Common technology and AR(1) instability

In panel data estimation of growth models, data for different countries is collected over a number of time periods. It is different from cross-sectional estimation which uses a single time period, and time series analysis which uses a single country. The estimation methods are different, too.

For analysis of panel data to be useful, there has to be some similarity across countries between the way in which the time series are generated. For example, two countries may each have their national income estimated as

capital in country^a * labour in the country^b

where a and b are parameters specific to each country. In many growth models, it is usual to assume that the parameters are also constant across countries. The assumption is known as the common technology assumption.

The last post argued that the autoregressive parameter in growth estimations is not stable across countries. Economists should be very cautious about specifying models where the autoregressive parameter cannot vary across countries. A common specification does just that however, taking the form

ln income per person = a.ln income last period + other terms assuming common parameters + country specific constant.

a is the same across countries and the country specific constant varies.

Suppose that the constant a is estimated as an average across countries, and that a particular country is found to have a positive specific constant. During the period of measurement, its real a was higher than estimated a, and the specific constant corrects for it. The constant corrects for an average discrepancy in income during the period, so roughly (since the average is over ln income, not income)

specific constant = a(real).lnaverageincome - a(estimated).lnaverageincome
= (a(real) - a(estimated)).lnaverageincome

Now, if country income is lower than its period average, the discrepancy is

(a(real) - a(estimated)).(lnaverageincome - something positive),

so that

specific constant - (a(real) - a(estimated)).(lnaverageincome - something positive)
= (a(real) - a(estimated)).something positive

The specific constant overestimates the income growth at small incomes. Similarly, it underestimates at large incomes. Out-of-sample predictions will usually occur for future time periods where countries are richer, so that the model will underestimate by increasing amounts over time.

The presence of time period specific constants corrects the problem a tiny bit, but the main difficulty is that the model remains misspecified. Time constants are moreover useless for future prediction unless they are subject to their own modelling.