Javier García-Enríquez and Josu Arteche are lecturer and associate professor of the departments Fundamentos del Análisis Económico II and Economía Aplicada III, respectively, in the University of the Basque Country UPV/EHU, Bilbao 48015, Spain. Javier Hualde is associate professor at the department of Economics of the Public University of Navarra, Pamplona 31006, Spain. Arantza Murillas-Maza is researcher at AZTI-Tecnalia, Technological Institute for Fisheries and Food, Marine Research Division, Sukarrieta 48395, Spain. E-mail: javier.garcia@ehu.es for correspondence. The authors thank the Editor and two anonymous referees for helpful comments which have led to significant improvements in the content of the paper. The first, third and fourth authors acknowledge funding from project ECO2010-15332 of the Spanish Ministry of Science and Innovation and from the ERDF. The second author gratefully acknowledges aid received from project ECO2011-24304 of the Spanish Ministry of Economy and Competitiveness. Javier García-Enríquez also wishes to thank the University of the Basque Country UPV/EHU for funding provided via its programme of aid for the training of research staff (2007–2010). Josu Arteche acknowledges financial support by the UFI 11/03 Sustainable Economics & Welfare of the University of the Basque Country UPV/EHU. Finally, Arantza Murillas-Maza acknowledges the funding of this work through SOCIOEC project by the EU (FP7-KBBE-2011-5) and the Basque Government. This paper is contribution no. 615 from AZTI-Tecnalia (Marine Research Division).
Original Article
Spatial Integration in the Spanish Mackerel Market
Article first published online: 29 MAY 2013
DOI: 10.1111/1477-9552.12020
© 2013 The Agricultural Economics Society
Additional Information
How to Cite
García-Enríquez, J., Hualde, J., Arteche, J. and Murillas-Maza, A. (2014), Spatial Integration in the Spanish Mackerel Market. Journal of Agricultural Economics, 65: 234–256. doi: 10.1111/1477-9552.12020
Publication History
- Issue published online: 26 JAN 2014
- Article first published online: 29 MAY 2013
- Manuscript Accepted: 20 FEB 2013
- Manuscript Revised: 11 JAN 2013
- Manuscript Received: 23 OCT 2012
Funded by
- Spanish Ministry of Science and Innovation. Grant Number: ECO2010-15332
- Spanish Ministry of Economy and Competitiveness. Grant Number: ECO2011-24304
- ERDF. Grant Number: ECO2010-15332
- University of the Basque Country UPV/EHU
- UFI 11/03 Sustainable Economics & Welfare of the University of the Basque Country
- EU. Grant Number: FP7-KBBE-2011-5
- Basque Government
- Abstract
- Article
- References
- Cited By
Keywords:
- Fisheries markets;
- fractional cointegration;
- long memory;
- mackerel;
- seasonality
Abstract
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
This article analyses the potential links between regional first-sale markets for mackerel in Spain using fractional cointegration techniques. The results indicate that this is not an integrated market, and we demonstrate that there are no links, at least in the long term, between any of Spain's five regional markets. This result has significant implications in policy terms, as local, regional and European authorities must take into account the need to apply distinct local policies.
1. Introduction
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
The usefulness of horizontal or spatial integration studies lies in their ability to delimit a market in geographical terms and thus enable policy recommendations to be made. In the context of an analysis of the North Sea cod fishery, Nielsen (2005) highlights that if markets are integrated then a reduction in quotas could have greater repercussions on prices, with bigger changes being observed than would be expected in a non-integrated market. This is because when there is integration, price elasticity is affected not only by changes in quantities at the local market level but also by changes elsewhere. Nielsen (2005) also asserts that a greater knowledge of market integration enables the effectiveness of regional market policies to be assessed. Indeed, if markets are perfectly integrated, regional policies are clearly ineffective as regional markets form part of a larger market, and it is there that action needs to be taken. It is therefore useful to analyse the extent of market integration before any fishery resource management measure, such as total allowable catches (TACs) or individual fishing quotas (IFQs), is introduced.
The European Commission (EC) acknowledges that the current Common Fisheries Policy (CFP), which is in force until 2013, does not seem to have achieved its economic objective of securing an economically viable fishing sector. Thus, after a long process of reflection, the EC has drawn up a new proposal for a far-reaching reform of the CFP, the Common Market Organisation and the European Maritime & Fisheries Fund, which shall be effective upon its publication (expected during 2013). Among other measures, this reform seeks to set up a more decentralised, more participative management of fisheries policies so that the measures proposed by the CFP can be applied and the various regulations involved can be implemented as close as possible to their points of application. Member States and stakeholders (e.g. the Regional Consultative Committees (RCCs)) will take on more responsibility for managing fisheries resources. Regionalisation is envisaged as involving active participation of fishermen in policy-making, which should result in a greater knowledge of regulations and thus in a higher level of compliance.
In this general context, this article seeks to analyse horizontal integration as a basis for determining what relationship, if any, exists between the prices of mackerel auctioned in the quayside fish markets of the various regional autonomous communities in Spain where there is a market for this species, that is the Basque Country, Cantabria, Asturias, Galicia and Andalusia. The objective is to identify whether these markets operate as distinct regional markets or as a single market for a homogenous product for the whole of Spain.
A widespread feature of non-integrated markets is that the price is almost the same in all markets, and they are influenced equally by all quantity shocks (Nielsen, 2005, cites a number of papers that confirm this finding for whitefish fisheries in Europe.) In view of this finding and the effect on first-sale prices for mackerel of the regulatory measures currently being implemented (mainly TACs and IFQs), these markets might at first sight be expected to be distinct, though prices may also remain unchanged for other reasons.
The concept of spatial integration, defined on the basis of the econometric concept of cointegration, is used to establish what level of integration exists between markets. Initially, most of the studies in this field were carried out under a bivariate approach, where only pairwise relationships were considered. Multivariate extensions were proposed by Asche et al. (1999) for the world salmon market, and later by González-Rivera and Helfand (2001) to analyse the spatial integration in the Brazilian rice market. Nielsen (2005) summarises this previous research with a definition of market integration from an econometric perspective. Under Nielsen's definition, n markets are said to be perfectly integrated if: (i) they are all cointegrated with one another (i.e. there are n − 1 cointegration relationships or a common trend, CT); and (ii) the ‘law of one price’ (LOP) is met. The LOP establishes that under the usual assumptions of perfect competition, that is, homogeneity of the good, perfect information and the absence of barriers to trade, the difference in prices between spatially separate markets cannot be greater than the transaction (arbitrage) costs. If this law is met then there are constant relative prices and a single market. If the LOP is not met but the assumption of cointegration is maintained, then markets are said to be partially integrated, which means that there is an incomplete transfer of changes from one price to another, resulting in deviations from equilibrium prices, especially in the short term. Such deviations can be explained by various factors, including the difficulties in arbitrage that may be caused by barriers to trade, incomplete information and risk aversion. Finally, if the number of cointegration relationships is lower than n − 1 the price series are not cointegrated two by two, but some markets are independent, so a sub-system with a CT must be sought that excludes independent markets.
Although the radial model of Ravallion (1986) can be considered one of the first dynamic models to analyse spatial integration, vector auto-regressive (VAR) models are frequently seen as its natural extension (see, e.g., Schroeder and Goodwin, 1990; or Goodwin and Schroeder, 1991a; for some applications). However, when price series are not stationary, the usual practice of modelling first differences of the series as a standard VAR leads to misspecification if the series are cointegrated. Thus, since the seminal paper of Ardeni (1989), studies of horizontal integration have conventionally used cointegrated models (see, e.g. Goodwin and Schroeder, 1991b; Gordon and Hannesson, 1996; Gil et al., 1996; Sanjuán and Gil, 2001; Asche et al., 2002; Setäla et al., 2008 or Nielsen et al., 2009).
In this vein, our aim in this article is to analyse the relationships among different series of prices of mackerel corresponding to five Spanish regional autonomous communities from a cointegration perspective. Unlike the traditional prescription of unit root observables and weak dependent cointegrating errors (see e.g. Asche and Tveteras, 2004; and Asche et al., 2013), one of the main contributions of the article is to allow for fractional integration and cointegration.
The main reasons for proposing a fractional approach are the following. First, this setting is very flexible, covering many different characterisations of observables and possible cointegrating errors, whereas at the same time including standard cointegration as a particular case. In particular, there are situations where the typical characterisation of observables as I(0) or I(1) is unsatisfactory. Specifically, if data are stationary (I(0)), external shocks have a short-term impact, whereas the data revert to the mean of the series at an exponential rate. In contrast, integrated variables (I(1)) are not mean reverting (so they do not move towards their means over time). ARIMA models do not account for the possibility that data can be mean reverting with a substantially slower mean reversion than an exponential decaying rate would imply. In fact, a slow mean reversion occurs when the effects of shocks disappear at a slow rate, and, unlike I(0) processes, fractional models can accommodate this possibility. Also, the use of a fractional approach appears to be appropriate when dealing with series which, like ours, have been constructed by aggregation (see Robinson, 1978; Granger, 1980). Finally, as will be seen below, our data appear as being characterised as fractional series, that is as I(d) with 0 < d < 1.
In a general fractional cointegration framework, one of the main challenges is to determine the cointegrating rank r (i.e. the number of linearly independent cointegrating relations). Here, the method we use to infer r has been developed by Hualde (2012), and has not been previously used in applied literature, with the exception of the very simple analysis in Hualde and Robinson (2010). This procedure identifies those variables that cointegrate with one another and those that are not cointegrated, so it is very useful in analyses (such as the one conducted here) in identifying which markets are integrated with one another. Furthermore, if the evidence provided by the data supports the possibility of r = n − 1, Hualde's methodology would lead to straightforward testing of the LOP by means of the Wald test statistic based on the generalised least squares estimators of the cointegrating parameters proposed by Hualde and Robinson (2010).
There are alternative procedures to infer the cointegrating rank in a fractional cointegration framework, although these have not been pursued in this article. These include Robinson and Yajima (2002), Robinson (2008), Nielsen (2010) and Johansen and Nielsen (2012). Of these methods, only Nielsen (2010) would be, in principle, general enough to cover the type of cointegration considered by Hualde (2012) (described in Section 'Methods' below). However, Nielsen (2010) requires all cointegrating errors to be stationary, which might be restrictive in practice. In addition, Nielsen's method is not straightforward to use in practice, because his proposed test statistic (which can be employed sequentially to estimate the rank) has a non-standard null limiting distribution which depends on the maximum integration order of the observables (which in general is not known).
The article is structured as follows. Section 'The Spanish mackerel fishery' gives an outline description of the mackerel fishery, its management and some of the main characteristics of the Spanish mackerel market. Section 'Methods' sets out the method used (described in detail in the Appendix). Section 'Empirical Analysis' presents the results of the empirical application.
2. The Spanish mackerel fishery
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
Mackerel (Scomber scombrus) is found throughout the Atlantic, from Norway to Portugal. It is distributed throughout both the Eastern (Baltic, Mediterranean & Black Sea) and Western areas of the North Atlantic.
The mackerel stock is managed annually via area-based TACs. The International Council for the Exploration of the Sea (ICES) provides advice on the admissible exploitation of the stock throughout its distribution as a whole, but that advice is transferred to two TACs: the Southern area (areas VIIIc & IXa in the ICES nomenclature), which corresponds to the northern and north-eastern coasts of Spain and the coast of Portugal, and the Western area (the rest of the distribution area).
The national fleets which have landed the most mackerel in the past five years are those of the UK, Norway, Spain and Ireland (Table 1).
2004 | 2005 | 2006 | 2007 | 2008 | Total | |
---|---|---|---|---|---|---|
| ||||||
UK | 172,785 | 152,801 | 95,815 | 133,688 | 112,145 | 667,234 |
Norway | 147,069 | 106,434 | 113,079 | 131,198 | 118,050 | 615,830 |
Spain | 34,455 | 52,753 | 54,136 | 62,946 | 64,637 | 268,927 |
Ireland | 60,631 | 45,687 | 40,664 | 49,260 | 44,759 | 241,001 |
Denmark | 25,665 | 23,212 | 24,219 | 25,223 | 26,726 | 125,045 |
Netherlands | 27,498 | 22,734 | 24,157 | 24,234 | 19,900 | 118,523 |
Germany | 23,244 | 19,040 | 16,608 | 18,214 | 15,502 | 92,608 |
France | 20,264 | 16,337 | 14,953 | 20,038 | 15,602 | 87,194 |
Faroe Islands | 12,379 | 9,739 | 12,067 | 13,151 | 11,166 | 58,502 |
The quota allocated to Spain in the Southern area (the main fishing area of the Spanish fleet) decreased from 33,120 tonnes in 2001 to 22,256 tonnes in 2008. As can be seen from Table 2, the Spanish fleet has caught quantities far in excess of the quota allocated under the TAC system in some years. One of the main reasons of this behaviour could be the existence of differentiated regional markets, because markets which are not linked may lead profit-maximising firms to avoid compliance with common regulations (in this case the Spanish TAC). The result may be excess catches in the different individual markets and thus for Spain, in general. Moreover, if regional and national daily markets are flooded with catches then prices will be lower, which might exacerbate the problem. When first-sale prices are low fishermen might seek to raise their profits by increasing catches (see García-Enríquez, 2012, for the mackerel fishery in the Basque Country). Thus, in the case of non-integrated markets, it seems advisable for regions to co-operate in matters of fishing. Attempts have been made to encourage this by giving more weight to the RCCs, whose agreements may be more strongly binding than European-wide regulations.
2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | |
---|---|---|---|---|---|---|---|---|
| ||||||||
TAC allocated to Spain (southern area) | 33,120 | 33,874 | 28,846 | 26,625 | 20,500 | 21,574 | 24,405 | 22,256 |
Spanish catches | 40,079 | 46,641 | 23,027 | 32,374 | 47,958 | 50,088 | 60,174 | 57,310 |
% TAC exceeded by Spain | 21 | 38 | −20 | 22 | 134 | 132 | 147 | 157 |
The different markets in the autonomous regions are clearly linked to the fleets that unload their catches at their fishing ports, and to the nature of the resource exploited. The behaviour of these fleets must therefore be described to understand how the various regional markets work. We note that not all vessels obtain the same first-sale prices for landing the same fish: prices depend, inter alia, on the type of fishing gear used, the area of operation, the on-board fish handling and conservation measures taken and even the way in which fish is unloaded at the port. All these and other factors affect the quality of fish, and thus its first-sale price. It is not unusual for there to be significant price differences according to the way in which fish are caught (see García-Enríquez, 2012, for the case of mackerel in the Basque Country).
The Spanish fleet comprises vessels from all the regional autonomous communities along the northern seaboard (Basque Country, Cantabria, Asturias and Galicia), plus some vessels from Andalusia. Most mackerel catches are landed by the artisanal fleet (mainly with hand lines), though some catches are also made with gillnets. The second biggest fleet in terms of fishing methods has traditionally been the purse seiner fleet, though its catches have dropped by 19% in recent years, reflecting a decrease in the number of vessels using this method. The decline in their relative contribution to overall catches by Spanish vessels has been even greater (34%) than the drop in their actual catches in percentage terms, reflecting the major increase in catches with trawl nets. The quantity of fish caught by bottom trawling has increased by 232% in the last decade.
In spring, the artisanal fleet catches mackerel with hand lines, then in summer it sets off to troll for tuna. It is this fleet that has the most stable catches: mackerel has accounted for between 60% and 70% of all the fish that it has landed in recent years.
Purse seine fishing on Spain's northern seaboard can be broken down into two main types: the first is that of the vessels of the Basque Country and Asturias, which spend their summers fishing for white and blue fin tuna with live bait. Mackerel accounts for between 20% and 33% of the total catches of these vessels, though with the recent crisis in anchovy fishing the figure has risen as high as 45% in some years. They generally account for over 66% of all the purse seine catches along the northern seaboard. All other purse seiners belong to the second group, which fishes with this method throughout the year, though some also use trolling lines in summer, and some occasionally use hand lines to catch mackerel. For this second group, made up mainly of vessels from Cantabria, Asturias and Galicia, mackerel fishing in spring is of considerably greater importance: indeed it is probably as important as it is for hand line and troll line vessels.
As mentioned previously, there has been a general increase in the amount of mackerel caught by bottom trawling all along Spain's northern seaboard since the year 2000. The trawling fleet has traditionally focused on other species with greater commercial value such as hake, whiting and horse mackerel, but diversifying its target species gives it more stability in its earnings. Moreover, the scarcity of hake in recent years and the plans for the recovery of the western and southern stocks of the species have led to its TACs being sharply curtailed, increasing the need for these vessels to seek earnings from other species. Thus, mackerel fishing is now providing a greater contribution to overall earnings in some periods of the year, particularly in winter.
Bottom trawls can be divided into two main types: otter trawls and pair trawls. Both have played their parts in the increase in mackerel landings, and the species has come to account for between 15% and 25% of the catches landed each year in the Basque Country by bottom trawlers. It is hard to ascertain just how important mackerel catches are at present for the trawler fleets operating out of Cantabria, Asturias and Galicia, but they probably do not exceed 20% of their annual earnings. For the trawler fleet operating out of the Basque Country, mackerel catches account for somewhere between 1% and 15% of annual earnings, with the figure varying from one vessel to another and in line with the amount of hake and other species also caught.
3. Methods
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
As mentioned in the introduction, our aim is to analyse the relationships among different price series from a cointegration perspective. These price series are obtained by aggregating information from different fishermen's guilds, so, given that Robinson (1978) and Granger (1980) justified that fractional integration could originate from aggregation of data exhibiting heterogeneous dynamic behaviour at the individual level, a fractional approach appears to be sensible.
As will be seen in the next section, this possibility is confirmed by the evidence provided by the individual estimates of the integration orders of the price series. Thus, if we believe that our series are indeed fractional, and we suspect that they might be cointegrated, the right approach to test for this possibility is to employ a procedure rooted in fractional cointegration, which is a natural extension of traditional cointegration. As noted above, the fractional cointegration model permits greater flexibility in representing equilibrium relationships among economic variables than the traditional I(1)/I(0) prescription. This greater level of generality allows modelling many situations which can be relevant in practice. For example, it is plausible that there exist long-run co-movements between non-stationary series which are not precisely unit roots. Also, there is usually no a priori reason for restricting the analysis to weakly dependent cointegrating errors, because the convergence to equilibrium of cointegrating relations could be much slower than the adjustment implied by, for example, a finite ARMA cointegrating error. Furthermore, we could also consider cointegration among (asymptotically) stationary variables, with some linear combinations producing cointegrating errors characterised by having weaker memory than that of the observed series. Finally, it could be that the cointegrating error is purely non-stationary but mean reverting, so that there is some long-run equilibrium among non-mean reverting observables.
In addition, relying on standard I(1)/I(0) techniques when facing fractional processes might be very misleading, as Gonzalo and Lee (1998) showed related to Johansen's LR type of tests. Thus, it is not just that a fractional cointegration approach has advantages due to its greater level of generality, but also that standard techniques are in many cases misspecified when applied to fractional processes, therefore leading to erroneous conclusions.
In a multivariate framework such as this, there are various definitions of cointegration (see, e.g. Robinson and Yajima, 2002) and, among the different alternatives, we choose the most general, as in Hualde (2012). So, we define cointegration as the situation where a linear combination of fractional processes is integrated1 of a strictly smaller order than the maximum order of the elements of the linear combination. This definition differs substantially from that of Engle and Granger (1987), as it permits components of the vector of observables to have different integration orders,2 and covers many situations of interest. Among these, we can find non-stationary but mean reverting cointegration errors, stationary cointegration, where the observables are stationary (but long memory) with ‘less-memoried’ (even short memory) cointegrating errors, and also more complicated possibilities, where for example two ‘high order’ observables cointegrate, leading to a cointegrating error which combines with a ‘low order’ observable to give another cointegrating error with even smaller order. This level of generality is attractive, but in a general multivariate setting, a complex cointegrating structure might occur, and even determining the cointegrating rank r, which is necessary to make inferences on cointegrating vectors, might be a very complicated issue.
The procedure applied here is designed to deal with such a level of generality and is based on the proposal by Hualde (2012), described in detail in the Appendix. With a p-dimensional vector it comprises at most p − 1 steps and its implementation is based on consistent estimators of the individual integration orders of the observables and consistent test statistics for the ‘no cointegration’ hypothesis. Possible choices for the estimators of the orders of integration are the log-periodogram (Robinson, 1995a) or the local Whittle (LW) (Robinson, 1995b) estimators, which avoid the possible inconsistency of parametric estimators if the model is misspecified. Although both are consistent for I(d) series with d ≤ 1 and asymptotically normal for d < 0.75 (Kim and Phillips, 1999a; Velasco, 1999, 2000; Phillips and Shimotsu, 2004), the LW is more efficient under less restrictive assumptions (e.g. Gaussianity of the series is not required). Some extensions have been proposed to cover the cases d ≥ 0.75 (Kim and Phillips, 1999b; Phillips and Shimotsu, 2005) without requiring tapering or any other data transformation, maintaining in that way the asymptotic variance of the estimator. However, because all the series analysed in the present article have an estimated memory degree well below 0.75, we use the simpler and more efficient LW to estimate the memory parameter of the individual series. With respect to the test statistics of the null of no cointegration, Robinson (2008) or Hualde and Velasco (2008) are good alternatives.
Finally, note that the key issue in the Hualde (2012) approach is the selection of CTs in every step of the procedure. Hualde (2012) justifies rigorously the properties of the estimator of the cointegrating rank r, which, in particular, do not depend on knowledge of true CTs, although due to the properties of the estimators of the orders and of the test statistics, the probability of making incorrect choices of CTs vanishes asymptotically (see Hualde, 2012, for specific details).
4. Empirical Analysis
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
This section analyses the prices of mackerel landed at the ports of those regional autonomous communities in Spain where this species is commercialised, that is, the Basque Country, Cantabria, Asturias, Galicia and Andalusia. Prices are indicated in euros per kilogram and correspond to landings handled by fishermen's guilds. Monthly series running from January 1997 to December 2008 are used, giving a total of 144 observations. The analysis is restricted to this period for several reasons: information on prices in Galicia is only available from January 1997 onwards; from 2009 onwards there are gaps in the observations; and a number of additional regulatory measures were introduced in 2009, with the introduction of limits (per fishermen and per vessel, based on their technology). These changes in regulations and lack of observations from 2009 onwards make it more difficult to analyse potential changes in price structures, so we exclude the more recent years from the study period.
All the analyses that follow are based on series expressed as logs. Figure 1 shows the changes in the logs of the different mackerel price series over the period analysed. As can be seen, the monthly series for the different regional autonomous communities do not seem to show any increasing trend. Indeed, a regression in first differences for each individual series on a constant shows no statistical significance in any case, supporting the absence of a deterministic trend.
The periodograms of the various series are drawn in Figure 2 to enable a detailed analysis of the degree of persistence of the series to be performed. In all cases, peaks can be observed both in the vicinity of frequency zero (indicating a possible long-term memory) and in some seasonal frequencies πj/6, j = 1, …, 6. The seasonal peaks are especially significant in the cases of the Basque Country and Cantabria, where they are higher than the peaks around frequency zero. However, this does not necessarily imply larger memory parameters, as it may also (as suggested by the valleys in the surrounding areas) be due to the presence of a deterministic component.
Given that this article seeks mainly to check for possible cointegration in the long-term components, consistent estimates need to be obtained for the memory parameters at frequency zero. The existence of significant seasonal peaks, as observed in Figure 2, could seriously distort such estimates, leaving only a very small number of usable frequencies, with the loss of efficiency that this entails. There would be a similar effect in checks for cointegration, where using only a small number of frequencies would have a significant effect on the power of the comparisons made (Robinson, 2008). It is therefore necessary first to deseasonalise the series so that the only significant peaks in the periodograms are those at frequency zero.
4.1. Deseasonalisation of series
The parametric deseasonalisation methods most widely used in the empirical literature tend to consider seasonality either in an exclusively stochastic or an exclusively deterministic way. Given the possibility that seasonal peaks here may be either stochastic or deterministic, both options are considered in this article. Moreover, to relax the unit root assumption traditionally used in the stochastic case, which may lead to over-differentiation (see, for instance, Arteche, 2007), each memory parameter of the seasonal frequencies is estimated semi-parametrically using the seasonal and cyclical LW estimator developed by Arteche and Robinson (1999, 2000). However, the limited number of observations available means that the bandwidth used in estimation is not very wide, so estimators with considerable variance emerge, making it complicated to choose the most suitable estimate. In an attempt to solve this problem, a strategy consisting of the following steps has been designed and applied series by series:
- The memory parameters for a series are estimated with different bandwidths (m hereafter) at the frequencies of the various cycles by the seasonal and cyclical LW estimator (Arteche and Robinson, 1999, 2000). m is set to between 3 and 10, as an estimate with only two frequencies is considered poor and bandwidths of more than 10 come too close to the next seasonal frequency. Once the memory parameters are estimated for m = 3, …, 10, a stable interim region is sought (as per Taqqu and Teverovsky, 1996) and the sample mean of the various estimates associated with that region is calculated. This mean is denoted as , with h = 1, …, 6 representing the individual seasonal frequencies (π/6, π/3, π/2, , and π, respectively).
- The persistent stochastic seasonality series is filtered as follows: , where ω_{j} represents the different seasonal frequencies in radians (except for frequency π, which is taken into account in ).
- The first differences in the series with no seasonal long memory () are regressed using Ordinary Least Squares (OLS) on the first differences of the dummy deterministic variables that represent the different seasonal cycles,3 that is, , where u_{t} is the random disturbance and is the matrix of seasonal variables. Each seasonal variable is defined in sine–cosine form: VF_{ht} = ( sin (ω_{j}t), cos (ω_{j}t))^{′}. The significant variables are identified via conventional t statistics, bearing in mind that heteroscedasticity and/or autocorrelation-consistent variance estimators are used in their construction. A deterministic cycle is considered to be significant if at least one of the two components of the seasonal variable by which it is represented (the sine form or the cosine form) is statistically significant at 5% significance level. A series clear of any persistent seasonality is then obtained (z_{it} hereafter), that is , where represents the significant variables.
- The periodogram for the series is used to validate the memory parameter estimates. If there is any remaining evidence of persistent seasonality or of overdifferentiation, another estimation in a different stable region is sought for each memory parameter not properly filtered, and the whole process is repeated. If there is no remaining evidence of seasonality, the process comes to an end; otherwise it is repeated as many times as necessary.
The stochastic nature of the cycle is modelled first, rather than the deterministic nature, because if seasonal dummy variables are used with series which are I_{ω}(1) at any seasonal frequency4 the likelihood of finding spurious links is very high (Abeysinghe, 1991). Indeed, Franses et al. (1995) show that unit roots can easily be confused with different seasonal means. Moreover, simulations carried out by Abeysinghe (1994) on small samples show that, if a series with seasonal unit roots is regressed on a set of seasonal dummy variables, the sample autocorrelation function of the residuals of that regression behaves like that of a stationary process, though the unit roots are not eliminated. Memory parameter estimates are not affected by the presence of any possible deterministic seasonality because, as shown by Arteche (2002), any such seasonality only affects the periodogram at the seasonal frequency (provided that the length of the series in question is a whole multiple of the number of observations per annum, as is the case here), and that frequency is not used in the estimation process.
Table 3 shows the range of values of m selected and the sample mean of the different estimates for each series. Table 4 shows the estimates of the parameters associated with each seasonal dummy variable and its P-value. A P-value of 0.05 or less means that the variable is significant and therefore the cycle that it represents must be extracted from the series.
Basque Country | Cantabria | Asturias | Galicia | Andalusia | ||||||
---|---|---|---|---|---|---|---|---|---|---|
ω | m | d | m | d | m | d | m | d | m | d |
π/6 | [9,10] | 0.12 | [4,10] | 0.00 | [7,9] | 0.25 | [6,7] | 0.18 | [3,10] | 0.00 |
π/3 | [7,9] | 0.25 | [9,10] | 0.26 | [3,10] | 0.00 | [8,10] | 0.18 | [7,8] | 0.17 |
π/2 | [3,10] | 0.00 | [6,8] | 0.13 | [8,10] | 0.19 | [5,7] | 0.24 | [7,8] | 0.47 |
2π/3 | [4,10] | 0.00 | [8,10] | 0.05 | [6,10] | 0.05 | [3,10] | 0.00 | [8,9] | 0.16 |
5π/6 | [4,5] | 0.16 | [4,6] | 0.13 | [3,10] | 0.00 | [7,9] | 0.11 | [3,10] | 0.00 |
π | [6,10] | 0.01 | [8,10] | 0.54 | [7,10] | 0.28 | [4,10] | 0.00 | [3,10] | 0.00 |
Basque Country | Cantabria | Asturias | Galicia | Andalusia | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Variable | Coeff. | P-value | Coeff. | P-value | Coeff. | P-value | Coeff. | P-value | Coeff. | P-value |
sin(πt/6) | −0.15 | 0.03 | −0.15 | 0.09 | 0.05 | 0.49 | 0.06 | 0.17 | −0.23 | 0.00 |
cos(πt/6) | 0.06 | 0.27 | 0.18 | 0.05 | −0.05 | 0.37 | −0.08 | 0.08 | 0.01 | 0.86 |
sin(πt/3) | 0.00 | 0.99 | 0.16 | 0.00 | −0.09 | 0.02 | 0.04 | 0.08 | 0.09 | 0.02 |
cos(πt/3) | 0.17 | 0.00 | 0.14 | 0.01 | 0.32 | 0.00 | 0.10 | 0.00 | −0.01 | 0.81 |
sin(πt/2) | 0.18 | 0.00 | 0.13 | 0.00 | 0.16 | 0.00 | 0.04 | 0.05 | 0.01 | 0.65 |
cos(πt/2) | 0.01 | 0.86 | 0.01 | 0.80 | −0.02 | 0.54 | 0.02 | 0.48 | −0.01 | 0.69 |
sin(2πt/3) | 0.02 | 0.51 | −0.07 | 0.03 | −0.01 | 0.63 | 0.03 | 0.20 | 0.06 | 0.09 |
cos(2πt/3) | 0.02 | 0.64 | −0.01 | 0.67 | −0.01 | 0.65 | −0.04 | 0.08 | −0.02 | 0.65 |
sin(5πt/6) | −0.01 | 0.72 | 0.01 | 0.74 | 0.04 | 0.12 | 0.02 | 0.47 | 0.12 | 0.00 |
cos(5πt/6) | 0.03 | 0.47 | −0.05 | 0.13 | −0.01 | 0.71 | −0.00 | 0.80 | −0.01 | 0.78 |
cos(π) | −0.02 | 0.35 | 0.00 | 0.89 | −0.01 | 0.50 | −0.07 | 0.00 | 0.04 | 0.21 |
As a check that the series have been successfully deseasonalised, Figure 3 shows the periodograms for the series free from any persistent seasonality (stochastic or deterministic). The periodograms now only show major peaks around frequency zero, indicating that there is a persistent component that needs to be analysed at that frequency.
4.2. Persistence analysis at frequency zero
As occurs in the seasonal case, the two types of persistent component (stochastic and deterministic) can coexist at frequency zero. However, as pointed out at the beginning of Section 'Empirical Analysis', the regression of the first differences for each series on a constant is not statistically significant in any case. This indicates that the hypothesis of the deterministic linear trend does not hold up, so the peaks observed in the vicinity of frequency zero can be seen as due exclusively to the existence of long-term memory. The next step is therefore to use an estimator – LW estimator in our case5 – to obtain a consistent estimate of the memory parameters for each deseasonalised series. Eliminating seasonal peaks makes it easier to estimate these parameters, as the choice of bandwidth is not as restricted as in the seasonal case. This article considers a range of values of m ∊ [10, 30], reducing variance and thus obtaining more stable estimates of the orders of integration (see Table 5). In general, orders of integration that fall within or very close to the non-stationarity region but well below one are observed. Moreover, a high level of homogeneity is observed between the orders for the different series. Overall, the highest order of integration seems to be that for the price series from Galicia, with estimates of around 0.60, though the estimates for Cantabria and Asturias are higher for some bandwidth values.
m | Basque Country | Cantabria | Asturias | Galicia | Andalusia |
---|---|---|---|---|---|
10 | 0.46 | 0.51 | 0.53 | 0.51 | 0.36 |
11 | 0.43 | 0.57 | 0.54 | 0.52 | 0.45 |
12 | 0.49 | 0.65 | 0.58 | 0.52 | 0.54 |
13 | 0.44 | 0.50 | 0.60 | 0.56 | 0.50 |
14 | 0.42 | 0.52 | 0.47 | 0.61 | 0.49 |
15 | 0.42 | 0.58 | 0.51 | 0.61 | 0.45 |
16 | 0.46 | 0.61 | 0.51 | 0.58 | 0.44 |
17 | 0.46 | 0.60 | 0.50 | 0.61 | 0.39 |
18 | 0.49 | 0.54 | 0.52 | 0.64 | 0.41 |
19 | 0.52 | 0.58 | 0.54 | 0.63 | 0.43 |
20 | 0.55 | 0.56 | 0.54 | 0.66 | 0.47 |
21 | 0.47 | 0.59 | 0.54 | 0.64 | 0.49 |
22 | 0.48 | 0.62 | 0.55 | 0.65 | 0.51 |
23 | 0.48 | 0.64 | 0.55 | 0.59 | 0.53 |
24 | 0.51 | 0.67 | 0.57 | 0.61 | 0.57 |
25 | 0.42 | 0.70 | 0.59 | 0.61 | 0.58 |
26 | 0.44 | 0.65 | 0.60 | 0.62 | 0.54 |
27 | 0.44 | 0.59 | 0.61 | 0.62 | 0.55 |
28 | 0.46 | 0.61 | 0.61 | 0.62 | 0.57 |
29 | 0.43 | 0.59 | 0.55 | 0.62 | 0.59 |
30 | 0.38 | 0.59 | 0.53 | 0.63 | 0.62 |
4.3. Cointegration analysis
The price series filtered for all types of seasonality for the Basque Country, Cantabria, Asturias, Galicia and Andalusia are denoted as: , , and , respectively, and a cointegration analysis is conducted. As outlined in Section 'Methods', the cointegrating rank of the five series can be inferred from a recursive procedure, which in our case is based on LW estimators of memory parameters and the Robinson's (2008) X* test statistics for the null of no cointegration. These are semi-parametric procedures, so the robustness of our results to the bandwidth choice will be illustrated by providing results for a wide set of bandwidths (specifically m = 10, …, 30).
The key aspect of Hualde (2012) procedure is to identify in every step a set of possible CTs. As in Hualde (2012), these CTs are chosen such that they present the highest evidence of being true CT. In particular, the CT in Step 1 is chosen to be the variable with highest estimated integration order, whereas in latter steps CTs are chosen according to the lowest value of corresponding test statistics (see Remark 1 in Hualde, 2012). Thus, the choice of CTs in every step depends on the bandwidth m, and this is clearly illustrated in Table 6, where the observables taking the role of CTs in every step vary with m.
Step | 1 | 2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
m | bas | can | ast | gal | and | bas | can | ast | gal | and |
10 | 0.01 | 0.20 | CT | 1.11 | 0.08 | CT | 0.37 | CT | 0.57 | 0.46 |
11 | 0.58 | CT | 0.63 | 0.10 | 0.24 | 0.36 | CT | 0.56 | CT | 0.00 |
12 | 0.50 | CT | 0.70 | 0.31 | 0.44 | 0.72 | CT | 0.86 | CT | 0.06 |
13 | 1.39 | 1.12 | CT | 0.54 | 0.13 | 2.10 | 0.59 | CT | 0.32 | CT |
14 | 0.70 | 0.44 | 1.64 | CT | 0.02 | 0.78 | 0.20 | 0.40 | CT | CT |
15 | 1.15 | 0.68 | 1.57 | CT | 0.18 | 0.81 | 0.30 | 1.02 | CT | CT |
16 | 0.41 | CT | 3.03 | 0.93 | 1.10 | CT | CT | 2.14 | 0.96 | 0.35 |
17 | 0.94 | 1.48 | 1.80 | CT | 0.00 | 0.03 | 0.30 | 0.95 | CT | CT |
18 | 0.66 | 1.47 | 1.85 | CT | 0.02 | 0.02 | 0.57 | 1.10 | CT | CT |
19 | 0.46 | 1.89 | 1.41 | CT | 0.10 | 0.01 | 0.75 | 1.14 | CT | CT |
20 | 0.35 | 1.61 | 1.14 | CT | 0.16 | 0.01 | 0.61 | 1.20 | CT | CT |
21 | 0.21 | 1.59 | 2.14 | CT | 0.05 | 0.13 | 0.35 | 1.74 | CT | CT |
22 | 0.26 | 1.89 | 2.83 | CT | 0.03 | 0.15 | 0.49 | 1.80 | CT | CT |
23 | 0.33 | CT | 1.87 | 1.71 | 2.80 | CT | CT | 1.77 | 1.62 | 0.09 |
24 | 0.27 | CT | 1.66 | 1.61 | 2.45 | CT | CT | 1.55 | 1.44 | 0.05 |
25 | 0.72 | CT | 1.64 | 1.43 | 2.28 | CT | CT | 2.64 | 1.74 | 0.14 |
26 | 0.73 | CT | 1.25 | 1.75 | 4.12 | CT | CT | 2.45 | 1.90 | 0.00 |
27 | 0.00 | 1.30 | 1.47 | CT | 0.09 | CT | 1.29 | 2.25 | CT | 0.19 |
28 | 0.00 | 1.23 | 1.34 | CT | 0.23 | CT | 1.36 | 2.17 | CT | 0.30 |
29 | 0.04 | 1.02 | 0.89 | CT | 0.19 | CT | 1.09 | 1.26 | CT | 0.18 |
30 | 0.01 | 1.29 | 1.41 | CT | 0.24 | CT | 1.60 | 2.14 | CT | 0.52 |
Step | 3 | 4 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
m | bas | can | ast | gal | and | bas | can | ast | gal | and |
10 | CT | CT | CT | 0.38 | 0.38 | CT | CT | CT | 0.25 | CT |
11 | 0.20 | CT | 0.07 | CT | CT | 0.47 | CT | CT | CT | CT |
12 | 0.36 | CT | 0.43 | CT | CT | CT | CT | 1.08 | CT | CT |
13 | 1.54 | 0.73 | CT | CT | CT | 1.10 | CT | CT | CT | CT |
14 | 0.76 | CT | 0.89 | CT | CT | CT | CT | 1.60 | CT | CT |
15 | 0.78 | CT | 1.43 | CT | CT | CT | CT | 2.08 | CT | CT |
16 | CT | CT | 2.23 | 0.93 | CT | CT | CT | 2.16 | CT | CT |
17 | CT | 0.43 | 1.68 | CT | CT | CT | CT | 1.86 | CT | CT |
18 | CT | 0.56 | 1.56 | CT | CT | CT | CT | 2.03 | CT | CT |
19 | CT | 0.72 | 1.29 | CT | CT | CT | CT | 2.05 | CT | CT |
20 | CT | 0.75 | 1.23 | CT | CT | CT | CT | 1.99 | CT | CT |
21 | CT | 1.17 | 3.53 | CT | CT | CT | CT | 3.84 | CT | CT |
22 | CT | 1.30 | 3.75 | CT | CT | CT | CT | 3.99 | CT | CT |
23 | CT | CT | 2.21 | 1.32 | CT | CT | CT | 3.66 | CT | CT |
24 | CT | CT | 1.84 | 1.12 | CT | CT | CT | 3.21 | CT | CT |
25 | CT | CT | 2.53 | 1.09 | CT | CT | CT | 3.51 | CT | CT |
26 | CT | CT | 1.80 | 0.79 | CT | CT | CT | 2.94 | CT | CT |
27 | CT | 0.73 | 3.16 | CT | CT | CT | CT | 2.82 | CT | CT |
28 | CT | 0.84 | 3.24 | CT | CT | CT | CT | 2.74 | CT | CT |
29 | CT | 0.67 | 2.27 | CT | CT | CT | CT | 2.24 | CT | CT |
30 | CT | 1.07 | 3.56 | CT | CT | CT | CT | 2.84 | CT | CT |
To illustrate how our procedure works, we will exemplify it for m = 20, although some of our comments below will apply more generally for other bandwidth choices. First, for m = 20, Table 5 suggests that is the series with the highest estimated order of integration, so this is our choice of CT. Then, for this particular bandwidth choice the first step in the procedure is to test:
against
where for :
Nicely, by Theorem 1 of Hualde (2012), if is a true CT (so it has the maximum integration order), then r < 4 if and only if H(1) holds, and r = 4 if and only if holds. Note that in Step 1 we check for cointegration two by two (specifically, between and a_{t} if m = 20), results for the various combinations of series and m being shown in block 1 of Table 6. Given the level of significance α = 0.05,6 the null hypothesis of no cointegration cannot be rejected for any pair (except for the relationship between and for m = 26), so the conclusion is that, for any bandwidth choice, there is statistical evidence in favour of r < 4.
Focusing again on m = 20, the results of the test suggest that is the ‘least cointegrated’ series with , so this information is used to design step two of the procedure, which involves the following checks:
against
where for ,
The results of the no cointegration test for different m are shown in block 2 of Table 6. Once again, the null hypothesis of no cointegration cannot be rejected for any of the groups of variables (in this case groups of three variables) and any m, so there is no statistical evidence to reject H(2), which would mean that r < 3.
Focusing again on m = 20, the least cointegrated variable with and is , so the sub-set () is selected for the design of step three of the procedure, which entails the following checks:
against
where, as before, for ,
In view of the results of these comparative checks (block 3 of Table 6), the null hypothesis cannot be rejected (for any m) with a significance level of 5%, so there is evidence in favour of r < 2. Also, for m = 20, the least cointegrated variable is (for m = 20), so the final step comprises the following check:
where
With the sole exception of the value associated with a bandwidth of m = 22, there is again insufficient evidence to reject the null hypothesis of no cointegration at the 5% significance level, so it can be inferred that r = 0, that is, there is no linear relationship between mackerel prices in the different regional autonomous communities. It is therefore concluded that the first-sale mackerel market in Spain is divided into five sub-markets, one for each regional autonomous community where the species is landed. Moreover, the price formation process in each of these sub-markets is unaffected by events at the others. The good is homogeneous, so the reasons for this finding may lie in non-competitive behaviour on the part of the markets, resulting in a lack of transparency and potential barriers to trade. In the Basque Country, in particular, there is evidence that the first-sale price formation process at ports is strongly conditioned by the high market power of a small group of agents (wholesalers, processors and retailers) who constitute an oligopsony that hinders the entry of new competitors. These agents act on the value chain by setting the first-sale price, and producers and consumers are price-takers (Mugerza et al., 2011). Another factor that has a great influence on price formation is the auction system (García et al., 2010). Thus, in countries such as France, the adoption of electronic auction systems and remote bidding in some ports has raised both price levels and volatility, guided by an increase in the number of buyers (Guillotreau and Jiménez-Toribio, 2006, 2011). On the other hand, in combination with the mentioned non-competitive behaviour of markets, the low profitability of the sector has led fishermen to breach regulations and systematically exceed TACs in order to maximise profits by increasing catches. This has helped to keep each market distinct from the others. It is worth mentioning here that in 2008 a voluntary quota per fishermen was introduced in the two Basque provinces where mackerel is landed – Bizkaia and Gipuzkoa – in an attempt to control overall landings in the Basque Country. However, this scheme proved ineffective because low profitability led one of the two provinces to breach the voluntary quota agreement. Only in 2011, when compulsory quotas were set and in-port inspections were increased in Basque ports, did fishermen stop exceeding the quota. This has resulted in an increase in prices (according to the AZTI-Tecnalia fishing database). Both the quota and the measures to check on its compliance are backed by the European Union (EU), which is aware that setting such maximum quotas may, if regulations are followed to the letter, lead to an economically sustainable European market. Accordingly, the new CFP envisages strong support for this system of regulation.
5. Conclusions
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
This article analyses possible relationships among the prices of mackerel on the first-sale markets in the ports of Spain's regional autonomous communities where the species is landed and auctioned.
The main objectives are firstly to determine the geographical limits of the first-sale market for mackerel in Spain, and secondly to check for interdependence and price transmission between the different regional markets. This is done by using fractional cointegration and, in particular, a novel procedure developed by Hualde (2012) that enables us to identify any regional sub-markets within the national market. Our findings show that the price formation processes in each individual market have no influence on the others, so we infer that it takes place at local level. In geographical terms, Spain can be thought of as having five independent regional markets for mackerel, in the Basque Country, Cantabria, Asturias, Galicia and Andalusia, that is one for each regional autonomous community where the species is landed.
The non-competitive behaviour of the mackerel market and the low prices obtained at auction in ports have led in recent years to a race to catch more and more fish on the part of the various regional fleets. This has resulted in systematic failure to comply with the TACs allocated to Spain, thus flooding the daily market for the species and preventing any increase in prices at ports. The existence of a joint TAC leads the various fleets to engage in a race to catch mackerel, because the fishery is closed when the quota is exhausted. This independence of action on the part of the different fleets also seems to have favoured the disconnection of local markets. However, the introduction of daily quotas per fishermen or per vessel and the introduction of strict checks on compliance in 2011 seem to have resulted in an increase in prices and a narrowing of the differences in price behaviour between the regions. However, only data for 2011 and 2012 are so far available, so it would be advisable to wait before conducting any in-depth analysis to determine whether there has been any change in the structure of first-sale mackerel prices.
Our findings suggest that the policies implemented to date at European level (through the CFP) with a view to influencing markets from fish are inadequate. With a centralised, top-down approach it is hard to adapt the CFP to the specific characteristics of the EU's different countries and regions. However, the EU seems to be aware of this problem, and the proposal for the reform of the CFP scheduled to come into force in 2013 includes decentralised management. Thus, at least from this point of view, the new CFP seems to be headed in the right direction as it will give Member States greater flexibility to design and implement specific policies in response to specific problems.
- 1
A series z_{it} is said to be integrated of order d or to have a degree of memory d, and is denoted as z_{it} ∼ I(d), if (1 − L)^{d}z_{it} is a process with a finite, non-null spectral density at frequency zero, where L is the lag operator so that L^{n}Z_{it} = Z_{i,t−n}.
- 2
- 3
Note that a difference analysis was conducted because it was initially not known whether was stationary or not. If it is not, and the regression is conducted in levels, the OLS estimators need not be consistent.
- 4
A series z_{it} is said to be I_{ω}(d) if (1 − 2L cos ω + L^{2})^{d}z_{it} is a process with a finite, non-null spectral density at frequency ω.
- 5
LW is consistent for d ≤ 1 and asymptotically normal for d < 3/4. As the degree of memory is not known beforehand, we have also implemented the exact LW estimator as proposed by Phillips and Shimotsu (2005), which allows for larger values of d, obtaining similar results (available upon request). Taking into account that this modification does not improve the properties of the LW for d < 3/4 and the range of estimates we obtain, we continue the analysis based on LW estimates.
- 6
When no level of significance is specified, α = 0.05 is used to calculate the corresponding critical values throughout this article.
- 7
The CT in this step is chosen to be the variable with highest estimated integration order.
- 8
In this and forthcoming steps CTs are chosen according to the lowest value of corresponding test statistics (less evidence of cointegration).
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- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
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Appendix: Hualde's Cointegration Analysis
- Top of page
- Abstract
- 1. Introduction
- 2. The Spanish mackerel fishery
- 3. Methods
- 4. Empirical Analysis
- 5. Conclusions
- References
- Appendix: Hualde's Cointegration Analysis
The procedure is based on Theorem 1 of Hualde (2012). This theorem states necessary and sufficient conditions for the cointegrating rank of a p-dimensional vector of observables, z_{t}, whose individual components have a maximum integration order δ_{p}. In particular, z_{t} has cointegrating rank r ∊ {1, …, p − 1} if and only if there exists a (p − r)-dimensional subvector of z_{t} [say z_{(b)t}] whose individual components are CTs [i.e. I(δ_{p}) and non-cointegrated variables], and all subvectors of z_{t} larger than p − r containing z_{(b)t} cointegrate.
Based on this result, Hualde (2012) proposes a recursive method to estimate the cointegrating rank r of the dimensional vector of observables. This estimator will be denoted as . The procedure consists in the following steps:
Step 1. Estimate the individual integration orders of the observables and choose as a possible CT variable () the one which, in view of its estimated integration order, shows the strongest evidence of being a CT.7 Next, reorder the variables in z_{t} so that in the new ordering (this reordering is irrelevant for the results, but simplifies subsequent notation substantially). Then, given the possible CT, z_{pt}, we test for , , where H_{p,i} : z_{pt}, z_{it} are not cointegrated and is not true, and set . This step checks whether all pairs of variables that contain z_{pt} are cointegrated. If they are, the process ends here because there is statistical evidence in favour of r = p – 1. However, if H(1) is not rejected (because there is no evidence of cointegration for a least one pair of series) the process continues to the next step. Also, note that if z_{pt} ∼ I({δ_{p}}) then by Hualde's Theorem 1, are equivalent to r < p − 1, r = p − 1, respectively, which justifies our estimator of r.
Step 2. If H(1) is not rejected, choose so the possible CTs are z_{pt}, . As in the previous step, is chosen such that the couple z_{pt}, shows the strongest evidence (in view of the test for no cointegration) of being a CT.8 Reorder again the variables so that , in the new ordering. Then we test for , (that is, we test whether sets of three variables containing the CTs are cointegrated or not) and estimate the rank by .
If H(2) is rejected the process comes to an end and it is concluded that . If not, the process continues, and checks for cointegration are made in vectors of four elements. Note that if z_{pt}, z_{p−1,t} are valid CTs, then H(1) ∩ H(2), are equivalent to r < p − 2, r = p − 2, respectively, which justifies .
Proceeding in this way, if none of the H(j), j = 1, …, p − 2, are rejected we arrive at the last step.
Step p − 1. If H(p − 2) is not rejected, choose . Note that in previous steps the variables have been reordered so that , so is chosen such that the variables show the strongest evidence (in view of the test for no cointegration) of being CTs. Reorder the variables so . Then test for H(p − 1):H_{p,p−1,…,2,1}, , and set if H(i), i = 1, …, p − 2, are not rejected and H(p − 1) is rejected, or set if H(i), i = 1, …, p − 1, are not rejected.