their relative success in estimating output multipliers. In doing so, we use each formula to regionalize the survey-based national inputoutput table for 1995 (Statistics Finland, 2000). We then estimate the type I output multiplier for each sector in each region. Finally, we compare these LQ-based estimates with the survey-based estimates for each region (ibid.). Our focus on output multipliers is motivated by their importance in regional analysis, along with the fact that many earlier studies have attempted to derive satisfactory estimates of such multipliers. This multiplier analysis will be complemented later in the paper by a detailed consideration of regional input coefficients. We now need to consider possible ways of assessing the accuracy of the estimated sectoral multipliers in a given region. The following measures will be examined here: μ1 = (100/n) Σj / mj (16) μ2 = (100/n) Σj (17) μ2* = 100 (18) μ3 = 100 Σj qj / mj (19) μ4 = 100 (20) μ5 = (1/n) Σj || / mj (21) sd = [(1/n) Σj {( || / mj ) μ5}2]0.5 (22) where is the estimated type I output multiplier for sector j (column sum of the LQ-based Leontief inverse matrix) in a given region, mj is the corresponding survey-based multiplier, qj is the proportion of regional output produced in sector j and n = 37 is the number of sectors. μ1 is a statistic that has been used in many earlier studies, thereby facilitating comparisons. Another useful characteristic of μ1 is its ability to measure bias. This is the main reason why μ1 is our preferred measure. Even so, it be argued that the inclusion of the unitary components (direct effects) in the formula tends to exaggerate the apparent precision of the estimated multipliers, as only the indirect effects need to be estimated.
satisfactory way of assessing the accuracy of estimated multipliers and coefficients (see, for example, Bonfiglio and Chelli, 2008; Sawyer and Miller, 1983). What is more, the negligible difference of 0.053 in the mean values of μ5 for δ = 0.25 and δ = 0.3 may mask some useful information. Hence a set of disaggregated results is presented in Table 4. Table 4 near here Table 4 strongly confirms the earlier finding that the optimal δ tends to rise with regional size. Indeed, five of the largest regions require δ = 0.35, whereas two of the smallest require δ = 0.15. Furthermore, compared with Table 3, there has been a shift towards the right in the distribution of results, with 14 of the 20 regions now clustered in the range 0.3 ± 0.05. However, only three regions are located in the centre of this range. One might ask whether this rightward shift is explicable in terms of the characteristics of the FLQ. To explore this issue, we examined the sectoral distribution of errors for each region when μ1 ≈ 0. We then increased δ by 0.025 to create a new distribution of errors. A fundamental property of the FLQ is that a rise in the value of δ decreases the extent to which any multipliers are overestimated, while increasing the amount of any underestimation. When μ1 is used as the criterion, overestimates can be offset by underestimates but this is not possible with μ5, as all errors of a given size are treated equally. We identified nine regions where the rise in the value of δ decreased the degree of overestimation by considerably more than it increased the degree of underestimation. For these regions, μ5 typically reached a minimum at a somewhat higher δ than it did for μ1. On the other hand, there were only two regions where there was a noticeable movement in the opposite direction. The overall shift rightwards in the distribution of results is, therefore, unsurprising. In the light of the above results, should we accept 0.3 as the best single value of δ? We would suggest not. Our justification for this view is that raising δ from 0.25 to 0.3 would introduce bias that was not previously there, yet yield only a minimal gain in terms of
inputoutput tables for 1995, which identify 37 separate sectors, as a benchmark to evaluate the relative performance of the FLQ and other LQ-based adjustment formulae in estimating input coefficients and type I sectoral output multipliers. We employed a wide range of statistical criteria to assess the results. As expected, the FLQ outperformed the conventional LQs by a wide margin, regardless of which criterion was used. Moreover, an analysis using Theil’s index of inequality revealed that the FLQ produced a much closer match between the estimated and survey-based input coefficients. This was true irrespective of whether one measured this match in terms of bias, variance or covariance. Using the mean proportional difference, μ1, as the criterion, we found that δ = 0.25 was the best single value for estimating multipliers. This criterion has the desirable property that it minimizes bias. When averaged across regions, the unweighted mean value of μ1 was 0.4%, compared with 1.9% for δ = 0.3. By contrast, when the results were weighted by size of region, δ = 0.3 gave the best outcome. There was a discernible tendency for the optimal value of δ to rise with regional size. It is also worth noting that the use of absolute values suggested that 0.3 rather than 0.25 might be the best single value of δ. Nevertheless, when averaged across regions, the outcomes for these two values of δ were very similar indeed. It is interesting that our findings are broadly consistent with of Bonfiglio and Chelli (2008), who used a Monte Carlo approach to generate 400,000 sectoral output multipliers. Their study indicated an optimal δ ≈ 0.3. On the other hand, using Scottish data, Flegg and Webber (2000) found that the FLQ with δ ≈ 0.15 yielded µ1 ≈ 0. This relatively low value of δ can probably be explained by the fact that almost half of the Scottish survey-based coefficients were larger than those in the parent table that was being used in the simulation. Although the FLQ was demonstrably better than the conventional LQs, its performance was not as good as we have wished. For instance, the unweighted mean absolute proportional error, with δ = 0.3, was 8.1% for multipliers. The weighted mean was 7.6%. As
0.008 of getting 0.6 ≤ δ ≤ 0.7 (ibid., Table 1). 39. Bonfiglio (2009) presents an alternative regression model for determining the value of δ. He includes variables to capture regional size (RSRP) and the propensity to import from other regions (PROP), as we do, but does not include a measure of the divergence between regional and national proportions of intermediate inputs. Unlike us, he finds an inverse relationship between the optimal δ and regional size. He also defines the import propensity differently. 40. Another way of adjusting for differences between regional and national proportions of intermediate inputs would be to apply Round’s ‘fabrication’ adjustment, whereby each column of the national coefficient matrix is scaled prior to applying LQs (Miller and Blair, 2009, pp. 356357, 361362; Sawyer and Miller, 1983). However, this adjustment presupposes that α = 0 in equation (30). Our preferred approach, which we believe to be both simpler and more comprehensive, is to use the regression model (23) to help determine an appropriate value of δ. 41. This statement is based on the a priori arguments presented in the Introduction. We intend to carry out an empirical analysis to substantiate it and to measure the extent of any improvements in accuracy. REFERENCES Bonfiglio A. (2009) On the parameterization of techniques for representing regional economic structures, Economic Systems Research 21, 115127. Bonfiglio A. and Chelli F. (2008) Assessing the behaviour of non-survey methods for constructing regional inputoutput tables through Carlo simulation, Economic Systems Research 20, 243258.