The development of literature

Past three decennaries witnessed really fast development in unit root literature. Possibly unit root and cointegration has been most debated issue in econometrics. However despite a batch of research, consensus on several of import issues and deductions has non emerged to day of the month ( Libanio, 2005 ) . Conflicting sentiments exist on the being of unit root in economic series being investigated by multiple research workers. The development of literature on unit root and cointegration is stimulated by two jobs:

  1. The consequences of unit root trials are frequently deceptive ; application of unit root trial requires figure of anterior specification determinations and improper pick of the specification determinations consequences in deceptive illation.
  2. Classical techniques for specification of economic theoretical account utilizing clip series informations are frequently deceptive ; In the presence of unit root all conventional illation processs and trials used for theoretical account specification are invalid and hence consequence in misdirecting illation.
  3. Specification of theoretical account before application of unit root trials is a major challenge in application of unit root trials. Performance of unit root trials depends on several specification determinations prior to their application e.g. whether or non to include a deterministic tendency and how to take the figure of the included slowdown in the theoretical account. Practitioners routinely make several arbitrary specification determinations to stipulate the theoretical account used for proving unit root. For the existent information series, arbitrary specification determinations are frequently indefensible and sometimes incompatible with informations [ see Andreou and Spanos ( 2003 ) , and Atiq-ur-Rehman and Zaman ( 2008 ) ] .

    The conventional steps of relationship between variables e.g. t-statistics and R-square depends on IID premise and these are frequently invalid for clip series particularly in the presence of unit root. It has been proven if two series contain unit root, the t-statistics and R-square are biased toward non-rejection of relationship between clip series and this prejudice increases asymptotically [ Granger and Newbold ( 1974 ) , Phillips ( 1987 ) ] . This job creates trouble in distinguishing between specious relation and echt economic relationship.

    This thesis makes three parts to work out the some of the specification jobs in unit root universe. First, we propose a method for specification of deterministic portion in a theoretical account to be used for proving unit root. Second, via extended Monte Carlo experiments, we measure capableness of assorted unit root trials to know apart between tendency and difference stationary theoretical accounts congruent to outdo fitting theoretical accounts of either category for existent clip series. We use GDP informations for a big pool of states for this intent and urge best trial on the footing of these experiments. Third, we propose a new step of association between two clip series which is robust to the type of stationarity and type of deterministic portion in the informations bring forthing procedure of the two series.

    Rest of Chapter is organized as follows: Section 2 of this chapter discusses importance of subject of unit root in economic sciences and econometrics. Deductions of unit root sing the economic theory, policy devising and the econometric patterns are discussed. Section 3 consists of brief history of major developments in unit root literature. Section 4 consists of treatment on inclusiveness of unit root argument and function of theoretical account specification. Section 5 describes part of the thesis in deciding jobs posed by theoretical account specification and the lineation of thesis. Why Unit Roots are of import:

    The presence or otherwise of unit root in a clip series play really of import function in finding statistical and economic belongingss of a clip series. Therefore the literature on unit root developed really quickly during past three decennaries. In the words of Haldrup and Jansson ( 19.. ) , Since the mid-1980s there has been a ‘veritable detonation ‘ of research on unit roots in the analysis of economic and other clip series informations. To the inquiry ‘why do we care about unit roots? ‘ Cribari-Neto ( 1996 ) provides the following interesting response: To a policymaker the reply could be: ‘Because the policy deductions are different. ‘ To a macroeconomic expert, it could be answered that ‘there are theoretical deductions on several theories and theoretical accounts. ‘ Finally, an econometrist would be satisfied with the reply: ‘Because the asymptotics are different. ‘

    Cribari-Neto ( 1996 ) really attractively summarized broad scope of deductions of unit roots. His position about importance of unit roots can be divided into three thoughts: ( I ) Policy deductions ( two ) Macroeconomic theories ( three ) Econometric deductions. Presence or otherwise of unit root well impact all three countries discussed. For illustration, see the econometric deduction of unit roots, the distribution of all conventional trial statistics, calculators and asymptotic belongingss are merely invalid when there is unit root in the clip series.

    Although our survey is non intended to study the doctrine of unit root testing, it is of import to advert some of the deductions of unit root to do it clear that the thought of unit root has deeper deductions than from a strictly statistical descriptive. In this subdivision we review some of the deductions of unit root sing econometric processs and the economic theory in non-technical and intuitive manner. Avoiding intricate algebraic item, we would concentrate on the chief constructs related to the deductions of unit root in empirical work. We would lucubrate the constructs utilizing AR ( 1 ) theoretical account, nevertheless similar logic applies to more complicated theoretical accounts.

    Deduction to economic theory:

    Being of unit root in a clip series has serious deductions sing economic theory. Economic theory frequently predicts that a variable should be stationary or otherwise. Here we mention few deduction of unit root in macroeconomic theories:

    Let us get down by a simple illustration ; suppose we are interested in researching whether or non there is long tally relationship between two clip series and. Suppose the relationship exist and it is given by: . If is unit root, than it can divert from its outlook for a long clip unboundedly, so that is unboundedly off from its long tally way for a long period of clip. This contradicts with the being of long tally relationship. Therefore the long tally relationship exists merely if there is a additive combination of two series which is stationary, so that the reply to strictly theoretical inquiry of long tally relationship depends on the being or otherwise of unit root. This line of statement led to the development of construct of cointegration. Loosely talking, cointegration occurs when a additive combination of some unit root series is stationary. Being of Buying Power Parity ( PPP ) is an illustration of this type of inquiry. If PPP exists the existent exchange rate should be stationary so that domestic currency ever has same buying power in the foreign markets.

    Here are some other economic theories whose cogency depends on being of unit root.

    Efficient markets theory of assets: efficient markets theory of plus pricing by Fama ( 1970 ) suggests that if future extra returns were predictable, they would supply opportunity of developments so that the monetary value ( or log monetary value ) would follow a random walk.

    Real Business Cycle theory: At first, grounds of unit roots in clip series ( Nelson and Plosser, 19882 ) was used to supply support for theories of fluctuations based on existent factors. Nelson and Plosser ( 1982 ) argue that most of the fluctuations in end product should be attributable to alterations in the tendency constituent, in a tendency versus cyclical decomposition. The being of unit roots implies that motions in end product are relentless. Since the cyclical constituent is assumed to be stationary, it follows that end product fluctuations are largely associated with the secular constituent. The statement is supported by the thought that pecuniary dazes are needfully impermanent and so can merely impact the cyclical constituent, and that the long tally way of the economic system is chiefly guided by existent factors such as gustatory sensations and engineering.

    New Keynesian Aggregate Fluctuation theoretical accounts: The first reactions to the decisions of Nelson and Plosser can be seen as an effort to advance new Keynesian theoretical accounts of aggregative fluctuations, in which GNP is expected to return to a long tally tendency, but in which the accommodation procedure can be really slow due to imperfectnesss in goods and labour markets. A figure of documents were published during the 1980s with different statements in this way. Campbell and Mankiw ( 1987 ) , McCallum ( 1986 ) and others present assorted statements in favour of/against the New Keynesian Model of aggregative fluctuations.

    Models for Income and Consumptions: If labour income has a unit root, so a simple version of the inter-temporal lasting income hypothesis implies that ingestion will besides hold a unit root and moreover that income subtraction ingestion ( nest eggs ) will non hold a unit root, so that ingestion and income are cointegrated ( Stock, 1995 ) .

    Many other illustrations of theoretical deductions of unit root can be found in Stock ( 1995 ) , Chinn ( 1992 ) and Libanio ( 2005 ) .

    Econometric Deductions:

    Point/Interval Appraisal:

    See the unit root procedure. The least square calculator for is given by Mann and Wald ( 1943 ) proved that under the premise of iid mistakes, converges in distribution to for. This suggests that expected value of the calculator is equal to the true and hence the calculator is indifferent. However when, the least square calculator is biased toward negativeness even asymptotically ( Phillips, 1987 ) . Hence if point estimations are of direct involvement, so the prejudice in the usual OLS calculator can be debatable. For illustration, if one ‘s object is calculating, so the usage of a colored calculator will ensue in median-biased conditional prognosiss. This has led to the development of median-unbiased calculators ( Stock, 1995 ) . Furthermore, the confining distributions of calculators are different for different specifications of deterministic tendency which makes appraisal more complicated. Normally the confining distributions are maps of Wiener procedure.

    Hypothesis Testing:

    Dickey and Fuller ( 1979 ) observed that the usual t-statistics for proving does non hold standard Student ‘s t distribution. They tabulated the quintiles of t-statistics for the unit root processes. Phillips ( 1987 ) proved that the distribution of t-statistics converges to some map of Wiener procedure. This means we can non utilize conventional statistical tools for hypothesis proving when there is unit root in the series and that hypothesis can non be tested by conventional hypothesis proving techniques.

    See the autoregressive theoretical account, the t-statistics for proving is. Under the void hypothesis if, and than Mann & A ; Wald theorem implies that the trial statistics has asymptotic normal distribution. But if, it follows that, but this does non makes any sense. Therefore the Mann & A ; Wald theorem is non valid for the unit root processes. Phillips ( 1987 ) proves that the t-statistics converges in distribution to a map of Wiener procedure. Further analysis output following interesting consequences:

  • Is super-consistent ; that is, it converges to more quickly than conventional calculators.
  • Is non asymptotically usually distributed and is non asymptotically standard normal. Its confining distribution is called the Dickey-Fuller ( DF ) distribution and does non hold a closed signifier representation. Therefore, quantiles of the distribution must be computed by simulation or by numerical estimate.

The above treatment implies that conventional hypothesis trials are non valid for the unit root procedure and different hypothesis proving techniques are needed when there is unit root in the clip series.

Specious Arrested development:

The most of import deduction of unit root is presence or otherwise of specious arrested development. A specious arrested development occurs when a brace of independent series, but with strong temporal belongingss, is found seemingly to be related harmonizing to standard illation in an OLS arrested development ( Granger, Hyung and Jeon, 1998 ) . Christmas ( 1926 ) showed that two irrelevant economic clip series might hold strong correlativity, but he was non certain of the grounds responsible for these specious consequences. He argues that specious consequences appear when we drop some theoretically relevant series from the theoretical account, in peculiar the additive tendency.

See two independent random walks:

See the arrested development, since the two series are independent, the true value of is zero and the estimated coefficient should be undistinguished in the arrested development end product. Similarly, growing in one series is unrelated to growing in the other series so that R-square should be closer to zero. However Granger & A ; Newbold ( 1974 ) observed that in the above apparatus frequence of rejection of ( true ) hypothesis is much greater for nominal 5 % significance degree and R-square is remarkably high.

The consequences of Granger & A ; Newbold ( 1974 ) were explained by Phillips ( 1987 ) who derived restricting distribution of arrested development coefficient and proved that this coefficient does non hold normal distribution as it is the instance for stationary variables.

In fact, there is ever some possibility of specious arrested development when there is some non-stationary variable involved in the arrested development.

Specious Regression unless the two series are cointegrated:

Therefore to set up cogency of arrested development end product, it is really of import to cognize whether or non the series is stationary.

Policy Deductions:

We can split the policy deductions of unit root into instances: indirect deductions and direct deductions. The two types of deductions are discussed below:

Direct Policy Deductions:

The direct deductions are due to characteristic of a individual economic clip series. Let us advert an intuitive illustration foremost. Suppose the GNP of a state contains unit root. This means the consequence of a negative cut in income will ne’er decease, therefore the authorities will be more loath to bring on a revenue enhancement cut to ease some industry. Hamilton and Flavin ( 1986 ) argued that the economic impression of sustainability translates to the statistical impression of stationarity of the budget shortage. That is, a stationary budgetary place is consistent with the thought that a authorities should run a sequence of discounted future non-interest budget excesss capable of countervailing the current outstanding debt/deficit.

Indirect Policy Deductions:

Indirect deductions are via the econometric theoretical accounts used in the policy devising. Ultimately assessment of alternate economic policies is the most ambitious usage of econometric theoretical accounts ( Hendry ) . Although `classical ‘ econometric theory by and large assumed stationary informations, peculiarly changeless agencies and discrepancies across clip periods, empirical grounds is strongly against the cogency of that premise ( Hendry… . ) . The stationary and non-stationary informations need different put of tools for developing a valid econometric theoretical account. As we have discussed in subdivision ( two ) all classical econometric processs are merely invalid in unit root government. Hence the economic policymaking eventually relies on the being of unit root in the information. Therefore the scope of indirect deductions of unit root in policy devising is every bit broad as the deductions of clip series itself. For inside informations reader is referred to Hendry and Juselius ( 1999 ) , Ericson Hendry and Mizon ( 1998 ) and Chinn ( 1991 ) .

Chinn ( 1991 ) examines several instances where tendency and difference stationary theoretical accounts lead to quite different policy deductions. In general it is assumed that depreciation in existent exchange rate will heighten the exports and will take down the trade shortage, but in 1985-87 US $ depreciated by about 40 % accompanied by a record trade shortage of $ 112 billion. Chinn argues that this happened because of belief of economic experts in J curve, which says that if exchange rate is decreased, the trade balance will deteriorate for short tally and so it will lift. But all econometric theoretical accounts back uping J curve are built on premise of stationarity, whereas the clip series used has been proven to be non-stationary. So he attributes failure of exchange rate policy to the theoretical account built on premise of stationarity.

Brief History of Development in Unit Roots Models and Trials:

This subdivision gives brief reappraisal of the history of application of unit root theoretical accounts in econometrics and the development of statistical theory/tests for unit root. The statistical theoretical accounts with autoregressive coefficient equal to or near unit root are familiar to econometrists since 1940 ‘s but the research in this country got impulse after survey of Nelson and Plosser ( 1982 ) . The country of statistical theory of autoregressive procedure with root near integrity has been active country of research during past three decennaries and a inundation of articles emerged so far. Several good reappraisals and commentaries of literature in this country are already available. For illustration Phillips ( 1988, 1992 ) , Campbell and Perron ( 1992 ) , Banerjee et Al. ( 1992 ) , Maddala and Kim ( 1998 ) , Stock ( 1995 ) and Perron ( 2006 ) supply first-class commentary of development in this country. Therefore a comprehensive study of the development in unit root would be unneeded and unwanted add-on to the already proliferated literature. We would merely travel through major development in unit root and the surveies that are more closely related to ours.

Models with high continuity are familiar to econometrists since 1940 ‘s. Orcutt ( 1948 ) found high consecutive correlativity in the econometric theoretical account of US economic system developed by Tinbergen ( 1939 ) . Orcutt examined figure of clip series and concluded that they are better described by the theoretical account, which is a unit root theoretical account. Mann and Wald ( 1943 ) developed theory of least square calculator of stationary autoregressive theoretical account which was extended to unit root and explosive theoretical accounts by White ( 1958 ) , Anderson ( 1959 ) and Rao ( 1961 ) . Harmonizing to Stock ( 1995 ) , it was customary in 1960 ‘s and 1970 ‘s to pattern economic relationship in differences which is appropriate method for unit root procedure.

Formal trial for unit root were developed by Fuller ( 1976 ) , Dickey ( 1976 ) and Dickey & A ; Fuller ( 1979 ) . Theses trials test the void hypothesis versus one sided option in one of following autoregression:

It was proved by Mann & A ; Wald ( 1943 ) that if and, than. Rubin ( 1950 ) proved that is consistent calculator for all values of. However the distribution of was unknown when. Dickey & A ; Fuller ( 1979 ) derived restricting distribution of when and. This distribution is nonstandard and converges to a map of Wiener procedure ( Phillips, 1987 ) . Dickey & A ; Fuller ( 1979 ) provided assorted statistics for proving presence of autoregressive unit root. These statistics are:

The asymptotic distribution of is nonstandard. The numerator is right skewed, but the ratio is left skewed.

Meanwhile Meese and Singleton ( 1982 ) and Nelson & A ; Plosser ( 1982 ) applied Dickey Fuller trial to assorted economic clip series and found that they were unable to reject presence of autoregressive unit root in most of the series. These findings enhanced the professional involvement in unit root trials since many econometric processs and economic theories hinge on the being of unit root. Finding of Nelson and Plosser was supported by many writers in following few old ages. This led to faster development in theory of unit root trials.

The restricting distribution of developed by Dickey & A ; Fuller ( 1979 ) depends on premise of independency of mistake construction. In instance of serially correlative mistakes, the distribution tabulated by Dickey & A ; Fuller ( 1979 ) is non valid. To acquire consistent calculator and trial with serially correlated mistake construction, either arrested development equation should be changed or the trial statistics should be modified.

Alteration to regression equation is due to Dickey & A ; Fuller ( 1981 ) and Said & A ; Dickey ( 1984 ) , whereas alteration to unit root trial statistics is due to Phillips ( 1987 ) and Phillips & A ; Perron ( 1988 ) . Sargan & A ; Bhargava ( 1983 ) generalise Durbin Watson ( DW ) trial and Brenblutt-Webb ( WB ) Test to utilize for unit root proving. Hall ( 1989, 1991 ) propose Instrumental variable ( IV ) trial for unit root in ARMA ( P, Q ) theoretical accounts. Further alteration to this trial is due to Li ( 1995 ) , Lee and Schmidt ( 1994 ) and Choi ( 1992 ) .

Leybourne ( 1995 ) propose trial based on forward and contrary Dickey Fuller arrested development. Forward Dickey Fuller trial statistics is the usual trial statistics proposed by Dickey & A ; Fuller ( 1979 ) . The contrary Dickey Fuller arrested development is obtained in following manner: Lashkar-e-Taiba, than the contrary Dickey Fuller arrested development is and is the DF statistics obtained from this autoregression. The trial statistics is. Leybourne ( 1995 ) investigates power belongingss of this trial and observes that this trial has power superior to Dickey Fuller trial.

Elliott, Rothenberg & A ; Stock ( 1996 ) , use King ( 1986 ) attack to develop best point optimum trial. They find out a trial whose power map is tangent to the power envelop and ne’er far below it. They so find a trial which has power map closest to this trial. This trial is based on GLS detrending. Detrending is a process of deducting deterministic portion from a clip series. There are several detrending techniques which differ in infinitesimal computational item and exhibit a assortment of features.

Another attack to distinguish between tendency and difference stationarity is to utilize stationarity as a nothing instead than the unit root. This type of trials are due to Tanaka ( 1991 ) , Park ( 1990 ) , Kwiatkowski et Al. ( 1992 ) and Leybourne and McCabe ( 1994 ) etc. Most popular trial is KPSS due to Kwaitkowsky, Phillips, Schmidt and Shin ( 1992 ) . See the representation of a clip series in footings of amount of unit root and stationary procedure:

Where is stationary and is random walk. If is tendency stationary than discrepancy of random walk constituent would be zero, hence trial for stationarity is tantamount to proving versus. Kwiatkowski et Al. ( 1992 ) usage LM statistics developed by Nabeya and Tanaka ( 1992 ) for proving stationarity. The trial statistics is:

Where denote the least square remainders from arrested development of on changeless and tendency, is discrepancy of remainders and.

KPSS trial can be considered as parallel of ADF trial for proving nothing of unit root. Leybourne and McCabe ( 1994 ) suggest a nonparametric trial for nothing of stationarity which may be considered as parallel of Phillips Perron trial.

Perron ( 1989 ) opened a new avenue in the theory of unit root proving. Perron ( 1989 ) proves that unit root trials are biased against stationarity if there is structural interruption in the deterministic portion of the series. Perron suggests that the strong grounds observed by Nelson & A ; Plosser ( 1982 ) and replacements was due to failure to account for structural interruptions in the series. Perron modified ADF trial to integrate structural interruptions and utilizing these trials, reversed the decision of Nelson & A ; Plosser ( 1982 ) for many clip series. Enormous literature emerged after the survey of Perron ( 1989 ) analysing impact of structural interruptions, methods to happen and prove the breakpoints and to plan powerful trials in presence of structural interruptions. Zivot and Andrew ( 1992 ) , Banejee et Al. ( 1992 ) proposed trials for unit root with endogenized structural alteration. Lumsdaine and Papell ( 1997 ) extend the Zivot-Andrew trial for two structural interruptions. Further development to unit root trials with known structural interruptions day of the month is due to Kunitomo and Sato ( 1995 ) , Amsler and Lee ( 1995 ) , Saikkonen and L & A ; uuml ; tkepohl ( 2001 ) , & A ; Lanne et Al. ( 2002 ) etc. Trial with known interruption day of the months are due to Perron and Vogelsang ( 1992a ) , Perron ( 1997a ) etc. Ohara ( 1995 ) , Kapetanios ( 2005 ) see trials with multiple structural interruptions. For a comprehensive study of literature see Perron ( 2006 ) .

Inconclusiveness of unit root argument and theoretical account specification:


Possibly the issue discussed most in the history of econometric literature is the argument on tendency versus difference stationarity, initiated by Nelson and Plosser ( 1982 ) . We have described in item the importance of information about being of unit root with respect to econometric process and economic theory. The empirical relevancy of unit root led to a immense sum of research in the past three decennaries, but consensus on several of import issues and deductions has non emerged to day of the month ( Libanio, 2005 ) . Even though huge Numberss of unit root trials have been proposed and studied, conflicting sentiments exist on the simplest of jobs. For illustration, here is a list of the decisions of writers who have studied the USA one-year GNP series:

  • Difference stationary ; Nelson and Plosser ( 1982 ) .
  • Trend Stationary ; Perron ( 1989 ) .
  • Trend Stationary ; Zivot and Andrews ( 1992 ) .
  • Do n’t cognize ; Rudebusch ( 1993 ) .
  • Trend stationary ; Diebold and Senhadji ( 1996 ) .
  • Difference stationary ; Murray and Nelson ( 2002 ) , Kilian and Ohanian ( 2002 ) .
  • Trend stationary ; Papell and Prodan ( 2003 ) .

Similarly, the alleged purchasing-power para is another contention in econometrics that led to purchasing-power para ( PPP ) mystifier. Purchasing-power para mystifier takes one of two signifiers. In its first signifier, early trials of the PPP-hypothesis failed to reject unit roots in existent exchange rates, therefore rejecting the hypothesis of PPP keeping in the long term. In the more recent literature, the literature on the PPP mystifier focused on stochastic existent exchange rate theoretical accounts that allow long term PPP to keep. One can happen and name a batch of controversial consequences on this issue with the decision that no unequivocal reply can be found so far. The undermentioned quotation mark by El-Gamal and Ryu ( 2003 ) reflects the ambiguity in the effect of PPP argument:

In peculiar, we show that it is possible to fit desired “ half-lives ” for any of the most popular non-linear theoretical accounts late proposed in the literature, at the disbursal of fiting their more general dynamic construction. We conclude that depending on the theoretical accounts and standards selected for look intoing the PPP-puzzle, the mystifier may be in the oculus of the perceiver.

Similarly, take any series that has been explored figure of times for stationarity, you will happen figure of conflicting decisions.

A major ground responsible for ambiguity in the illation of unit root trials is the theoretical account specification prior to application of unit root trials. Performance of unit root trials depends on several specification determinations prior to application of unit root trial e.g. whether or non to include a deterministic tendency and how to take the order of the included slowdown in the theoretical account. Practitioners routinely make several arbitrary specification determinations either implicitly or explicitly. Since Monte Carlo surveies take these initial determinations as valid background information, such surveies frequently overestimate the public presentation of trials on existent informations. In Monte Carlo, when the experiments status on some inexplicit specification, the design of informations bring forthing procedure supports the inexplicit premises. But for the existent information series, inexplicit assumptions/arbitrary specification determinations are frequently indefensible and sometimes incompatible with informations.

For some of these determinations some of determination there exits good documented and analyzed processs and techniques e.g. choice of slowdown length and presence of structural interruption. The pick of slowdown length and presence of structural interruptions have a batch of literature in their recognition ; see Ng and Perron ( 2001 ) and Perron ( 2005 ) for elaborate studies. However existing literature does non supply satisfactory solution for pick of deterministic portion in a theoretical account used for proving unit root ( Elder and Kennedy, 2001 ) . A portion of this thesis is devoted to discourse the process for specification of deterministic portion in the theoretical account used for proving unit root.

Second, the public presentations of conventional steps for relationship between two variables remain disbelieving when they were applied to clip series. Since Christmas ( 1926 ) , it is known that t-statistics and R-square overestimate strength relationship between two clip series. It was discovered much later by Granger and Newbold ( 1974 ) that if two economic clip series contain unit root, t-stat and R-square are biased toward non-rejection of relationship between the two series. R-square is considered to be a good descriptive to mensurate strength of relationship between variables when they do non incorporate outlier. However for incorporate clip series informations there is no cogency of R-square. Therefore a portion of this thesis is devoted to discourse an surrogate descriptive for association between clip series which is robust to type of stationarity and type of deterministic tendency in the theoretical account.

Outline of the thesis:

Rest of the thesis is organized as follows:

Chapter 2 consists of brief debut of constructs, definitions and trials repeatedly used in the thesis. This chapter starts by presenting basic nomenclature in clip series that is often used in the survey. Next is treatment on categorization of clip series theoretical accounts, categorization of autoregressive theoretical accounts and their belongingss. Finally we discuss the unit root trials that are utilized in the thesis.

Chapter 3 consists of brief reappraisal of literature most relevant to our survey.

Chapter 4 is devoted to suggest new process for the specification of deterministic in autoregressive theoretical account. First, groundss are presented that the determination of deterministic portion is really of import to find end product of unit root trial. Next it is discussed that bing methods and techniques are incapable of stipulating deterministic portion in autoregressive theoretical account even with stationarity. Than we present new scheme for pick of deterministic portion. The public presentation of this scheme is measured via extended Monte Carlo experiment.

In Chapter 5 we aim to suggest best trial when we are skeptic about the true signifier of informations bring forthing procedure. In peculiar we compare assorted unit root trials for their ability to know apart between best suiting tendency stationary and best adjustment difference stationary theoretical accounts. After explicating the job statement, the methodological analysis and Monte Carlo design is described. Consequences are explained at the terminal of Chapter.

Chapter 7 proposes a new step of association between two clip series. Conventional step of association is the correlativity coefficient invented by Francis Galton, developed and popularized by Pearson and Yule. The correlativity coefficient has been proven to neglect in mensurating grade of association between two incorporate series. Granger et Al. ( 1998 ) find that correlativity coefficient may be misdirecting for stationary informations every bit good. We propose new steps of association every bit valid for stationary and non-stationary series. This new steps is robust to specification the deterministic portion, distribution of error term and order and strength of autoregression.

Chapter 7 nowadayss existent application of the three proposed processs. The extent to which proposed techniques are helpful in work outing specification issue is discussed. Conclusion, recommendations and waies for new research are presented in the last.