考量起始值下的共整合檢定

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(1)國立中山大學經濟學研究所碩士論文. 考量起始值下的共整合檢定 Tests for Cointegration and the Initial Conditions. 研究生：洪大為撰指導教授：李慶男博士. 中華民國九十六年六月.

(2) 謝辭「真理不會欺騙我們，只是我們的錯誤正受到檢驗」。在追求真理、檢驗自我的道路上，很慶幸，因為很多人的鼓勵、支持及幫助，讓自己能走的不那麼顛簸。最感謝的是我的指導教授李慶男老師，因為老師的關係，讓我認識了時序及數理之美，也讓我不後悔自己因為喜歡計量而決定繼續攻讀研究所的決定；在老師身上，我不僅見識一個理論學者追求真知時的執著與堅持，也瞭解到對於消除疑問過程的永不妥協是鞭策自己不斷前進的最大動力。雖然，未來不會步上學術的道路，但是這兩年跟在老師身旁一起討論問題、一起消除心中的疑問所學習到的態度，將會是我人生道路上最大的資產。謝謝兩年來陪我一起努力奮鬥的同窗好友們：立佳學長、家齊學長、宗逸、政翔、又誠、冬威、靖惠、小白、駿傑、書銘、萬來、書弘、謙柔、瑪莎、遠靜以及可愛又認真的學弟妹們，還有勞苦功高、常常幫忙不懂行政程序又常出紕漏的我的所辦秘書：秀燕姐、育萍姐及憲馥，很高興在學生生涯的最後階段能有大家作伴。因為你們，讓我的碩士生涯多了許多色彩和驚奇，也是你們無時無刻的加油打氣，讓我能一路努力撐過，謝謝你們。感謝爸媽及瓊瑩的包容，原諒我退伍後仍要繼續唸書這個任性的要求，是你們的支持讓我圓了自己想專研經濟學的小小夢想，是你們的支持，讓我能無後顧之憂的埋首書堆、享受追求學問的樂趣，也是你們的支持，幫助我熬過一個又一個的難關。未來，讓我成為你們的支柱，支持你們吧。最後，謝謝 Annie 老妹，愛叫我阿頭的你總是能讓心情低落的我馬上回復動力。波蘭詩人 Szymborska 說過「寫作是作家對抗時間最好的武器」，我不是作家，但我盡力為自己這段精彩的碩士生涯留下痕跡，為自己見證一段認真活過的歲月. 洪大為謹誌于中山大學經濟學研究所中華民國九十六年仲夏.

(3) 摘要一反傳統推導共整合檢定統計量時假設起始值為零的做法，本文嘗試將起始值參數化，在推導檢定統計量的過程中將起始值條件納入考慮。如此推導下的檢定統計量，不只可幫助我們了解起始值如何影響共整合檢定，也可幫助我們判別兩組資料間是否存在共整合關係。而此種依據判別函式以及誤差檢定所構成的檢定統計量也可讓我們依據手中握有的資訊來模擬出所需要的查表值。關鍵字：起始值條件, 共整合, 判別函式, 誤差檢定.

(4) Contents 1 Introduction. 3. 2 Hypothesis Testing And Initial Condition. 7. 2.1. Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. The Discriminant Function . . . . . . . . . . . . . . . . . . . . . .. 11. 2.3. The Statistic for Cointegration Test . . . . . . . . . . . . . . . . .. 14. 3 Simulation. 19. 4 Conclusion. 25. 1.

(5) CONTENTS. Abstract In stead of assuming the starting observations as zero, the cointegration test statistic derived in this paper takes the initial conditions into consideration. The statistic helps us understand how the initial conditions affect the test and helps us recognize whether the cointegration relationship exists in the data generation process or not. Beside, with this statistic derived with the concept of discriminant function and residual based test, we can simulate our own critical value table in according to what starting observations we have in hand, what significant level we want and what value of ρ we meet under H1 . KEYWORDS : Initial conditions, cointegration, discriminant function, residual based test. 2.

(6) 1 INTRODUCTION. 1. Introduction. There are by now many tests for examining whether or not a time series has unit root. However, few papers which discuss the unit root test explicitly address the role of the beginning of the data series, either do those discussing about tests for cointegration. According to Elliott (1999, 2003), his studies note us that the initial conditions exactly have the effects on the form and power of the unit root test and differing performance of unit root tests for different initial values allows us choose the starting date to obtain the desired outcome. However, it is long known that the power of unit root tests depends on the initial condition in small samples, see Evans and Savin (1981, 1987), for instance. Typical asymptotic results, by contrast, imply that the power of various test statistics remains unaffected by the initial condition for large samples, whether the initial condition is assumed fixed or random but bounded in probability. But the influence of the impact that initial condition brought in has been noticed in recent years. Elliott (1999) pointed that the statistic needn’t be the most powerful test under the assumption that initial observation is drawn from its unconditional distribution under the alternative hypothesis | ρ |< 1, because the assumption would result in an unconditional variance of σ 2 /(1 − ρ2 ) for ρ less than one, where σ 2 is the variance of the innovation process, and the assumption would also make the initial condition involve the unknown parameter of interest ρ. Differing the assumption on initial condition rather than fixing it changes the asymptotically 3.

(7) 1 INTRODUCTION optimal test. Moreover, it is clear that initial condition profoundly affects both power and the form of the unit root tests when Elliott and M¨ uller (2003) shed the light on the impact of the initial condition. In order to capture the effects of the initial condition, they develop tests which are invariant to initial condition, which is treated as the nuisance parameter ξ, by maximizing implicit weighted average power over various values of ξ. Besides, they also explore the relationship of the optimal test based on their literature with other unit root tests suggested and recommended in other literature to make it clear that “choices between statistics in practice” comes down to “what types of initial conditions are likely for the application at hand”. With their studies, researchers are able to gain a much fuller understanding of the relationships between initial condition and those ubiquitous tests. In most situations, a researcher can make the highly plausible assumption that the absolute value of the initial condition is relatively small or even be equivalent to zero. But, in other situations, especially in the test of cointegration for the model in which the beginning dates of the time series data are different, the assumptions that make the beginning observation of the time series data zero are not only unsuitable, but also neglect the impact induced by the starting data. For example, under the assumption that there are just two sets of Data Generating Process(DGP), if we want to test whether or not the cointegration relationship exactly exists between these two time series data while two starting data don’t happen at the same time in this system, the model derived in Elliott and 4.

(8) 1 INTRODUCTION M¨ uller (2003) with just one parameter of initial condition can not fit into this situation. Often the data collected and available for testing has a beginning date dictated by how far back records can be found, or they date back to periods when governments started collecting and disseminating data on a regular basis. When we meet the two sets of DGP which have different lengths in the test model for cointegration, we have to take two initial conditions into consideration because we have to sacrifice some data of the longer DGP happening earlier than the date of starting data in the shorter one while running cointegration test. After truncation, the new starting data of the longer DGP we use in the testing model has accumulated those information of those truncated part. Instead of following the model that Elliott and M¨ uller (2003) built to separately distinguish the existence of unit root in single time series data and then combined together to test whether there is any cointegration relationship or not, we manage to simultaneously consider two initial observations and focus on their impact on cointegration testing. We follow the ideas Elliott and M¨ uller (2003) used, but we rearrange their model. In this paper, we suppose there are only two sets of time series observations and try to test whether the cointegration relationship exists between them. In according to the suggestion in Phillips and Ouliaris (1990), the Zt test derived in Phillips (1987) and Phillips and Perron (1988) should have superior power properties due to the slower rates of divergence under cointegration, so we try to compare the statistic we derived in this paper with the PP test in the cointegration model. we pick up residual based tests as the presence of cointegration 5.

(9) 1 INTRODUCTION for the reasons that the residual based test is not only easily combined with the concept of the initial condition, but also is the procedure which has attracted the attention of empirical researchers. This is partly because of the recommendations of Engle and Granger (1987), partly because the tests are so easy and convenient to apply, and partly because what they set to test is clear intuitively. In next section, we build the cointegration model which considers two initial conditions and try to derive the statistic with the concept of discriminant function. Section three exhibits small sample Monte Carlo evidence and Section four concludes.. 6.

(10) 2 HYPOTHESIS TESTING AND INITIAL CONDITION. 2. Hypothesis Testing And Initial Condition. 2.1. Preliminary. We consider the following cointegration model in this paper: yt xt x0 y0. = = = =. αxt + ut , t = 0, 1, 2, . . . , T xt−1 + ηt , t = 1, 2, . . . , T ζ ξ. where both x0 and y0 are the initial conditions. Because we use the residual based analysis to discuss the cointegration test, so we rearrange our model with the least squared estimator and the model becomes yt uˆt x0 y0. = = = =. α ˆ xt + uˆt , t = 0, 1, 2, . . . , T ρˆ ut−1 + υt , t = 1, 2, . . . , T ζ ξ. (1). where α ˆ is the least squared estimator and we are interested in distinguishing the two hypotheses ½. H0 : yt and xt are without Cointegration H1 : yt and xt are with Cointegration. Before calculating α ˆ , we define the symbols used in this paper y = (y1 , . . . , yT )0 , x = (x1 , . . . , xT )0 , u = (u1 , . . . , uT )0 ˜ = (x0 , x1 , . . . , xT )0 , u y˜ = (y0 , y1 , . . . , yT )0 , X ˜ = (u0 , u1 , . . . , uT )0 the linear regressions of all T + 1 observations is. ˜ +u y˜ = αX ˜ 7. (2).

(11) 2 HYPOTHESIS TESTING AND INITIAL CONDITION and the ordinary least squared estimator of α ˆ is ˜ 0 X) ˜ −1 (X ˜ 0 y˜) α ˆ = (X. At next step, we assume that the T × T matrix V is the variance-covariance matrix of υ = (υ1 , · · · , υT )0 , and then, we manage to bring the information of initial condition ξ and ζ into the residual based test model by rearranging the residual terms. uˆt = ρˆ ut−1 + υt , t = 1, 2, . . . , T. t = 0 , uˆ0 = y0 − α ˆ x0 = ξ − α ˆ ζ , where uˆ0 contains two initial conditions t = 1 , uˆ1 = ρˆ u0 + υ 1 t = 2 , uˆ2 = ρˆ u1 + υ2 = ρ(ρˆ u0 + υ1 ) + υ2 = ρ2 uˆ0 + ρυ1 + υ2 .. . t = T , uˆT = ρˆ uT −1 + υT = ρT uˆ0 + ρT −1 υ1 + ρT −2 υ2 + · · · + ρυT −1 + υT So    u ˆ= . uˆ1 uˆ2 .. .. . .      = uˆ0   . uˆT. ρ1 ρ2 .. .. . .     +  . ρT. 1. 0 ··· 0 . . . ρ1 . . . . .. .. . . . . . . 0 . T 1 ρ ··· ρ 1.     . υ1 υ2 .. ..     . υT. for simplification, let’s define the column R(ρ) = (ρ1 , ρ2 , · · · , ρT )0 and two T × T matrices A(ρ)−1 and A(ρ) as. 8.

(12) 2 HYPOTHESIS TESTING AND INITIAL CONDITION    A(ρ)−1 =  . 1. 0 ··· 0 .. 1 ... ... ρ . .. . . . . . . 0 . T 1 ρ ··· ρ 1.     . . 1.   −ρ1 and A(ρ) =  ..  . 0.  0 ··· 0 . . . . . . ..  .   .. .. . . 0  · · · −ρ1 1. then we can describe our linear model y = α ˆx + u ˆ with replacing u ˆ by u ˆ = uˆ0 R(ρ)+A(ρ)−1 υ and using 0 as a T ×1 vector of zeros. The linear model become y=α ˆ x + uˆ0 R(ρ) + A(ρ)−1 υ υ ∼ iid N ( 0 , V ) where E[ˆ u] = uˆ0 R(ρ) and the variance-covariance matrix Σ(ρ) of u ˆ satisfies A(ρ)Σ(ρ)A(ρ)0 = V . At the third step, we will use tildes to denote vectors and matrices which contain all T + 1 observations. we define ˜ 0 as a (T + 1) × 1 vector of zeros, ˜ ˜ R(ρ) = (ρ0 , R(ρ)0 )0 and u ˆ = (ˆ u0 , u ˆ0 )0 . We can then write the linear model for all ˜ ˜ +u T + 1 observations as y˜ = α ˆX ˆ . With υ ˜ = (υ0 , υ 0 )0 , we rewrite the model ˜ described above for all (T + 1) samples by replacing the u ˆ and get. µ ˜ ˜ + uˆ0 R(ρ) y˜ = α ˆX +. 0 00 0 A(ρ)−1. ¶ υ ˜. υ ˜ ∼ iid N ( ˜ 0 , V˜ ) ˜ ˜ where the variance-covariance matrix of u ˆ be given by Σ(ρ) = diag(0, Σ(ρ)) ˜ because A(ρ)Σ(ρ)A(ρ)0 = V , so the variance-covariance matrix of u ˆ satisfies µ. 0 00 0 A(ρ)−1. ¶. µ V˜. 0 00 0 A(ρ)−1 9. ¶. µ =. 0 00 0 Σ(ρ). ¶.

(13) 2 HYPOTHESIS TESTING AND INITIAL CONDITION Before we derive our statistic for cointegration test, we like to introduce the concept of discriminant function. In Rao (1971, 1973), this concept is very helpful in recognizing which hypotheses that the samples we have in hand support that the population would be. Besides, this concept is also helpful in deriving the statistics when there are singular variance–covariance matrices in our probability density function. So, we introduce the basic theory of the discriminant function in subsection 2.2 and then, we derive some useful general–inverse formulations to help us derive the statistic in subsection 2.3 .. 10.

(14) 2 HYPOTHESIS TESTING AND INITIAL CONDITION. 2.2. The Discriminant Function. Before deriving our statistic for the cointegration test, we use the concept introduced in Rao, C. (1973) and Rao, C. and Mitra (1971) to determine the discriminant function (ratio of likelihoods) for assigning an individual as a member of the particular population to which it may belong. In Rao, C. R.(1973), when we consider the problem of assigning an individual to one of a finite number of groups to which it may belong on the basis of a set of characteristics observed, this kind of problem is one of deciding among a number of alternative hypothses and not of testing any particular hypothesis against a set of alternatives and is often referred to as one of “classification” or “discrimination”. In this kind of problem, it is useful to have an explicit expression for density as in the construction of a discriminant function. Before we discuss the problem of identification, classification or discrimination, first we have to get the discriminant score of the ith population. For a satisfactory solution, we need to know the following:. 1. The probability density, Pi (U ), for a given set of measurements U . 2. Prior probability, πi , for the population, which is a relative frequency of individual of the total populations in the composite population from which an individual to be identified has been observed.. Then, given an individual with measurements U , the discriminant score for the ith population is computed as 11.

(15) 2 HYPOTHESIS TESTING AND INITIAL CONDITION. Si = πi Pi (U ). Let us consider the case where the distribution of U is p–variate normal in each of the population. Choose Pi (U ) as the normal density. (2π)−p/2 | Σi |−1/2 exp[−. 1 (U − µi )0 Σi −1 (U − µi )] 2. that is, with mean µi and dispersion Σi for the ith population. Taking the logarithm of πi Pi (U ) and omitting the factor (2π)−p/2 common to all i, we see that the equivalent discriminant score for the ith population is. 1 1 Si = − log | Σi | − (U − µi )0 Σi −1 (U − µi ) + log πi 2 2 involving the mean µi and dispersion Σi for the ith population. The function Si is quadratic in U and may be called a quadratic discriminant score. An individual is assigned to that population for which the quadratic discriminant score has the highest value. If the populations do not differ in the dispersion matrices, the terms. 1 1 − log | Σi | − U 0 Σi −1 U 2 2 are common to all Si and also can be omitted, then, the equivalent discriminant score for the ith population may be written. 12.

(16) 2 HYPOTHESIS TESTING AND INITIAL CONDITION. 1 Si = (µ0i Σi −1 )U − µ0i Σi −1 µi + log πi 2 which is linear in U and may be called a linear discriminant score It may be seen that when there are only two populations, only one comparison is involved and the decision can be taken by computing the difference S1 − S2 , which is L( U ) − C where L( U ) = (µ01 − µ02 )Σi −1 U C =. 1 (µ01 Σ1 −1 µ1 2. − µ02 Σ2 −1 µ2 ) + log π2 − log π1 .. The function L( U )−C is the linear discriminant function and the last two items of C could be zero because πi is 1 while the perfect discrimination is available under the condition of µ1 6= µ2 . The decision rule may then be expressed in the form assign to the population 1 if L( U ) ≥ c, assign to the population 2 if L( U ) ≤ c.. 13.

(17) 2 HYPOTHESIS TESTING AND INITIAL CONDITION. 2.3. The Statistic for Cointegration Test. In this subsection, we try to develop the statistic for the cointegration test which consider two initial conditions under finite sample. We assume that υt is the Gaussian disturbances and the nonsingular (T + 1) × (T + 1) variance-covariance matrix V˜ of υ ˜ is known. In order to construct our statistic by the methods of discriminant function, we try to get two probability density functions under H0 and H1 . For the purpose of just concentrating on the term which brings the ˜ 0 , where I˜ is ˜ = I˜ − X( ˜ X ˜ 0 X) ˜ −1 X information of the initial conditions, we use M ˜ , we the (T + 1) × (T + 1) identity matrix. Let our linear model multiply by M ˜ from the model and get the probability density functions from can eliminate α ˆX ˜ y˜ because M ˜ ˜ y˜ = α ˜X ˜ +M ˜u M ˆM ˆ ˜ ˜u = M ˆ ˜ ˜ (ˆ = M u0 R(ρ) + diag(0, A(ρ)−1 )˜ υ) ˜ ˜ Then, under H1 , we use Σ(ρ) = diag(0, Σ(ρ)) and under H0 , we use Σ(1) = ˜ ˜ ˜ 0 and Σ ˜ 1 in diag(0, Σ(1)) . Due to simplification, we will replace Σ(1), Σ(ρ) by Σ ˜ sequence and use e˜ instead of R(1) because e˜ is a (T + 1) × 1 ones vector, which ˜ is equivalent to R(1). Under H0 and for any ξ and ζ, we can get ˜ ˜ y˜ ∼ N ( uˆ0 M ˜ R(1), ˜ Σ(1) ˜ M ˜) M M. (3). On the other hand, under H1 and for any ξ and ζ, ˜ ˜ y˜ ∼ N ( uˆ0 M ˜ R(ρ), ˜ Σ(ρ) ˜ M ˜) M M. 14. (4).

(18) 2 HYPOTHESIS TESTING AND INITIAL CONDITION The density function we have derived in (3) and (4), involve the inverse of the variance–covariance (dispersion) matrix, which doesn’t necessitate the assumption that the variance–covariance matrix is nonsingular. Then, we need to use the method of generalized inverse which is useful in defining this kind of density function and in extending some of the results developed for the nonsingular case to the singular distribution. So, before we derived our statistic for cointegration test by the concept of discriminant function, we introduce some tools of generalized inverse to help us solve the problems resulting from the singular matrices while deriving the statistic. ˜ y˜ is proportional to Under H0 , the density of M exp{−. 1 ˜ (M ˜ Σ(1) ˜ M ˜ )− M ˜ (˜ (˜ y − uˆ0 e˜)0 M y − uˆ0 e˜)} 2. and, under H1 , it is proportional to exp{−. 1 0 ˜ ˜ ˜ ˜ Σ(ρ) ˜ M ˜ )− M ˜ (˜ (˜ y − uˆ0 R(ρ)) M (M y − uˆ0 R(ρ))} 2. where ( )− is any generalized inverse and both ξ and ζ are fixed to any value. ˜ i− − Σ ˜ i − X( ˜ X ˜ 0Σ ˜ i − X) ˜ −1 X ˜ 0Σ ˜ i − are the Now, we try to prove that Gi = Σ ˜Σ ˜ iM ˜ , where Σ ˜ i , i = 0, 1 , respectively indicated to generalized inverses of M ˜ 0 and Σ ˜ 1 to replace Σ(1), ˜ ˜ H0 and H1 . It means that we use Σ Σ(ρ) in the following derivation. ˜ i − is the generalized In according to the formulation in Rao, C. R. (1973, p. 33), Σ ˜ i and can be showed as inverse of Σ −. ˜i = Σ. µ. 1 + e0 Σi −1 e −e0 Σi −1 Σi −1 −Σi −1 e 15. ¶.

(19) 2 HYPOTHESIS TESTING AND INITIAL CONDITION where e is a T × 1 vector of ones and Σi , i = 0, 1 is respectively indicated to the T × T variance-covariance matrices of u ˆ under H0 and H1 . ˜Σ ˜ iM ˜ , we first show In order to check that Gi is the generalized inverses of M ˜ 0Σ ˜ i − X) ˜ is necessarily nonsingular. Following the suggestion in Phillips that (X and Ouliaris(1990), if the m × m matrix Ω ( where m = 1 + n ) is  Ω=. 1. n. .  1 a b   n c d. we can know that the det(Ω) = a∗ det(d), where a∗ = a − bd−1 c . By this formulation, we know −. ˜ i ) = det(Σ−1 )det[1 + e0 Σi −1 e − e0 Σi −1 e] 6= 0. det(Σ. ˜ 0Σ ˜ i − X) ˜ is nonsingular and (X ˜ 0Σ ˜ i − X) ˜ −1 in the This outcome supports that (X Gi exists exactly. ˜Σ ˜ iM ˜ and there Next, we start proving that Gi is the general inverse of M are some useful properties which can help us prove it smoothly. ˜ =X ˜ 0 Gi = 0. Because First, Gi X ˜ ˜ −1 X ˜ 0Σ ˜ i − ]X ˜ X ˜ 0Σ ˜ i − X) ˜ i − X( ˜ = [Σ ˜ i− − Σ Gi X ˜ ˜ −Σ ˜ i−X ˜ i−X = Σ = 0 ˜ Gi = Gi M ˜ = Gi . Because Gi X ˜ =X ˜ 0 Gi = 0 and Second, M 16.

(20) 2 HYPOTHESIS TESTING AND INITIAL CONDITION. ˜ Gi = [I˜ − X( ˜ X ˜ 0 X) ˜ −1 X ˜ 0 ]Gi M ˜ X ˜ 0 X) ˜ −1 X ˜ 0 Gi = Gi − X( = Gi Third, we use a property of generalized inverse. The property shows that a generalized inverse G of the matrix A has the property GAG = G. With these three properties, we hence find. ˜ i Gi ˜ Gi = Gi Σ ˜Σ ˜ iM Gi M ˜ i− − Σ ˜ i − X( ˜ X ˜ 0Σ ˜ i − X) ˜ −1 X ˜ 0Σ ˜ i − ]Σ ˜ i [Σ ˜ i− − Σ ˜ i − X( ˜ X ˜ 0Σ ˜ i− = [Σ ˜ i−] ˜ −1 X ˜ 0Σ X) ˜ i−Σ ˜ iΣ ˜ i− − Σ ˜ i−Σ ˜ iΣ ˜ i− − Σ ˜ i − X) ˜ −1 X ˜ 0Σ ˜ i−Σ ˜ iΣ ˜ i− ˜ i − X( ˜ X ˜ 0Σ = Σ ˜ i−Σ ˜ iΣ ˜ i−X ˜ ˜ i− + Σ ˜ i − X) ˜ −1 X ˜ 0Σ ˜ i − X) ˜ −1 X ˜ 0Σ ˜ i − X( ˜ X ˜ 0Σ ˜ X ˜ 0Σ X( 0 − 0 − ˜i ˜ i X) ˜ −1 X ˜Σ ˜Σ (X because of ˜ i−Σ ˜ iΣ ˜ i− = Σ ˜ i− Σ. so. ˜Σ ˜ iM ˜ Gi = Σ ˜ i− − Σ ˜ i − X( ˜ X ˜ 0Σ ˜ i − X) ˜ −1 X ˜ 0Σ ˜ i− Gi M = Gi. ˜Σ ˜ iM ˜ and we can replace It is clear that Gi is the generalized inverse of M ˜Σ ˜ iM ˜ )− by Gi in our statistic. (M. Now, let’s put everything together to construct the equivalent discriminant function and just retain only the terms depending on y˜. We can get. 17.

(21) 2 HYPOTHESIS TESTING AND INITIAL CONDITION LR =. =. =. ˜Σ ˜ 1M ˜ )− M ˜ (M ˜ (y˜−û0 R(ρ))} 0M ˜ ˜ exp{− 12 (y ˜−û0 R(ρ)) ˜ (M ˜Σ ˜ 0M ˜ )− M ˜ (y˜−û0 e˜)} exp{− 12 (y ˜−û0 e˜)0 M. 0 G (y ˜ ˜ exp{− 12 (y ˜−û0 R(ρ)) u0 R(ρ))} 1 ˜−ˆ. exp{− 12 (y ˜−û0 e˜)0 G0 (y˜−û0 e˜)}. ˜ ˜ 0 G1 y ˜ 0 G1 R(ρ)−2ˆ u0 R(ρ) ˜)} ˜0 G1 y˜)} exp{− 12 (û20 R(ρ) exp{− 12 (y 0. 0. 0. exp{− 12 (y ˜ G0 y˜)}exp{− 12 (û20 e˜ G0 e˜−2û0 e˜ G0 y˜)}. ˜ 0 G1 R(ρ) ˜ y }exp{− 12 [ˆ u20 (R(ρ) − e˜0 G0 e˜) = exp{− 21 y˜0 (G1 − G0 )˜ ˜ 0 G1 − e˜0 G0 )˜ −2ˆ u0 (R(ρ) y ]}. Then, we take logarithms to construct the discriminant function and we can get the following Theorem . Theorem: Consider the data generating process (1) when the disturbance vector υ ˜ is assumed to be multivariate Gaussian N (0, V˜ ). ln LR = S(ρ, uˆ0 ) is the test of H0 against H1 . H0 is non–rejected if statistic is smaller than the critical value or H1 is rejected for small value of the statistic. ˜ 0 G1 R(ρ) ˜ ˜ 0 G1 − e˜0 G0 )˜ S(ρ, uˆ0 ) = y˜0 (G1 − G0 )˜ y + uˆ20 (R(ρ) − e˜0 G0 e˜) − 2ˆ u0 (R(ρ) y. The form of the statistic for cointegration test enables us to discuss the testing problem in both the ρ and uˆ0 dimension.. 18.

(22) 3 SIMULATION. 3. Simulation. Simulations were run to assess the adequacy of the new tests derived in this paper and to evaluate their performance in comparison with the procedure suggested by Phillips and Perron (1988). By developing the statistic with regards to testing cointegration under various possible value of the initial conditions, we can understand more deeply the merits of the coinitegration test with respect to their handling of the initial condition. With the statistic derived in section 2.2 , we can simulate tables containing percentiles for the null distributions and the critical value. Because our statistic has its own critical value for rejecting the null hypothesis under various value of the initial conditions, so we can focus on those combinations resulting in serious size distortion. It is long known that Phillips and Perrons’ (1988) Zt test is frequently used in the test of cointegration. When we compute the size of Zt test, there are some assumptions described below for simplification. In the subsection 2.1 , we assume the innovation process in our model are all independent and identity distribution of Gaussian but the PP test allows autocorrelation happen in the variance–covariance matrices of the innovation process. In order to fit the property of the PP test into our model and avoid employing the long autoregression, the data are generated with no moving average errors before simulation. The lag truncations are all fixed in 12 . The innovation process is independent and identical distribution of N (0, 1) and the result of the size dis19.

(23) 3 SIMULATION tortion of PP test for the different values of two initial conditions are exhibited below. Size Distortion of PP Test y0 x0. 0. 0. 0.0795. 5. 5. 10. 50. 100. 0.0661 0.0558. 0.1932. 0.2617. 0.0827. 0.0710 0.0548. 0.1521. 0.1529. 10. 0.0831. 0.0733 0.0612. 0.1165. 0.0940. 50. 0.0815. 0.0837 0.0743. 0.0425. 0.0191. 100. 0.0823 0.0810 0.0829. 0.0378. 0.0134. By the means of size distortion, we not only understand how the impact of initial condition profoundly affect the traditional cointegration test , but also get the conclusions similar with those pointed by Elliott and M¨ uller (2003). When we regressions Yt on Xt in our linear regressions model, it is obvious to see that the starting observations of Yt affects the size more than that of Xt dose, especially when the starting observation of Yt is more larger than that of Xt . However, the value of the starting data of Xt seems not to have significant effect on the size distortion, even though it is very large. Another interesting discovery is that the size performs well when both values of the starting observations are very close. Moreover, the larger both initial condi20.

(24) 3 SIMULATION tions are, better the size performs. This result is similar to the conclusion Elliott and M¨ uller (2003) made. They explore the relationship of their optimal statistic which considers the initial condition with other unit root tests suggested in other literature and classify those statistics for unit root test by their asymptotic distributions. In according to Elliott and M¨ ullers’ categories, the statistic suggested by Dickey and Fuller (1979) as well as those suggested by Phillips (1987) and Phillips and Perron (1988) are all belong to the same class, which places very high weight in extremely large values of the initial condition. Next, we can simulate the tables of critical values for the cointegration test based on the optimal statistic derived in section 2 . Because each combinations of two initial conditions needs a table and the number of combinations is infinite, so we just exhibit some tables with the interesting combination, especially those values which make serious size distortion. All the tables are run by MATLAB Program, with replication 20,000. In our table, the number of observations, T, are restricted to be no more than 300 for the purpose of fitting our assumption that our statistic are derived under the condition of finite sample.. 21.

(25) 3 SIMULATION. Table 1 ξ = 0, ζ = 0 α = 0.05 ρ T. 0.9. 0.95. 0.975. 0.99. 25. -2.5920. -1.4327. -0.7619. -0.3096. 50. -4.2295. -2.4842. -1.3366. -0.5620. 100. -6.3550. -4.0547. -2.3149. -1.0182. 200. -7.1187. -6.3241. -4.0620. -1.870. 300. -7.8663. -6.4220. -4.3101. -2.2205. Table 2 ξ = 100, ζ = 5 α = 0.05 ρ T. 0.9. 0.95. 0.975. 0.99. 25. -163.6830. -146.5482. -92.3229. -41.3281. 50. 131.7036. -96.2468. -96.8201. -52.6086. 100. 1024.5. 124.4323. -38.3054. -44.8958. 200. 2945.2. 576.5019. 78.3340. -18.1721. 300. 5713.8. 742.6. 164.445. 19.2460. 22.

(26) 3 SIMULATION. Table 3 ξ = 50, ζ = 5 α = 0.05 ρ T. 0.9. 0.95. 0.975. 0.99. 25. -48.9543. -39.2604. -24.2838. -10.8583. 50. 17.0203. -31.5893. -22.7796. -14.2000. 100. 205.4159. 14.5741. -15.6511. -13.0752. 200. 572.0189. 103.2629. 5.6271. -8.3591. 300. 765.2601. 205.0623. 23.6606. 1.8523. Table 4 ξ = 0, ζ = 10 α = 0.05 ρ T. 0.9. 0.95. 0.975. 0.99. 25. -3.2886. -1.7831. -0.9304. -0.4015. 50. -4.6237. -2.8426. -1.5091. -0.6557. 100. -6.8909. -2.7927. -2.5163. -1.1182. 200. -7.5045. -6.8874. -4.2848. -1.9537. 300. -7.5805. -7.8223. -5.3057. -2.4380. 23.

(27) 3 SIMULATION. Table 5 ξ = 5, ζ = 3 α = 0.05 ρ T. 0.9. 0.95. 0.975. 0.99. 25. -3.8248. -2.2260. -1.1964. -0.5102. 50. -5.1892. -3.1968. -1.7868. -0.7692. 100. -6.9576. -4.7473. -2.7143. -1.2340. 200. -6.1569. -6.6485. -4.3565. -2.0587. 300. -6.5002. -7.0255. -5.0558. -2.2550. Table 6 ξ = 5, ζ = 50 α = 0.05 ρ T. 0.9. 0.95. 0.975. 0.99. 25. -3.6311. -2.0491. -1.1443. -0.4688. 50. -5.1033. -3.4062. -1.9648. -0.8515. 100. -7.0585. -5.1247. -3.1651. -1.4341. 200. -7.2270. -7.1813. -4.7193. -2.3999. 300. -7.4203. -7.8455. -5.8046. -2.8071. 24.

(28) 4 CONCLUSION. 4. Conclusion. In this paper, we manage to show that for more assumptions on the initial conditions, we can derive an cointegration test statistic which considers two initial conditions of two data and understand that it is different than that derived by Elliott and M¨ uller with only one parameter of the initial condition. By dealing with two initial conditions at the same time, the statistic derived from discriminant function would help us recognize the differences between null hypotheses and alternative hypotheses. Beside, the statistics make it possible to simulate our own critical value table in according to what starting observations we have in hand, what significant level we want and what value of ρ we meet under H1 . However, there are some weaknesses of the statistic derived in this paper. Even though we take two starting observations of two sets of time series into consideration and define two initial conditions as two different parameters, they would be amalgamated into an initial residual, uˆ0 . This outcome makes it a little difficult to identify the impact of each starting observations separately and clearly. Next, from the simulation in section 3, under the assumption of both starting data known, the critical value used in hypothesis test while we regression y on x is different than that used while we regression x on y. It is very similar with the situation we meet while dealing with the role of Normalization in cointegration model. So, it is crucial that we choose carefully which variable to call y and which one to call x. However, we still can’t exactly distinguish which variables should 25.

(29) 4 CONCLUSION be set as y and which one should be set as x. Third, the value of ρ discussed in the alternative hypotheses is hard to be observed in practice and it would make our statistic less useful. If we can derive the asymptotic distribution of our statistic, maybe we can introduce the concept of local-to-unity under the condition of asymptotic analysis to solve the problem. There are some extendable directions following the results of this paper. First, in this paper, we just assume that {νt }+∞ −∞ are independent and identical to the Gaussian distribution. It is possible to derived a family of tests, if we derived the statistics under every possible condition of innovation process {νt }+∞ −∞ . Second, it is difficult to derived the asymptotic distribution of the statistics we derived in this paper. But, if we can break through the barrier, maybe we can find the relationship which is similar with what Elliott and M¨ uller found between our statistic and other well-known statistics for cointegration tests. Third, a multivariate analog of unit root testing is testing for cointegration rank. Such tests usually assume away the initial condition as is often the case in the univariate literature.. 26.

(30) REFERENCES. References [1] Dickey, D. and W. Fuller (1979) “ Distribution of the Estimators for Autoregressive Time Series with a Unit Root ”, Journal of the American Statistical Association, 74 , 427–431. [2] Dickey, D. and W. Fuller (1981) “ Likelihood Ration Statistics for Autoregressive Time Series with a Unit Root ”, Econometrica, 49 , 115–143. [3] Elliott, G., T. Rothenberg, and J. Stock (1996) “ Efficient Tests for an Autoregressive Unit Root ”, Econometrica, 64 , 813–836. [4] Elliott, G. (1999) “ Efficient Tests for a Unit Root When the Initial Observation Is Drawn from its Unconditional Distribution ”, International Economic Review, 40 , 767–783. [5] Elliott, G. and K. M¨ uller (2003) “ Tests for Unit Roots and the Initial Condition ”, Econometrica, 71 , 1269–1286. [6] Elliott, G. and K. M¨ uller (2006) “ Minimizing the Impact of the Initial Condition on Testing for Unit Roots ”, Journal of Econometrics, 135 , 285–310. [7] Engle, F. and C. W. J. Granger (1987) “ Co–Integration and Error Correction : Representation, Estimation, and Testing ”, Econometrica, 55 , 251–276. [8] Evans, G. and N. Savin (1981) “ Testing for Unit Roots : 1 ”, Econometrica, 49 , 753–779. [9] Evans, G. and N. Savin (1984) “ Testing for Unit Roots : 2 ”, Econometrica, 52 , 1241–1269. 27.

(31) REFERENCES [10] Hamilton, James D. (1994) Time Series Analysis , Princeton University Press, Princeton, N. J.. [11] Juhl, T. and Z. Xiao (2003) “ Power Functions and Envelopes for Unit Root Tests ”, Econometric Theory, 19 , 240–253. [12] Lehmann, E. (1986) Testing Statistical Hypotheses , Wiley, New York, second edn. [13] Maddala, G. S. and Kim In–Moo (1998) Unit Roots Cointegration and Structural Change , Cambridge University Press, New York. [14] Phillips, P. C. B. (1986) “ Understanding Spurious Regressions in Econometrics ”, Journal of Econometrics, 33 , 311–340. [15] Phillips, P. C. B. (1987a) “ Time Series Regression with a Unit Root ”, Econometrica, 55 , 277–301. [16] Phillips, P. C. B. (1987b) “ Toward a Unified Asymptotic Theory for Autoregression ”, Biometrika, 74 , 535–547. [17] Phillips, P. C. B. (1988) “ Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations ”, Econometric Theory, 4 , 528–533. [18] Phillips, P. C. B. and P. Perron (1988) “ Testing for Unit Root in Time Series Regression ”, Biometrika, 75 , 335–346. [19] Phillips, P. C. B. and S. Ouliaris (1990) “ Asymptotic Properties of Residual Based Tests for Cointegration ”, Econometrica, 58 , 165–193. 28.

(32) REFERENCES [20] Rao, C. R. (1973) Linear Statistical Inference and its Applications , Wiley, New York. [21] Rao, C. R. and S. Mitra (1971) Generalized Inverse of Matrices and its Applications , Wiley, New York. [22] Schott, James R. (1997) Matrix Analysis For Statistics , Wiley, New York.. 29.

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