# A Statistical Test of Single Firm Market Power.

1. Introduction

There has been a huge number of studies of the profitability-concentration relationship using cross-section data, especially in the United States (US) (for a fairly recent survey, see Schmalensee, 1989). Several of these studies have searched for a level of seller concentration, associated with the emergence of oligopoly, at which competition may first be significantly lessened and profitability enhanced (see, for example, Meehan and Duchesneau, 1973; Dalton and Penn, 1976; White, 1976; Geroski, 1981; and Bradburd and Over, 1982). The notion of a critical concentration ratio arose from the seminal work of Bain (1951), who found that a linear fit to his data was inferior to one based on segmentation at an eight-firm concentration ratio of 70 per cent. Subsequent US studies, with one or two exceptions,(1) typically have found a critical level for the four-firm concentration ratio of between 45 and 60 per cent. This supports Chamberlain's (1956) theory that as the number of firms in an industry falls, the recognition at some point of mutual interdependence causes an abrupt change from competitive behaviour to oligopolistic co-ordination. Geroski (1988, p.171) has also argued strongly that there is little evidence to suggest that linear associations best describe profitability-concentration data, and that modifications like the critical concentration ratio often improve the fit.

Apart from theoretical expectations that competition will be lessened with interdependence, the oligopoly focus of overseas research probably reflects the ready availability in the US of manufacturing industry census data with four- and eight-firm concentration ratios, which facilitate testing for the group exercise of market power. However, in the search for the critical concentration level, the limitations of using such arbitrary ratios as measures of the distribution of firm sizes was emphasised by Kwoka (1979). Using disaggregated census-like data for the US, he found that industry profitability was unaffected until market share by the leading one or two firms reached 25-30 per cent, and that a sufficiently large third-ranked firm had a depressing effect on margins. He concluded that "three-firm co-ordination problems are so severe as to make a large third firm more likely a rival", although this finding has been questioned by Mueller and Greer (1984) (see also Kwoka, 1984). In a similar vein, a recent New Zealand (NZ) study based on manufacturing industry census data by Pickford and Wai (1995) found that the profitability of the first-ranked firm, as measured by its price-cost margin, was inter alia positively related to its own market share and relative efficiency, and negatively related to the market share of number two firms.

These results, by directing attention away from group market power towards the market power exercised by leading firms, raise a further question: is there a critical market share of the leading firm in a market, beyond which its profit rate expands more rapidly with rising share? The purpose of the present study is to test this issue empirically. In so doing, it breaks new ground in various respects. Firstly, it makes use of an efficient and novel computation method to provide tests of a breakpoint in the regression relationship with price-cost margin for the largest firm which do not assume that the breakpoint is known. In the only previous New Zealand study, Ratnayake (1996) found several critical four-firm concentration ratios when using a single equation model, but none when a simultaneous equations model was used with the same data set. In the latter case the concentration variable was both insignificant and negative. However, Ratnayake's t statistics evaluated to determine a break at various concentration ratios (unlike the methods used in this paper) are conditional on the breakpoint being known, and are thus inclined to give tests which appear significant when they are not. This may be the reason for his "perverse results" when a single equation model is compared with two stage least squares. It is also the reason for the choice here of cusum and cusum of square tests in preference to Chow F statistics or its equivalent t test.

Secondly, no studies seem to have searched for a leading firm critical market share (although Kwoka's technique had the potential to reveal one). Economic theory provides relatively little guidance as to the threshold for single firm market power, and economists disagree over the minimum market share required. Rule of thumb figures range from 40 per cent by Stigler (1947) and Geroski (1986), through 50 per cent by Shepherd (1982), to persistence of a 60 per cent level by Williamson (1972). Finally, critical concentration studies typically have not incorporated efficiency effects as independent variables, yet these have an important impact on profitability in NZ manufacturing industries (Ratnayake, 1994; Pickford and Wai, 1995), and in US profit-concentration studies where such variables have been included (e.g., Chappell and Cottle, 1985; Martin, 1988).(2) Their omission could have led to serious specification errors in earlier studies.

In the balance of the paper we proceed as follows. The model and variables are outlined in section 2. The statistical tests employed to search for a critical market share are discussed in section 3. The results are given in section 4, and in the final section we draw together our conclusions.

2. The Model and Data

The model developed by Pickford and Wai (1995) is used as the basis for the empirical testing. They estimated the following equation by OLS multiple regression analysis for number one firms in a cross-section of manufacturing industries:

(1) PCM1 = [Alpha] + [[Beta].sub.1]RP1 + [[Beta].sub.2]S1 + [[Beta].sub.3]S2 + [[Beta].sub.4]S4 + [[Beta].sub.5]PCM2 + [[Beta].sub.6]PCM3N + [[Beta].sub.7]ASR + [[Beta].sub.8]KSR + [[Beta].sub.9]MESM + [[Beta].sub.10]MGR + [[Beta].sub.11]REG + [Epsilon]

where the variables are defined in columns (1)-(3) of Table 1; [Alpha], [[Beta].sub.1] ... [[Beta].sub.11] are coefficients to be estimated; and [Epsilon] is an error term with zero mean and constant variance.(3) It was hypothesised that the size of the price-cost margin (PCM1) would be influenced by three groups of factors: the firm's relative efficiency (RP1); various market power influences (S1, S2, S4, PCM2, and PCM3N); and other industry-wide elements (ASR, KSR, MESM, MGR, and REG).(4)

Note: superscripts a, b, and c denote significance at the 1, 5, and 10 per cent levels respectively.

PCM1 is expected to be positively associated with the firm's relative efficiency (RP1), and with its own market share (S1), and negatively associated with the market shares of rival number two firms (S2) and of smaller firms (proxied by S4, to avoid multicollinearity problems). PCM1 is also hypothesised to be influenced by strategic groups within the industry. The theory holds (Porter, 1979) that the market power exercised by one group, as reflected in the PCM, should be greater (all else remaining the same) the more that market power is enjoyed by other groups, leading to the inference that PCM1 should be positively associated with both PCM2 and PCM3N.(5)

The final group of variables control for industry-wide differences between industries. Advertising intensity (ASR) is generally considered to give rise to an entry barrier, and thus to be positively linked with PCM1. The capital-sales ratio (KSR) is included to control for variations in PCMs caused by differing capital intensities in production, with a positive association expected. The relative plant scale variable (MESM) can be seen either as measuring the scale economy entry barrier, or as indicating scale-induced efficiency, but in both cases a positive link is expected. Market growth (MGR) was included to capture any inflation of profit margins resulting from disequilibrium states caused by a fast growth in demand, faster growth being linked with higher margins. Finally, a regional market dummy (REG) was used to capture the geographical dispersion of the activities of the two leading firms, such that dispersion was associated with enhanced market power and hence higher PCMs.

The results of the model are shown in column (4) of Table 1. They indicate that all of the relative efficiency and market power variables have the expected signs, and are statistically significant at the five per cent level or better, with the exception of S4, which has the correct sign but is insignificant. The five industry-wide variables are generally not significant statistically, except for MESM, which has a strong, positive influence on PCM1. The REG variable is significant at almost the five percent level but has the wrong sign.(6) Pickford and Wai (1995) concluded that both market power and efficiency factors are important in explaining inter-industry variations in leading firm profitability in New Zealand manufacturing industries in the late 1970s.

A potential problem with using S1 and S2 together in the regressions follows from the inequality S1 [is greater than or equal to] S2.(7) This implies that a plot of S2 against S1 should reveal an inverted `U'-shape relationship, since the value of S2 will tend to be constrained when that of S1 is both low and high. A regression incorporating a squared S1 term confirms this:(8)

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This opens the possibility that the negative coefficient of the S2 variable in column (4) of Table 1 is a statistical artefact without economic significance. However, the segmentation of the data below on the basis of the S1 values reveals that the S2 regression coefficient is often significantly negative, even for those regression equations involving only low values of S1 where on definitional grounds a positive relationship might be expected. The restraining influence on the leading firm's profitability of competition from the second-ranked firm thereby seems to be supported.

The question now is whether there may be a discontinuity in the profitability-market share association which relates to single firm market power.

3. The Methodology

Various methods have been used to test for discontinuities in regression data with uncorrelated errors. One somewhat discredited technique is the 'differential slope-intercept test' (Dalton and Penn, 1976). This method assesses whether there has been a change in intercept and/or slope, where in this case slope changes are limited to changes in the regression coefficient for market share (S1), and changes in intercept occur at some level of S1. The point at which the regression is allowed to change is varied. The significance level for each change point is usually assessed as if the breakpoint had been known a priori and is consequently not conservative. In other words, the nominal significance level understates the actual significance level, if the test statistic is treated as if the change point were known in advance rather than as being that for the maximum of a sequence of possible tests, and more apparently significant results are found than are actually significant.

A generalisation of this technique is a particular type of switching regression, which allows all the coefficients in the regression to alter at the change point (Quandt, 1982). Such a procedure has been used previously in the context of concentration ratio studies, for example by Uri and Coate (1987), under the assumption that errors within and between regressions are uncorrelated (also see Meehan and Duchesneau, 1973; Rhoades and Cleaver, 1973). Again the change point is not known a priori, which complicates significance testing, whether a Chow test (Chow, 1960; Gujurati, 1978 pp. 295-298), or a likelihood ratio test (Quandt, 1958; White, 1976), is used. For the Chow test, the test, being one in a sequence, is for the maximum of a number of correlated F statistics, while a transformation of the likelihood ratio yields a sequence of correlated chi-square random variables. In both cases the actual distribution of the maximum is complicated (see Feder, 1975 for a discussion of the likelihood ratio case); there is some suggestion (Freeman, 1983) that the Pearson type III (i.e., a gamma distribution) serves as an approximation to the distribution of the negative log transformed likelihood ratio. A Bayesian technique is also available (see Kim, 1991). Note that if the possible change point had been set, rather than being allowed to vary, the Chow test would have an F distribution, and the transformed likelihood ratio a chi-square distribution, under the null hypothesis of no change.

An alternative set of tests have been developed which make use of recursive residuals (Brown, Durbin and Evans, 1975) to derive cusum and cusum of square tests (see Uri and Coate, 1987 for an application). These techniques can be extended to the case where regression errors are correlated, and additional data beyond a change point are added not one at a time but simultaneously (Haslett, 1985). In the case of the PCM data described in the previous section, regression errors are taken to be uncorrelated, but the device of first fitting a regression to the minimum number of ordered data points (the ordering being by S1) and adding the remaining data points in one step, if applied separately to data ordered by both ascending and descending values of S1, yields a computationally efficient device for Chow tests, likelihood ratio tests, and for backward and forward cusum tests and cusum of squares tests. For the uncorrelated error case, the benefits are purely computational; for the correlated error case with known covariance matrix, the method provides an extension to current methods of studying critical concentration rates. A further brief explanation is given later in this section.

The updating procedure is given in detail in Haslett (1985), and is summarised below for the particular uncorrelated error case relevant here. Suppose an initial regression has been fitted to a subset of the data

(3) [y.sub.1] = [X.sub.1] [Beta] + [e.sub.1]

where: [Beta] is a p x 1 vector of parameters possibly including an intercept; [y.sub.l] is p x 1 vector of data points; and [X.sub.1] is a full rank p x p matrix of explanatory variables, including the p smallest or p largest values of S1. (Note that increasing or decreasing S1 determine natural ordering of the data for assessing a possible break point at some particular value of S1.)

It follows that

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the superscript T denotes matrix transpose and -1 a matrix inverse, provides an initial estimate of [Beta], assuming the components of [e.sub.1] are uncorrelated and of equal variance.

Then if a further grouping of (n-p) data points are added, the updated regression parameter estimates are given by

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with residual sum of squares (RSS) given by

RSS = [w.sup.T]w

where w is a set of (n-p) recursive residuals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[I.sub.n-p] is the (n-p) x (n-p) identity matrix

and the full model is now

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and var(e) = [[Sigma].sup.2] [I.sub.n] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [I.sub.n] is the n x n identity matrix.

Consider the Choleski decomposition, which decomposes a positive definite matrix A into A = [LL.sup.L] where L is lower triangular (see Seber, 1984). Then if [([C.sub.1] + [C.sub.2).sup.-1/2] is determined by taking the Choleski decomposition of ([C.sub.1] + [C.sub.2]) followed by inversion it will be lower triangular. Then the successive recursive residuals in w, taken in order, correspond to the recursive residuals that would have been obtained if the additional data points had been added one at a time. Further, the residual sum of squares can be decomposed in the same way, as can the updates to [[Beta].sub.1] (provided the n-p individual columns of [[G.sub.2][[([C.sub.1]+ [C.sub.2]).sup.-1/2]].sup.T], when multiplied by the corresponding element of w, are added sequentially).

Although [[Sigma].sup.2] is unknown, it can be estimated in the usual way from the n data points as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This procedure can be applied to the data ordered by S1 in both ascending and descending orders. To each break point i in the overall regression, there then corresponds two subset regressions, the first using i data points in the forward direction, the second using (n-i) data points in the backward direction.

Taken together these two regressions provide the information necessary to calculate the Chow ([F.sub.c]) and likelihood ratio ([Lambda]) statistics for breakpoint i.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[RSS.sub.f,i] = residual sum of squares for regression using the i data points corresponding to the i smallest values of S1.

[RSS.sub.b,n-i] = residual sum of squares for the regression for the remaining n-i data points ordered by descending S1.

[RSS.sub.n] residual sum of squares for the regression = applied to all n data points. = [RSS.sub.f,n] = [RSS.sub.b,n]

As the change point is varied from following data point i to following data point i + 1, the updating formula for the forward regression gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for the backward direction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [W.sub.f, i+1-p] is the [(i+1-p).sup.th] component of w for the forward regression, [W.sub.b, n-i-p] is the [(n-i-p).sup.th] component of rev(w) for the forward regression, and where rev(w) is w with its components in reverse order. Note where rev(w) is w with its components in reverse order. Note that w for the forward and backward regressions are not equivalent.

With some modifications detailed in Haslett (1985) the method can be extended to the case of correlation between data points within but not between subset regressions (where the subsets are formed at each breakpoint i), and to the case where data are correlated both within and between subset regressions. The first of these corresponds to a block diagonal correlation structure that changes with each i. The second requires a preliminary Choleski or possibly a spectral or eigen decomposition of the covariance matrix for all n data points.

The updating methodology can be further extended, even in the correlated error cases, to testing for a statistically significant change point in a stochastically varying parameter vector (see Haslett, 1996 for the theoretical details).

4. The Results

The computation devices explained in section 3 were programmed in PROC IML (interactive matrix language) in SAS. With the data ordered by S1, a `minimal' regression was fitted to the first twelve data points with the lowest values of S1 (corresponding to the number of regression parameters including an intercept), and the remaining 80 data points were added in a single step to produce 80 `forward' recursive residuals. The data were then reversed, a `minimal' regression fitted to the twelve data points with the largest values of S1, and the remaining 80 data points added in a single step to produce 80 `backward' recursive residuals. Chow tests, likelihood ratio tests, and cusum and cusum of square tests for forward and backward data, were then carried out for each possible breakpoint (from after the twelfth, to after the eightieth data point). Note that because of the number of points required for the minimal regression at each end of the data, only 0.1070 [is less than] S1 [is less than] 0.5277 was testable.

Preliminary checking for a breakpoint included visual inspection of the graphs of -2log [Lambda], [F.sub.c], and the cusum of squares tests for the reversed data. These graphs, shown in part in Figures 1 to 4, revealed similar features, namely evidence of a breakpoint at S1 = 0.1671, following the 35th data point. Note here that -2log [Lambda] is minus two times the natural logarithm of [Lambda], the likelihood ratio statistic, and that on the graphs the first twelve data points and/or the last twelve data points from the regressions produce nothing to graph since they are necessary for the `minimal fit' (i.e, where the number of data points in the regression equals the number of parameters including an intercept).

[Figures 1-4 ILLUSTRATION OMITTED]

Formal tests found that the backward cusum of squares test was statistically significant at [Alpha] = 0.02 at this change point. The distribution of -2log [Lambda], and [F.sub.c] for varying breakpoints being unknown, the significance levels could not be accurately assessed. Although the tests are not conservative, both statistics were nominally significant at [Alpha] [is less than] 0.001 if -2log [Lambda] was treated as having a [[Chi].sup.2] distribution, and [F.sub.c] as having a F distribution (each under the null hypothesis of no breakpoint at S1 = 0.1671, with the possible breakpoint specified a priori).

For the breakpoint S1 = 0.1671 the regression coefficients for each part of the data are given in Table 2. The changes in the parameter estimates are most pronounced for REG (regional markets) and PCM3N (price-cost margin of all firms except the largest two). Given the breakpoint, these changes are statistically significant at [Alpha] [is less than] 0.01. Changes in the regression coefficients for MGR (market growth) and PCM2 (price-cost margin of number two firms) are marginally significant. Note that there is some correlation between the parameter estimates for a given breakpoint so the tests for parameter differences are not statistically independent.

Table 2. Regression Coefficients for Subset Regressions at the Most Significant Breakpoint

Note : significance levels for difference tests the significance of the difference between parameter estimates for each variable given a change point occurring in the range 0.1671 [is less than or equal to] S1 [is less than or equal to] 0.1676. This corresponds to a breakpoint following the 35th of the 92 data points ordered by S1.

A comparison of the sub-set regressions reveals that some variables which are significant in the regression for all 92 data points are not significant in the segmented regressions. This may reflect differential effects of particular variables in the two segments and the correlation between parameter estimates. Overall, however, the model provides a better fit to the data for S1 [is greater than or equal to] 0.1671, which covers those Census industries where single firm market power appears to exist. The efficiency variables - the leading firm's relative productivity index (RP1) and the industry relative scale economy measure (MESM) - are statistically more significant in explaining the leading firm's profitability (PCM1). Of the market power variables, S1 has a strong positive influence, and S2 a strong negative influence, on PCM1. The presence of mobility barriers, inferred from the statistical significance of PCM2 and PCM3N, also contribute positively to PCM 1. In the sub-set regression for S1 [is greater than or equal to] 0.1671, the results suggest that the profitability of leading firms is the product of both efficiency and of market power effects.

There was also the possibility of a second change point at S1 = 0.2360 when this was fitted without the first change point also being fitted. However, neither of the cusum or cusum of squares tests for either the forward or backward regressions were significant at [Alpha] = 0.05 at this possible change point. The non-conservative Chow and likelihood ratio tests were, however, significant at the nominal [Alpha] = 0.05 level (although these tests assume an a priori change point to be tested and consequently have nominal significance levels). The evidence for the breakpoint at S1 = 0.2360 is consequently much weaker than that for a breakpoint at S1 = 0.1671. Evidence for other breakpoints was weaker still.

As a final technical point, the likelihood ratio statistic was unstable for change points near either end of the data. This is a direct consequence of the divisor for variance estimates being i or n-i, rather than the number of degrees of freedom i-p or n-i-p. This is discussed further in Phillips (1991).

5. Conclusions

The purpose of this paper has been to use a statistical model to test for a structural break in the profitability-concentration relationship associated with single firm market power. This is, as far as we know, the first time that this issue has been tested empirically. Moreover, unlike previous discontinuity studies focussing on oligopoly market power, our model has the advantage of incorporating efficiency explanations, as well as market power explanations, of profitability. In our view, the inclusion of the former seems desirable in the oligopoly focus, and essential in the single firm one, especially in New Zealand where the combination of a very small economy by OECD standards and of scale economies is likely to promote concentrated markets.

While various techniques have been used to test for discontinuities in regression data with uncorrelated errors, in the testing of the significance level for each breakpoint, some suffer from the test being conducted as if the change point were known a priori. As a result, more apparently significant effects are found than are warranted. Our approach is based on the use of recursive residuals to derive cusum and cusum of square tests following Brown, Durbin and Evans (1975), and the extension proposed by Haslett (1985) for the case where regression errors are correlated, and where additional data beyond a change point are added simultaneously rather than one at a time. While we take the regression errors to be uncorrelated with the PCM data, the approach, when applied separately to data ordered both by ascending and descending values of the market share of the leading firm (S1), is a computationally efficient way of conducting Chow tests, likelihood ratio tests, and backward and forward cusum tests and cusum of square tests.

Given that a `minimal' regression required 12 data points at both ends of the ordered data, corresponding to the number of regression parameters (including an intercept), the range of testable values of S1 was restricted to 0.1070 [is less than] S1 [is less than] 0.5277. Graphs of -2log [Lambda], [F.sub.c], and the cusum of squares tests, all provided clear evidence of a breakpoint following the 35th of the 92 data points, where S1 = 0.1671. The backward cusum of squares test at this breakpoint was statistically significant at [Alpha] =0.02. The distribution of -2log [Lambda] and [F.sub.c] were both nominally significant at [Alpha]= 0.001 on the basis of non-conservative tests. This suggests that there is a significant discontinuity in the data at the point where the leading firm's market share is about 16-17 per cent. A second possible break point at about 24 per cent was found, but this proved to be much weaker statistically. The estimate does not reflect the usual reservations about using market share as an indicator of market power because other relevant factors, such as entry barriers and relative efficiency effects, have been incorporated into the regression model. Moreover, given the "Albania of the South Pacific" tag often accorded the prederegulated New Zealand economy, it is perhaps surprising that the profitability-concentration relationship is found to hold up in the late 1970s.

The sub-set regressions show that the model for S1 [is greater than or equal to] 0.1676 where single firm market power is found to exist, provides a superior fit. Note that S1 is unlikely to be acting as a proxy for oligopolistic market power because S2 has a strongly negative impact on the top firm's profit margin. Here the leading firm's profitability strongly reflects both market power and efficiency effects, suggesting the existence of a trade-off between the advantages of the latter, and the detriments associated with the former. This highlights a particular difficulty faced by competition policy in small economies. In contrast S 1 and S2 become irrelevant, and the efficiency variables RP1 and MESM much less significant, when the top firm's market share is "low".

The New Zealand data for 1978-79 thus suggest that there is a structural break in the profitability-concentration relationship at a `critical' level of market share for the leading firm. Two caveats are warranted, however. Firstly, since the late 1970s the formerly heavily protected New Zealand economy has undergone radical reform, including import liberalisation. Domestic market power may now be restrained by the potential for import competition. It is thus an open question as to whether more recent Census data (which is not currently available in the individual firm detail required) would confirm, or require a modification of, this conclusion. Secondly, heeding the caution of Geithman, Marvel and Weiss (1981) relating to oligopoly studies involving a broad range of industries, the ability to exercise single firm market power may depend upon a number of product and buyer characteristics, which may be difficult to incorporate as independent variables in regression analysis. The estimated critical market share is thus likely to be an average of a wide range of cases. Finally, the leading firm market share figure of 16 - 17 per cent should not be taken to be the threshold of dominance in either the economic or legal senses. Rather, it is the point at which a positive association with profits appears to `kick in' in the late 1970s. Non-empirical (mainly legal) assessments of the threshold level of market dominance set dominance at a much higher level of market share, albeit in the context of much more narrowly defined markets.

(1) Exceptions include Sleuwaegen and Dehandschutter (1986) and Uri and Coate (1987).

(2) Uri and Coate (1987, p.1050) note this in passing but deliberately keep to the `standard model', which excludes efficiency effects, in order to test the reliability of the results from previous studies.

(3) The sample comprised 92 of the 145 five-digit industries defined in the New Zealand Standard Industrial Classification (NZSIC) for the Census of Manufacturing, 1978-79. Fifty-three industries were omitted, mainly because of data disclosure restrictions and economically inappropriate definitions. Data from the 1978-79, rather than a more recent, Census were used because of the prior access to unpublished and confidential market share information, which would not be released for later censuses. Efforts to obtain similar information for the 1995 Census of Manufacturing proved unsuccessful. Statistics New Zealand offered two possible solutions: either they could approach the respondents for permission to release the relevant data, or they could undertake the statistical calculations. Both seemed to us to be impractical, the first because of the number of firms involved, and their likely unwillingness to release what they regard as sensitive information, especially to an employee of the Commerce Commission; and the second because we doubted the availability of sufficiently senior Statistics New Zealand staff for the complex econometric analysis required. In either case, it was unlikely that we could have met the charges involved.

(4) The model was drawn up following a review of the mainly US literature, and allowing for the nature of the New Zealand manufacturing sector, with choice of variables being somewhat constrained by data availability. For example, PCMs are commonly used with Census data. In keeping with most other studies, a single equation rather than a simultaneous equation model was used, although it is recognised that the latter could produce useful results. In 1978-78 the New Zealand economy was a relatively closed one, and so the inclusion of international trade variables was thought to be unnecessary, and certainly not possible using trade data keyed to the NZSIC census industry classification. For a fuller discussion of the model specification, as well as of the data, testing and results, see Pickford and Wai (1995).

(5) One referee argued against the inclusion of PCMs for competitors as independent variables on two grounds: that if those variables were driven by industry considerations they would affect the estimation of the control variables, as appeared to have happened because KSR in unexpectedly negative and insignificant; and that the strategic group argument is unconvincing because firm strategies are determined by factors other than market share. The first objection was tested by rerunning the equation with the two "offending" variables being omitted. KSR remained negative and insignificant, suggesting that the unexpected result is intrinsic to that variable, possibly because of the proxy measure for K used in its construction. In similar fashion the ASR variable is insignificant, in contrast to most US studies. With respect to the second objection, it is accepted that the statistical modelling of the strategic group argument is imperfect, but the approach used follows overseas precedents and is rooted in the relevant theory. In addition, the two PCM independent variables are statistically significant, implying that to omit them would cause bias in the regression coefficient estimates for other variables.

(6) REG is a dummy variable which focuses on the hypothesis that market power might have emerged in each of the North and South Islands, given the high cost of transporting goods between the two in the 1970s. To capture that effect, REG = 1 if the top two firms had head offices in different islands, and zero if not, with a positive association expected. The negative correlation found implied a different interpretation for the variable: that collusion between those firms was more difficult when they were separately located. As an alternative, a coefficient of localisation variable similar to that favoured in

US studies was tried in the earlier study but it was not statistically significant.

(7) See Collins and Preston (1969, pp.275-76) and Miller (1971) for analysis based on the four-firm concentration ratio.

(8) The brackets contain the t-ratios; `a' denotes statistical significance at the one per cent level.

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Pickford, M. and M. Wai (1995), "The Profitability-Concentration Relationship in New Zealand Manufacturing Industry", New Zealand Economic Papers, 29(2), 195-213.

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Rhoades, S. A. and J. M. Cleaver (1973), "The Nature of the Concentration-Price Cost Margin Relationship for 352 Manufacturing Industries: 1967", Southern Economic Journal, 40, 90-102.

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Seber, G.A.F. (1984), Multivariate Observations, New York: John Wiley and Sons.

Shepherd, W.G. (1982), "Causes of Increased Computation in the US Economy, 1939-1980", Review of Economics and Statistics, 64, 613-26.

Sleuwaegen, L. and W. Dehandschutter (1986), "The Critical Choice Between the Concentration Ratio and the H-Index in Assessing Industry Performance", Journal of Industrial Economics, 35, 193-208.

Stigler, G. J. (1947), "The Kinky Oligopoly Demand Curve and Rigid Prices", Journal of Political Economy, 55,432-449.

Uti, N. D. and M. B. Coate (1987), "The Search for a Critical Concentration Revisited", Applied Economics, 19, 1049-57.

White, L. J. (1976), "Searching for the Critical Industrial Concentration Ratio: An Application of the 'Switching of Regimes' Technique", in: S. Goldfeld and R. E. Quandt (eds.), Studies in Non-Linear Estimation, Cambridge: Ballinger, 6175.

Williamson, O. E. (1972), "Dominant Firms and the Monopoly Problem: Market Failure Considerations", Harvard Law Review, 85, 1512-31.

Michael Pickford, Chief Economist, Commerce Commission, Wellington, New Zealand.

Stephen Haslett, Associate Professor of Statistics, Statistics Research and Consulting Centre, Massey University, Palmerston North, New Zealand. The authors would like to thank Alan Bollard, the editor and three referees for their helpful comments. The usual disclaimers apply. The views expressed here are not necessarily shared by the Commerce Commission.

There has been a huge number of studies of the profitability-concentration relationship using cross-section data, especially in the United States (US) (for a fairly recent survey, see Schmalensee, 1989). Several of these studies have searched for a level of seller concentration, associated with the emergence of oligopoly, at which competition may first be significantly lessened and profitability enhanced (see, for example, Meehan and Duchesneau, 1973; Dalton and Penn, 1976; White, 1976; Geroski, 1981; and Bradburd and Over, 1982). The notion of a critical concentration ratio arose from the seminal work of Bain (1951), who found that a linear fit to his data was inferior to one based on segmentation at an eight-firm concentration ratio of 70 per cent. Subsequent US studies, with one or two exceptions,(1) typically have found a critical level for the four-firm concentration ratio of between 45 and 60 per cent. This supports Chamberlain's (1956) theory that as the number of firms in an industry falls, the recognition at some point of mutual interdependence causes an abrupt change from competitive behaviour to oligopolistic co-ordination. Geroski (1988, p.171) has also argued strongly that there is little evidence to suggest that linear associations best describe profitability-concentration data, and that modifications like the critical concentration ratio often improve the fit.

Apart from theoretical expectations that competition will be lessened with interdependence, the oligopoly focus of overseas research probably reflects the ready availability in the US of manufacturing industry census data with four- and eight-firm concentration ratios, which facilitate testing for the group exercise of market power. However, in the search for the critical concentration level, the limitations of using such arbitrary ratios as measures of the distribution of firm sizes was emphasised by Kwoka (1979). Using disaggregated census-like data for the US, he found that industry profitability was unaffected until market share by the leading one or two firms reached 25-30 per cent, and that a sufficiently large third-ranked firm had a depressing effect on margins. He concluded that "three-firm co-ordination problems are so severe as to make a large third firm more likely a rival", although this finding has been questioned by Mueller and Greer (1984) (see also Kwoka, 1984). In a similar vein, a recent New Zealand (NZ) study based on manufacturing industry census data by Pickford and Wai (1995) found that the profitability of the first-ranked firm, as measured by its price-cost margin, was inter alia positively related to its own market share and relative efficiency, and negatively related to the market share of number two firms.

These results, by directing attention away from group market power towards the market power exercised by leading firms, raise a further question: is there a critical market share of the leading firm in a market, beyond which its profit rate expands more rapidly with rising share? The purpose of the present study is to test this issue empirically. In so doing, it breaks new ground in various respects. Firstly, it makes use of an efficient and novel computation method to provide tests of a breakpoint in the regression relationship with price-cost margin for the largest firm which do not assume that the breakpoint is known. In the only previous New Zealand study, Ratnayake (1996) found several critical four-firm concentration ratios when using a single equation model, but none when a simultaneous equations model was used with the same data set. In the latter case the concentration variable was both insignificant and negative. However, Ratnayake's t statistics evaluated to determine a break at various concentration ratios (unlike the methods used in this paper) are conditional on the breakpoint being known, and are thus inclined to give tests which appear significant when they are not. This may be the reason for his "perverse results" when a single equation model is compared with two stage least squares. It is also the reason for the choice here of cusum and cusum of square tests in preference to Chow F statistics or its equivalent t test.

Secondly, no studies seem to have searched for a leading firm critical market share (although Kwoka's technique had the potential to reveal one). Economic theory provides relatively little guidance as to the threshold for single firm market power, and economists disagree over the minimum market share required. Rule of thumb figures range from 40 per cent by Stigler (1947) and Geroski (1986), through 50 per cent by Shepherd (1982), to persistence of a 60 per cent level by Williamson (1972). Finally, critical concentration studies typically have not incorporated efficiency effects as independent variables, yet these have an important impact on profitability in NZ manufacturing industries (Ratnayake, 1994; Pickford and Wai, 1995), and in US profit-concentration studies where such variables have been included (e.g., Chappell and Cottle, 1985; Martin, 1988).(2) Their omission could have led to serious specification errors in earlier studies.

In the balance of the paper we proceed as follows. The model and variables are outlined in section 2. The statistical tests employed to search for a critical market share are discussed in section 3. The results are given in section 4, and in the final section we draw together our conclusions.

2. The Model and Data

The model developed by Pickford and Wai (1995) is used as the basis for the empirical testing. They estimated the following equation by OLS multiple regression analysis for number one firms in a cross-section of manufacturing industries:

(1) PCM1 = [Alpha] + [[Beta].sub.1]RP1 + [[Beta].sub.2]S1 + [[Beta].sub.3]S2 + [[Beta].sub.4]S4 + [[Beta].sub.5]PCM2 + [[Beta].sub.6]PCM3N + [[Beta].sub.7]ASR + [[Beta].sub.8]KSR + [[Beta].sub.9]MESM + [[Beta].sub.10]MGR + [[Beta].sub.11]REG + [Epsilon]

where the variables are defined in columns (1)-(3) of Table 1; [Alpha], [[Beta].sub.1] ... [[Beta].sub.11] are coefficients to be estimated; and [Epsilon] is an error term with zero mean and constant variance.(3) It was hypothesised that the size of the price-cost margin (PCM1) would be influenced by three groups of factors: the firm's relative efficiency (RP1); various market power influences (S1, S2, S4, PCM2, and PCM3N); and other industry-wide elements (ASR, KSR, MESM, MGR, and REG).(4)

Table 1. Definition of Variables and Summary of Results Variable Title Definition (1) (2) (3) PCM1 Price-cost margin Firm's sales revenue minus of leading firm. labour costs, divided by revenue. Constant n/a n/a RP1 Relative Firm's value added per productivity index worker divided by industry for leading firm. value added per worker. S1 Market share of Sales revenue of leading leading firm. firm divided by industry revenue. S2 Market share of Sales revenue of number two number two firm. firm divided by industry revenue. S4 Market share of Sales revenue of number number four firm. four firm divided by industry revenue. PCM2 Price-cost margin Firm's sales revenues minus of number two labour costs, divided by firm. revenue. PCM3N Price-cost margin Firm's sales revenues minus of all firms except labour costs, divided by largest two. revenues. ASR Industry Industry advertising advertising-sales expenses divided by sales ratio. revenue. KSR Capital-sales ratio. Industry depreciation divided by sale revenue. MESM Relative size of Sales of minimum efficient minimum efficient scale plant divided by scale plant. sales revenue. MGR Market growth. Growth in industry sales revenue, 1975/76-1978/79. REG Regional markets. Index of regional markets. Adj.[R.sup.2] F-stat n Results (4) n/a 0.20017 [(2.64).sup.a] 0.08506 [(5.05).sup.a] 0.12038 [(2.55).sup.b] -0.2682 [(-2.00).sup.b] -0.2903 (-1.08) 0.24074 [(2.60).sup.b] 0.4490 [(4.04).sup.a] -0.00441 (-1.00) -0.6680 (-1.02) 0.027891 [(3.11).sup.a] 0.000053 (0.23) -0.02998 [(1.92).sup.c] 0.554 [11.30.sup.a] 92

Note: superscripts a, b, and c denote significance at the 1, 5, and 10 per cent levels respectively.

PCM1 is expected to be positively associated with the firm's relative efficiency (RP1), and with its own market share (S1), and negatively associated with the market shares of rival number two firms (S2) and of smaller firms (proxied by S4, to avoid multicollinearity problems). PCM1 is also hypothesised to be influenced by strategic groups within the industry. The theory holds (Porter, 1979) that the market power exercised by one group, as reflected in the PCM, should be greater (all else remaining the same) the more that market power is enjoyed by other groups, leading to the inference that PCM1 should be positively associated with both PCM2 and PCM3N.(5)

The final group of variables control for industry-wide differences between industries. Advertising intensity (ASR) is generally considered to give rise to an entry barrier, and thus to be positively linked with PCM1. The capital-sales ratio (KSR) is included to control for variations in PCMs caused by differing capital intensities in production, with a positive association expected. The relative plant scale variable (MESM) can be seen either as measuring the scale economy entry barrier, or as indicating scale-induced efficiency, but in both cases a positive link is expected. Market growth (MGR) was included to capture any inflation of profit margins resulting from disequilibrium states caused by a fast growth in demand, faster growth being linked with higher margins. Finally, a regional market dummy (REG) was used to capture the geographical dispersion of the activities of the two leading firms, such that dispersion was associated with enhanced market power and hence higher PCMs.

The results of the model are shown in column (4) of Table 1. They indicate that all of the relative efficiency and market power variables have the expected signs, and are statistically significant at the five per cent level or better, with the exception of S4, which has the correct sign but is insignificant. The five industry-wide variables are generally not significant statistically, except for MESM, which has a strong, positive influence on PCM1. The REG variable is significant at almost the five percent level but has the wrong sign.(6) Pickford and Wai (1995) concluded that both market power and efficiency factors are important in explaining inter-industry variations in leading firm profitability in New Zealand manufacturing industries in the late 1970s.

A potential problem with using S1 and S2 together in the regressions follows from the inequality S1 [is greater than or equal to] S2.(7) This implies that a plot of S2 against S1 should reveal an inverted `U'-shape relationship, since the value of S2 will tend to be constrained when that of S1 is both low and high. A regression incorporating a squared S1 term confirms this:(8)

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This opens the possibility that the negative coefficient of the S2 variable in column (4) of Table 1 is a statistical artefact without economic significance. However, the segmentation of the data below on the basis of the S1 values reveals that the S2 regression coefficient is often significantly negative, even for those regression equations involving only low values of S1 where on definitional grounds a positive relationship might be expected. The restraining influence on the leading firm's profitability of competition from the second-ranked firm thereby seems to be supported.

The question now is whether there may be a discontinuity in the profitability-market share association which relates to single firm market power.

3. The Methodology

Various methods have been used to test for discontinuities in regression data with uncorrelated errors. One somewhat discredited technique is the 'differential slope-intercept test' (Dalton and Penn, 1976). This method assesses whether there has been a change in intercept and/or slope, where in this case slope changes are limited to changes in the regression coefficient for market share (S1), and changes in intercept occur at some level of S1. The point at which the regression is allowed to change is varied. The significance level for each change point is usually assessed as if the breakpoint had been known a priori and is consequently not conservative. In other words, the nominal significance level understates the actual significance level, if the test statistic is treated as if the change point were known in advance rather than as being that for the maximum of a sequence of possible tests, and more apparently significant results are found than are actually significant.

A generalisation of this technique is a particular type of switching regression, which allows all the coefficients in the regression to alter at the change point (Quandt, 1982). Such a procedure has been used previously in the context of concentration ratio studies, for example by Uri and Coate (1987), under the assumption that errors within and between regressions are uncorrelated (also see Meehan and Duchesneau, 1973; Rhoades and Cleaver, 1973). Again the change point is not known a priori, which complicates significance testing, whether a Chow test (Chow, 1960; Gujurati, 1978 pp. 295-298), or a likelihood ratio test (Quandt, 1958; White, 1976), is used. For the Chow test, the test, being one in a sequence, is for the maximum of a number of correlated F statistics, while a transformation of the likelihood ratio yields a sequence of correlated chi-square random variables. In both cases the actual distribution of the maximum is complicated (see Feder, 1975 for a discussion of the likelihood ratio case); there is some suggestion (Freeman, 1983) that the Pearson type III (i.e., a gamma distribution) serves as an approximation to the distribution of the negative log transformed likelihood ratio. A Bayesian technique is also available (see Kim, 1991). Note that if the possible change point had been set, rather than being allowed to vary, the Chow test would have an F distribution, and the transformed likelihood ratio a chi-square distribution, under the null hypothesis of no change.

An alternative set of tests have been developed which make use of recursive residuals (Brown, Durbin and Evans, 1975) to derive cusum and cusum of square tests (see Uri and Coate, 1987 for an application). These techniques can be extended to the case where regression errors are correlated, and additional data beyond a change point are added not one at a time but simultaneously (Haslett, 1985). In the case of the PCM data described in the previous section, regression errors are taken to be uncorrelated, but the device of first fitting a regression to the minimum number of ordered data points (the ordering being by S1) and adding the remaining data points in one step, if applied separately to data ordered by both ascending and descending values of S1, yields a computationally efficient device for Chow tests, likelihood ratio tests, and for backward and forward cusum tests and cusum of squares tests. For the uncorrelated error case, the benefits are purely computational; for the correlated error case with known covariance matrix, the method provides an extension to current methods of studying critical concentration rates. A further brief explanation is given later in this section.

The updating procedure is given in detail in Haslett (1985), and is summarised below for the particular uncorrelated error case relevant here. Suppose an initial regression has been fitted to a subset of the data

(3) [y.sub.1] = [X.sub.1] [Beta] + [e.sub.1]

where: [Beta] is a p x 1 vector of parameters possibly including an intercept; [y.sub.l] is p x 1 vector of data points; and [X.sub.1] is a full rank p x p matrix of explanatory variables, including the p smallest or p largest values of S1. (Note that increasing or decreasing S1 determine natural ordering of the data for assessing a possible break point at some particular value of S1.)

It follows that

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the superscript T denotes matrix transpose and -1 a matrix inverse, provides an initial estimate of [Beta], assuming the components of [e.sub.1] are uncorrelated and of equal variance.

Then if a further grouping of (n-p) data points are added, the updated regression parameter estimates are given by

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with residual sum of squares (RSS) given by

RSS = [w.sup.T]w

where w is a set of (n-p) recursive residuals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[I.sub.n-p] is the (n-p) x (n-p) identity matrix

and the full model is now

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and var(e) = [[Sigma].sup.2] [I.sub.n] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [I.sub.n] is the n x n identity matrix.

Consider the Choleski decomposition, which decomposes a positive definite matrix A into A = [LL.sup.L] where L is lower triangular (see Seber, 1984). Then if [([C.sub.1] + [C.sub.2).sup.-1/2] is determined by taking the Choleski decomposition of ([C.sub.1] + [C.sub.2]) followed by inversion it will be lower triangular. Then the successive recursive residuals in w, taken in order, correspond to the recursive residuals that would have been obtained if the additional data points had been added one at a time. Further, the residual sum of squares can be decomposed in the same way, as can the updates to [[Beta].sub.1] (provided the n-p individual columns of [[G.sub.2][[([C.sub.1]+ [C.sub.2]).sup.-1/2]].sup.T], when multiplied by the corresponding element of w, are added sequentially).

Although [[Sigma].sup.2] is unknown, it can be estimated in the usual way from the n data points as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This procedure can be applied to the data ordered by S1 in both ascending and descending orders. To each break point i in the overall regression, there then corresponds two subset regressions, the first using i data points in the forward direction, the second using (n-i) data points in the backward direction.

Taken together these two regressions provide the information necessary to calculate the Chow ([F.sub.c]) and likelihood ratio ([Lambda]) statistics for breakpoint i.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[RSS.sub.f,i] = residual sum of squares for regression using the i data points corresponding to the i smallest values of S1.

[RSS.sub.b,n-i] = residual sum of squares for the regression for the remaining n-i data points ordered by descending S1.

[RSS.sub.n] residual sum of squares for the regression = applied to all n data points. = [RSS.sub.f,n] = [RSS.sub.b,n]

As the change point is varied from following data point i to following data point i + 1, the updating formula for the forward regression gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for the backward direction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [W.sub.f, i+1-p] is the [(i+1-p).sup.th] component of w for the forward regression, [W.sub.b, n-i-p] is the [(n-i-p).sup.th] component of rev(w) for the forward regression, and where rev(w) is w with its components in reverse order. Note where rev(w) is w with its components in reverse order. Note that w for the forward and backward regressions are not equivalent.

With some modifications detailed in Haslett (1985) the method can be extended to the case of correlation between data points within but not between subset regressions (where the subsets are formed at each breakpoint i), and to the case where data are correlated both within and between subset regressions. The first of these corresponds to a block diagonal correlation structure that changes with each i. The second requires a preliminary Choleski or possibly a spectral or eigen decomposition of the covariance matrix for all n data points.

The updating methodology can be further extended, even in the correlated error cases, to testing for a statistically significant change point in a stochastically varying parameter vector (see Haslett, 1996 for the theoretical details).

4. The Results

The computation devices explained in section 3 were programmed in PROC IML (interactive matrix language) in SAS. With the data ordered by S1, a `minimal' regression was fitted to the first twelve data points with the lowest values of S1 (corresponding to the number of regression parameters including an intercept), and the remaining 80 data points were added in a single step to produce 80 `forward' recursive residuals. The data were then reversed, a `minimal' regression fitted to the twelve data points with the largest values of S1, and the remaining 80 data points added in a single step to produce 80 `backward' recursive residuals. Chow tests, likelihood ratio tests, and cusum and cusum of square tests for forward and backward data, were then carried out for each possible breakpoint (from after the twelfth, to after the eightieth data point). Note that because of the number of points required for the minimal regression at each end of the data, only 0.1070 [is less than] S1 [is less than] 0.5277 was testable.

Preliminary checking for a breakpoint included visual inspection of the graphs of -2log [Lambda], [F.sub.c], and the cusum of squares tests for the reversed data. These graphs, shown in part in Figures 1 to 4, revealed similar features, namely evidence of a breakpoint at S1 = 0.1671, following the 35th data point. Note here that -2log [Lambda] is minus two times the natural logarithm of [Lambda], the likelihood ratio statistic, and that on the graphs the first twelve data points and/or the last twelve data points from the regressions produce nothing to graph since they are necessary for the `minimal fit' (i.e, where the number of data points in the regression equals the number of parameters including an intercept).

[Figures 1-4 ILLUSTRATION OMITTED]

Formal tests found that the backward cusum of squares test was statistically significant at [Alpha] = 0.02 at this change point. The distribution of -2log [Lambda], and [F.sub.c] for varying breakpoints being unknown, the significance levels could not be accurately assessed. Although the tests are not conservative, both statistics were nominally significant at [Alpha] [is less than] 0.001 if -2log [Lambda] was treated as having a [[Chi].sup.2] distribution, and [F.sub.c] as having a F distribution (each under the null hypothesis of no breakpoint at S1 = 0.1671, with the possible breakpoint specified a priori).

For the breakpoint S1 = 0.1671 the regression coefficients for each part of the data are given in Table 2. The changes in the parameter estimates are most pronounced for REG (regional markets) and PCM3N (price-cost margin of all firms except the largest two). Given the breakpoint, these changes are statistically significant at [Alpha] [is less than] 0.01. Changes in the regression coefficients for MGR (market growth) and PCM2 (price-cost margin of number two firms) are marginally significant. Note that there is some correlation between the parameter estimates for a given breakpoint so the tests for parameter differences are not statistically independent.

Table 2. Regression Coefficients for Subset Regressions at the Most Significant Breakpoint

S1 [is less [is less 0.1671 than or than or equal to equal to] [0S.sub.1] Variable Parameter Standard Estimate Error INTERCEPT 0.09466 0.13871 RP1 0.06442 0.02954 S1 0.32571 0.56988 S2 -0.76100 0.73466 S4 -1.09435 0.90499 PCM2 0.00233 0.18151 PCM3N 0.93666 0.22364 ASR 0.00209 0.01275 KSR 0.07404 1.36221 MESM 0.03427 0.02057 MGR -0.00040 0.00041 REG -0.08934 0.02925 S1 [S.sub.1] [is less 0.1671 than or equal to] Variable Parameter Standard Significance Estimate Error level for difference INTERCEPT 0.13734 0.08435 0.7926 RP1 0.10524 0.01803 0.2382 S1 0.16041 0.05160 0.7727 S2 -0.31366 0.10855 0.5469 S4 0.10889 0.25378 0.2005 PCM2 0.37483 0.08928 0.0656 PCM3N 0.26126 0.10284 0.0061 ASR -0.00232 0.00371 0.7397 KSR -0.92606 0.58358 0.4998 MESM 0.03705 0.00839 0.9002 MGR 0.00063 0.00024 0.0284 REG -0.00225 0.01491 0.0080

Note : significance levels for difference tests the significance of the difference between parameter estimates for each variable given a change point occurring in the range 0.1671 [is less than or equal to] S1 [is less than or equal to] 0.1676. This corresponds to a breakpoint following the 35th of the 92 data points ordered by S1.

A comparison of the sub-set regressions reveals that some variables which are significant in the regression for all 92 data points are not significant in the segmented regressions. This may reflect differential effects of particular variables in the two segments and the correlation between parameter estimates. Overall, however, the model provides a better fit to the data for S1 [is greater than or equal to] 0.1671, which covers those Census industries where single firm market power appears to exist. The efficiency variables - the leading firm's relative productivity index (RP1) and the industry relative scale economy measure (MESM) - are statistically more significant in explaining the leading firm's profitability (PCM1). Of the market power variables, S1 has a strong positive influence, and S2 a strong negative influence, on PCM1. The presence of mobility barriers, inferred from the statistical significance of PCM2 and PCM3N, also contribute positively to PCM 1. In the sub-set regression for S1 [is greater than or equal to] 0.1671, the results suggest that the profitability of leading firms is the product of both efficiency and of market power effects.

There was also the possibility of a second change point at S1 = 0.2360 when this was fitted without the first change point also being fitted. However, neither of the cusum or cusum of squares tests for either the forward or backward regressions were significant at [Alpha] = 0.05 at this possible change point. The non-conservative Chow and likelihood ratio tests were, however, significant at the nominal [Alpha] = 0.05 level (although these tests assume an a priori change point to be tested and consequently have nominal significance levels). The evidence for the breakpoint at S1 = 0.2360 is consequently much weaker than that for a breakpoint at S1 = 0.1671. Evidence for other breakpoints was weaker still.

As a final technical point, the likelihood ratio statistic was unstable for change points near either end of the data. This is a direct consequence of the divisor for variance estimates being i or n-i, rather than the number of degrees of freedom i-p or n-i-p. This is discussed further in Phillips (1991).

5. Conclusions

The purpose of this paper has been to use a statistical model to test for a structural break in the profitability-concentration relationship associated with single firm market power. This is, as far as we know, the first time that this issue has been tested empirically. Moreover, unlike previous discontinuity studies focussing on oligopoly market power, our model has the advantage of incorporating efficiency explanations, as well as market power explanations, of profitability. In our view, the inclusion of the former seems desirable in the oligopoly focus, and essential in the single firm one, especially in New Zealand where the combination of a very small economy by OECD standards and of scale economies is likely to promote concentrated markets.

While various techniques have been used to test for discontinuities in regression data with uncorrelated errors, in the testing of the significance level for each breakpoint, some suffer from the test being conducted as if the change point were known a priori. As a result, more apparently significant effects are found than are warranted. Our approach is based on the use of recursive residuals to derive cusum and cusum of square tests following Brown, Durbin and Evans (1975), and the extension proposed by Haslett (1985) for the case where regression errors are correlated, and where additional data beyond a change point are added simultaneously rather than one at a time. While we take the regression errors to be uncorrelated with the PCM data, the approach, when applied separately to data ordered both by ascending and descending values of the market share of the leading firm (S1), is a computationally efficient way of conducting Chow tests, likelihood ratio tests, and backward and forward cusum tests and cusum of square tests.

Given that a `minimal' regression required 12 data points at both ends of the ordered data, corresponding to the number of regression parameters (including an intercept), the range of testable values of S1 was restricted to 0.1070 [is less than] S1 [is less than] 0.5277. Graphs of -2log [Lambda], [F.sub.c], and the cusum of squares tests, all provided clear evidence of a breakpoint following the 35th of the 92 data points, where S1 = 0.1671. The backward cusum of squares test at this breakpoint was statistically significant at [Alpha] =0.02. The distribution of -2log [Lambda] and [F.sub.c] were both nominally significant at [Alpha]= 0.001 on the basis of non-conservative tests. This suggests that there is a significant discontinuity in the data at the point where the leading firm's market share is about 16-17 per cent. A second possible break point at about 24 per cent was found, but this proved to be much weaker statistically. The estimate does not reflect the usual reservations about using market share as an indicator of market power because other relevant factors, such as entry barriers and relative efficiency effects, have been incorporated into the regression model. Moreover, given the "Albania of the South Pacific" tag often accorded the prederegulated New Zealand economy, it is perhaps surprising that the profitability-concentration relationship is found to hold up in the late 1970s.

The sub-set regressions show that the model for S1 [is greater than or equal to] 0.1676 where single firm market power is found to exist, provides a superior fit. Note that S1 is unlikely to be acting as a proxy for oligopolistic market power because S2 has a strongly negative impact on the top firm's profit margin. Here the leading firm's profitability strongly reflects both market power and efficiency effects, suggesting the existence of a trade-off between the advantages of the latter, and the detriments associated with the former. This highlights a particular difficulty faced by competition policy in small economies. In contrast S 1 and S2 become irrelevant, and the efficiency variables RP1 and MESM much less significant, when the top firm's market share is "low".

The New Zealand data for 1978-79 thus suggest that there is a structural break in the profitability-concentration relationship at a `critical' level of market share for the leading firm. Two caveats are warranted, however. Firstly, since the late 1970s the formerly heavily protected New Zealand economy has undergone radical reform, including import liberalisation. Domestic market power may now be restrained by the potential for import competition. It is thus an open question as to whether more recent Census data (which is not currently available in the individual firm detail required) would confirm, or require a modification of, this conclusion. Secondly, heeding the caution of Geithman, Marvel and Weiss (1981) relating to oligopoly studies involving a broad range of industries, the ability to exercise single firm market power may depend upon a number of product and buyer characteristics, which may be difficult to incorporate as independent variables in regression analysis. The estimated critical market share is thus likely to be an average of a wide range of cases. Finally, the leading firm market share figure of 16 - 17 per cent should not be taken to be the threshold of dominance in either the economic or legal senses. Rather, it is the point at which a positive association with profits appears to `kick in' in the late 1970s. Non-empirical (mainly legal) assessments of the threshold level of market dominance set dominance at a much higher level of market share, albeit in the context of much more narrowly defined markets.

(1) Exceptions include Sleuwaegen and Dehandschutter (1986) and Uri and Coate (1987).

(2) Uri and Coate (1987, p.1050) note this in passing but deliberately keep to the `standard model', which excludes efficiency effects, in order to test the reliability of the results from previous studies.

(3) The sample comprised 92 of the 145 five-digit industries defined in the New Zealand Standard Industrial Classification (NZSIC) for the Census of Manufacturing, 1978-79. Fifty-three industries were omitted, mainly because of data disclosure restrictions and economically inappropriate definitions. Data from the 1978-79, rather than a more recent, Census were used because of the prior access to unpublished and confidential market share information, which would not be released for later censuses. Efforts to obtain similar information for the 1995 Census of Manufacturing proved unsuccessful. Statistics New Zealand offered two possible solutions: either they could approach the respondents for permission to release the relevant data, or they could undertake the statistical calculations. Both seemed to us to be impractical, the first because of the number of firms involved, and their likely unwillingness to release what they regard as sensitive information, especially to an employee of the Commerce Commission; and the second because we doubted the availability of sufficiently senior Statistics New Zealand staff for the complex econometric analysis required. In either case, it was unlikely that we could have met the charges involved.

(4) The model was drawn up following a review of the mainly US literature, and allowing for the nature of the New Zealand manufacturing sector, with choice of variables being somewhat constrained by data availability. For example, PCMs are commonly used with Census data. In keeping with most other studies, a single equation rather than a simultaneous equation model was used, although it is recognised that the latter could produce useful results. In 1978-78 the New Zealand economy was a relatively closed one, and so the inclusion of international trade variables was thought to be unnecessary, and certainly not possible using trade data keyed to the NZSIC census industry classification. For a fuller discussion of the model specification, as well as of the data, testing and results, see Pickford and Wai (1995).

(5) One referee argued against the inclusion of PCMs for competitors as independent variables on two grounds: that if those variables were driven by industry considerations they would affect the estimation of the control variables, as appeared to have happened because KSR in unexpectedly negative and insignificant; and that the strategic group argument is unconvincing because firm strategies are determined by factors other than market share. The first objection was tested by rerunning the equation with the two "offending" variables being omitted. KSR remained negative and insignificant, suggesting that the unexpected result is intrinsic to that variable, possibly because of the proxy measure for K used in its construction. In similar fashion the ASR variable is insignificant, in contrast to most US studies. With respect to the second objection, it is accepted that the statistical modelling of the strategic group argument is imperfect, but the approach used follows overseas precedents and is rooted in the relevant theory. In addition, the two PCM independent variables are statistically significant, implying that to omit them would cause bias in the regression coefficient estimates for other variables.

(6) REG is a dummy variable which focuses on the hypothesis that market power might have emerged in each of the North and South Islands, given the high cost of transporting goods between the two in the 1970s. To capture that effect, REG = 1 if the top two firms had head offices in different islands, and zero if not, with a positive association expected. The negative correlation found implied a different interpretation for the variable: that collusion between those firms was more difficult when they were separately located. As an alternative, a coefficient of localisation variable similar to that favoured in

US studies was tried in the earlier study but it was not statistically significant.

(7) See Collins and Preston (1969, pp.275-76) and Miller (1971) for analysis based on the four-firm concentration ratio.

(8) The brackets contain the t-ratios; `a' denotes statistical significance at the one per cent level.

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Michael Pickford, Chief Economist, Commerce Commission, Wellington, New Zealand.

Stephen Haslett, Associate Professor of Statistics, Statistics Research and Consulting Centre, Massey University, Palmerston North, New Zealand. The authors would like to thank Alan Bollard, the editor and three referees for their helpful comments. The usual disclaimers apply. The views expressed here are not necessarily shared by the Commerce Commission.

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Author: | Pickford, Michael; Haslett, Stephen |
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Publication: | New Zealand Economic Papers |

Article Type: | Statistical Data Included |

Geographic Code: | 8NEWZ |

Date: | Dec 1, 1999 |

Words: | 6946 |

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