Considerations in Extending Shift-Share Analysis: Note

Considerations in Extending Shift-Share Analysis: Note

KINGSLEY E. HAYNES ZACHARY B. MACHUNDA

ABSTRACT One of the limitations to the widespread use of the Esteban- Marquillas shift-share extension has been the Stokes (1974) proof of the lack of regional additivity of the Esteban-Marquillas competitive components. Since the Arcelus extension (1984) of the traditional shift-share decomposition is a continuation of the logical framework by Esteban-Marquillas (1972), the Stokes criticism would at first appear to have equal relevance to this new work. This note demonstrates that the relationship between the rate of growth of the larger region and the rates of growth of its constituent subregions is ignored in the Stokes criticism and in the subsequent critique by Beaudry and Martin (1979). By explicitly incorporating the relationship between the rate of growth of the larger region and the rates of growth of the constituent subregions, we also demonstrate that the recent shift-share extensions by Esteban-Marquillas and Arcelus share with the classical shift-share decomposition the desirable additive properties under regional disaggregation of the data.

N RECENT YEARS, a resurgence of theoretical interest in shift-share I as a method for the analysis of sources of regional growth has brought forth two classes of important contributions.' In one category, authors have turned to other methods in order to develop analytic procedures that simultaneously overcome certain inherent shortcomings of the traditional shift-share method and perform similar functions. Sakashita (1973), for ex- ample, utilizes a multi-regional growth model developed by Sakashita and

Kingsley E . Haynes is Director of the Regional Economic Development Institute, and a professor of public and environmental affairs and of geography at Indiana Uniuersity. Zachary B . Machunda is a research associatefor the Institute and a doctoral candidate in regional analysis and planning at Indiana Uniuersity. The authors wish to express their thanks to the anonymous referees for their comments and suggestions, and to say that any errors of interpretation or presentation are the responsibility of the authors.

70 GROWTH AND CHANGE, SPRING 1987

Kamoike (1973) in order to provide a theoretical justification for shift-share analysis and to derive a shift-share variant that is devoid of the asymmetry problem that arises when traditional shift-share is employed to analyze sources of growth differentials between a pair of regions. This asymmetry problem, which manifests itself as a problem of choosing appropriate weights, was first identified and thoroughly discussed by Klaassen and Paelinck (1972). Indeed, it could be argued that the contribution of Klaassen and Paelinck (1972) to some extent motivated the work of Sakashita (1973) and the later work by Theil and Gosh (1980). In their paper, Theil and Gosh (1980) turned to information theory in order to justify the use of the RAS method within the shift-share framework. The authors argue that RAS does not possess the above-mentioned asymmetry problem but performs functions that are similar to those of shift-share analysis. The shortcoming of the RAS method, as the authors correctly point out, is that this method, unlike shift-share, does not provide an exhaustive decomposition of regional employment/output levels .z

A second class of theoretical contributions, and one that is the subject of this note, refers to alternative extensions of the traditional shift-share decomposition. These extensions were intended to address those criticisms of shift-share that are different from the ones that have attracted the attention of Sakashita (1973) and Theil and Gosh (1980). In particular, the Esteban-Marquillas (1972) extension is intended to address the Rosenfeld (1959) criticism that the original shift-share formulation of the competitive effect does not really measure what it is supposed to, since its magnitude is influenced by and interwoven with factors operating outside the regional economy, such as changes in external demand for a region’s products.

More recently, the Arcelus (1984) shift-share extension, while taking in- to account the Rosenfeld criticism, is intended to overcome the criticism that the traditional shift-share formulation unrealistically assumes that all sectors operate on a national market basis. One of the barriers to the wide adoption of these new extensions was articulated by Stokes (1974) and has since then been reiterated by Richardson (1978) , Herzog and Olssen (1979) , and Dawson (1982). Examining the Esteban-Marquillas extension, Stokes (1974) pointed out that when shift-share is extended as suggested by Esteban- Marquillas (1972) it loses the property of region-to-region additivity, which holds for the traditional shift-share decomposition. According to Stokes (1974), the Esteban-Marquillas shift-share extension loses this aggregation- disaggregation symmetry precisely because the Esteban-Marquillas modified competitive effect of a larger region is not equal to the sum of the contributions of each subregion. The property of aggregation-disaggregation symmetry has always been one of the attractive qualities of the shift-share method, providing ease of validity of results and providing flexibility in ap- plication. Since the Arcelus extension is a continuation of the logical framework developed by Esteban-Marquillas (1974), the Stokes (1974) argument would at first appear to have equal relevance to this new work. The purpose of this note is to explore this additivity issue in hopes of understanding the full limits and possibilities of these new extensions.

EXTENDING SHIFT-SHARE ANALYSIS 71

A Review of Previous Evaluations for Additivity In order to be empirically useful, it is desirable that any decomposition

of growth rates be independent of the level of disaggregation of the data. In the case of shift-share analysis, such a requirement implies that the answers to the following questions need to be on the affirmative side. For a given industry, will the sum of the shift-share components for all individual sub-regions belonging to region i be equal to the shift-share component for region i? For a given region, will the shift-share component of industry i be equal to the sum of the shift-share components for all the sub-industries belonging to industry i? In the case of the traditional shift-share decomposition, the answer to the first question has been on the affirmative side; but the second question has received mixed responses. The existing testimony is that in the case of the traditional shift-share decomposition, each of the traditional shift-share components satisfies region-to-region additivity but only the reference area (national growth) effect satisfies industry-to-industry additivity (Ashby 1968, Houston 1967, and Stokes 1974). Industry-to- industry additivity is what is implied in the second question above.

In his critique of the Esteban-Marquillas (1972) modification and extension of the traditional shift-share equation, Stokes (1974) remarks that both the Esteban-Marquillas national growth effect and his industry-mix effect satisfy the region-to-region additive property. Stokes also demonstrates with a mathematical proof that the Esteban-Marquillas modified competitive effect does not satisfy the desired additive property under regional disaggregation. In the light of this result from his proof, he doubts the potential usefulness of the Esteban-Marquillas contribution to shift-share analysis.

To examine whether the Esteban-Marquillas modified competitive effect of a larger region is equal to the sum of its sub-regional contributions, Stokes seeks to find out if equality holds between the right-hand side of the following expression and its left-hand side:

where bii- is the homothetic employment of industry i in region j at time t - 1, yii is the rate of growth of employment in industry i in region j ; yio is the national rate of growth of employment of industry i; bGlt-l is the homothetic employment level of industry i of subregion 1 at time t - 1; biz,- is the homothetic employment level of industry i of subregion 2 at time t - 1; yi.l is the rate of growth of employment in industry i of subregion indexed 1; and yijz is the corresponding rate of growth for subregion indexed 2. Both subregions belong to region i .

It is argued here that utilizing expression (1) in order to examine the region-to-region additivity of the Esteban-Marquillas modified competitive effect is misleading, because the explicit relationship between the rate of growth of the larger region (yii) and the rates of growth of its subregions ( Y , ~ ~ ) and ( y i i z ) is omitted. Consequently, the formula above ignores ad-


ditional terms for evaluating the region-to-region additivity of the Esteban- Marquillas modified competitive effect.

It can be shown to be true that if we partition region j into its component sub-regions, then the rate of growth of the larger region (yii ) is equal to a weighted average of the rates of growth of the individual subregions belonging to the larger region. Stokes fails to incorporate this fact in his proof. Consequently, he identifies a wrong expression to use in order to evaluate the region-to-region additive property of the Esteban-Marquillas modified competitive effect.

In their paper, Beaudry and Martin (1979) argue that unlike the traditional competitive effect, the Esteban-Marquillas competitive effect for industry i assumes a non-zero value at the national level. On the basis of this argument, the authors recommend that the Esteban-Marquillas modification be rejected since “the possibility of having a non-zero value for the competitive position” of industry i “at the national level implies economic nonsense; which considerably reduces the ability of the method to describe and predict the regional dimensions of national growth.” (Beaudry and Mar- tin, 1979, p. 390).

It is argued in this paper that the logic used by Beaudry and Martin (1979) to analyze the region-to-region additive property of the Esteban-Marquillas competitive effect is also incorrect. By ignoring additional terms implied by regional disaggregation in their formula for evaluating the region-to- region additive property of the Esteban-Marquillas modified competitive effect, Beaudry and Martin commit the same error as Stokes. Consequent- ly, the recommendation based on their result may be unwarranted.

Additivity Properties of the Arcelus Extension

By employing the concept of homothetic employment, Arcelus (1984) is able to extend the traditional shift-share equation into a more comprehen- sive decomposition. His extension captures the growth effects of differences in the size of local (regional) markets and of interregional differences (a) in industry mixes and hence (b) in income elasticities of demand and (c) in input-output relationships. Surely, the analytical wealth that the Arcelus extension brings to shift-share analysis provides a strong case for its exploita- tion by researchers. But before potential users of the extension seize upon the opportunity to avail themselves of this extension, it would be useful if they knew whether or not the Arcelus extension also possesses meaningful additivity properties under regional disaggregation. Besides familiarity with the enhanced analytical content of the Arcelus shift-share equation, knowledge of its region-to-region additive properties is also crucial in deter- mining the relative advantages and disadvantages of the extension. However, such knowledge is still lacking. It is the purpose of this research to remedy this situation by acquainting the reader with the additivity properties of the Arcelus shift-share extension under regional disaggregation and by implication indicate the usefulness and generality of both the Esteban- Marquillas and Arcelus shift-share extensions.


Property 1: Additivity of the Arcelus national growth effect. To examine whether the property stated above is held by the Arcelus national growth effect, we need to show that both its homothetic part and its differential component can be consistently disaggregated into their respective subregional elements.

Suppose we partition region j into n subregions, i , , i z , . . . j k , , . ., i,. On the basis of the foregoing assumption, it can be demonstrated that the Arcelus homothetic national growth of region i is equal to a simple sum of the homothetic national growth effects of all subregions belonging to region i . This result is formally expressed by the following equation:

where b,',, is the homothetic level of employment in industry i of subregion k; you is the national rate of growth of employment in all industries; and b( is the homothetic employment level of industry i in region

Furthermore, the differential component of the Arcelus national growth effect of a larger region permits the following additive property under regional disaggregation of the given data:

i .

where bii is the effective employment level of industry i in region j and the rest of the notations are as defined above.

Property 2: Additivity of the Arcelus national industry mix effect. Equa- tion (4) below indicates that the homothetic component of the Arcelus national industry mix effect of industry i in region j is equal to the sum of the homothetic national industry mix effects of all subregions belonging to region i :

Given equation (4), it is a relatively easy matter to show that the differential component of the Arcelus national industry mix of industry i in region j is always equal to a simple sum of each differential component of the Arcelus national industry mix effect in industry i of subregion k .

Property 3: Additivity of the Arcelus Regional Growth Effect. To examine whether or not the Arcelus regional growth effect possesses the property of region-to-region additivity, we first focus our attention on the homothetic component of the Arcelus regional growth effect and then on the differential component. One of the implications of regional disaggregation that is


relevant to the Arcelus regional growth effect is that the rate of growth of total employment for all industries in region j is equal to a weighted average of the rates of growth of total employment for all industries in all subregions of region i . Mathematically, this implication is

n

where yoi is the rate of growth of employment in all industries in region j; yojk is the rate of growth of employment in all industries in subregion k or region j ; and the term f0ik denotes the ratio of employment in all industries in subregion j k to employment in all industries in region j .

Using equation (5 ) we obtain the following result: n

Equation (6) indicates that the homothetic component of the Arcelus regional growth effect of industry i in region j is always equal to the sum of the homothetic components of all subregions of j for industry i .

Employing the results in equations (5) and (6), it can be demonstrated that

where the term b;k denotes that expected employment in industry i of subregion k given that the industry structure of subregion k is identical to that of the region to which it belongs.

The result in equation (7) indicates that the differential component of the regional growth effect of region j can be consistently disaggregated into a sum of its subregional growth effects. Furthermore, we notice in equation (7) that each of the subregional growth effects reflects the influence of the regional industry structure independent of the influence of the national industry structure.

Suppose we consider the following inequality:

( 8 ) (bij - b;) (TOj - 700) f 2 (hik - qk) (YO& - Too) k = l

The inequality of (8) holds as long as subregion k is more (or less) specialized in industry i than in region i . In other words, the inequality in (8) holds only when the location quotient of subregion k differs from one.

That the inequality in (8) holds only when the location quotient of subregion k differs from one can easily be seen when equation (7) is rewritten as


where l i j k is the location quotient of subregion k. This location quotient is computed by using the region as the reference area.

It may be argued that the result in (8) provides a reason for doubting the argument presented in this paper that the differential component of the Arcelus regional growth effect is equal to the sum of its subregional components. But a closer look at expression (8) indicates that the right-hand side of the expression is obtained when it is assumed that some of the resulting implications of regional disaggregation are: (a) the effective employment level of industry i in region j is equal to the sum of the effective employment level of industry i in subregion k, for k = 1, 2, . . . , n; (b) the rate of growth of employment in all industries of region j is equal to the weighted sum of the rate of growth of employment in all industries of subregion k, for k = 1, . . . , n. The weights are (fii,, k = 1, . . . , n). Each of these weights denotes the employment of industry i of subregion k as a fraction of employment in region j in the same industry. It is argued in this note that this second implication is equivalent to assuming that each subregion is as specialized in industry i as is the region. Such an assumption is without theoretical justification and it also constitutes a statistical error. The source of the error is the weighting structure used to relate the rate of growth of employment in all industries in region j to the corresponding growth rates for all subregions of i . The appropriate weighting structure to use is, as we have indicated above, f 0 j k , for k = 1, . . , , n instead of fii,. Hence, the inequality presented in (8) does not provide any valid grounds for doubting the argument presented in this paper that the differential component of the Arcelus regional growth effect of region j can be consistently disaggregated into a sum of its subregional contributions.

Property 4: Additivity of the Arcelus regional industry mix effect. The homothetic component of the Arcelus regional industry mix effect may be rewritten as

It was shown above in equation (6) that the second term on the right-hand side of equation (10) can be consistently disaggregated so that the homothetic component of the Arcelus regional growth effect of region j is equal to the sum of the subregional contributions.

The remaining term on the right-hand side of equation (10) is indeed the Esteban-Marquillas modified competitive effect. To show that this remaining term can also be consistently disaggregated we proceed as follows. If we partition region j into n subregions, then the rate of growth of employment in industry i of region j is always equal to a weighted average of the rates of growth of employment in industry i for all constituent subregions. Employing this relationship yields the following result:


with

(12) Z i i k = (biik/b;k).

It is clear from the result in equation (11) that the Esteban-Marquillas modified competitive effect, itself part of the Arcelus extension, can also be consistently disaggregated into its subregional contributions. Further- more, the result in equation (11) indicates that the Esteban-Marquillas modified competitive effect of industry i of region j is always equal to a weighted sum of the subregional homothetic competitive effects, the weights being the location quotients of the constituent subregions.

If, in equation (l), we substitute f o k k for fiik 7 we obtain the inequality given below, as long as fo ik + f i i k for all k:

As it was argued above, the inequality in (13) is also unjustifiable, since it is based on the wrong use of the weighting structure implied by regional disaggregation. Hence, the inequality in (13) does not provide any valid grounds for doubting the argument that the Esteban-Marquillas modified competitive effect, itself part of the Arcelus extension, can also be consistently disaggregated into its subregional contributions.3

From the discussion given above it can be concluded that the homothetic component of the regional industry mix effect of a larger region is also equal to the sum of the subregional contributions. This result implies that the differential component of the industry mix effect of a larger region is always equal to the sum of the differential industry mix effects of all subregions of the larger region. Indeed, it can be shown that the equality between the differential regional industry mix effect of sector i of region i and a weighted sum of differential subregional industry mix effects of all subregions of j can be represented by the following equation:

Furthermore, equation (14) indicates that the differential component of the Arcelus regional industry mix effect is actually the difference between the Esteban-Marquillas allocation effect and the Arcelus differential component of the regional growth effect. Equation (14) also indicates that the


Esteban-Marquillas allocation effect of a larger region is always equal to the sum of the contributions of all subregions.

Conclusion One of the limitations to the widespread use of the Esteban-Marquillas

shift-share extension was raised by Stokes (1974), who, in his contribution, concluded that the usefulness of the Esteban-Marquillas extension was limited by the failure of the Esteban-Marquillas competitive effect to possess the region-to-region additivity property. This additivity property has been one of the attractive and highly desirable qualities of the traditional shift- share method. Since the recent Arcelus extension of the traditional shift- share decomposition is a continuation of the logical framework developed by Esteban-Marquillas (1972), the Stokes (1974) criticism would at first appear to have equal relevance to this new work. The purpose of this paper was to explore the additivity issue with respect to the Arcelus extension in the hopes of understanding the full limits and possibilities of the important extensions of Esteban-Marquillas (1972) and Arcelus (1984). Our main result is that contrary to prior expectations, the Arcelus shift-share extension satisfies the property of aggregation-disaggregation symmetry, which holds for the traditional shift-share method. It has been shown that when the full implications of regional disaggregation are taken into account, each of the Arcelus shift-share components of the larger region is always equal to the sum of the contributions of all subregions belonging to the larger region.

It has also been demonstrated that when the implication of regional disaggregation on the relationship between the rate of growth of the larger region and the growth rates of the constituent subregions is ignored, it is possible to commit a mathematical and interpretive error in an effort to evaluate the additive properties of the Arcelus extension of the traditional competitive effect. It is the ignoring of this issue that caused Stokes (1974) and Beaudry and Martin (1979) to raise their concern about the additive properties of the Esteban-Marquillas shift-share extension. On the basis of our analysis in this paper, the recent shift-share extensions by Esteban-Marquillas (1972) and Arcelus (1984) share with the traditional shift-share method the desirable property of aggregation-disaggregation symmetry. Since these recent extensions also do have an analytical superiority over the traditional shift-share equation, they prove to be very important contributions to shift-share analysis. Accordingly, it is strongly recommended that the analytical powers these extensions bring to shift-share analysis should be judiciously exploited by researchers.

N O T E S

1. For recent reviews of the use of shift-share in forecasting and in policy analysis, see Stevens and Moore (1980), Bartels et al. (1982), Berzeg (1984), and Tervo and Okko (1983).


2. The RAS method also ignores the contribution of the interaction between industry effects and region effects. Furthermore, Berzeg (1984) has demonstrated that the Theil-Gosh (1980) approach is, except for differences in loss function, identical to a two-way ANOVA procedure without interactions.

3. It is on the basis of this arbitrary and theoretically unjustified argument that Stokes (1974), challenges the Esteban-Marquillas extension and by implication the Arcelus advances in shift-share analysis applications.

R E F E R E N C E S

Arcelus, F. J . 1984. An extension of Shift-share analysis. Growth and Change 15( 1):3-8.

Ashby, L. D. 1968. The shift-share analysis: A reply. Southern Economic Journal 34(3):423-5.

Bartels, C. P. A., et al. 1982. Estimating the impact of regional policy: A review of applied research methods. Regional Science and Urban Economics 12( 1):3-41.

Beaudry, R., and F. Martin. 1979. Shift-share analysis revisited: The allocation effect and the stability of regional structure, a comment. Journal of Regional Science 19(3):389-90.

Berzeg, K . 1984. A note on statistical approaches to shift-share analysis. Journal of Regional Science 24(2):277-85.

Dawson, J . 1982. Shift-share analysis: A bibliographic review of technique and applications. Monticello, IL: Vance Bibliographies.

Esteban-Marquillas, J . M. 1972. A reinterpretation of shift-share analysis. Regional and Urban Economics 2(3):249-55.

Herzog, H. W. Jr., and R. J . Olsen. 1979. Shift-share analysis revisited: The allocation effect and the stability of regional structure, a reply. Journal of Regional Science 19(3):393-5.

Houston, D. B. 1967. The shift-share analysis of regional growth: A critique. Southern Economic Journal 33(4):577-81.

Klaassen, L . H., and J . H. P. Paelinck. 1972. Asymmetry in shift-share analysis. Regional Science and Urban Economics 2(3):256-61.

Richardson, H. W. 1978. The state of regional economics: A survey article. Znter- national Regional Science Review 3(1): 1-48.

Rosenfeld, F. 1959. Commentaire l’expose de M. Dunn. Economie AppliquBe 4:531-4,

Sakashita, N. 1973. An axiomatic approach to shift-share analysis. Regional and Urban Economics 3(3):263-72.

Sakashita, N . , and 0. Karnoike. 1973. National growth and regional income inequality: A consistent model. International Economic Reuiew 14(2):373-82.

Stevens, B. H., and C. L. Moore. 1980. A critical review of the literature on shift- share as a forecasting technique. Journal of Regional Science 20(4):419-38.

Stokes, H. K. 1974. Shift-share once again. Regional and Urban Economics 4:57-60. Tervo, H., and Okko. 1983. A note of shift-share as a method of estimating the

employment effects of regional economic policy. Journal of Regional Science 23(1):115-21.

Theil, H . , and R. Gosh. 1980. A comparison of shift-share and the RAS adjust- ment. Regional Science and Urban Economics 10: 175-80.

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Considerations in Extending Shift-Share Analysis: Note