Relative Prices and Inflation: New Evidence from Different Inflationary Contexts

Keywords: inflation, relative price variabilty, inflation regimes, inflation volatility, expected and unexpected inflation

This paper analyzes the relationship between inflation and relative price variability, in the direction of the latter, in two countries with very different inflationary experiences: Argentina and Spain. To address this objective, using disaggregated price indexes (the Wholesale Price Index for Argentina and the Consumer Price Index for Spain), we delimitate different inflationary regimes and compute a set of regressions for each country. Our results suggest evidence in favour of the non-neutrality of inflation (mostly in hyperinflation periods) and do not support either the menu costs or the signal extraction approaches. We also detect significant structural changes in the relationship depending on the inflationary regime.

Empirical evidence thus supports a positive relationship between RPV and inflation, but there is no consensus about the causal mechanism that generates this relationship. A variety of theories can explain this empirical fact; the prevalent ones are menu costs and signal extraction models. The first emphasizes the role of expected inflation while the second is focused on the effects of unexpected inflation.

On the one hand, the menu costs model assumes that price adjustments are costly, which implies that firms respond using a (S,s) pricing rule. According to this rule, the firm holds its nominal price constant and lets inflation erode the real price of its products until it reaches the lower bound s. Then nominal price is adjusted such that the new real price is equal to the upper bound S. If inflation increases, the firms will widen the distance between the optimal s and S.

Moreover, if menu costs are different among firms, or firms experience specific shocks, staggered price setting will arise, exacerbating the effect of higher inflation on RPV. Therefore, menu costs models suggest a positive correlation between RPV and expected inflation.

The signal extraction model, on the other hand, states that as inflation is not always anticipated correctly, it creates "misperceptions" of absolute and relative prices generating an increase in RPV. Hence, increased unexpected inflation will raise RPV.

Empirical evidence in this area is mixed, in the sense that there are studies supporting the menu costs model, the signal extraction model, both or neither of them. The existing literature is huge, in particular for the US. The results of some of the most relevant papers are summarized in the following table :   Table 1 Finally, some authors attend to the inflation volatility as a determinant of RPV, especially in inflationary economies. In this sense, [START_REF] Logue | A Note on the Relation between the Rate and Variability of Inflation[END_REF] F o r P e e r R e v i e w 3 and [START_REF] Blejer | Inflation Variability in Latin America: A Note on the Time-Series Evidence[END_REF] for several Latin American countries, [START_REF] Moura | Inflaçao e Preços Relativos: Medidas de Dispersao[END_REF] for Brazil and [START_REF] Dabús | Inflationary Regimes and Relative Price Variability: Evidence from Argentina[END_REF] for Argentina find a positive relationship between RPV and different measures of inflation volatility. This result holds for stable economies, as has been shown for US by [START_REF] Chang | Further Evidence on the Variability of Inflation and Relative Price Variability[END_REF], and as we will show in this paper for Spain.

In short, there is a consensus on the positive correlation between inflation and RPV, but not on which factor is generating this correlation. This paper presents new evidence on this relationship, trying to see if the mechanisms that are causing it differ depending on the inflationary history of the economy analyzed. In order to address this, we study two economies with very different inflationary experiences: Spain, for the 1985-2001 period, and Argentina, for the 1960-1991 period. The first economy has been historically stable in the last fifty years, especially during the period studied in this paper, in which the monthly inflation rate oscillated in a narrow range between zero and 2%. Argentina, on the contrary, shows a very rich inflationary history: in the last forty years its monthly inflation rate has fluctuated between deflation and hyperinflation.

The empirical results obtained in this paper indicate that the relationship between inflation and RPV changes in extreme inflation; in fact RPV exploded for the hyperinflation period in Argentina. For both countries inflation volatility and unexpected inflation increase RPV, although this result must be interpreted in a different way in each case, as we shall see later. On the contrary, expected inflation positively affects RPV in Argentina but is never significant in the case of Spain, denoting that inflation expectations play different roles in explaining the non-neutrality of inflation depending on the macroeconomic environment. Moreover, the inflation-RPV correlation exhibits significant structural changes across the different inflationary regimes, which verifies that The remainder of the paper is organized as follows. The next section presents the main variables used in our analysis and the empirical methodology followed. Section 3 explains the method applied in order to determine the inflation regimes for each country. In section 4 we specify the main features of the data used. In section 5 we give a brief description of the statistics, and we report the results concerning the inflation-RPV relationship, as well as the results referring to the structural change observed in this relationship.

Finally, section 6 concludes.

Definition of variables and empirical methodology

Definition of variables

Our study is based on the monthly inflation rate, which is used to define different measures of inflation volatility and RPV. Moreover, the inflation rate is separated into expected and unexpected inflation using a forecast equation of inflation.

On the one hand, for inflation volatility, three variables have been defined:

1 - - = t t t IN IN DIN (1) 1 - - = t t t IN IN ABDIN (2) ( ) ∑ - = - * - = 3 3 7 / 1 i i t t t IN IN VARIN (3)
where IN t is the inflation rate at time t. As it can be seen, DIN is the difference between the inflation rates of two consecutive months, ABDIN is the absolute value of DIN, and VARIN is a centred moving average of seven months of inflation. VARIN tries to capture transitory deviations of current inflation from a F o r P e e r R e v i e w 5 certain inflationary environment. In order to choose the lags included in VARIN, we defined the variable with three, four until twelve lags, we ran the regression specified in equation ( 7), and we compared the R 2 and the significance of the explanatory variables obtained in each estimation. As the results were very similar for all of them, we included the minimum number of lags, given the fluctuations observed for inflation in Argentina. 3 RPV is a measure of the non-uniformity of the variations of individual prices, relative to the average inflation rate. It is obtained in quadratic terms, using the weighted sum of the monthly inflation rate of individual prices. RPV can be defined as follows:

( ) 2 ∑ - = i t it it t IN IN w RPV

Empirical methodology

As was stated in the introduction, the goal of this paper is to analyze the links between inflation and RPV. Firstly, in order to study the effects of the inflation rate and inflation volatility on RPV, a set of equations is estimated:

t t t t e DIN b IN b a RPV + + + = 2 1 (5) t t t t e ABDIN b IN b a RPV + + + = 3 1 (6) t t t t e VARIN b IN b a RPV + + + = 4 1 (7)
According to the results obtained in previous contributions (see references in the introduction), the inflation rate and inflation volatility affect RPV positively, therefore b1, b2, b3 and b4 are expected to be positive.

As far as expected and unexpected inflation is concerned, the following equation is estimated:

t t t t e INO b INE b a RPV + + + = 6 5 (8)
In this case, there is neither theoretical nor empirical consensus on their effects on inflation. The sign and the significance of the coefficients can be interpreted as evidence in favour of the menu costs model -if b5 is positive and significant-or the signal extraction model -if b6 is positive and significant.

Inflationary regimes classification

This section classifies the different periods of inflation in several regimes.

For Argentina, following a version of the criterion suggested by Leijonhufvud (1990) for high inflation economies, we have distinguished four regimes: moderate inflation when the monthly inflation rate is lower than 1%-2%, high inflation for the 2%-10% range, very high inflation for the 10%-50% range, and hyperinflation for values beyond 50%.

The methodology to determine such regimes is based on a procedure that divides the total period into different sub-periods of inflation. 5 A "smoothed-out" series from the original series of inflation is obtained as follows:

( )

100 1 12 / 1 6 / 1 12 1 *               -             = ∑ = - i i t t t PI PI SIN (10)
where SIN is the "smoothed-out" series of monthly inflation rate and PIt is the monthly price index at period t. Discontinuities are detected in this series when variations of the "smoothed-out" inflation are larger than three standard deviations from the moving average of inflation, as follows:

( )

2 / 1 12 1 2 12 , 1 1 12 , 1 12             - = ∑ - - - - - t t t t t MAVIN SIN SDESVIN (11)
where SDESVIN is the standard deviation of the moving average of inflation, and MAVIN is the yearly moving average of inflation rate for the twelve months previous to the discontinuity. Thus, this procedure captures only persistent changes, disregarding transitory variations in inflation levels. The next step is to detect changes in the regime of inflation if the following conditions are fulfilled: 1) SINt > MAVINt-1,t-12 + 3SDESVINt-1,t-12, 2) the discontinuity holds for three or more consecutive months, and 3) the average inflation rates between two periods separated by such discontinuity are significantly different, which is checked by a simple test of difference of means. When discontinuities are detected, the months including the "critical points" that fulfil these conditions must be identified in the original inflation rate series.

Finally, once the periods of inflation are obtained, they are grouped in different regimes.

According to the aforementioned methodology, in Argentina each regime contains the following periods:

Table 2 Following the same criterion, for Spain the whole period is classified as moderate inflation -the test of difference of means was not significant in any case. However, for a low inflation country like Spain, some changes in the criterion must be made. In fact, sustained changes of the inflation level can be observed at the beginning of 1992, when the condition SINt > MAVINt-1,t-12 + SDESVINt-1,t-12 was fulfilled. In order to consider the change in inflation in 1992, the total period has been divided into two periods, high and low inflation, for the September 1985-March 1992 and April 1992-December 2001 periods respectively.

Once the inflationary regimes were determined, we have looked for the most significant breaks in the inflation rate history for both countries. According to our methodology these breaks are: February 1975 for Argentina and March 1992 for Spain. Finally, the total period has been divided into two sub-periods. 6

Data

The analysis is based on monthly data. For each country the price index available at the highest degree of disaggregation has been chosen, which is an advantage in order to calculate RPV. 7 According to this criterion, we have used the Wholesale Price Index (WPI) for Argentina and the Consumer Price Index (CPI) for Spain. For the Spanish case, the data cover the period September 1985-December 2001, and have been drawn from the Instituto Nacional de Estadística. We have used 57 categories of the CPI.

The methodology for the collection of data differs for both countries. In Spain most of the price data are collected between the 1 st and the 22 nd day of each month. This methodology can generate a spurious correlation between RPV and inflation in a high inflation context, but not under price stability. 8

Fortunately, in our high inflation case (Argentina), most prices -the prices of industrial and imported products 9 -are collected the same day of each month (the 15 th ), or are sampled as a monthly average from daily (or nearly daily) information -the prices of agricultural products. Hence, for Argentina, correlation between RPV and inflation should not be "contaminated" by the methodology of price collection.

As said, for both countries monthly data are used. For Argentina, as we have chosen WPI, there are no seasonality problems, because most prices, and especially the prices of industrial and imported products, do not present a seasonal component. On the contrary, for the Spanish case, CPI presents an important seasonal component. In order to remove this, an X-12 ARIMA method is applied. Thus, all the estimation results presented in the paper refer

Results

Preliminary analysis

The two countries studied in this paper show very different inflationary experiences. On the one hand, Spain is a stable economy, with a monthly inflation rate ranging between zero and 2% approximately. On the other hand, Argentina is a very unstable economy, with sundry inflationary episodes, going from the moderate inflation of the sixties to the extreme inflation periods of the late eighties (see Figures 1 and2, for Spain and Argentina respectively).

Nevertheless, both economies share a common pattern: higher inflation is associated with higher RPV, a relationship which is even more evident for Argentina.

Figure 1

Figure 2 For the Spanish economy, as was stated in section 3, two slightly different periods of high and low inflation can be distinguished: September 1985-March 1992 and April 1992-December 2001 respectively. Both inflation and RPV are lower in the second period. In Argentina, RPV is clearly increasing in inflation, in particular when the inflation rate increases suddenly. This is verified in the inflationary accelerations of 1962, 1975-76, 1985, and, especially, On the one hand, periods with gradual increases in inflation -for example in Argentina during the 1981-85 period. This evidence seems to be consistent with the intuition that abrupt changes of the inflation rate are required to increase RPV. High inflation volatility should increase the range between prices that adjust jointly with the general inflation and those that are indexed taking into account past values of the inflation rate. In short, the lack of synchronization in price adjustments increases RPV. On the contrary, even at high levels, a gradual increase in inflation would allow the agents to adapt to it, perhaps by means of indexation mechanisms, which may avoid staggering in price adjustments and therefore, even when inflation is increasing, RPV will be decreasing.

On the other hand, the second case refers to periods of stability in which RPV remains at very low levels, for example in the Spanish lower inflation period and during the sixties in Argentina. 10 In short, higher inflation seems to be related to a more volatile and less predictable inflation rate, and to a higher RPV: the behaviour of relative prices and inflation changes at different inflation levels. This can be checked by examining the average values of these variables for each inflationary regime (see Table 3). For all cases RPV is, on average, systematically higher at higher inflation, especially in Argentina. These results show an interesting difference from previous findings. Firstly, unlike [START_REF] Van Hoomissen | Price Dispersion and Inflation: Evidence from Israel[END_REF] for Israel, [START_REF] Palerm | Market Structure and Price Flexibility[END_REF] for Mexico and [START_REF] Tommasi | Inflation and Relative Prices: Evidence from Argentina[END_REF] for Argentina, we find a non-concave relationship between RPV and inflation. Moreover, price dispersion explodes in extreme inflation, therefore there is no evidence of unifying forces of price revisions at hyperinflation. This can be due to the high volatility and inflationary surprises verified in these situations. Indeed, inflation volatility (ABDIN and

The inflation-relative price relationship: the regression analysis

There are some issues to point out before running the regressions.

Firstly, stationarity of the series has been checked by means of the ADF test, 11 for the total period and for the lower and higher inflation periods. It was applied to the original series in Argentina, and to the seasonally adjusted series in the case of Spain. In all cases we found that the series are stationary, except for Spanish RPV when the whole period is considered. In order to deal with this result, we include lags of RPV in the estimations, as is shown in Table 5. In turn, as the ADF test results show the presence of a deterministic trend in Spanish RPV, for the total and lower inflation periods, a trend term has been included in the respective estimations.

In second place, for Argentina the White test shows the presence of heteroskedasticity in some regressions. This problem has been solved using the White heteroskedasticity-consistent variances and standard errors;

nevertheless, the results concerning the significance of the regressors did not change.

Thirdly, in order to tackle autocorrelation in the residuals, we have included lags of the endogenous variable in those estimations where they are required. However, the Breusch-Godfrey (BG) test shows that for some estimations the introduction of lags is not enough to remove autocorrelation. 12 Nevertheless, the BG test results are very sensitive to the number of lags selected, so that the evidence of autocorrelation problems is not conclusive (see footnotes in Tables 4 and5). and VARIN for Argentina. Therefore, in these cases, the results must be taken cautiously.

For both countries we carry out three kinds of estimations: for the total period, and for the lower and higher inflation periods. Recall that, for Argentina, the lower inflation period goes from January 1960 to January 1975, and the higher one from February 1975 to February 1991. For Spain the higher inflation period covers the September 1985-March 1992 period, and the lower one the April 1992-December 2001 period.

Table 4

Table 5 Tables 4 and5 present the results of the regressions for equations (5) to (9). In contrast to previous literature, we offer comparative evidence from different inflationary experiences. It can be seen that there are interesting differences between the two countries at different inflation levels. In Argentina, for all regressions R 2 coefficients and the significance of the explanatory variables are higher in the higher inflation period. On the contrary, for the Spanish case, R 2 is generally higher in the lower inflation period, which is due to the significance of the negative trend; in addition to this, the inflation rate is significant only for the total period. These results suggest the existence of structural changes in the inflation-RPV relationship for both countries across different inflation regimes, as we will test in the next sub-section. There are also some similarities between the two countries. Firstly, as has been shown in previous literature, inflation volatility affects RPV positively. 13 VARIN appears to be the best volatility measure to explain RPV, suggesting that changes in inflationary environment, rather than transitory variations in the inflation rate, affect RPV.

Secondly, regressions including both expected and unexpected inflation indicate that the latter is generally significant, and with the expected sign, except for the Spanish lower inflation period. Moreover, ABINO is also significant, thereby not only unexpected inflation but its magnitude increase RPV. Thus, inflationary surprise seems to be a relevant factor in order to explain RPV, regardless of the average inflation rate, as has been obtained for other countries. However, from our point of view, the interpretation of these results must be different in each case. In this sense, they can support the signal extraction approach in the Spanish higher inflation case, but they seem to be related to high economic instability in the chronic Argentine high inflation, and especially in extreme inflation. indicate the presence of important problems in order to forecast current inflation, as well as high macroeconomic volatility is making increasingly complex the price decisions for economic agents. 14 In fact, RPV exploded in the Argentine hyperinflation, which suggests that there are no successful mechanisms to avoid the impact of inflation on relative prices, like indexation or a "good" model to make expectations on current inflation. In other words, our analysis suggests that agents cannot find an adaptive mechanism to minimize the inflationary surprise associated with those episodes.

In short, our findings are similar to those obtained in previous literature, in the sense that they suggest clearly that inflation is non-neutral, but they go further in that they suggest that non-neutrality is more evident at higher inflation, and particularly in extreme inflation. Moreover, the plausible causes for non-neutrality are different depending on the macroeconomic environment: for stable economies the explanations based on the signal extraction or the menu costs models can be meaningful, but they do not seem to be suitable for unstable economies. Finally, our results appear to support the signal extraction model rather than the menu costs approach, as has been shown for other stable economies like Germany -see [START_REF] Miszler | Inflation and Relative Price Variability in a Low Inflation Country: Empirical Evidence from Germany[END_REF].

Structural change

The different results obtained in the estimations for the whole period and for lower and high inflation periods suggest that there can be structural changes in the RPV-inflation relationship. Firstly, applying the recursive residual estimation and the CUSUM test, we check for both countries if there is one or more structural changes. Once the breaks have been detected, the Chow test is applied in order to verify the results. Finally, we compare if the breaks detected in the RPV-inflation relationship with this methodology are the same found by means of the method used in section 3 to delimitate the inflationary We check for all cases that there is a structural change in that relationship when there is a change in the inflationary regime -recall that the most relevant breaks that we found were February 1975 for Argentina and March 1992 for Spain-as is shown in Table 6.

Table 6 As can be seen in Table 6, structural changes are significant at 1% level of confidence, which seems to prove our hypothesis that the effects of inflation, its volatility and the components of expected and unexpected inflation are different at low and high inflation. In fact, as has been mentioned, in general these changes are associated with higher significance of the explanatory variables, which supports the idea that the non-neutrality of inflation is more evident with higher economic instability.

In conclusion, the structural change seems to support the hypothesis that the determinants of RPV -and their relevance-change at different inflation levels, even in a narrow range of inflation, as in the Spanish case.

Concluding remarks

This paper analyzes the relationship between inflation and relative prices for two economies with very different inflationary history: Argentina and Spain.

While the former shows high price instability, the latter is characterized by a low and stable inflation. Our findings support the hypothesis of the non-neutrality of inflation, which is more evident at high inflation, and particularly at extreme inflation (for example, in both Argentine hyperinflations). Moreover, there is a non-concave relationship between RPV and inflation in Argentina, denoting that inflation affects RPV more than proportionally beyond a certain threshold of inflation.

The main determinants of RPV are inflation, inflation volatility and unexpected inflation. RPV, which suggests that changes in the inflationary environment have a greater effect than transitory deviations of the inflation rate from its trend on relative prices. Our results can support the signal extraction approach in low inflation; however, in extreme inflation they seem to show that there are notable problems in forecasting the current inflation rate, as well as in taking price decisions.

Comparing the results for the two economies we find an interesting difference: expected inflation is only significant in the high inflation country (Argentina). Apparently, this result neither supports the menu costs approachthere are no nominal rigidities in high inflation-nor the signal extraction approach. Instead, this latter approach seems to be suitable for Spanish results, where only unexpected inflation is, in general, significant.

Finally, the inflation-RPV relationship exhibits significant structural changes across the different inflationary regimes, which seems to prove the hypothesis that the determinants of RPV, and their relevance, are different at different inflation levels.

This research can be developed in several directions. A first extension is to include new economies in the sample, in order to check if our results hold.

Another extension is to analyze the role of higher moments of the price change distribution, like skewness and kurtosis. 16 This extension could determine if the inflation-RPV relationship is influenced by them, as suggested by [START_REF] Bryan | Inflation and the Distribution of Price Changes[END_REF]

Notes

(1) Likewise, an important strand of the literature focuses on intramarket RPV.

This variable can be defined as the standard deviation of relative price changes of a given product across stores around the average inflation rate of that product. A number of authors have found evidence supporting a positive correlation between intramarket RPV and inflation: [START_REF] Domberger | Relative Price Variability and Inflation: a Disaggregated Analysis[END_REF] for the United Kingdom, [START_REF] Lach | The Behaviour of Prices and Inflation: An Empirical Analysis of Disaggregated Price Data[END_REF] for Israel, [START_REF] Amano | Menu Costs, Relative Prices, and Inflation: Evidence for Canada[END_REF] for Canada, and [START_REF] Parsley | Inflation and Relative Price Variability in the Short and Long Run: New Evidence from the United States[END_REF] for some cities of United States (US).

However, when economies are experiencing very high inflation rates, intramarket RPV can decrease when inflation increases, as has been shown by [START_REF] Dazinger | Inflation, Fixed Cost of Price Adjustment, and the Measurement of Relative Price Variability: Theory and Evidence[END_REF] and [START_REF] Van Hoomissen | Price Dispersion and Inflation: Evidence from Israel[END_REF] for Israel, [START_REF] Tommasi | Inflation and Relative Prices: Evidence from Argentina[END_REF] for Argentina and [START_REF] Caglayan | Nonlinear Impact of Inflation on Relative Price Variability[END_REF] for Turkey.

(2) Nevertheless, [START_REF] Fischer | Relative Shocks, Relative Price Variability, and Inflation[END_REF][START_REF] Fischer | Relative Price Variability and Inflation in the United States and Germany[END_REF] and [START_REF] Bomberger | Inflation and Relative Price Variability: Parks´ Study Reexamined[END_REF] assert that the relationship between inflation and RPV found for US is dominated by energy and food price shocks.

(3) In fact, the key factor is to determine how many months "around" the current inflation make a homogeneous period of inflation. As has been said, for Argentina we select only three months, because given the great changes observed in its inflation rate, adding more lags could lead us to include months belonging to different inflationary contexts, and therefore there would be a risk of overestimating this measure. For Spain this problem is not so important, but as the results of the estimations are similar with three or more lags, we have chosen the same number of lags as for Argentina in order to compare the results. (4) For Argentina a slight variation of equation ( 4) is used, because, as was stated in [START_REF] Dabús | Inflación y Precios Relativos: Estudio del Caso Argentino[END_REF], equation ( 4) is not the best measure for RPV in high inflation economies. In this context, the estimation of the coefficient of variation of the price change distribution is required, instead of the simple variance, because at high inflation the latter is spuriously correlated with the mean of the distribution -the inflation rate. To avoid this problem, RPV for Argentina is defined as follows:

( )

( ) 2 2 1 t i t it it t IN IN IN w RPV + - =

∑

(5) For a more detailed explanation of this methodology see [START_REF] Dabús | Inflación y Precios Relativos: Estudio del Caso Argentino[END_REF]. An alternative approach to delimitate inflationary regimes could be based on changes in the variance of the inflation series, for example using the ICSS algorithm developed by [START_REF] Inclan | Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance[END_REF] and [START_REF] Sansó | Testing for Changes in the Unconditional Variance of Financial Time Series[END_REF]. In further research we intend to compare both approaches for data from several countries.

(6) Given the size of each sub-period for Argentina, it is not possible to develop the analysis for each one.

(7) The degree of disaggregation can affect the estimations; therefore, in order to compare the results for both countries, homogeneity in the degree of disaggregation is required, otherwise different results can be obtained due just to the different kind of data used. Another distortion can be introduced in the results by the fact that we are using two different price indexes. On the one hand, this problem is not avoidable because the same price index, with a similar degree of disaggregation, is not available for both countries. On the other hand, [START_REF] Caraballo | Inflation and Relative Prices. Empirical Evidence for the Spanish Economy[END_REF] carry out a similar study for Spain using two price indexes, Producer Price Index (PPI) and Consumer Price Index (CPI), with a similar degree of disaggregation (25 categories for PPI and 33 Moreover, the estimations using CPI with different degree of disaggregation yield more relevant changes. Taking these arguments into account, it seems to be more appropriate to use different price indexes with a similar degree of disaggregation.

(8) For example, if two prices are always equal, and every month one of those prices is sampled the first day and the other the last day, the true variability of relative prices is zero. At low inflation a low RPV should be detected, but at high inflation a higher variability will be detected, which would be just the consequence of the periodicity of price collection.

(9) Data include 77 industrial and imported good prices, from a total of 87, for the 1960-1984 period, and 55, from a total of 64, for the 1984-1991 period.

(10) In other countries different results have been obtained. For instance, for Turkey, during the 1948-1997 period, [START_REF] Caglayan | Nonlinear Impact of Inflation on Relative Price Variability[END_REF] find a lower effect of inflation on relative prices during the higher inflationary period.

(11) In order to select the number of lags, the Akaike criterion has been applied.

(12) This result is observed especially in high inflation periods. It denotes that there are variables affecting RPV that have not been included in our equations, in particular real variables related to high economic volatility, like changes in real exchange rate and real wages -see, for example, [START_REF] Fischer | Relative Shocks, Relative Price Variability, and Inflation[END_REF] and [START_REF] Dabús | Inflación y Precios Relativos: Estudio del Caso Argentino[END_REF] for further details.

(13) As we stated in the introduction, this result has been obtained both for inflationary economies -see [START_REF] Logue | A Note on the Relation between the Rate and Variability of Inflation[END_REF], [START_REF] Blejer | Inflation Variability in Latin America: A Note on the Time-Series Evidence[END_REF], [START_REF] Moura | Inflaçao e Preços Relativos: Medidas de Dispersao[END_REF] and [START_REF] Dabús | Inflationary Regimes and Relative Price Variability: Evidence from Argentina[END_REF]-and stable economies -see [START_REF] Chang | Further Evidence on the Variability of Inflation and Relative Price Variability[END_REF]. (1946)(1947)(1948)(1949) see [START_REF] Tang | On Relative Price Variability and Hyperinflation[END_REF].

(15) Recent papers show that sectoral factors have an important impact on RPV as well -see, for example, [START_REF] Nath | Relative Importance of Sectoral and Aggregate Sources of Price Changes[END_REF].

(16) In general, a positive association inflation-skewness is supported by the data -see, for example, [START_REF] Ball | Relative-Price Changes as Aggregate Supply Shocks[END_REF] for the US, Amano and Macklen (1997) for Canada, [START_REF] Hall | Are There Downward Nominal Rigidities in Product Markets?[END_REF] for the United Kingdom, [START_REF] Aucremanne | Inflation, Relative Prices and Nominal Rigidities[END_REF] for Belgium, [START_REF] Döpke | Inflation and Skewness of the Distribution of Relative Price Changes: Empirical Evidence for Germany[END_REF] for Germany, [START_REF] Assarsson | Inflation and Relative-Price Changes in the Swedish Economy[END_REF] for Sweden and Caraballo andUsabiaga (2004a, 2004b) for Spain. [1981][1982][1983][1984][1985][1986][1987][1988][1989] Coefficients for unexpected inflation are generally statistically significant and negative. When expected inflation is significant, the response of RPV to its changes is greater where expected inflation is rising [START_REF] Miszler | Inflation and Relative Price Variability in a Low Inflation Country: Empirical Evidence from Germany[END_REF] Germany , 1991-2003 Only unexpected inflation affects positively RPV. The impact of expected inflation disappears if a credible monetary policy stabilizes inflationary expectations * For Tables 5A and5B, lags of the endogenous variable were included when the unit root test indicated the presence of non-stationarity, and to deal with autocorrelation. The trend term (TREND) was included when it was significant according to that test.

Following the parsimony principle we include the lowest number of lags required in each case.

  , and their relevance, are different at different inflation levels.

  denotes the weight of price i in the price index, IN it the inflation rate of price i and IN t the general inflation rate at time t. 4 On the other hand, expected inflation (INE) is the inflation rate forecasted by the agents for the current period, and is estimated by means of the ARMA model that fits the "best forecast" of inflation. The ARMA structure has been selected according to the Akaike-Schwarz criterion -the estimation with the lowest value of both tests. The results suggest an AR(1) model for Argentina, and an ARMA(10,10) model for Spain. From our point of view these results are plausible, in the sense that in a stable economy like Spain agents could use long-run information to forecast current inflation, while in an unstable and changing environment like Argentina only short-run information should be useful. Finally, unexpected inflation (INO) is the forecasting error, and can be defined as the difference between current and expected inflation (INO=IN-INE).

  the Argentina price series is concerned, they have been drawn from the statistical bulletins of the Instituto Nacional de Estadísticas y Censos, covering the period January 1960-February 1991. At the three-digit level of disaggregation in the International Standard Industrial Classification, we use 87 individual prices for the January 1960-June 1984 period and 64 for the July 1984-February 1991 period -the structure of WPI in Argentina changed in July 1984.

  seasonal variables for Argentina and seasonally adjusted variables for Spain.

  in the hyperinflations of[1989][1990]. Indeed, RPV increases strongly in these cases, and reaches the highest values in the months of highest inflation. The months of hyperinflation seem to show a collapse of the price system, which implies evidence in favour of the hypothesis of non-neutrality of inflation.There are two cases in which price variability decreases, denoting a coordination in the individual price adjustments.

  unexpected inflation increase systematically with the level of inflation, and particularly at hyperinflation. Table3

  , multicollinearity problems can appear in estimations including inflation rate and inflation volatility measures, because both variables are closely related. As the correlation between explanatory variables can help us to understand the relevance of this problem, we have calculated the correlation coefficient, obtaining that only for two cases is it bounded between 0.50 and 0.75 -the correlation betweenIN and DIN for Spain and between IN

  ), and there are not remarkable changes in the estimations.

  The same result was obtained for the Chinese hyperinflation

  Figures and tables

  15 VARIN is the best inflation volatility variable to explain

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  . Finally, an interesting branch of research could study if the causality of the relationship changes with the regime of inflation. Applied Economics. The remaining errors are ours.

	Jimeno and anonymous referees of Centro de Estudios Andaluces (working
	paper E2004/71) and
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Table 1 .

 1 Empirical evidence

	Author		Data	Results
	Parks	US, 1930-1975	RPV increases mainly with unexpected inflation
	(1978)
	Fischer	US, 1930-1979	RPV increases with expected inflation and positive
	(1981,1982)			unexpected inflation but not with negative unexpected
				inflation
	Tang and	China, 1946-1949	RPV increases with expected inflation as well as with the
	Wang	(hyperinflation	absolute value of unexpected inflation
	(1993)	period)
	Aarstol	US, 1947-1997	Expected inflation and unexpected inflation, in particular
	(1999)			when positive, are all positively related to RPV
	Silver and	Nine	European
	Ioannidis	countries,
	(2001)

Table 2 .

 2 Inflation regimes. Argentina

	Inflation regime	Period
	Moderate inflation	January 1960-April 1970
	High inflation	May 1970-January 1975, May 1976-June 1982, July
		1985-June 1987, September1988-March 1989, August
		1989-November 1989, April 1990-February 1991
	Very high inflation	February 1975-April 1976, July 1982-June 1985, July
		1987-August 1988
	Hyperinflation

Table 3 .

 3 A. Average values by regime of inflation. Argentina and Spain Autocorrelation is due to the presence of outliers in the abrupt increases of inflation rate of 1975 and both hyperinflations. Fortunately, autocorrelation problems in Argentina do not obey the structure of the residuals. For all tables of results presented in this paper, regressions were made by means of OLS. Values in brackets refer to the p-value obtained when testing the null hypothesis that the coefficient estimated is significantly different from zero. R 2 refers to adjusted R 2 . The specification test p-values reported are the BG test results for the presence of first order serial correlation.

			Spain			Argentina
	Regimes/	Higher	Lower		Lower	Higher
	Variables	inflation	inflation	inflation	inflation
			period	period	period	period
			(1985.09-	(1992.04-	(1960.01-	(1975.02-
			1992.03)	2001.12)	1975.01)	1991.02)
	IN		0.53	0.27		1.9	14.3
	ABDIN		0.30	0.13		1.5	7.3
	VARIN		0.19	0.10		1.1	7.1
	ABINO		0.36	0.18		1.7	11.1
	RPV		2.20	1.17		0.4	1.0
	Table 3.B. Average values by regime of inflation. Argentina
	Regimes/	Moderate	High	Very	high	Hyperinflation July 1989 *
	Variables	Inflation	inflation	inflation
	IN		1.4	5.5	18.3	96.8	209.1
	ABDIN		1.3	4.6	6.4		36.6	75.6
	VARIN		1.0	4.1	5.9		43.2	135.2
	ABINO		1.2	2.4	5.9		10.1	16.7
	RPV		0.3	0.5	1.2		6.4	25.7
	* The highest inflation month of the Argentine hyperinflations.

* *

Table 5 .

 5 A. Regression results. Spain dependent variable: RPV, explanatory variables: volatility measures

			Total period: 1985.09-2001.12
	Regression 1		Regression 2		Regression 3
	R 2	0.56	R 2	0.56	R 2	0.60
	BG	0.02	BG	0.01	BG	0.09
	CONSTANT	1.72	CONSTANT	1.75	CONSTANT	1.77
		(0.00)		(0.00)		(0.00)
	RPV(-1)	0.19	RPV (-1)	0.20	RPV(-1)	0.18
		(0.00)		(0.00)		(0.01)
	TREND	-0.01	TREND	-0.01	TREND	-0.01
		(0.00)		(0.00)		(0.00)
	IN	0.42	IN	0.26	IN	0.27
		(0.01)		(0.06)		(0.05)
	DIN	-0.11	ABDIN	0.14	VARIN	0.47
		(0.37)		(0.33)		(0.03)
		Higher inflation period: 1985.09-1992.03
	Regression 4		Regression 5		Regression 6
	R 2	0.01	R 2	0.01	R 2	0.07
	BG	0.09	BG	0.07	BG	0.29
	CONSTANT	1.76	CONSTANT	1.83	CONSTANT	1.83
		(0.00)		(0.00)		(0.00)
	RPV(-1)	0.11	RPV(-1)	0.10	RPV(-1)	0.09
		(0.35)		(0.39)		(0.43)
	IN	0.39	IN	0.20	IN	0.17
		(0.13)		(0.35)		(0.46)
	DIN	-0.10	ABDIN	0.168	VARIN	0.56
		(0.58)		(0.39)		(0.09)
		Lower inflation period: 1992.04-2001.12
	Regression 7		Regression 8		Regression 9
	R 2	0.18	R 2	0.19	R 2	0.20
	BG	0.11	BG	0.15	BG	0.14
	CONSTANT	1.79	CONSTANT	1.74	CONSTANT	1.72
		(0.00)		(0.00)		(0.00)
	TREND	-0.01	TREND	-0.01	TREND	-0.01
		(0.00)		(0.00)		(0.00)
	IN	0.25	IN	0.12	IN	0.18
		(0.29)		(0.52)		(0.35)
	DIN	-0.15	ABDIN	0.41	VARIN	0.74
		(0.48)		(0.10)		(0.05)
	Note:

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Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 5 .

 5 B. Regression results. Spain dependent variable: RPV, explanatory variables: expected and unexpected inflation

	Submitted Manuscript	Page 32 of 33
	Total period: 1985.09-2001.12
	Regression 1		Regression 2
	R 2	0.57	R 2	0.56
	BG	0.15	BG	0.36
	CONSTANT	1.84	CONSTANT	1.82
		(0.00)		(0.00)
	RPV(-1)	0.23	RPV (-1)	0.21
		(000)		(0.00)
	TREND	-0.00	TREND	-0.01
		(0.00)		(0.00)
	INE	-0.02	ABINE	-0.07
		(0.84)		(0.57)
	INO	0.37	ABINO	0.51
		(0.00)		(0.00)
	Higher inflation period: 1985.09-1992.03
	Regression 3		Regression 4
	R 2	0.04	R 2	0.02
	BG	0.21	BG	0.39
	CONSTANT	2.13	CONSTANT	2.05
		(0.00)		(0.0)
	INE	0.01	ABINE	-0.02
		(0.92)		(0.92)
	INO	0.38	ABINO	0.49
		(0.04)		(0.08)
	Lower inflation period: 1992.04-2001.12
	Regression 5		Regression 6
	R 2	0.18	R 2	0.20
	BG	0.05	BG	0.06
	CONSTANT	2.01	CONSTANT	1.93
		(0.00)	(0.00)
	TREND	-0.00	TREND	-0.00
		(0.00)	(0.00)
	INE	-0.21	ABINE	-0.25
		(0.33)	(0.26)
	INO	0.01	ABINO	0.43
		(0.97)	(0.11)

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 6 .

 6 Structural change. Argentina and Spain

		Argentina	Spain
	Regressions/Regimes	Lower to higher inflation	Higher to lower inflation
		(1960.01-1975.01 to	(1985.09-1992.03 to
		1975.02-1991.02)	1992.04-2001.12)
	RPV=f(IN)	1	1
	RPV=f(IN, DIN)	1	1
	RPV=f(IN,ABDIN)	1	1
	RPV=f(IN,VARIN)	1	1
	RPV=f(INE,INO)	1	1
	RPV=f(ABINE,ABINO)	1	1
	Note:
	1: Structural change was verified at the 1% level.

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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Submitted Manuscript

Acknowledgements

We acknowledge the financial support from Centro de Estudios Andaluces (Project ECO 17), as well as the useful comments from the participants in the Latin American Meeting of the Econometric Society (2004), members of the Macroeconomics Group (2004) of Centro de Estudios Andaluces, Juan F.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Michael G Arghyrou
email: michael.arghyrou@brunel.ac.uk

Estimation < C5 -Econometric Modeling < C -Mathematical and Quantitative Methods, C52 -Model Evaluation and Testing < C5 -Econometric Modeling < C -Mathematical and Quantitative Methods, E31 -Price Level|Inflation|Deflation < E3 -Prices, Business Fluctuations, and Cycles < E -Macroeconomics and Monetary Economics, F15 -Economic Integration < F1 -Trade < F -International Economics

Keywords: price level, structural breaks , non linear adjustment, Euro price level, structural breaks, non-linear adjustment, Euro JEL Classification: C51, C52, E31, F15

We test the widely-held hypothesis that accession to the EMU has caused a structural increase in Greek consumer prices. We find no econometric evidence of such an effect. We also find strong evidence of (a) multiple structural breaks in the process driving Greek equilibrium consumer prices and (b) non-linear price adjustment. Our findings explain the post-EMU accession acceleration in Greek prices as normal, equilibrium-restoring behaviour. They also have important policy implications for the countries planning to join the Euro in the foreseeable future.

INTRODUCTION

The accession of Greece to the European Economic and Monetary Union (EMU) in January 2001 was followed by a noticeable increase in the level of Greek consumer prices. This has given rise to a widely-held held hypothesis according to which the Euro has caused a structural shift in the Greek price level. Such a hypothesis can theoretically be supported by the price-transparency properties of the single currency. These are typically regarded to be among the benefits of the latter, as they are expected to promote price competition in the traded sector, reduce price differentials associated with exchange rate variability; and limit the ability of firms to follow pricing-to-market policies. 1 However, in countries like Greece, with large public and services sectors, the aggregate price level is significantly determined by the non-traded component. 2 A widely accepted stylised fact is that in such countries, where productivity and per capita income are lower than in more developed ones, prices in the non-traded sector, as well as the aggregate price level, are relatively low when expressed in terms of a common currency. 3 In addition, and according to the [START_REF] Balassa | The purchasing power parity doctrine: a reappraisal[END_REF]- [START_REF] Samuelson | Theoretical notes on trade problems[END_REF] hypothesis, when less developed countries achieve productivity gains, their aggregate price level tends to increase. 4 In the case of Greece, the above imply that by making comparisons of nontraded prices more transparent (contributing to upward rather than downward price equalisation), and by providing domestic firms with an extra motivation to increase their productivity due to the loss of the competitiveness-inducing option of currency depreciation (giving rise to a Balassa-Samuelson type of effect), accession to the Euro may have caused a structural increase in Greek consumer prices. In this paper, we test this hypothesis and soundly reject it. We then look for alternative that end, we model Greek equilibrium consumer prices and the process of short-run price adjustment following the methodology used by Arghyrou, [START_REF] Martin | Modelling monetary policy: inflation targeting in practice[END_REF].

The theoretical literature on non-linear (state-dependent) price adjustment (see e.g. [START_REF] Deveraux | State dependent pricing and business cycles asymmetries[END_REF] and the references therein) suggests that menu costs and other market imperfections result in firms adjusting prices only when they differ from their equilibrium level enough to render adjustment profitable. Not accounting for such non-linearities, when they exist, may lead to biased econometric inference. Similar biases arise if the relationship determining equilibrium prices is treated as a structurally stable process when in reality it has been subject to structural breaks (see Madalla and Kim, 1998).

Our econometric analysis accounts for both sources of bias mentioned above. We test a joint hypothesis, according to which the process driving long-run equilibrium prices is structurally stable and, at the same time, characterised by a linear adjustment process. We reject both constituents of this joint hypothesis as we find evidence of both multiple structural breaks and non-linear price adjustment.

The identified structural breaks are found not to be related to the accession of Greece to the EMU. With regard to non-linearities, we find Greek consumer prices to converge always to their steady-state; however convergence is found to be much faster when prices are away rather than close to their equilibrium.

These findings help us to explain the post-EMU accession increase in Greek consumer prices on the following grounds. First, we argue that consumer prices increased at the early stages of EMU participation because they had taken values increasingly below their equilibrium level prior to EMU accession. As a result, F o r P e e r R e v i e w 3 restoring equilibrium was bound to involve accelerated price increases. Second, we claim that the post-accession increase in prices was sharp because the pre-accession under-pricing episode was big enough to set in motion the high speed of price adjustment identified by our non-linear price adjustment analysis. We conclude that the post-EMU accession behaviour of Greek prices is not surprising: It represents normal equilibrium-restoring behaviour. These findings have important policy implications for the countries that have recently joined the EU and plan to join the single currency in the foreseeable future.

The remainder of this paper is structured as follows: Section 2 discusses our methodology and presents our empirical results. Section 3 discusses our findings.

Finally, section 4 summarises and offers some concluding remarks.

METHODOLOGY AND EMPIRICAL RESULTS

Benchmark models of equilibrium prices and price adjustment

Following the analysis in Arghyrou, [START_REF] Martin | Modelling monetary policy: inflation targeting in practice[END_REF], our first step is to estimate the benchmark models for Greek equilibrium prices and price adjustment respectively described by equations ( 1) and (2) below: 5

p t = α + β 1 ulc t + β 2 pw t + u t (1) ∆p t = ∑ = k i 1 β i ∆p t-i + ∑ = m i 0 γ i ∆p* t-i + δ (p-p*) t-1 + ν t (2)
In ( 1) and (2), p denotes the actual (observed) price level; ∆p t = p t -p t-1 the change in the actual price level (our measure of price adjustment); ulc unit labour cost; pw the log of world prices expressed in terms of domestic currency; p* the equilibrium price level; and u t and ν t white noise error terms. All variables are 1), denoted by p ˆ*, provide an estimate for the equilibrium price level p*, with the cointegrating vector CV t = p t -α -β 1 ulc tβ 2 pw t , measured by the estimated residual term t u ˆ, corresponding to the disequilibrium term (p-p*) t in (2). Substituting (p-p*) t by CV t modifies equation (2) as per (3) below.

∆p t = ∑ = k i 1 β i ∆p t-i + ∑ = m i 0 γ i ∆ * p t-i + δ (p-p ˆ*) t-1 + ν t (3)
In (3), the parameter δ describes the speed at which the actual price level p t convergences to its estimated equilibrium * ˆt p . 6 For our estimations we use quarterly series. Data availability limits our sample to cover the period 1971Q4 to 2003Q3. Our data source is the International Financial Statistics Databank provided by DataStream. Our measure of p is the index of consumer prices (CPI). This is the headline Greek measure of prices, for which the Bank of Greece used to announce policy objectives during the 1990s and the one widely assumed to have been subject to a structural shift following Greece's accession to the EMU. World prices expressed in domestic currency (pw) are calculated multiplying the index series of world consumer prices (expressed in US dollars) by the drachma/dollar exchange rate 7 . Unit labour cost (ulc) is approximated by the index of unit labour cost in the Greek manufacturing sector. 8

Following [START_REF] Hendry | Modelling UK Inflation, 1875-1991[END_REF], we estimate (1) using the [START_REF] Engle | Cointegration and error-correction: representation, estimation and testing[END_REF] methodology. 9 The results are reported in We now estimate (3) using CV t , the estimated residuals of (1), as a measure of the error correction term (p-p ˆ*). We follow a general-to-specific approach, initially including twelve lags for ∆p and ∆ p ˆ* and gradually eliminating insignificant terms. The results are reported in

Tests and models of structural breaks in long-run price determination

We now test the hypothesis that the accession of Greece to the EMU in 2001, and/or the physical introduction of the Euro in 2002, have caused a structural increase in Greek equilibrium consumer prices. Such a test will only be credible if it takes into account any structural breaks preceding those events. To identify such breaks, we use the methodology of [START_REF] Gregory | Residual-based test for cointegration in models with regime shifts[END_REF] to identify multiple structural breaks in cointegrating relationships endogenously. To do so, we estimate:

p* t = α + β 1 ulc t + β 2 pw t + d 0 D t + d 1 D t ulc t + d 2 D t pw t + u t ( 4
)
where [START_REF] Quintos | Sustainability of the Deficit Process with Structural Shifts[END_REF] The null hypothesis H 0 : d i = 0 describes structural stability for each of the coefficients in equation (1). 13 Structural breaks are identified in those dates for which the estimated Wald statistic is higher than the 5% critical value of χ 2 (1). Given that structural breaks cannot fall too close, we follow [START_REF] Quintos | Sustainability of the Deficit Process with Structural Shifts[END_REF] and treat all those falling within two years as representatives of the same structural shift. The exact timing of the break is then selected to be the observation with the highest test value. Assuming j structural breaks for each of the coefficients in (1), the augmented equilibrium price process is then given by ( 7) below:

D t = 1 if t ∈ (1,..., T) (5) = 0 if t ∈
p* t = α + β 1 ulc t + β 2 pw t + ∑ = j i it i D d 1 0 + ∑ = k i it i D d 1 1 ulc t + ∑ = l i it i D d 1 2 pw t + u t (7)
In ( 7), T i is the date at which the ith (in a total of j, k and l respectively) identified structural break for each of the three coefficients in (1) (α, β 1 and β 2 respectively) has been found to occur. D it is defined to be 0

if t ∈ (1,..., T i ). If on F o r P e e r R e v i e w
7 the other hand t ∈ (T i +1, ..., N), D it is respectively defined to equal α, ulc t or pw t according to the variable which has been found to be subject to a structural break at period T i . Equation ( 7) picks up the long-run effect of structural breaks on each of the coefficients of the long-run equilibrium price relationship. The cointegrating vector augmented for such breaks is given by CVBR

t = p* t -α -β 1 ulc t -β 2 pw t -∑ = j i it i D d 1 0 -∑ = j i it i D d 1 1 ulc t -∑ = j i it i D d 1 2 pw t .
Finally, if structural breaks exist, In the benchmark price adjustment model in (4) has to be estimated using CVBR t (which accounts for the breaks) rather than CV t (which does not) to replace for the error correction term (p-p ˆ*) t- 1Figure 1 plots the values of the sequentially estimated Chi-square tests for each of the coefficients of equation ( 1), against the 5 and 1 per cent critical values.

We obtain strong evidence of structural instability for each coefficient. For the constant term α, stability is rejected for a number of observations in the early 1970s and the 1980s; for β 1 in the early 1980s; and for β 2 in the mid-1980s and the mid-1990s. Using the criterion of the highest F-statistic for statistically significant values falling close to each other, we find structural breaks of similar timing for the three coefficients, namely 1974Q1 and 1985Q3 for the constant term; 1982Q1 for and 1974Q1, 1985Q3, 1996Q1 and 1998Q2 for β 2 .

β 1 ;
We now estimate equation ( 7) which accounts for the long-run effect of the identified breaks on the price determination relationship. We start from a general specification where all breaks are included in the analysis. The results are reported in Table 1, column (b). This suggests that not all break dummies are significant.

Eliminating gradually the insignificant terms, we obtain the parsimonious specification reported in Table 1, col. (c). We conclude that the price determination in 1974, 1985 and 1996. 14 Crucially, for our analysis, we find that the accession of Greece to the EMU in January 2001, and the physical introduction of the Euro in January 2002 have not caused a structural shift in Greek equilibrium consumer prices.

As a robustness check, we superimpose on the data such an EMU-induced break and test for its statistical significance. We do so by adding to the parsimonious equation reported in Table 1, 3). The results are reported in Table 1, columns (d) and (e). None of the EMU or Euro dummy variables is statistically significant. These remain insignificant when each of the dummies is added to the model on an individual basis. 15 We confirm that there exists no econometric evidence to support the hypothesis that Greece's accession to the EMU and/or the introduction of the Euro in physical form have caused any structural increase in equilibrium Greek prices.

Finally, we use the estimated residuals of our preferred long-run equation reported in Table 1, col. (c) as a measure for CVBR to replace for (p-p ˆ*) t-1 and reestimate the price adjustment model in (3). The results are reported in Table 2, column (b). The estimated speed of adjustment is statistically significant and, compared to the model in column (a), almost four times as fast (14.4 per cent). This highlights the bias dangers involved in empirical papers of price adjustment that do not account for structural breaks in the long-run price determination equation.

Allowing for breaks also improves the model's fit, as manifested by the reduction in the standard error of the regression.

Tests of non-linear price adjustment

We now test formally for the existence of non-linear price adjustment using the procedure described in [START_REF] Saikonnen | Lagrange multiplier tests for testing nonlinearities in time series models[END_REF], [START_REF] Luukkonen | Testing linearity against smooth transition autoregressive models[END_REF], Granger and[START_REF] Granger | Modelling Non-linear Economic Relationships[END_REF]Teräsvirta (1994). This involves estimating t u ˆ=γ 00 + ( )

∑ = - - - - - - - + + + φ γ γ γ γ 1 3 3 2 2 1 0 ĵ d t j t j d t j t j d t j t j j t j u u u u u u u +γ 4 2 ˆd t u -+γ 5 3 ˆd t u -+ν t (8)
where t u ˆ is the estimated cointegrating vector obtained by (1), or (7), depending on whether (1) has been found to be subject to structural breaks. The parameter d is the delay parameter of the transition function to be used and v(t) ∼ niid (0,σ 2 ).

Linearity implies the null hypothesis H 0 : [γ 1j = γ 2j = γ 3j =γ 4 =γ 5 = 0] for all j ∈(1,2...φ). This can be tested using an LM-type test.

A linear model of size effects in price adjustment

One way to capture non-linear price adjustment is to use the model proposed by [START_REF] Escribano | Investigating the relationship between gold and silver prices[END_REF] described by equation ( 9) below:

∆p t = ∑ = k i 1 β i ∆p t-i + ∑ = m i 0 γ i ∆ * p t-i +δ 1 (p-p ˆ*) t-1 +δ 2 (p-p ˆ*) 2 t-1 +δ 3 (p-p ˆ*) 3 t-1 +ν t (9)
In equation ( 9), the error correction term enters the price adjustment equation not only in its first power but also in the second and the third. This allows prices to revert to their equilibrium faster for large rather than small deviations from the latter, thus allowing for size effects in the process of price adjustment. We estimate equation ( 9) using a general-to-specific approach, starting with general specification including twelve lags for ∆p t and ∆ * p and gradually reducing it to a parsimonious one. Table 2, column (c) presents the results. The terms capturing size effects through (p-p ˆ*) 3 t is statistically significant with a negative sign, suggesting that the speed of adjustment increases with the size of the deviation from equilibrium. In addition, the model produces an improvement in fit, compared to the models in columns (a) and (b), as manifested by a significant reduction in the regression's standard error.

A non-linear model of price adjustment

The Escribano-

∆p t = θ t R It + (1-θ t ) R Ot + ε t (10) R It = α 1 + ∑ = - ∆ n i i t i I p 1 β + i t p i i I p - = ∆ ∑ * 0 γ + δ I (p-p ˆ*) t-1 + ε It (11) R Ot = α 2 + ∑ = - ∆ n i i t i O p 1 β + i t p i i O p - = ∆ ∑ * 0 γ + δ O (p-p ˆ*) t-1 + ε Ot ( 12
)
θ t = pr { τ L ≤ d t p p - -*) ( ≤ τ U } = 1 - ] *) ][( *) [( 1 1 U d t L d t p p p p e τ τ σ - - - - - - - + (13)
The QL-STECM involves two regimes, the inner (R I ) and the outer (R O ), described by equations ( 11) and ( 12) respectively. These are linear price adjustment equations similar to (2) and respectively describe price adjustment for small and large deviations from equilibrium. The inner regime is defined by two threshold values, the upper (τ U ) and the lower (τ L ). The thresholds define a band within which the speed of price adjustment takes a value different to the one prevailing outside the band. Equation ( 13) models the probability that the transition variable

d t p p - -*) (
takes values within the inner regime using the quadratic logistic function. Menu costs and other market imperfections imply that reversion of prices to their equilibrium is faster when deviation from the latter is large rather than small, i.e. δ O >δ I . The QL-STECM also captures a certain type of sign effects in price adjustment if the regime thresholds are asymmetric (τ L + τ U ≠0).

This asymmetry, if validated by the data, allows one form of disequilibrium (positive or negative) to be sustained longer than the other before the outer regime is reached. The results from estimating the QL-STECM model using the cointegrating vector accounting for structural breaks (CVBR) are reported in Table 2, column (d). The error correction coefficient is significant in both regimes. In the inner regime, the quarterly speed of adjustment to equilibrium is estimated to be 13.2 per cent. In the outer regime, however, adjustment is more than three times as fast quite fast (45.3 per cent).

Finally, the threshold symmetry restriction reported at the end of the Table is not significant, so the inner regime's thresholds are found to be symmetric (τ L = τ U ). With the point estimates for τ L and τ U being equal to -2.28 and 1.93 respectively, the model suggests that the inner regime is defined to be within two percentage points from the equilibrium price level. We have tested the robustness of these findings by estimating a number of models allowing for sign effects in the (2002). In all cases, we evidence of size but no sign effects in the process of shortrun price adjustment. In addition, all these models produced a higher regression standard error when compared to the Escribano-Granger and QL-STECM models reported in Table 2. 17

EMU ACCESSION INCREASE IN GREEK CONSUMER PRICES

Our analysis in the previous section rejected the widely-held hypothesis according to which joining the single currency is the factor underlying the increase in Greek consumer prices observed following the accession of Greece to the EMU.

So why did these take place? Figure 2, which plots the estimated disequilibrium term accounting for structural breaks (CVBR) versus the estimated regime thresholds of the QL-STECM model, provides some explanations.

First, Figure 2 shows that during the run-up to EMU accession in 1998-2000, the Greek price level took values increasingly below its equilibrium level. This is a period during which Greek authorities introduced a number of pricerestraining measures, including freezing or increasing only marginally the prices of public utilities and concluding agreements with unions of Greek private firms, involving restraints on the latter's pricing policy. These aimed to ensure meeting the inflation criterion set by the Maastricht Treaty so that Greece would be able to join the EMU in 2001. This target was eventually met but, as Figure 2 suggests, these measures meant that for a period of two years prior to EMU accession, consumer prices were falling more and more behind their equilibrium level.

Following Greece's accession to the EMU in 2001, which eliminated the basis on which the measures were taken, for prices to catch-up with their equilibrium level their rate of growth was bound to accelerate. Note that this conclusion stands even if the speed of price adjustment were to be a linear process.

The second factor explaining post-EMU accession price behaviour in Greece relates to our finding that price adjustment in that country is, in fact, a nonlinear process. Figure 2 suggests that at the eve of Greece's accession to the EMU , the Greek price level was not only lower than its equilibrium level, but it was also low enough to have hit, and marginally surpassed, the lower threshold of the inner regime (τ L ). Under such conditions, the QL-STECM model predicts that prices would not only increase; but they would also increase fast to correct the existing large disequilibrium. Indeed, as Figure 2 shows, this is what has happened on all previous occasions when prices took values in the outer regime below their equilibrium (1973, 1977-78, 1985-86 and 1989-90) and, in line with these episodes, immediately after the under-pricing episode of 2000.

To summarise, our findings suggest that at the early stages of EMU participation (2001)(2002) Greek consumer prices did not accelerate because of the introduction of the Euro; they accelerated because of the existence of strong underpricing pressures prior to the adoption of the Euro. From that point of view, there is nothing surprising in their behaviour. Their movements represent normal equilibrium-restoring price behaviour.

Figure 2 suggests that following their restoration to equilibrium level in mid-2002, consumer prices continued to increase so that by the end of 2003, they

were above their equilibrium level (in one occasion, in the outer regime). One explanation for this overshooting is that this is an over-reaction to the under-pricing episode discussed above, similar to the one that took place following all previous under-pricing episodes depicted in Figure 2. In addition, there exist another two, non-mutually exclusive, possible explanations.

The first is a supply-based hypothesis, according to which Greek firms have deliberately been increasing prices beyond equilibrium in pursuit of higher profits.

This hypothesis is not testable without using micro data, but it is not unreasonable, The second is a demand-based hypothesis according to which the increase in Greek prices beyond their equilibrium level in 2003 reflects an incompatibility between the domestic requirements of the Greek economy and the monetary policy followed by the European Central Bank. According to this hypothesis, during 2001-2003, a period when the Greek rate of output growth was four times higher than the EMU average, Greece needed relatively high interest rates to prevent highdemand conditions resulting in significant price increases. Instead, Greece experienced, along with the rest of the EMU members, interest rate reductions, aiming to revive economic activity in the face of subdued economic growth in the EMU area (see ECB, 2004 pp. 91-92). 18 We have tested this incompatibility hypothesis in another paper, belonging to the same research project the present paper belongs, and have found significant econometric evidence in support of it.

SUMMARY AND CONCLUDING REMARKS

This paper tested the widely-held hypothesis according to which accession to the EMU has caused a structural increase in Greek consumer prices. We find that joining the single currency did not cause such an effect. We also find strong evidence of (a) multiple structural breaks in the process driving equilibrium prices and (b) non-linear price adjustment. Our findings explain the post-EMU accession acceleration in Greek prices as normal, equilibrium-restoring price behaviour.

Our findings also have important policy implications for the new EU members planning to join the EMU in the foreseeable future. These present a number of similarities with Greece before the latter's accession to the Euro. Despite the well-established productivity gains these countries have achieved in the 1990s, their productivity remains below that of the EMU average; they have large non-traded sectors; and their aggregate price level, when expressed in Euros, is below the Euro-zone average (see, [START_REF] Taylor | Real exchange rate dynamics in transition economies: a non-linear analysis[END_REF], [START_REF] Dibooglu | Sources of Real Exchange rate Fluctuations in Transition Economies: The case of Poland and Hungary[END_REF] and the references therein).

The Greek experience shows that these countries will not necessarily pay an "admission cost" upon entering in the EMU in the form of an upward structural shift in their consumer price level. However, joining the single currency may in practice be accompanied by significant price increases, if pre-accession inflation convergence is not entirely market-achieved but also administratively enhanced.

The Greek experience also suggests that if the new EU members do not achieve adequate synchronization of their business cycle with that of the EMU average, EMU entry may lead to incompatibility between the single monetary policy and their domestic economic conditions. To avert this eventuality, prior to entering the single currency they should aim to achieve a higher degree of real convergence than the one achieved by Greece in the 1990s.

ACKNOWLEDMENTS

This paper presents some of the findings of the research project titled The effects of joining the Euro on the Greek Economy: Initial estimates and future prospects, commissioned and funded by the Centre of Planning and Economic Research (KEPE) of the Ministry of Finance of the Hellenic Republic. I wish to thank KEPE and the Hellenic Ministry of Finance for commissioning and supporting financially this project. I would also like to thank the officials of KEPE and the Hellenic Ministry of Finance for the presentation of the project's findings at KEPE, in November 2004, and providing me with useful comments and suggestions. Thanks are also due to Christopher Martin for comments on a previous draft of this paper.

The usual disclaimer applies. van Dijk, D., T. Teräsvirta, and P.H. Franses (2002), "Smooth transition autoregressive models -a survey of recent developments", Econometric Reviews, 21, 1-47.

ENDNOTES

1 For a thorough survey of empirical studies on price distortions caused by exchange rate variability, marking-to-market pricing policies and other trade barriers, see [START_REF] Sarno | The economics of exchange rates[END_REF].

2 For a justification of this argument, see the detailed analysis on the determinants of the Greek aggregate price level in 1990s in [START_REF] Mourmouras | Monetary Policy at the European Periphery: Greek Experience and Lessons for EU Candidates[END_REF], chapter 5. 3 See, for example, [START_REF] Rogoff | The Purchasing Power Parity Puzzle[END_REF]. 4 For a theoretical justification of the Balassa-Samuelson hypothesis see, among others, [START_REF] Froot | Perspectives on PPP and long-run real exchange rates[END_REF]. For a survey of related theoretical models and empirical evidence see Asea and Corden (1994). 5 For a theoretical justification of these econometric specifications, and other references in which they have been adopted, see [START_REF] Arghyrou | Non-linear Inflationary Dynamics: Evidence from the UK[END_REF]. 6 An alternative methodology consists of substituting ( 2) into (1) and estimate ∆p t = Σβ i ∆p t-i + Σγ i ∆p* t-i + δ (p-β′ z) t-1 + ν t . However, working with equation ( 3) is preferable when estimating models with a large number of parameters like the non-linear model we estimate below. 7 For the post-EMU accession period (2001 onwards), we calculated the drachma value of world prices first by calculating their euro value (by multiplying the world consumer prices index expressed in dollars by the Euro/US dollar exchange rate; and then multiplying the resulting product by the irrevocable Drachma/Euro exchange rate (340.75) 8 This index is taken from the OECD Historical Data Databank provided by DataStream. 9 Preliminary unit root analysis (not reported here but available upon request) suggests that ulc is integrated of order one. The order of integration of p and pw is more ambiguous. Ambiguity for the order of integration of price-level series is a theme frequently encountered in empirical literature, with the prevailing view being that inflation is a stationary. Following this, we treat p and pw as I(1) variables (for further discussion on this point, see [START_REF] Martin | Modelling monetary policy: inflation targeting in practice[END_REF] and the references therein). 10 As a robustness check, we have estimated equation (1) using the Generalised Dynamic OLS method (see [START_REF] Stock | A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems[END_REF]. This method adds to the right hand side of equation (1) a number of lead and lag terms relating to the independent variables aiming to account for any possible feedback effects between the left-and right-hand side terms. The results we obtained are very similar to those reported in (1993) proposes trimming the initial and final parts of the sample when testing for structural stability. We have tested for structural breaks trimming our sample as suggested by Andrews and found results very similar to those reported below (these are available upon request). However, we prefer to use the results without trimming because the date of the accession of Greece to the EMU (January 2001) falls into the final part of the sample which is trimmed. 14 The timing of the identified breaks can be related to macroeconomic events of significance for the Greek economy. The break in 1974 can be related to the first oil shock. The break of 1985 may be related to the fiscal and monetary expansion that took place in Greece in the first half of the 1980s. Finally, the break of 1996 can be related to stabilisation effort undertook in the context of the Greek convergence programmes implemented in the 1990s. For a thorough discussion of Greek macroeconomic policy and performance during the period 1974-1995, see Alogoskoufis, 1995. The EMU and Euro dummies remain insignificant when they are added to the parsimonious equation accounting for the structural breaks derived when, in line with [START_REF] Andrews | Tests for Parameter Instability and Structural Change With Unknown Change Point[END_REF] we trim the initial and final parts of the sample. The results are available upon request.

16 [START_REF] Granger | Modelling Non-linear Economic Relationships[END_REF] and Teräsvirta (1994) advise against choosing φ using an information criterion such as the Akaike, since this may induce a downward bias. 17 To preserve space, these results are not reported here but are available upon request. 18 This hypothesis is consistent with one of the main predictions of the Theory of Optimum Currency Areas formulated as early as the 1960s (see [START_REF] Mundell | A Theory of Optimal Currency Areas[END_REF][START_REF] Mckinnon | Optimum Currency Areas[END_REF] and [START_REF] Kenen | The Theory of Optimum Currency Areas: an Eclectic View[END_REF]), according to which joining a monetary union without having achieved significant synchronisation with the business cycle of the rest of the union members may result in pro-cyclical and destabilising monetary policy. 50.00 1 9 7 1 -4 1 9 7 2 -4 1 9 7 3 -4 1 9 7 4 -4 1 9 7 5 -4 1 9 7 6 -4 1 9 7 7 -4 1 9 7 8 -4 1 9 7 9 -4 1 9 8 0 -4 1 9 8 1 -4 1 9 8 2 -4 1 9 8 3 -4 1 9 8 4 -4 1 9 8 5 -4 1 9 8 6 -4 1 9 8 7 -4 1 9 8 8 -4 1 9 8 9 -4 1 9 9 0 -4 1 9 9 1 -4 1 9 9 2 -4 1 9 9 3 -4 1 9 9 4 -4 1 9 9 5 -4 1 9 9 6 -4 1 9 9 7 -4 1 9 9 8 -4 1 9 9 9 -4 2 0 0 0 -4 2 0 0 1 -4 2 0 0 2 -4 1 9 7 1 -4 1 9 7 2 -4 1 9 7 3 -4 1 9 7 4 -4 1 9 7 5 -4 1 9 7 6 -4 1 9 7 7 -4 1 9 7 8 -4 1 9 7 9 -4 1 9 8 0 -4 1 9 8 1 -4 1 9 8 2 -4 1 9 8 3 -4 1 9 8 4 -4 1 9 8 5 -4 1 9 8 6 -4 1 9 8 7 -4 1 9 8 8 -4 1 9 8 9 -4 1 9 9 0 -4 1 9 9 1 -4 1 9 9 2 -4 1 9 9 3 -4 1 9 9 4 -4 1 9 9 5 -4 1 9 9 6 -4 1 9 9 7 -4 1 9 9 8 -4 1 9 9 9 -4 2 0 0 0 -4 2 0 0 1 -4 2 0 0 2 -4

CVBR tU tL

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Submitted Manuscript

  post-EMU accession increase in Greek consumer prices. To

  of logarithms. Provided that equation (1) is cointegrated, the fitted values of equation (

  subject to three structural breaks,

  col. (c) dummy variables covering the post-2001 (post-EMU accession) and post-2002 (post Euro-circulation) period respectively, for all three coefficients in equation (

  and models of non-linear price adjustment

  behaviour of Greek prices, i.e. models which allow different speed of adjustment for positive and negative deviations from equilibrium. More specifically, we estimated the Granger-Lee (1989) model, a composite model combining the Granger and Lee specification with that by Escribano and Granger in (9), the Logistic STECM model, and the 3-stage STECM model discussed van Dijk et al.

  of the profitability losses Greek firms may have suffered during the prolonged under-pricing episode of the late 1990s.

  1994), "Specification, estimation and evaluation of smooth transition autoregressive models", Journal of the American Statistical Association 89, pp. 208-218.

Figure

  Figure 1: Chi-square tests for structural stability

Figure 2 :

 2 Figure 2: Cointegrating vector accounting for structural breaks v estimated thresholds of inner price-adjustment regime

Table 1

 1

	, col. (a). 10 The estimates

Table 2

 2

	, col. (a). The error

Table 3

 3 reports the results of the non-linearity tests on CVBR. We obtain evidence of non-linearity, for a number of d values, with the highest F-score observed for d=7. We conclude that price adjustment in Greece presents non-linear behaviour, which we proceed to model below.

	F o
	r
	P e
	e
	r
	R e
	v i
	e w

  Granger model discussed above captures non-linear effects in price adjustment within a linear econometric estimation framework. From that point of view, it can only approximate, rather than model precisely, the non-

	model, which is a generalisation of the ESTAR model, is described by equations
	(10) to (13) below (see van Dijk et al., 2002):
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	11

linearities involved in the process of price adjustment. To do so, we estimate the Quadratic Logistic Smooth Threshold Error Correction Model (QL-STECM). This

  Table 1, col. (a) with the long-run equation being estimated as p t = -0.445 + 0.540 ulc t + 0.306 pw t + u t .11 The lag structure of the ADF statistics reported in Table1is determined using the Akaike information criterion.12 This equation, and all equations in Table2, are estimated with two intercept dummies taking the value of unity for 1979Q1 and 1985Q4 respectively, and zero otherwise. Their inclusion does not change the nature of results but ensures that the equations pass the reported misspecification tests.

13

Andrew

Table 3 Tests for non-linear adjustment of the cointegrating vector accounting for breaks

 3 : Numbers in parentheses denote standard errors; p-values in square brackets, unless other wise stated; AR-F is the Lagrange Multiplier F-test for residual serial correlation; ARCH-F is a general misspecification test of Autoregressive Conditional Heteroscedasticity; Normality Chi-sq is the Bera-Jarque test for residual non-normality Hetero-F is an F-test for heteroskedasticity. : The Table reports the F-scores of the LM test described in equation (8); * and ** denote statistical significance at the 5 and 1 per cent level respectively

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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Notes: Standard errors in parentheses; ** denotes statistical significance at the 1 per cent level NOTESNoteEditorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript F o r P e e r R e v i e w

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UKSubmitted Manuscript


Philip Hans Franses
email: franses@few.eur.nl

Lotje Kruithof

Richard Paap

Govert

Econometric and Statistical Methods: Special Topics < C -Mathematical and Quantitative Methods

teaching and research institutions in France or abroad, or from public or private research centers.

In this paper I put forward a statistical model that can be used to estimate how many postal stamps are, or better must be, around amongst collectors. One possible use of this model is that it allows to examine the fairness of quoted prices at auctions. Indeed, imputing an observed quoted price into the model results in an estimate of the current, but unobserved, stock of that particular postal stamp. Domain-specific knowledge can then be used to evaluate the reliability of this estimate. The model is calibrated using data for over 1000 Dutch postal stamps that have been issued since World War II, but it can naturally be considered for stamps of other countries, or for collectors' items like banknotes, coins, copies of paintings and perhaps other items of art.

The model requires three ingredients. The first is the price of the stamp when it was issued. Conveniently, the price is printed on each postal stamp. At the time of issue, this price was the same for all individuals. Also, postal stamps can either be used for sending postal mail or for collection. Except for misprints all issued stamps have the same quality, and preferences for certain stamps over others issued at the same time do not exist. The second ingredient of the model is the amount of stamps that were issued. Often, stamps come in series with a certain theme (summer, anniversaries). Each series usually contains a stamp with a value that, at that time, equals the price for sending a regular letter. Usually, more of these particular stamps are issued.

The third, and very important, ingredient of the model is the current price of the stamps, either fresh (not used) or stamped (used). Each year the average current prices are reported in a catalogue, which is available to the general public. These prices reflect a consensus value and are set by informed traders and collectors. These prices concern regular versions of these stamps, so misprints, rare prints or special envelopes are not taken into account. Also, all stamps are assumed to be of good quality. This is a relevant feature of stamps, in contrast to products like old cars or early vinyl recordings, where quality might decrease over time due to its usage. Indeed, used stamps could have been used only once. With these three components, and a few assumptions about the functional form of the model, it is possible to estimate the current stock of (even rare) postal stamps.

In Section 2, I discuss the model. In Section 3, I present the data and the estimation results. Section 4 concludes with a concise discussion of the findings.

The model

Consider a postal stamp i, with i = 1, 2. . . . , N . Denote the face value of the stamp at the time of issue, which is equal to its price which then had to be paid for it, as p i,0 . The issued quantity at that time is denoted by q i,0 . Finally, the current price is equal to p i,1 , as it appears in a most recent philately catalogue. Obviously, the unobserved variable of interest here is q i,1 , which is unknown, and it is unlikely that it can ever be observed.

It seems reasonable to assume that q i,1 is a fraction of q i,0 , with an upper bound fraction 1 (more likely for very recent stamps) and a lower bound 0 (perhaps more likely for very old stamps). Below we will see that aspects of the stamps other than age may also effect this fraction.

One way to describe this relation is to assume that

q i,1 = exp(Xβ) 1 + exp(Xβ) q i,0 , (1)
where X contains an intercept and variables like time, type of stamps, the amount issued, and so on.

The second component of the model concerns the relation between price and quantity, which, due to the latency of q i,1 , is most conveniently written as

log p i,1 -log p i,0 = µ + γ(log q i,1 -log q i,0 ) + ε i , ( 2
)
where log is the natural logarithm, and with µ capturing inflation and other common factors concerning philately, and with γ measuring the price elasticity and ε i is assumed to be a normal error term.

Dividing both sides of (1) by q i,0 , taking logs, and then after substituting (1) in

(2), one obtains the final model specification The parameters of (3) can be estimated using non-linear least squares (NLS). The catalogue provides information on p i,1 , p i,0 and q i,0 , where p i,1 concerns prices for either fresh or used stamps. Additionally, the variables in X can be the number of the stamp (as a measure of recency), any surplus value (usually for charity purposes), a dummy variable for a children's series, the amount of stamps within a particular series, the rank of a stamp with that series, whether the stamp value corresponds with the regular price of sending a common letter, the number of days of issue, and the time (in days) to the end of validity for actual use. For the latter variable there turn out to be many missing observations (as many stamps can still be used), and hence this variable is not included in further analysis. For most variables, one would expect to find a positive effect on the decay factor, and hence on the change in price, as they either would imply a heavier use and a higher value for collectors. The sample concerns 1413 stamps. However, for only 1006, I can obtain full information on all relevant variables. To facilitate the process of NLS parameter estimation, all variables are taken in deviations from their mean.

log p i,1 -log p i,0 = µ + γ(Xβ -log(1 + exp Xβ)) + ε i . ( 3
The estimation results appear in Table 1. This table only reports the parameter estimates that are significant at the 1 per cent level as there are many observations.

Clearly, all parameters obtain the expected sign, except perhaps the series size and the duration of issue. The R 2 of the models for fresh and used stamps are 0.841 and 0.853, respectively, which suggests an exceptional fit for a cross-sectional database.

With the parameter estimates, I compute the decay factors as in (1), as well as the estimated values of q i,1 , see Table 2. As could be expected, the spread of the decay factor for fresh stamps is larger than that of used stamps. This is also 3 If one would take the estimates of q i,1 as given, then the four stamps (their numbers), which according to the model and the resultant parameter estimates must be most rare, are given in Table 3. Interestingly, I find that about the same stamps, out of two particular series, are the most rare, either new or used. Figures 1 and2 give pictures of these two series.

Discussion and conclusion

Even though collecting stamps is enjoyed worldwide by millions of individuals, surprisingly little research has been done towards understanding aspects of philately.

I could trace two studies which relate amateur stamp trade with a particular type of auctions, see [START_REF] Thiel | Bidding behavior in 2nd-price auctions -Rare stamp sales, 1923-1937[END_REF] and [START_REF] Luckin-Reiley | Vickrey auctions in practice: From nineteenth0century philately to twenty-first-century e-commerce[END_REF]. Furthermore, an interesting ethnological study appears in van de [START_REF] Van Der | Passion and profit -The world of amateur traders in philately[END_REF].

Despite the active amateur trade, nowhere could I find any study of price-setting behavior of fresh and used stamps. Obviously, the officially quoted price must reflect an estimate of what informed individuals believe is the quantity available. This quantity is not observed, but with a few assumptions it is possible to get an estimate of the remaining stock of each stamp. For observations on over 1000 stamps, I found that stamps belonging to two particular series must be most rare. Although the point estimates lie within confidence regions, it seems safe to conclude that of the rarest

  1

3

  Post World War II Dutch postal stamps In this section I consider model (3) for Dutch postal stamps that have been issued in the years 1945 to and including 2001 (May), covering 1413 different stamps. All information is based on the Special Catalogue 2002 of the NVPH (ISBN 90-73646-33-2). Prices in Dutch guilders are converted into euros, using the standard conversion rate (2.20371).

  reflected in the estimates of the remaining stock. Apparently, for some stamps only little stock exists, although one should of course discount for parameter uncertainty.

	Submitted Manuscript
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Table 2 :

 2 Dutch stamps there are less than 100 around. Intriguingly, this estimate cannot be matched with actual data, but what could be done is comparing the model estimates with those obtained from interviewing informed traders. Properties of the decay factor and the remaining stock

	Other further research topics concern an analysis of similar data for other coun-
	tries and an evaluation of panel data. Indeed, the present analysis was based on the
	2002 catalogue, but looking at data from catalogues of other years could shed light
	on dynamic patterns in the price-setting of stamps.
	4

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Figure 1: Series with stamps 550 to 555 8 Page 10 of 12 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Figure 2: Series with stamps 556 to 560 9 Page 11 of 12 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK