The linear nature of the PRP and WRP curves encourages the research for the presence of a hyper-basic industry (or a few) embedded in the structure of the economy bearing most of the weight that gives rise to the quasi linearity of the curves. This is something very similar to the idea of one- commodity world or, what amounts to the same thing, the idea that of a single industry containing enough information to display trajectories that are similar to those that we would have derived had we had the information from all the sectors comprising the entire economy. It goes without saying, that a hyper-basic industry does not exist in reality, in the sense that it can be located in input—output data along with many other usual industries; however, such a hypothetical industry, if constructed may contain enough information, which is extracted, ideally, from the basic industries of the economy. The construction of such a hyper-basic industry bears some similarity to the pivotal role of the Sraffian standard industry that has been utilized in our analysis.
From the above discussion, it follows that the matrix H| although a linear approximation nevertheless is a pretty good approximation of matrix H , as this can be judged by an inspection of the graphs in Figures 7.11 and 7.12 as well as the statistics of deviation which are minimal. In the sense that both matrices give rise to price trajectories not very different from each other, we can decompose the matrix H| into two matrices (factors). As in Chapter 6, we apply the Schur factorization (Meyer 2001, 508—509, see also Mariolis and Tsoulfidis 2016b, 2018) which gives the matrix of eigenvalues S, a diagonal matrix whose maximal eigenvalue is one and the right-hand eigenvector U|of matrix Hi associated with it.
The column vector U] of the matrix Hj will be
which when replaces the first column of the identity matrix, gives the new matrix U of dimensions 5x5. The matrix S can be now estimated as follows:
The first row of the resulting matrix Бщ is the economically significant vector, which is displayed in Table 7.3.
We observe that the first element of the so-estimated vector S is (approximately) equal to one and the rest on the first row are all positive or rather of the same sign. Finally, all the other elements of matrix S are in principle zero."
Thus, with the aid of the Schur decomposition, we managed to transform the matrix Hi into the matrix S. Since matrix H| gives rise to trajectories of prices quite similar to those of matrix H, it follows that matrix H| is not very different from H. Therefore, the matrices H] and H are quite similar to the matrix S. This similarity among the three matrices essentially amounts to the cells of the first row corresponding to a single industry of the matrix S. The composition of this so to speak hyper-basic industry, as the direct result of a similarity transformation is designed to encompass essentially the structural properties characterizing the entire economy.
This approximation of the matrix H through the first term of its eigende- composition, that is, the matrix H] is further enhanced, the lower the spectral ratio. In our case, the eigenvalue gap is large enough ensuring a relative accurate approximation, as this is discussed in Chapter 6 and it is depicted in the graphs of Figures 6.10 and 6.11 and Table 6.3. In short, if the second eigenvalue tends to zero, the better will be the approximation. This is particularly pronounced in the fixed capital model.
In the case of the circulating capital model, the application of Schur decomposition to the matrix HR gives the same matrix U| as that obtained from the matrix H|, of course except by multiplication by a scalar; the latter does not affect the results when the first column of the matrix U) replaces the first column of the identity matrix, which multiplied by the matrix H gives
The first row of the matrix S gives us the economically significant vector displayed in Table 7.4.
One wonders to what extent, if any, the two row vectors—displayed in Tables 7.3 and 7.4 derived from the rank 1 of matrix Бщ and the rank 5 of matrix Sh, respectively—relate close enough to each other. For this reason, we plot them in a scatter graph along with their linear regression line (Figure 7.13). The R-square of the two vectors is quite high at 92.7%
Table 7.3 The matrix Sjj|, Schur method
Table 7.4 The first row of the matrix SH
Figure 7.13 First row of matrix S j-j us. approximation Sjp
indicating that the two estimated vectors move pretty much together. The MAD of the two estimated basic vectors is only 6.02% and the d-statistic is 21.23%. Of course, the number of observations is very small to claim unquestionable proximity; however, we know that the derived results are in line with those found for similar but larger dimension matrices in Chapter 6.
Lastly and for the same purpose, we tried the singular value decomposition (SVD) whose results not surprisingly were approximately the same with those of Schur method. Let us start with the application of the SVD to the matrix H] that gives us the following economically meaningful vector
that replaces the first column of the identity matrix. Subsequently, we apply the similarity transformation Бщ = U-IH|U and the matrix S^p will be
Table 7.5 The matrix SH[, SVD method
The nominal rank of the above matrix is equal to 5, but in fact is equal to one in that all the information of the matrix is contained in its first row that forms the economy’s hyper-basic industry. The results from the matrix HR gives the economically meaningful vector
which as above, it replaces the first column of the identity matrix. Subsequently, we apply our usual similarity transformation and get
Table 7.6 The first row of the matrix SH, SVD method
The differences between the two methods of constructing the hyper-basic industry are minimal, and there is no particular reason to prefer one method over the other.
Summary and conclusions
In this chapter, we carried out most of our estimations with the use of an aggregated into five sectors input—output table of the USA of the year 2014. We chose this particular input—output table of 54 industries because of the necessity for our estimations data. The purpose of this aggregation was to enable the reader to evaluate visually the arguments presented in Chapters 4—6 focusing on manageable dimensions matrices.
We utilized two models: a circulating capital model and a fixed capital model in which the matrix of fixed capital has been estimated according to two methods. As expected, the fixed capital model, in both of its versions, gave more definitive results than the circulating capital that has been used extensively in similar studies. This by no means implies that the results in the two models are so much different; both models are very good approximations to reality and the fixed capital model is preferred to circulating. However, the lack of sufficient data of investment flows matrices, as well as the vectors of capital stock and turnover times limits the applicability of fixed capital model. The circulating capital model is usually used precisely because its data requirements are not too demanding.