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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 907, October 2007--- Screen Reader
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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 finance literature.

Keywords: Bonferroni test, cointegration, near unit roots

JEL classification: C12, C22

Cointegration tests have been among the most important and influential 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 finite 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 definitive 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
different 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 different 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.

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
specified such that under the null hypothesis of a zero slope
coefficient, the dependent variable is a stationary process. Tests
on the slope coefficient therefore become joint tests of
cointegration as well, and the issue of spurious regressions never
occurs. Although this is a useful specification, 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 specific 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 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 find significant support for a long-run equilibrium relationship between nominal interest rates and inflation; 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 finance literature. We find 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.

Let be an vector of nearly integrated processes, such that the data generating process satisfies

(1) |

where is an matrix with and , and is the sample size. That is, each component process in is generated as a near unit root process with individual local-to-unity parameters , . The initial conditions are set at and is assumed randomly distributed with finite variance. Although none of the formal results depend upon it, we will work under the assumption that for all , which rules out explosive processes. The innovations satisfy a general linear process.

*2.
is iid with mean zero, variance
matrix
, and finite fourth-order
moment.*

By standard results, e.g. Phillips and Solo (1992), , where is a Brownian motion with covariance matrix . Partition such that is a scalar and is an vector . Let , , and be conformable partitions of , , and , respectively. We assume that and write . Denote an vector standard Brownian motion as , and it follows that . Further, as , . Partition conformably with and let

We consider residual-based tests of the null of no cointegration using the regression residuals, , from the following empirical regression:

(2) |

We focus on the traditional Augmented Engle-Granger 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
and
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 defined as the statistic for
from the regression
. The below result follows from the results in Phillips and
Ouliaris (1990) and the results for near-integrated processes in
Phillips (1987,1988).

(3) |

*where*

(4) |

*and*

(5) |

*are the
projection residuals of
and on
the spaces spanned by
and
respectively.*

For a known , the above test is trivial to use
once critical values for the asymptotic distribution are obtained.
Unfortunately, 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.

Consider confidence intervals for , , of the shape with an overall coverage rate equal to percent. Let be the set of parameter values in this confidence region for which the critical value of the asymptotic distribution of the statistic is most conservative, for some given percent level (e.g. five percent). If the statistic is evaluated using this conservative critical value, calculated at the percent level, the size of the resulting cointegration test will be less than or equal to , by Bonferroni's inequality.

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,
confidence intervals of the local-to-unity parameter can be
obtained by inverting a unit root test statistic (Stock, 1991). In
the dimensional case, a confidence region for
could be obtained by inverting individual
unit root test statistics in order to obtain confidence intervals
,
, each with coverage rate
. The overall confidence level
of
is at least
percent,
again by Bonferroni's inequality. Although theoretically sound,
such an approach suffers from 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,
, is less than in the original data,
. As seen in equations (3) and (4), the critical
values of the test depend on both the persistence in the
`dependent' variable, , and the regressors,
, denoted and
respectively. However, it seems
reasonable to conjecture that the main determinant of the
asymptotic distribution will be , rather than
. Thus, using
for some
, rather than
, to form critical values might not cause a large size distortion
in the test. Although this conjecture is difficult 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
, whereas the critical values are
almost constant across . Additional evidence
supporting this conclusion is provided by simulations in the
following section.

Furthermore, if is used to calculate the critical values for the asymptotic distribution in Theorem 1, the cointegration test will be more conservative as the value of decreases; that is, as becomes more negative, so do the corresponding critical values, as shown in Table A3. Only the lower bound on , say , is therefore of interest in constructing a conservative test; for a given confidence level, such a lower bound can be obtained from a one-sided confidence interval for , .

By restricting the attention to the parameter , and calculating critical values based on
, it now becomes easy to implement the Bonferroni method. The lower
confidence bound for ,
, is obtained by inverting
a unit root test statistic for the variable .
Based on this lower bound of , the test is
evaluated using the corresponding critical value for
. If the lower bound
has confidence level
and the *AEG* test is
evaluated at the
level, the resulting test will
have a size no larger than
.^{3}

In general, Bonferroni's inequality is strict, and the size of the test will be less than . To obtain a correctly sized test of size , which is distinct from , we first fix at some level and then find such that the resulting test has size . Finding is effectively a trial and error exercise. In the simulations below, we let and show that setting equal to percent will approximately result in an overall five percent test. Thus, by effectively using a median unbiased estimate of , 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*.'s (1996) DF-GLS unit
root test statistic to obtain a lower bound for . Table A1 provides the
lower 95th, 75th, 50th, 25th, and 5th percent confidence bounds of
, given a value of the DF-GLS test
statistic.^{4} For instance, the lower confidence
bound that corresponds to
is given in the
column. Table A2 provides the
corresponding bounds when a trend is allowed for in the DF-GLS
regression. Table A3 tabulates the
five percent critical values for the *AEG* statistic, for
to ,
assuming that
; values for one to
five 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 specification of the value of
when necessary. Unless otherwise
noted, we let
.

We analyze the finite-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 drawn from a multivariate normal distribution such that and . The sample size is set to either or and the number of regressors, , is equal to either one or three. The regression

(6) |

is estimated, which is a spurious regression given the above DGP, and the cointegration tests are applied to the fitted residuals, . Each simulated dimensional time-series is thus partitioned as , as described previously. When all components in are ex-ante identical, i.e. have the same persistence , the first component series is set to and the remainder to . When varies between each series, we describe explicitly which series are set as and . All tests are performed at the five percent significance level and are evaluated using the critical values given in Table A3. The results are based on 10,000 repetitions.

In the first round of simulations, we let the local-to-unity matrix for be given by , so that all the series have identical persistence. The local-to-unity parameter varies from 0 to -30.

Figure 2 shows the size properties for the traditional
cointegration test, which by definition
is evaluated at , as a function of the
local-to-unity parameter . The nominal size of the
test is five percent, and for close to zero, the
actual rejection rate is also close to five percent. However, as
decreases in value, the test starts
over-rejecting and the rejection rates already approach ten percent
for . The rejection rates become even larger
and approach one as becomes even smaller. It
should be stressed that this is not a small-sample bias, but a
reflection of the inconsistency of the test when . Since the autoregressive root of the residual in
equation (6)
is less than one for , the *AEG* test,
evaluated under the assumption of , will
reject the null of a unit root in the residuals more frequently
than its nominal size. For time series that do not necessarily have
a unit root, standard cointegration tests can thus be highly
misleading. This raises questions regarding previous studies that
have relied on cointegrating methods, despite having found evidence
of stationarity of the included variables; see, for example,
Crowder and Hoffman (1996).

We next consider the size properties of the Bonferroni
*AEG* test using a conservative estimate of .
As discussed in the previous section, we use
where
is the lower bound on the
persistence in . A direct application of the
Bonferroni method suggests choosing
such that the one-sided
confidence interval
has confidence level
percent, and then evaluating the *AEG* test-statistic at the
percent level for a total size of
percent. In
practice, however, such an approach will deliver extremely
conservative tests. For instance, if
, the rejection
rate for the resulting test is virtually identical to zero in the
simulations considered here. Instead, we follow the approach
outlined above and fix
and choose
such that the size of the overall
test is close to five percent. In particular, we consider setting
and . That is,
is chosen as the lower
bound in one-sided confidence intervals with confidence levels
equal to , , and percent, respectively. To obtain these values for
, the DF-GLS unit root
test-statistic is inverted, using the values in Table A1.^{5}

Figure 3 shows the results for the Bonferroni *AEG* test
using these different estimates of
. It is immediately
apparent that for small values of , the test
tends to over-reject when
, and under-reject when
. For
, the test still tends to
under-reject somewhat, except for small values of
in the case of and ,
where there is instead a slight over-rejection. Overall, however,
for
, the rejection rate is
typically between two and five percent. One could achieve rejection
rates that are somewhat closer to the nominal size by letting
vary with
in some manner, but at the
cost of a substantially more cumbersome procedure. Using a fixed
value of
, for all
values of
, yields a very simple test
to implement. The procedure would simply be given as:

*(i)*- Obtain the value of the
*AEG*test statistic from a standard implementation of the Engle and Granger test. *(ii)*- Calculate the DF-GLS unit root statistic for the variable and obtain the corresponding value of from Table A1 or A2.
*(iii)*- Compare the
*AEG*test statistic to the critical value corresponding to in Table A3.

It may seem surprising that using, for instance, a lower bound with only a 25 percent confidence level, does not result in a larger size distortion. Figure 4 helps shed some light on this puzzle. The results in the figure are based on simulations of a univariate local-to-unity process, with local-to-unity parameter , normal innovations and sample size . It shows estimates of the lower bounds of , with confidence levels of 25, 50, and 75 percent, using the inversion of the DF-GLS statistic in Table A1. The panels in Figure 4 show the densities for the lower-bounds estimates for and . As expected, the bounds estimates at the 75 percent confidence level are furthest to the left. However, the densities are far from symmetric, especially for close to zero; the density for the 25 percent confidence bound is also less symmetric than the density for the 75 percent bound. Thus, although the density is shifted further to the right as the confidence level decreases, which leads to estimates of closer to zero, the shift is not symmetric and the risk of vastly over-estimating is not increased dramatically. This explains, to some extent, why the rejection rates in the cointegration test only increase slowly as the confidence level of the lower bound is decreased.

In the last set of size simulations, shown in Figure 5, we
analyze the properties of the Bonferroni *AEG* test when the
local-to-unity parameters , are not identical; i.e. when the processes in
do not have the same persistence. Two
different cases are considered. In the first case, there are two
regressors with persistence parameters equal to and . In the second setup, there are
three regressors with persistence parameters , and . In both cases, it is assumed
that the persistence in , , varies between 0 and . Thus, in
the first case,
, and in
the second case
. The
same methods as in the case with identical
are used and the results for
and , are shown. Overall, the results in Figure 5 are very
similar to those in Figure 3. Using
and a nominal size of five
percent results in actual rejection rates around three percent.
Given the results shown previously in Figure 1, it is not
surprising that the test also performs well when the are not identical.

In summary, the proposed procedure for tests of cointegration in data with an unknown appears to work well in finite samples, once the confidence level of the lower bound is chosen appropriately. Additional fine tuning of this confidence level could be done to bring the actual size even closer to the nominal size, but at the cost of adding some complexity.

We next perform a second Monte Carlo simulation to evaluate the finite-sample rejection rates under the alternative of cointegration. The `independent' variable is still generated according to equation (1) using standard normal innovations. However, the `dependent' variable , is now generated as

(7) |

where is an
process with an
auto-regressive root ; the innovations to this
process are standard
normal. is set to an -vector
of ones. The same empirical regression, including the constant, as
in the size simulations is estimated, and the Bonferroni *AEG*
test with
is applied to the estimated
residuals
. The critical values that are
used are thus for the case with a constant in the regression. Two
different sample sizes, and , and and regressors,
are considered. In the case of one regressor, the persistence in
is set equal to either
or . In the
case of three regressors, it is assumed that
.

Figure 6 shows the results in four sub-plots corresponding to
the different combinations of sample size and number of regressors.
The vertical axes of the graphs show the power of the Bonferroni
*AEG* test plotted against the persistence in the error term . In the case
of , results for
are shown and
for the , results for
are shown. As is
to be expected, power is a monotone and declining function of the
persistence, . It should be noted that for very
large values of , we expect the test to have low
power; for example, in the bivariate case, a residual that is less
persistent than cannot be generated by
regressing on when
. For most values of
, however, the test appears to exhibit
good power properties and appears sufficiently powerful that it
would be a useful tool in many empirical applications, including
those with relatively small sample sizes.

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.

It is well known that both nominal interest rates and inflation
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
Hoffman (1996). However, the assumption made in most of these
studies of exact unit roots in both nominal interest rates and
inflation 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 -period nominal interest rate () is related to the real interest rate () and inflation ( ) according to

(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

(9) |

where the constant has the interpretation
of the (constant) equilibrium real interest rate, the error term
is assumed to be a stationary ARMA
process and , in the most traditional
interpretation, should be equal to unity.^{7}

Monthly data on the short nominal interest rate - given by the
three month treasury bill - and CPI inflation 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 ,
denoted , and a 90 percent confidence
interval for ; the estimates and confidence
intervals of 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 inflation, on the
other hand, the evidence is more mixed since the DF-GLS test
rejects a unit root but the KPSS test rejects stationarity.^{9}

Table 1: Unit root tests.

Unit root tests for the monthly short interest rate and CPI inflation in the U.S. The sample spans January 1955 to October 2006. Result for the DF-GLS and KPSS tests are presented, along with the median unbiased estimate of *c* and a 90 percent confidence interval for *c*.

Test | ||
---|---|---|

DF-GLS | -1.40 | -2.54 |

KPSS | 0.53 | 0.52 |

-3.40 | -12.91 | |

90% CI for | [-9.06, 2.00] | [-21.37, -3.46] |

Notes: * indicates significance at the five percent level.

The cointegration tests are conducted using a significance level
of five percent. For the Bonferroni *AEG* test, based on the
simulation results in the previous section, we set
; thus
, 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 specification in equation
(9) are given in
Table 2.^{10}Asymptotic critical values are used
for both the standard Engle-Granger test (denoted ) and the Bonferroni *AEG* test (denoted ) and are provided in Table 2; the
critical value is obtained from
Table A3 and
linear interpolation.

Table 2: Cointegration tests.

Cointegration tests between the short interest rate and the CPI inflation in monthly U.S. data spanning January 1955 to October 2006. The outcome of the augmented Engle-Granger test statistic along with the standard critical values and the Bonferroni critical values are presented.

Test statistic | -3.43 |
---|---|

Critical value | -3.47 |

Critical value | -3.34 |

Notes: Nominal size is 0.05.

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 inflation 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 the Fisher hypothesis. Having looked at a traditional
application from the macroeconomic literature, we next turn to a
recent issue from financial economics.

Many studies argue that financial 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 financial markets, Lettau and Ludvigson (2001) argue that consumption is a function of aggregate wealth. Based on this claim, they suggest that aggregate consumption (), asset holdings () and labour income () are cointegrated and that the deviation from equilibrium is useful in terms of predicting both excess stock returns and real stock returns. The empirical specification used by Lettau and Ludvigson accordingly takes its starting point in a cointegrating relationship of the type

(10) |

where the error term is assumed to be a stationary ARMA process which has predictive power for future returns.

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's
results when the uncertainty regarding the persistence in the data
is taken into account. Quarterly data on US consumption, asset
holdings and labour income ranging from the first quarter 1952 to
the fourth quarter 2006 were obtained from Professor Ludvigson's
web page;^{12} all variables are given by the
natural logarithm of real, per 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
, , as well as
90 percent confidence intervals.^{13} The evidence for unit
roots in consumption and labour income seems strong, whereas it is
mixed for asset holdings.

Table 3: Unit root tests.

Unit root tests for U.S. aggregate consumption, asset holdings and labour income. The data are on quarterly frequency and spans quarter 1, 1952, to quarter 4, 2006. Result for the DF-GLS and KPSS tests are presented, along with the median unbiased estimate of *c* and a 90 percent confidence interval for *c*.

Test | |||||
---|---|---|---|---|---|

DF-GLS | -1.95 | -2.54 | -0.78 | ||

KPSS | 0.36 | 0.20 | 0.38 | ||

-4.06 | -9.98 | 2.32 | |||

90% CI for | [-12.28, 3.35] | [-19.63, 2.32] | [-2.18, 4.44] |

Notes: * indicates significance at the five percent level.

As in the previous application, we choose a significance level
of five percent for the cointegration tests and set
. 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.

Cointegration tests between aggregate consumption, asset holdings and labour income in U.S. quarterly data spanning quarter 1, 1952, to quarter 4, 2006. The outcome of the augmented Engle-Granger test statistic along with the standard critical values and the Bonferroni critical values are presented.

Test statistic | -4.03 |
---|---|

Critical value | -3.86 |

Critical value | -3.77 |

Notes: Nominal size is 0.05.

For many economic time series, it is difficult 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.

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 finite 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's (1988,1991) VAR-based framework extended to a setting with near-integrated variables.

Campbell, J.Y. and M. Yogo, 2006. Efficient Tests of Stock
Return Predictability, *Journal of Financial Economics* 81,
27-60.

Casares, M., 2007. The New Keynesian Model and the Euro Area
Business Cycle, *Oxford Bulletin of Economics and Statistics*
69, 209-244.

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Table A1: Lower confidence bounds for *c* based on the DF-GLS statistic.

For a given value of the DF-GLS statistic, without a time trend included, the following columns give lower confidence bounds of the local-to-unity parameter *c* with confidence levels of 95, 75, 50, 25, and 5 percent, respectively.

DF-GLS | 95% | 75% | 50% | 25% | 5% |
---|---|---|---|---|---|

1.0 | -0.29 | 0.72 | 1.47 | 2.39 | 4.23 |

0.9 | -0.40 | 0.65 | 1.41 | 2.34 | 4.19 |

0.8 | -0.50 | 0.57 | 1.35 | 2.29 | 4.15 |

0.7 | -0.63 | 0.49 | 1.29 | 2.24 | 4.12 |

0.6 | -0.76 | 0.40 | 1.23 | 2.19 | 4.08 |

0.5 | -0.91 | 0.30 | 1.15 | 2.13 | 4.04 |

0.4 | -1.07 | 0.20 | 1.07 | 2.07 | 4.00 |

0.3 | -1.25 | 0.09 | 0.99 | 2.02 | 3.95 |

0.2 | -1.46 | -0.03 | 0.90 | 1.94 | 3.90 |

0.1 | -1.66 | -0.17 | 0.80 | 1.87 | 3.85 |

0.0 | -1.89 | -0.31 | 0.70 | 1.79 | 3.80 |

-0.1 | -2.14 | -0.46 | 0.59 | 1.71 | 3.75 |

-0.2 | -2.41 | -0.63 | 0.48 | 1.62 | 3.69 |

-0.3 | -2.72 | -0.82 | 0.34 | 1.53 | 3.63 |

-0.4 | -3.05 | -1.03 | 0.18 | 1.42 | 3.57 |

-0.5 | -3.45 | -1.29 | -0.02 | 1.30 | 3.51 |

-0.6 | -3.84 | -1.60 | -0.23 | 1.15 | 3.40 |

-0.7 | -4.31 | -1.94 | -0.47 | 0.98 | 3.28 |

-0.8 | -4.87 | -2.32 | -0.75 | 0.78 | 3.17 |

-0.9 | -5.44 | -2.78 | -1.10 | 0.54 | 3.06 |

-1.0 | -6.04 | -3.27 | -1.47 | 0.28 | 2.91 |

-1.1 | -6.73 | -3.79 | -1.90 | -0.06 | 2.71 |

-1.2 | -7.45 | -4.37 | -2.35 | -0.39 | 2.49 |

-1.3 | -8.19 | -4.97 | -2.88 | -0.76 | 2.29 |

-1.4 | -9.04 | -5.66 | -3.40 | -1.18 | 2.01 |

-1.5 | -9.90 | -6.33 | -3.97 | -1.65 | 1.74 |

-1.6 | -10.82 | -7.05 | -4.60 | -2.15 | 1.38 |

-1.7 | -11.75 | -7.85 | -5.23 | -2.72 | 1.03 |

-1.8 | -12.78 | -8.65 | -5.94 | -3.27 | 0.59 |

-1.9 | -13.84 | -9.51 | -6.69 | -3.86 | 0.22 |

-2.0 | -14.90 | -10.41 | -7.44 | -4.51 | -0.31 |

-2.1 | -15.95 | -11.37 | -8.25 | -5.21 | -0.88 |

-2.2 | -17.14 | -12.35 | -9.09 | -5.94 | -1.36 |

-2.3 | -18.34 | -13.38 | -9.97 | -6.74 | -1.98 |

-2.4 | -19.57 | -14.42 | -10.92 | -7.48 | -2.55 |

-2.5 | -20.84 | -15.48 | -11.89 | -8.32 | -3.28 |

-2.6 | -22.15 | -16.61 | -12.91 | -9.19 | -3.95 |

-2.7 | -23.53 | -17.78 | -13.95 | -10.06 | -4.69 |

-2.8 | -24.93 | -18.98 | -15.03 | -11.06 | -5.52 |

-2.9 | -26.34 | -20.20 | -16.13 | -12.06 | -6.28 |

-3.0 | -27.71 | -21.49 | -17.29 | -13.08 | -7.14 |

-3.1 | -29.27 | -22.81 | -18.45 | -14.12 | -7.97 |

-3.2 | -30.86 | -24.17 | -19.62 | -15.21 | -8.88 |

-3.3 | -32.44 | -25.53 | -20.87 | -16.33 | -9.84 |

-3.4 | -34.06 | -26.94 | -22.15 | -17.52 | -10.83 |

-3.5 | -35.78 | -28.39 | -23.49 | -18.70 | -11.80 |

-3.6 | -37.43 | -29.87 | -24.85 | -19.87 | -12.87 |

-3.7 | -39.09 | -31.44 | -26.24 | -21.16 | -13.96 |

-3.8 | -40.85 | -32.98 | -27.65 | -22.48 | -15.09 |

-3.9 | -42.69 | -34.55 | -29.11 | -23.82 | -16.25 |

-4.0 | -44.52 | -36.22 | -30.62 | -25.18 | -17.52 |

-4.1 | -46.35 | -37.87 | -32.17 | -26.55 | -18.71 |

-4.2 | -48.24 | -39.50 | -33.70 | -27.93 | -19.87 |

-4.3 | -50.14 | -41.27 | -35.31 | -29.44 | -21.22 |

-4.4 | -52.14 | -43.07 | -36.94 | -30.94 | -22.57 |

-4.5 | -53.96 | -44.86 | -38.58 | -32.45 | -23.89 |

-4.6 | -56.08 | -46.68 | -40.23 | -34.00 | -25.21 |

-4.7 | -58.20 | -48.54 | -41.95 | -35.67 | -26.59 |

-4.8 | -60.27 | -50.39 | -43.70 | -37.29 | -28.05 |

-4.9 | -62.38 | -52.31 | -45.50 | -38.90 | -29.53 |

Table A2: Lower confidence 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 linear time trend, the following columns give lower confidence bounds of the local-to-unity parameter *c* with confidence levels of 95, 75, 50, 25, and 5 percent, respectively.

DF-GLS | 95% | 75% | 50% | 25% | 5% |
---|---|---|---|---|---|

1.0 | 2.20 | 2.63 | 3.07 | 3.72 | 5.24 |

0.9 | 2.16 | 2.60 | 3.04 | 3.69 | 5.20 |

0.8 | 2.12 | 2.57 | 3.01 | 3.65 | 5.16 |

0.7 | 2.09 | 2.53 | 2.97 | 3.62 | 5.13 |

0.6 | 2.05 | 2.50 | 2.93 | 3.58 | 5.09 |

0.5 | 2.02 | 2.46 | 2.90 | 3.55 | 5.05 |

0.4 | 1.97 | 2.42 | 2.86 | 3.51 | 5.01 |

0.3 | 1.93 | 2.38 | 2.82 | 3.47 | 4.97 |

0.2 | 1.88 | 2.34 | 2.78 | 3.42 | 4.93 |

0.1 | 1.83 | 2.30 | 2.74 | 3.38 | 4.88 |

0.0 | 1.78 | 2.26 | 2.70 | 3.33 | 4.84 |

-0.1 | 1.72 | 2.22 | 2.65 | 3.29 | 4.79 |

-0.2 | 1.64 | 2.17 | 2.61 | 3.24 | 4.75 |

-0.3 | 1.56 | 2.12 | 2.56 | 3.20 | 4.70 |

-0.4 | 1.47 | 2.07 | 2.52 | 3.15 | 4.64 |

-0.5 | 1.32 | 2.02 | 2.47 | 3.10 | 4.59 |

-0.6 | -0.81 | 1.95 | 2.42 | 3.05 | 4.54 |

-0.7 | -1.58 | 1.89 | 2.36 | 3.01 | 4.49 |

-0.8 | -2.29 | 1.82 | 2.31 | 2.95 | 4.43 |

-0.9 | -2.95 | 1.75 | 2.26 | 2.89 | 4.37 |

-1.0 | -3.70 | 1.61 | 2.18 | 2.82 | 4.31 |

-1.1 | -4.43 | 1.45 | 2.10 | 2.76 | 4.25 |

-1.2 | -5.15 | -0.72 | 2.03 | 2.69 | 4.17 |

-1.3 | -6.01 | -1.85 | 1.92 | 2.60 | 4.09 |

-1.4 | -6.83 | -2.75 | 1.80 | 2.52 | 4.01 |

-1.5 | -7.74 | -3.62 | 1.63 | 2.42 | 3.91 |

-1.6 | -8.69 | -4.46 | 1.36 | 2.31 | 3.81 |

-1.7 | -9.67 | -5.33 | -1.56 | 2.19 | 3.69 |

-1.8 | -10.65 | -6.22 | -2.69 | 2.06 | 3.56 |

-1.9 | -11.76 | -7.17 | -3.64 | 1.89 | 3.42 |

-2.0 | -12.90 | -8.15 | -4.56 | 1.69 | 3.27 |

-2.1 | -14.04 | -9.15 | -5.54 | 1.28 | 3.12 |

-2.2 | -15.26 | -10.17 | -6.48 | -2.10 | 2.95 |

-2.3 | -16.52 | -11.28 | -7.47 | -3.22 | 2.76 |

-2.4 | -17.85 | -12.40 | -8.49 | -4.26 | 2.58 |

-2.5 | -19.14 | -13.55 | -9.59 | -5.30 | 2.39 |

-2.6 | -20.49 | -14.77 | -10.67 | -6.33 | 2.19 |

-2.7 | -21.97 | -16.04 | -11.80 | -7.41 | 1.96 |

-2.8 | -23.44 | -17.35 | -12.98 | -8.47 | 1.61 |

-2.9 | -24.97 | -18.67 | -14.20 | -9.62 | -1.55 |

-3.0 | -26.55 | -20.02 | -15.47 | -10.75 | -3.10 |

-3.1 | -28.14 | -21.48 | -16.78 | -11.91 | -4.27 |

-3.2 | -29.86 | -22.97 | -18.10 | -13.19 | -5.55 |

-3.3 | -31.64 | -24.49 | -19.51 | -14.48 | -6.68 |

-3.4 | -33.42 | -26.05 | -20.96 | -15.80 | -7.91 |

-3.5 | -35.21 | -27.67 | -22.45 | -17.15 | -9.12 |

-3.6 | -37.09 | -29.37 | -23.95 | -18.53 | -10.30 |

-3.7 | -38.99 | -31.09 | -25.56 | -19.93 | -11.62 |

-3.8 | -40.97 | -32.85 | -27.19 | -21.42 | -12.96 |

-3.9 | -43.06 | -34.64 | -28.85 | -22.97 | -14.34 |

-4.0 | -45.18 | -36.50 | -30.56 | -24.57 | -15.79 |

-4.1 | -47.18 | -38.45 | -32.34 | -26.18 | -17.31 |

-4.2 | -49.36 | -40.35 | -34.13 | -27.89 | -18.77 |

-4.3 | -51.66 | -42.37 | -36.01 | -29.56 | -20.19 |

-4.4 | -53.91 | -44.46 | -37.90 | -31.31 | -21.83 |

-4.5 | -56.27 | -46.60 | -39.83 | -33.15 | -23.44 |

-4.6 | -58.74 | -48.74 | -41.89 | -35.04 | -25.00 |

-4.7 | -61.20 | -50.98 | -43.94 | -36.99 | -26.66 |

-4.8 | -63.78 | -53.32 | -46.07 | -38.96 | -28.52 |

-4.9 | -66.25 | -55.64 | -48.29 | -40.95 | -30.28 |

Table A3 - Panel 1: No constant: Five percent critical values for the *AEG* statistic

This table gives the critical values for the *AEG* statistic at the five percent level, for different values of *c* under the assumption that *c*_{1} = ... = *c _{m}* =

c | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

0 | -2.77 | -3.30 | -3.73 | -4.09 | -4.41 |

-1 | -2.80 | -3.32 | -3.74 | -4.09 | -4.42 |

-2 | -2.88 | -3.35 | -3.75 | -4.10 | -4.42 |

-3 | -2.96 | -3.39 | -3.78 | -4.12 | -4.44 |

-4 | -3.05 | -3.45 | -3.82 | -4.15 | -4.45 |

-5 | -3.15 | -3.51 | -3.86 | -4.17 | -4.46 |

-6 | -3.23 | -3.58 | -3.91 | -4.21 | -4.51 |

-7 | -3.32 | -3.64 | -3.96 | -4.26 | -4.53 |

-8 | -3.41 | -3.72 | -4.01 | -4.30 | -4.57 |

-9 | -3.50 | -3.79 | -4.07 | -4.34 | -4.61 |

-10 | -3.58 | -3.86 | -4.13 | -4.40 | -4.65 |

-11 | -3.68 | -3.93 | -4.19 | -4.45 | -4.69 |

-12 | -3.75 | -4.01 | -4.26 | -4.50 | -4.73 |

-13 | -3.84 | -4.08 | -4.33 | -4.56 | -4.78 |

-14 | -3.92 | -4.16 | -4.38 | -4.60 | -4.83 |

-15 | -4.01 | -4.23 | -4.44 | -4.66 | -4.88 |

-16 | -4.06 | -4.30 | -4.50 | -4.72 | -4.94 |

-17 | -4.15 | -4.37 | -4.57 | -4.78 | -4.98 |

-18 | -4.22 | -4.43 | -4.63 | -4.83 | -5.03 |

-19 | -4.29 | -4.50 | -4.69 | -4.89 | -5.07 |

-20 | -4.37 | -4.57 | -4.76 | -4.94 | -5.13 |

-21 | -4.44 | -4.62 | -4.81 | -5.00 | -5.19 |

-22 | -4.51 | -4.69 | -4.87 | -5.04 | -5.23 |

-23 | -4.57 | -4.76 | -4.93 | -5.11 | -5.28 |

-24 | -4.65 | -4.81 | -5.00 | -5.16 | -5.34 |

-25 | -4.71 | -4.88 | -5.04 | -5.22 | -5.39 |

-26 | -4.77 | -4.93 | -5.11 | -5.27 | -5.44 |

-27 | -4.84 | -5.00 | -5.16 | -5.32 | -5.49 |

-28 | -4.90 | -5.06 | -5.21 | -5.38 | -5.53 |

-29 | -4.96 | -5.12 | -5.28 | -5.43 | -5.59 |

-30 | -5.02 | -5.18 | -5.32 | -5.47 | -5.64 |

-31 | -5.09 | -5.23 | -5.39 | -5.53 | -5.69 |

-32 | -5.14 | -5.30 | -5.44 | -5.59 | -5.74 |

-33 | -5.20 | -5.35 | -5.49 | -5.63 | -5.79 |

-34 | -5.26 | -5.41 | -5.54 | -5.69 | -5.83 |

-35 | -5.32 | -5.46 | -5.60 | -5.75 | -5.89 |

-36 | -5.38 | -5.51 | -5.65 | -5.80 | -5.94 |

-37 | -5.44 | -5.56 | -5.71 | -5.84 | -5.98 |

-38 | -5.50 | -5.63 | -5.76 | -5.90 | -6.03 |

-39 | -5.55 | -5.68 | -5.81 | -5.94 | -6.07 |

-40 | -5.60 | -5.73 | -5.86 | -5.99 | -6.12 |

-41 | -5.65 | -5.78 | -5.91 | -6.04 | -6.17 |

-42 | -5.71 | -5.84 | -5.96 | -6.09 | -6.22 |

-43 | -5.77 | -5.89 | -6.01 | -6.13 | -6.26 |

-44 | -5.81 | -5.93 | -6.06 | -6.18 | -6.30 |

-45 | -5.87 | -5.98 | -6.11 | -6.23 | -6.36 |

-46 | -5.92 | -6.04 | -6.16 | -6.28 | -6.41 |

-47 | -5.96 | -6.09 | -6.20 | -6.33 | -6.44 |

-48 | -6.01 | -6.14 | -6.26 | -6.37 | -6.49 |

-49 | -6.07 | -6.19 | -6.30 | -6.43 | -6.54 |

-50 | -6.13 | -6.23 | -6.36 | -6.47 | -6.59 |

-51 | -6.17 | -6.29 | -6.40 | -6.51 | -6.63 |

-52 | -6.22 | -6.33 | -6.44 | -6.56 | -6.67 |

-53 | -6.27 | -6.39 | -6.49 | -6.60 | -6.71 |

-54 | -6.31 | -6.43 | -6.54 | -6.65 | -6.76 |

-55 | -6.38 | -6.47 | -6.58 | -6.69 | -6.80 |

-56 | -6.41 | -6.52 | -6.62 | -6.75 | -6.85 |

-57 | -6.46 | -6.57 | -6.68 | -6.78 | -6.89 |

-58 | -6.51 | -6.62 | -6.72 | -6.83 | -6.93 |

-59 | -6.56 | -6.66 | -6.77 | -6.87 | -6.98 |

-60 | -6.60 | -6.71 | -6.82 | -6.92 | -7.02 |

Table A3 - Panel 2: Constant: Five percent critical values for the *AEG* statistic

c | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

0 | -3.34 | -3.77 | -4.10 | -4.42 | -4.72 |

-1 | -3.37 | -3.76 | -4.12 | -4.43 | -4.73 |

-2 | -3.40 | -3.78 | -4.12 | -4.44 | -4.73 |

-3 | -3.45 | -3.82 | -4.15 | -4.46 | -4.75 |

-4 | -3.50 | -3.86 | -4.17 | -4.47 | -4.76 |

-5 | -3.56 | -3.89 | -4.21 | -4.49 | -4.77 |

-6 | -3.62 | -3.94 | -4.24 | -4.53 | -4.80 |

-7 | -3.68 | -4.00 | -4.28 | -4.56 | -4.81 |

-8 | -3.75 | -4.05 | -4.33 | -4.60 | -4.86 |

-9 | -3.82 | -4.11 | -4.37 | -4.64 | -4.89 |

-10 | -3.89 | -4.16 | -4.43 | -4.68 | -4.92 |

-11 | -3.97 | -4.22 | -4.47 | -4.72 | -4.95 |

-12 | -4.03 | -4.29 | -4.52 | -4.76 | -4.99 |

-13 | -4.10 | -4.34 | -4.58 | -4.81 | -5.03 |

-14 | -4.18 | -4.41 | -4.64 | -4.85 | -5.07 |

-15 | -4.25 | -4.47 | -4.69 | -4.90 | -5.11 |

-16 | -4.30 | -4.54 | -4.74 | -4.96 | -5.16 |

-17 | -4.38 | -4.60 | -4.80 | -5.00 | -5.20 |

-18 | -4.44 | -4.66 | -4.85 | -5.05 | -5.25 |

-19 | -4.50 | -4.72 | -4.90 | -5.11 | -5.29 |

-20 | -4.58 | -4.77 | -4.97 | -5.16 | -5.34 |

-21 | -4.65 | -4.83 | -5.01 | -5.21 | -5.39 |

-22 | -4.70 | -4.89 | -5.07 | -5.25 | -5.42 |

-23 | -4.77 | -4.96 | -5.13 | -5.30 | -5.47 |

-24 | -4.83 | -5.00 | -5.19 | -5.35 | -5.52 |

-25 | -4.89 | -5.06 | -5.23 | -5.41 | -5.57 |

-26 | -4.95 | -5.11 | -5.29 | -5.45 | -5.62 |

-27 | -5.01 | -5.18 | -5.34 | -5.50 | -5.67 |

-28 | -5.08 | -5.23 | -5.39 | -5.55 | -5.70 |

-29 | -5.13 | -5.29 | -5.45 | -5.60 | -5.76 |

-30 | -5.19 | -5.34 | -5.49 | -5.64 | -5.81 |

-31 | -5.25 | -5.39 | -5.55 | -5.70 | -5.85 |

-32 | -5.30 | -5.46 | -5.61 | -5.75 | -5.90 |

-33 | -5.36 | -5.51 | -5.66 | -5.80 | -5.94 |

-34 | -5.42 | -5.56 | -5.70 | -5.85 | -5.99 |

-35 | -5.47 | -5.61 | -5.76 | -5.90 | -6.04 |

-36 | -5.53 | -5.66 | -5.81 | -5.95 | -6.09 |

-37 | -5.58 | -5.71 | -5.86 | -5.99 | -6.13 |

-38 | -5.64 | -5.77 | -5.91 | -6.04 | -6.18 |

-39 | -5.69 | -5.82 | -5.95 | -6.09 | -6.22 |

-40 | -5.74 | -5.87 | -6.00 | -6.13 | -6.27 |

-41 | -5.79 | -5.92 | -6.05 | -6.18 | -6.31 |

-42 | -5.84 | -5.97 | -6.10 | -6.22 | -6.35 |

-43 | -5.90 | -6.02 | -6.15 | -6.27 | -6.40 |

-44 | -5.94 | -6.06 | -6.19 | -6.32 | -6.44 |

-45 | -6.00 | -6.11 | -6.24 | -6.36 | -6.49 |

-46 | -6.04 | -6.17 | -6.29 | -6.41 | -6.54 |

-47 | -6.09 | -6.22 | -6.33 | -6.45 | -6.58 |

-48 | -6.14 | -6.26 | -6.38 | -6.49 | -6.61 |

-49 | -6.19 | -6.32 | -6.43 | -6.55 | -6.67 |

-50 | -6.25 | -6.36 | -6.48 | -6.59 | -6.72 |

-51 | -6.29 | -6.41 | -6.53 | -6.63 | -6.76 |

-52 | -6.35 | -6.45 | -6.56 | -6.68 | -6.80 |

-53 | -6.39 | -6.50 | -6.62 | -6.72 | -6.83 |

-54 | -6.43 | -6.55 | -6.66 | -6.76 | -6.88 |

-55 | -6.49 | -6.59 | -6.70 | -6.81 | -6.92 |

-56 | -6.52 | -6.64 | -6.74 | -6.86 | -6.97 |

-57 | -6.58 | -6.68 | -6.80 | -6.89 | -7.01 |

-58 | -6.62 | -6.73 | -6.83 | -6.94 | -7.05 |

-59 | -6.67 | -6.77 | -6.89 | -6.98 | -7.09 |

-60 | -6.71 | -6.82 | -6.93 | -7.03 | -7.13 |

Table A3 - Panel 3: Constant and trend: Five percent critical values for the *AEG* statistic

c | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

0 | -3.79 | -4.14 | -4.44 | -4.73 | -5.00 |

-1 | -3.79 | -4.14 | -4.46 | -4.72 | -5.01 |

-2 | -3.82 | -4.16 | -4.45 | -4.74 | -5.01 |

-3 | -3.86 | -4.18 | -4.48 | -4.76 | -5.03 |

-4 | -3.89 | -4.21 | -4.50 | -4.78 | -5.03 |

-5 | -3.94 | -4.24 | -4.53 | -4.79 | -5.05 |

-6 | -3.98 | -4.27 | -4.56 | -4.81 | -5.07 |

-7 | -4.03 | -4.32 | -4.58 | -4.85 | -5.09 |

-8 | -4.08 | -4.36 | -4.63 | -4.89 | -5.11 |

-9 | -4.14 | -4.41 | -4.66 | -4.91 | -5.16 |

-10 | -4.19 | -4.44 | -4.71 | -4.93 | -5.18 |

-11 | -4.26 | -4.51 | -4.74 | -4.98 | -5.21 |

-12 | -4.32 | -4.56 | -4.79 | -5.02 | -5.23 |

-13 | -4.37 | -4.60 | -4.84 | -5.06 | -5.28 |

-14 | -4.44 | -4.66 | -4.88 | -5.09 | -5.30 |

-15 | -4.50 | -4.71 | -4.93 | -5.14 | -5.34 |

-16 | -4.55 | -4.78 | -4.97 | -5.19 | -5.39 |

-17 | -4.61 | -4.82 | -5.02 | -5.23 | -5.42 |

-18 | -4.67 | -4.88 | -5.07 | -5.27 | -5.46 |

-19 | -4.73 | -4.94 | -5.12 | -5.32 | -5.50 |

-20 | -4.79 | -4.99 | -5.18 | -5.37 | -5.55 |

-21 | -4.85 | -5.04 | -5.21 | -5.41 | -5.59 |

-22 | -4.91 | -5.08 | -5.27 | -5.45 | -5.62 |

-23 | -4.97 | -5.15 | -5.33 | -5.49 | -5.67 |

-24 | -5.03 | -5.19 | -5.38 | -5.54 | -5.72 |

-25 | -5.08 | -5.25 | -5.42 | -5.59 | -5.76 |

-26 | -5.14 | -5.29 | -5.47 | -5.64 | -5.81 |

-27 | -5.19 | -5.36 | -5.53 | -5.69 | -5.85 |

-28 | -5.25 | -5.41 | -5.56 | -5.72 | -5.88 |

-29 | -5.30 | -5.47 | -5.63 | -5.77 | -5.93 |

-30 | -5.36 | -5.52 | -5.66 | -5.82 | -5.98 |

-31 | -5.41 | -5.57 | -5.72 | -5.86 | -6.02 |

-32 | -5.47 | -5.62 | -5.76 | -5.92 | -6.07 |

-33 | -5.52 | -5.67 | -5.82 | -5.95 | -6.11 |

-34 | -5.57 | -5.72 | -5.86 | -6.01 | -6.15 |

-35 | -5.62 | -5.76 | -5.91 | -6.05 | -6.20 |

-36 | -5.67 | -5.82 | -5.96 | -6.10 | -6.24 |

-37 | -5.72 | -5.87 | -6.01 | -6.14 | -6.28 |

-38 | -5.79 | -5.92 | -6.06 | -6.19 | -6.33 |

-39 | -5.84 | -5.97 | -6.10 | -6.24 | -6.36 |

-40 | -5.88 | -6.01 | -6.15 | -6.28 | -6.41 |

-41 | -5.93 | -6.06 | -6.20 | -6.32 | -6.46 |

-42 | -5.99 | -6.11 | -6.24 | -6.37 | -6.49 |

-43 | -6.04 | -6.16 | -6.28 | -6.41 | -6.54 |

-44 | -6.08 | -6.20 | -6.33 | -6.45 | -6.58 |

-45 | -6.13 | -6.25 | -6.38 | -6.49 | -6.63 |

-46 | -6.18 | -6.30 | -6.42 | -6.55 | -6.68 |

-47 | -6.22 | -6.35 | -6.46 | -6.59 | -6.71 |

-48 | -6.27 | -6.40 | -6.51 | -6.63 | -6.75 |

-49 | -6.32 | -6.44 | -6.56 | -6.68 | -6.80 |

-50 | -6.37 | -6.48 | -6.60 | -6.72 | -6.84 |

-51 | -6.42 | -6.54 | -6.65 | -6.76 | -6.89 |

-52 | -6.47 | -6.58 | -6.69 | -6.81 | -6.92 |

-53 | -6.51 | -6.63 | -6.74 | -6.85 | -6.95 |

-54 | -6.55 | -6.67 | -6.78 | -6.89 | -7.00 |

-55 | -6.60 | -6.71 | -6.82 | -6.93 | -7.04 |

-56 | -6.64 | -6.76 | -6.86 | -6.98 | -7.09 |

-57 | -6.69 | -6.80 | -6.91 | -7.01 | -7.13 |

-58 | -6.74 | -6.85 | -6.95 | -7.06 | -7.16 |

-59 | -6.78 | -6.88 | -7.00 | -7.10 | -7.21 |

-60 | -6.82 | -6.93 | -7.04 | -7.15 | -7.25 |

Figure 1: Critical values at the five percent level for the *AEG* test as a function of *c*_{1} and *c*_{2}.

The top panel shows the surface describing the five percent critical values of the *AEG* test, in the case of an intercept and one regressor, when *c*_{1} and *c*_{2} are non-identical. The bottom panel shows the corresponding contour plot. The values are based on 10,000 repetitions with *T* = 1,000.

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 the different 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.

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 *α*_{1} = 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.

Figure 4: Estimates of the lower bounds of *c*.

The graphs show the density of the estimates of the lower bounds of *c*, with confidence 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.

Figure 5: Size properties of the Bonferroni *AEG* test when *c _{i}* is not identical for all

The graphs show the average rejection rates for the Bonferroni AEG test, under the null hypothesis of no cointegration, for *α*_{1} = 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* (*c*_{1}, -10, -20), and for *n* = 3, *C* = *diag* (*c*_{1}, 0, -10, -20), where *c*_{1} varies between 0 and -30. The results are based on 10,000 repetitions.

Figure 6: Power properties of the Bonferroni *AEG* test.

The graphs show the average rejection rates of the Bonferroni AEG test, for *α*_{1} = 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 *C*_{2} = -2, -10, or -20. The right column gives the results for the case with three regressors and *C*_{2} = *diag* (0,-10,-20). The results are based on 10,000 repetitions.

* We have benefitted 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 reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Österholm gratefully acknowledges financial support from Jan Wallander's and Tom Hedelius' Foundation. Return to text

^{†} Division of International Finance,
Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA.
email: erik.hjalmarsson@frb.gov Phone: +1 202 452
2426 Return to text

^{‡} Department of Economics, Uppsala
University, Box 513, 751 20 Uppsala, Sweden. email:
par.osterholm@nek.uu.se Phone: +1 202 378 4135 Return to text

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. Return to text

2. Although, perhaps, less obvious, the same also holds true for non-residual-based tests, such as those of Johansen (1988,1991); see Hjalmarsson and Österholm (2007). Return to text

3. Since 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 . The confidence level of the lower bound is therefore rather than , as discussed above. Return to text

4. Note that, for instance, the two lower confidence bounds at the 5 percent and 95 percent level provide a two-sided confidence interval with confidence level 90 percent. Return to text

5. The number of lags included in the DF-GLS test is chosen using the Schwarz (1978) information critierion, with a maximum number of two allowed in order to keep the simulation times managable. The same number of lags is also included in the regression; that is, in . Return to text

6. See, for example, Wu and Zhang (1996), Culver and Papell (1997), and Wu and Chen (2001). Return to text

7. Note that in the estimations below, time inflation is given as future inflation between and . This can be motivated by assuming rational expectations; see, for example, Mishkin (1992). Return to text

8. Regarding the specification of deterministic terms in the unit root tests, it should be noted that we test for mean reversion around a constant level. Return to text

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. Return to text

10. As in the DF-GLS test, lag length in the test equation is determined using the Schwarz (1978) criterion. Return to text

11. 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 finds 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. Return to
text

12. http://www.econ.nyu.edu/user/ludvigsons/ Return to text

13. Note that in this application, the unit root tests have both constant and trend included in the specification. Thus, the estimates and confidence intervals of are derived using the values in Table A2; again, linear interpolation is used. Return to text

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