Journal of Policy Modeling
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Price Instability, Trade and Futures Markets*

C.W. Morgan


The author is a member of CREDIT and lecturer in the Department of Economics, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. Tel: (0602)515473 or fax (0602) 514159.

* An earlier draft of this paper was presented at seminars at the Claremont Graduate School, and the University of Exeter. The author is grateful to participants at those seminars for helpful comments. He would also like to thank Professor David Greenaway for his constructive comments although all errors are the author's alone.


This paper models the degree to which free trade can cause price volatility to increase in a market for a soft commodity particularly when there has been a policy shift from barrier induced autarky to completely free trade. Given the possibility that it may increase volatility, can the introduction of a futures market for that commodity provide a means for reducing price volatility. Using the experience of the British maincrop potato market and the London Potato Futures Market as examples, this paper shows that whilst trade may indeed lead to increased volatility, the futures market does reduce intra-seasonal price volatility but has little impact on inter-seasonal volatility.

Key Words: commodity trade, price instability, futures markets

JEL Classification: G13, F14, Q18

I    Introduction

The completion of the Uruguay Round of the GATT in December 1993 marked a significant point on the path towards greater trade liberalisation and a move away from protectionist polices for many products. A key area of the agreement was the agricultural sector which has been characterised by high levels of price support and protection in most of the major developed economies, such as the EU, USA and Japan. One reason for high protection in those economies was that governments believed that instability was to be avoided and that producers benefited from price stabilisation (Tyers, 1990). The hidden assumption was that free trade was viewed as destabilising prices. The GATT has sought to adjust these policies to promote greater trade and lower consumer prices (Rayner, Hine and Ingersent, 1993). However, the reforms also imply less certainty for the individual producer and possibly greater price volatility as guaranteed prices are removed as trade barriers fall.

Other broader measures aimed at price stabilisation have also declined in popularity as the political environment changes. Marketing boards, for example, are viewed as hindering market forces (although they can be a source of funds in developing economies (Ridler, 1988) and the use of international commodity agreements has come in to disrepute due to an inherent lack of belief in their ability to stabilise prices, and also due to cheating by member nations.1 It is apparent, therefore, that alternative means of price stabilisation should be sought perhaps with an emphasis on individual rather than collective schemes and also on schemes that do not distort underlying market forces. The answer may lie with the use of futures markets.

This paper addresses these issues. In Section Two, a theoretical framework outlining the impact of free trade on price volatility in a previously protected market is developed. In Section Three a brief review of futures markets will provide the necessary background to the empirical study in Section Four which will centre on Great Britain's maincrop potato market. Finally, Section Five will offer some conclusions and discuss the wider implications of the work.

II    Liberalisation of Trade and Price Volatility

The theoretical model is that of Greenaway, Morgan, Rayner and Reed (1993). They devise a two country trade model that analyses the impact of a shift in policy regime from an autarchic position to one of free trade, in terms of price volatility in both economies. Two key components of the model are the relative sizes of production and also the correlation between production levels. These turn out to determine the impact of liberalisation upon the importing economy.

Assume that the two countries have identical linear and non-stochastic demand curves, with prices being denominated in the currency of country 1 2


but that their supply curves differ. Specifically, assume that output is of a crop with stochastic yield per hectare and that cultivated areas are not price-responsive. We then have


where Hj is the area planted, j is the mean yield, and ej is the disturbance to that yield. Let the mean yield be the same in both countries, and let the yield disturbances have zero mean, identical variances of 2e , and correlation coefficient . If the area planted in country 2 is greater than that in country 1 by a factor of z (z > 1) then we may write the supply functions as



where H is the area in country 1, u = He1, and v = He2. It follows that the variance of u is , and that the variance of v is

The self-sufficiency prices in the two countries will be




which will have means (A-G)/ and (A-zG)/ and variances



respectively. Since z > 1, country 2 will be the exporter 'on average', i.e. when u = v = 0. The common price under free trade is

which has variance

The change in the price variance on the move from self-sufficiency to free trade may be measured for the importing country by

and for the exporting country by

Note that these changes depend only on the 'scale parameter' z and the correlation between the disturbances, . With must always be greater then one: the move from self-sufficiency to free trade will always reduce the variance of prices in the exporting country. The value of is however indeterminate; for example, with high positive values of and high values of z free trade will increase the price variance in the importing country, but with negative values of and moderate values of z free trade will decrease the price variance. The combinations of and z that leave the price variance in country 1 unchanged are given by

We shall assume in what follows that a reduction in the price variance is indeed a gain to the country concerned (shifts in the mean price may be obtained by various instruments that do not affect the variance).

The partial derivatives of with respect to and z are both negative, as is the partial derivative of with respect to . The partial derivative of with respect to z however is negative only if   > -1/z. The signs of the two partial derivatives with respect to the correlation are intuitively plausible. The correlation obviously does not affect the price variance under self-sufficiency, so the changes in and are due solely to changes in the price variance under free trade. Other things being equal, the more negatively correlated are the output disturbances, the lower will be the variability of total output and so of the free trade price. The sign of the partial derivative of with respect to the scale parameter is equally simple to explain, since changes in the scale parameter do not affect the importing country's self-sufficiency price variance. Provided that the output disturbances are not perfectly positively correlated, the importing country will enjoy more stable prices under free trade when the difference in size is relatively small. However, if the exporting country is large then the variability in its output will dominate the variability in the importer's output; even if the correlation is negative, the net effect may still be to increase price variability in the importing country.

That the sign of the partial derivative of with respect to the scale parameter should change as z increases when is sufficiently negative is not quite so obvious. The reason for the sign change is that a change in z changes both the numerator and the denominator. As z increases then so do the price variances under both self-sufficiency and free trade. When the correlation is positive or weakly negative then the increase in price variance is always proportionately larger under self-sufficiency than under free trade. However, when the correlation is negative and the scale difference is small, the impact of a change in z on the free trade price variance will be relatively small compared with the effect on the self-sufficiency variance.3

That is, other things being equal, a higher correlation between the supply disturbances will reduce the importing country's gain (if the difference in scale is small) or increase its loss (if the scale difference is large), and will always reduce the exporting country's gain. For a given correlation, a greater difference in scale will reduce the importing country's gain or increase its loss, and will reduce the exporting country's gain unless the output disturbances are sufficiently negative correlated.

Figure 1 shows the outcomes for the importer and exporter for a range of values of p and z. There are four possibilities. In area A the move to free trade increases the price variance faced by country 1, and reduces the price variance in country 2. If a reduction in variance is a 'good thing' then the importing country loses and the exporting country gains. The greater the difference in scale, the greater are both changes. The higher is however, the greater the loss for the importing country but the less the gain for the exporting country. In area B the move to free trade again increases the price variance faced by country 1. The more negative is , the smaller the loss for the importing country and the greater the gain for the exporting country. A greater difference in scale increases the loss made by the importer while reducing the gain made by the exporter. In area C the move to free trade decreases the price variance faced by both countries - that is, both gain. The smaller the difference in scale, and the more negative is the correlation, the more both countries gain. Finally, in area D the move to free trade again leads to gains for both countries. An increase in the difference in scale will increase the gain to the importing country but decrease the gain to the exporting country. The more the supply variations in the two countries move together, the lower the gains to both.

The key conclusion to be drawn from this analysis is that the introduction of trade liberalisation will have an uncertain impact on price volatility in the importing market, the uncertainty being a function of production correlation and the difference in the area planted to a crop. The implication of this is that some form of stabilisation scheme may be appropriate. However, as noted above, the majority of such schemes involve market intervention and as such may generate significant by product distortions. An alternative solution is to introduce a market based policy, and that could be a futures market.

III    Futures Markets

The development of futures markets and their utilisation since 1945 has been quite striking (Yamey, 1984), not only in unprocesed physical commodities but also in financial markets and their derivatives. In this paper we shall only be interested in the former 4. Futures markets exist for many soft commodities, the trade in which affects many different economies. Some futures markets are established for commodities where there is little or no trade such as pork bellies, potatoes and wool. Any commodity can in principle be traded on a futures market. However, as Young (1978)demonstrated, several key features are a pre-requisite to the establishment of a futures market, these are; price instability in the underlying spot market; uncertainty of sources and scale of supply; the possibility of standardised grading; a lack of market information; finally the scale of potential trade must be large enough to attract non-trade interest (i.e. speculators). Basically, the need for futures markets arises from a failure of forward contracts to provide the degree of fungibility most traders desire.

Trade on a futures market is carried out through the medium of exchange of highly standardised contracts which implies a fungible market with the possibility of closing out positions 5. Fundamental to the efficient operation of any futures market is the relationship between the spot and futures price, known as the basis. The futures price must reflect the underlying spot market conditions and as such it is a prediction of the spot price for some future date taking in to account storage costs. If this link is not maintained then arbitrage between the markets will force the basis back to the expected level or the market will collapse as Paul, Kahl and Tomek (1988) showed for the New York potato market. A well functioning basis allows a futures market to perform its two main roles as defined by Telser (1981), price insurance and price formation. Price insurance implies the trader swapping spot price risk for basis risk by strategic use of closing out. Generally, the mean and variance of the basis are less than the spot price thus implying the trader will neither suffer losses but nor will he receive windfall gains. The price insurance role has been examined by inter alia Telser (1981), Kamara (1982), Stiglitz (1983) and Newbery (1983).

Underlying the insurance role is the closing of the basis over the life of the contract which implies that the futures price reacts to information changes, or in other word the market is performing its price discovery role effectively. The literature on this has centred on two major themes - Goss (1992), Kawai (1983), MacDonald and Taylor (1988) inter alia have examined rational expectations within a futures market framework whereas Goss (1990), Leuthold (1979), Burns (1983), Ennew, Morgan and Rayner (1992) have tested individual futures markets for the efficient markets hypothesis. There are various degrees of efficiency and as Telser (1981) shows, there is little or no likelihood of futures and spot prices moving in parallel, suggesting that market "noise" prevents perfect efficiency.

Provided, however, that the market is essentially efficient, then it will provide a price formation function that will underpin the price insurance role. The latter then suggests that the individual will be able to reduce price and income risk without recourse to a guaranteed price from a marketing board or commodity agreement by using a market based mechanism which does not affect the fundamentals of the market. To illustrate how this may arise, we will now examine the British maincrop potato market and investigate how the establishment of a futures market has affected price volatility.

IV    Empirical Analysis

Studying the British maincrop potato market is interesting for three reasons. First, it is a market where supply control and support buying are operated by a compulsory marketing organisation, the Potato Marketing Board 6. Second, until 1979, these policies were carried out in a closed market as trade was banned. However, the European Commission ruled this to be anti-competitive and obliged Britain to lift the ban in 1979. Third, in 1980, the London Potato Futures Market (LPFM) was established, thus providing a source of potential price stability for traders in the physical commodity.7

Referring to Section II above, it is difficult to identify empirically which of the situations applies in the case of the British potato market, the relatively high correlation in yield disturbances (Morgan, 1991) and the size of the British market relative to the EC as a whole (around 14% of EC production) suggests that the British market may provide an example of a situation in which free trade might be expected to increase price variability. That is to say, from a theoretical perspective, the removal of the import ban on potatoes in 1979 will have increased price variability in the British market and reduced price variability in European markets.

For a futures market to be an effective mechanism for risk reduction, it must perform the price discovery function. The ability of the market to act as a predictor of future spot prices depends upon the quality of information flows between the two markets. This in turn will be influenced by both the volume of trade and the nature of the futures commodity relative to the spot commodity. Adequate trading volumes are necessary to eliminate the dramatic price fluctuations which often exist in thin markets and a close relationship between the futures commodity and the spot commodity is required if agents in the market are to view the futures price as representative of the spot price 8.

Given the importance of price discovery, the first stage in any analysis of the effectiveness of the LPFM requires an investigation into its ability to perform the price discovery function. If a futures market is performing its price discovery role efficiently, then any fluctuations in spot and futures prices should be positively correlated to reflect this flow of information. In the case of the LPFM, empirical work to date has tested the efficient markets hypothesis (Sheldon, 1987) and the effectiveness of hedging strategies (Entwhistle, 1987) but not, the ability of the market to perform its price discovery role.

The concept of price discovery implies that the futures price embodies all available information and thus is the best predictor of the future spot price. Evidence on the extent to which future prices respond to market information is provided by Ennew, Morgan and Rayner (1993) who show that market prices are responsive to relevant information such as quota decisions and yields. This analysis can be extended to examine more explicitly the ability of the futures market price to predict future spot price. This analysis is conducted over the period of the markets operation, using 35 data points (9 years by 4 delivery dates except May 1981 where no trading was operative). The specific objective is to examine the ability of the futures price 4 months prior to delivery (FT-4,t) in month T in the marketing year t to predict the spot price at contract maturity (ST,t).Univariate analyses of the time series shows that futures and spot prices had comparable time series properties. Specifically, they both contained deterministic "seasonal" components (relating to delivery) and neither contained either a non-seasonal unit root or a unit root at a seasonal lag 9. In other words, both series could be represented as comprising seasonal means plus stochastic components that were stationary in levels and across delivery dates. Thus, a necessary condition for market efficiency, that of comparable time series properties, is met by the futures and spot price series.

The deterministic component was removed from both series and the stochastic component of ST,t, (EST,t) regressed on the stochastic component of FT-4,t, (EFT-4,t)10. The predicted values from the regression were added to the relevant deterministic components to give the predicted spot price values SHT,t. Figure 2 graphs these against the actual spot prices ST,t. It is clear that there are two aberrant periods - April and May of 1983/4 and of 1987/8 - but that otherwise the predictive ability of the information contained in the futures price is quite reasonable11.

In 1983/4 spot prices in April-May were substantially above levels that could be predicted from futures contracts struck earlier in the season but this might have arisen from an abnormally low yield giving rise to high prices in that season. For 1987/8, spot prices in April and May were substantially below levels that could be predicted from prior futures contracts. This divergence was sufficiently large as to provoke an investigation into trading behaviour and calls for the suspension of trading due to fears that the market had been cornered. Subsequent investigation revealed no irregularities. A possible explanation for the occurrence of this situation lies with the reduced volume of trade in that season, in combination with imports which were higher than normal. If this was the case, then it is interesting to note that the higher level of imports was not reflected in lower futures prices and this may suggest some problems in relation to information flows onto the futures market. While the analysis above suggests that futures prices on the LPFM have generally acted as reasonable predictors of future spot prices, further analysis of the risk management role of the market is required. In particular, there is a need to consider the overall impact of the LPFM on price variability as well as a more specific evaluation of the hedging performance of the market.

As argued in section II, there is a strong theoretical case to suggest that inter-seasonal price instability would increase in the British market as a consequence of liberalising trade in potatoes. At the same time, it can be argued that the introduction of futures trading might be expected to reduce inter-seasonal price instability. Since both of the events occurred virtually simultaneously, it is difficult to disentangle the relative effects of futures trading and free trade. Nevertheless, a comparison of inter-seasonal price instability pre and post 1980 is instructive. Furthermore, supporting evidence may be gleaned from an examination of price variability in Europe, given that theoretical arguments suggest that an increase in price variability in Britain would be accompanied by a reduction in price variability in Europe. Table 1 reports a simple coefficient of variation for prices for five countries for the periods before and after trade liberalisation. The table also reports the ratio of the coefficient of variation in price to the coefficient of variation in yield. The latter measure may be more reliable since it controls for the effects of yield variability l2 in a systematic fashion, whereas the simple coefficient of variation only controls for yield variability by the exclusion of outlying observations. Hypothesising that liberalising trade would increase variability in Britain and reduce variability in Europe, but that the introduction of futures trading in Britain may offset the trade induced increase in variability, we would expect to find evidence of a reduction in price variability in Europe and also in Britain. The results from an examination of the simple coefficient of variation on real price are counter to what might have been expected. The figures indicate a reduction in variability in Britain in the period following the removal of the import ban and the introduction of futures trading, and an increase in variability elsewhere in Europe. However, when the ratio of price variability to yield variability is considered, there is some support for the basic hypothesis in that there is evidence of a reduction in price variability in Britain, Germany and Holland, with only Belgium showing signs of increased price variability.

It may be worth noting that given the existence of area control in the British market, inter-seasonal pricing will be underpinned by the knowledge that there are unlikely to be substantial variations in plantings and thus the role of the futures market in guiding price formation across seasons may be somewhat limited. Accordingly, it might be expected that the impact of futures trading on inter-seasonal price variability will be relatively small.

However, the futures market may play a more significant role in relation to intra-seasonal price variability. Price instability within seasons arises partly as a consequence of variable demand patterns, but primarily as a consequence of variations in quantities available to the market at any one point. The impact of trade liberalisation on intra-seasonal price variability is unclear. However, given that free trade removes any certainty about the total supply to be moved in any given season, there is an intuitive case for arguing that liberalising trade may increase price variability. Furthermore, given that the production cycle is basically the same for all northern european producers, the opportunities for imports to reduce price variability by smoothing out supply changes across the season is limited. By the same token, it is arguably in relation to intra-seasonal pricing that the futures market for a perishable commodity will display its most significant risk reducing properties.

Through the process of arbitrage, a futures market will be linked to the cash market for a commodity. However, the introduction of futures trading will not necessarily decrease price volatility in the spot market. Conceptually, speculators add liquidity to the market so allowing spot traders to hedge and price variability to be smoothed. Speculation itself can stabilise prices if speculators buy when prices are low and sell when prices are high (Friedman 1953). However, if speculators respond to price movements, buying/selling only after prices have changed then volatility may be increased (Baumol 1957). The introduction of a futures market might be expected to improve the price discovery mechanism - the processing of information with regard to price determination. Consequently, the price efficiency of the spot market might be expected to increase. However, spot market volatility might increase if futures markets are distorted by technical factors or manipulation, or suffer from a lack of liquidity, or agents investing in futures are not well-informed (Figlewski 1983).

Provided we are prepared to assume that liberalising trade does not reduce intra-seasonal price variability, the risk reducing properties of the LPFM can be examined through an analysis of the behaviour of spot market prices pre and post 1980. The data set consisted of weekly cash prices from September to May for each of the production years 1969-91. The null hypothesis of a unit root for the spot price within the marketing season was not rejected for practically all years. 14

A monthly volatility series was constructed based on the Figlewski (1983) measure: 15

where sit is the (Figlewski measure of) volatility in month i in (production) year t

pitj is the price in week j in month i in (production) year t
nit is the number of weeks in month i in (production)year t

Given substantial inter-seasonal price variability, this series was divided by average monthly price to give a 'normalised' price volatility measure akin to a coefficient of variation, vit:

vit = sit/pit
where pit is average price in month i in (production)yeart

Table 2 lists mean values for vit by month for the period 1969-79 prior to the futures trading and the period 1980-91 after the introduction of futures trading. It is apparent from these figures that volatility varies across the different months of the marketing season, being higher at the beginning when information concerning the size of the crop is under evaluation and at the end when imports of 'new' potatoes affect the market. However, the aggregate picture indicates that normalised spot price volatility is lower in the latter period.

The simple analysis of average levels of volatility may only give a partial insight into the degree of price variability pre and post 1980 and this again is due to the need to incorporate the effects of yield variability in a systematic fashion. There tends to be an association between yield and the quality/storability of the crop. Years in which yields are low (and prices high) may be characterized by a higher degree of price volatility than years in which yields are high, simply because of the poor quality of the crop and the greater potential for deterioration in store. Thus to accommodate such effects it may be appropriate to examine the relationship between volatility and the price level and the a priori expectation would be that a positive relationship would exist between the two. In order to investigate this, the logarithm of volatility was regressed on the logarithm of price and monthly dummy variables;

where i = 1..9 denotes each month in the marketing season from September to May. Table 3 presents estimates of this equation for the full sample period 1969-91 whilst Tables 4 and 5 present estimates for the two sub-samples 1969-79 and 1980-91. The F statistic is 3.9 for the Chow test of the null hypothesis that the coefficients are identical in the two sub-samples. The 5 per cent (1 per cent) critical value from the F-distribution with (10,187) degrees of freedom is 1.83 (2.32) so that the null is rejected and the models for the two sample periods are considered to be superior to a single model for the whole period.

The sub-sample regressions are acceptable in that the relevant diagnostics do not reject the null hypotheses of no autocorrelation and homoscedasticity.l6 Both suggest that there is a significant and positive relationship between the level of price volatility and the level of price. A comparison of the regression parameters across the sub-samples indicates that price volatility is lower in the period 1980-81 than in the period 1969-79. The response of volatility to price is also diminished in the second sub-period compared to the first sub-period.

V    Conclusions

Futures trading provides an alternative to forward contracting or government intervention as a means of managing the risk associated with producing and trading in soft commodities. The risk management and risk reduction functions of a futures market are, however, dependent on the ability of that market to provide a forum for price discovery. The price discovery features of a futures market are generally considered to be greater for storable than for non-storable commodities, although this does not mean that futures markets for non-storable commodities cannot perform forward pricing and resource allocation functions. An analysis of the forward pricing and risk reducing properties of the London Potato Futures Markets provides evidence of some success in forward pricing and risk reduction, although in the latter case, the effect is most noticeable within rather than between seasons.

However, the LPFM is by no means an ideal market. Some of the more obvious market distortions have been discussed in this paper and there is a body of evidence that highlights problems in the market relating to quality specification (Anderson, 1989), production and storage hedging (Ennew, Morgan and Rayner, 1992) and institutional mystique/lack of awareness (Ennew and Morgan, 1991). Despite these problems, the evidence presented in this paper suggests that in aggregate terms, futures trading has had some success in risk reduction. In that context, there may be a role for futures trading as a mechanism for stabilization in relation to a variety of other soft commodities. However, if such a system is being considered as an alternative to intervention or forward contracting then it is clearly important to create an institutional framework which encourages clear links between spot and futures prices. Furthermore, given the mystique surrounding futures trading (Paul, Kahl and Tomek, 1981), there is a clear need to provide active encouragement to growers and merchants to use futures markets in order to ensure the necessary volume of trade to allow the market to function effectively.

Much of the discussion of futures markets relates to developing countries - are there any lessons to be learned here? One in fact has to be cautious. Futures markets are not simple in their operation or their role particularly in countries where financial markets are not well developed. Use is costly and the lack of efficiently functioning credit markets and their limited scope implies that their potential benefits are far greater than their actual benefits if markets were to be established. However, this may suggest that the authorities in developing economies should use existing futures markets and trade on behalf of their producers, providing that any benefits are distributed to the producers. As such, these markets may yet play a significant role in the policy portfolio of developing economies.


1Personal communication with C.L. Gilbert, March 1994.

2The subsequent analysis assumes that the exchange rate is fixed.

3The gains to the exporter are 'high' when there is a strong negative correlation. However, the gains relative to self-sufficiency diminish as production from the exporter becomes relatively larger since the exporter has to absorb proportionately more of its own supply fluctuations.

4The basic features and operation of all futures markets are essentially the same. Differences usually only arise in specification of the contract and settlement of the contract at maturity e.g. whether it is cash settlement or physical delivery.

5For an introduction to futures markets and their operation see inter alia Goss and Yamey (1982)

6For a discussion of PMB policy see Marsh (1985) or Morgan (1991).

7It is important to note at this stage that this paper is not seeking to determine the stabilising impact of the PMB's policies other than the change to a free market environment. Indeed, it has been shown by Ennew, Jennings, Rayner and Reed (1985) that the PMB has had little success in stabilising prices for producers.

8Differences in quality between spot and futures markets are a cause of concern in relation to the LPFM. The more stringent quality specifications employed by the futures market will inevitably drive a wedge between futures and spot price. However, if the quality premium is constant, this wedge need not detract from the price discovery and risk management functions of the market.

9A variable Xt is said to be integrated of order (d, D)denoted by Xt ~ I(d, D) if the series requires differencing d times and seasonally differencing D times to make it stationary. The Augmented Dickey-Fuller (ADF)unit root test and the Dickey-Hasza-Fuller (DHF) seasonal unit root test were employed to test for stationarity. The test statistics are available from the authors.

10The prior removal of seasonal means enforces a zero constant on the regression. The regression corrected for first order serial correlation (rho = 0.3) was:

11 This presupposes that traders would be informed as to the deterministic components (delivery date means).

12 Since a policy of area control operates in the GB market, the main source of price variability is the variability in yield.

13 Both 1975 and 1976 were characterised by severe drought in Europe, with the result that potato yields were unusually low and prices were unusually high. However, given that Eurostat data is provided only on an annual rather than a crop year basis, only 1976 was identified as an outlier.

14The hypothesis that the weekly price series within the marketing season had a unit root was rejected at the 10 percent level for only the 1973, 1975 and 1986 production years.

15 If the price series is a random walk, pitj = pitj-1 + uitj, then the measure of volatility is simply the standard deviation of the residuals.

16The series Invit and Inpit were also examined for non stationarity for each sub-sample. Residual series were constructed from regressions of Invit and Inpit on the nine monthly dummy variables. The relevant Dickey-Fuller statistics for the residual series were:

The residual series are stationarity and relevant regressions are:

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