تعصبات ادغام در تولید توابع : تجزیه و تحلیل داده ها از مدل ترانسلوگ

The value of panel data for micro-units in exploring aggregated relationships empirically is substantial, since such data are not only useful for estimation at both the micro- and the macro-level, but also allow comparison of properties of relationships aggregated formally from the micro-level with those resulting from aggregation by analogy in an informal way. Using panel data from manufacturing plants, we consider biases in the aggregation of the Translog production function, which does not generally satisfy the strict mathematical conditions for perfect aggregation. Linear and log-linear aggregation in the presence of random coefficients are considered. Under linear aggregation across plants, departures between geometric and arithmetic means of inputs as well as correlation between log-inputs, contribute substantially not only to biases in the output volume, but also to instability in derived input and scale elasticities and biases in total factor productivity growth. Overall, the biases under log-linear aggregation are smaller.

Macro-economists, macro-econometricians, constructors of empirical macro-models and policy makers who utilize micro-based theories for macro-economic purposes, very often rely on a ‘representative agent’ interpretation of the relationships describing the theory, i.e., they aggregate the relationships informally, ‘by analogy’. This way of treating, e.g. production functions, investment demand functions, and labour supply functions is far from being satisfactory. From the formal theory of aggregation1 it is well known that micro-relationships can only be aggregated to functional relationships if certain restrictive assumptions with respect to the form of the micro-relationship(s), the aggregation procedure for the micro-variables, and/or the distribution of the macro-variables across micro-units are satisfied. One familiar case is linear aggregation of linear relationships with identical coefficients across micro-units. Another case is linear aggregation of linear relationships with varying coefficients across micro-units when all micro-variables move proportionally over time. Still, the stability of aggregates of non-linear functions is of great interest for macro-economic research, modelling, and policy purposes. With respect to, for example, the aggregate production function, Jorgenson remarks that ‘The benefits of an aggregate production model must be weighted against the costs of departures from the highly restrictive assumptions that underly the existence of an aggregate production function’ (Jorgenson, 1995, p. 76). Interesting questions then are: Which are the most important sources of aggregation bias and instability? Will aggregation by analogy, in which estimated micro-parameter values are inserted directly into the macro-relationships, perform satisfactorily?

I n this article, using panel data, we consider aggregation of Translog production functions from the plant to the industry level. Plant specific heterogeneity is represented by random intercepts and first-order coefficients. We show how exact aggregation in different contexts should be carried out and derive exact formulae for the scale and input elasticities. In the empirical part of the paper, the main issue is to estimate aggregation biases in expected output, scale and input elasticities, and rates of TFP growth when exact formulae are compared with formulae typically used at the macro-level in analogy to the micro-level. Three sources of aggregation bias can be distinguished. Considering aggregation in terms of expectations of logarithms (geometric aggregation) and disregarding correlation between log-inputs and their coefficients, only one source is of relevance, that is the correlation between log-inputs. If we consider aggregation in terms of expectations of non-transformed variables (arithmetic aggregation), which most closely resembles the aggregation by analogy often used by practitioners, two more sources should be taken into account. The relative discrepancy between the logs of the arithmetic and the geometric means of the inputs and the output give important contributions. The means of the estimated macro-scale elasticity over this 22-year period varies between 1 and 1.25 across the two industries, depending on the aggregation procedure. There is a substantial difference in the size of the aggregation bias depending on whether geometric or arithmetic aggregation is performed. A primary reason for this is that the genuine disturbance in the Translog function plays a different role. In the former case, it has no influence on the estimated aggregation bias in expected output, since it has zero expectation, whereas in the latter, it will contribute to the bias not only in expected output, but also in the scale and input elasticities, owing to the non-linearity of the transformation from geometric to arithmetic means. When using aggregation by analogy in macro-economics, one is in general forced to neglect this bias.