The Literature on unit root related issues is so huge and can non be covered in several volumes of a book. There exist figure of good sum-ups, reappraisals and commentaries on unit root literature. Interested reader is referred to these beginnings for item. Libanio ( 2005 ) , Chinn ( 1991 ) , and Stock ( 1995 ) discourse the deductions of unit root in economic theory, policy deductions and econometric processs. Stock ( 1995 ) and Patterson ( 2003 ) provide reappraisal of history of unit root theoretical accounts in economic sciences and econometrics. Hatnaka ( ) presents an overview of theory of unit root proving. Maddala and Kim ( 1998 ) supply first-class overview of modern-day attacks to unit root proving and cointegration. There are several studies on specialised subjects in unit roots e.g. Ng and Perron ( 1996, 2001 ) provide sum-up of available methods for pick of slowdown length. Perron ( 2005 ) provides detail reappraisal of literature on unit root in concurrence with structural interruptions.
As we have discussed in Introduction, despite immense literature on theory related to autoregressive procedure near or equal to integrity, there is no lucidity on several of import issues and deductions sing unit root. We have illustrated this by utilizing illustration of widely studied GNP series of United States. Every writer reaches a different decision after probe of moral force of the series.
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. Before the application of unit root trials, a research worker has to do figure of specification determinations ( implicit or explicit ) e.g. determination about choice of slowdown length, presence of structural interruption and deterministic portion used in the theoretical account. The pick of slowdown length and presence of structural interruptions have a batch of literature in their recognition and good documented methods exist for doing these determination, see Ng and Perron ( 2001 ) and Perron ( 2005 ) for item. 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 ) . Chapter 4 and 5 of this thesis are dedicated to show systematic process for specification of deterministic tendency in a theoretical account to be used for proving unit root. Therefore sufficient item of literature related to this issue is provided in subdivision 2.
Finally, Chapter 6 of this thesis is devoted is to suggest a new step of association between two clip series. Conventional steps of association between variables i.e. correlativity coefficient and t-statistics fail to supply a valid consequences when one or more series in the informations under probe are integrated. We propose a new step that is robust to existence or otherwise of unit root in the clip series. The relevant literature is reviewed in subdivision 3.
Section 2: Model specification ; the specification of deterministic portion in unit root theoretical accounts
The survey of Dickey and Fuller ( 1979 ) is a Prime Minister in unit root proving. They developed assorted statistics for unit root proving and computed distributions of these statistics via Monte Carlo simulations. The used three different equations for calculating unit root trial statistics. These three equations are:
Without impetus, tendency
With impetus, but no tendency
( 1 )
With impetus and tendency
The parametric quantities of involvement in these equations is value of but the distribution of trial statistics for proving versus any alternate depends on the nuisance parametric quantities and. This fact was realized by Dickey and Fuller ( 1979 ) , hence they present three different sets of quintiles of distributions of statistic corresponding to the theoretical account used for proving unit root. Calculation of critical value assumes that proving equation is congruous with informations bring forthing procedure. However, Dickey and Fuller do non supply and systematic process to take between three trial equations for existent informations sets.
The pick among three equations has really serious impact on the end product of unit root. Campbell and Perron ( 1991 ) study following belongingss of unit root trials with respect to pick of deterministic portion:
When the estimated arrested development includes at least all deterministic elements in the existent informations bring forthing procedure, the distribution of trial statistics is non normal under the nothing of unit root. The distribution itself varies with the set of parametric quantities included in the estimating equations.
If the estimated arrested development includes deterministic regressors that are non in the existent informations bring forthing procedure, power of unit root trial against a stationary alternate lessenings as extra deterministic regressors are added.
If the estimated arrested development omits an of import deterministic trending variable nowadays in the true informations bring forthing procedure, such as additive deterministic tendency, the power of t-test goes to zero as the sample size additions. If the estimated arrested development omits a non-trending variable, ( mean or a alteration in the mean ) , t-statistics is consistent but finite sample power is adversely affected and decreases as the magnitude of coefficient of omitted constituent additions ( Campbell and Perron, 1992 ) .
However despite recognizing importance of deterministic tendency, in the early old ages of development of unit root trial processs, we can non happen any systematic process for specification of deterministic portion in a theoretical account used for proving unit root.
Nelson and Kang ( 1984 ) found that if conventional t-statistics for proving coefficient additive tendency in Dickey Fuller arrested development is to a great extent biased toward non-rejection. They conducted a Monte Carlo experiment in which simple random walks generated by theoretical account. Than they estimated following arrested development equation from the series therefore generated: . Harmonizing to standard statistical theory, there is no predictable relationship between clip way of series and additive deterministic tendency constituent ; therefore its coefficient should be undistinguished. But NK84 found that in 87 % of the arrested development appeared to be important. This determination creates incredulity about usage of classical hypothesis proving process for deterministic constituent in the unit root theoretical accounts.
First systematic process for specification for the specification of deterministic portion is presented by Perron ( 1988 ) frequently known as consecutive proving scheme. This scheme starts from most general theoretical account including impetus and additive tendency and figure of deterministic regressors is reduced by consecutive testing.
Minor alterations to the consecutive testing scheme were proposed by Dolado, Jenkinson, and Sosvilla-Rivero ( 1990 ) , Holden and Perman ( 1994 ) , Enders ( 1995 ) , and Ayat and Burridge ( 2000 ) . All these schemes start by the most general theoretical account ( Elder and Kennedy, 2001 ) and so cut down the theoretical account in several stairss. The latest version of consecutive proving scheme is described in item by Ender ( 2004 ) and is described in item in the appendix.
Elder and Kennedy ( 2001 ) oppose usage of consecutive proving schemes to be used in pattern. His statements against usage of these schemes are as follows:
They do non work anterior cognition of the growing position of the variable under trial, coercing their schemes to cover all possibilities. For illustration, unemployment clearly does non hold a long-term growing tendency, and so for this variable, unit-root testing should get down by puting the tendency coefficient equal to zero but these schemes ne’er do so.
They worry about results that are non realistic, for illustration, coincident being of a unit root and a tendency. This is thought to be unrealistic as noted for illustration by Perron ( 1988 ) and by Holden and Perman ( 1994, 63 ) .
They double- and triple-test for unit root as they start by most general theoretical account and so proving is done at each measure of decrease.
Elder and Kennedy ( 2001 ) proposes an surrogate scheme, which differ from consecutive proving scheme in followers:
They recommend to get down mold by the graphical analysis of the informations
They discard to see some theoretical accounts which they think to economically implausible. In peculiar they discard the possibility of a theoretical account with coincident unit root and tendency.
The consecutive testing scheme and the EK2001 scheme have many things common. Particularly, the two schemes utilize Dickey Fuller F-test in theoretical account decrease. Therefore these schemes are non applicable to stipulate theoretical account for trials other than those based on Dickey Fuller arrested development equation.
Section 3: The Association between two Variables
Possibly the first statistical step of association between two variables is the correlativity coefficient. In instance of two variables, correlativity coefficient is the covariance of two variables weighted by their standard divergences. In instance of multiple variables, it is ration of explained discrepancy to the entire discrepancy.
Sir Francis Galton, a biometrician and cousin of Charles Darwin, is the individual responsible for the early development of coefficient of correlativity. Galton was transporting out an extended research on the heredity and wanted to demo that intelligence of one coevals was correlated with the intelligence of the old coevals. Galton ( 1888 ) introduces the term ‘correlation as:
Two variable variety meats are said to be co-related when the fluctuation of the 1 is accompanied on the norm by more or less fluctuation of the other, and in the same way.
Galton was interested in the correlativity coefficient because it was utile to foretell divergences in one variable given the fluctuation in other variable. Weldon, W. F. R. ( 1892 ) applied Galton ‘s correlativities coefficient to the measuring of physical features of runt.
Karl Pearson provided the mathematical model for calculation of correlativity coefficient with which we are familiar today. Pearson of a co-worker of Weldon in University College London and they both were influenced by the scholarly part of Galton. They started printing Biometrika in 1901. Pearson published a series of articles entitled ‘Contribution to Mathematical Theory of Evolution ‘ , and in his paper published in 1896, he modified the correlativity the correlativity coefficient introduced by Galton and adopted a new model that is known as merchandise minute correlativity.
Pearson is besides responsible for the debut of construct of ‘spurious correlativity ‘ and partial correlativity. Number of writers including Pearson, W.S. Gosset and R.A. Fisher attempted to pull the distribution of R-square as an estimation of true population correlativity, but Fisher succeeded in making so and he presented his consequences in 1915 and 1921.
Pearson is besides responsible for the development of construct of specious correlativity. Aldrich ( 2001 ) studies first brush of Pearson with the specious correlativity during a survey of personal judgement in 1897. Elderton ( 1907 ) published a book entitled ‘Frequency-Curve and Correlation ‘ , which is assumed to be representative of Pearsonian School of Thought. In his book, he writes:
It is possible to obtain important value of correlativity coefficient when in world two maps are perfectly uncorrelated ( Cited in Aldrich ) .
Aldrich reports that readings of correlativity as governed by the common cause go a common instance in Psychology. However a different reading became a criterion elsewhere: one variable causes ( at least in portion ) the other variable.
Correlation, Spurious and Genuine
Specious correlativities occurs if two variables appear to be correlated ( or extremely correlated ) but in fact there is no ( or really hebdomad ) relationship between the two variables.
Assorted types of happenings of specious correlativity has been discovered first three decennaries of 20th century e.g. ( I ) specious correlativity due to utilize of ratios ( Yule,1987 ) , ( two ) specious correlativity due to blending of races ( Pearson, Lee and Bramley-Moore, 1989 ) , Yule ‘s illusory association that does non bespeak causal relation but common dependance on 3rd variable ( Yule,1910, 1922, Simon, 1954 ) , and Yule ‘s none-sense arrested development of clip series.
However despite all these, as reported by John Aldrich ( 1995 ) , ‘a different reading has become standard elsewhere, one variable causes ( at least in portion ) the other variable ‘ .
R-square and Unit Root
Since the survey of Yule ( 1926 ) it was known to the econometrists that if two clip series are regressed on to each other, they might bring forth the excess normally high correlation.. However there was no lucidity on the grounds behind happening of this phenomenon. Christmas ( 1926 ) and Simon ( 1954 ) thought that it is some losing variable which is responsible for such inordinately high correlativity. Granger & A ; Newbold ( 1974 ) observed that if two series incorporating unit root are regressed onto each other, they are most likely to bring forth high R-square and important t-statistics. Nelson and Plosser ( 1982 ) observed that unit root theoretical account is more appropriate for most of economic clip series. These two surveies present an alternate account of the phenomena observed by Yule a half century ago i.e. it ‘s the unit root responsible for being of specious correlativity. Phillips ( 1987 ) and Phillips ( 1988 ) provided analytical cogent evidence of the happening of specious correlativity and other diagnostic for the incorporate series.
It turned into belief of most of econometrician that the being of unit roots is the ground for being for specious correlativity. Although it has been proven that two independent stationary series when regressed onto each other, may besides bring forth specious consequences ( Granger et al. 1998 ) , most of econometric text editions present being of unit root an parallel of the specious arrested development [ see Gujrati, Greene, Ender for illustration ] .
The cogency of alternate account creates several inquiries Markss in the application econometric methodological analysis to the economic clip series. See the two clip series:
And see the undermentioned arrested development equation:
The arrested development may bring forth really high R-square and R-square does non meet to 0, the true step of association between two series even in boundlessly big samples.
The distribution of t-statistics for proving versus is non-standard and does non meet to Student ‘s t-distribution even in big samples.
A additive combination of two incorporate series is besides integrated so that is besides integrated. Therefore the discrepancy of arrested development remainders is boundless and remainders may hold infinite discrepancy. This means the series may float off from each other for boundlessly big distance and there may non be any long tally relationship between the series.
The development of literature on unit root is largely stimulated by the fact that the regardless of the sample size, a arrested development of unit root series will bring forth specious correlativity. This determination created uncertainty about the classical illation in economic clip series. Engle and Granger observed that it is possible for two incorporate series to hold a stationary additive combination so that these two series ne’er far off from each other. This means that there may be long tally relationship between two incorporate series. This determination originated another watercourse of literature on the cointegration. However unit root testing is a pre-requisite for the cointegration testing, because the cointegration is possible between the series which are integrated of same order. Anyhow, the literature on unit root remain spread outing because the of the inquiry grade on the cogency of classical illation including R-square for the integrate series.
R-square in Time Series and in Cross Sections
The correlativity coefficient was introduced to analyse the cross-sectional information. The nature of clip series is different from the cross sectional informations. Most of import characteristic of the clip series informations is the consecutive dependance of the observations. Due to the consecutive dependance, distribution of assorted calculators differs for clip series than from transverse subdivision. This consecutive dependance is one of the beginnings of specious correlativity among two informations sets.
Statisticians remain disbelieving about the grounds of specious correlativity in the clip series informations. The general construct or misconception about such specious correlativity was that it is due to losing clip factor. However there was no lucidity about how the clip factor influences the correlativity among clip series. Yule ( 1926 ) writes ;
aˆ¦..now it has been said that to construe such correlativities as connoting causing is to disregard the common influence of clip factoraˆ¦.. I can non see clip per se a causal factor ; and the words merely suggest that there is some 3rd measure changing with the clip to which the alterations in both ascertained variables are dueaˆ¦ .
He farther expects that if the observations are collected for a long continuance of clip, this type of correlativity will vanish:
aˆ¦.. in non-technical footings it is merely a good luck, and if we had or could hold experience of the two variables over a much long period of clip, we could non happen any appreciable correlativity between them.
The undermentioned decisions can be drawn from the treatment of Yule ( 1926 ) ;
The evident correlativity is due to some missing clip factor clip itself is non a causal factor but it is an indicant that there is some relevant variable that is losing from the theoretical account
If the clip series observations are collected for a longer period of clip, such correlativity will vanish
The paper of Simon ( 1954 ) is besides an advocator of similar perceptual experience about the nature of specious correlativity among clip series. Simon ( 1954 ) writes:
aˆ¦To trial whether the correlativity between two variables is specious or echt, extra variables or equation must be introduced and sufficient premises must be made to place the parametric quantities of this wider system. If two variables are causally related in wider system, the correlativity is ‘genuine ‘ .
Granger and Newbold ( 1974 ) , in an influential paper, extended our apprehension of specious arrested developments. They gave a simple theoretical analysis which showed how the arrested development of one random walk or another would give rise to specious consequences. Granger and Newbold ( 1974 ) show some illustrations of unnaturally generated informations where two series are generated by independent random Numberss so that there is no echt relationship between the series. But the arrested development of two series output high R-square, t-statistics and important other diagnostic statistics.
The survey of Granger and Newbold clears some misconceptions about the being of specious correlativity. First, the two series may give high correlativity even if there is no concealed clip factor or causal factor. Second, it is non necessary that the specious correlativity will vanish if we collect informations for a long span of clip. If the series are generated by independent random walk processes, even though there is no concealed clip factor nowadays, the arrested development will give high R-square and this will non vanish even if we collect informations for a really long span of clip.
Phillips ( 1986 ) provided analytical cogent evidence of being of specious correlativity. Phillips ( 1986 ) developed the econometric theory that explains the Granger-Newbold consequences. Phillips showed that if two independent random walks are regressed onto each other, The R-square converges to 1, opposed to the apprehension of Yule ( 1926 ) who was presuming that correlativity among irrelevant clip series should meet to zero.
These surveies created incredulity about the cogency of arrested development consequences for the clip series with unit root. However it was assumed that distribution of calculators for the clip series with stationary roots have asymptotic belongingss fiting with the asymptotic belongingss of matching calculators of IID series. Therefore the accent was to develop alternate asymptotic theory for the clip series incorporating unit root. Most of import development is the construct of cointegration, which suggests even if two series are integrated, there is possibility to acquire utile illation from the arrested development of two series.
In a missive survey Granger, Hyung, and Jeon ( 1998 ) found that the phenomenon of specious arrested development is non restricted to the non-stationary variables. Granger et Al. ( 1998 ) found grounds of specious correlativity between positively autocorrelated stationary series. Chapter 6 of this thesis provides grounds of specious correlativity between autocorrelated series utilizing assorted Monte Carlo experiments.