Business Cycles in the Contemporary World pp 91-108 | Cite as

# An Aggregation Problem for Linear Models?

## Abstract

This part of the monograph analyzes the generation of macro economic cycles from sectoral cycles on the basis of US investment data. The focus of the study is on the variable new real (base year 1987) capital spending, including permanent additions and major alterations to plant structures along with new machinery and equipment, taken from the NBER-CES/Census US Manufacturing Productivity Database. The dataset consist s of 450 SIC^{1} 4-digit industrial series, covering annual observations for the period 1958 to 1992. For further detail, see Bartelsm an and Gray (1996). I test t hese 450 SIC 4-digit series, covering the period from 1958 to 1992, for cyclicity and estimate their central spectral features. I find that almost all of the disaggregated series, as well as the aggregate series, are strongly and significantly autoregressive. These series have complex roots that imply business cycles with traditional periods centered around five years. In this context alternat ive VAR specifications are tested. Their ability to explain the spectral features of constituent and aggregate series is examined in simulation experiments. I find that there is a substantial and statistically significant feedback effect from the aggregate activity on the sectoral series. In t he VAR simulations the aggregate feedback acts as a common shock impinging on the sectors. The residuals, which I interpret as idiosyncratic shocks, are only weakly correlated among the sectors. In general I find that the VAR model is able to describe the cyclical characteristics of the individual time series both accurately and parsimoniously. The linear models are only partially successful in modelling the aggregation process. This points to the desirability of a nonlinear aggregation model. This is also suggested by the empirical finding that the phases and periods of the cycles are drawn together over time, cf. the preceding chapter of the present study.

## Keywords

Monte Carlo Business Cycle Aggregation Problem Common Shock Aggregate Activity## Preview

Unable to display preview. Download preview PDF.

## Notes

- 1.
*Standard Industrial Classification*SystemGoogle Scholar - 2.Indicating that the total private investment series, slightly leads the manufacturing investment business cycle component.Google Scholar
- 3.Despite for the MFI series, where we consider the slightly longer period from 1958 to 1992.Google Scholar
- 4.Note the SIC-codes are given in 1972 definitions. For concordance with 1987 definitions, see http://www.nber.org/nberces/sic_72.txtGoogle Scholar
- 5.For the not displayed results for the higher aggregated constituent series, i.e. the 143 SIC3 and the 20 SIC2 manufacturing subsectors, the respective motor vehicles group contributes the highest part also for the HP(100)-filtered series. Furthermore, the ranking pattern is quite robust for both filtering methods for these higher aggregated sectoral definitions, too.Google Scholar
- 6.I implicitly assume that the constituent series bear a similar autoregressive structure as the resulting aggregate series and therefore fix the order for all series at
*p*= 2. It is also the average order, recommended by the Bayesian (V)AR-order determination procedure suggested by Heintel (1998).Google Scholar - 7.Note, there is only one industry SIC 3334 ‘primary aluminium’ that lies in the first half of the ranking
*and*shows a standardized standard deviation > 1%.Google Scholar - 8.In general the periodicities at maximum of coherency (aggregate to disaggregate) are distributed between 2.42 and 14.08 years with a median of 4.68 and a mean of 4.72 years. The distribution’sstandard deviation equals 1.18.Google Scholar
- 9.In 160 cases there exist two complex solutions and thereby two implied periodicities for the VAR(2)-models. For these cases Figure 6.2 displays the implied periodicity of that root that correponds to the relative higher sc between respective disaggregate and aggregate series. On average these second roots show a sc-value that is less than half that of the first root. Furthermore, the implied period lengths of the second roots are more than three times as dispersed as the ones of the first roots.Google Scholar
- 10.For an overview of simulation-based inference in econometrics, see Mariano. Schuermann and Weeks (2000).Google Scholar
- 11.There is a body of (mainly) statistical and methodological literature concerned with the fundamental problem of the linear aggregation of economic dynamics. It includes the contributions by Theil (1965), who coined the term
*non possumum*results for linear aggregation, van Daal and Merkies (1984), Goodwin (1984), Schlicht (1985), Barker and Pesaran (1990), Quah (1994), Uhlig (1996) and Forni and Lippi (1997).Google Scholar - 12.While changes of phase shifts
*over time*can be accounted for in linear models (for example by means of aggregate shocks setting idiosyncratic cyclicalities in phase), varying periodicties over time can, by no means, be explained in the framework of linear models. In a similar context Haxholdt et al. (1995) note: “For linear systems the principle of superposition applies. The behavior of any variable in the system is a sum of distinct modes, or elementary excitations, where the frequencies and attenuation rates are determined by the eigenvalues of the Jacobian matrix. Different modes can exist and develop independently of one another, and the sensitivity of any mode to an external disturbance will be the same, irrespective of the phases of the others. Mode-locking cannot occur in linear systems, because the individual oscillatory modes do not affect one another,” p. 179-180. The decisive point is that the modes do*not*interact in linear systems.Google Scholar - 13.Actually, in the RBC framework these are linearized behavioral equations of representative agents or sectors in a Ramsey model’senvironment.Google Scholar