People's Republic of China b Beijing Key Laboratory of Emergency Support Simulation

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1 This article was downloaded by: [Haitao Zheng] On: 01 July 2015, At: 12:13 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Click for updates Economic Systems Research Publication details, including instructions for authors and subscription information: UPDATING INPUT OUTPUT TABLES WITH BENCHMARK TABLE SERIES Huiwen Wang ab, Cheng Wang a, Haitao Zheng a, Haoyun Feng c, Rong Guan d & Wen Long e a School of Economics & Management, Beihang University, Beijing, People's Republic of China b Beijing Key Laboratory of Emergency Support Simulation Technologies for City Operations, Beijing, People's Republic of China c Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA d School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, People's Republic of China e Research Center on Fictitious Economy and Data Science, Chinese Academy of Sciences, Beijing, People's Republic of China Published online: 29 Jun To cite this article: Huiwen Wang, Cheng Wang, Haitao Zheng, Haoyun Feng, Rong Guan & Wen Long (2015): UPDATING INPUT OUTPUT TABLES WITH BENCHMARK TABLE SERIES, Economic Systems Research, DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

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3 Economic Systems Research, UPDATING INPUT OUTPUT TABLES WITH BENCHMARK TABLE SERIES HUIWEN WANG a,b, CHENG WANG a, HAITAO ZHENG a, HAOYUN FENG c, RONG GUAN d and WEN LONG e a School of Economics & Management, Beihang University, Beijing, People s Republic of China; b Beijing Key Laboratory of Emergency Support Simulation Technologies for City Operations, Beijing, People s Republic of China; c Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA; d School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, People s Republic of China; e Research Center on Fictitious Economy and Data Science, Chinese Academy of Sciences, Beijing, People s Republic of China (Received 29 October 2013; in final form 19 May 2015) Numerous methods have been proposed to update input output (I O) tables. They rely on the assumption that the economic structure will not change significantly during the interpolation period. However, this assumption may not always hold, particularly for countries experiencing rapid development. This study attempts to combine forecasting with a matrix transformation technique (MTT) to provide a new perspective on updating I O tables. Under the assumption that changes in the trend of an economic structure are statistically significant, the method extrapolates I O tables by combining time series models with an MTT and proceeds with only the total value added during the target years. A simulation study and empirical analysis are conducted to compare the forecasting performance of the MTT to the Generalized RAS (GRAS) and Kuroda methods. The results show that the comprehensive performance of the MTT is better than the performance of the GRAS and Kuroda methods, as measured by the Standardized Total Percentage Error, Theil s U and Mean Absolute Percentage Error indices. Keywords: Input output table series; Matrix transformation; Forecasting; Updating 1. INTRODUCTION As the core of the input output (I O) technique (Leontief, 1936), the I O table has long been widely used to analyse a range of research topics, such as the socio-economic and environmental impacts of globalization, international trade and regional integration, productivity and efficiency, innovation and research and development spillovers, and migration (Temurshoev et al., 2011; Kazemier et al., 2012). The timeliness of I O tables is a major concern for both academics and policy-makers (Dietzenbacher et al., 2013). Numerous methods have been proposed to update I O tables. Among them, many non-survey methods have been presented and extensively applied (Pavia et al., 2009; Wood, 2011; Lindner et al., 2012). Most methods use a combination of statistical and optimization techniques. Two methods are popular: the naive method and the RAS method, which is also known as the iterative proportional method (Khan, 1993). *Corresponding author. zhenghaitao@buaa.edu.cn c 2015 The International Input Output Association

4 2 H. WANG et al. Below, we will present a brief review of various methods (for details, see the original sources). The naive method is divided into the final demand and the transactions proportional to the value-added methods (Buetre and Ahmadi-Esfahani, 2000). The former approach uses the inter-industry relationship from a known base year to forecast the target year s gross output. The latter approach assumes that the proportion of industrial transactions to value added remains unchanged and heavily depends on the economic assumption that the ratio of intermediate inputs to labour and capital inputs for a given commodity production process remains fixed from the base year to the target year. The most well-known and widely used method is RAS (Stone, 1961; Stone and Brown, 1962; Bacharach, 1970), including its variants such as the Generalized RAS (GRAS) algorithm (Günlük-Senesen and Bates, 1988; Junius and Oosterhaven, 2003; Lahr and De Mesnard, 2004). The basic idea of RAS and extensions thereof lies in minimizing the dissimilarity between the prior table and the target table. The RAS modelling process is based on a constraint that the industry-level gross output, intermediate output totals and intermediate input totals in the target year are known. Many techniques also employ a broad range of partial information (Gilchrist and St Louis, 2004; Lenzen et al., 2006; Mueller and Domínguez, 2008; Lenzen et al., 2009; Wood, 2011; Lenzen et al., 2013; Tukker et al., 2013). The RAS method has long been widely applied to forecast I O tables, and considerable development in I O analysis has taken place since its introduction by Stone (1961) and Stone and Brown (1962). The RAS procedure interprets changes in I O coefficients as inter-industry or inter-commodity substitution effects and productivity effects. Under this assumption, a series of candidate solution I O tables are derived through biproportional adjustments to the original tables. However, a prominent defect of RAS is that it can only address non-negative tables. To estimate tables that contain both positive and negative entries, Günlük-Senesen and Bates (1988) proposed a modified RAS. This work received limited acknowledgement until Junius and Oosterhaven (2003) re-discovered and renamed it GRAS. The GRAS method has been further revised by correcting its biased objective function (Lenzen et al., 2007) and, more recently, by relaxing the assumption that for the GRAS to be balanced, every row and column of a matrix must have at least one positive element (Temurshoev et al., 2013). In updating an I O table, it is likely that, besides the row and column totals of the target matrix, additional information is available. Thus, Gilchrist and St Louis (1999) developed the Two-stage RAS (TRAS) method to utilize such information in addition to the row and column totals to produce updated matrices that are consistent with the known cells and the aggregates of the known cells. Gilchrist and St Louis (2004) reformulated TRAS, and the incremental gains from these TRAS adjustments are statistically significant. Because neither TRAS nor GRAS satisfactorily addresses uncertainty and data conflicts, Lenzen et al. (2006, 2007, 2009) developed a generalized iterative scaling method called KRAS ( Konfliktfreies RAS ), which is able to balance and reconcile I O tables under conditions of conflicting external information and inconsistent constraints. Wood (2011) documented the development of a series of Australian I O tables using KRAS. Lenzen et al. (2014) also developed a variant of the RAS generalized iterative scaling method that is able to change signs between successive iterations and thus satisfies signchanging constraint that existing RAS variants could not.

5 UPDATING INPUT OUTPUT TABLES 3 Friedlander (1961) and Matuszewski et al. (1964) proposed the pioneering approaches of normalized squared differences (NSD) and normalized absolute differences (NAD), respectively, concentrating on the coefficient pairs of the prior and target years, NSD minimizes the sum of squared differences, whereas NAD attempts to identify the minimal sum of absolute differences. Thereafter, Almon (1968) introduced the Squared Differences approach (SD), which solves for the least-squared differences in coefficient pairs. Lecomber (1975) presented a comparison of the RAS, NSD, NAD and SD approaches. Seeking to minimize the weighted squared differences, Kuroda (1988) and Wilcoxen (1989) proposed an intermediate coefficients solution using the Lagrange multiplier method. Harthoorn (1988) also used the Lagrange multiplier method to adjust I O tables. Subsequently, Golan et al. (1994) applied the cross entropy method to update I O tables, while Jackson and Murray (2004) extended the research of Lecomber (1975) and provided a comparison of various methods. Moreover, they proposed several alternative minimands, including absolute differences, weighted absolute differences, weighted squared differences (Lahr, 2001), global change constant, and sign-preserving squared differences, which all minimize the distance between the target and prior tables under particular constraints. Most recently, Lenzen et al. (2012) presented a new method for constructing I O table series from incomplete data by averaging over alternate forward and backward sweeps across the time series period. They also solved the problem of hysteresis, which causes the forecast and backcast table estimates to differ. A concise summary of the aforementioned methods for updating I O tables is provided in Table 1. It should be emphasized that the existing methods per se and their implementation are two separate concerns. The existing methods per se are pure techniques to minimize the distance between an initial table and a target table and are used primarily for balancing an initial estimate rather than pure updating. The implementation of the method refers to the choice of the starting point and how the required conditions are achieved. When implementing a method, problems arise because of various initial estimate options and different required conditions. First, in practical applications, most existing methods rely heavily on the assumption that the transaction coefficients will not change dramatically from the prior year to the target year. However, the research of Sonis and Hewings (1992) and Israilevich et al. (1997) has revealed that this assumption may not hold, particularly when an economy is experiencing structural changes. Consider, for example, the US I O tables from 1947 to 1977 (Miller and Blair, 1985). We can observe fluctuations of specific transaction coefficients in Figure 1. 1 Second, in practical applications, these methods cannot proceed without access to economic data for the target years, including gross outputs, intermediate output totals and intermediate input totals by industry. Finally, optimization methods are typically computationally complex and potentially unable to derive globally optimal solutions under certain circumstances (Jackson and Murray, 2004). Therefore, further study of forecasting an I O table for a target year is warranted. This paper intends to investigate a method for forecasting I O tables referred to as the matrix transformation technique- (MTT) based forecasting, and the main contributions of 1 Industrial definitions: 1, Agriculture, Forestry, and Fishing; 2, Metal Mining; 3, Petroleum and Natural Gas Mining; 8, Wood Products and Furniture; and 10, Chemicals and Chemical Products.

6 4 H. WANG et al. TABLE 1. Short description of the methods for updating I O tables. Method Naive method RAS Source Buetre and Ahmadi-Esfahani (2000) Stone (1961), Stone and Brown (1962), and Bacharach (1970) Generalized RAS (GRAS) Günlük-Senesen and Bates (1988), Junius and Oosterhaven (2003), and Temurshoev et al. (2013) Normalized Squared Differences Friedlander (1961) (NSD) Normalized Absolute Matuszewski et al. (1964) Differences (NAD) Squared Differences approach Almon (1968) (SD) Absolute differences (AD) Jackson and Murray (2004) Weighted absolute differences Jackson and Murray (2004) (WAD) Weighted squared differences Jackson and Murray (2004) (WSD) and Lahr (2001) Global change constant (GCC) Jackson and Murray (2004) Sign-preserving squared Jackson and Murray (2004) differences (SPSD) Kuroda s method Kuroda (1988) and Wilcoxen (1989) Cross entropy method (CE) Golan et al. (1994) TRAS Gilchrist and St Louis (1999, 2004) KRAS Lenzen et al. (2006, 2007, 2009), Wood (2011) A non-sign-preserving RAS Lenzen et al. (2014) variant Cycling method for constructing Lenzen et al. (2012) I O table time series Data requirements for the target year(s) Transaction coefficients Gross outputs Intermediate output totals Intermediate input totals by industry Information in addition to row and column totals Incomplete data this paper are as follows. First, time series analysis is used to forecast I O tables for the target years. MTT considers more of the available information in a series of I O tables than do other approaches. According to Tilanus (1966) and Barker (1977), considering the linear trend of 10 transaction coefficient tables yields worse forecasting results than using only the most recent table. We have doubts about this conclusion because the constraints inherent to transaction coefficient tables were not removed before forecast modelling (for details, please see Appendix A). In this paper, we will demonstrate the merits of using a linear (or nonlinear) trend of historical I O tables for forecasting. Moreover, MTT is a nonsign-preserving method because time series analysis will always be applicable regardless of the presence of positive or negative entries. Second, MTT always proceeds and requires only the aggregate industry value added or final demands in the target years, that is, it

7 UPDATING INPUT OUTPUT TABLES 5 FIGURE 1. The fluctuation of specific transaction coefficients. proceeds regardless of the availability of economic data for the industries in question for the target years. When the target years occur after the current fiscal year, the unavailable economic data can be endogenously derived through MTT. Moreover, a derivative of the MTT method, called MTT II, was developed to interpolate I O tables when the economic data are already known. Third, the calculation of MTT is much simpler and more convenient in practise. MTT is a matrix calculation method rather than an iterative procedure, as employed in other methods. It should be emphasized that the MTT and other techniques have different areas of application. When there is more than one table, we should employ all of the information obtained from the MTT to ensure that the structural changes are more accurately considered. When we have only the most recent table or the assumption of statistically significant trend does not hold, MTT cannot proceed; however, existing methods may work well. The possibility of zero final demand in known tables is a weakness of the MTT approach in practise. Although we have a derivative version of the MTT method, called MTT I, to address this situation, there are nevertheless special cases in which most MTT methods have no solution, except MTT II which requires gross outputs, intermediate input and output totals in the target years, unless the tables are aggregated. In general, the more disaggregated the I O tables, the greater the likelihood of zero total final demand (or zero final demand, excluding net exports). The remainder of this paper is organized as follows. In Section 2, we describe the preliminaries, define the problem studied in this paper, and introduce the techniques of matrix transformation and inverse computation. Section 3 proposes the MTT method. Next, in Section 4, a simulation experiment for the MTT method is performed. Then, in Section 5, we compare the MTT method and other existing methods based on an empirical analysis of 23-industry US I O tables. Finally, in Section 6, we discuss the application of the MTT method and draw conclusions. 2. PRELIMINARIES We first illustrate how to transform an I O table into a simplified I O matrix. Thereafter, we will provide a brief introduction of the MTT.

8 6 H. WANG et al I O Table and I O Matrix An n-industry I O table can be described as a matrix, as depicted in Figure 2. Quadrant I reflects the technical economic ties of interdependence and mutual restriction among different industries. For the sake of simplicity, we leave Quadrant I unchanged and transform Quadrants II and III into row summations and column summations, respectively. Then, we obtain a simplified I O matrix X: X = x 11 x 1n x 1(n+1) x n1 x nn x n(n+1) x (n+1)1 x (n+1)n x (n+1)(n+1) [ ] Z f = v. (1) μ The element x ij (i = 1,..., n, j = 1,..., n) in Z represents the industrial intermediate input and output in Quadrant I; f = (x 1(n+1),..., x n(n+1) ) is the industrial total final demand in Quadrant II; and v = (x (n+1)1,..., x (n+1)n ) denotes the industrial total value added in Quadrant III. Because Quadrant IV is null, we set it as a constant μ to facilitate the subsequent derivation. It is denoted as μ = f i = v i, (2) where i = (1...1) and μ is the summation of all industries value added. Given a series of I O tables, we can apply Equations 1 and 2 to obtain a series of simplified I O matrices X 1, X 2,, X T. Thus, the I O table forecasting problem is converted as follows. Problem Given a series of I O matrices X 1, X 2,..., X T, forecast matrices ˆX T+1,..., ˆX T+L (L 1) of target years. To guarantee the establishment of forecasting models, we hereby assume a statistically significant trend for each entry series x t ij (1 i, j n + 1, t = 1, 2,..., T). This assumption implies that there is a rule governing changes in economic data. We consider this a realistic assumption. According to Sonis and Hewings (1992) as well as Israilevich et al. (1997), the transaction coefficients are not fixed from the prior year to the target year. This paper FIGURE 2. Input output table.

9 UPDATING INPUT OUTPUT TABLES 7 FIGURE 3. The fluctuation of specific entries in matrix X. can completely exploit the statistically significant trend of each entry because a series of I O matrices is added to the model. For example, consider a data set of 23-industry US I O tables for the years 1947, 1958, 1963, 1967, 1972, and The changes in specific elements are depicted in Figure 3. It is apparent that the trend in each entry is statistically significant Matrix Transformation Techniques The elements in X are subject to certain constraints and thus cannot be freely forecasted (Aitchison, 1982). The constraint inherent to matrix X is that the summation of each row must equal the corresponding column summation, which in matrix form is Zi + f = Z i + v, f i + μ = v i + μ. (3) Considering all of these constraints, we obtain the degrees of freedom for X as n (n + 1). The most intuitive approach to completing the forecasting task described in Section 2.1 is to forecast the elements in X individually, which is equivalent to the approach employed by Tilanus (1966). However, it may lead to inconsistencies such that the constraints in Equation 3 cannot be satisfied. That is, the forecast outcomes may no longer be an I O table. This paper presents an alternative MTT to solve this problem. To begin, we will describe the technique below. Assume that each industry s total final demand and the summation the value added for all industries are never equal to zero (i.e. x i(n+1) 0 (i = 1, 2,..., n) and μ 0). Then, X is transformed into Y by T = ˆf 1 Z, d = v μ, (4)

10 8 H. WANG et al. FIGURE 4. Fluctuations of specific entries in matrix T. where ˆf = diag(f). Then: Y = [ ] T i d. (5) 1 We employ the same data set to determine the fluctuation of t ij (i, j = 1, 2,..., n). The fluctuations of the specific elements in matrix T are depicted in Figure 4. The results indicate that they can be used for forecast modelling because their trends are statistically significant. Remark t ij (i, j = 1, 2,..., n) in matrix T is the proportion of the monetary value of the transactions between a pair of industries (from Industry i to Industry j) to the total final demand of Industry i. Because t ij is unconstrained, we are free to choose the forecast models for it. Because d concerns the ratio of Industry j s value added accounting for the summation of all industries values added, the sum of the elements in vector d always equals 1. Thus, before forecasting, this constraint should be removed via another transformation, which will be detailed in Section 2.3. Once matrix Y in the target year is obtained, we need to back-transform it into matrix X using an inverse computation. We describe the process of inverse computation as follows. Step 1 Assuming that μ is known, we calculate v according to d by Step 2 The solution of f can be given by where v = μd. (6) f = B 1 v (7) B = diag(ti + i) T. (8) The derivation process is described in Appendix B. Obviously, as a strictly diagonally dominant matrix, B is an invertible matrix.

11 UPDATING INPUT OUTPUT TABLES 9 Step 3 The elements in Quadrant I (i.e. Z) can be calculated by Z = ˆfT. (9) Now, all of the entries x ij (i, j = 1,..., n + 1) in matrix X have been calculated from matrix Y. Given n-industry I O matrices X 1, X 2,..., X T chronologically collected for T years, we divide the forecasting process for an I O matrix into three stages. (1) Transformation stage: remove the constraints of the I O matrices X 1, X 2,..., X T using Equation 4 and obtain Y 1, Y 2,..., Y T. (2) Forecasting stage: extrapolate the matrix series Y 1, Y 2,..., Y T to Ŷ T+1,..., Ŷ T+L (L 1). (3) Inverse-transformation stage: obtain the forecasted I O matrices ˆX T+1,..., ˆX T+L from Ŷ T+1,..., Ŷ T+L using inverse computation The Unit-sum Constrained Vector Transformation Technique According to Equation 2, we find that d is subject to the unit-sum constraint (i.e. d i = 1). Thus, the degrees of freedom of d decline to (n 1). Because the first n columns in matrix Y correspond to the n parallel industries, without loss of generality, assuming that y (n+1)n 0 and the vector d can be transformed into m = (m 1, m 2,..., m n 1,1), where for j = 1, 2,..., n: m j = y (n+1)j. (10) y (n+1)n Because the vector m is no longer subject to the unit-sum constraint, we are now free to choose forecasting models for m j (j = 1, 2,..., n 1). Having obtained vector m for the target year, we must back-transform m to d through inverse computation. The process of inverse computation is summarized as follows: 3. THE I O MATRIX FORECAST MODELLING d = m mi. (11) 3.1. MTT Method In the event that the gross outputs, intermediate output totals and intermediate input totals by industry in the target year of T + l(1 l L) are unknown, the value added and final demand of each industry (i.e. v T+l and f T+l ) must also be forecasted. The specific forecasting process is described as follows. (1) Transformation Stage. For t = 1, 2,..., T, transform X t into Y t using Equation 4. (2) Forecasting Stage.

12 10 H. WANG et al. Given a sequence of matrices Y 1,Y 2,...,Y T, forecast Ŷ T+l. (a) For i=1, 2,..., n and j = 1, 2,..., n, establish forecasting models for y t ij, that is, y t ij = f ij(t) + ε ij (12) Taking t = T + l as the target year, Equation 12 enables us to forecast ŷ T+l ij Ẑ T+l. in matrix (b) Transform d t into m t with Equation 10. (c) Establish a forecasting model for m t j (j = 1, 2,..., n 1), similar to Equation 12. (d) Transform ˆm T+l into ˆd T+l using Equation 11. (3) Inverse-Transformation Stage. Given Ŷ T+l, estimate the I O matrix ˆX T+l by inverse computation. First, regarding the summation of the value added of all industries μ as an exogenous variable, we establish a model as follows: μ t = f (n+1)(n+1) (t) + ε (n+1)(n+1). (13) For the target year, T + l, forecast ˆμ T+l by calculating f (n+1)(n+1) (T + l). Next, estimate ˆv T+l with Equation 6, ˆf T+l with Equation 7, and Ẑ T+l with Equation 9. Consolidating all previous forecast results yields the I O matrix ˆX T+l. Note. The MTT assumes that each industry s total final demand and the summation of all of industries values added are not equal to zero. However, this does not hold under certain circumstances. Accordingly, a derivative version of the MTT method, the MTT I, addresses this situation. Compared with the MTT method, the MTT I requires more exogenous information for the target years. To maintain clarity, we provide the details of MTT I in Appendix C. Though the MTT I can mitigate the problem of zero final demand to some extent, there remain special cases in which MTT I is not effective. It may be the case that a product is only exported and not used for other final demand purposes or is used only for intermediate purposes and not for exporting or other final demand purposes. For example, in the 2005 Spanish I O table, 2 Product 64 Market activities of membership organization n.e.c. has positive intermediate use, but both net exports and other final demands are zeroes. This problem represents a potential weakness of the MTT approach in practise. In this case, only the MTT II, presented in Section 3.2, will be effective because it requires gross output as well as intermediate input and output totals for the target years. 2 The table is available at

13 UPDATING INPUT OUTPUT TABLES 11 TABLE 2. Known conditions and application scenarios of MTT and MTT II. MTT Past years I O tables X 1, X 2,..., X T Known Known Gross outputs Zi + for Z i + v Unknown Known Intermediate output totals Z i Unknown Known Intermediate input totals Zi Unknown Known Application scenario To estimate I O tables after the current fiscal year MTT II To update I O tables from the latest benchmark year to the current fiscal year 3.2. MTT II Method The MTT method presented in Section 3.1 is intended to extrapolate I O tables. However, in practise, it is also commonly employed to interpolate I O tables. Interpolation implies that some information for the target years is already known, including gross outputs, intermediate output totals, and intermediate input totals by industry. Therefore, MTT II is provided (see Appendix D). To describe the difference between the MTT and MTT II, Table 2 lists their conditions and application scenarios, which reveals that MTT II is a special case of MTT. 4. SIMULATION EXPERIMENT FOR THE MTT METHOD To validate the effectiveness of the MTT method, this section will conduct a simulation experiment using synthetic data. This method is compared with the GRAS and Kuroda methods. In the experiment, we will randomly generate a set of T matrices of I O tables, select the first T 1 matrices to implement the MTT model, and evaluate its performance. Then, we apply the GRAS and Kuroda methods to forecast the Tth I O matrix using the (T 1)th I O matrix and evaluate these models performance Evaluation Indices Three indices are commonly employed to assess I O table forecasting performance: (1) Standardized Total Percentage Error (STPE; Miller and Blair, 1985): (2) Theil s U (Theil, 1971): STPE = 100 n+1 i=1 U = n+1 i=1 [ n+1 n+1 i=1 n+1 i=1 n+1 j=1 x ij x ij n+1 j=1 x. (14) ij j=1 (x ij x ] ij) 2 1/2. (15) n+1 j=1 x2 ij

14 12 H. WANG et al. (3) Mean absolute percentage error (MAPE; Butterfield and Mules, 1980): 1 n+1 n+1 [ x ] ij x ij MAPE = (n + 1) 2, (16) x ij i=1 where x ij and x ij denote the actual and estimated values in the I O matrix, respectively. Additionally, for x ij =0, we set MAPE to zero Experimental Procedure j=1 The experiment is run N=10, 000 times. In each replication, we conduct a four-step procedure as follows. Step 1 Generating a series of synthetic I O matrices X 1, X 2,..., X T The detailed process is described in Figure 5. First, randomly generate a matrix Y, as shown in Equation 5. Then, convert it into an n-industry I O matrix X through inverse transformation. The generation of matrix Y can be divided into two parts. (a) Generation of Quadrant I of matrix Y. Specify the time series of entry y ij (t)(i, j = 1, 2,, n) at time t as a cubic polynomial: y ij (t) = p ij0 + p ij1 t + p ij2 t 2 + p ij3 t 3 + ε ij (t). (17) The coefficient p ijk (i, j = 1, 2,..., n, k = 0, 1, 2, 3) follows the standard normal distribution. To unify the scopes of p ijk and t, we perform a range standardization transformation of p ijk : p ijk min p ijk p ijk = k=0,1,2,3 max p ijk min p. (18) ijk k=0,1,2,3 k=0,1,2,3 Similarly, we conduct a maxima standardization transformation to t as (t/t). The random disturbance, ε ij (t)(i, j = 1, 2,..., n, t = 1, 2,..., T), is subject to a normal distribution and the selected signal-to-noise ratio (SNR). FIGURE 5. The process of generating synthetic I O matrices.

15 (b) Generation of Quadrant III of matrix Y. UPDATING INPUT OUTPUT TABLES 13 Because the entries in Quadrant III are subject to the unit-sum constraint, we first randomly generate a set of time series data, m j (t) (j = 1, 2,..., n 1, t = 1, 2,..., T), asin the process of generating y ij (t), and then conduct an inverse transformation (see Equation 11) to obtain d t. Next, we continue to generate synthetic data of μ t in Quadrant IV of matrix X. Regarded as an exogenous variable, μ t can be substituted by GDP in the tth year. Equipped with a full matrix Y t together with μ t, we finally obtain T I O matrices X 1, X 2,..., X T through inverse computation. Step 2 Using MTT to establish the forecasting model Divide the synthetic data into two parts: the first T 1 matrices function as the training data set and the second set provides the testing data set. The training data set is used to establish the model. Note. The forecasting models and polynomials for generating the data in the I O tables are independent. Step 3 Computing the evaluation indices Using the established model, we estimate I O matrices in both the training data set and the testing data set and compute the mean and standard deviation of the STPE, U and MAPE of the estimations. Step 4 Applying the GRAS and Kuroda methods to the simulated data Using the GRAS and Kuroda methods, we estimate the Tth I O matrix based on the (T 1)th I O matrix and compute the evaluation indices. The parameters of the experiment are set as follows: n = 17, T = 16, SNR = Results For each estimated I O matrix produced by these three methods, the mean and standard deviation of STPE, U and MAPE are listed in Table 3. The averaged index values demonstrate the merits of the MTT method. When forecasting an out-of-sample I O matrix (the 16th I O matrix), the MTT method exhibits excellent performance, although it is not as high as its fitting performance (the first 15 I O matrices). Additionally, the indices reveal that MTT outperforms GRAS, while GRAS outperforms the Kuroda method. Therefore, MTT is accurate and valid. Moreover, we conducted this experiment 10,000 times, which indicates that MTT is also robust. TABLE 3. Results of the simulation experiment. Method No. of I O tables STPE U MAPE MTT (0.002) 0.023(0.002) 0.286(0.746) (0.001) 0.013(0.001) 0.011(0.001) (0.002) 0.035(0.002) 0.030(0.002) GRAS (0.002) 0.043(0.002) 0.036(0.002) Kuroda (0.166) 0.045(0.002) 0.037(0.002) Notes: (1) The numbers in parentheses are standard deviations. (2) The MATLAB procedure for the GRAS is obtained from Temurshoev et al. (2013). Hereinafter, the procedure is the same.

16 14 H. WANG et al. 5. EMPIRICAL ASSESSMENT In this section, we will use 23-industry US I O tables for the years 1967, 1972, and 1977 to further evaluate the forecasting performance of the proposed methods. Because the implementation of MTT requires at least two historical I O matrices, our empirical analysis will concentrate on forecasting the 1977 table based on the 1967 and 1972 tables. When updating I O tables, we will consider whether prior information has been given. If the gross outputs, intermediate output totals, and intermediate input totals of industries have been published, MTT II will be used. If not, we will adopt the MTT method. Consequently, our empirical analysis will be conducted under these two scenarios. Because we lack a longer data series, we will select the simplest linear model. We stress that more complex models that satisfy the development rule of the data will be considered if more historical I O matrices are available. Scenario A: Assume that gross outputs, intermediate output totals, and intermediate input totals by industry in 1977 are given. MTT differs in two ways from the other algorithms. First, MTT uses the development information between 1967 and 1972 to predict the 1977 table. The other methods use only the 1972 table. Second, MTT uses a new calculation method. We should determine which difference leads to the different performance of the MTT and the other methods. Two types of empirical analyses are considered here. (1) Using time series analysis in GRAS and Kuroda. The specific forecast process is described as follows. Step 1 An I O structure for 1977 is predicted by establishing a linear model using the I O tables for 1967 and To capture the economic meaning of the entries in the predicted structure, the model is established under the restriction that the signs of the entries cannot be changed. In that case, the predicted entries are assumed to be zeroes. Step 2 Based on the predicted I O structure, that is, the row and column totals of the 1977 I O table, we implement GRAS and Kuroda to update the I O table for We term these methods extended GRAS and extended Kuroda, which we implement to compare their performance using the STPE, U and MAPE indices (Table 4). The results in Table 4 indicate that the extended GRAS and extended Kuroda are inferior to GRAS and Kuroda with respect to all three indices, which means that their performance is not improved after addressing the changing trend in the transaction coefficients. (2) Comparing MTTII with the GRAS and Kuroda methods. TABLE 4. Results of forecasting the 1977 I O table. STPE U MAPE GRAS Extended GRAS Kuroda Extended Kuroda

17 UPDATING INPUT OUTPUT TABLES 15 TABLE 5. Results of forecasting the 1977 I O table. STPE U MAPE Methods Index value Rank Index value Rank Index value Rank GRAS Kuroda MTT II TABLE 6. Results of forecasting the 1977 I O table using the MTT model. Method STPE U MAPE MTT In the above empirical analysis, GRAS and Kuroda outperformed the extended GRAS and extended Kuroda. Therefore, we compare the MTT II results to those of GRAS and Kuroda (Table 5). Table 5 reveals a slight disparity across the various evaluation indices. The MTT II method is ranked first for both the STPE and U indices, although it is ranked third with respect to the MAPE index. The poor performance of this method on the MAPE index is due primarily to the poor results given by specific entries, such as Industry 1 Industry 20, Industry 10 Industry 22, and Industry 10 Industry 19; 3 Furthermore, the MAPE index substantially overstates the error when the actual values are small (Tayman and Swanson, 1999). The overall performance of the indices indicates that the MTT II method yields superior forecasting performance when updating I O tables to the current fiscal year. These two empirical analyses reveal the advantages of the MTT. Both the changing trend of the Y tables and the new calculation methods improve the performance of the MTT method, but the latter is the primary reason. From another perspective, the results also demonstrate the rationality of selecting Y rather than the transaction coefficients when considering the changing trend of the I O structure. Scenario B: Assuming that the 1977 data are not given. In the scenario without the economic data for the target years, the MTT method is implemented. However, GRAS and Kuroda cannot be used. Note that one can first forecast gross outputs, intermediate output totals, and intermediate input totals and then apply GRAS or Kuroda methods to update I O tables, which is similar to scenario A. As indicated in Table 6, the performance of the MTT method is also satisfactory. The results demonstrate the merits of the MTT method in forecasting I O tables after the current fiscal year. In summary, based on the preceding empirical analysis, we can conclude that the MTT method has a broad scope of application. Regardless of the availability of gross outputs, intermediate output totals, and intermediate input totals for the target years, MTT provides 3 Industrial definitions: 1-Agriculture, Forestry, and Fishing; 10-Chemicals and Chemical Products; 19- Transportation and Trade; 20-Electric, Gas, and Sanitary Services; and 22-Government Enterprises.

18 16 H. WANG et al. excellent forecasting performance. However, MTT and other techniques have different areas of application. Whereas the MTT method requires at least two benchmark tables for forecast modelling, other methods require only one. 6. CONCLUSIONS AND REMARKS In this paper, we combine forecasting with a MTT to update I O tables. The key idea is to remove constraints from the I O matrix entries using the MTT. Given a series of unconstrained matrices obtained by this technique, we can update I O tables to the current fiscal year or forecast beyond the current fiscal year. To validate the performance of the MTT method, we conduct both a simulation experiment and an empirical analysis. According to the results of the simulation experiment, MTT is accurate and robust. The empirical application includes two parts. First, when we know industry values added and final demands, the MTT II method displays the best performance. Additionally, under the assumption that the industries economic data for the target years are unknown, the MTT method demonstrates its merits. Consequently, we conclude that the MTT method may be preferred for updating purposes. In the real world, when target years come after the current fiscal year, one can always obtain previous years I O tables, but it is not possible to acquire industry economic data for future target years. In such a context, one can use the MTT method to forecast I O tables for the target years by simply estimating the aggregate of all industries values added or final demands in the target years. When target years come before the current fiscal year, one can obtain past years I O tables and the industries economic data for the target years. In this case, one can interpolate more realistic I O tables using the MTT method. In addition, once the industries values added and final demands are revised, official I O tables can be revised accordingly using the MTT method. In general, the MTT is a non-iterative method to arrive at results satisfying the requirement that row totals equal column totals. The method combines forecasting with matrix transformation, while the extended GRAS and extended Kuroda combine forecasting with existing updating methods. Using precise forecasts based on historical data as a starting point will also improve other methods, but further research is needed in this area. When constructing a sequence of I O tables using the MTT method, the following points deserve special attention. Data and quality requirements. It is necessary to obtain a series of I O tables that contain at least two tables for the same industry categories. Length of time. From a practical perspective, if the duration of the I O table sequence is extremely long, one should devote particular attention to the benchmark tables of both the immediate predecessor and the immediate successor in the interpolation process while considering series trend changes within the overall context when extrapolating future I O tables. Economic structure. One should be watchful for changes in the economic structure because these changes may affect the accuracy of I O table forecasts. The prior table used. Both transaction value tables and coefficient tables can be used to establish time series models. Which table is selected in practise depends on whether the changing trend is statistically significant. The tables with more significant changing trend will be chosen.

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