Xuemei Jiang
email: jiangxuem@gmail.com

Erik Dietzenbacher
email: h.w.a.dietzenbacher@rug.nl

Bart Los
email: b.los@rug.nl

D57 -Input-Output Analysis < D5 -General Equilibrium and Disequilibrium < D -Microeconomics, R11 -Regional Economic Activity

Keywords: Non-survey methods, cross-regional methods, regional input-output tables, China. 基于多地区集成的地区投入产出表非调查编制方法研究 摘要:基于已有统计数据进行非调查的地区投入产出表的编制一直是投入产出学界 的关注焦点

Many regional input-output tables are estimated by means of non-survey methods. Often, information on the margins of the projected table is complemented by full information on intermediate inputs from tables for other regions. This paper compares the performance of four of such 'cross-regional' methods. Two of these were proposed in the literature before, whereas the other two are based on recent advances in regression analysis. The methods are not only tested against each other, but also against traditional methods that do not employ cross-regional information. To this end, 27 regional input-output tables for China in 1997 and 2002 are used.

Introduction

Fueled by the policy relevance of regional input-output analysis, a vast literature on the construction of regional input-output tables has emerged. Especially hybrid methods, which combine non-survey approaches with superior survey-based data, have become popular. 1 This does not mean, however, that non-survey methods are not being employed anymore. On the contrary, non-survey techniques still receive considerable attention, if only since they are at the heart of the first step of hybrid methods (see e.g. [START_REF] Lahr | A review of the literature supporting the hybrid approach to constructing regional input-output models[END_REF][START_REF] Lahr | A strategy for producing hybrid regional input-output tables[END_REF][START_REF] Okamoto | Non-survey methods for estimating regional and interregional input-output multipliers[END_REF][START_REF] Bonfiglio | Assessing the behaviour of non-survey methods for constructing regional input-output tables through a Monte Carlo simulation[END_REF].

A number of non-survey techniques to estimate an 'object table' (or, a table for the 'object year') have been developed over the past decades. These techniques, like all methods introduced an analyzed in this study, have in common that row and column totals (like sectoral gross output) are known, but that the block of intermediate inputs has to be estimated.

Updating the latest available survey-based input-output table by iteratively rescaling rows and columns to known margin totals of the object table, i.e. the so-called RAS technique, is still a very popular method. In terms of estimation performance, it is hard to beat if no supplementary information is available [START_REF] Oosterhaven | Theory and practice of updating regional versus interregional interindustry tables[END_REF][START_REF] Polenske | Current uses of the RAS technique: a critical review[END_REF][START_REF] Jackson | Alternative input-output matrix updating formulations[END_REF]. Alternatively, regionalization using location quotients is an often used method if a survey-based national table for the object year is available (see, e.g. [START_REF] Flegg | On the appropriate use of location quotients in generating regional input-output tables[END_REF]. In case survey-based tables for other regions are available for the object year, substituting input coefficients from a table for the region that is similar according to some yardstick is also widely used (see, for example, [START_REF] Rueda-Cantuche | A symmetric input-output table for EU27: Latest progress[END_REF], who use information for Belgium to construct import tables for Luxembourg). These methods have in common that estimated coefficients are based on information contained in a single survey-based table. 2 We feel that much less experience has been gained with regional IO table construction based on information contained in several other regional tables, although some methods have been proposed (see, e.g., [START_REF] Jensen | The study of regional economic structure using input-output tables[END_REF][START_REF] Midmore | Input-output in agriculture: a review[END_REF].

In this study, we aim at providing information to practitioners about how to take full advantage of the information on intermediate inputs included in a cross-section of other regional tables in estimating a regional object table. Next to methods based on classical linear regression analysis as applied by [START_REF] Jensen | The study of regional economic structure using input-output tables[END_REF][START_REF] Midmore | Input-output in agriculture: a review[END_REF], we also study methods grounded in more recent contributions to regression analysis, such as robust regression [START_REF] Rousseeuw | Unmasking multivariate outliers and leverage points[END_REF]) and threshold regression [START_REF] Hansen | Sample splitting and threshold estimation[END_REF]. We apply these advanced methods after having shown that data contained in regional input-output tables can have distributional characteristics that render classical regression methods less appropriate. The methods we analyze are empirically compared on the basis of a collection of survey-based input-output tables for Chinese provinces in 1997 and 2002, covering 27 regions and 31 industries. 3,4 Our choice for these Chinese tables is suggested by two considerations (see Appendix 1 for a brief description of the Chinese provincial tables and their compilation). First, the Chinese set of regional IO tables is unique in the sense that it is the largest available set of harmonized tables expressed in one single currency. Second, the well-known characteristic of large geographical disparities in China adds to the attraction of our analysis; the vast majority of regions are clearly not representative for the nation and heterogeneity abounds.

The plan of the paper is as follows. Section 2 briefly reviews 'traditional' approaches of constructing non-survey regional technical tables, which do not rely on the identification of cross-regional patterns. Section 3 proposes the four cross-regional methods that will be employed. Section 4 presents a comparison of the estimation results obtained using the cross-regional approach to those generated by the traditional methods.

Section 5 systematically tests the robustness of our comparison results if the available cross-regional sample would be smaller and contain much fewer than 26 tables. These experiments provide guidelines on which method to use in a variety of situations regarding data availability. In section 6 we summarize our findings and conclude.

Non-Survey Methods Based on the Coefficients of a Single Input-Output Table

Before starting our review of methods, we should first delve a little bit deeper into the nature of the Chinese regional input-output tables at hand. It should be emphasized that Chinese regional tables only provide information on intermediate deliveries including This makes that some parts of the literature on the construction of regional IO tables, which focus on estimation of intra-regional inputs only, is not relevant for the situation at hand. Since we focus on what [START_REF] Boomsma | A double-entry method for the construction of bi-regional input-output tables[END_REF] coined "technical tables", we do not have to deal with the estimation of location quotients (alternatively called regional purchase coefficients). 5 Location coefficients [START_REF] Flegg | On the appropriate use of location quotients in generating regional input-output tables[END_REF][START_REF] Flegg | Regional size, regional specialization and the FLQ formula[END_REF][START_REF] Tohmo | New developments in the use of location quotients to estimate regional input-output coefficients and multipliers[END_REF][START_REF] Riddington | Comparison of gravity model, survey and location quotient-based local area tables and multipliers[END_REF] indicate what share of a regional industry's inputs are sourced domestically. Sizes of regions and transport costs of specific inputs are just two of the main variables that are often supposed to play an important role in the determination of location coefficients. We can abstain from these issues.

Intertemporal Updating

The 'RAS' technique developed by [START_REF] Stone | A Computable Model of Economic Growth[END_REF] has been acknowledged as one of the most widely-used ways to update tables, based on the input-output structure of an older survey-based table and information on the margins (such as total intermediate input use and total intermediate inputs supplied by industry) for the object table. Many variations of the original RAS updating techniques exist, however (see e.g. [START_REF] Morrison | Nonsurvey input-output techniques at the small area level: an evaluation[END_REF][START_REF] Sawyer | Experiments in regionalization of a national inputoutput table[END_REF][START_REF] Polenske | Current uses of the RAS technique: a critical review[END_REF][START_REF] Jalili | Comparison of two methods of identifying input-output coefficients for exogenous estimation[END_REF][START_REF] Jackson | Alternative input-output matrix updating formulations[END_REF]. RAS can be seen as a method that tries to reconcile the old intermedite input structure as well as possible with the new column and row totals. Despite regular complaints about the poor performance of RAS, reviews of empirical results such as [START_REF] Polenske | Current uses of the RAS technique: a critical review[END_REF] and [START_REF] Jackson | Alternative input-output matrix updating formulations[END_REF] tend to conclude that RAS results are seldom outperformed by alternatives using the same type of information.

In the context of the present analysis, information on total interindustry sales and total interindystry purchases taken from a 2002 table and the input coefficients taken from the 1997 table for the same region allows us to apply the RAS method to update all 27 regional tables to 2002. Next, the quality of these estimates by updating can be assesed

Regionalization of National Tables

Updating techniques, however, cannot be used if IO tables have not been constructed for the object region before. For regional analyses (as opposed to country-level studies), the literature recognizes many alternative approaches to produce non-survey IO tables, but most of these focus on the domestic sourcing issue that is not relevant to us, as we explained above.

As far as technical tables (the cells of which contain both domestically produced and imported inputs) are concerned, national tables are most often regionalized by RAS methods [START_REF] Boomsma | A double-entry method for the construction of bi-regional input-output tables[END_REF]. The national input coefficients are taken as a starting point and information on the row and column sums of the regional intermediate deliveries matrix is taken as constraints. Iterated rescaling of rows and columns then generates a table with estimated technical coefficients for the object region.

Exchanging coefficients

Instead of using a national table to reflect the economic characteristics of a particular region r, one might use information from an existing table for another region, r'.

Especially if r and r' are thought to be economically and technologically similar, the estimation error is likely to be small [START_REF] Miller | Input-Output Analysis: Foundations and Extensions[END_REF]. [START_REF] Hewings | Evaluating the possibilities for exchanging regional input-output coefficients[END_REF] gave an example of coefficient exchange at the regional level, estimating a table for the state of Kansas 1965 borrowing input coefficients from the table of Washington State for 1963.

Finally, RAS was used to balance the Kansas table obtained in this way.

A problem arises if several regional tables are available to choose from: which of the regions is defined to be most similar to r (the object region), in particular in a situation in which the input coefficients of the object table are unknown? This issue has hardly been discussed in the literature. In this paper, we propose to use the vector of input coefficients for each sector in 1997 to represent the input technology of the corresponding region and sector. 6 Then the similarity index SI j rk for 1997 is calculated for a pair of region r and k for each and every sector j:

F o r P e e r R e v i e w O n l y 2 / 1 1 1 2 2 1 ) ( ) ( ) ( ) (       ⋅ ⋅ = ∑ ∑ ∑ = = = n i n i ij ij n i ij ij rk j k a r a k a r a SI (1)
in which a ij denotes the input coefficients for a region. The expression in the right hand side is the cosine between the two input coefficients vectors of r and k. [START_REF] Jaffe | Technological opportunity and spillovers of R&D: evidence from firms' patents, profits, and market value[END_REF] proposed such a measure (which is bounded by zero and one given the nonnegativity of input coefficients) based on shares of technology classes in the patent portfolios of firms. 7

For each industry j, we consider the region k which has the highest SI j rk with the object region r as the most similar region. Consequently, its coefficients for 2002 have been inserted in the corresponding column of the object table. We repeated this experiment for all sectors, after which application of RAS ensured a balanced estimated table for 2002.

Non-Survey Methods Using Cross-Regional Information

As opposed to the methods described in the previous section, methods using information from a multitude of regional tables have barely been evaluated. The availability of comparable regional input-output tables for as many as 27 Chinese regions allows for a systematic analysis along these lines. We will compare estimated tables against the survey-based tables, as well as to the more traditional estimates based on information from a single region. In this section, we present two commonly used cross-regional approaches. These are based on regression analysis. We find, however, that assumptions essential to classical linear regression are violated in our dataset. Hence, we also propose two novel methods that deal with these problems.

The idea to use information from other available regional tables in constructing regional IO tables is not entirely new. [START_REF] Imansyah | An efficient method for constructing regional input-output table: a horizontal approach in Indonesia[END_REF], for example, proposed the "averaging" method, which computes the average input coefficients of the other regions, multiplies these with the the industry's gross output level and balances the resulting table using RAS method, to generate the objective table.

Another well-known way to produce a matrix of deliveries for the object region from a cross-regional perspective starts from the notion of the Fundamental Economic Structure (FES), as proposed by [START_REF] Jensen | The study of regional economic structure using input-output tables[END_REF][START_REF] Midmore | Input-output in agriculture: a review[END_REF]. By regressing the intermediate deliveries on an independent variable that represents the regional economic 'size', the concept of FES provides a cross-regional insight into the estimation of intermediate deliveries, as the following equation shows:

) ( ) ( ) ( r r X r X ij ij ij ij ε β α + + =
(2)

X ij (r) represents the intermediate deliveries for the rth region and X(r) is an indicator of the economic size of the rth region. α ij and β ij are cell-specific parameters to be estimated, ε ij is considered to be random noise. Based on a series of IO tables for ten regions of Queensland, [START_REF] Jensen | The study of regional economic structure using input-output tables[END_REF] found highly significant estimates for the parameters for many cells X ij , though not for all. Jensen et al. consider the cells for which Equation 2 has a high explanatory power to be part of the Fundamental Economic Structure and indicated that such a FES could be used in a compilation of regional tables. [START_REF] Van Der Westhuizen | Towards developing a hybrid method for input-output table compilation and identifying a fundamental economic structure[END_REF] and [START_REF] Thakur | Structure and structural changes in India: a fundamental economic structure approach[END_REF] actually tried to use the FES technique to compile regional IO tables: They also estimated regression equations that related the intermediate delivery X ij (r) to alternative region-specific variables, such as total population, total value-added, gross output for sector i and gross output for sector j in the region. Next, they estimated cells for the object table based on the parameters and corresponding independent variables, and applied RAS for balancing.

We can show that [START_REF] Imansyah | An efficient method for constructing regional input-output table: a horizontal approach in Indonesia[END_REF] averaging method represents a special case of the FES approach, if we limit Jensen et al.'s (1988) FES method to regressions with regional sectoral gross output levels X j (r) as the independent variable. Denoting the input coefficients by a ij , Equation 2 can then be written as

) ( ) ( ) ( ) ( ) ( r u r X r X r X r a ij ij j ij j ij ij + + = = β α
If α ij is set to zero, the averaging method produces identical estimates as this FES equation. The actual differences between the estimates depend on the extent to which returns to scale are non-constant. If regions with large sectors can use their inputs more 10 efficiently, α ij will be significantly positive. In our analyses below, we will start from an equivalent regression equation that has the advantage of being linear in X j (r):

) ( ) ( ) ( r e r X r a ij j ij ij ij + + = λ κ (3)
All four cross-regional methods discussed below have Equation 3 as their point of departure and can therefore be seen as originating from the FES approach. 8

Averaging coefficients

Our first cross-regional approach amounts to estimating Equation 3 for all a ij s with the restriction that λ ij is equal to zero. The sample consists of all Chinese regional inputoutput tables for 2002 in our dataset, with the exception of the object table. Next, the estimated input coefficients are multiplied by the values of X j of the object region to arrive at estimated intermediate input flows for 2002. These are reconciled with the available margin totals for the object table by means of simple RAS. The method is exactly identical to what [START_REF] Imansyah | An efficient method for constructing regional input-output table: a horizontal approach in Indonesia[END_REF] proposed.

Ordinary Least Squares regression

Estimating Equation 3 by means of Ordinary Least Squares (OLS) without any restrictions on the two parameters comes close to the procedures advocated by [START_REF] Jensen | The study of regional economic structure using input-output tables[END_REF]. The samples are identical to the samples used for the averaging method. After having obtained the estimates for the a ij s, the remaining steps in the procedure are exactly the same as those for the averaging method.

Figure 1 graphically depicts two situations. In the left panel (which relates to the inputs of "textiles" per unit of gross output of the "wearing apparel" industry), OLS regression yields an almost flat line that nearly coincides with the line produced by the averaging method. Apparently, the use of textiles in the wearing apparel manufacturing industry is not subject to economies of scale. The right panel of Figure 1 shows an example of an input coefficient for which OLS regression and averaging yield completely different results. In this case, which refers to the inputs of "electronic and telecommunications equipment" in the "wearing apparel" industry, the OLS regression line is clearly upward sloping. Most probably, the input of such high-tech equipment (instead of labor or more traditional equipment) is only commercially attractive when high volumes are produced.

INSERT FIGURE 1 ABOUT HERE

As is well-known, linear regression by means of the method of least squares leads to estimates with desirable properties if a number of assumptions are met. One of these assumptions is 'homogeneity of the data-generating process'. This assumption can be violated in several ways. An example is the occurrence of outliers, which are often generated by differences in parts of the data-generating process related to variables that are omitted from the regression equation. Another type of violation emerges if the true value of parameters included in the regression equation varies with ranges of values of the explanatory variables. Taking Equation 3as an example, one might think that increasing returns to scale do not play a role for relatively small values of X j , but might set in for larger values (or the other way round). If so, the relationship between the variables cannot be represented by a single set of parameters and one should allow for parameter heterogeneity. So far, the limited body of literature proposing cross-regional methods has not addressed these potential problems.

The right panel of Figure 1 above suggests that violations of the homogeneity assumption may play a role indeed. The very high input coefficient of almost 0.006 in the upper right corner of the diagram, for example, is either an outlier or points towards a relation between a ij and X j that is different between low and high values of X j . Below, we will propose two advanced regression approaches that address these problems. The robust regression approach explicitly deal with the potentially disturbing effects of outliers, whereas the threshold regression approach allows for parameter heterogeneity.

Robust regression

Outliers can have substantial impacts on OLS estimates of parameters in a regression equation. If such estimates are not accurate, the estimates of the object tables will be inaccurate as well. The potential effects of outliers can be illustrated by means of Figure 12 1 (right panel). The very high regional input coefficient of just below 0.006 associated with a sectoral total input of approximately 160 millions RMB is a clear outlier. Since this outlier is located at one of the extremes in the horizontal dimension, it tilts the OLS regression line anticlockwise. This single observation (which is called a 'bad leverage point' in the terminology of [START_REF] Rousseeuw | Unmasking multivariate outliers and leverage points[END_REF]) has much more impact on the estimated coefficients than the observations that are closer to the centre of the cloud of observations. The second highest regional input coefficient of about 0.004 is located much closer to the center of this cloud. Hence, this outlier does not have much of an impact on the estimated slope. Its effect largely remains limited to the estimated intercept.

In order to reduce the effects of outliers and bad leverage points, several robust regression techniques have been developed. In our robust regression approach to estimating Equation 3, we use the procedure that underlies the robustfit algorithm in the Matlab programming language. This algorithm uses an iteratively reweighted least squares sequence. 9 In this algorithm, observations that yield a large residual in the first iteration, get a small weight in the weighted least square estimation in the next iteration.

Hence, the impact of outliers is severely reduced. In our application weights are determined according to a bisquare weighting function (see [START_REF] Beaton | The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data[END_REF].

After having obtained estimates for the parameters of Equation 3 in this way, an a ij for the object region is predicted based on the total sectoral inputs X j (r). If the sample for a specific a ij does not contain outliers, the weights in the iteratively reweighted least squares do not deviate much from each other and the estimates using robust regression will not be very different from those obtained using OLS. After all the input coefficients for the object table have been estimated using robust regression, the RAS algorithm is used to align the corresponding table of intermediate input flows to the marginal totals.

Threshold regression

If the relation between the dependent variable and the explanatory is characterized by strong parameter heterogeneity, estimating parameters as if they were identical for the entire sample is likely to lead to undesirable results. One of the simplest approaches to [START_REF] Hansen | Sample splitting and threshold estimation[END_REF]. 10 In the context of this paper, the point of departure is the following set of equations

) ( ) ( ) ( ) ( ) ( ) ( 2 2 1 1 r u r X r a r u r X r a ij j ij ij ij ij j ij ij ij + + = + + = λ κ λ κ ij j ij j r X r X γ γ > ∀ ≤ ∀ ) ( ) ( (4)
where

n j i m r ,..., 1 , ; ,..., 1 = =
. Equations 4 can be seen as a generalization of Equation 3:

For regions with large total inputs of sector j (X j ), the linear relationship between the intermediate input coefficient a ij and X j is characterized by different values of κ and λ than for regions with small total inputs. γ is the threshold between the two 'regimes'. It is endogenously estimated, by taking the sample value for which the reduction in the sum of squared residuals (SSR) attained by allowing for two sets of parameters is largest [START_REF] Hansen | Sample splitting and threshold estimation[END_REF]. 11 A likelihood ratio test, the outcome of which depends on the degree to which SSR is reduced by allowing for two sets of parameters, leads to the decision about whether the split is significant or not. 12,13 If it is significant, the estimation of a coefficient of the object table depends on the size group to which the corresponding total inputs belong. If not, Equation 3 is estimated for all the observations and the estimate for the input coefficient in the object table is based on the estimates for the coefficients in this equation. Figure 2 describes the entire procedure underlying the threshold approach.

INSERT FIGURE 2 ABOUT HERE

Figure 3 gives two empirical examples of comparisons between the relationships between input coefficients and total sectoral inputs as found by applying three of our crossregional estimation methods: OLS, robust regression and threshold regression. The two cases are identical to those depicted in Figure 1, in which the results for the averaging coefficients method and OLS regression were compared. In the left panel, the results for each of the techniques are very much alike. The absence of outliers leads to results for OLS and robust regression that are virtually identical. The threshold regression approach yields two line segments that are slightly upward sloping, but do not show a clear threshold (it did not turn out to be significant at 10%). Following the procedure depicted in Figure 2, we would use OLS in this case.

INSERT FIGURE 3 ABOUT HERE

A completely different situation emerges from the right panel. The importance of bad leverage points such as the one in the far northeast of the diagram is reduced in the robust regression approach, which leads to a much flatter regression line than in the case of regular OLS regression. This implies that robust regression points towards much less pronounced decreasing returns to scale than OLS, since the required inputs per unit of gross output appear to depend much less on total output levels. The results for the threshold regression approach are also very different from the OLS results. For sectoral total inputs below the estimated threshold of 0.5x10 8 RMB, the regression results are much flatter than for OLS. The estimated intercepts are very different however. For the subsample of four observations above the threshold size, the intercept is considerably higher than for the remaining observations associated with regions with small wearing apparel-manufacturing sectors.

The fact that the results across the three cross-regional methods are very different from each other does not offer proof that adopting more advanced methods is worthwhile.

If samples like the one depicted in the right panel of Figure 3 would be very rare in regional input-output tables, not much could be won. Unfortunately, the robust regression analysis procedure does not make a dichotomous distinction between outliers and regular observations. As we explained above, the algorithm recomputes weights for all observations. For the threshold regression approach, we can provide more evidence.

When considering observations for all 27 regions, we found splits for 112 out of the 961 cells, which amounts to a share of 11.7%. 14 This share of cells seems sufficiently large to warrant further consideration.

The number of cells for which the estimated slopes (λ ij in Equation 3) are significant is small. For about 3% of the cells we find an R 2 of at least 0. Table 2 compares the frequencies of estimated slopes for the subset of cells for which threshold regression yielded a split into subsamples corresponding to low and high values of sectoral output X j (see Equation 4) significant at a level of 10%.

INSERT TABLE 2 ABOUT HERE

The results show that positive slopes larger than 0.005 are found slightly more often for the subsamples associated with large sectoral output levels. The differences are not very marked, though.

In the next section, we will compare the estimating performance of the cross-regional methods introduced above not only to each other, but also to the more traditional methods based on single tables as discussed in Section 2.

Comparison of Estimation Results

In this section, we compare the deviations between the survey-based ("true") regional IO tables for 2002 and the estimated tables obtained by applying the procedures outlined in the previous section. Thoughout the empirical analysis, we will employ the WAPE (Weighted Absolute Percentage Error) as our measure of deviation: The WAPE has been used in a large number of studies, since the weighted average of deviations is taken in such a way that large cells receive a larger weight than small cells (see, for example, Oosterhaven et al., 2007). We decided to compare the deviations for individual cells of the Leontief inverse (instead of, for exampe, the values of intermediate input deliveries or input coefficients), because the cells of the Leontief inverse constitute the building blocks of multipliers used in traditional impact analyses.

INSERT TABLE 3 ABOUT HERE

Table 3 presents the WAPEs for each region and method. The last row ("count")

indicates the number of regions for which the methods of the associated columns have the highest accuracy. In a similar vein, the row "average" presents the unweighted averages of WAPEs over regions for the seven methods considered.

The WAPEs documented in Table 3 appear to be high, but it is well-known that applications of unmodified RAS generally lead to inaccurately estimated object tables (see, e.g. [START_REF] Lynch | An assessment of the RAS method for updating input-output tables[END_REF][START_REF] Polenske | Current uses of the RAS technique: a critical review[END_REF]. For the dataset we study here, most WAPEs would decline sharply to 0.1-0.2 when the 5% most important cells would be replaced by the true, survey-based values (see [START_REF] Jiang | Targeting the collection of superior data for the estimation of regional input-output tables[END_REF]. The actual construction of regional input-output tables is often done by means of such 'hybrid' methods.

A first and very important finding is that cross-regional methods yield far better results than single-table methods. The estimations made with the cross-regional models produce overall average WAPEs between 0.291 and 0.295, while the corresponding range is 0.329-0.385 for traditional methods. At the level of individual regions, cross-regional methods also show a clear superiority over single-table methods, since only for Jilin the minimum WAPE is found for a single-table method. We also observe that for most regions the WAPEs for cross-regional methods are very close to each other. In 21 out of 27 regions the worst cross-regional method still scores better than the best single-table method.

Second, it appears that regionalization based on the national table generates the best estimations among the class of single-table methods, followed by updating, while exchanging coefficients with the most similar region performs worst. This is a rather surprising result since updating a recent table is one of the most popular techniques to compile regional input-output tables. It is also striking that the exchanging coefficients procedure is outperformed by regionalization, in spite of the fact that it uses information from all other regions in selecting the regional production structure that was most similar in 1997. A reasonable explanation for these results might be that input coefficients for regions undergoing rapid development are far less stable than ones for developed countries. 15 [START_REF] Dietzenbacher | Coefficient stability and predictability in inputoutput models: a comparative analysis for the Netherlands[END_REF], for example, examined the stability of input coefficients based on a time series of annual input-output tables for the Netherlands, covering the period 1948-1984. They found that 80% of the cells had coefficients of variation below 0.3. For a set of Chinese survey-based national tables covering the period 1987-2002, we find that not a single input coefficients features a coefficient of variation smaller than 0.5, and the proportion of input coefficients with a coefficient of variation below 0.8 is a mere 30%.

Third, turning our attention to the cross-regional methods, we can conclude that the four methods perform very close to each other on average, but that there are some marked differences at the level of individual regions. The robust regression method performs best for 12 regions, while the averaging coefficients method appears superior for 8 regions.

Although the WAPEs for ordinary least squares regression are similar to the WAPEs for robust regression if averaged over provinces (see last row of 1 in this empirical application for China. Threshold regression emerges as an approach to this issue that should not be preferred. It yields substantially more accurate estimations for only two Western regions (Guizhou and, particularly, Yunnan).

Instead, using the averaging approach (which imposes constant returns to scale) and robust regression turn out to be promising approaches.

The overall average WAPE of the averaging method is slightly lower than the WAPE for the robust regression approach. Robust regression, however, is superior to averaging in the majority of cases (15 out of 27). This paradox is mainly due to two regions with "extreme" results: Shandong and, to a somewhat lesser extent, Zhejiang. For these regions, the robust approach yields far worse accuracies than averaging. The results in Table 4 are based on robust regressions of Equation 3 with all 27 provinces included in the sample. The regression was run 961 times, i.e. for each of the (i,j)-pairs. The colum "numbers of outliers" shows how often an observation for the corresponding region was found to be an outlier. 16 Shandong and Zhejiang are special indeed, in the sense that the numbers of cells considered as outliers is very high. About 12% of the 961 input coefficients in each of these two regions are located very far from the main cloud of observed input coefficients. In the robust regression approach, such observations get a very low weight, as a consequence of which the regression line is relatively often very far away from the observation. Hence, it is not surprising that predictions for regions with large numbers of outliers are relatively bad.

INSERT TABLE 4 ABOUT HERE

Robustness test of comparison results

The most important conclusion so far is that cross-regional methods systematically generate better estimations than more traditional methods using information from just one table. It should of course be borne in mind that we obtain these results for a set of If we want to judge our results in a more general context of practitioners in need of regional tables without the funds or time to construct survey-based material, it is a rather strong assumption that as many as 26 tables are available as inputs for cross-regional inputs. In this section, we will investigate whether the superiority of cross-regional methods as reported in the previous section carries over to situations in which less tables can be used.

For reasons of space, we focus on the averaging coefficients and robust regression methods as representatives of the cross-regional methods. Further analyses of Ordinary Least Squares and threshold regression will be omitted, because these were most often outperformed. For similar reasons, we drop the exchanging coefficients method from the set of approaches based on coefficients from a single input-output table . In this class of methods, we scrutinize the performance of intertemporal updating and regionalization of national tables.

Experiments with sets of random samples

The relative performance of the estimation methods under consideration is likely to depend on the regional tables making up the sample. In the previous section, randomness did not play any role, because all 26 tables (27 minus the object table) were automatically included in the sample. Now, we intend to have a closer look at how the estimation methods perform if, for example, the averaging method is based on just 10 observations.

In principle, we could study results for all 26!/(10!•16!)=5,311,735 possible distinct samples, but we decided for a different approach. In an experiment with strong similarities to bootstrapping, we randomly drew 1,000 samples for each region and sample size studied. 17 Next, we computed WAPEs for each sample. We summarize the empirical distribution of WAPEs as obtained in this way by means of the most straightforward statistic: the average WAPEs for the methods (as computed over 1,000

WAPEs). Finally, to facilitate bilateral comparisons of methods, we computed the percentage of random samples for which one method yielded lower WAPEs than for the other.

Comparison between averaging and robust method

First, we study the relative performance of the two cross-regional methods for situations with samples of 10, 15 and 20 observations, respectively. The results are presented in Table 5. The average WAPEs for the averaging and robust regression methods are listed in the first two columns, for each number of observations. The percentages in the rightmost columns denote the percentage of random samples for which averaging yielded a higher accuracy (lower WAPE) than robust regression. 18 For example, 34% in the first row and third column indicates that for the Anhui region, averaging coefficients outperformed robust regression for only 34% of the random samples.

INSERT TABLE 5 ABOUT HERE

First, we observe that WAPEs increase when less observations are available, irrespective of the method adopted. This implies that as many tables as possible should be used when applying cross-regional methods. We also find, however, that the WAPEs increase remarkably slowly when sample sizes are reduced, which is a reassuring result.

With respect to the comparison between the averaging coefficients and robust regression methods, we find that the advantage of the latter over the first in terms of the number of regions for which it performs better (see Table 3) switches to a disadvantage when the numbers of observations in the sample decline. For the case of 20 observations, the robust regression method performs better in 13 regions according to average WAPE, while these numbers drop to only 10 and 5 in the cases of 15 and 10 observations, respectively (see the bottom line of Table 5). In a similar vein, we find that the advantage of the averaging coefficients method in terms of the unweighted average of average for averaging. Apart from this case, the average WAPE appears to be a statistic that captures the entire empirical distribution well, at least for the purposes of our analysis.

WAPEs as documented in

The result that the performance of the robust regression method worsens with lower numbers of observations can be explained by the fact that robust regression is less capable to identify outliers, if the cloud of "regular observations" is small (see [START_REF] Rousseeuw | Unmasking multivariate outliers and leverage points[END_REF]. Consequently, observations that are outliers for the entire set of 27 provinces as counted in Table 4 are often treated as almost regular observations if sample sizes are small and the empirical differences between robust regression and Ordinary Least Squares regression vanish. In Table 3, we already found that OLS regression performs systematically worse than the averaging method. An important intermediate conclusion we draw is that if only a few regional tables are available, the use of the averaging coefficients method is recommended.

Comparison of averaging against traditional methods

In this section, we compare the performance of the averaging coefficients method (a cross-regional method) to those of the intertemporal updating and regionalization of national tables techniques. If only few regional tables are available, one might expect that the clear advantage of the cross-regional methods (as presented in Table 3) disappears and that the use of single table methods should be favored. Like in the previous subsection, we will first compare the accuracies based on average WAPEs over 1,000 random samples. The results are documented in Table 4.

INSERT TABLE 6 ABOUT HERE

For all regions except Jilin and the cities of Beijing and Shanghai, we find that the crossregional method beats both single-table methods as long as the number of available regional tables is eight or more. Apparently, small numbers of contemporaneous tables already yield sufficient information to compensate for the fact that the tables relate to regions different from the object region. Only if regions have very special structures, such as city-provinces, regionalization of national tables (and to a lesser extent) intertemporal updating methods may prove superior also for larger samples. 22 Again, we also present results for another statistic (the percentage of random samples for which regionalization of national tables leads to higher accuracies than averaging), which provides more insight into the empirical distribution of relative WAPEs. 19 The results are presented in Table 7.

INSERT TABLE 7 ABOUT HERE

Table 7 by and large confirms the results obtained for the average WAPEs. For a sample size as small as seven, we find that in as many as 16 of 27 regions regionalization is more accurate than averaging for less than 10% of the random samples. For smaller samples, the percentage of samples for which regionalization beats averaging coefficients increases. For a sample size of five, for example, we find that this happens in eight regions for more than half of the randomly drawn samples.

Again, Beijing and Shanghai are the main exception to the rule of superiority of cross-regional methods like averaging. That is, regionalization not only performs better for moderately small samples, but also for large samples. The production structures of these metropolitan cities are apparently better reflected in national tables (which incorporate the structures of Shanghai and Beijing) than in regional tables for other regions. We also find that some other coastal and central regions with a highly developed manufacturing sector, such as Fujian, Hebei, Hubei, Jilin, Liaoning and Zhejiang, tend to have lower accuracies of the averaging coefficients method in a relatively large fraction of the set of samples.

Conclusions

This paper presents four cross-regional non-survey methods to estimate regional IO tables (using as much data for other regions as possible) and tests these methods against three more traditional non-survey methods that rely on information contained in a single regional We first argued that [START_REF] Imansyah | An efficient method for constructing regional input-output table: a horizontal approach in Indonesia[END_REF] averaging coefficients method can be seen as a special case of the Ordinary Least Squares regression approach that [START_REF] Jensen | The study of regional economic structure using input-output tables[END_REF] advocated (inspired by the notion of the Fundamental Economic Structure). Next, we introduced two alternative regression-based methods, which deal with outliers and bad leverage points. In the present context, these are regions with a production structure that is very different from the production structures in many other regions (a situation that is frequently encountered in China). The robust regression method assumes that there is a single 'law' governing the size of input coefficients and gives low weight to observations that do not appear to obey this relationship. Threshold regression, however, supposes that two 'laws' could prevail, one of which relates to small sectors and the other to regions in which the sector is large. If evidence for two laws is found, the sample is split into two and subsample-specific estimates are obtained.

We find that cross-regional methods have systematically better performance than the traditional methods based on a single table. This result carries over to situations in which fewer regional tables are available. In most cases the availability of seven or eight regional tables is sufficient to render averaging coefficients more accurate than regionalization of national tables and intertemporal updating. Among the group of crossregional methods, averaging coefficients and robust regression generally turn out to be slightly more accurate than OLS and threshold regression. The accuracy of the robust regression technique as compared to averaging is relatively weak if the number of available regional tables is rather small. In such cases, simple averaging of coefficients appears to be the preferred method.

The results obtained in this paper should be considered carefully, because they cannot be generalized to all situations practitioners might face. The Chinese data we used are attractive for the purpose of this study because sets of 27 harmonized, survey-based regional input-output tables are very rare. This dataset allowed us to compare the estimation performance of a number of techniques as if samples of different sizes were available. One should take into account, however, that this dataset is also rather specific in at least two respects. First, the Chinese economy is both very heterogeneous and dynamic. Some regions are very backward, while other regions (especially those in the coastal zone) have been developing very rapidly. Regionalization of national tables might perform much better for regions that are part of a country without the differences in production structures associated with the Chinese regional inequality. A similar argument goes for the bad estimation performance of intertemporal updating in our study. If inputoutput tables are estimated for regions that do not develop as quickly as many of the coastal and central regions in China, production structures as reflected in input coefficients are likely to be much more stable over time. This would enhance the quality of estimates obtained by intertemporal updating significantly.

Secondly, given the nature of our Chinese data, the study focuses on the estimation of technical coefficients, which are defined as intermediate inputs (both domestically produced and imported) divided by gross output. Often, however, practitioners are interested in estimating input coefficients, defined as domestically produced intermediate inputs divided by gross output levels. If cross-regional methods would be used to estimate input coefficients, some additional steps seem to be necessary, including the estimation of location quotients to correct for differences in economic size of regions: large regions will purchase relatively much from domestic sources, while small regions will import relatively much. This will be reflected in different sets of input coefficients, even if the production technologies would be identical. An account of the relative qualities of the (adapted) cross-regional methods discussed in this study and more traditional methods based on information contained in a single Sector Inputs Input coefficients(High variation) Observations Averaging FES regression * Both panels contain observations for all 27 provinces for which input-output tables are available. In the empirical analyses, Equation 3 is estimated on the basis of only 26 observations, since the object table is assumed to be unknown. The parameter estimates are used to estimate the input coefficients of the object table. Since each of the 27 tables can arbitrarily be selected as the object table, this figure depicts observations for the entire population of tables. * Percentages indicate the share of random samples for which the intertemporal updating method yielded a higher accuracy than averaging coefficients. Shaded cells represent shares less than 10%.

1 See, e.g., [START_REF] Jensen | Regional Economic Planning: Generation of Regional Input-Output Analysis[END_REF], [START_REF] Greenstreet | A conceptual framework for constructing of hybrid regional inputoutput models[END_REF], [START_REF] West | Regional trade estimation: a hybrid approach[END_REF], [START_REF] Midmore | Input-output in agriculture: a review[END_REF], [START_REF] Jackson | Regionalizing national commodity-by-industry accounts[END_REF], [START_REF] Madsen | Make and use approach to regional and interregional accounts and model[END_REF]Jensen-Butler (1999) and[START_REF] Lahr | A strategy for producing hybrid regional input-output tables[END_REF].

2

Purists would be right in arguing that exchanging or substituting coefficients first uses information from multiple regional tables to identify the most similar region. Next, however, it disregards all information contained in tables for regions that might have a high degree of similarity to the object region, but are not the most similar. The mainland of China is administratively divided into 31 regions, including 22 provinces, 5 autonomous regions and 4 centrally administrative municipalities. We do not have data for Hainan province and the autonomous regions Tibet, Qinghai and Xinjiang.

4

In order to make the tables comparable for 1997 and 2002, the industries in our data set were aggregated into 31 industries. See Table A1 in Appendix 1 for the classification.

5

See [START_REF] Round | Nonsurvey techniques: a critical review of the theory and the evidence[END_REF] and [START_REF] Lahr | A review of the literature supporting the hybrid approach to constructing regional input-output models[END_REF]2001) for thorough overviews of these techniques.

6

This approach is inspired by [START_REF] Leontief | Input-output data base for analysis of technological change[END_REF], who viewed columns of input coefficients as lists of ingredients for sectoral 'cooking recipes'.

7

See [START_REF] Oksanen | An alternative factor-analytic approach to aggregation of input-output tables[END_REF] and [START_REF] Los | The empirical performance of a new inter-industry technology spillover measure[END_REF] for applications of the cosine measure as a similarity measure of two vectors of input coefficients.

8

We conducted many estimation experiments using other explanatory variables than regional sectoral gross output and nonlinear forms (for example using quadratic forms or allowing for parameter heterogeneity between "poor" regions and "rich" regions). Equation 3, however, consistently led to the estimated tables that most closely resembled the true object tables. Results obtained with other explanatory variables and other functional forms are not included in this paper, but are available upon request. 9

An early overview of alternative robust regression techniques was given by [START_REF] Holland | Robust regression using iteratively reweighted least-squares[END_REF]. In principle, more than two sets of parameters might govern the relationship between the input coefficients and the total sectoral inputs. In this study, we focus on a situation with two subsamples only. We have two reasons for this decision. First, the estimation theory for multiple sample splits has not been developed thoroughly, and second, the numbers of observations in ours samples are not very high, as a consequence of which we would lose many degrees of freedom when estimating multiple splits.

12 In this study, we adopt a significance level of 10%.

13

The threshold regression approach advocated by [START_REF] Hansen | Sample splitting and threshold estimation[END_REF] requires the minimum size of both subsamples to be set exogenously ('trimming'). We followed [START_REF] Hansen | Sample splitting and threshold estimation[END_REF][START_REF] Hansen | The new econometrics of structural change: Dating breaks in U.S. labor productivity[END_REF] convention to set this value to 10-15% of the sample size. This implies that the minimum size of each subsample is 3 observations. 14 This result obtained for Chinese regions does not necessarily generalize to other sets of national or regional input-output tables, particularly because the Chinese economy is characterized by a strong variation of regional specialization patterns. Formally, the iteratively reweighted least squares procedure does not yield a clear distinction between outliers and regular observations. However, if the algorithm leads to observations with a very small weight after the final iteration, this is a sign that the corresponding observation is located well above or below the robust regression line. Here, we rather arbitrarily denote observations with a final weight smaller than 0.00005 as outliers.

17

In theory, it is possible that the sets of 1,000 samples contained duplicates, since we drew samples with replacement. Please note that a single sample could not contain a regional table more than once, since we drew regions, in each sample, without replacement.

18

The percentages depend on the draw of the 1,000 samples and are therefore random variables themselves.

19

For the large majority of Chinese regions, regionalization of national tables performs better than intertemporal updating. Hence, we benchmark averaging to regionalization. Readers interested in a comparison of the averaging coefficients method and intertemporal updating are referred to Table A2 in Appendix 2. From a practitioner's point of view, the results in Table A2 might be very relevant, because national tables are sometimes unavailable while an old regional table might exist.

20

The 31-industry classification used in this study (see Table A1) is slightly more aggregated than the input-output tables published for 1997 and 2002, which contain 40 and 42 industries, respectively.

Aggregation was needed to have tables with an identical industry classification.

21

There might be minor adjustments in terms of the sample percentages, made by local regional statistic bureaus. It might be the case, for example, that some regions do not have large-scale enterprises in a certain industry. In that case, the regional statistic bureau tend to increase the sample percentages for medium-scale and small-scale enterprises in the industries (Qi, 2007).

  means that the intermediate delivery X ij expresses the total input of products from industry i by industry j in region r, irrespective of the location of industry i.

  updated tables to the true 2002 tables, by yardsticks that will be discussed below.

  problems is threshold estimation, pioneered by

  b ˆ and ij b denote the estimated and true values of the Leontief inverse B = (I-A) -1 , respectively.

  China, of which some are undergoing rapid changes in production structure.

Figure

  Figure 1. Two cases: Averaging coefficients vs. OLS regression*

Figure 2 :

 2 Figure 2: Threshold estimation procedure

  presents a very accessible introduction to a strongly related approach used to identify structural breaks in time series. In an input-output context,[START_REF] Yamakawa | Environmental input-output analysis: Using timeseries to measure uncertainty[END_REF] apply both robust regression and (slightly different) sample-splitting techniques to study input coefficient stability over time. 11

  concern. Our primary interest lies in accuracy of the projections, for which a comparison of the estimated slopes is much more important. Table1provides a comparison of frequencies of classes of slopes as estimated by means of Ordinary Least Squares regression and robust regression. Table 1 clearly shows to what extent corrections for the presence of outliers change the estimation results. The estimated slopes are generally closer to zero. The share of cells with an absolute value of λ ij larger than 0.005 is 53% for Ordinary Least Squares, and 43% for for robust regression.

	INSERT TABLE 1 ABOUT HERE
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

25 for the univariate regressions, which indicates that deviations from constant returns to scale are generally not very strong. It should be noted, however, that statistical significance is not our main

Table 3

 3

	), OLS and threshold

least accurate) of the four methods, averaged over the 27 regions. These are 2.30, 2.72, 2.26 and 2.72, for averaging, OLS regression, robust regression and threshold regression,

Table 5

 5

	also grows if fewer observations are available (from

  analysis is done on the basis of two series of Chinese survey-based regional input-output tables for 1997 and 2002. They cover 27 regions and 31 industries.

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

table: intertemporal updating, regionalization of a national table and exchanging coefficients with a table for the most similar region in the previous period. The empirical

  table if input coefficients rather than technical coefficients are to be estimated is a subject for future work. Chinese provincial tables should not be used. In such cases, assumptions about the origin of inputs are needed to adapt the tables in a suitable way.

	F o F o F o F o F o F o Intertemporal Updating o F multiplier analyses), Appendix 2: Relative Performance of Averaging Coefficients and FIGURES AND TABLES
	r r r r r r INSERT TABLE A2 ABOUT HERE r
	P P P P P P P
	e e r e e r e e r e e r e e r e e r e e r
	R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w
	O n O n O n O n O n O n O n
	l y l y l y l y l y l y l y
	30

Table 1 . Frequencies of estimated slopes for OLS and robust regression

 1

	Estimated slope	Frequency (Ordinary	Frequency (Robust
		Least Squares)	Regression
	λ < -0.005	2	0
	-0.005 ≤ λ < 0	21	44
	0 ≤ λ < 0.0025	277	382
	0.0025 ≤ λ < 0.005	156	124
	0.005 ≤ λ < 0.01	133	114
	0.01 ≤ λ < 0.025	155	125
	0.025 ≤ λ < 0.05	114	90
	λ ≥ 0.05	103	82
	Total	961	961

Table 2 . Frequencies of estimated slopes for various regression approaches (subset of cells with threshold significant at 10%)

 2

	Estimated slope	Ordinary Least	Robust	Threshold	Threshold
		Squares	Regression	regression (small	regression (small
				sectoral output)	sectoral output)
	λ < -0.005	0	0	6	3
	-0.005 ≤ λ < 0	1	2	13	10
	0 ≤ λ < 0.0025	31	35	27	25
	0.0025 ≤ λ < 0.005	15	16	12	8
	0.005 ≤ λ < 0.01	8	13	10	11
	0.01 ≤ λ < 0.025	28	23	15	26
	λ ≥ 0.025	29	23	29	29
	Total	112	112	112	112

Table 5 . Comparison of the accuracy of the averaging coefficients and robust regression methods, with different numbers of observations *

 5 Number of regions with percentage of WAPEa<WAPEr smaller than 50%.

					Number of observations
			20			15			10
	Object	Average	Average	Percent	Average	Average	Percent	Average	Average	Percent
	Region	WAPE	WAPE	of	WAPE	WAPE	of	WAPE	WAPE	of
		of	of	WAPEa	of	of	WAPEa	of	of	WAPEa
		averaging	robust	<	averaging	robust	<	averaging	robust	<
		coeff.	regress.	WAPEr	coeff.	regress.	WAPEr	coeff.	regress.	WAPEr
	Anhui	0.242	0.240	34%	0.246	0.246	49%	0.252	0.256	68%
	Beijing	0.339	0.336	31%	0.341	0.344	56%	0.345	0.356	74%
	Chongqing	0.375	0.367	4%	0.378	0.370	13%	0.382	0.377	31%
	Fujian	0.352	0.363	99%	0.355	0.366	94%	0.360	0.371	86%
	Gansu	0.304	0.290	0%	0.306	0.294	4%	0.312	0.304	21%
	Guangdong	0.285	0.287	51%	0.288	0.301	73%	0.291	0.327	94%
	Guangxi	0.319	0.327	92%	0.322	0.329	82%	0.329	0.336	78%
	Guizhou	0.335	0.343	95%	0.338	0.344	80%	0.344	0.350	75%
	Hebei	0.204	0.207	62%	0.208	0.216	73%	0.214	0.231	81%
	Henan	0.308	0.305	31%	0.311	0.312	50%	0.317	0.327	70%
	Heilongjiang	0.217	0.216	33%	0.222	0.220	42%	0.230	0.231	56%
	Hubei	0.210	0.226	100%	0.215	0.230	99%	0.223	0.239	95%
	Hunan	0.258	0.260	76%	0.262	0.265	74%	0.268	0.273	78%
	Jilin	0.396	0.382	19%	0.400	0.392	36%	0.405	0.410	45%
	Jiangsu	0.273	0.279	71%	0.276	0.290	82%	0.280	0.308	95%
	Jiangxi	0.262	0.248	0%	0.265	0.254	5%	0.269	0.266	36%
	Liaoning	0.221	0.217	20%	0.224	0.222	32%	0.230	0.234	56%
	Neimeng	0.337	0.344	87%	0.341	0.346	71%	0.347	0.352	67%
	Ningxia	0.315	0.313	28%	0.318	0.320	61%	0.324	0.333	79%
	Shaanxi	0.285	0.279	9%	0.287	0.283	20%	0.292	0.291	44%
	Shandong	0.350	0.429	100%	0.353	0.428	100%	0.358	0.429	100%
	Shanxi	0.374	0.385	91%	0.376	0.385	83%	0.380	0.387	73%
	Shanghai	0.238	0.223	3%	0.240	0.234	26%	0.245	0.254	61%
	Sichuan	0.250	0.259	99%	0.254	0.263	93%	0.261	0.273	90%
	Tianjin	0.331	0.332	58%	0.332	0.334	62%	0.335	0.341	74%
	Yunnan	0.288	0.283	18%	0.292	0.287	31%	0.299	0.296	42%
	Zhejiang	0.256	0.290	100%	0.258	0.298	100%	0.264	0.308	100%
	Average **	0.293	0.297	52.3%	0.297	0.303	58.9%	0.302	0.313	69.2%
	Count	14	13	13 #	17	10	10 #	22	5	6 #
	* Shaded cells indicate the method with the highest accuracy for a region.
	** Unweighted averages over regions.

#

Table 6 . Comparison of averaging with different numbers of observations to single- table methods (1,000 random samples).

 6 * For a sample size of 26, only one sample can be constructed, containing all regional tables except the object table.Thus, the reported WAPE is the same as in Table1.

	Object Region	26 **	20	Num of Obs. for Averaging 15 10 8	5	3	Update	Regiona-lization
	Anhui	0.240 0.242 0.246 0.252 0.256 0.267	0.286	0.347	0.281
	Beijing	0.337 0.339 0.341 0.345 0.348 0.356	0.368	0.346	0.339
	Chongqing	0.374 0.375 0.378 0.382 0.385 0.394	0.408	0.452	0.423
	Fujian	0.350 0.352 0.355 0.360 0.363 0.372	0.384	0.444	0.370
	Gansu	0.301 0.304 0.306 0.312 0.317 0.327	0.345	0.377	0.365
	Guangdong	0.284 0.285 0.288 0.291 0.295 0.302	0.317	0.328	0.309
	Guangxi	0.317 0.319 0.322 0.329 0.333 0.345	0.364	0.407	0.381
	Guizhou	0.332 0.335 0.338 0.344 0.348 0.360	0.379	0.429	0.399
	Hebei	0.202 0.204 0.208 0.214 0.218 0.231	0.251	0.264	0.228
	Henan	0.306 0.308 0.311 0.317 0.320 0.329	0.345	0.385	0.335
	Heilongjiang	0.214 0.217 0.222 0.230 0.235 0.250	0.271	0.297	0.263
	Hubei	0.207 0.210 0.215 0.223 0.228 0.243	0.265	0.239	0.236
	Hunan	0.256 0.258 0.262 0.268 0.272 0.283	0.301	0.364	0.290
	Jilin	0.394 0.396 0.400 0.404 0.405 0.426	0.451	0.362	0.405
	Jiangsu	0.272 0.273 0.276 0.280 0.284 0.291	0.304	0.337	0.306
	Jiangxi	0.260 0.262 0.265 0.269 0.274 0.284	0.302	0.352	0.301
	Liaoning	0.218 0.221 0.224 0.230 0.235 0.247	0.264	0.304	0.239
	Neimeng	0.334 0.337 0.341 0.347 0.352 0.363	0.381	0.455	0.401
	Ningxia	0.313 0.315 0.318 0.324 0.328 0.337	0.353	0.389	0.362
	Shaanxi	0.283 0.285 0.287 0.292 0.295 0.304	0.318	0.335	0.313
	Shandong	0.348 0.350 0.353 0.358 0.362 0.371	0.387	0.391	0.401
	Shanxi	0.373 0.374 0.376 0.380 0.383 0.390	0.401	0.383	0.407
	Shanghai	0.237 0.238 0.240 0.245 0.247 0.257	0.272	0.256	0.226
	Sichuan	0.248 0.250 0.254 0.261 0.265 0.278	0.299	0.329	0.306
	Tianjin	0.330 0.331 0.332 0.335 0.336 0.344	0.352	0.433	0.349
	Yunnan	0.285 0.288 0.292 0.299 0.303 0.314	0.333	0.434	0.365
	Zhejiang	0.254 0.256 0.258 0.264 0.268 0.279	0.295	0.320	0.275

* * Shaded cells indicate that the averaging coefficients method outperforms both single table-based methods. *

Table 7 . Comparison of accuracies of averaging and regionalization methods for different numbers of observations *

 7

	Num of obs.	20	15	10	8	7	6	5	3
	Anhui	0%	0%	0%	0%	1%	5%	14%	63%
	Beijing	53%	64%	70%	75%	78%	80%	84%	91%
	Chongqing	0%	0%	0%	0%	0%	0%	0%	14%
	Fujian	0%	0%	10%	19%	28%	41%	51%	81%
	Gansu	0%	0%	0%	0%	0%	0%	2%	16%
	Guangdong	0%	0%	5%	13%	19%	25%	38%	62%
	Guangxi	0%	0%	0%	0%	0%	0%	1%	15%
	Guizhou	0%	0%	0%	0%	0%	0%	2%	16%
	Hebei	0%	0%	8%	23%	31%	42%	58%	87%
	Henan	0%	0%	5%	13%	21%	26%	39%	66%
	Heilongjiang	0%	0%	0%	0%	2%	5%	13%	66%
	Hubei	0%	0%	3%	17%	30%	51%	69%	96%
	Hunan	0%	0%	0%	0%	3%	8%	26%	74%
	Jilin	25%	35%	44%	47%	48%	50%	52%	60%
	Jiangsu	0%	0%	0%	2%	4%	9%	16%	49%
	Jiangxi	0%	0%	0%	0%	1%	6%	14%	44%
	Liaoning	0%	0%	16%	33%	46%	56%	68%	89%
	Neimeng	0%	0%	0%	0%	0%	0%	1%	17%
	Ningxia	0%	0%	0%	0%	1%	5%	11%	38%
	Shaanxi	0%	0%	1%	4%	8%	13%	23%	58%
	Shandong	0%	0%	0%	0%	0%	1%	4%	22%
	Shanxi	0%	0%	0%	2%	4%	7%	14%	36%
	Shanghai	99%	96%	95%	94%	93%	96%	96% 100%
	Sichuan	0%	0%	0%	0%	0%	0%	1%	35%
	Tianjin	0%	0%	8%	16%	18%	27%	33%	53%
	Yunnan	0%	0%	0%	0%	0%	0%	1%	13%
	Zhejiang	0%	0%	12%	28%	35%	46%	65%	88%

* Percentages indicate the share of random samples for which the regionalization method yielded a higher accuracy than averaging coefficients. Shaded cells represent shares less than 10%.

Table A1 . Industry classification

 A1

		Sector		Sector
	01 Agriculture		17 Electric equipment and machinery
	02	Coal mining, Crude petroleum and natural gas extraction	18 Electronic and telecommunication equipment
	03 Metal ore mining		19	Instruments, meters, cultural and office machinery
	04 Nonmetal mineral mining		20 Other manufacturing products
	05	Manufacture of food products and tobacco processing	21 Electricity, gas and water production and supply
	06 Textile goods 07 Wearing apparel, leather, furs, down and related products 08 Sawmills and furniture 09 F o Paper and products, printing and record medium reproduction r	22 Construction 23 Transport and storage, post and telecommunication 24 Wholesale and retail trade, catering trade 25 Finance and insurance
	10 Petroleum processing and coking 11 Chemicals 12 Nonmetal mineral products P 13 Metals smelting and pressing 14 Metal products e e r	26 Real estate 27 Social services 28 Health services, sports and social welfare 29 Education, culture and arts, radio, film and television 30 Scientific research and general technical services
	15 Machinery and equipment		31 Public administration and other sectors
	16 Transport equipment	R e v i e w
				O n
				l y

Table A2 . Comparison of accuracies of averaging and intertemporal updating methods for different numbers of observations (1,000 random samples) *

 A2

	Num of obs.	20	15	10	8	7	6	5	3
	Anhui	0%	0%	0%	0%	0%	0%	0%	0%
	Beijing	8%	23%	42%	55%	59%	64%	72%	84%
	Chongqing	0%	0%	0%	0%	0%	0%	0%	0%
	Fujian	0%	0%	0%	0%	0%	0%	0%	0%
	Gansu	0%	0%	0%	0%	0%	0%	0%	9%
	Guangdong	0%	0%	0%	0%	2%	3%	7%	30%
	Guangxi	0%	0%	0%	0%	0%	0%	0%	1%
	Guizhou	0%	0%	0%	0%	0%	0%	0%	1%
	Hebei	0%	0%	0%	0%	0%	0%	2%	25%
	Henan	0%	0%	0%	0%	0%	0%	0%	4%
	Heilongjiang	0%	0%	0%	0%	0%	0%	0%	10%
	Hubei	0%	0%	1%	7%	19%	36%	59%	93%
	Hunan	0%	0%	0%	0%	0%	0%	0%	0%
	Jilin	100%	96%	92%	89%	87%	88%	89%	92%
	Jiangsu	0%	0%	0%	0%	0%	0%	0%	4%
	Jiangxi	0%	0%	0%	0%	0%	0%	0%	1%
	Liaoning	0%	0%	0%	0%	0%	0%	0%	3%
	Neimeng	0%	0%	0%	0%	0%	0%	0%	0%
	Ningxia	0%	0%	0%	0%	0%	0%	0%	13%
	Shaanxi	0%	0%	0%	0%	0%	0%	1%	17%
	Shandong	0%	0%	0%	1%	2%	4%	11%	38%
	Shanxi	0%	11%	39%	48%	52%	61%	65%	84%
	Shanghai	0%	2%	18%	28%	36%	41%	49%	71%
	Sichuan	0%	0%	0%	0%	0%	0%	0%	5%
	Tianjin	0%	0%	0%	0%	0%	0%	0%	0%
	Yunnan	0%	0%	0%	0%	0%	0%	0%	0%
	Zhejiang	0%	0%	0%	0%	0%	0%	0%	10%

http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nlRegional Studies

Acknowledgements

The authors thank two anonymous referees for their constructive comments.

Growth, Development, and Changes < R1 -General Regional Economics < R -Urban, Rural, and Regional Economics -output tables, China

Appendix 1: Construction of Chinese Provincial Input-Output Tables

The construction of survey-based provincial input-output tables became a regularly activity since 1987, at five-year intervals. During each survey year, the national statistical agency (NBS) establishes methods of conducting the survey and explains these to the provincial statistical bureaus. The guidelines include the survey forms, the tabulation method and the industry classifications. 20 In each region, provincial officials firstly train accountants of enterpirses in the methods of filling out the forms. Enterprises provide information of intermediate consumption for production and the generation of output by means of these forms. All large-scale enterprises are surveyed, while random sampling is done for medium-scale and small-scale enterprises. For 1997 and 2002, the proportions of sampled firms were set at 20% and 8% for the medium-scale and small-scale enterprises respectively. 21 Subsequently, the forms are submitted to provincial statistical offices, who forward the data to NBS for their national tabulation and adjustments, whereas they also use the data to tabulate their own regional tables. The standardized method of data survey ensures the quality of data collection over space. For the purposes of this study, it is essential that provincial tables are not derived from a national table (which would imply that production technologies of a province would be considered as similar to those of the country), but purposefully constructed from data at the provincial level.

INSERT TABLE A1 ABOUT HERE

It should be noted that the Chinese input-output survey defines intermediate inputs as including both domestically produced materials and materials that have been imported (either from an other province or from abroad). This procedure results in a subtle but important difference between Chinese provincial input-output tables and the internationally more common regional tables. Input coefficients derived from Chinese provincial input-output tables could be considered as "technical coefficients", which represent production technologies well. For studies of the impact of policy measures or changes in consumption or investment behavior on provincial economies (e.g. via