Here, we review three recent studies that offer alternatives to the pure LQbased methods discussed previously.^{Footnote 5} The first is an innovative study by Fujimoto (2019), who examines official surveybased data for nine Japanese regions in 2005, the most recent year available in a series of official tables published quinquennially since 1960. This study’s focus is on crosshauling and its primary aim is to determine which of four alternative assumptions is most appropriate. Each assumption is associated with a particular modelling approach, as follows:

1.
There is no crosshauling (LQ approach);

2.
Crosshauling depends on regional size (FLQ approach);

3.
Crosshauling is proportional to its potential, as measured by output or demand (RCHARM);

4.
Crosshauling is proportional to its potential, as measured by the volume of trade (MCHARM).
To represent the ‘LQ approach’, Fujimoto rejects the SLQ in favour of the scaling formula
$$ {\text{DSLQ}}_{i}^{r} = {{\frac{{X_{i}^{r} }}{{D_{i}^{r} }}} \mathord{\left/ {\vphantom {{\frac{{X_{i}^{r} }}{{D_{i}^{r} }}} { \frac{{X_{i}^{n} }}{{D_{i}^{n} }}}}} \right. \kern\nulldelimiterspace} { \frac{{X_{i}^{n} }}{{D_{i}^{n} }}}}, $$
(13)
where X denotes output and D denotes demand. The rationale for using this alternative formula is to overcome aggregation bias (Fujimoto 2019, p. 113).
For the second approach, Fujimoto employs the formula:
$$ {\text{FLQ}}_{i}^{r} \equiv {\text{DSLQ}}_{i}^{r} \times \left[ {\log_{2} (1 + v^{r} /v^{n} )} \right]^{\delta } , $$
(14)
where v^{r}/v^{n} is the ratio of total regional to total national valueadded payments. However, the author does not explain why \({\text{DSLQ}}_{i}^{r}\) is used instead of \({\text{CILQ}}_{ij}^{r}\) nor why value added is used as a proxy for regional size rather than superior measures such as output or employment. It is, therefore, misleading to refer to this approach as the ‘FLQ approach’. The third approach is the refined version of CHARM developed by Többen and Kronenberg (2015), while the fourth is the modified version of CHARM proposed by Fujimoto.
Fujimoto (2019, p. 115) remarks that ‘[t]he FLQ approach has a problem in addition to the difficulty of [specifying] a value for δ: the crosshauling caused in interregional trade depends not only on regional size.’ To demonstrate this, he derives the following equation for \({\text{DSLQ}}_{i}^{r}\) < 1:
$$ \Delta E_{i}^{r} = \Delta M_{i}^{r} = {\text{DSLQ}}_{i}^{r} (1  \lambda )\left( {1  m_{i}^{n} } \right) D_{i}^{r} , $$
(15)
where E and M denote exports and imports, respectively, \(m_{i}^{n}\) is the national propensity to import from abroad and D is demand. Fujimoto adds that there is ‘no reason why crosshauling [should] depend on \({\text{DSLQ}}_{i}^{r}\) and \(m_{i}^{n}\) [and] this dependence causes serious bias …’ (p. 116).
However, a straightforward interpretation can be given to the role of both \({\text{DSLQ}}_{i}^{r}\) and \(m_{i}^{n}\) in Eq. (15). Since it is assumed that \(m_{i}^{r}\) = \(m_{i}^{n}\), the term (1 − \(m_{i}^{n}\)) captures the proportion of regional demand \( D_{i}^{r}\) that is met by domestic suppliers, some of which are located in region r and the rest in other regions. As expected, there is a negative relationship between \(m_{i}^{n}\) and Δ\(M_{i}^{r}\), ceteris paribus.
To explain the positive relationship between \({\text{DSLQ}}_{i}^{r}\) and Δ\(M_{i}^{r}\), we note that Δ\(M_{i}^{r}\) represents the difference between the extra imports generated by using \({\text{FLQ}}_{i}^{r}\) and those generated by applying \({\text{DSLQ}}_{i}^{r}\). Although this difference is invariably positive, its magnitude increases as \({\text{DSLQ}}_{i}^{r}\) rises. This is due to the inclusion of λ in \({\text{FLQ}}_{i}^{r}\). The upshot is that there is a positive relationship between \({\text{DSLQ}}_{i}^{r}\) and Δ\(M_{i}^{r}\).^{Footnote 6}
The above discussion suggests that Fujimoto has failed to identify a genuine problem with the FLQ approach. Nonetheless, after performing various statistical tests, using data for 106 sectors and nine regions, he rejects this approach on the grounds (i) that it yields biased estimates of regional imports and (ii) that the appropriate value of δ is unknown and depends on each case.
However, since Fujimoto does not use the correct FLQ formula, his findings do not constitute a valid test of the FLQ approach. It is also likely that superior estimates could have been obtained by using a different δ for each Japanese region. In particular, the islands of Hokkaido and Okinawa may well require different values of δ from the mainland regions. The approach discussed later in this paper affords a way of generating such regionspecific values.
As regards the other approaches, the scatter diagrams of estimated and surveybased import propensities (Fujimoto 2019, figure 2) reveal an almost identical pattern for the DSLQ and RCHARM methods, with evidence of substantial and widespread underestimation. By contrast, the diagram for MCHARM suggests a more random distribution, albeit with a greater variance and some heteroscedasticity. Even so, using the mean absolute error as the criterion, RCHARM invariably outperforms MCHARM (Fujimoto 2019, table 2). The DSLQ is clearly in third place.^{Footnote 7}
In another recent study, PereiraLópez et al. (2020) focus on the AFLQ rather than the FLQ. They argue persuasively that regional specialization can have different effects on the columns (cost structure) and rows (selling structure) of a regional coefficient matrix. For instance, a region that is specialized in the extraction of mining products may sell most of its output to processing sectors such as metal industries located in other regions. The AFLQ has the limitation that it presumes that specialization only affects purchasing sectors (columns). The authors thus propose a bidimensional procedure, the 2DLQ, whereby a parameter α is used to adjust the rows, while another parameter β is applied to the columns. The regionalization employs the following relationships:
$$ \hat{a}_{i j}^{r} = \left\{ {\begin{array}{*{20}ll} {({SL Q}_{\text{i}} )^\alpha a_{ij}^{n} (x_{j}^{\;r} /x_{j}^{\;n} )^{\beta} } & {{\text{if}}\;{SLQ}_{i} \le 1} \\ {[0.5 \tan \text{h} ({SLQ}_{\text{i}}  1) + 1]^\alpha a_{ij}^{n} (x_{j}^{\;r} /x_{j}^{\;n} )^{\beta} } & {{\text{if}}\;{SLQ}_{\text{i}} > 1} \\ \end{array} } \right., $$
(16)
where \((x_{j}^{r} /x_{j}^{n} )\) measures the relative size of purchasing sector j. The hyperbolic tangent function (tanh) allows the estimated regional coefficients to be ‘slightly higher’ than the corresponding national coefficients if \({\text{SLQ}}_{i} > 1\) (PereiraLópez et al. 2020, p. 480). They make the interesting observation that the CILQ, FLQ and AFLQ are all nested within the 2DLQ: \({\text{FLQ}}_{ij} = 2{\text{DLQ}}_{ij}\) if α = (ln λ \({\text{SLQ}}_{i}\)/ln \({\text{SLQ}}_{i}\)) and β = 1, \({\text{FLQ}}_{ij} = {\text{AFLQ}}_{ij}\) if \({\text{SLQ}}_{j} \le 1\), and \({\text{FLQ}}_{ij} = {\text{CILQ}}_{ij}\) if δ = 0.
Equation (16) has the same data requirements as the AFLQ but the authors claim that it makes more efficient use of such data and consequently yields more accurate results. To test this claim, they use the Eurostat IO database for 2005 to extract symmetric 59 × 59 domestic coefficient matrices for:

1.
Austria, Belgium, France, Germany, Italy and Spain (the ‘observed’ matrices);

2.
the European Area 17 (EA17) (the parent table).
Two estimated coefficient matrices are derived for each country by applying the 2DLQ and AFLQ formulae to the parent table. These estimates are then compared by using the weighted absolute percentage error (WAPE) and two other statistics, U and U*, which consider the number of cells (n^{2}) and the number of nonempty cells (n^{2} − \(z\)), respectively. WAPE is defined as follows:^{Footnote 8}
$$ {\text{WAPE}} = {{100 \sum \limits_{i = 1}^{n} \sum \limits_{j = 1}^{n} \left {\hat{a}_{ij}^{r}  a_{ij}^{r} } \right} \mathord{\left/ {\vphantom {{100 \sum \limits_{i = 1}^{n} \sum \limits_{j = 1}^{n} \left {\hat{a}_{ij}^{r}  a_{ij}^{r} } \right} { \sum \limits_{i = 1}^{n} \sum \limits_{j = 1}^{n} a_{ij}^{r} }}} \right. \kern\nulldelimiterspace} { \sum \limits_{i = 1}^{n} \sum \limits_{j = 1}^{n} a_{ij}^{r} }}. $$
(17)
The results for coefficients show that using the 2DLQ method reduces the WAPE for all countries, most noticeably for Austria and Italy (PereiraLópez et al. 2020, table 2). However, there is little change in the outcomes for Belgium, France and Germany. Spain is an intermediate case. On average, the WAPE is lowered by 4.5%. The U and U* measures yield similar results. The authors also assess the performance of the 2DLQ in terms of multipliers. Here the 2DLQ method gives better results than the AFLQ for all countries apart from Belgium. The average improvement is 3.55%.^{Footnote 9}
Of the two studies reviewed thus far, the 2DLQ method seems the most promising, when evaluated in terms of its theoretical foundations, empirical performance and ease of application. Even so, some caveats should be borne in mind. The first concerns the need to determine suitable values of α and β. Here the authors provide some reassurance that the range of suitable values of these parameters is relatively small and that analysts would not go far wrong by choosing an α of 0.1 or 0.15 and a β in the range 0.8 to 1.2 (PereiraLópez et al. 2020, table 5).
The second caveat concerns the fact that countries rather than regions are used in the testing process, which poses some potential problems. While the authors are right to stress the quality and consistency of the Eurostat data they employ, it is not true to say that using countries instead of regions is the only possible way to proceed. For instance, suitable surveybased official regional and national data are available for Finland (Flegg and Tohmo 2013), South Korea (Jahn et al. 2020) and Japan (Fujimoto 2019). It would be instructive to reexamine the 2DLQ method by using one or more of such data sets to perform a sensitivity analysis.
The study by Lahr et al. (2020) represents a radical departure from those discussed hitherto. The authors examine data for 2014 from the World Input–Output Database, pertaining to 23 manufacturing sectors in 28 EU countries. A novel feature of this study is its use of quasibinomial econometric models, in which the regional purchase coefficient (RPC) is regressed on various variables.^{Footnote 10} Each country is treated as a ‘region’ and the aggregate of all 28 countries as the ‘nation’. Accordingly, \({\text{RPC}}_{i}^{c}\) is the proportion of national requirements of industry i supplied by firms located within country c.
The following regressors were found to be statistically significant (p = 0.001, twotailed test):
In addition, six industryspecific binary variables were included. R^{2} = 0.660. By contrast, a model with the SLQ alone gave R^{2} = 0.142. This worse fit is unsurprising, since the SLQ rules out the key factor of crosshauling and cannot allow for the peculiarities of specific industries.
Lahr et al. also carry out a test of what is described as the FLQ method. This is done by regressing \({\text{RPC}}_{i}^{c}\) on \({\text{SLQ}}_{i}^{c}\) and the ‘employment share’. R^{2} = 0.195. However, this test is inconsistent in several respects with the FLQ approach. Most importantly, the FLQ has a crossindustry foundation, which cannot be captured in a rowsonly estimation. Consequently, no account is taken of the likelihood that purchasing industries would differ in their use of particular inputs. This aspect is captured in \({\text{CILQ}}_{ij}^{c}\) and hence in \({\text{FLQ}}_{ij}^{c}\) but not in \({\text{SLQ}}_{i}^{c}\). We should note too that the FLQ formula is multiplicative, whereas the regression model is additive. It is also unclear how the regressor ‘employment share’ was measured.^{Footnote 11} Finally, whereas output was used to calculate \({\text{SLQ}}_{i}^{c}\), employment was used to measure regional size. That would affect the results to the extent that productivity differed across EU countries.
The authors compare the performance of their econometric approach with that of the SLQ, SDR and CHARM. The procedures are judged in terms of their ability to estimate RPCs and to replicate each country’s coefficient matrix, Leontief inverse and output multipliers. As expected, the SLQ and SDR yield similar results and, on average, both methods greatly overstate input coefficients (Lahr et al. 2020, table 5). By comparison, the regressionbased approach necessarily yields a mean error of zero. The authors find that RCHARM performs ‘somewhat better’ than the SLQ and SDR, yet it still systematically overstates RPCs, with a mean error of 0.240 (Lahr et al. 2020, p. 1594). By contrast, MCHARM yields negative RPCs for 337 of 1568 national industries. Therefore, both variants of CHARM have serious demerits as a means of estimating RPCs.
A crucial consideration when selecting a nonsurvey method is its ability to yield unbiased estimates of input coefficients. A regressionbased approach can be relied upon to perform very well in that respect but it is also possible to obtain unbiased estimates via the FLQ approach, so long as an appropriate value of the parameter δ is used.
A drawback of the RPC approach visàvis the FLQ is its more demanding data requirements.^{Footnote 12} In the model discussed above, for instance, it would be challenging to find data for some of the regressors, whereas the FLQ only requires figures for output (or employment) in each regional and national industry.
On the other hand, the FLQ has often been criticized on the basis that the results obtained from one country or region are not necessarily transferable elsewhere, since the optimal δ would differ. This problem is addressed in the present paper via a procedure whereby countryspecific and regionspecific values of δ can be derived. Lahr et al. (2020, p. 1591) note that econometric approaches face a similar challenge in terms of transferability of results.
While the model constructed by Lahr et al. sheds some helpful light on the determinants of RPCs in EU countries, it would be interesting to see how well it would perform when constructing a RIOT for, say, Catalonia from a Spanish NIOT.