Let us aggregate widgets. They are exactly the same. Then someone changes the definition of "widget" but we keep aggregating. Then someone uses a different estimator but we keep aggregating. How can anyone explain what results? Do you trust macro statistics?
There is a large, but mostly ignored and forgotten, literature on this topic. Aggregating labour was generally shown to be ok. Aggregating capital (to be used as an input) has been shown, by some, to be without issues only in the most mundane of situations. However, the consequences of mis-aggregation have not been clearly spelled out to my knowledge and the difficulties in obtaining and estimating models from dis-aggregated data appears to have let the inertia of aggregate production models to win out. The consensus seems to be that this is a problem with macro applications rather than micro applications although i do not see why personally as many micro applications have heavily aggregated indices of heterogeneous capital. The aggregation of capital is a problem of some difficulty in getting around, hence the continuation in use of aggregate production functions. If your aim is to emply them for a standard growth accounting excercise then you have a lot of previous research work to fall back on. With regards to your question over widgets - most aggregate production functions divide by some function of value to obtain the capital 'levels'. If you can obtain the valuation basis for the widgets in each period you should be able to create a uniform value index of your widgets.
References include:
"The existence of aggregate production functions" Fisher, F 1969.
"The existence of input and output aggregates in aggregate production functions" Blackorby and Schworm 1988.
"Aggregate production functions, neoclassical growth models and the aggregation problem" Felipe and Fisher 2006.
Daniel Gregg,
Thank you very much for your comments, and these citations. I have a tendency to trust most whatever comes first to economists' minds, and you've put some of your best there, which leads me to believe you've thought more than I have about this. Oh, I know there's an old literature to all this, but I feel the need to reprise the problem in the context of a data glut and coninued reliance on large N. So it's the applied angle I;m worried about. These production and I/O ideas are decidedly macro - so they may somehow take a defensive posture, and I couched the problem in those terms because I generally think about the loss of information when dealing with aggregates.I think not enough people think about variance.
Anyways, I appreciate this input!
Oh, I forgot to mention the representative agent problem, which is a condition for aggregating log-linear models, and maybe Cobb Douglas. Lewbel's "Aggregation with Linear Models" .
The aggregation of heterogeneous goods is one problem, but there is another aspect to the aggregation problem: Aggregating outside of equilibrium. If markets do not clear or prices adjust slowly it makes no sense to calculate excess supply or excess demand because the prices do not reflect market values. At the micro level Richard Quandt and colleagues have several articles on the econometrics of disequilibrium situations. At the macro level it was Barro and Grossman. But they assume markets are adjusting instantaneously toward the market equilibrium, or not at all. economies where market prices adjust slowly, or there are controlled prices the problem is very difficult. At the macro level aggregating outside of equilibrium is meaningless. As a practical matter I have no suggestions but would welcome ideas.
As someone who spends most days working with data from the BEA, BLS and the Fed, I have thought a lot about this topic. To answer the question I think we first have to ask why are we looking at some particular data report. I think the first point is that fewer sins are made when you aggregate nominal (or current price) data than when you aggregate real (or inflation adjusted data). The aggregation of nominal variables has an obvious interpretation and I can add across different classes of goods or services or people. Once you start adding up real things, then the trouble begins. Unless all prices are rising at the same uniform rate, there is no unique was to even calculate the answer conceptually (base weighted, chain weighted etc.) In that sense, calculating total retail sales commits far fewer sins than calculating industrial production. I am not even tacking modeling and forecasting issues here--just measurement issues. Regardless of the sins in aggregation, we need to do it because we simply cannot understand and follow the micro detail. So my purpose in following data is to try to predict things that matter to the markets (such as inflation, employment growth, monetary policy etc.) Not only do we have aggregation issues but also measurement issues in general (such as sampling errors). I find in general that we have plenty of data on manufacturing and housing (except perhaps a good quality-adjusted house price series) but far too little data on services (which is the largest part of the economy). U.S. data focuses too much on spending and not enough on output based measures. As to reliability, I am very sceptical of the usefulness of the GDP data in the U.S. I would prefer to track the overall economy from employment trends and business surveys (such as the ISM) The nature of the GDP calculation--especially real GDP--which involves backing out imports often gives some very strange results when there are large swings in trade prices (such as the early releases of GDP in 2008 Q2, which put growth around 3% when we were arguing that the economy was in recession but growth in that quarter is now estimated at 1.3%. In the end, we have to work with what we have regardless of the sin committed!
The effects of aggregation can be substantial in case of strong structural change. In a study of wage share in Indian Industry, we found that while the aggregate wage share was declining throughout, if we stabilised industrial composition, the decline in the wage share was arrested for almost half of the period (1963-87) and resumed until later (see figure 2, attachment). In other words aggregation was responsible for the decline in the wage share in Indian Industry for the first fifteen years under study, and other factors only thereafter.
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