I'm using the WIOD database to compute GHG emissions embodied in imports to Israel. While calulating the a-matrix (technical coefficients) I have to decide how to handle total ouputs with 0 value, e.g. sector P: households with employees, at most regions. That is in order to prevent NAN values, an to ensure Leontief Inverse can be computed.
Currently I assigned a relatively small values to those cells, i.e. 0.00000000001 Million USD, but I was wondering if it is indeed the correct way to handle it, or should I apply zeros to NAN values in the tecnical coefficient matrix?
Dear Dor,
I think you can check the sensitivity of the number you think you can input. I attach a page from the Nobel laureate, Walter Lewis on this matter. Hope it helps.
Thank @LallB.Ramarattan. I posted the question, exactly for the reason of an "undetected" effect of small errors on my resutls, through feedback loops within the model. Following the suggestion above, I did some iterations with changing values, and suprisingly have found no changes at all for the mean, median, range and stdev.
Further checking this issue, I've came to understand that this value has no special significance. That is due to the following:
Computing a-coefficients is done as a division of each sector's input by its total output; thus for sectors with total output == 0; all inputs, by definition (under the case that there are no negative values within the IOT), equal zero. To conclude, if the numerator (i.e. inputs) is zero, the value of the denominator (i.e. total output), does not take effect on the result, as long as it is un-equal to zero.
Dear Dor,
I am glad you have found out the answer.Good luck in your research.
Dear Dor Fridman,
I am not a specialist working on IIOT. Perhaps, you have completed your research and you are not interested in this question now.
I recently studied how IIOT by WIOD is organized. I have found that IIOT and the new theory of international values has a close relations between them. See, for the details, section 15 in my paper below.
I have a question I want to ask you. Are you calculating A-matrix for each "region"? In my understanding, you have to calculated A-matrix by dividing each entry by the total world output. If the total output is positive, there is no problem. It is inconceivable that you have total world output 0. In such a case, the IO table designer will omit such "industry". Am I wrong?
Working Paper The New Theory of International Values: An Overview
Dear Dor,
I have used many times WIOD database in my research. In order to derive A and L matrices if one row of total production vector x is equal to 0, I suggest to put a 1 in the corresponding element of the final demand vector. This way, the corresponding element in the total production vector will return 1, and the matrices A and L can be computed without affecting the values of the others economic sectors.
Moreover, for large multi-regional matrices such as the WIOT, I suggest to calculate matrix L using the Power Series approximation in a recursive way: L = (I-A)^(-1) = I + A + AA + AAA + ... . The stop criteria can be set as the equality between the total environmental impact (sum of the elements in the environmental intervention vector B) and the sum of the embodied environmental impact in final demand products (E = B*L), since the environmental impact is conserved.
I hope this may contribute in answering your question.
Regards,
Matteo Rocco
Dear Yoshinori and Dear Matteo,
Thank you for your kind answers. Indeed I have already completed this research and moved on. Yet I think I should comment back.
First, Yoshinori, I don't estimate a different A-matrix for each region. The WIOT is one input-output table, thus it should be treated as one table. Each row corresponds to all outputs (sales) of sector, i, within country, r. Each column corresponds to inputs. So technically rows sums are regional and sectoral, and not global. If one sector`s production is set to zero, and by-logic, this sector's inputs will be set to zero as well, it can be removed from the IOT. However, the authors have decided to leave it that way. A matrix is calculated by dividing each column by the corresponding row's sum, and produces NA if the row's sum is zero. In that case, you can either assign some value (e.g. 1, 0.1, etc) to the total output vector (x) or to the final demand vector as suggested by Matteo Rocco. Alternatively, you can calculate the A-matrix and than assign zero, to all NA values.
Second, Matteo. I agree that it can be done as you suggest. It can also be done by altering the total output directly. I personally prefer to assign zero to all NA, after producing the A matrix.
I would appreciate if you could comment on the reason for using the approximation of the Power Series for the WIOD. I am aware that calculating an inverse takes some time, yet using R for that purpose I was able to calculate a time-series of L matrices at a reasonable time, and with a modest, desktop machine.
Best,
Dor Fridman
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