Board of Governors of the Federal Reserve System
International Finance Discussion Papers Number 907 October 2007
A Residual-Based Cointegration Test for Near Unit Root Variables
Erik Hjalmarsson and Pr sterholm
NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.
A Residual-Based Cointegration Test for Near Unit Root Variables
Erik Hjalmarssony Pr sterholmz
October 9, 2007
Abstract Methods of inference based on a unit root assumption in the data are typically not robust to even small deviations from this assumption. In this paper, we propose robust procedures for a residual-based test of cointegration when the data are generated by a near unit root process. A Bonferroni method is used to address the uncertainty regarding the exact degree of persistence in the process. We thus provide a method for valid inference in multivariate near unit root processes where standard cointegration tests may be subject to substantial size distortions and standard OLS inference may lead to spurious results. Empirical illustrations are given by: (i) a re-examination of the Fisher hypothesis, and (ii) a test of the validity of the cointegrating relationship between aggregate consumption, asset holdings, and labor income, which has attracted a great deal of attention in the recent …nance literature. JEL classi…cation: C12, C22. Keywords: Bonferroni test; Cointegration; Near unit roots.
We have bene…tted from comments by Meredith Beechey, David Bowman, Mike McCracken, Chris Erceg, Dale Henderson, Lennart Hjalmarsson, Randi Hjalmarsson, George Korniotos, Rolf Larsson, Andy Levin, Johan Lyhagen, John Rogers, Jonathan Wright, and seminar participants at the Federal Reserve Board. The views in this paper are solely the responsibility of the authors and should not be interpreted as reecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. sterholm gratefully acknowledges …nancial support from Jan Wallander' and Tom Hedelius'Foundation. s y Division of International Finance, Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA. email: erik.hjalmarsson@frb.gov Phone: +1 202 452 2426 z Department of Economics, Uppsala University, Box 513, 751 20 Uppsala, Sweden. email: par.osterholm@nek.uu.se Phone: +1 202 378 4135
1
Introduction
Cointegration tests have been among the most important and in uential tools in empirical economics since their introduction over two decades ago. In essence, cointegration tests attempt to identify common driving factors in stochastically trending data, thus identifying long-run equilibrium relationships between economic variables. The most common cointegration tests are based on the assumption that the individual variables are unit root processes. The unit root assumption, however, is often hard to fully justify for actual economic data. In …nite samples, many economic variables appear highly, but not totally, persistent; that is, the largest autoregressive root is close to, but not necessarily equal to, unity. Unfortunately, inferencial procedures designed for unit root data tend not to be robust to even small deviations from the unit root assumption. For instance, Elliot (1998) shows that large size distortions can occur when performing inference on the cointegrating vector in a system where the individual variables follow near unit root processes rather than pure unit root processes. Unit root tests go some way toward alleviating the uncertainty regarding the persistence in a given time series but do not provide a de…nitive answer. Since unit root tests have low power against local alternatives, a failure to reject the null hypothesis of a unit root does not rule out the possibility of a root slightly di¤erent from unity. On the other hand, rejecting the null of a unit root does not rule out that the process is still fairly persistent and leaves open the possibility of spurious regressions. It is thus far from obvious how to deal with a multivariate near unit root process: Standard cointegration tests will not be valid under deviations from the pure unit root assumption and the possibility of spurious regressions invalidates standard OLS inference.1 The aim of this paper is to design a test of cointegration that is robust to deviations from the pure unit root assumption. In particular, we extend the standard framework to the case where the original data possess autoregressive roots that are local-to-unity, rather than identically equal to unity. The methods developed here are useful from two di¤erent perspectives. First, they provide a robustness check to standard cointegration tests in the typical situation where it is not known with certainty that there is an exact unit root in the data. Second, and just as importantly, the test procedures in this paper allow for valid inference in the case when the data is likely not a pure unit root process, but still highly persistent.
1 In most cointegration studies, the regressors are endogenous, in which case OLS inference would be further complicated and invalid even in the strictly stationary case. Stock (1997) provides a detailed discussion on many of the issues that arise in inference with near unit root variables.
1
While there is a large literature on cointegrating regressions with near unit root regressors, the focus has been on inference on the slope parameter in these regressions, rather than actual tests of cointegration; see, for example, Cavanagh et al. (1995), Elliot (1998), Campbell and Yogo (2006) and Jansson and Moreira (2006). Typically, the models in this literature have been speci…ed such that under the null hypothesis of a zero slope coe cient, the dependent variable is a stationary process. Tests on the slope coe cient therefore become joint tests of cointegration as well, and the issue of spurious regressions never occurs. Although this is a useful speci…cation, for instance, in tests of stock-return predictability which motivated much of this literature, it is less convenient in most typical economic applications where both dependent and independent variables are near-integrated. The closest related literature to the current paper is the work on stationarity tests (Leybourne and McCabe, 1993, and Shin, 1994) and the work by Wright (2000). In particular, Wright (2000) develops a joint test of a speci…c hypothesis regarding the cointegrating vector and a test of the null hypothesis of cointegration that is robust to deviations from the pure unit root framework. We focus on a residual-based test of cointegration. Following the work of Phillips and Ouliaris (1990), we extend the asymptotic results for a residual-based test to the case of near-integrated processes. Unlike the pure unit root case, the asymptotic distribution of the test statistic now depends on an unknown nuisance parameter; the local-to-unity root. Since this parameter is not consistently estimable, feasible tests cannot be directly constructed from the asymptotic distribution. Instead, we propose to replace the unknown parameter value for the local-to-unity root with a conservative estimate. In order to understand the intuition behind our procedure, it is useful to consider the potential errors when applying a standard, pure unit root case, cointegration test to a set of near unit root variables. A residual-based cointegration test evaluates whether the residuals from the empirical regression contain a unit root. Now, if the original data are in fact near-integrated, with a root less than unity, the test will over-reject since the residuals will not contain a unit root even if there is no cointegration. But, by instead using critical values based on a conservative estimate of the local-to-unity root in the original data, a valid test is obtained. Intuitively, if one views a residual-based test of cointegration as a test of whether there is less persistence in the residuals than in the original data, then this test is only valid if the persistence of the original data is not overstated.2 In a spirit similar to the Bonferroni
perhaps, less obvious, the same also holds true for non-residual-based tests, such as those of Johansen (1988,1991); see Hjalmarsson and sterholm (2007).
2 Although,
2
methods proposed by Cavanagh et al. (1995), we show how an appropriately conservative estimate of the local-to-unity root is obtained. The rest of the paper is organized as follows. Section 2 outlines the modelling assumptions and the theoretical results. Section 3 describes the Bonferroni methods. In Section 4, the proposed procedure is evaluated using Monte Carlo simulations. We show that once the conservative estimate for the local-to-unity parameter is chosen appropriately, the resulting test has both good size and power properties. This is in contrast to standard cointegration tests, based on the unit root assumption, which are shown to severely over-reject as the data generating process deviates from a pure unit root setup. As an illustration of the method, two empirical applications are considered in Section 5. First, we re-examine the Fisher hypothesis and show that using the robust methods proposed in this paper, one can no longer …nd signi…cant support for a long-run equilibrium relationship between nominal interest rates and in ation; using standard unit root based cointegration tests on the other hand, the null hypothesis of no cointegration is rejected. In a second illustration, we consider the robustness of the long-run relationship between aggregate consumption, asset holdings, and labor income, which was initially studied by Lettau and Ludvigson (2001) and has since received a great deal of attention in the …nance literature. We …nd that after controlling for the unknown persistence in the variables, there is still strong evidence of cointegration between the three variables. Section 6 concludes and the Appendix contains tables of critical values for the test statistic.
2
2.1
Theoretical framework
Model and assumptions
1
Let fzt g0 be an m vector of nearly integrated processes, such that the data generating process satis…es zt = Azt
1
+ ut
(1)
where A = I + C/ T is an m
m matrix with A = diag (a1 ; :::; am ) and C = diag (c1 ; ::; cm ), and T is
the sample size. That is, each component process in zt is generated as a near unit root process with individual local-to-unity parameters ci , i = 1; :::; m. The initial conditions are set at t = 0 and z0 is assumed randomly distributed with …nite variance. Although none of the formal results depend upon it, we will work under the assumption that ci 0 for all i, which rules out explosive processes. The 3
innovations ut satisfy a general linear process. Assumption 1 1. ut = D (L) 2.
t
t
=
is iid with mean zero, variance matrix
P1
j=0
Dj
t j;
, and …nite fourth-order moment.
1=2
P1
0
j jjDj jj 0 and write from the following empirical regression:
We consider residual-based tests of the null of no cointegration using the regression residuals, vt , ^
yt =
0
xt + vt :
(2)
2.2
The test statistic
We focus on the traditional Augmented Engle-Granger t test (Engle and Granger, 1987) of the null of no cointegration, which is probably the most commonly used residual-based test of cointegration. Our analysis could easily be extended to cover the Z and Zt cointegration tests proposed by Phillips and Ouliaris (1990), but for brevity we restrict ourselves to the Augmented Engle-Granger test (henceforth denoted AEG test). The AEG test is de…ned as the t statistic for ^ from the regression vt = ^ vt ^
1+
wt . The below result follows from the results in Phillips and Ouliaris (1990) and the results for nearintegrated processes in Phillips (1987,1988). Theorem 1 Let the data generating process satisfy equation (1) for some given C = diag (c1 ; :::; cm ), and let Assumption 1 hold. Suppose that the autoregressive order in the AEG regression satis…es
Pp
i=1
'i vt i + ^
4
p ! 1 as T ! 1 such that p = o T 1=3 . Then, under the null of no cointegration, as T ! 1, AEG ) c1 Z
1 1=2 2 W J1 2;C
+
0
R1
0
R1
0
W J1 2;C dW1 2 W J1 2;C 2 1=2
(3)
where
W J1 2;C
(r) =
W J1;c1
(r)
Z
1 W0 W J1;c1 J2;C2
0
Z Z
1 W0 W J2;C2 J2;C2
1 W J2;C2 (r)
(4)
0
and W1 2 (r) = W1 (r)
Z
1 0 W1 W2
1 0 W2 W2
1
W2 (r) ;
(5)
0
0
W W are the L2 projection residuals of J1;c1 and W1 on the spaces spanned by J2;C2 and W2 respectively.
Remark 1.1 The limiting distribution of the AEG statistic depends on the unknown parameter C, but is otherwise free of nuisance parameters. For a given C, the asymptotic distribution can thus easily be tabulated. The next section describes a feasible implementation of the test when C is unknown. Remark 1.2 E¤ectively, the AEG test evaluates whether the persistence in the residuals is less than that predicted under the null hypothesis of no cointegration. However, since the original data is not necessarily a unit root process, the critical values re this fact. In the special case of C = 0, the ect limiting distribution reduces to the usual one for pure unit root variables. Remark 1.3 In empirical work, a constant or a constant and a linear trend are typically included in the empirical regression (2). As in standard cointegration analysis, this will a¤ect the limiting distribution in a straightforward manner (e.g. Phillips and Ouliaris, 1990) and thus the critical values used, but will otherwise not alter the analysis.
3
Feasible implementation
For a known C, the above test is trivial to use once critical values for the asymptotic distribution are obtained. Unfortunately, C is typically not known. We therefore consider a Bonferroni test approach, which is similar to that used by Cavanagh et al. (1995) and Campbell and Yogo (2006) in their pursuit of inference in predictive regression with near-integrated variables.
5
Consider con…dence intervals for ci , i = 1; :::; m, of the shape f[ci ; ci ]gi=1 with an overall coverage rate equal to 100 (1
1)
m
percent. Let fei 2 [ci ; ci ]gi=1 be the set of parameter values in this c percent level (e.g. …ve percent). If the AEG statistic is evaluated
2 2,
m
con…dence region for which the critical value of the asymptotic distribution of the AEG statistic is most conservative, for some given
2
using this conservative critical value, calculated at the cointegration test will be less than or equal to =
1
percent level, the size of the resulting by Bonferroni' inequality. s
+
However, relative to the Cavanagh et al. (1995) and Campbell and Yogo (2006) studies, there is an additional complication in the current setup. In those papers, there is only one local-to-unity process, whereas here there are at least two in the simplest case with just one regressor. In the univariate case, con…dence intervals of the local-to-unity parameter can be obtained by inverting a unit root test statistic (Stock, 1991). In the m dimensional case, a con…dence region for C could be obtained by inverting individual unit root test statistics in order to obtain con…dence intervals [ci ; ci ], i = 1; :::; m, each with coverage rate 1
1 =m.
The overall con…dence level of f[ci ; ci ]gi=1 is at least 100
m
(1
1)
percent, again by Bonferroni' inequality. Although theoretically sound, such an approach su¤ers from s the practical disadvantage that it would be virtually impossible to tabulate the critical values for the asymptotic distribution beyond the simple two-dimensional case. We therefore propose a simpler approach that allows for tabulation of critical values and seems to give up little in robustness. Intuitively, the AEG test evaluates whether the persistence, or autoregressive root, in the regression residuals, vt , is less than in the original data, yt . As seen in equations (3) and (4), the critical values of the test depend on both the persistence in the ' dependent'variable, yt , and the regressors, xt , denoted c1 and C2 respectively. However, it seems reasonable to conjecture that the main determinant of the ^ ^ asymptotic distribution will be c1 , rather than C2 . Thus, using C = C1 = diag (^1 ; :::; c1 ) for some c1 , c ^ ^ ~ rather than C = diag (~1 ; c2 ; :::; cm ), to form critical values might not cause a large size distortion in c ~ ~ the test. Although this conjecture is di cult to evaluate analytically, extensive simulation evidence supports it. For instance, Figure 1 shows the critical values for the AEG test in the two-dimensional case with an intercept in the empirical regression. As is evident, the primary changes come from changing c1 , whereas the critical values are almost constant across C2 . Additional evidence supporting this conclusion is provided by simulations in the following section. Furthermore, if C1 = diag (c1 ; :::; c1 ) is used to calculate the critical values for the asymptotic distribution in Theorem 1, the AEG cointegration test will be more conservative as the value of c1
6
decreases; that is, as c1 becomes more negative, so do the corresponding critical values, as shown in Table A3. Only the lower bound on c1 , say c1 , is therefore of interest in constructing a conservative test; for a given con…dence level, such a lower bound can be obtained from a one-sided con…dence interval for c1 , [c1 ; +1). By restricting the attention to the parameter c1 , and calculating critical values based on C 1 = diag (c1 ; :::; c1 ), it now becomes easy to implement the Bonferroni method. The lower con…dence bound for c1 , c1 , is obtained by inverting a unit root test statistic for the variable yt . Based on this lower bound of c1 , the test is evaluated using the corresponding critical value for C 1 = diag (c1 ; :::; c1 ). If the lower bound c1 has con…dence level 1 resulting test will have a size no larger than
1
and the AEG test is evaluated at the
1
2
level, the
=
+
3 2.
In general, Bonferroni' inequality is strict, and the size of the test will be less than . To obtain s a correctly sized test of size ~ , which is distinct from then …nd
1
=
1
+
2,
we …rst …x
2
at some level and
such that the resulting test has size ~ . Finding
2
1
is e¤ectively a trial and error exercise.
1
In the simulations below, we let ~ =
= 0:05 and show that setting
equal to 50 percent will
approximately result in an overall …ve percent test. Thus, by e¤ectively using a median unbiased estimate of c1 , an approximately correctly-sized test is obtained. These results are discussed more extensively in conjunction with the Monte Carlo simulations in the next section. In terms of practical implementation, we follow Campbell and Yogo (2006) and invert Elliot et al.' (1996) DF-GLS unit root test statistic to obtain a lower bound for c1 . Table A1 provides the s lower 95th, 75th, 50th, 25th, and 5th percent con…dence bounds of c1 , given a value of the DF-GLS test statistic.4 For instance, the lower con…dence bound that corresponds to 100 (1
1) % 1
= 0:05 is given in the
= 95% column. Table A2 provides the corresponding bounds when a trend is allowed
for in the DF-GLS regression. Table A3 tabulates the …ve percent critical values for the AEG statistic, for c1 = 0 to c1 = 60, assuming that c1 = c2 = ::: = cm ; values for one to …ve regressors are provided
for the cases of no intercept, intercept, and intercept and a linear trend in the empirical regression. Henceforth, we will refer to the cointegration test constructed in the manner above as the Bonferroni AEG test, with the additional speci…cation of the value of we let
3 Since
1
when necessary. Unless otherwise noted,
2
= 0:05.
C2 is assumed not to play an important role in the distribution of the test-statistic, the only uncertainty regarding the persistence of the data comes from uncertainty regarding c1 . The con…dence level of the lower bound C 1 is therefore 1 m 1 rather than 1 1 , as discussed above. 4 Note that, for instance, the two lower con…dence bounds at the 5 percent and 95 percent level provide a two-sided con…dence interval with con…dence level 90 percent.
7
4
4.1
Finite-sample properties
Size properties
We analyze the …nite-sample properties of the proposed test procedure through a series of Monte Carlo simulations. Starting with the size properties, it is assumed that the data generating process (DGP) is given by equation (1), with the innovations ut drawn from a multivariate normal distribution such that E [ut ] = 0 and E [ut u0 ] = I. The sample size is set to either T = 100 or 500 and the number of t regressors, n = m 1, is equal to either one or three. The regression
yt =
+
0
xt +
t
(6)
is estimated, which is a spurious regression given the above DGP, and the cointegration tests are applied to the …tted residuals, vt . Each simulated m dimensional time-series zt is thus partitioned ^ as zt = (yt ; x0 ) , as described previously. When all components in zt are ex-ante identical, i.e. have t the same persistence c, the …rst component series is set to yt and the remainder to xt . When ci varies between each series, we describe explicitly which series are set as yt and xt . All tests are performed at the …ve percent signi…cance level and are evaluated using the critical values given in Table A3. The results are based on 10; 000 repetitions. In the …rst round of simulations, we let the local-to-unity matrix for zt be given by C = diag (c; :::; c), so that all the series have identical persistence. The local-to-unity parameter c varies from 0 to 30.
0
Figure 2 shows the size properties for the traditional AEG cointegration test, which by de…nition is evaluated at c = 0, as a function of the local-to-unity parameter c. The nominal size of the test is …ve percent, and for c close to zero, the actual rejection rate is also close to …ve percent. However, as c decreases in value, the test starts over-rejecting and the rejection rates already approach ten percent for c = 5. The rejection rates become even larger and approach one as c becomes even smaller. It
should be stressed that this is not a small-sample bias, but a re ection of the inconsistency of the test when c < 0. Since the autoregressive root of the residual in equation (6) is less than one for c 1 + C2 =T . For most values of
, however, the test appears to exhibit good power properties and appears su ciently powerful that it would be a useful tool in many empirical applications, including those with relatively small sample sizes.
5
Empirical illustrations
To illustrate the empirical use of the Bonferroni AEG test, we next consider two applications where the variables in question are all fairly persistent, but not necessarily pure unit root processes. As a comparison to the robust methodology proposed in this paper, we will also conduct the traditional AEG test.
11
5.1
The Fisher hypothesis
It is well known that both nominal interest rates and in ation are fairly persistent in most countries. Accordingly, cointegration techniques have been a popular approach to test the Fisher hypothesis in more recent years; see, for example, Mishkin (1992), Wallace and Warner (1993), Evans and Lewis (1995), and Crowder and Ho¤man (1996). However, the assumption made in most of these studies of exact unit roots in both nominal interest rates and in ation can be questioned on both theoretical and empirical grounds.6 It is therefore worth re-interesting this issue using the Bonferroni AEG test. A common formulation of the Fisher hypothesis is that the m-period nominal interest rate (im ) is t
m related to the real interest rate (rt ) and in ation ( m t )
according to
m im = Et (rt ) + Et ( t
m t ):
(8)
Relying on the commonly made assumption of a constant or mean-reverting real interest rate, an empirical version of the Fisher hypothesis can be written as
im = t where the constant
+
m t
+ vt ;
(9)
has the interpretation of the (constant) equilibrium real interest rate, the error , in the most traditional interpretation,
term vt is assumed to be a stationary ARMA process and should be equal to unity.7
Monthly data on the short nominal interest rate – given by the three month treasury bill – and CPI in ation from January 1955 to October 2006 in the United States were provided by the Board of Governors of the Federal Reserve System. Table 1 shows the results from the DF-GLS unit root test and the KPSS stationarity test, as well as the median unbiased estimate of c, denoted c, and ^ a 90 percent con…dence interval for c; the estimates and con…dence intervals of c are derived using the values in Table A1 and linear interpolation.8 As can be seen, the evidence for a unit root in the interest rate appears reasonably strong; the DF-GLS test fails to reject the null of a unit root whereas the KPSS test rejects the null of stationarity. For in ation, on the other hand, the evidence is more
for example, Wu and Zhang (1996), Culver and Papell (1997), and Wu and Chen (2001). that in the estimations below, time t ination is given as future ination between t and t + m. This can be motivated by assuming rational expectations; see, for example, Mishkin (1992). 8 Regarding the speci…cation of deterministic terms in the unit root tests, it should be noted that we test for mean reversion around a constant level.
7 Note 6 See,
12
mixed since the DF-GLS test rejects a unit root but the KPSS test rejects stationarity.9 Table 1: Unit root tests. it t DF-GLS KPSS c ^ 90% CI for c -1.40 0.53 -3.40 [-9.06, 2.00] -2.54 0.52 -12.91 [-21.37, -3.46]
Notes: * indicates signi…cance at the …ve percent level.
The cointegration tests are conducted using a signi…cance level of …ve percent. For the Bonferroni AEG test, based on the simulation results in the previous section, we set
1
= 0:5; thus c = ^
3:40,
the median unbiased estimate for the nominal interest rate, is used to establish the critical value in the Bonferroni AEG test. The results from the cointegration tests based on the speci…cation in equation (9) are given in Table 2.10 Asymptotic critical values are used for both the standard Engle-Granger test (denoted AEG) and the Bonferroni AEG test (denoted AEGC ) and are provided in Table 2; the AEGC critical value is obtained from Table A3 and linear interpolation. Table 2: Cointegration tests. Test statistic Critical value AEGC Critical value AEG
Notes: Nominal size is 0.05.
-3.43 -3.47 -3.34
As can be seen, the null hypothesis of no cointegration is rejected if the standard method is used, as the test statistic is smaller than the critical value for the traditional AEG test. However, when the Bonferroni AEG test is used, the null hypothesis is not rejected. Thus, performing inference using robust methods, there is no strong evidence of cointegration, or co-movement, between the nominal interest rate and in ation in U.S. data. This raises doubts about the validity of the Fisher hypothesis, and also illustrates the importance of controlling for the unknown degree of persistence in the data; assuming unit roots in the data, the cointegration test would have resulted in evidence favorable of
9 Lag length in the DF-GLS test was determined using the Schwarz (1978) information criterion. For the KPSS test, a Newey-West estimator was employed to correct for serial correlation. 1 0 As in the DF-GLS test, lag length in the test equation is determined using the Schwarz (1978) criterion.
13
the Fisher hypothesis. Having looked at a traditional application from the macroeconomic literature, we next turn to a recent issue from …nancial economics.
5.2
Consumption, aggregate wealth and stock returns
Many studies argue that …nancial valuation ratios such as the dividend- and earnings-price ratios may have predictive power for excess stock returns over the risk-free rate. In a novel attempt to tie macroeconomic variables more closely to …nancial markets, Lettau and Ludvigson (2001) argue that consumption is a function of aggregate wealth. Based on this claim, they suggest that aggregate consumption (kt ), asset holdings (at ) and labour income (yt ) are cointegrated and that the deviation from equilibrium is useful in terms of predicting both excess stock returns and real stock returns. The empirical speci…cation used by Lettau and Ludvigson accordingly takes its starting point in a cointegrating relationship of the type
kt =
+ at + yt +
t;
(10)
where the error term future returns.
t
is assumed to be a stationary ARMA process which has predictive power for
However, there is no strong a priori reason to assume that the above variables contain pure unit roots.11 We therefore investigate the sensitivity of Lettau and Ludvigson' results when the uncertainty s regarding the persistence in the data is taken into account. Quarterly data on US consumption, asset holdings and labour income ranging from the …rst quarter 1952 to the fourth quarter 2006 were obtained from Professor Ludvigson' web page;12 all variables are given by the natural logarithm of real, per s capita data. Table 3 shows the results from unit root tests and stationarity tests for all variables and also provides the median unbiased estimates of c, c, as well as 90 percent con…dence intervals.13 The ^
1 1 As was shown above, the persistence of the dependent variable is of special importance when using the AEG test. The assumption of a unit root in consumption is thus of particular interest. Although this conjecture …nds some support – see, for example, Hall (1978) and Gali (1993) – the opinion in the litterature is far from unanimous. For instance, the vast literatue that uses linear trends to detrend consumption – see, for example, Cooper and Ejarque (2000) and Casares (2007) – implicitly or explicitly assumes that consumption is trend stationary rather than generated by a unit root process. Furthermore, it has been argued that consumption and output should be integrated of the same order. Thus, if output is trend stationary (e.g. Flavin, 1981 and Diebold and Senhadji, 1996) then consumption should be as well. 1 2 http://www.econ.nyu.edu/user/ludvigsons/ 1 3 Note that in this application, the unit root tests have both constant and trend included in the speci…cation. Thus, the estimates and con…dence intervals of c are derived using the values in Table A2; again, linear interpolation is used.
14
evidence for unit roots in consumption and labour income seems strong, whereas it is mixed for asset holdings. Table 3: Unit root tests. at -1.95 0.36 -4.06 [-12.28, 3.35] -2.54 0.20 -9.98 [-19.63, 2.32]
kt DF-GLS KPSS c ^ 90% CI for c
yt -0.78 0.38 2.32 [-2.18, 4.44]
Notes: * indicates signi…cance at the …ve percent level.
As in the previous application, we choose a signi…cance level of …ve percent for the cointegration tests and set
1
= 0:5. The results from the AEG and Bonferroni AEG cointegration tests are shown
in Table 4. The null hypothesis of no cointegration is rejected regardless of which test is used. The robust cointegration methods developed here thus support the conclusion of Lettau and Ludvigson that US consumption, asset holdings and labour income are cointegrated. Table 4: Cointegration tests. Test statistic Critical value AEGC Critical value AEG
Notes: Nominal size is 0.05.
-4.03 -3.86 -3.77
6
Conclusion
For many economic time series, it is di cult to justify theoretically that they are generated by unit root processes. This is problematic from an empirical point of view since cointegration tests may be misleading when the data follow near-integrated, rather than pure unit root, processes. The size distortions of cointegration tests relying on the unit root assumption – combined with the fact that standard OLS inference could lead to spurious results –makes it unclear how to analyze a multivariate time series of near-integrated variables.
15
In this paper, we have extended a standard residual-based cointegration test to allow for an unknown local deviation from the unit root assumption. This more robust test is easy to implement and Monte Carlo simulations show that it works well in …nite samples. Unlike standard cointegration tests, the methods developed in this paper thus provide a means of performing valid inference on a multivariate near unit root process. The framework suggested in this paper therefore takes another step towards addressing the problems associated with inference when variables are near-integrated. The methods presented here take their starting point in the work of Engle and Granger (1987). In future research it would also be of interest to see Johansen' (1988,1991) VAR-based framework extended to a setting s with near-integrated variables.
16
References
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[26] Schwarz, G., 1978. Estimating the Dimension of a Model, Annals of Statistics 6, 461– 464. [27] Shin, Y., 1994. A Residual-Based Test of the Null of Cointegration Against the Alternative of No Cointegration, Econometric Theory 10, 91-115. [28] Stock, J.H., 1991. Con…dence intervals for the largest autoregressive root in U.S. economic timeseries. Journal of Monetary Economics 28, 435-460. [29] Stock, J.H., 1997. Cointegration, Long-Run Comovements, and Long-Horizon Forecasting, in D. Kreps and K.F. Wallis (eds), Advances in Econometrics: Proceedings of the Seventh World Congress of the Econometric Society, vol. III. Cambridge: Cambridge University Press, 34-60. [30] Wallace, M. and J. Warner, 1993. The Fisher E¤ect and the Term Structure of Interest Rates: Test of Cointegration. Review of Economics and Statistics 75, 320-324. [31] Wright J.H., 2000. Con…dence Sets for Cointegrating Coe cients Based on Stationarity Tests, Journal of Business and Economic Statistics 18, 211-222. [32] Wu, J.-L. and S.-L. Chen, 2001. Mean Reversion of Interest Rates in the Eurocurrency Market, Oxford Bulletin of Economics and Statistics 63, 459-474. [33] Wu, Y. and H. Zhang, (1996). Mean Reversion in Interest Rates: New Evidence from a Panel of OECD Countries, Journal of Money, Credit and Banking 28, 604-621.
19
for c based on the DF-GLS statistic. For a given value of the DF-GLS statistic, without a time trend lower con…dence bounds of the local-to-unity parameter c with con…dence levels of 95; 75; 50; 25; and 5
20
Table A1: Lower con…dence bounds included, the following columns give percent, respectively. DF-GLS 95% 1.0 -0.29 0.9 -0.40 0.8 -0.50 0.7 -0.63 0.6 -0.76 0.5 -0.91 0.4 -1.07 0.3 -1.25 0.2 -1.46 0.1 -1.66 0.0 -1.89 -0.1 -2.14 -0.2 -2.41 -0.3 -2.72 -0.4 -3.05 -0.5 -3.45 -0.6 -3.84 -0.7 -4.31 -0.8 -4.87 -0.9 -5.44 -1.0 -6.04 -1.1 -6.73 -1.2 -7.45 -1.3 -8.19 -1.4 -9.04 -1.5 -9.90 -1.6 -10.82 -1.7 -11.75 -1.8 -12.78 -1.9 -13.84 75% 0.72 0.65 0.57 0.49 0.40 0.30 0.20 0.09 -0.03 -0.17 -0.31 -0.46 -0.63 -0.82 -1.03 -1.29 -1.60 -1.94 -2.32 -2.78 -3.27 -3.79 -4.37 -4.97 -5.66 -6.33 -7.05 -7.85 -8.65 -9.51 50% 1.47 1.41 1.35 1.29 1.23 1.15 1.07 0.99 0.90 0.80 0.70 0.59 0.48 0.34 0.18 -0.02 -0.23 -0.47 -0.75 -1.10 -1.47 -1.90 -2.35 -2.88 -3.40 -3.97 -4.60 -5.23 -5.94 -6.69 25% 2.39 2.34 2.29 2.24 2.19 2.13 2.07 2.02 1.94 1.87 1.79 1.71 1.62 1.53 1.42 1.30 1.15 0.98 0.78 0.54 0.28 -0.06 -0.39 -0.76 -1.18 -1.65 -2.15 -2.72 -3.27 -3.86 5% 4.23 4.19 4.15 4.12 4.08 4.04 4.00 3.95 3.90 3.85 3.80 3.75 3.69 3.63 3.57 3.51 3.40 3.28 3.17 3.06 2.91 2.71 2.49 2.29 2.01 1.74 1.38 1.03 0.59 0.22 DF-GLS -2.0 -2.1 -2.2 -2.3 -2.4 -2.5 -2.6 -2.7 -2.8 -2.9 -3.0 -3.1 -3.2 -3.3 -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -4.0 -4.1 -4.2 -4.3 -4.4 -4.5 -4.6 -4.7 -4.8 -4.9 95% -14.90 -15.95 -17.14 -18.34 -19.57 -20.84 -22.15 -23.53 -24.93 -26.34 -27.71 -29.27 -30.86 -32.44 -34.06 -35.78 -37.43 -39.09 -40.85 -42.69 -44.52 -46.35 -48.24 -50.14 -52.14 -53.96 -56.08 -58.20 -60.27 -62.38 75% -10.41 -11.37 -12.35 -13.38 -14.42 -15.48 -16.61 -17.78 -18.98 -20.20 -21.49 -22.81 -24.17 -25.53 -26.94 -28.39 -29.87 -31.44 -32.98 -34.55 -36.22 -37.87 -39.50 -41.27 -43.07 -44.86 -46.68 -48.54 -50.39 -52.31 50% -7.44 -8.25 -9.09 -9.97 -10.92 -11.89 -12.91 -13.95 -15.03 -16.13 -17.29 -18.45 -19.62 -20.87 -22.15 -23.49 -24.85 -26.24 -27.65 -29.11 -30.62 -32.17 -33.70 -35.31 -36.94 -38.58 -40.23 -41.95 -43.70 -45.50 25% -4.51 -5.21 -5.94 -6.74 -7.48 -8.32 -9.19 -10.06 -11.06 -12.06 -13.08 -14.12 -15.21 -16.33 -17.52 -18.70 -19.87 -21.16 -22.48 -23.82 -25.18 -26.55 -27.93 -29.44 -30.94 -32.45 -34.00 -35.67 -37.29 -38.90 5% -0.31 -0.88 -1.36 -1.98 -2.55 -3.28 -3.95 -4.69 -5.52 -6.28 -7.14 -7.97 -8.88 -9.84 -10.83 -11.80 -12.87 -13.96 -15.09 -16.25 -17.52 -18.71 -19.87 -21.22 -22.57 -23.89 -25.21 -26.59 -28.05 -29.53
21
Table A2: Lower con…dence bounds for c based on the DF-GLS statistic with a linear time trend included. For a given value of the DF-GLS statistic, allowing for a linear time trend, the following columns give lower con…dence bounds of the local-to-unity parameter c with con…dence levels of 95; 75; 50; 25; and 5 percent, respectively. DF-GLS 95% 75% 50% 25% 5% DF-GLS 95% 75% 50% 25% 5% 1.0 2.20 2.63 3.07 3.72 5.24 -2.0 -12.90 -8.15 -4.56 1.69 3.27 0.9 2.16 2.60 3.04 3.69 5.20 -2.1 -14.04 -9.15 -5.54 1.28 3.12 0.8 2.12 2.57 3.01 3.65 5.16 -2.2 -15.26 -10.17 -6.48 -2.10 2.95 0.7 2.09 2.53 2.97 3.62 5.13 -2.3 -16.52 -11.28 -7.47 -3.22 2.76 0.6 2.05 2.50 2.93 3.58 5.09 -2.4 -17.85 -12.40 -8.49 -4.26 2.58 0.5 2.02 2.46 2.90 3.55 5.05 -2.5 -19.14 -13.55 -9.59 -5.30 2.39 0.4 1.97 2.42 2.86 3.51 5.01 -2.6 -20.49 -14.77 -10.67 -6.33 2.19 0.3 1.93 2.38 2.82 3.47 4.97 -2.7 -21.97 -16.04 -11.80 -7.41 1.96 0.2 1.88 2.34 2.78 3.42 4.93 -2.8 -23.44 -17.35 -12.98 -8.47 1.61 0.1 1.83 2.30 2.74 3.38 4.88 -2.9 -24.97 -18.67 -14.20 -9.62 -1.55 0.0 1.78 2.26 2.70 3.33 4.84 -3.0 -26.55 -20.02 -15.47 -10.75 -3.10 -0.1 1.72 2.22 2.65 3.29 4.79 -3.1 -28.14 -21.48 -16.78 -11.91 -4.27 -0.2 1.64 2.17 2.61 3.24 4.75 -3.2 -29.86 -22.97 -18.10 -13.19 -5.55 -0.3 1.56 2.12 2.56 3.20 4.70 -3.3 -31.64 -24.49 -19.51 -14.48 -6.68 -0.4 1.47 2.07 2.52 3.15 4.64 -3.4 -33.42 -26.05 -20.96 -15.80 -7.91 -0.5 1.32 2.02 2.47 3.10 4.59 -3.5 -35.21 -27.67 -22.45 -17.15 -9.12 -0.6 -0.81 1.95 2.42 3.05 4.54 -3.6 -37.09 -29.37 -23.95 -18.53 -10.30 -0.7 -1.58 1.89 2.36 3.01 4.49 -3.7 -38.99 -31.09 -25.56 -19.93 -11.62 -0.8 -2.29 1.82 2.31 2.95 4.43 -3.8 -40.97 -32.85 -27.19 -21.42 -12.96 -0.9 -2.95 1.75 2.26 2.89 4.37 -3.9 -43.06 -34.64 -28.85 -22.97 -14.34 -1.0 -3.70 1.61 2.18 2.82 4.31 -4.0 -45.18 -36.50 -30.56 -24.57 -15.79 -1.1 -4.43 1.45 2.10 2.76 4.25 -4.1 -47.18 -38.45 -32.34 -26.18 -17.31 -1.2 -5.15 -0.72 2.03 2.69 4.17 -4.2 -49.36 -40.35 -34.13 -27.89 -18.77 -1.3 -6.01 -1.85 1.92 2.60 4.09 -4.3 -51.66 -42.37 -36.01 -29.56 -20.19 -1.4 -6.83 -2.75 1.80 2.52 4.01 -4.4 -53.91 -44.46 -37.90 -31.31 -21.83 -1.5 -7.74 -3.62 1.63 2.42 3.91 -4.5 -56.27 -46.60 -39.83 -33.15 -23.44 -1.6 -8.69 -4.46 1.36 2.31 3.81 -4.6 -58.74 -48.74 -41.89 -35.04 -25.00 -1.7 -9.67 -5.33 -1.56 2.19 3.69 -4.7 -61.20 -50.98 -43.94 -36.99 -26.66 -1.8 -10.65 -6.22 -2.69 2.06 3.56 -4.8 -63.78 -53.32 -46.07 -38.96 -28.52 -1.9 -11.76 -7.17 -3.64 1.89 3.42 -4.9 -66.25 -55.64 -48.29 -40.95 -30.28
22
Table A3: Five percent critical values for the AEG statistic. This table gives the critical values for the AEG statistic at the …ve percent level, for di¤erent values of c under the assumption that c1 = ::: = cm = c, and for one to …ve regressors. The …rst set of values provide the critical values when no intercept is included in the cointegrating regression. The second set provides the values when an intercept, but no time trend is included and the third set of values represent the case with both an intercept and a linear time trend. The values are based on 100; 000 repetitions with T = 1; 000. No constant Constant Constant and trend c 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0 -2.77 -3.30 -3.73 -4.09 -4.41 -3.34 -3.77 -4.10 -4.42 -4.72 -3.79 -4.14 -4.44 -4.73 -5.00 -1 -2.80 -3.32 -3.74 -4.09 -4.42 -3.37 -3.76 -4.12 -4.43 -4.73 -3.79 -4.14 -4.46 -4.72 -5.01 -2 -2.88 -3.35 -3.75 -4.10 -4.42 -3.40 -3.78 -4.12 -4.44 -4.73 -3.82 -4.16 -4.45 -4.74 -5.01 -3 -2.96 -3.39 -3.78 -4.12 -4.44 -3.45 -3.82 -4.15 -4.46 -4.75 -3.86 -4.18 -4.48 -4.76 -5.03 -4 -3.05 -3.45 -3.82 -4.15 -4.45 -3.50 -3.86 -4.17 -4.47 -4.76 -3.89 -4.21 -4.50 -4.78 -5.03 -5 -3.15 -3.51 -3.86 -4.17 -4.46 -3.56 -3.89 -4.21 -4.49 -4.77 -3.94 -4.24 -4.53 -4.79 -5.05 -6 -3.23 -3.58 -3.91 -4.21 -4.51 -3.62 -3.94 -4.24 -4.53 -4.80 -3.98 -4.27 -4.56 -4.81 -5.07 -7 -3.32 -3.64 -3.96 -4.26 -4.53 -3.68 -4.00 -4.28 -4.56 -4.81 -4.03 -4.32 -4.58 -4.85 -5.09 -8 -3.41 -3.72 -4.01 -4.30 -4.57 -3.75 -4.05 -4.33 -4.60 -4.86 -4.08 -4.36 -4.63 -4.89 -5.11 -9 -3.50 -3.79 -4.07 -4.34 -4.61 -3.82 -4.11 -4.37 -4.64 -4.89 -4.14 -4.41 -4.66 -4.91 -5.16 -10 -3.58 -3.86 -4.13 -4.40 -4.65 -3.89 -4.16 -4.43 -4.68 -4.92 -4.19 -4.44 -4.71 -4.93 -5.18 -11 -3.68 -3.93 -4.19 -4.45 -4.69 -3.97 -4.22 -4.47 -4.72 -4.95 -4.26 -4.51 -4.74 -4.98 -5.21 -12 -3.75 -4.01 -4.26 -4.50 -4.73 -4.03 -4.29 -4.52 -4.76 -4.99 -4.32 -4.56 -4.79 -5.02 -5.23 -13 -3.84 -4.08 -4.33 -4.56 -4.78 -4.10 -4.34 -4.58 -4.81 -5.03 -4.37 -4.60 -4.84 -5.06 -5.28 -14 -3.92 -4.16 -4.38 -4.60 -4.83 -4.18 -4.41 -4.64 -4.85 -5.07 -4.44 -4.66 -4.88 -5.09 -5.30 -15 -4.01 -4.23 -4.44 -4.66 -4.88 -4.25 -4.47 -4.69 -4.90 -5.11 -4.50 -4.71 -4.93 -5.14 -5.34 -16 -4.06 -4.30 -4.50 -4.72 -4.94 -4.30 -4.54 -4.74 -4.96 -5.16 -4.55 -4.78 -4.97 -5.19 -5.39 -17 -4.15 -4.37 -4.57 -4.78 -4.98 -4.38 -4.60 -4.80 -5.00 -5.20 -4.61 -4.82 -5.02 -5.23 -5.42 -18 -4.22 -4.43 -4.63 -4.83 -5.03 -4.44 -4.66 -4.85 -5.05 -5.25 -4.67 -4.88 -5.07 -5.27 -5.46 -19 -4.29 -4.50 -4.69 -4.89 -5.07 -4.50 -4.72 -4.90 -5.11 -5.29 -4.73 -4.94 -5.12 -5.32 -5.50 -20 -4.37 -4.57 -4.76 -4.94 -5.13 -4.58 -4.77 -4.97 -5.16 -5.34 -4.79 -4.99 -5.18 -5.37 -5.55 -21 -4.44 -4.62 -4.81 -5.00 -5.19 -4.65 -4.83 -5.01 -5.21 -5.39 -4.85 -5.04 -5.21 -5.41 -5.59 -22 -4.51 -4.69 -4.87 -5.04 -5.23 -4.70 -4.89 -5.07 -5.25 -5.42 -4.91 -5.08 -5.27 -5.45 -5.62 -23 -4.57 -4.76 -4.93 -5.11 -5.28 -4.77 -4.96 -5.13 -5.30 -5.47 -4.97 -5.15 -5.33 -5.49 -5.67 -24 -4.65 -4.81 -5.00 -5.16 -5.34 -4.83 -5.00 -5.19 -5.35 -5.52 -5.03 -5.19 -5.38 -5.54 -5.72 -25 -4.71 -4.88 -5.04 -5.22 -5.39 -4.89 -5.06 -5.23 -5.41 -5.57 -5.08 -5.25 -5.42 -5.59 -5.76 -26 -4.77 -4.93 -5.11 -5.27 -5.44 -4.95 -5.11 -5.29 -5.45 -5.62 -5.14 -5.29 -5.47 -5.64 -5.81 -27 -4.84 -5.00 -5.16 -5.32 -5.49 -5.01 -5.18 -5.34 -5.50 -5.67 -5.19 -5.36 -5.53 -5.69 -5.85 -28 -4.90 -5.06 -5.21 -5.38 -5.53 -5.08 -5.23 -5.39 -5.55 -5.70 -5.25 -5.41 -5.56 -5.72 -5.88 -29 -4.96 -5.12 -5.28 -5.43 -5.59 -5.13 -5.29 -5.45 -5.60 -5.76 -5.30 -5.47 -5.63 -5.77 -5.93 -30 -5.02 -5.18 -5.32 -5.47 -5.64 -5.19 -5.34 -5.49 -5.64 -5.81 -5.36 -5.52 -5.66 -5.82 -5.98
23
c -31 -32 -33 -34 -35 -36 -37 -38 -39 -40 -41 -42 -43 -44 -45 -46 -47 -48 -49 -50 -51 -52 -53 -54 -55 -56 -57 -58 -59 -60
1 -5.09 -5.14 -5.20 -5.26 -5.32 -5.38 -5.44 -5.50 -5.55 -5.60 -5.65 -5.71 -5.77 -5.81 -5.87 -5.92 -5.96 -6.01 -6.07 -6.13 -6.17 -6.22 -6.27 -6.31 -6.38 -6.41 -6.46 -6.51 -6.56 -6.60
Table A3: Critical values for the AEG statistic (continued). No constant Constant 2 3 4 5 1 2 3 4 5 1 -5.23 -5.39 -5.53 -5.69 -5.25 -5.39 -5.55 -5.70 -5.85 -5.41 -5.30 -5.44 -5.59 -5.74 -5.30 -5.46 -5.61 -5.75 -5.90 -5.47 -5.35 -5.49 -5.63 -5.79 -5.36 -5.51 -5.66 -5.80 -5.94 -5.52 -5.41 -5.54 -5.69 -5.83 -5.42 -5.56 -5.70 -5.85 -5.99 -5.57 -5.46 -5.60 -5.75 -5.89 -5.47 -5.61 -5.76 -5.90 -6.04 -5.62 -5.51 -5.65 -5.80 -5.94 -5.53 -5.66 -5.81 -5.95 -6.09 -5.67 -5.56 -5.71 -5.84 -5.98 -5.58 -5.71 -5.86 -5.99 -6.13 -5.72 -5.63 -5.76 -5.90 -6.03 -5.64 -5.77 -5.91 -6.04 -6.18 -5.79 -5.68 -5.81 -5.94 -6.07 -5.69 -5.82 -5.95 -6.09 -6.22 -5.84 -5.73 -5.86 -5.99 -6.12 -5.74 -5.87 -6.00 -6.13 -6.27 -5.88 -5.78 -5.91 -6.04 -6.17 -5.79 -5.92 -6.05 -6.18 -6.31 -5.93 -5.84 -5.96 -6.09 -6.22 -5.84 -5.97 -6.10 -6.22 -6.35 -5.99 -5.89 -6.01 -6.13 -6.26 -5.90 -6.02 -6.15 -6.27 -6.40 -6.04 -5.93 -6.06 -6.18 -6.30 -5.94 -6.06 -6.19 -6.32 -6.44 -6.08 -5.98 -6.11 -6.23 -6.36 -6.00 -6.11 -6.24 -6.36 -6.49 -6.13 -6.04 -6.16 -6.28 -6.41 -6.04 -6.17 -6.29 -6.41 -6.54 -6.18 -6.09 -6.20 -6.33 -6.44 -6.09 -6.22 -6.33 -6.45 -6.58 -6.22 -6.14 -6.26 -6.37 -6.49 -6.14 -6.26 -6.38 -6.49 -6.61 -6.27 -6.19 -6.30 -6.43 -6.54 -6.19 -6.32 -6.43 -6.55 -6.67 -6.32 -6.23 -6.36 -6.47 -6.59 -6.25 -6.36 -6.48 -6.59 -6.72 -6.37 -6.29 -6.40 -6.51 -6.63 -6.29 -6.41 -6.53 -6.63 -6.76 -6.42 -6.33 -6.44 -6.56 -6.67 -6.35 -6.45 -6.56 -6.68 -6.80 -6.47 -6.39 -6.49 -6.60 -6.71 -6.39 -6.50 -6.62 -6.72 -6.83 -6.51 -6.43 -6.54 -6.65 -6.76 -6.43 -6.55 -6.66 -6.76 -6.88 -6.55 -6.47 -6.58 -6.69 -6.80 -6.49 -6.59 -6.70 -6.81 -6.92 -6.60 -6.52 -6.62 -6.75 -6.85 -6.52 -6.64 -6.74 -6.86 -6.97 -6.64 -6.57 -6.68 -6.78 -6.89 -6.58 -6.68 -6.80 -6.89 -7.01 -6.69 -6.62 -6.72 -6.83 -6.93 -6.62 -6.73 -6.83 -6.94 -7.05 -6.74 -6.66 -6.77 -6.87 -6.98 -6.67 -6.77 -6.89 -6.98 -7.09 -6.78 -6.71 -6.82 -6.92 -7.02 -6.71 -6.82 -6.93 -7.03 -7.13 -6.82 Constant and 2 3 -5.57 -5.72 -5.62 -5.76 -5.67 -5.82 -5.72 -5.86 -5.76 -5.91 -5.82 -5.96 -5.87 -6.01 -5.92 -6.06 -5.97 -6.10 -6.01 -6.15 -6.06 -6.20 -6.11 -6.24 -6.16 -6.28 -6.20 -6.33 -6.25 -6.38 -6.30 -6.42 -6.35 -6.46 -6.40 -6.51 -6.44 -6.56 -6.48 -6.60 -6.54 -6.65 -6.58 -6.69 -6.63 -6.74 -6.67 -6.78 -6.71 -6.82 -6.76 -6.86 -6.80 -6.91 -6.85 -6.95 -6.88 -7.00 -6.93 -7.04 5 -6.02 -6.07 -6.11 -6.15 -6.20 -6.24 -6.28 -6.33 -6.36 -6.41 -6.46 -6.49 -6.54 -6.58 -6.63 -6.68 -6.71 -6.75 -6.80 -6.84 -6.89 -6.92 -6.95 -7.00 -7.04 -7.09 -7.13 -7.16 -7.21 -7.25 trend 4 -5.86 -5.92 -5.95 -6.01 -6.05 -6.10 -6.14 -6.19 -6.24 -6.28 -6.32 -6.37 -6.41 -6.45 -6.49 -6.55 -6.59 -6.63 -6.68 -6.72 -6.76 -6.81 -6.85 -6.89 -6.93 -6.98 -7.01 -7.06 -7.10 -7.15
3
Critical Value
3.5 4 4.5 5 5.5 0
5
10
15
20
25
30
30
25
20
15
10
5
0
c2
c1
30 25 20
3.6
3.8
3.4
4.4
4.8
4.6
4
2
15 10 5 0
5
10
15 c
1
4.2
20
25
5
c
30
Figure 1: Critical values at the ve percent level for the AEG test as a function of c1 and c2 . The top panel shows the surface describing the ve percent critical values of the AEG test, in the case of an intercept and one regressor, when c1 and c2 are non-identical. The bottom panel shows the corresponding contour plot. The values are based on 10, 000 repetitions with T = 1, 000.
1
0.9
0.8
0.7
0.6
Size
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15 c
20
25
30
T=100, n=1
T=100, n=3
T=500, n=1
T=500, n=3
Figure 2: Size properties of the Engle and Granger (1987) test of cointegration, as a function of the local-to-unity parameter c. The graph shows the average rejection rates under the null hypothesis of no cointegration for the Engle and Granger test of cointegration, i.e. the standard AEG test evaluated under the assumption that c = 0, for dierent true values of c. The sample size is equal to either T = 100 or 500, and the number of regressors equal to either n = 1 or 3. The true persistence in the data is equal to C = diag (c, ..., c), where c varies between 0 and 30. The results are based on 10, 000 repetitions.
T=100, n=1 0.1 0.08 0.06
Size
T=100, n=3 0.1 0.08 0.06
Size
0.04 0.02 0
0.04 0.02 0
0
5
10
15 c
20
25
30
0
5
10
15 c
20
25
30
T=500, n=1 0.1 0.08 0.06
Size
T=500, n=3 0.1 0.08 0.06
Size
0.04 0.02 0
0.04 0.02 0
0
5
10
15 c
20
25
30
0
5
10
15 c
20
25
30
α = 0.75
1
α = 0.50
1
α = 0.25
1
Figure 3: Size properties of the Bonferroni AEG test when the variables all have equal persistence. The graphs show the average rejection rates for the Bonferroni AEG test, under the null hypothesis of no cointegration, for α1 = 0.75, 0.50, and 0.25. The sample size is equal to either T = 100 or 500, and the number of regressors is equal to either n = 1 or 3. The true persistence in the data is equal to C = diag (c, ..., c), where c varies between 0 and 30. The results are based on 10, 000 repetitions.
c=5 0.12 0.15 0.1 0.1 0.08 0.06 0.05 0.04 0.02 0 20 0 25
c=10
15
10
5 c
0
5
10
20
15
10 c
5
0
5
c=20 0.08 0.06 0.05 0.06 0.04 0.04 0.03 0.02 0.02 0.01 0 50 0 60
c=30
40
30 c
20
10
0
50
40
30 c
20
10
0
75 Percent
50 Percent
25 Percent
Figure 4: Estimates of the lower bounds of c. The graphs show the density of the estimates of the lower bounds of c, with condence levels of 75, 50, and 25 percent, based on inversion of the DF-GLS statistic. The results are obtained from 10, 000 simulations of a univariate local-to-unity process, with local-to-unity parameter c, iid normal innovations and sample size T = 500.
T=100, n=2 0.12 0.1 0.08
Size
T=100, n=2 0.12 0.1 0.08
Size
0.06 0.04 0.02 0
0.06 0.04 0.02 0
0
5
10
15 c
20
25
30
0
5
10
15 c
20
25
30
T=500, n=3 0.12 0.1 0.08
Size
T=500, n=3 0.12 0.1 0.08
Size
0.06 0.04 0.02 0
0.06 0.04 0.02 0
0
5
10
15 c
20
25
30
0
5
10
15 c
20
25
30
α1 = 0.75
α1 = 0.50
α1 = 0.25
Figure 5: Size properties of the Bonferroni AEG test when ci is not identical for all i. The graphs show the average rejection rates for the Bonferroni AEG test, under the null hypothesis of no cointegration, for α1 = 0.75, 0.50, and 0.25. The sample size is equal to either T = 100 or 500, and the number of regressors is equal to either n = 2 or 3. For n = 2, the true persistence in the data is equal to C = diag (c1 , 10, 20), and for n = 3, C = diag (c1 , 0, 10, 20) , where c1 varies between 0 and 30. The results are based on 10, 000 repetitions.
Equal Persistence: T=100, n=1
Different Persistence: T=100, n=3
1
1
0.75
Power
0.75
Power
0.5 0.25
0.5 0.25
0 0.5
0.6
0.7
ρ
0.8
0.9
1
0 0.5
0.6
0.7
ρ
0.8
0.9
1
Equal Persistence: T=500, n=1
Different Persistence: T=500, n=3
1 0.75
Power
1 0.75
Power
0.5 0.25 c=2 c=10 c=20 0.9 ρ 1
0.5 0.25
0 0.8
0 0.8
0.9 ρ
1
Figure 6: Power properties of the Bonferroni AEG test. The graphs show the average rejection rates of the Bonferroni AEG test, for α1 = 0.50, under the alternative of cointegration. The power is plotted as a function of ρ, the AR (1) persistence parameter in the cointegrating residuals. The sample size is set equal to either T = 100 or 500. The left column gives results for the case of one regressor with persistence C2 = 2, 10, or 20. The right column gives the results for the case with three regressors and C2 = diag (0, 10, 20). The results are based on 10, 000 repetitions.