Appendix 6.A: Digression in the effective rank

Roy and Vetterli (2007) are from the first that proposed a metric for the estimation of the effective rank of a matrix. For this purpose, they suggested the application of the SVD method to the matrix H. The singular values О differ from the eigenvalues of the same matrix H, in that they are the positive square roots of the eigenvalues of the matrix HTl, which are no different from those of the matrix HH'. If the rank of the matrix H is equal to к (< и), then only the к singular values are positive (with zero all the rest). In case that the singular values are relatively small in that they contain very little of the information governing the behavior of the economic system most of which is compressed by the uppermost singular values, we can ignore them (by setting them equal to zero) and reconstruct the new matrix Hj whose rank is k. This new matrix Hj constructed by fewer singular values is considered a very good approximation of the matrix H. The SVD allows an exact representation of any matrix, and makes it easy to eliminate its less important terms and, by so doing, to give rise to an approximate configuration with any desired number of dimensions. It goes without saying that the fewer the dimensions, the less accurate will be the approximation. In order to pass judgment on the accuracy of a representation, Roy and Vetterli (2007) suggest the concept of the effective rank.

In order to find the required number of terms to be included in the representation, they employ Shannon’s (1948) entropy index or the spectral entropy defined as

where <7, stands for the singular values of the matrix H.^l(> Assuming that 01n0=0, the effective rank (erank) becomes

erank(H) =e^s

In applying this formula to our 2018 BEA 65x65 matrix of vertically integrated circulating capital coefficients, we get an erank = 27. Of course, such a high rank is way far from what can be called “desired approximation”.

From a mathematical point of view, the idea of the estimation of the effective rank in the above formula is sound, in that, it examines the extent to which the singular values are different from the majority of eigenvalues. Consequently, just a few singular values compress a lot more information than the rest of them. However, the estimates of effective rank according to the suggested formula were not of any great help in our eliminating process. We only got the assurance that the above estimated effective rank is not equal to the nominal and there is a significant difference between nominal and effective rank, a result that is particularly pronounced in the case of fixed capital model. In our example of the U.S. economy, the matrix of capital stock coefficients of 2018 gave a nominal rank 27, which is no different from the rank of the vertically integrated capital stock coefficients. Furthermore, in estimating the number of zeros of our 65x65 matrix K, we found 39 rows containing zero elements together with the zeros scattered to the rest of cells with zeros constitutes the 61% of total figures of the matrix K, without counting the near-zero or not very different from zero elements.

In short, the applied factorization method revealed that the structure of the economies is simpler than originally thought and a lot of information is compressed in the maximal eigenvalue of the system matrices while the remaining eigenvalues add little additional information. Thus, by limiting ourselves to a first-order term, we obtain a very good approximation of the price trajectories in the face of a change in income distribution. In so doing, we end up with the view that the economies, although they are not similar to a single commodity world, the capital intensities of industries comprising the economy would be exactly equal to each other; so, the system’s matrix would be of nominal and effective rank equal to one.

Notes

• 1 Shaikh’s (1984) article was available at least two years before its publication.
• 2 See https://library.duke.edu/rubenstein/findingaids/solowrm/
• 3 It is important to note that the discussion of Brody’s kind of description of the economy is limited to circulating capital models. The fixed capital model is not investigated in this particular literature, which is usually left to a later stage of the analysis. Our treatment of fixed capital follows that suggested by Leontief (1953) and Brody (1970).
• 4 In general, the spectral gap is defined as the difference between the moduli (the absolute values) of the two largest eigenvalues of a matrix.
• 5 In experimenting with the US input—output data of dimensions of 71x71 and the years 1958, 1962, 1967, 1972, 1977, 1982 and 1987, Clnlcote (1997, pp. 176-177) concludes that the depreciation matrix plays an important role in the measurement of DP and brings them 20% to 25% closer to MP as this can be judged by the various statistics of deviation. The matrix of fixed capital improved the predictability of PP. However, the inclusion of turnover time, capacity utilization and we could add indirect business taxes exert only a relatively minor effect to either direction.
• 6 For the differences and the implications between the net and gross fixed capital stock, see Shaikh (2016, pp. 801-806), Malikane (2017) and Tsoulfidis and Pai- taridis (2019). For the joint production treatment of fixed capital, see Schefold (1971), and for its critique, see Shaikh (2016, pp. 804—806) and Congliano et al. (2018, ch. 11).

• 7 The nominal or numerical rank of a matrix is equal to the number of nonzero eigenvalues. The effective rank is defined as the dimensionality of the matrix determined by the number of eigenvalues or singular values that exert most of the influence in a matrix and these may end up to be only very few.
• 8 From the very early attempts to arrive at a metric of effective rank is in Roy and Vetterli (2007). For further discussion, see the Appendix of this chapter.
• 9 In the case of circulating capital model, we found that the more detailed the Leontief inverse (by including the workers’ consumption expenditures coefficients), the smaller the number of singular values required to approximate the 90 percent borderline. The idea is that the matrix of workers consumption coefficients, bl, as the product of two vectors, is linearly dependent and its presence in the Leontief inverse increases the maximal eigenvalue and its gap with the subdominant eigenvalues.
• 10 For further discussions of eigenvalue distribution when changing the aggregation level, see Mariolis and Tsoulfidis (2014, 2016a), Gurgul and Wojtowicz (2015). Shaikh and Nassif-Pirez (2018) and Shaikh et al. (2020) show that the spectral ratio decreases with the disaggregation level.
• 11 The spectral flatness (SF) is indicated in the last row of Table 6.3 and it can be at most equal to one, since the geometric mean will be always less or equal to arithmetic mean. Of course, the closer to one the more distinct are the eigenvalues. In Table 6.3, we find that the SF varies from 0.416 for GRC 2000 (the lowest) to 0.636 for CAN 2000 (the highest).
• 12 We used the Matlab for the estimation of the vector Uj, which is no different in the case of the estimation of the l.h.s. eigenvector of the matrix H|or of similar experiment with the SVD method (see Section 6.5.2).
• 13 Practically, this might not be exactly the case; the other rows of S may contain cells containing near zero numbers most likely due to rounding. In the subroutine of Gauss (or Matlab) that we utilized in our estimations, we found that the elements in the rows other than the first were trivial in size. Thus, the rank of H| although we know is equal to 1, the answer that we derive from Gauss or Matlab may be different.
• 14 We present the estimates of the l.h.s. eigenvector corresponding to the maximum singular value. The results with the application of the Schur method were the same with the difference in the 39th element. The reason perhaps is that the Schur gives the maximal eigenvalue not necessarily as its first eigenvalue and respective eigenvector; thus, we need to make the appropriate multiplications and then interchange with the first element which is zero in our case with the 39th element which is equal to 1. This is the reason that we opted to display the SVD estimations whose 39th element is equal to 0.014 and all other elements are equal in both estimations.
• 15 Samuelson (1962) thanked Garegnani ‘for saving me from asserting the false conjecture that my extreme assumption of equi-proportional inputs in the consumption and machine trades could be lightened and still leave one with many of the surrogate propositions’.
• 16 It follows that the more similar are the singular values, the higher is the entropy which is maximized when S,_mx = -n * in * ln(in = X O',/,,, and in our case for n = 65 are of the same value, the entropy will be rn = 0.157117. The exponential of the term S_mex gives an effective rank of approximately equal to one depending on the number of terms, whereas the maximal nominal rank might be и, that is the number of linearly independent rows or columns. In the case of a random matrix, its effective rank will be 1! Although the nominal rank may be quite higher, because the subdominant eigenvalues tend to becoming negligibly small near but not necessarily zero.