required may also change with different variables. GMM instruments are a generalisation of classical instruments.

Does the augmented Solow growth model work for oil producers?

Oil producing countries are usually excluded from economic growth estimations based on the augmented Solow model. The reason is that much of their economic growth is not from accumulation of capital, but rather from sales of a natural asset.

I tested the augmented Solow model on a panel of 24 oil producing countries, with a GMM System estimator and a variety of instrument sets of lagged variables to correct for endogeneity. The fit of the model, measured by parameter uncertainty, was quite good. The parameters on the returns to capital and education were both far lower than for estimates of all countries, with returns to education particularly low by comparison. So it looks like it is sensible to exclude oil producers when analysing growth models, but the Solow model could have merit in describing part of growth in oil producers too. It's probably a small amount though.

Democracy protects against war?

It is sometimes stated that two democracies have never gone to war. If true now and in the future, the world would have no wars if all countries were democracies.

One trouble with the step from democracies have never fought each other to democracies will never fight each other is that the sample set of democracies in the past were mainly Western European and North American, who had little incentive to fight and were closely tied in other ways too. So the second hypothesis may not apply outside these countries.

I am not sure whether the democracies have never fought each other assertion is true in the past, not having information about all conflicts. There have been two recent conflicts, one in the Middle East and now one in the Caucasus, where democracies seem to have fought each other. There are questions about how democratic the parties involved were, but all the same, it doesn't seem that mutual democracy is a perfect protection against conflict.

Biopic of an African hurdler

There was an extended television program on last night, presenting the athletics career of the 400 metre hurdler John Akii-Bua. The form of its presentation meant that one saw how his life and career interacted at the highest levels with some of the major figures and events of recent world and athletics history. It was most enjoyable to watch.

Logging firms, the Congos, and tax

There's an accusation at the moment that logging firms in Congo Kinshasa and Congo Brazzaville are evading taxes. It has been rejected in strong terms by one of the companies concerned.

It will be interesting to see how their governments respond. The DR Congolese government has been a canny manager of the economy generally and foreign companies in particular, so the logging firms may be checking nervously whether their records are indeed clean.

Three competing explanations for authoritarianism in oil producing states

Many major oil producing countries are non-democratic. Here are three plausible explanations from political economy:

1. The economy and incomes in oil producing states are created mainly by sales of a natural commodity, so that the conditions are not conducive of growth through capital accumulation. Thus the class structures which either lead to or support democracy are not present.

2. External powers support corrupt and repressive leaders because they find it more profitable to buy oil from them than from institutions and companies in free democracies. Oil is such an important commodity that most external powers act immorally when dealing with it.

3. The relation between oil and authoritarianism is an accident, with coincidence of geographical clusters of oil and states which are non-democratic for other reasons.

I've seen explanation one a couple of times in theoretical economic literature, and I am fairly sympathetic to it. Recently, events in Africa suggest that explanation two may have merits as well. I do not have strong opinions about explanation three.

Using lagged income to proxy for a single accumulating factor

This post is a small extension to last Thursday's "Lagged income and the omitted accumulating factors" post. In studies of technology's effect on growth, the change in lagged income is sometimes used as a proxy for the change in technology. If we look at the formulas presented in the previous post, it follows that the lagged term is capturing the accumulation of all variables, including capital and education which are usually already present in the empirical specification. Thus, collinearity of variables will be increased and interpreting coefficients will be more difficult. Moreover, technology is identified with all excluded accumulating variables, so must be defined in a broad sense for the approximation to work.

Using Wald tests in checking GMM estimated growth models

Under the misspecified AR(1) model, GMM estimates for two sets of time series with mixed autoregressive parameters in each set will tend to select autoregressive parameters near the top of the ranges for the single estimated autoregressive parameters. Then if a Wald (Chow) test of parameter equality is performed, as is sometimes done in academic papers, the test statistic will become arbitrarily small if the two highest parameters in the two sets coincide. Thus, the Wald test will imply equality of the parameters, but most of the parameters in the first set could be far away from most of the parameters in the second set. I am not sure how Lagrange multiplier tests and likelihood ratio tests would handle things, but given their relations with the Wald test (which is most likely to reject in certain circumstances), I suspect they would have the same problem.

So here's something that works: a series of Wald tests between OLS and robust estimated parameters for the individual time series in the first set with the estimated GMM autoregressive parameter of the second set. If most of the tests reject, then the parameters are not equal. If the original test of parameter equality between the two sets has accepted, then we also have evidence of instability in the parameters.

The asymptotic distribution of the autoregressive parameter in a misspecified AR(1) model under GMM estimation

I have written many times on this blog about estimation using the generalised method of moments of the equation

y(i,t)=a(i)*y(i,t-1)+error

where the i indicates the ith group and t indicates time. This equation is important in growth theory estimation. The misspecification is that a(i) is assumed to be a constant across groups, but really varies. Under GMM System, GMM Difference, and OLS estimators, the estimated a coefficient tends to be near the top of the real range of the a(i)s.

The distribution of the estimated coefficient around the limiting value for a is normal with mean a and a standard error taking the same form as in a correctly specified model. The proof follows similarly to the original proof for the correctly specified model. I am reasonably sure this works - I've followed things through - but haven't worked everything out in detail, so there might be some small issues left.