David Maddison
email: d.maddison@ucl.ac.uk

Anders Jul-Pedersen

D42 -Monopoly < D4 -Market Structure and Pricing < D -Microeconomics, D44 -Auctions < D4 -Market Structure and Pricing < D -Microeconomics, E30 -General < E3 -Prices, Business Fluctuations, and Cycles

Keywords: Art Prices, Durable Monopoly, Auctions Art Prices, Auctions, Durable Monopolist

Analysing a panel of paintings by Danish painters suggests that the conditional life expectancy of the artist at the time of sale has a negative impact on the sale price. This is consistent with the idea that artists share some of the characteristics of durable monopolists and that the ageing and ultimately the death of the artist represent acceptable forms of commitment not to 'overproduce'. In addition interest in an artist's work begins to wane after their death.

In order to answer this question however requires an index of art prices over time.

It is difficult to provide such an index because art objects are heterogenous and not traded very often. One cannot simply take an average price of all works traded within a particular period or take a basket of art works and calculate the cost of purchasing that basket. This being so two alternative approaches are typically taken to computing the rate of return on paintings over time. The first of these is the hedonic approach in which differences in the characteristics of paintings are controlled for using regression analysis. Variables include things like the dimension of the painting, the materials used, the date of the painting etc. The other technique is the resale method and is a more direct method of calculating the rate of return to art. This approach waits until a painting has been sold twice and then computes the rate of return over the intervening period.

From [START_REF] Baumol | Unnatural Value or Art Investment as a Floating Crap Game[END_REF] onwards the general finding is that the rate of return on art is lower than that obtained on risk free financial assets. The reason for this is that individuals presumably derive a non-pecuniary benefit from possessing art that is quite independent of the financial rate of return that they expect to achieve when the time comes to sell the painting [START_REF] Frey | On the Rate of Return in the Art Market: Survey and Evaluation[END_REF].

The difference between the financial rate of return on the painting and the rate of return on alternative financial asset therefore reflects the benefit that the individual derives from possessing the painting. Unless one actually likes the picture, ownership of a painting is therefore not a good investment. Neither can it be said that art is a particularly liquid asset since the costs of buying and selling are high. [START_REF] Chanel | Is Art Market Behaviour Predictable?[END_REF] demonstrates that art markets are linked to financial markets but [START_REF] Ginsburgh | Long-term Comovements in International Markets for Paintings[END_REF] argue that there is only a short run relationship between the two.

A second reason that economists have studied the market for art is in order to test the efficient market hypothesis and the law of one price. The efficient market hypothesis does not however receive much support from the analysis of art markets. It would appear that one is able to predict future price movements on the basis of past price movements and that there is significant serial correlation in the data. Furthermore it is frequently said that old masters enjoy a higher rate of return than contemporary artists of the works of lesser-known artists (although a sequence of papers suggests that the opposite might be true). There also appear to be deviations from the law of one 2 price in which identical prints are sold for different prices days apart with no intervening release of information. Some auction houses appear to obtain consistently higher prices for auctioned paintings, and there is evidence to suggest that art prices are systematically lower in some months than in others. [START_REF] Ashenfelter | Auctions and the Price of Art[END_REF] provide a recent survey of the literature.

One characteristic of artists that has not so far received much analytical attention is that they suffer the same problems encountered by a monopoly producer of durable goods. The problem of the durable goods monopolist first described by [START_REF] Coase | Durability and Monopoly[END_REF] is that the monopoly producer cannot sign a contract limiting future production. Then the best thing he can do after having sold a unit is to try and sell another for as high a price as he can obtain. This goes on until the price falls to the marginal cost of production. But consumers will anticipate the fall in price and be unwilling to pay more than the competitive price for the goods. Thus the monopolist forfeits his monopoly power.

The artist shares the same difficulty as any other durable monopolist in that he or she lacks any credible commitment not to 'overproduce' in future time periods (e.g. [START_REF] Grampp | Pricing the Priceless: Art, Artists and Economics Basic Books[END_REF]. The word 'overproduce' should however be extended to include anything that might jeopardise the market value of paintings already sold to collectors. It seems that the only credible commitment that the artist can make not to over-produce in future time periods is either to die or at least grow old to the point where the ability to overproduce is limited by a short remaining life expectancy and declining health. This phenomenon might explain why artists often enjoy greater success dead than they did alive. The problem appears elsewhere in the field of culture as well. There is for example, the problem of the book publisher who would like to persuade individuals to buy an expensive hardback copy of a book rather than waiting for the paperback edition. Once individuals have bought the hardback version however the incentives of the publisher change.

A number of solutions to the problem of the durable monopolist have been suggested in the context of industrial economics [START_REF] Bulow | Durable-Goods Monopolists[END_REF]. One possibility is to rent the product rather than to sell it.

Another possibility is to make binding commitments about future production, issue price guarantees or to make the product less durable than it otherwise would be. These do not seem to help the artist much. One possibility is that the artist who makes lithographs can destroy the plates. There is nevertheless one mechanism by which the artist might be able to enjoy the financial benefits of their work prior to their death. This is practice of 'droit de suite' [START_REF] Solow | An Economic Analysis of Droit de Suite[END_REF]. Under this practice the artist has the right to a percentage of the selling price every time one of his paintings is sold. This right is inalienable and continues for a period even after the artist's death. It clearly burdens the artist with some of the risk of uncertain future price changes. This might lead to the artist not producing copies of original work or doing anything that might lower his stock as an artist such as a failure to censor sub-standard pieces of work. Note that this practice is illegal in all the United States except for California, but more common in Europe. From January 2006 a European Union directive will harmonise the practice throughout member states leading to concern over the possible relocation of trade to locations such as Zurich and New York.

The impact of the changed supply conditions associated with the death of the artist has been examined by [START_REF] Ekelund | The Death-Effect in Art Prices: A Demand-Side Exploration[END_REF] whose study involves 21 Latin American artists whose works have been sold at major auction houses in USA. The conclusion of the paper is that there is a measurable "death effect". Prices rise around the time of the artist's death but fall again.

Unfortunately the results use a questionable variable for representing the death effect "years after death" squared (i.e. the year of sale minus the year of death of the artist squared so as to ensure that the variable takes only positive values). Ekelund et al attribute their results to the existence of durable monopoly. It is not immediately clear however why the death of the artist should result in a symmetric response in prices around the time of their death, nor why the effect should be identical for all artists irrespective of whether they die prematurely or survive until old age.

Recently, [START_REF] Matheson | Death Effect on Collectible Prices[END_REF] have suggested that any death effect in art prices and in collectibles generally is a transient phenomenon caused by renewed interest in the artist's work following news of their death. They refer to this as the "nostalgia effect" and demonstrate its existence using time series data on the price of sports cards for 13 famous baseball players. The interesting aspect of their study is that the baseball cards are already fixed in supply following the retirement of the player and that changed perceptions regarding the supply of the cards around the time of the players death cannot therefore account for any observed price dynamics. The authors find that there is an increase in the price of cards around the time of a player's death but that it is short-lived. But whilst Matheson and Baade's work demonstrates the existence of a "nostalgia The purpose of this paper is examine closely the price dynamics of an artist's work around the time of their death. More specifically we argue that although temporary renewed interest and a change in perceived supply conditions might both cause an increase in prices following the death of the artist, only concern over supply conditions can explain why the conditional life expectancy of the artist at the time when the painting is sold should be reflected in the price. We use of the artist's conditional life expectancy at the time of sale to represent supply conditions because from the perspective of the collector, an ageing artist means that supply conditions are improving all the time. For the same reason any death effect should be most pronounced for those who die prematurely since supply conditions for an artist in their nineties are already quite favourable to the collector. Furthermore, prices should not change after the death of an artist since supply conditions cannot change and conditional life expectancy falls to zero at this point. To anticipate the results, it appears that using conditional life expectancy of the artist at the time of sale as a proxy for supply conditions, the durable monoply hypothesis has considerable empirical content even when variables are included to account for any temporary renewed interest in the artist following news of their death. The identity of the artist is treated as a fixed effect in what follows accounting for the style as well as the reputation of the artist. The exact day of birth and death of the artists were found via a database for Danish artists.2 Because the day of the sale is also known it is possible to create a dummy variable (DEAD) indicating whether the artist was dead or alive at the time of the sale.

Another variable was created to represent the time elapsed between the death of the artist and the time of the sale (YEARS_AFTER_DEATH). This variable takes the value zero for paintings sold while the artist is still alive. Finally, combining knowledge of the artist's date of birth as well as the date of sale makes it possible, via a database on Danish demographic statistics, to compute the artist's conditional life expectancy at the time of sale (COND_LIFE_EXPECT).3 This variable is obviously zero for those artists who were already dead at the time of sale. There is a great deal of variation in the data set: the youngest artist to die lived until 38 whilst the oldest artist to die survived until 100. Table 1 gives information on conditional life expectancy in Denmark for the decade 1990-1999.

Apart from the personal details of the artist, a number of other characteristics are contained in the data set. These include whether the painting is attributed to the artist as opposed to being signed Note that the effect of the artist's age at time of production is not included because it is not systematically recorded in the data set. The artist's age at the time of production might explain some of the variance in the price of paintings. However, the timing of any "peaks" and "slumps" of the artist's career will differ markedly from artist to artist. By contrast, the effect of conditional life expectancy on perceived supply conditions is, at the time of sale, the same for all artists.

Furthermore there is no relationship between the age of the artist at the time at which a painting is sold and the age of the artist when the work was created, other than that the former necessarily exceeds the latter. 4

4 [START_REF] Gallenson | The Careers of Modern Artists Evidence from Auctions of Contemporary Art[END_REF] published a study looking at the age of the artist on the day of the paintings execution and the value of his work. The study looks at 42 contemporary America artists, and finds that artists born before 1920 were most likely to produce the most valuable work late in their careers and those born in the 1920s and 30s early in their careers. Some artists however peak at age 66 or 84 and overall there is great variation.

Empirical Analysis

In order to reveal price dynamics associated with the ageing and death of the artist it is first necessary to control for autonomous trends in the price of paintings. This is achieved by creating a series of linear spline functions with knots placed at yearly intervals. These spline functions are included in all subsequent regressions, but the coefficients are not displayed in the table of results.

Table 2 contains the results of three models of increasing complexity. In each model the dependent variable PRICE is taken in the logarithms following attempts at a Box-Cox transformation. In Model 1 the variable COND_LIFE_EXPECT is included as a proxy for anticipated supply conditions. The coefficient on this variable is negative suggesting that conditional life expectancy of the artist at the time of sale has a significant depressing effect on art prices. Turning to the remaining variables, a painting that is merely attributed to a particular artist rather than actually signed attracts a lower price. Paintings sold by the two major auction houses Kunsthalle and Rasmussen attract a higher price whilst paintings sold in Copenhagen, Aarhus and Vejle all attract lower prices. These variables likely serve as a proxy for the characteristics of particular pieces leading them to be sold in particular places and by particular auction houses. 5 The size of the painting is statistically significant but the increase in price per square inch of canvas varies as the size of the painting increases.

Model 2 includes the dummy variable DEAD along with the variable YEARS_AFTER_DEATH.

The purpose of this specification is to represent a situation in which the death of the artist generates renewed interest in his work that gradually ebbs away (or depending on the sign of the coefficient even increases). The results suggest that prices jump following the artist's death but the effect is significant only at the five percent level of confidence. The price of an artist's work does however fall significantly in the years following their death.

Model 3 combines the previous two models. As before there is evidence of the existence of a durable monopoly, at least in so far as the fact that the artist's conditional life expectancy at the time of the sale is highly significant. The dummy variable denoting the period following the death 5 The most important Danish paintings are obviously sold outside of Denmark.

Conclusions

The results contained in this paper suggest that including information on the artists' age and whether they are dead or alive at the time of sale can explain better the price dynamics of paintings.

Such findings are consistent with the view that artists share some of the problems encountered by monopoly producers of durable goods. Certainly it is hard to think of an alternative reason why the artist's conditional life expectancy at the time of sale should be of such importance to collectors. In addition it appears that interest in an artist's work generally dwindles after their death. There is however no sudden leap in art prices following news of the death of the artist other than that attributable to the resolution of the problem of the durable monopolist.

  studied the art market in an attempt to answer the perennial question: Is art a good investment?

  does not say anything about the existence of a death effect associated with the problem of durable monopoly precisely because changed supply conditions are absent in their example.

  used to compare the "durable monopolist" and "nostalgia effect" explanation of price dynamics around the time of death of the artist comprises the price of oil paintings by contemporary Danish artists who died during the period 1983 to 2003. The price data was extracted from Hislop's Art Price Index. The first sale took place in 1971 and the last recorded sale was in 2003. The paintings included in the data set were all sold at auction, reflecting the most visible segment of the art market. This means that the study does not reflect the entire art market, but it rather the largest part with the highest prices. The data set contains 4,857 transactions of paintings by 93 artists. No bought-in paintings (i.e. paintings that failed to reach their reservation price) are included in the data set. The price of the paintings (PRICE) is in current USD.

(

  ATTRIBUTED). Two different auction houses are separately identified: KUNSTHALLE and RASMUSSEN. The latter auction house includes paintings sold by Bruun who merged with the biggest in Scandinavia and the world's 10th largest auction house. Shortly after the final sale recorded in the data set Kunsthallen too was purchased by Bruun Rasmussen. The remaining paintings were sold by miscellaneous auctioneers, including the Museumsbygningen (the art museum in Copenhagen), Sothebys and Christies etc. Selling at auction at the Museumsbygningen in Copenhagen costs 16 percent of the realised value of the painting indicating how difficult it is to take account of opportunities for arbitrage. Three different places of sale are identified: COPENHAGEN, AARHUS (Denmark's second largest city) and VEJLE near the German border (many collectors are German). Other places of sale not separately identified include Stockholm, London, and Amsterdam etc. The vast majority of the paintings were sold within Denmark. The size of the painting (SIZE) is measured in square inches.

F

Table 1 . The Conditional Life Expectancy of the Danish 1990-1999

 1

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

  is however, not significant. There is no surge in interest in an artist's work following their death. Only the time trend measuring years after death is significant.Note: The regression includes 32 spline functions whose coefficients are not displayed in the table.T-statistics are heteroscedastic-consistent.

	of the artist Submitted Manuscript Table 2. Regression Analysis R 2 0.1843 0.1983	0.2040
	Dependent Variable = Log PRICE
	Method = Fixed Effects Panel Data
			Model 1			Model 2	Model 3
	ATTRIBUTED		-0.592833			-0.6558729	-0.6149806
			(-4.07)			(-4.58)	(-4.24)
	KUNSTHALLE		0.2283518		0.2258041	0.2254439
			(3.94)			(3.92)	(3.91)
	RASMUSSEN COPENHAGEN	F o r F o r 0.1452296 (2.56) -0.4371322 (-5.85) F o F o r r		0.1393776 (2.48) -0.4318278 (-5.79)	0.1396537 (2.48) -0.4320232 (-5.79)
	AARHUS		P e P e -0.5177402 (-8.36) P e P e		-0.516128 (-8.32)	-0.5171571 (-8.34)
	VEJLE		-0.602433	e e e e	-0.5935193	-0.5898853
			(-7.54)	r r r r	(-7.43)	(-7.38)
	SIZE SIZE 2		0.0012067 (18.57) -3.00E-07 (-7.60)	R e v i R e v i 0.0012048 (18.46) -3.02E-07 (-7.58) R e R e v i v i	0.0012034 (18.61) -2.99E-07 (-7.61)
	SIZE 3 COND_LIFE_EXPECT	2.53E-11 (4.26) -0.0134608		e w e w 2.57E-11 (4.31) e w e w	2.51E-11 (4.29) -0.0150265
			(-5.75)			(-4.34)
	DEAD					0.0642483	-0.0666766
						(2.14)	(-1.53)
	YEARS_AFTER_DEATH				-0.0234978	-0.021138
						(-5.01)	(-4.54)
	Number of Observations	4857			4857	4857
	Zero Slopes		F(42, 4722) = 71.07 F(43, 4721) = 67.75 F(44, 4720) = 64.61
	9 10 11 12

Page 12 of 18 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

The databse on Danish artists can be found at: www.kid.dk.

Information on conditional life expectancies for Denmark for the decade 1990-1999 can be found at: www.mortality.org.

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


George Emm Halkos
email: halkos@econ.uth.gr

Ilias S Kevork

UNIT ROOT, POWER, MONTE Carlo Arima

CRITICAL VALUES Simulations

UNIT ROOT UNDER THE ALTERNATIVE HYPOTHESIS OF ARIMA (0,2,1)

Keywords: Unit root -ARIMA(0, 2, 1

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

 [START_REF] Dickey | Distribution of the estimators for autoregressive time series with a unit root[END_REF]in their pioneer work, considered the random walk model with drift as a special case of the Gaussian AR(1), . Empirical results from Monte-Carlo simulations, although they support the outcome of first differences, they do not agree with a white noise in second differences.

Figure 1 displays the sample autocorrelation (ACF) and partial autocorrelation (PACF) functions of first and second differences of a typical realisation of size 250 observations from the random walk with 2 = µ , 0 y o = , and 1 = . Details about the adopted random number generator can be found in [START_REF] Kevork | Confidence Interval Methods for Discrete Event Computer Simulation: Theoretical Properties and Practical Recommendations[END_REF]. For first differences, the plots confirm a white noise process. On the contrary, the sample ACF and PACF of second differences, combined with a stationary time series plot, indicate an MA(1) process, as a significant negative autocorrelation occurs at lag 1, and the partial autocorrelations decay exponentially to zero.

Figure 1 about here

The findings of Monte-Carlo simulations are shown analytically by first rewriting the second differences of the random walk model as . Thus, overdifferencing a random walk model with drift leads to ARIMA (0,2,1) with parameter 1 = . But also the random walk model can be considered as a special case of ARIMA (0,2,1) with 1 = . Rewriting ARIMA (0,2,1) as in (2), we take eventually the random walk model with drift.

To investigate further the behaviour of ARIMA (0,2,1) for greater but close to -1, we generated 1000 replications of 600 observations each, from the population model with 0 y o = . Figures 2a up to 2c display the time series plots of first differences, as well as, the sample ACF of first and second differences, for a typical realisation of 600 observations. As is approaching to -1 + , the time series plots of first differences are changing gradually from an obvious non-stationary pattern to a stationary one. The corresponding plots of ACF and PACF have the representative pattern of a nonstationary process for greater but quite far away from -1, but for close to -1, they indicate a white noise process. Finally, the graphs in figure 2c indicate an MA(1) in second differences, even when is not close to -1.

Figure 2a-c about here

The behaviour of ARIMA (0,2,1) for close to -1 leads us to consider it as the alternative hypothesis for evaluating and comparing the power of unit root test methods. Four such general methods are considered in this paper. The first method tests in second differences the null hypothesis H o : 1 = , against the alternative H o : 1 > , using a new set of critical values, which Halkos and Kevork generated for testing a unit root in finite samples from an MA(1). The second method is based on testing the statistical significance of the estimated coefficient of the lagged dependent variable in the right hand side of the "intercept" and the "trend-and-intercept" models, using Mac-Kinnon critical values. The third method includes common regression F-tests introduced by Dickey and Fuller, which are known as tests. Finally the last method concerns the non-parametric Phillips-Perron (PP) test, which is applied after estimation of the "intercept" model".

The previous discussion makes the structure of the paper to be as follows: In the next section we review the relevant literature on over-differencing empirical non-stationary series, and validity of unit root tests. In section 3, we revise the theoretical background for exploring the existence of a random walk by testing for a unit root in ARIMA (0,2,1). The power of such a test is also presented for different combinations of sample sizes and values of . In section 4, we estimate the power of t, , and PP tests on ARIMA (0,2,1). The method, based on testing for a unit root in ARIMA (0,2,1), attains low power for small samples and values of close to -1. For such cases, in section 5, we explore the consequences of a wrong decision, namely to accept a random walk model for making predictions for future values of the true process. Finally, section 6 summarises the main findings of this research.

LITERATURE REVIEW

The literature review focuses on two basic issues: over-differencing empirical nonstationary series and validity of unit root tests. Specifically, in empirical research applications of Box-Jenkins ARIMA (p,d,q) models for making valid predictions, we have to identify correctly the proper ARIMA model, which governs the behaviour of the empirical time series (hereafter TS). For a non-stationary time series before identifying the parameter p and q we must identify the times the series should be differenced.

The number of times that the TS under consideration must be differenced is determined intuitively by using the autocorrelation or/and partial autocorrelation functions of the differenced series. Model identification is complicated especially if the TS under consideration is seasonal or periodic. For non-seasonal TS, manual identification may be achieved by using the autocorrelation or/and partial autocorrelation functions, the extended autocorrelation function and the smallest canonical correlation table (Tsay and Tiao, 1984, 1985[START_REF] Box | Time Series Analysis: Forecasting and Control. Holden Say[END_REF][START_REF] Box | Time Series Analysis: Forecasting and Control[END_REF][START_REF] Pankratz | Testing for unit roots in autoregressive moving average models[END_REF]. The above methods seem to be ineffective in seasonal TS. In this case the identification may be performed using a filtering method [START_REF] Liu | Identification of seasonal ARIMA models using a filtering method[END_REF][START_REF] Liu | Forecasting and time series analysis using the SCA Statistical System[END_REF][START_REF] Liu | Forecasting and time series analysis using the SCA[END_REF]. This method is effective for automatic identification of ARIMA models for both seasonal and non-seasonal TS. [START_REF] Koreisha | New approaches for determining the degree of differencing necessary to induce stationarity in ARIMA models[END_REF] presented two methods for determining the degree of differencing in order to achieve stationarity in the data. Using simulation of different model structures, they confirmed the results. [START_REF] Hall | Testing for a unit root in the presence of moving average errors[END_REF] proposes a test for unit root relying on an instrumental variable (IV) estimator, which was applied in the case where the series is generated by an ARIMA(0,1,q) process. Pantula and Hall (1991) extended Hall's framework to the case of a series generated by an ARIMA(p,1,q) model. To obtain the asymptotic distributions, they F o r P e e r R e v i e w 5 assumed that either p or q was known, and using simulations they provide evidence that the finite-sample distributions of their test statistics were well approximated by the Dickey-Fuller distributions even in the case of over-specifying p and q. [START_REF] Reilly | Experiences with an automatic Box-Jenkins modeling algorithm[END_REF] and [START_REF] Reynolds | Box-Jenkins forecast model identification[END_REF] develope automatic methods for identifying ARIMA models for TS. The method developed by [START_REF] Reynolds | Box-Jenkins forecast model identification[END_REF] employs a neural network approach and is restricted to non-seasonal TS, while the method developed by [START_REF] Reilly | Experiences with an automatic Box-Jenkins modeling algorithm[END_REF] works properly for non-seasonal TS but it is less effective in the case of seasonal TS.

The above-mentioned methods require the existence of long TS, which are used for model development and validation before we proceed to parameter estimation and predictions. The ARIMA approach for TS predictive model development is justified in both theoretical and statistical grounds. But [START_REF] Makridakis | Forecasting: Methods and applications[END_REF] claim the complexity of these models has been an obstacle for their adoption as a forecasting tool in organisations. The one-step ahead forecast for an ARIMA (0,1,1) model is equivalent to forecasting using an exponential smoothing method, when the smoothing constant leads to minimum mean square error forecast [START_REF] Abraham | Statistical methods for forecasting[END_REF].

A unit root in the moving average polynomial can be interpreted in various ways depending on the modeling application. Testing for a unit root in the moving average polynomial is equivalent to test that the series is over-differenced [START_REF] Brockwell | Introduction to Time Series and Forecasting[END_REF]. A difficulty with the null hypothesis H 0 : =1 is that estimating a moving average model with a unit root is an irregular problem. The asymptotic distribution of the maximum likelihood estimator of a noninvertible MA parameter is unknown but there is a positive probability that a local maximum is attained by the likelihood function at a point of a unit root [START_REF] Anderson | Why do noninvertible estimated moving averages occur?[END_REF]Takemura, 1986, Tanaka andSatchell, 1989). This implies that the development of LR and Wald tests is "intractable". Lagrange multiplier tests can be obtained, as they require the estimation of the model under the null hypothesis. [START_REF] Ahtola | Some aspects of parameter inference for nearly stationary and nearly noninvertible ARMA models II[END_REF] prove it in the case of an MA(1) model with a zero mean value, while Tanaka (1990) obtains a general score-type test for the MA unit root hypothesis. [START_REF] Phillips | Time series regression with a unit root[END_REF] and [START_REF] Phillips | Testing for a unit root in time series regression[END_REF] extends this work on autoregressive unit root tests.

Similarly Saikkonen and Luukkonen (1993) derive two tests for the MA unit root hypothesis. In the case of serially uncorrelated errors these can be motivated by local optimality arguments. Halkos and Kevork (2005c) A large literature has been recently developed for analyzing TS regression with difference stationary processes. [START_REF] Dickey | Estimation and hypothesis testing for non-stationary time series[END_REF] and [START_REF] Dickey | Estimation and hypothesis testing for non-stationary time series[END_REF]Fuller (1976, 1981) in their seminal papers examine the OLS estimation when the innovations in the unit root process are i.i.d. [START_REF] Phillips | Time series regression with a unit root[END_REF] extends these results to a more general setting for the innovation process in such a way as to allow both time dependence and heterogeneity. [START_REF] Phillips | Testing for a unit root in time series regression[END_REF] explore data generating mechanisms with drift and trend. [START_REF] Phillips | Time series regression with a unit root and infinite variance errors[END_REF] and [START_REF] Chan | On the first-order autoregressive process with infinite variance[END_REF] have explored the estimation of the autoregressive parameter and tested for a unit root when the random walk process has errors, which obey to a stable law. [START_REF] Phillips | Time series regression with a unit root and infinite variance errors[END_REF] generalises this case using a semi-parametric modification of the usual t-ratio. [START_REF] Leybourne | The behaviour of Dickey Fuller and Phillips Perron tests under the alternative hypothesis[END_REF] using simple theoretical calculations, confirm simulation evidences that probabilities of rejecting the null hypothesis of the Dickey Fuller and the Phillips-Perron tests differ substantially when the true generating process is the stationary second order autoregression. Halkos and Kevork (2005b) using certain estimates from Monte-Carlo simulations and considering the random walk as the true model, derived the probability the prediction interval to include any future value s T y + of AR(1). [START_REF] Ahn | Unit root tests with infinite variance errors[END_REF] analyse both asymptotically and in finite sample the properties of some unit root test, when the errors obey to a stable law. They consider a number of test statistics (such as the Dickey Fuller and the Lagrange Multiplier), when the data generating process is a driftless random walk and the regression model matches exactly the data generation process. [START_REF] Gallegari | Asymptotic inference in time series regressions with a unit root and infinite variance errors[END_REF], in a similar analysis, characterize as limited both the behavior of OLS estimators of regression coefficients and the DF tests under the data generating processes usually encountered in the unit root literature (random walk with and without drift and the associated regression models with constant term, without deterministic component and with constant and time trend terms). They also investigate the consequences of the 'local to finite' variance analysis assessing that the size distortion of the DF test as the departure from the standard finite variance set up tends to decrease as the sample size tends to infinite. [START_REF] Dickey | Distribution of the Estimators for Autoregressive Time-Series with Unit Root[END_REF] based their analysis on the asymptotic properties of the OLS estimator. Important variations of the DF tests are their extensions to other estimation methods such as Maximum Likelihood (Shin andLee, 2000, Skin andFuller, 1998), the generalised least squares detrending under a fixed local alternative [START_REF] Elliott | Efficient tests for an autoregressive unit root[END_REF], Xiao and Phillips, 1998[START_REF] Hwang | Alterative methods of detrending and the power of unit root tests[END_REF] and the weighted symmetric estimator [START_REF] Park | Alternative estimators and unit root tests for the autoregressive process[END_REF]Fuller, 1995, Fuller, 1996). [START_REF] Hassler | On the power of unit root test against fractional alternatives[END_REF] claim that the Augmented Dickey Fuller (hereafter ADF) compared to fractional alternatives loses considerable power when augmented terms are added.

On the other hand, [START_REF] Krämer | Fractional integration and the augmented Dickey Fuller test[END_REF] finds that ADF is consistent if the order of autoregression does not tend to infinity too fast. [START_REF] Bisaglia | On the power of the Augmented Dickey-Fuller test against fractional alternatives bootstrap[END_REF], using Monte Carlo simulations, clarify this contradiction and find that the ADF bootstrap works in general better than the ADF even if the power of the test is quite low, especially if the data generating process is a nonstationary fractional integrated one.

On the contrary, Halkos and Kevork (2005a) Finally, a number of researchers have developed tests for a single structural break with unknown break points in various dynamic models [START_REF] Andrews | Tests for parameter instability and structural change with unknown change point[END_REF][START_REF] Perron | Nonstationary and level shifts with an application to purchasing power parity[END_REF]Sen, 2004;[START_REF] Blot | Convergence of fiscal policies in EMU: a unit-root tests analysis with structural break[END_REF]. In most cases, these tests were either designed to test for a structural change in regression coefficients with stationary series or for a unit root against a stationary alternative with an unknown single break point. The applications of these tests were extremely successful in analysing breaking points in variables like real exchange rates, real GNP and other integrated processes [START_REF] Banerjee | Recursive and sequential tests of the unit root and trend break hypothesis: Theory and international evidence[END_REF][START_REF] Perron | Nonstationary and level shifts with an application to purchasing power parity[END_REF]Zivot and Andrews, 1992;[START_REF] Charemza | Is inflation stationary?[END_REF][START_REF] Harvey | Evidence for common features in G7 macroeconomic time series[END_REF].

TESTING FOR A UNIT ROOT IN ARIMA (0,2,1)

For the non-invertible MA(1),

1 t t t y + = µ
, given a sample of size T, the ln of the exact likelihood function is given by ( ) ( )

( ) = = = T 1 t tt 2 t 2 T 1 t tt 2 2 y 21 2 1 ln 2 T 2 ln 2 T , L ln (3)
where Hamilton, 1994). So, whenever ~ is less than -1, the test is applied by taking the reciprocal of ~.

( ) ± = + ± = 1 , t 1 t 1 , 1 1 t 2 1 t 2 tt ( ) ± = ± = 1 , 1 1 , 1 1 ~1 1 2 1 2 µ µ t t t t
~ = = (
In the introductory section we illustrated the dual relationship between the random walk model and ARIMA (0,2,1). That is, overdifferencing the random walk model leads to ARIMA (0,2,1) with parameter 1 = , as well as, the random walk model can be considered as a special case of ARIMA (0,2,1) with 1 = . This means that applying the test H o : 1 = , against H 1 : 1 > , to second differences of a time series realisation is equivalent of testing in the same series for the existence of a unit root. The only problem, which occurs in such a case is that taking second differences, the actual sample size is reduced from T to T-2. For this reason, we applied a linear interpolation to the initially suggested critical values of Halkos and Kevork, resulting to those presented in . The initial conditions, as well as, comments about the validity of the random number generator have been already discussed in the introductory section. In each replication, and for nominal T=25, 50, 100, 250, 500, after subtracting the mean of second differences, we fit an MA(1) without a constant term to second differences in each replication, by maximising (3). The maximisation process was performed in E-VIEWS.

In each replication, to update and 2 in successive iterations, we used the Marquard first derivative method, where first derivatives were evaluated analytically. The computation of

( ) ( ) 2 , L ln
was repeated until the improvement between iterations was less than 0.001. The Marquardt algorithm modifies the Gauss-Newton by adding a correction matrix to the Hessian approximation. This ridge correction copes with numerical problems in case the outer product is near singular, improving the convergence rate. Estimation using the analytical evaluation of first derivatives was preferred as it involves fewer function evaluations, and therefore is faster than evaluating the derivatives analytically.

As starting values for and 2 , we used those provided by E-VIEWS, using the method of backcasting (Box and Jenkins;[START_REF] Dickey | Estimation and hypothesis testing for non-stationary time series[END_REF]. EVIEWS backcasts MA terms by computing the unconditional residuals and using the backward recursion to compute backcast values. To start this recursion the values for the innovations beyond the estimation sample are set to zero and a forward recursion is used for estimating the values of the innovations using the backcasted values of the innovations and the actual residuals. The sum of squared residuals is minimized, after having being formed as a function of and relying on the fitted values of the lagged innovations.

The backcast step, forward recursion and the minimization of SSR are repeated till the convergence of estimates of .

The standard errors of the estimated and 2 were computed from the Gauss-Newton Hessian. Gauss-Newton follows Newton-Raphson replacing the negative of the Hessian using for and the outer product is necessarily positive and semi-definite. But in case that it is away from the maximum, we may approximate poorly the overall shape of the function. This implies the need for more iterations for convergence. , the power remains at very low levels, even with a sample of size 500. Further, for all the examined cases, the probability of being in uncertainty ranges below 4%. The cases where low power is observed will be further discussed in section 5, where we shall investigate, from the forecasting point of view, the consequences of not rejecting the null hypothesis on the validity of prediction intervals generated using the random walk model. Pindyck and Rubinfeld (1988) as an F-ratio test.

Regarding the three tests, adding lagged difference terms of t y to the right-hand side of (4) and ( 5), the sample size is reduced. For this reason, before the implementation of the tests, an appropriate linear interpolation was applied again to the corresponding DF critical values.

Additionally to this, adding j t y terms to the right-hand side of ( 4) and ( 5), not rejecting the unit root null hypothesis in tests leads to ARIMA (P,1,0). On the contrary, not including such lagged difference terms of the dependent variable in the testing models, the rejection of the null leads to the rejection of the random walk model. where t and s are respectively the t-statistic and the standard error from applying OLS regression to (6), ˆ the standard error of the regression,

( ) [ ] = + + = q 1 j j o 2 1 q j 1 2 ˆ, T T 1 j t j t t j + = =
, and q the truncation lag defined as the largest integer not exceeding

( ) 9 2 T 01 . 0 4
. In this test, the null hypothesis is rejected whenever PP t is less than the corresponding Mac-Kinnon critical values.

Tables 3 up to 7 present the results from the application of all the previous tests on ARIMA (0,2,1), for different sample sizes T=25, 50, 100, 250, 500, as is approaching to -1 + .

Considering t INT , t TREND , and PP tests, the first interesting remark is a non-negligible probability to estimate a positive , something which contradicts the nature of the alternative hypothesis, <0, in these tests (see table 3). For t INT , and PP, and for every combination of T and , this probability ranges close to 0.50, while no significant differences are observed between the two tests. Besides, adding lagged difference terms of the dependent variable to the right hand side of (4), no considerable changes are observed, while increasing the sample size these probabilities rise slightly in both tests. On the other hand, t TREND statistic displays lower probabilities of a positive compared to the other two tests. Furthermore, the probability related to t TREND statistic is rather low either for small samples or for quite close to -1. Increasing the sample size, this probability is getting larger and larger, approaching =1, the probability declines, as well as, adding j t y terms to the right hand side of (5), the probability declines again.

For the previous three tests, tables 4 displays their "conditional" power, at nominal level of significance 10%. The term conditional implies that the power has been estimated as a percentage of replications where the null hypothesis is rejected, from the subset of replications, where a negative -estimate is obtained. For the three tests, the estimated conditional power increases as T is getting larger, but decreases as is approaching -1 + . However, the most important feature, regarding t INT and t TREND tests, is that for large T (e.g T 100), when is not very close to -1, adding more j t y terms in (4) and ( 5), trying to correct autocorrelation in the error term, we lead the tests to lower power. Comparing now the performance between the three 13 tests, which one of the three attains the highest power depends upon the sample size and how close is to -1. For example, for small samples (e.g. T 50) and not close to -1, PP attains the highest power, while for T=500, t INT succeeds the highest power, even when = -0.98.

Tables 5, 6 and 7 present the power (again at nominal level of significance 10%) of the tests, for which the problem of a negative -estimate is not met. For this reason, the power was estimated with reference to the total number of 1000 replications for each case. For P=0, the power of 3 statistic is higher than the power of 2 and 3 . Besides, the power of 3 test, (a) declines as takes values closer to -1, (b) increases by drawing a larger sample, and (c) has a downward trend for not very close to -1, by adding more j t y terms in the right-hand side of (5). Regarding now 1 and 2 tests, adding lagged difference terms of the dependent variable to the right hand side of ( 4) and ( 5), we observe extremely high powers. Especially, for not so close to -1, the maximum power is attained in most of the cases at P=1. But we should not forget that for models (4) and ( 5) with P 1, rejecting Ho in 1 and 2 leads not only to reject the null hypothesis but also the hypothesis that the population model is integrated of order 2.

Tables 3-7 about here

FORECASTING ARIMA (0,2,1) USING THE RANDOM WALK MODEL

In table 2, testing a unit root on ARIMA (0,2,1), there are cases (combinations of T and ) where the power remains at low levels. In such cases, therefore, it is very likely to accept that the process of generating the data is the random walk model, when in fact this is not true. The consequences of such a wrong decision are investigated in the current section, by evaluating the validity of the prediction interval for a future value ( )

l T y
of ARIMA (0,2,1), using, however, the random walk prediction equation and error variance for the period l forecast, computed as The first criterion of evaluation is the coverage of the prediction interval ( )

( ) µ + = ŷ y ˆT T l l , and
( ) ( ) ( ) 2 v 2 T T ŷ ŷ V = l l l respectively, where 1 T y y 1 T y ˆ1 T T 2 t t = = µ = ( ) 2 T ŷ ˆT2 t 2 t 2 v µ = =
l l l l l v T T v T 2 ŷ y 2 ŷ + µ + µ + (7)
namely, the percentage the interval to include the true value, with reference all those replications where the random walk null hypothesis cannot be rejected. Table 8 presents the coverage, but only for those combinations of T and of table 2, where the method based on testing the unit root on ARIMA (0,2,1) gave low power. The general remark is that keeping l close to T, accordingly to the values of T and , coverage is acceptable, as it is over 90%. On the other hand, increasing l , coverage declines reaching low levels, especially using very large samples.

The second criterion concerns the precision of prediction interval ( 7), expressed as the ratio of its average half length (computed as the mean of half length of prediction intervals constructed from those replications where the random walk null hypothesis cannot be rejected) over the true value of the half length. It is known for ARIMA models that, given information up to time T, the conditional distribution of ( )

l T y is normal with mean ( ) l T y ˆ and variance ! ! " # $ $ % & + = 1 1 j 2 j 2 1 l .
For ARIMA (0,2,1), the weights are computed recursively from

( ) 1 1 j j + + =
. Using the last relationship, the variance of the period l forecast error for ARIMA (0,2,1) is given by

( ) ( ) ( ) ( ) ( )( ) ( )( ) ' ( ) + + + + = 1 1 6 1 2 1 1 1 y ŷ E 2 2 2 T T l l l l l l (8)
Thus the true value of the half-length, in the context of our simulation experiments, is computed by multiplying (8) by 2, replacing 1 2 = , and using the appropriate value for . Table 9 displays this ratio, again at those cases of table 2, where we meet low power. The ratio although is above one in all cases, indicating a smaller precision than it should be expected, it reduces by increasing l , resulting however in smaller coverage. Summarising, therefore, using the random walk model to predict future values of ARIMA (0,2,1) when is close to -1, for l close to T, coverage is ranged at satisfactory levels, but we take as a penalty a lower precision.

SUMMARY AND CONCLUSIONS

In this paper we show that taking second differences in the random walk model generates an ARIMA (0,2,1) with parameter 1 = . The reverse argument also holds, as a random walk model can be also considered as a special case of ARIMA (0,2,1) with 1 = . Using this dual relationship between a random walk model and an ARIMA (0,2,1), we illustrate that the behaviour of the latter model in levels and first differences, when is close but greater to -1, resembles the behaviour of either the random walk model with drift or the behaviour of the stationary AR(1) with autoregressive coefficient close to unity [Halkos & Kevork(2005b)]. The last finding leads us to suggest a new alternative hypothesis, in the form of replications from ARIMA (0,2,1) with close but greater to -1, to estimate and to compare the power of alternative methods of unit root tests, using Monte Carlo simulations. Considering the first method, for any greater than -0.95, a sample over 250 observations generates acceptable power, while for very close to -1, unfortunately, only an enormously large sample might lead to satisfactory levels of power. Regarding now the two tests based on t-statistics together with the PP test, which one of the three attains the highest power depends upon the sample size, and how close is to -1. Finally, comparing the power of tests without including lagged difference terms of the dependent variable to the right hand side of the corresponding models, the 3 statistic performs better than the other two. However, with the inclusion of lagged difference terms, the 1 and 2 statistics produce extremely high powers,

rejecting not only the unit root null hypothesis but also the hypothesis that the population model is ARIMA (0,2,1). Two other interesting findings should be also reported for some of the tests tests, adding more and more lagged difference terms of the dependent variable to the right hand side of the testing models, in an effort to eliminate autocorrelation in the error term, unfortunately the power shows a downward trend.

Up to now, the most known disadvantage of unit root tests has been the extremely low power when the parameter(s) of the model representing the alternative hypothesis take values close to limiting situations. On the other hand, this paper brings forward an additional problem concerning the selection of the appropriate unit root test, even when in the alternative hypothesis model its parameter(s) are not taking values close to a certain limit. For example, the "trend and intercept model" with an autoregressive coefficient not very close to unity cannot be easily differentiated and identified from an ARIMA (0,2,1) with close but greater to -1. This happens because in finite samples, both models display similar patterns in the plots of levels and in the sample autocorrelation and partial autocorrelation functions of first differences. And for the "trend and intercept model", Halkos & Kevork (2005a) showed that the 3 statistic should be preferred, while this paper suggests for ARIMA (0,2,1) the null hypothesis H o : 1 = being tested to second differences of the series.

We believe that the research on unit root tests, apart from developing new methods, should also take a new direction on exploring the consequences of reaching a wrong conclusion due to the low power of a specific unit root test. This is something that we also explore in the current paper. More specifically, regarding the first method, in cases where it attains low power, we study the validity of the prediction interval for a future value of ARIMA (0,2,1), when, however, we use the prediction equation and the error variance for the period l forecast of the random walk. Besides, the validity is explored only to those replications where the random walk hypothesis cannot be rejected. The results indeed are not disappointed, looking at the estimated probability (coverage) the prediction interval to include a future value of ARIMA (0,2,1) periods l ahead. Keeping l low, the prediction interval with half-width 2 times the square root of the random walk error variance attains coverage more than 90%. For such cases, however, the average half-length of the prediction interval might reach to be four times more than its true value.

F

25

-0,7358 -0,7243 -0,6775 -0,6578 -0,4675 -0,3685 50 -0,8287 -0,8222 -0,7980 -0,7891 -0,7087 -0,6692 100 -0,8933 -0,8897 -0,8765 -0,8720 -0,8387 -0,8248 250 -0,9422 -0,9403 -0,9340 -0,9316 -0,9176 -0,9127 500 -0,9635 -0,9625 -0,9587 -0,9573 -0,9493 -0,9464 -0.90 0,451 0,447 0,457 0,019 0,029 0,036 0,456 -0.95 0,430 0,440 0,451 0,012 0,016 0,029 0,437 -0,98 0,426 0,434 0,452 0,007 0,013 0,027 0,442 50 -0.90 0,471 0,477 0,476 0,054 0,050 0,057 0,474 -0.95 0,461 0,460 0,459 0,015 0,012 0,019 0,463 -0,98 0,434 0,446 0,460 0,006 0,005 0,006 0,442 100 -0.90 0,498 0,500 0,496 0,493 0,174 0,154 0,144 0,133 0,496 -0.95 0,497 0,491 0,495 0,500 0,059 0,049 0,047 0,048 0,497 -0,98 0,483 0,483 0,483 0,481 0,004 0,008 0,006 0,008 0,485 250 -0.90 0,493 0,495 0,494 0,489 0,489 0,398 0,368 0,348 0,321 0,299 0,493 -0.95 0,474 0,476 0,473 0,481 0,481 0,251 0,233 0,223 0,219 0,201 0,473 -0,98 0,463 0,466 0,469 0,465 0,467 0,051 0,048 0,044 0,044 0,048 0,464 500 -0.90 0,563 0,563 0,558 0,555 0,546 0,538 0,467 0,446 0,416 0,397 0,368 0,344 0,561 -0.95 0,505 0,504 0,504 0,503 0,505 0,504 0,383 0,375 0,367 0,352 0,342 0,325 0,505 -0,98 0,511 0,508 0,504 0,508 0,507 0,507 0,184 0,178 0,173 0,173 0,167 0,162 0,511 -0.90 0,071 0,058 0,057 0,077 0,104 0,080 0,097 -0.95 0,046 0,030 0,044 0,083 0,102 0,080 0,066 -0,98 0,030 0,027 0,038 0,099 0,109 0,083 0,050 50 -0.90 0,168 0,140 0,124 0,072 0,072 0,072 0,192 -0.95 0,050 0,050 0,039 0,085 0,098 0,092 0,073 -0,98 0,012 0,023 0,020 0,100 0,108 0,096 0,034 100 -0.90 0,458 0,388 0,347 0,292 0,054 0,037 0,040 0,043 0,405 -0.95 0,187 0,175 0,141 0,138 0,055 0,054 0,060 0,059 0,209 -0,98 0,029 0,033 0,033 0,029 0,089 0,096 0,078 0,080 0,039 250 -0.90 0,724 0,671 0,617 0,569 0,524 0,223 0,177 0,147 0,125 0,097 0,659 -0.95 0,532 0,515 0,488 0,489 0,464 0,072 0,072 0,051 0,047 0,041 0,518 -0,98 0,194 0,200 0,185 0,170 0,173 0,054 0,043 0,042 0,046 0,050 0,209 500 -0.90 0,828 0,778 0,719 0,667 0,606 0,545 0,484 0,394 0,320 0,264 0,217 0,175 0,752 -0.95 0,770 0,746 0,718 0,704 0,669 0,645 0,271 0,234 0,202 0,188 0,173 0,151 0,731 -0,98 0,444 0,431 0,417 0,407 0,400 0,396 0,067 0,062 0,060 0,052 0,056 0,055 0,431 -0,90 0,934 0,921 0,893 0,872 0,866 -0,95 0,931 0,925 0,915 0,899 0,891 -0,98 0,935 0,934 0,927 0,904 0,899 50 -0,90 0,941 0,920 0,869 0,860 0,855 0,729 -0,95 0,944 0,949 0,907 0,925 0,917 0,877 -0,98 0,951 0,953 0,929 0,941 0,944 0,929 100 -0,90 0,919 0,909 0,861 0,834 0,805 0,681 0,601 -0,95 0,936 0,941 0,930 0,912 0,893 0,858 0,807 -0,98 0,941 0,950 0,943 0,939 0,941 0,932 0,920 250 -0,95 0,950 0,930 0,892 0,884 0,869 0,756 0,667 0,616 -0,98 0,956 0,960 0,948 0,944 0,928 0,899 0,884 0,862 500 -0,98 0,947 0,933 0,918 0,923 0,920 0,871 0,834 0,794 0,672

  evaluated simple versions of the Dickey-Fuller test under the null hypothesis of a random walk model or an alternative non-stationary mean reverted process. Through Monte Carlo simulations they show that, apart from few cases, testing the existence of a unit root, using both McKinnon critical values and an F test, recommended by Pindyck and Rubinfeld, they obtain actual type I error and power very close to their nominal levels.

  an appropriate simulation strategy after maximising (3),Halkos and Kevork (2005c) offer a new set of critical values for testing H o : 1 = , against H 1 : 1 > , in samples of size T = 25, 50, 100, 250, 500. The case of an estimate of less than -1 does not contradict the alternative hypothesis, as (3) is maximised at 1

  approximation of the sum of the outer product of the gradient vectors and its contribution to the objective function. In this way we have just to evaluate the first derivatives,

  Four such general methods are considered in the current research. The first is based on testing H o : 1 = to second differences, rejecting the random walk null hypothesis using a set of critical values suggested byHalkos & Kevork (2005c) for finite samples. The second method refers to tests based on the t-statistic of the estimated coefficient of the lagged dependent variable in the right hand side of the "intercept" and the "trend-and-intercept" models. The third method includes tests based on the known statistics, which are calculated by estimating the unrestricted and the restricted forms of the "intercept model" ( 1 test) and the trend-and-intercept model ( 2 and 3 tests). Finally the last method concerns the non-parametric Phillips-Perron (PP) test, which is applied after again the estimation of the "intercept" model, without the presence of lagged difference terms of the dependent variable in the right hand side.

  The first concerns the t and PP test statistics, where a positive value might occur, something that contradicts the nature of the alternative hypothesis. Second, for the t and 3

Figure 1 :

 1 Figure 1: Sample autocorrelation and partial autocorrelation functions of first and second differences from the random walk model t 1 t t y 2 y + + =

  , using the exact maximum likelihood estimator of from

	the MA(1), and a certain simulation strategy, estimate appropriate percentiles, together with their
	standard errors, offering a new set of critical values for testing in finite samples H o :	1 = ,
	against H 1 :	1 > . In this way, appropriate regions for rejecting the null or being in uncertainty
	are defined.
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Table 1 .

 1 The critical values of table 1 can be used for testing a unit root in ARIMA (0,2,1) at nominal sample sizes T=25, 50, 100, 250 and 500, or actual ones T =

	23, 48, 98, 248, and 498, leading to rejection of H o when	ˆ >	U	, and to uncertainty whenever
	L	<	ˆ <	U	.
			To estimate the power of the unit root test in ARIMA (0,2,1) using the critical values of
	table 1, for each value of = -0.90, -0.95, -0.98, we generated 1000 replications of size 600
	observations from model	y	t	=	2	y	t	1	y	t	2	+	t	+	t	1
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Table 1

 1

	about here

Table 2 about here 4. POWER OF UNIT ROOT TESTS ON ARIMA (0,2,1)

 2 , concerns the null hypothesis H o : µ=0, =0, and is applied by estimating first (4), and then its restricted form under H o . The null hypothesis is rejected whenever the calculated The third test is based again on the t TREND statistic of the estimated . The unit root null hypothesis, H o : =0, is rejected in favour of the alternative H 1 : <0, whenever t TREND is less than the corresponding Mac-Kinnon critical value. Furthermore, the next two tests are the common regression "F-tests" (as 1 was), known as 2 and 3 . For 2 , the null hypothesis H o : µ=0, =0,

	In this section, we evaluate traditional unit root tests on ARIMA (0,2,1), using the 1000
	replications already generated from model	y	t	=	2	y	t	1	y	t	2		+	t	+	t	1	, for
	= -0.90, -0.95, -0.98. In the following lines, we present first the theoretical properties of each
	test separately, and the way of implementing it in each replication from ARIMA (0,2,1).
	The first two tests are carried out by estimating using ordinary least squares (OLS) the
	"intercept" model
	y	t	=	µ	+		y	t	1	+		1 = P j	j		y	t	j	+	t	(4)

The first test is based on the t INT statistic of the estimated parameter . Whenever t INT is less than the corresponding Mac-Kinnon critical value, we reject H o : =0, in favour of H 1 : <0. The second critical value of table V. In a similar manner, for 3 , when the ratio defined in 1 test is greater than the DF critical value of table VI, the null hypothesis H o : µ=0, =0, is rejected. The latter test is referred by

Table 1 :

 1 Critical values for a unit root test in ARIMA (0,2,1)

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	Submitted Manuscript
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Table 2 :

 2 Probability of rejecting a unit root in ARIMA(0,2,1); = 0.10

		Sample size
		T=25		T=50		T=100		T=250		T=500
		Uncertainty	Reject	Uncertainty	Reject	Uncertainty	Reject	Uncertainty	Reject	Uncertainty	Reject
	-0.90	0,025	0,160	0,024	0,197	0,037	0,367	0,013	0,872	0,000	0,994
	-0.95	0,029	0,143	0,024	0,149	0,024	0,192	0,036	0,406	0,019	0,808
	-0.98	0,024	0,141	0,026	0,131	0,017	0,158	0,022	0,204	0,025	0,277

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Table 3 :

 3 Probability of a positive t INT , t TREND , and PP

					t INT					t TREND		PP
	T	P=0	P=1	P=2	P=3	P=4	P=5	P=0	P=1	P=2	P=3	P=4	P=5
	25

Table 4 :

 4 Conditional power of t INT , t TREND , and PP test statistics

					t INT					t TREND		PP
	T	P=0	P=1	P=2	P=3	P=4	P=5	P=0	P=1	P=2	P=3	P=4	P=5
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Table 5 :

 5 Power of Dickey-Fuller 1 statistic

	Page
	T		P=0	P=1	P=2	P=3	P=4	P=5
	25	-0.90	0,042	0,964	0,740
		-0.95	0,017	0,972	0,766
		-0,98	0,011	0,972	0,771
	50	-0.90	0,136	0,999	0,986
		-0.95	0,036	1	0,994
		-0,98	0,012	1	0,996
	100	-0.90	0,404	0,997	0,995	0,992
		-0.95	0,134	1	1	1
		-0,98	0,016	1	1	1
	250	-0.90	0,719	0,981	0,976	0,967	0,956
		-0.95	0,502	1	1	1	1
	500	-0,98 -0.90 -0.95 -0,98	0,138 0,843 F 0,745 o 1 0,971 0,999 0,391 1	1 0,959 0,998 1	1 0,943 0,998 1	1 0,930 0,997 1	0,915 0,997 1
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Table 6 :

 6 Power of Dickey-Fuller 2 statistic

								Page
	T		P=0	P=1	P=2	P=3	P=4	P=5
	25	-0.90	0,068	0,935	0,703
		-0.95	0,059	0,956	0,744
		-0,98	0,055	0,966	0,760
	50	-0.90	0,149	0,999	0,975
		-0.95	0,073	1	0,991
		-0,98	0,057	1	0,991
	100	-0.90	0,364	0,998	0,997	0,990
		-0.95	0,130	1	1	1
		-0,98	0,060	1	1	1
	250	-0.90	0,712	1	0,999	0,997	0,989
		-0.95	0,469	1	1	1	1
		-0,98	0,128	1	1	1	1
	500	-0.90	0,850	0,996	0,993	0,989	0,981	0,968
		-0.95	0,733	1	1	1	1	1
		-0,98	0,377	1	1	1	1	1

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Table 7 :

 7 Power of Dickey-Fuller 3 statistic

	T		P=0	P=1	P=2	P=3	P=4	P=5
	25	-0.90	0,111	0,127	0,094
		-0.95	0,092	0,123	0,084
		-0,98	0,098	0,118	0,086
	50	-0.90	0,191	0,162	0,147
		-0.95	0,118	0,111	0,113
		-0,98	0,095	0,107	0,101
	100	-0.90	0,420	0,354	0,305	0,274
		-0.95	0,176	0,150	0,138	0,133
		-0,98	0,104	0,099	0,095	0,089
	250	-0.90	0,725	0,683	0,635	0,598	0,553
		-0.95	0,516	0,487	0,461	0,445	0,417
		-0,98	0,173	0,163	0,166	0,158	0,144
	500	-0.90	0,861	0,819	0,779	0,723	0,677	0,632
		-0.95	0,747	0,724	0,706	0,676	0,643	0,623
		-0,98	0,418	0,411	0,408	0,395	0,385	0,371

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Table 8 :

 8 Coverage of prediction intervals

		l	period	forecast
	T	1	2	3	4	5	10	15	20	40
	25

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Table 9 :

 9 Ratio of the estimated average half-length of the prediction interval over its true one

		LAG
	T	1	2	3	4	5	10	15	20	40
	25	-0,90 2,37	2,26	2,15	2,05	1,96
		-0,95 2,36	2,30	2,24	2,19	2,14
		-0,98 2,36	2,34	2,32	2,29	2,27
	50	-0,90 2,53	2,41	2,29	2,19	2,09	1,71
		-0,95 2,37	2,31	2,25	2,20	2,15	1,92
		-0,98 2,34	2,32	2,29	2,27	2,25	2,14
	100	-0,90 3,44	3,28	3,12	2,98	2,85	2,33	1,96
		-0,95 2,53	2,47	2,41	2,35	2,30	2,05	1,85
		-0,98 2,40	2,38	2,35	2,33	2,31	2,20	2,10
	250	-0,95 3,67	3,58	3,49	3,41	3,33	2,97	2,68	2,44
		-0,98 2,57	2,55	2,52	2,50	2,47	2,36	2,25	2,15
	500	-0,98 2,89	2,86	2,83	2,80	2,77	2,64	2,52	2,41	2,05

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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