直接測(cè)試之間的協(xié)整論分析-A DIRECT TEST FOR COINTEGRATION BETWEEN
08-09, 2014
格蘭杰在1981年開(kāi)創(chuàng)自己的工作，而恩格爾是在1987年，這個(gè)協(xié)作研究的主題是經(jīng)濟(jì)學(xué)的時(shí)間序列問(wèn)題。在此大量的理論發(fā)展會(huì)隨后報(bào)道出來(lái)，考慮到協(xié)作的可能性以及相關(guān)錯(cuò)誤模式要重新更正，并且努力使它成為時(shí)間序列經(jīng)濟(jì)學(xué)中一個(gè)完整的范例。我們可以讓Y1T與2T被處理成一個(gè)完整的部分，這個(gè)序列是非穩(wěn)定性的，兩者的第一個(gè)差異就是靜止性。其次這個(gè)序列也可以理解成為一個(gè)共合體，又或者會(huì)出現(xiàn)一個(gè)標(biāo)量B，這些是靜止的一種表現(xiàn)。我們?cè)谶@里測(cè)試的也是一種一般形式，它們應(yīng)用在序列處理過(guò)程之中，可以被看作是一個(gè)相對(duì)較低的自然回歸現(xiàn)像，它也在第三章節(jié)中有所體現(xiàn)。
Following the pioneering work of Granger (1981) and Engle and Granger (1987), the topic of cointegration has been at the heart of time series econometrics. In addition to the substantial theoretical developments that have subsequently been reported, considering the possibility of cointegration, and the associated errorcorrection model representation, has become an integral part of the standard paradigm of applied time series econometrics. In this paper we shall discuss a specific problem - testing the null hypothesis of no cointegration between a pair of time series. Let y1t and y 2t both be generated by processes which are integrated of order one, I (1) ; that is, the series are nonstationary, but their first differences are stationary, or I (0) . Then, the series are cointegrated if and only if there exists a scalar b, such that ( y1t − by2 t ) is stationary. The general form of our test, applicable when the process generating the series can be wellapproximated by a relatively low order vector autoregression, is introduced in Section 3.
Simulation evidence reported there suggests that the new test has superior power compared with the most commonly applied extant test. Although generalisation of our test to the case of several series would be very difficult, the case of two series is of considerable interest in its own right: see, for example, Ng and Perron (1997). Section 2 of the paper is, in effect, an extended introduction to, and motivation for, the general test procedure. In that section, we analyse a very simple generating model, for which our test has a straightforward and intuitively appealing structure. We argue that, for this simple model, the new test has desirable properties when compared with the two tests that have most often been applied in this case. Finally, in Section 4, we briefly discuss an empirical example, where the new test finds strong evidence of cointegration when one of its competitors does not, in a situation where cointegration might reasonably be expected on a priori grounds .
Regression on these residuals, with no intercept or lagged first differences. Critical values were obtained from the response surfaces of MacKinnon (1991). The JT test was based on the vector autoregression, with intercept, using critical values given by Osterwald-Lenum (1992). Here, and in the more general case of the next section, we tried also the maximal-eigenvalue variant of the Johansen test. However, the results were invariably very similar to those from the trace test, and so will not be reported. The results of these experiments on the powers of the two tests are summarised in Table 1. The most striking feature of these results is that, while the power of the JT test is unaffected by the parameter ρ , the power of the EGDF test depends dramatically on that parameter. For example, for ρ close to 1, the EGDF test is much more powerful than the JT test, whereas for ρ close to –1 the performance of EGDF is relatively very poor. It is certainly interesting that the relative powers of the two tests depend so dramatically on a single parameter in such a simple model. In fact, it is straightforward to show, for the particular model considered here, that the distribution of the JT test statistic is invariant to ρ .
By contrast, the distribution of the EGDF test statistic is clearly influenced by the parameter ρ . This test statistic is based initially on least squares estimates from the regression model y1t = a + by2 t + ut (2) Its distribution therefore depends on the sampling distribution of the least squares estimator b . Table 2 reports simulation evidence on the mean and standard deviation ˆ for series of 100 observations generated from (1). For large positive ρ this of b estimator has mean close to 1, with small standard deviation. However, as ρrapidly increases, the extent of that bias being quite startling decreases, the bias in b for large negative ρ . It is certainly not surprising that the EGDF test has low power in these circumstances. Gonzalo (1994) discusses in some detail bias in this and other estimators of the cointegrating parameter. Our interest lies in the impact of that bias on the EGDF test. The results in Table 1 suggest that, for ρ close to –1, the performance of the EGDF test is unacceptably poor. It is perhaps worth asking if such results would be obtained in practice, for when b is poorly estimated the residuals ut from (2) may not appear to be first order autoregressive.
Consequently, the user may apply the augmented version of the Dickey-Fuller test to these residuals. We simulated series of 100 observations from the model (1), applying now the augmented version of the EGDF test. We allowed a maximum of six lagged differences in the Dickey-Fuller regressions, selecting the number of lags actually employed each time through general-to-specific testing at the 0.05 level, in line with proposals of Ng and Perron (1995). In fact, the power of the EGDF test was not improved, and was slightly lower for ρ close to –1. We conclude then that there exist circumstances - those in which the parameter of the cointegrating regression is poorly estimated - when EGDF has very poor performance. Unfortunately, in practical applications, which are likely to involve models that are different from and more elaborate than (1), it would be practically impossible to identify cases where such circumstances arose. Although the performance of the EGDF test can be very poor, the results of Table 1 suggest also that there are cases where substantial gains in power can be achieved through using EGDF rather than the Johansen test. It is worth asking whether at least some of those gains can be captured through a modified test whose power does not fall dramatically where the power of EGDF does. We explore here a test motivated directly by the definition of cointegration, continuing to employ Dickey-Fuller-type regressions, but omitting the estimation of b through the regression (2), since this seems to be the source of the difficulties apparent in Table 1. Let y1t and y2 t be a pair of I(1) time series. Then, these series are cointegrated if and only if there exists a scalar b such that ( y1t − by2 t ) is stationary. We consider Dickey-Fuller regressions for all possible b . Specifically, we estimate regressions of the form ?( y1t − by 2 t ) = a + c( y1,t −1 − by 2,t −1 ) + ω t Let τ b be the t-ratio associated with the estimate of c.
where ℑ is a compact interval. Viewing τ b as a function of the scalar b , τ * can easily be found numerically. Table 3 shows critical values of the τ * statistic at the standard significance levels, obtained through simulations based on independent driftless random walks. We shall discuss the asymptotics in the next section, but note for now that the critical values appear to settle down quite quickly with increasing sample size. It is not necessary to verify the robustness of these critical values to correlation between the innovations generating the random walks. As we shall see, for the model (1), the finite sample distribution of τ * is invariant to ρ for any γ . We next show that, to assess the power of the τ * test applied to the model (1), it is only necessary to consider the case of ρ = 0 . We can write (1) as (1 − φL)( y1t − y2 t ) = {2(1 − ρ )}1/ 2 e1t (1 − L)( y1t + y2 t ) = {2(1 + ρ )}1/ 2 e2 t where L is the lag operator and the white noise innovations (e1t , e2t )′ have identity covariance matrix. Then, for any b , it follows that (1 − φL)(1 − L)( y1t − by2 t ) = {(1 − ρ ) / 2}1/ 2 (1 + b)(1 − L)e1t + {(1 + ρ ) / 2}1/ 2 (1 − b)(1 − φL)e2 t (3).
The required results follow directly from (3). Specifically we shall show that, in the case where ρ = 0 , there exists b* such that ( y1t − b* y2 t ) is precisely the same series as ( y1t − by2 t ) in the ρ ≠ 0 case. We need only consider |b| ≤ 1, as for |b| > 1 , we might just as well analyse ( y2 t − dy1t ) for | d | < 1. The ρ = 0 variant of (3) is given by (1 − φL)(1 − L)( y1t − b* y2 t ) = 2 −1/ 2 {(1 + b* )(1 − L)e1t + (1 − b* )(1 − φL)e2 t } (4) First, if b = ±1, the right-hand side of (3) is a constant multiple of the right-hand side of (4) with b* = ±1. In the more general case, the right-hand side of (3) is a constant multiple of the right-hand side of (4) if.
Then, since k > 0 , it follows that for every b, with |b| < 1, there exists a corresponding b * , with |b * | < 1, and vice versa. This establishes that the set of τ b Dickey-Fuller statistics is the same, whatever the value of ρ , for a specific realisation of the white noise generating series (e1t , e2 t )′ . Therefore, both the size and power of the τ * test are invariant to ρ for series generated from the model (1). Table 4 shows powers of the τ * test for model (1). Comparing these with the results of Table 1, we find that τ * has appreciably higher power than the JT test. In fact, the new test has about the same power as the EGDF test when ρ = 0 . For ρ close to 1, EGDF is noticeably more powerful than τ * . This is to be expected, however, since in that case, as we see from Table 2, b is quite precisely estimated through the regression (2). There is a sense in which the relatively strong performance of the EGDF test is fortuitous when ρ is not very close to one. For example, in the case ρ = 0 , EGDF is noticeably more powerful than the Johansen test, and about as powerful as the τ * test. Yet, in that case, as can be seen from Table 2, the least squares estimator of b in (2) is quite severely biased. To see why the EGDF test nevertheless has respectable.
Agiakloglou and Newbold 1992) that Dickey-Fuller regressions lead to frequent rejections of the unit root hypothesis for I(1) generating models with large positive moving average parameters. The τ * statistic performs impressively for the model (1), which is of course a very special case of cointegration.
In the next section we discuss an augmented 6 variant of the τ * test, appropriate for general vector autoregressive generating models. However, even in the non-augmented case, the simulation results of Table 4 apply to a much broader set of models than simply (1). transformation x1t = cy1t + dy 2 t , x 2 t = (c − 1) y1t + (d + 1) y 2 t Then ( x1t − x 2t ) = ( y1t − y 2t ) . Also, the set of linear functions of ( x1t , x 2t ) is the same as the set of linear functions of ( y1t , y 2t ) , so the distribution of the τ * test statistic is invariant to this transformation. From (1), we then have ?x1t = −1 / 2(c − d )(1 − φ )( x1,t −1 − x 2,t −1 ) + u1t ?x 2 t = 1 / 2(d − c + 2)(1 − φ )( x1,t −1 − x 2,t −1 ) + u 2t where the white noise innovations (u1t , u 2t )′ have covariance matrix ? = M ∑ M ′, d ? ? c M =? ? ?(c − 1) (d + 1)? Starting with (1), consider the Then, all values of (c, d, ρ) in this model yield τ * statistics with identical sampling distributions. In particular, unidirectional casual models result from setting c = d. The usual error-correction interpretation in this model requires 0 ≤ (c − d ) < 2 .
Some sample moments which are O p (1) . In this situation the argument in the proof still goes through without any major modification. The main feature of the above approach is that it avoids weaknesses inherent with the variants of τ * based on augmented Dickey-Fuller style regressions. As a simple example, suppose p = 1 so that we have the model ?y t = A1 ?y t −1 + η t Then ?xt = β ′A1 ?y t −1 + ω t ≠ a1 ?xt −1 + ω t except in the particular case when A1 = a1 I . So, in general, ?xt will not follow an AR(p) process and hence ADF-style corrections to unit root tests are, strictly speaking, not appropriate. Our suggested procedure, however, corrects for autocorrelation by fitting models explicitly to ?y t rather than ?xt and so avoids this problem. Moreover, since the actual process followed by ?xt depends on b, so too will the degree of any autoregressive approximation used for ADF tests. Thus, we would need to allow the numbers of lagged terms in ?xt included in the ADF regression to vary with b which would make finding the appropriate minimum ADF test over b computationally very expensive. On the other hand, in our approach, b does not influence the order of the model which needs to be estimated. In this more general setup, we need to assess the effects of estimating the Ai and Σ on the finite sample size and power of the test z * p . To this end, we simulated the model (1) with γ = 0 , assuming ν t follows.
As a simple example of our test procedure, we test for cointegration between short and long term U.K. interest rates. The short term rate ( y1t ) is the 91 day U.K. Treasury Bill rate and the long term rate ( y2 t ) is the yield on 20 year U.K. gilts. The data are quarterly from 1952Q1-1988Q4 (148 observations), and were obtained from Mills (1993). Augmented Dickey-Fuller tests applied to each series suggest that both are I(1) without drift, not rejecting the I(1) null even at the 0.10 significance level (we omit the detailed results here). Table 8 gives the values of the cointegration test statistics z * p and JTp for p = 0,1,...,5 , where p is the order of the VAR model fitted 12 to ?yt . Using standard t-ratios to assess significance of the estimated coefficients on lagged terms in ?yt suggested p = 3 as the preferred order. Also shown, as bmin in the table, is the value of b found to minimize z p (τ b ) , for each p. The Johansen test fails to reject at the 0.10-level the null hypothesis of no cointegration between the interest rates for any of the lag lengths considered here, whereas z * p rejects non- cointegration at the 0.05-level with p = 1,2,...,4 , and at the 0.10-level with p = 0,5 . For this particular data set then, the z * p test suggests strong evidence of cointegration between short and long term interest rates, while the JTp test does not.
In this paper we have introduced a new test of the null hypothesis of no cointegration between a pair of I(1) time series. This test was motivated in Section 2 by observing the extremely variable power of the Engle-Granger/Dickey-Fuller (EGDF) test, relative to the Johansen trace (JT) test, over a wide range of values of a single parameter in a simple model. It was seen that the very poor performance of the EGDF test is associated with the severe bias in the least squares estimator of the parameter in the cointegrating regression. We therefore proposed circumventing the estimation of the cointegrating regression by, in effect, applying the Dickey-Fuller test to all linear combinations of the two time series. Our test statistic, in the simple case of Section 2, is then the Dickey-Fuller t-ratio that is least favourable to the null hypothesis. We saw, for the simple model, that neither the JT test nor the new test exhibit the volatility in power characteristic of the EGDF test. Moreover, simulation evidence indicated that the new test was a good deal more powerful than the JT test. A subsidiary benefit of our test, compared with EGDF, is that the test outcome does not depend on the arbitrary choice of one series as the dependent variable in the cointegrating regression. Of course, for our test to have much practical value, it must be applicable in far more general situations than that of Section 2, where every linear combination of the series is a random walk under the null hypothesis, and a particular linear combination is a stationary first order autoregression under the alternative.
An obvious possible 13 extension would be to base the new test on augmented Dickey-Fuller regression, by analogy with EGDF. However, we prefer an alternative generalisation, for two reasons. First, we wanted a direct comparison with the JT test for finite order vector autoregression. However, linear combinations of series generated by such models generally have infinite autoregressive order, requiring some truncation rule for fixing the number of lags in the Dickey-Fuller regressions. Second, since we are considering all possible linear combinations of the series, choosing the number of lags in this way is both problematic and computationally burdensome. Different linear combinations of the series will have different generating models. This suggests the necessity either of a different number of lags for every linear combination, which is practically impossible, or the fixing for every possibility of a very high number of lags, which would inevitably reduce the power of the test. In Section 3 we proposed a general version of our test, which incorporates the test of Section 2 as a special case, based on the prior fitting of vector autoregressions to first differences of the time series. Simulation results confirmed that this general test has satisfactory size properties - more satisfactory in some cases than the JT test. Moreover, the relative superiority in power of the new test, observed in the simple case of Section 2, continues to hold for the vector autoregressions examined in a simulation experiment in Section 3. Finally, in Section 4 we applied the new test and the JT test to series of short-and long-term interest rates. While the former suggested strong evidence of cointegration, the latter did not, in a situation where several authors, including Engle and Granger (1987), have suggested that cointegration might reasonably be expected on a priori grounds.
References
AGIAKLOGLOU, C. and NEWBOLD, P. (1992), Empirical evidence on DickeyFuller-type tests, Journal of Time Series Analysis, 13, 471-483. BANERJEE, A., DOLADO, J.J., HENDRY, D.F. and SMITH, G.W. (1986), Exploring equilibrium relationships in econometrics through static models: Some Monte Carlo evidence, Oxford Bulletin of Economics and Statistics, 48, 253-277. BERGE, C. (1963), Topological Spaces, tr. By Patterson, New York, Macmillan, (French original, 1959). BLANGIEWICZ, M. and CHAREMZA, W. (1990), Cointegration in small samples: Empirical percentiles, drifting moments and customized testing, Oxford Bulletin of Economics and Statistics, 52, 303-315. CHAN, N.H. and WEI, C.Z. (1988), Limiting distributions of least squares estimates of unstable autoregressive processes, Annals of Statistics, 16, 367-401. ENGLE, R.F. and GRANGER, C.W.J. (1987), Cointegration and error correction: Representation, estimation and testing, Econometrica, 55, 251-276. ENGLE, R.F. and YOO, B.S. (1987), Forecasting and testing in cointegrated systems, Journal of Econometrics, 35, 143-159. GRANGER, C.W.J. (1981), Some properties of time series data and their use in econometric model specification, Journal of Econometrics, 16, 121-130. GONZALO, J. (1994), Five alternative methods of estimating long-run equilibrium relationships, Journal of Econometrics, 60, 203-233. HAMILTON, J.D. (1994), Time Series Analysis. Princeton, New Jersey: Princeton University Press. HANSEN, B.E and PHILLIPS, P.C.B. (1990), Estimation and inference in models of cointegration: A simulation study, Advances in Econometrics, 8, 225-248. JOHANSEN, S. (1988), Statistical analysis of cointegration vectors, Journal of Economic Dynamics and Control, 12, 231-254. JOHANSEN, S. (1991), Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models, Econometrica, 59, 1551-1580. JOHANSEN, S. and JUSELIUS, K. (1990), Maximum likelihood estimation and inference on cointegration - with applications to the demand for money, Oxford Bulletin of Economics and Statistics, 52, 169-210. MacKINNON, J.G. (1991), Critical values for cointegration tests, in Engle, R.F. and Granger, C.W.J. (eds), Long-Run Economic Relationships (Oxford: Oxford University Press). MILLS, T.C. (1993), The Econometric Modelling of Financial Time Series. Cambridge: Cambridge University Press. NG, S. and PERRON, P. (1995), Unit root tests in ARMA models with datadependent methods for the selection of the truncation lag, Journal of the American Statistical Association, 90, 268-281. NG, S. and PERRON, P. (1997), Estimation and inference in nearly unbalanced nearly cointegrated systems, Journal Econometrics, 79, 53-81. OSTERWALD-LENUM, M. (1992), A note with fractiles of the asymptotic distribution of the maximum likelihood cointegration rank test statistics: Four cases, Oxford Bulletin of Economics and Statistics, 54, 461-472.
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Following the pioneering work of Granger (1981) and Engle and Granger (1987), the topic of cointegration has been at the heart of time series econometrics. In addition to the substantial theoretical developments that have subsequently been reported, considering the possibility of cointegration, and the associated errorcorrection model representation, has become an integral part of the standard paradigm of applied time series econometrics. In this paper we shall discuss a specific problem - testing the null hypothesis of no cointegration between a pair of time series. Let y1t and y 2t both be generated by processes which are integrated of order one, I (1) ; that is, the series are nonstationary, but their first differences are stationary, or I (0) . Then, the series are cointegrated if and only if there exists a scalar b, such that ( y1t − by2 t ) is stationary. The general form of our test, applicable when the process generating the series can be wellapproximated by a relatively low order vector autoregression, is introduced in Section 3.
Simulation evidence reported there suggests that the new test has superior power compared with the most commonly applied extant test. Although generalisation of our test to the case of several series would be very difficult, the case of two series is of considerable interest in its own right: see, for example, Ng and Perron (1997). Section 2 of the paper is, in effect, an extended introduction to, and motivation for, the general test procedure. In that section, we analyse a very simple generating model, for which our test has a straightforward and intuitively appealing structure. We argue that, for this simple model, the new test has desirable properties when compared with the two tests that have most often been applied in this case. Finally, in Section 4, we briefly discuss an empirical example, where the new test finds strong evidence of cointegration when one of its competitors does not, in a situation where cointegration might reasonably be expected on a priori grounds .
Regression on these residuals, with no intercept or lagged first differences. Critical values were obtained from the response surfaces of MacKinnon (1991). The JT test was based on the vector autoregression, with intercept, using critical values given by Osterwald-Lenum (1992). Here, and in the more general case of the next section, we tried also the maximal-eigenvalue variant of the Johansen test. However, the results were invariably very similar to those from the trace test, and so will not be reported. The results of these experiments on the powers of the two tests are summarised in Table 1. The most striking feature of these results is that, while the power of the JT test is unaffected by the parameter ρ , the power of the EGDF test depends dramatically on that parameter. For example, for ρ close to 1, the EGDF test is much more powerful than the JT test, whereas for ρ close to –1 the performance of EGDF is relatively very poor. It is certainly interesting that the relative powers of the two tests depend so dramatically on a single parameter in such a simple model. In fact, it is straightforward to show, for the particular model considered here, that the distribution of the JT test statistic is invariant to ρ .
By contrast, the distribution of the EGDF test statistic is clearly influenced by the parameter ρ . This test statistic is based initially on least squares estimates from the regression model y1t = a + by2 t + ut (2) Its distribution therefore depends on the sampling distribution of the least squares estimator b . Table 2 reports simulation evidence on the mean and standard deviation ˆ for series of 100 observations generated from (1). For large positive ρ this of b estimator has mean close to 1, with small standard deviation. However, as ρrapidly increases, the extent of that bias being quite startling decreases, the bias in b for large negative ρ . It is certainly not surprising that the EGDF test has low power in these circumstances. Gonzalo (1994) discusses in some detail bias in this and other estimators of the cointegrating parameter. Our interest lies in the impact of that bias on the EGDF test. The results in Table 1 suggest that, for ρ close to –1, the performance of the EGDF test is unacceptably poor. It is perhaps worth asking if such results would be obtained in practice, for when b is poorly estimated the residuals ut from (2) may not appear to be first order autoregressive.
Consequently, the user may apply the augmented version of the Dickey-Fuller test to these residuals. We simulated series of 100 observations from the model (1), applying now the augmented version of the EGDF test. We allowed a maximum of six lagged differences in the Dickey-Fuller regressions, selecting the number of lags actually employed each time through general-to-specific testing at the 0.05 level, in line with proposals of Ng and Perron (1995). In fact, the power of the EGDF test was not improved, and was slightly lower for ρ close to –1. We conclude then that there exist circumstances - those in which the parameter of the cointegrating regression is poorly estimated - when EGDF has very poor performance. Unfortunately, in practical applications, which are likely to involve models that are different from and more elaborate than (1), it would be practically impossible to identify cases where such circumstances arose. Although the performance of the EGDF test can be very poor, the results of Table 1 suggest also that there are cases where substantial gains in power can be achieved through using EGDF rather than the Johansen test. It is worth asking whether at least some of those gains can be captured through a modified test whose power does not fall dramatically where the power of EGDF does. We explore here a test motivated directly by the definition of cointegration, continuing to employ Dickey-Fuller-type regressions, but omitting the estimation of b through the regression (2), since this seems to be the source of the difficulties apparent in Table 1. Let y1t and y2 t be a pair of I(1) time series. Then, these series are cointegrated if and only if there exists a scalar b such that ( y1t − by2 t ) is stationary. We consider Dickey-Fuller regressions for all possible b . Specifically, we estimate regressions of the form ?( y1t − by 2 t ) = a + c( y1,t −1 − by 2,t −1 ) + ω t Let τ b be the t-ratio associated with the estimate of c.
where ℑ is a compact interval. Viewing τ b as a function of the scalar b , τ * can easily be found numerically. Table 3 shows critical values of the τ * statistic at the standard significance levels, obtained through simulations based on independent driftless random walks. We shall discuss the asymptotics in the next section, but note for now that the critical values appear to settle down quite quickly with increasing sample size. It is not necessary to verify the robustness of these critical values to correlation between the innovations generating the random walks. As we shall see, for the model (1), the finite sample distribution of τ * is invariant to ρ for any γ . We next show that, to assess the power of the τ * test applied to the model (1), it is only necessary to consider the case of ρ = 0 . We can write (1) as (1 − φL)( y1t − y2 t ) = {2(1 − ρ )}1/ 2 e1t (1 − L)( y1t + y2 t ) = {2(1 + ρ )}1/ 2 e2 t where L is the lag operator and the white noise innovations (e1t , e2t )′ have identity covariance matrix. Then, for any b , it follows that (1 − φL)(1 − L)( y1t − by2 t ) = {(1 − ρ ) / 2}1/ 2 (1 + b)(1 − L)e1t + {(1 + ρ ) / 2}1/ 2 (1 − b)(1 − φL)e2 t (3).
The required results follow directly from (3). Specifically we shall show that, in the case where ρ = 0 , there exists b* such that ( y1t − b* y2 t ) is precisely the same series as ( y1t − by2 t ) in the ρ ≠ 0 case. We need only consider |b| ≤ 1, as for |b| > 1 , we might just as well analyse ( y2 t − dy1t ) for | d | < 1. The ρ = 0 variant of (3) is given by (1 − φL)(1 − L)( y1t − b* y2 t ) = 2 −1/ 2 {(1 + b* )(1 − L)e1t + (1 − b* )(1 − φL)e2 t } (4) First, if b = ±1, the right-hand side of (3) is a constant multiple of the right-hand side of (4) with b* = ±1. In the more general case, the right-hand side of (3) is a constant multiple of the right-hand side of (4) if.
Then, since k > 0 , it follows that for every b, with |b| < 1, there exists a corresponding b * , with |b * | < 1, and vice versa. This establishes that the set of τ b Dickey-Fuller statistics is the same, whatever the value of ρ , for a specific realisation of the white noise generating series (e1t , e2 t )′ . Therefore, both the size and power of the τ * test are invariant to ρ for series generated from the model (1). Table 4 shows powers of the τ * test for model (1). Comparing these with the results of Table 1, we find that τ * has appreciably higher power than the JT test. In fact, the new test has about the same power as the EGDF test when ρ = 0 . For ρ close to 1, EGDF is noticeably more powerful than τ * . This is to be expected, however, since in that case, as we see from Table 2, b is quite precisely estimated through the regression (2). There is a sense in which the relatively strong performance of the EGDF test is fortuitous when ρ is not very close to one. For example, in the case ρ = 0 , EGDF is noticeably more powerful than the Johansen test, and about as powerful as the τ * test. Yet, in that case, as can be seen from Table 2, the least squares estimator of b in (2) is quite severely biased. To see why the EGDF test nevertheless has respectable.
Agiakloglou and Newbold 1992) that Dickey-Fuller regressions lead to frequent rejections of the unit root hypothesis for I(1) generating models with large positive moving average parameters. The τ * statistic performs impressively for the model (1), which is of course a very special case of cointegration.
In the next section we discuss an augmented 6 variant of the τ * test, appropriate for general vector autoregressive generating models. However, even in the non-augmented case, the simulation results of Table 4 apply to a much broader set of models than simply (1). transformation x1t = cy1t + dy 2 t , x 2 t = (c − 1) y1t + (d + 1) y 2 t Then ( x1t − x 2t ) = ( y1t − y 2t ) . Also, the set of linear functions of ( x1t , x 2t ) is the same as the set of linear functions of ( y1t , y 2t ) , so the distribution of the τ * test statistic is invariant to this transformation. From (1), we then have ?x1t = −1 / 2(c − d )(1 − φ )( x1,t −1 − x 2,t −1 ) + u1t ?x 2 t = 1 / 2(d − c + 2)(1 − φ )( x1,t −1 − x 2,t −1 ) + u 2t where the white noise innovations (u1t , u 2t )′ have covariance matrix ? = M ∑ M ′, d ? ? c M =? ? ?(c − 1) (d + 1)? Starting with (1), consider the Then, all values of (c, d, ρ) in this model yield τ * statistics with identical sampling distributions. In particular, unidirectional casual models result from setting c = d. The usual error-correction interpretation in this model requires 0 ≤ (c − d ) < 2 .
Some sample moments which are O p (1) . In this situation the argument in the proof still goes through without any major modification. The main feature of the above approach is that it avoids weaknesses inherent with the variants of τ * based on augmented Dickey-Fuller style regressions. As a simple example, suppose p = 1 so that we have the model ?y t = A1 ?y t −1 + η t Then ?xt = β ′A1 ?y t −1 + ω t ≠ a1 ?xt −1 + ω t except in the particular case when A1 = a1 I . So, in general, ?xt will not follow an AR(p) process and hence ADF-style corrections to unit root tests are, strictly speaking, not appropriate. Our suggested procedure, however, corrects for autocorrelation by fitting models explicitly to ?y t rather than ?xt and so avoids this problem. Moreover, since the actual process followed by ?xt depends on b, so too will the degree of any autoregressive approximation used for ADF tests. Thus, we would need to allow the numbers of lagged terms in ?xt included in the ADF regression to vary with b which would make finding the appropriate minimum ADF test over b computationally very expensive. On the other hand, in our approach, b does not influence the order of the model which needs to be estimated. In this more general setup, we need to assess the effects of estimating the Ai and Σ on the finite sample size and power of the test z * p . To this end, we simulated the model (1) with γ = 0 , assuming ν t follows.
As a simple example of our test procedure, we test for cointegration between short and long term U.K. interest rates. The short term rate ( y1t ) is the 91 day U.K. Treasury Bill rate and the long term rate ( y2 t ) is the yield on 20 year U.K. gilts. The data are quarterly from 1952Q1-1988Q4 (148 observations), and were obtained from Mills (1993). Augmented Dickey-Fuller tests applied to each series suggest that both are I(1) without drift, not rejecting the I(1) null even at the 0.10 significance level (we omit the detailed results here). Table 8 gives the values of the cointegration test statistics z * p and JTp for p = 0,1,...,5 , where p is the order of the VAR model fitted 12 to ?yt . Using standard t-ratios to assess significance of the estimated coefficients on lagged terms in ?yt suggested p = 3 as the preferred order. Also shown, as bmin in the table, is the value of b found to minimize z p (τ b ) , for each p. The Johansen test fails to reject at the 0.10-level the null hypothesis of no cointegration between the interest rates for any of the lag lengths considered here, whereas z * p rejects non- cointegration at the 0.05-level with p = 1,2,...,4 , and at the 0.10-level with p = 0,5 . For this particular data set then, the z * p test suggests strong evidence of cointegration between short and long term interest rates, while the JTp test does not.
In this paper we have introduced a new test of the null hypothesis of no cointegration between a pair of I(1) time series. This test was motivated in Section 2 by observing the extremely variable power of the Engle-Granger/Dickey-Fuller (EGDF) test, relative to the Johansen trace (JT) test, over a wide range of values of a single parameter in a simple model. It was seen that the very poor performance of the EGDF test is associated with the severe bias in the least squares estimator of the parameter in the cointegrating regression. We therefore proposed circumventing the estimation of the cointegrating regression by, in effect, applying the Dickey-Fuller test to all linear combinations of the two time series. Our test statistic, in the simple case of Section 2, is then the Dickey-Fuller t-ratio that is least favourable to the null hypothesis. We saw, for the simple model, that neither the JT test nor the new test exhibit the volatility in power characteristic of the EGDF test. Moreover, simulation evidence indicated that the new test was a good deal more powerful than the JT test. A subsidiary benefit of our test, compared with EGDF, is that the test outcome does not depend on the arbitrary choice of one series as the dependent variable in the cointegrating regression. Of course, for our test to have much practical value, it must be applicable in far more general situations than that of Section 2, where every linear combination of the series is a random walk under the null hypothesis, and a particular linear combination is a stationary first order autoregression under the alternative.
An obvious possible 13 extension would be to base the new test on augmented Dickey-Fuller regression, by analogy with EGDF. However, we prefer an alternative generalisation, for two reasons. First, we wanted a direct comparison with the JT test for finite order vector autoregression. However, linear combinations of series generated by such models generally have infinite autoregressive order, requiring some truncation rule for fixing the number of lags in the Dickey-Fuller regressions. Second, since we are considering all possible linear combinations of the series, choosing the number of lags in this way is both problematic and computationally burdensome. Different linear combinations of the series will have different generating models. This suggests the necessity either of a different number of lags for every linear combination, which is practically impossible, or the fixing for every possibility of a very high number of lags, which would inevitably reduce the power of the test. In Section 3 we proposed a general version of our test, which incorporates the test of Section 2 as a special case, based on the prior fitting of vector autoregressions to first differences of the time series. Simulation results confirmed that this general test has satisfactory size properties - more satisfactory in some cases than the JT test. Moreover, the relative superiority in power of the new test, observed in the simple case of Section 2, continues to hold for the vector autoregressions examined in a simulation experiment in Section 3. Finally, in Section 4 we applied the new test and the JT test to series of short-and long-term interest rates. While the former suggested strong evidence of cointegration, the latter did not, in a situation where several authors, including Engle and Granger (1987), have suggested that cointegration might reasonably be expected on a priori grounds.
References
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