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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 928, April 2008 --- Screen Reader
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Abstract:

This paper analyzes the asymptotic properties of long-horizon estimators under both the null hypothesis and an alternative of predictability. Asymptotically, under the null of no predictability, the long-run estimator is an increasing deterministic function of the short-run estimate and the forecasting horizon. Under the alternative of predictability, the conditional distribution of the long-run estimator, given the short-run estimate, is no longer degenerate and the expected pattern of coefficient estimates across horizons differs from that under the null. Importantly, however, under the alternative, highly endogenous regressors, such as the dividend-price ratio, tend to deviate much less than exogenous regressors, such as the short interest rate, from the pattern expected under the null, making it more difficult to distinguish between the null and the alternative.

Keywords: Predictive regressions, long-horizon regressions, stock return predictability

JEL classification: C22, G1

Long-run regressions were made popular by influential articles
such as Fama and French (1988) and Campbell and Shiller (1988).
When attempting to predict stock returns over longer horizons,
often covering several years, rather than on a month-to-month
basis, the evidence in favour of predictability generally appears
much stronger. For instance, the estimated regression coefficients
tend to increase almost linearly with the forecasting horizon.
However, in a recent paper, Boudoukh, Richardson, and Whitelaw
(2006) (BRW hereafter) show that this pattern is exactly the one to
be expected in the *absence* of predictability. Although it is
well known that the estimated slope coefficient will increase with
the horizon when there *is* predictability, BRW appear to be
the first ones to observe that the same is also true under the null
hypothesis. They interpret their findings as a strong critique of
the widespread belief that long-horizon regressions provide solid
evidence of predictability (e.g. Cochrane, 2001).^{1}

The aim of the current paper is to understand in more detail the properties of long-horizon estimates under an alternative of predictability. The practical purpose of this is to establish the pattern of estimated coefficients that may be expected both under the null and the alternative, across different forecasting horizons. Although the generally increasing pattern of coefficients in long-horizon regressions is well established, the exact asymptotic sampling properties of long-run estimators under an alternative of predictability are not previously well understood.

I derive the asymptotic distribution of the long-run OLS estimator, with overlapping observations, under the assumptions that the true data generating process is given by the standard linear predictive regression model and that the regressors are highly persistent variables. Under the alternative of predictability, the sampling properties of the long-run estimator are fundamentally different than under the null hypothesis, and the limiting distribution is highly non-standard. From a practical perspective, this result is of individual interest. It shows that confidence intervals for long-run estimates, based on inverting a test statistic that is valid under the null hypothesis, will not be correctly sized under the alternative, given the non-standard distribution.

The theoretical results allow for an exact characterization of
the conditional distribution of the long-run estimator, given the
short-run estimate. Under the null hypothesis, the long-run
estimator is, asymptotically, completely determined once the
short-run estimate is given. Importantly, however, this is not true
under the alternative of predictability. In fact, the degree to
which the long-run estimate can vary independently of the short-run
estimate is determined by the degree of endogeneity of the
regressors.^{2} Long-run estimates for highly
endogenous regressors, such as the dividend-price ratio, are also
almost completely pinned down by the short-run estimate under the
alternative hypothesis. On the other hand, for near exogenous
regressors such as the short interest rate, the long-run estimator
has a relatively large independent component given the short-run
estimate.^{3} Given the short-run estimate, under
the alternative of predictability, the implications are that: (i)
the highly endogenous regressor will have a more predictable
pattern for long-horizons than the near exogenous one, and (ii)
this pattern will closely resemble that under the null
hypothesis.

These results provide an alternative interpretation of the empirical findings in BRW. BRW interpret the findings that the dividend-price ratio has a pattern very similar to that predicted under the null, whereas the short interest rate does not, as evidence against the predictive ability of the dividend-price ratio and in favor of the predictive ability of the short rate. Given the results in this paper, however, their findings could merely reflect the fact that there is more independent variation in the long-run estimates for fairly exogenous regressors.

Let denote the one period stock return from to and let be the corresponding period return from to . The standard long-run forecasting regression is specified as follows,

(1) |

where long-run future returns are regressed onto a one period predictor, . Let the OLS estimator of in equation (1), using overlapping observations, be denoted by . The primary focus of interest will be the properties of for different values of , and in particular the relationship between and for .

In order to formally analyze the sampling properties of , the data generating process for and need to be explicitly specified. Following Nelson and Kim (1993) and Campbell (2001), I assume that and satisfy:

(2) | ||

(3) |

Thus, the one-period regression in equation (1) coincides with
the true data generating process and for any horizon, will be the optimal forecaster of returns given current
information at time .^{4} To capture the near
persistence found in most forecasting variables, such as interest
rates or valuation ratios, it is further assumed that the
auto-regressive root, , is close to one in a
local sense. In particular, it is assumed that
, where is some
finite parameter and is the sample size with
. This captures the near unit-root,
or highly persistent, behavior of many predictor variables, but is
less restrictive than a pure unit-root assumption. The near
unit-root construction, where the autoregressive root drifts closer
to unity as the sample size increases, is used as a tool to enable
an asymptotic analysis where the persistence in the data remains
large relative to the sample size, also as the sample size
increases to infinity. That is, if is treated
as fixed and strictly less than unity, then as the sample size
grows, the process will behave as a
strictly stationary process asymptotically and the standard first
order asymptotic results will not provide a good guide to the
actual small sample properties of the model. For , the usual unit-root asymptotics apply to the model,
but this is clearly a restrictive assumption for most potential
predictor variables. Instead, by letting
, the effects from the high
persistence in the regressor will appear also in the asymptotic
results, but without imposing the strict assumption of a unit root.
Cavanagh et al. (1995), Lanne (2002), Valkanov (2003), Torous et
al. (2004), and Campbell and Yogo (2006) all use similar models,
with a near unit-root construct, to analyze the predictability of
stock returns.

The error processes are assumed to satisfy a martingale difference sequence with finite fourth order moments. That is, let and be the filtration generated by . Then , , , and

By standard arguments, where denotes a two dimensional Brownian motion and denotes weak convergence of the associated probability measures. Further, as , and an analogous result holds for the demeaned variables , with the limiting process replaced by (Phillips, 1987, 1988). Let and be the standardized Brownian motions, with unit variance and correlation , that correspond to and , respectively. By the properties of conditional normal distributions, , where is a Brownian motion with unit variance and orthogonal to . Further, let be the standardized version of .

To ease the notation, define

and | (4) |

and write where .

The foundations for the subsequent analysis is given in the following theorem, which outlines the asymptotic properties of under both the null hypothesis of no predictability and the alternative of predictability.

*1. Under the null hypothesis that ,
as
,*

(5) |

*2. Under the alternative hypothesis that
, as
,*

(6) |

*where
with
, and
. For , it holds exactly that
, and
.*

**Remark 1.1** Note that the asymptotic
distribution of
is identical under the null
and the alternative. That is, for any value of ,

(7) |

This follows both from standard asymptotic theory applied to the one-period OLS estimator, but also from plugging in in equations (5) and (6).

**Remark 1.2** Under the null hypothesis,
the asymptotic distribution of
is identical to that of
, apart from a scaling factor.
This result follows from the persistent nature of the regressors;
the intuition behind it is discussed in more detail in Hjalmarsson
(2007). For the purposes of this paper, the implications are that
the short-run and long-run estimators are perfectly correlated
asymptotically. In fact, from equation (5) it follows
that,

and | (8) |

where is used to denote an approximate distributional equivalence. Asymptotically, therefore, the conditional distribution of given satisfies . Given , the estimator is thus asymptotically a deterministic linear function of the forecasting horizon. This is similar to the result in BRW, which is derived under the assumption of a fixed autoregressive root that is strictly less than unity.

**Remark 1.3** Under the alternative
hypothesis, the long-run estimator converges to the parameter
,
which for
implies that
. That
is, the 'true' parameter value, , as well as
the estimator
, grows approximately
one-to-one with the forecasting horizon.

**Remark 1.4** Under the alternative
hypothesis of predictability, the distribution of
is quite different than
under the null hypothesis. To understand the intuition behind this
result, note first that the true model is given by equations
(2) and (3). The long-run
regression equation is thus a fitted regression, rather than the
data generating process. As shown in Appendix A, the long-run
returns
actually satisfy the
following relationship when ignoring the constant, derived from
equations (2) and
(3):

(9) |

There are now two error terms, the usual plus the additional term , which stems from the fact that at time there is uncertainty regarding the path of for . That is, since the true model is given by equations (2) and (3), there is uncertainty regarding both the future realizations of the returns as well as of the predictor variable when forming period ahead forecasts. The first error term, , corresponds to the asymptotic term in the limiting distribution and the second error term, , corresponds to the term. For large , the second error term in (9) will clearly dominate the asymptotic properties since it is of an order of magnitude larger than the first one, a result reflected in the weights on and in equation (6). However, the weight on in (6) also depends on . Thus for close to zero, will still be important for relatively large .

**Remark 1.5** Following the analysis in
BRW, the results are derived under the assumption that is fixed as
. However, it is easy to
show that the results remain similar if increases
with the sample size , but at a slower pace, such
that
, as
. Under this assumption,
it follows easily under the null hypothesis that
. Under the alternative hypothesis,
, where the first term in (6) now disappears
asymptotically. As discussed in the previous remark, however, the
first term in (6) will
still be important for relatively large , provided
is small; this can be achieved also
for asymptotically large by treating as small in a local sense. All the results in this
paper therefore hold also under the general assumption that
grows with the sample size but at a slower
pace. As shown in Hjalmarsson (2007), asymptotic results derived
under this assumption seem to provide good approximations of the
finite sample properties of
for forecasting horizons
spanning upwards of to
percent of the sample size. For completeness, however, Appendix
B presents the
results for the case where is asymptotically
large relative to , in a manner such that
, as
; i.e. when grows at the same pace as the sample size.

In the analysis of BRW, it follows that under the null
hypothesis,
(see equation 6 in BRW), rather than
, as
found here, where
is defined in
Theorem 1.^{5}
However, under the current assumption of
, it follows that
. Thus, for
the local-to-unity specification of that is
used here,
and
are asymptotically indistinguishable, and replacing
by
does not affect the
asymptotic arguments but merely provides a finite sample
adjustment. As the analysis of BRW implies, along with simulation
results that are not reported here, the rate of growth of
under the null hypothesis
seems to correspond best to
, rather than
, in finite samples.

Likewise, in Part 2 of Theorem 1, the factor in front of can be replaced by , since this multiplier arises in an identical manner to the one in Part 1. That is, under the alternative hypothesis, one can write,

(10) |

In the analysis in the next section, I use these finite sample adjusted results. This does not qualitatively change any of the results, and for it holds exactly that .

The results in the previous section provide the necessary building blocks for understanding the properties of, and relationship between, the long- and short-run estimators both under the null hypothesis and under the alternative of predictability. In this section, I consider the implications of these results through an informal analysis. For ease of notation, it is assumed that .

Under both the null and the alternative, the short-run estimator satisfies, , and one can write informally,

(11) |

Similarly, under the null with , the long-run estimator satisfies

(12) |

Thus, as noted above, under the null-hypothesis, and are perfectly asymptotically correlated.

Under the alternative of predictability,

(13) |

The distribution of is now a function of , as well as an additional term. Note, however, that given , the random variable is fixed, and the only independent information in , given , derives from the variable.

To better understand the properties of under the alternative hypothesis, it is useful to consider the two special cases of and close to . The case of close to will be symmetrical to that of close to , but the latter is much more common in stock return applications. To more easily understand the variation in , Figure 1 shows the density plots for , for different values of , and for the case of ; the density of is identical to that of .

As is seen in Figure 1, , and hence , is almost always negative, a fact which will be used in the discussion below. To see this analytically, consider the case when and note that one can then write . Since is distributed as a variable, there is an approximately two-thirds probability that the first term in the numerator will be negative. The second term will also tend to be negative, since the correlation between and is positive. Further, when is large, the denominator will also be large, skewing the distribution further to the left. The ratio will therefore be negative most of the time and have a negative mean. A similar argument can be made for .

When ,

(14) |

where the second term is stochastically independent of . Since , as is apparent from Figure 1 and the discussion above, will tend to be below the curve ; the 5th, 50th and 95th percentile of , for , are given by , , and , respectively. To further understand the relationship between and in this case, consider a simple example. Suppose , , and , which is a typical estimate of the short-run slope parameter in a regression with monthly standardized data such that (e.g. Campbell and Yogo, 2006). If the true value of is equal to zero, then asymptotically, . On the other hand, if , so that the short-run estimate is equal to the true value, then conditional on , the 5th, 50th and 95th percentiles of are equal to , , and , respectively, based on equation (14) and the percentiles of .

As

(15) |

For close to minus one, is almost always positive, and will tend to be smaller than . Note also, that once is determined, there is no additional variance left in the estimator . That is, since , for a given and , is pinned down, and hence as well.

Consider a similar thought experiment to that above. Again, suppose , , and . If , then , which implies that and Now, if , then implies that , and . If , then , and To the extent that is greater than or equal to zero, a large negative correlation severely limits the range of probable values that can attain once is fixed. (The 5th, 50th and 95th percentile of for and , are given by , , and , respectively.)

Thus, when the predictor is exogenous, so that , somewhat substantial deviations in
from that predicted under
the null are possible and to some extent expected. When the
regressors are highly endogenous, and is
close to minus one, the range of possibilities also under the
alternative is more restricted and large deviations from that
predicted under the null are not likely.^{6}

Figure 2 further illustrates this last point. Using equations (14) and (15), it plots potential outcomes of , given , for four different values of the true . The same parameters as in the examples above are used, with and . For , once and are fixed, the outcome of is fully determined and there is thus no range of possibilities. For , there is independent variation left in , given and , in the form of . The graphs for show the lower bound of , based on the 5th percentile of . The upper bound, based on the 95th percentile of is virtually identical to , since the 95th percentile of is almost equal to zero.

The graphs clearly demonstrate the limited range of plausible outcomes for given a typical one-period estimate of , when the regressor is highly endogenous. Indeed, when the estimate is in fact identical to the true , the outcome is observationally equivalent to that under the null hypothesis. When the predictor is exogenous, the range of outcomes is obviously much larger, and there is a fair chance of detecting patterns that deviate substantially from those expected under the null.

In their empirical analysis, BRW show that the coefficients for
the dividend-price ratio, which is highly endogenous, are nearly
linear in the forecasting horizon whereas those for the short
interest rate, which is nearly exogenous, grow at a much slower
pace. In light of Figure 2, these findings are suggestive of
predictive ability in the short interest rate, but can say little
or nothing regarding the predictive ability of the dividend-price
ratio.^{7}

To sum up, under the null hypothesis, the long-run estimator is asymptotically completely determined by the one-period estimate and the persistence in the regressor. Under the alternative hypothesis, the degree to which the long-run estimates can vary independently of the one period ones is determined by the degree of endogeneity in the regressors. Nearly exogenous predictors, such as the short interest rate, allow for more independent variation than highly endogenous predictors such as the earnings-price ratio. Unfortunately, long-run estimates therefore provide additional information in cases where short-run inference is relatively straightforward but adds little in the case of endogenous regressors where short-run inference is fraught with difficulties (i.e. Stambaugh, 1999, and Campbell and Yogo, 2006).

Finally, it is worth pointing out, that the asymptotic framework used in this paper delivers an asymptotically degenerate distribution of given , under the null hypothesis. This prevents the construction of formal, and asymptotically meaningful, tests on the joint distribution of and under the null. BRW devise a joint test of under the assumption of a fixed strictly less than one. However, under these assumptions, the standard asymptotic distribution of the test is not likely to be well satisfied due to the standard complications in inference with endogenous and persistent variables. Monte Carlo simulations not reported in this paper confirm that for a large negative , the BRW test will tend to severely over reject. It is also interesting to note that the Wald statistic of BRW is scaled by , which diverges to infinity as . The degenerate case encountered in the current asymptotics thus follows as a limiting case in their analysis. The construction of an asymptotically valid and correctly sized joint test of and for close to unity is thus left unresolved.

**Proof.** For ease of notation the case with no
intercept is treated. The results generalize immediately to
regressions with fitted intercepts by replacing all variables by
their demeaned versions. Part 1 is proved in Hjalmarsson (2007),
but is repeated here for completeness.

1. Under the null hypothesis,

By standard arguments, as , since for any , . Therefore,

2. By summing up on both sides in equation (2),

where . Thus,

and

Observe that for all . Let and it follows that, as , . By the results in Part 1., , as , and the desired result follows. Note that,

Some of the literature on long-horizon regressions has analyzed
the case where is asymptotically large relative
to , such that
, as
. In the context of this
study, such asymptotics are less useful because the long-run OLS
estimator will *not* converge to a properly defined long-run
coefficient. Since the current focus is on the distribution of the
long-run estimator conditional on the short-run estimator, it makes
more sense to consider the case when the long-run estimator does
converge. Nevertheless, it is still interesting to see if any of
the results derived in the main text continue to hold under this
assumption.

Again treating the case without an intercept, Valkanov (2003) shows that under the null hypothesis, with as , where . Under the alternative, , where Thus, no longer converges to a constant and it is therefore not surprising that the strong connection between the asymptotic distributions for the short-run and long-run estimators is no longer apparent. Indeed, given the highly non-standard limiting distributions, it is difficult to get a grasp of the properties of . A very rough approximation, however, can provide some guidelines.

Note that and . Under the null hypothesis, it follows that , and under the alternative hypothesis, . Thus, also for , it would appear that will grow with the forecasting horizon. Under the null hypothesis, there is still some indication of the relationship between and , but under the alternative hypothesis the more subtle connections between the short-run and the long-run estimator are no longer evident, as might be expected given the lack of a consistent long-run estimator.

Let be the coefficient of determination from the period regression in equation (1). Using the same arguments as in the proof of Theorem 1, it follows that under the null hypothesis, . Similarly, using the result that , which is derived in Hjalmarsson (2007). Thus, asymptotically, under the null hypothesis, .

Because is a near-integrated regressor, it follows easily that as . This is not very useful from a practical perspective and it is more interesting to analyze the properties under a local alternative, . Under this alternative, using similar arguments as before, and Standardizing so that , and using the approximation that , it follows that and .

Boudoukh J., M. Richardson, and R.F. Whitelaw, 2006. The myth of long-horizon predictability, Review of Financial Studies, forthcoming.

Campbell, J.Y., 2001. Why long horizons? A study of power against persistent alternatives, Journal of Empirical Finance 8, 459-491.

Campbell, J.Y., and R. Shiller, 1988. Stock prices, earnings, and expected dividends, Journal of Finance 43, 661-676.

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

Cavanagh, C., G. Elliot, and J. Stock, 1995. Inference in models with nearly integrated regressors, Econometric Theory 11, 1131-1147.

Cochrane, J., 2001. Asset Pricing, Princeton, Princeton University Press.

Fama, E.F., and K.R. French, 1988. Dividend yields and expected stock returns, Journal of Financial Economics 22, 3-25.

Goetzman W.N., and P. Jorion, 1993. Testing the Predictive Power of Dividend Yields, Journal of Finance 48, 663-679.

Hansen, L.P., and R.J. Hodrick, 1980. Forward Exchange Rates as Optimal Predictors of Future Spot Rates: An Econometric Analysis, Journal of Political Economy 88, 829-853.

Hjalmarsson, E., 2007. Inference in Long-Horizon Regressions, Working Paper, Federal Reserve Board.

Hodrick, R.J., 1992. Dividend Yields and Expected Stock Returns: Alternative Procedures for Inference and Measurement, Review of Financial Studies 5, 357-386.

Lanne, M., 2002. Testing the Predictability of Stock Returns, Review of Economics and Statistics 84, 407-415.

Kirby, C., 1997. Measuring the Predictable Variation in Stock and Bond Returns, Review of Financial Studies 10, 579-630.

Nelson, C.R., and M.J. Kim, 1993. Predictable Stock Returns: The Role of Small Sample Bias, Journal of Finance 48, 641-661.

Phillips, P.C.B, 1987. Towards a Unified Asymptotic Theory of Autoregression, Biometrika 74, 535-547.

Phillips, P.C.B, 1988. Regression Theory for Near-Integrated Time Series, Econometrica 56, 1021-1043.

Richardson, M., 1993. Temporary Components of Stock Prices: A Skeptic's View, Journal of Business and Economics Statistics 11, 199-207.

Richardson, M., and T. Smith, 1991. Tests of Financial Models in the Presence of Overlapping Observations, Review of Financial Studies 4, 227-254.

Richardson, M., and T. Smith, 1994. A Unified Approach to Testing for Serial Correlation in Stock Returns, Journal of Business 67, 371-399.

Richardson, M., and J.H. Stock, 1989. Drawing Inferences from Statistics Based on Multiyear Asset Returns, Journal of Financial Economics 25, 323-348.

Stambaugh, R., 1999. Predictive regressions, Journal of Financial Economics 54, 375-421.

Torous, W., R. Valkanov, and S. Yan, 2004. On Predicting Stock Returns with Nearly Integrated Explanatory Variables, Journal of Business 77, 937-966.

Valkanov, R., 2003. Long-horizon regressions: theoretical results and applications, Journal of Financial Economics 68, 201-232.

Figure 1: Density plots of
and for *c* = 0.

The graphs show the densities for the Brownian functionals , for different values of , and , obtained by kernel estimation of simulated data using repetitions and a sample size of in each repetition. The shape of the density of is identical to that of

Figure 2: Possible outcomes of given and .

The graphs show potential outcomes of , given that , for . The left hand side panel shows the case for exogenous regressors with and the right hand side shows the case with highly endogenous regressors. The plots are formed using equations (14) and (15), letting and . For , once and are fixed, the outcome of is fully determined and there is thus no range of possibilities. The graphs for show the lower bound of , based on the 5th percentile of . The upper bound, based on the 95th percentile of , is virtually identical to ; i.e. the line for .

* This paper has benefitted from comments by Lennart Hjalmarsson, Randi Hjalmarsson and Jonathan Wright, as well as an anonymous referee. Tel.: +1-202-452-2426; fax: +1-202-263-4850; email: erik.hjalmarsson@frb.gov. The views in this paper are solely the responsibility of the author 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. Return to text

1. Earlier studies discussing other inferential issues in long-horizon regressions include Hansen and Hodrick (1980), Richardson and Stock (1989), Richardson and Smith (1991, 1994), Hodrick (1992), Goetzman and Jorion (1993), Nelson and Kim (1993), Richardson (1993), Kirby (1997), and Valkanov (2003). Return to text

2. A predictive regressor is generally referred to as endogenous if the innovations to the returns are contemporaneously correlated with the innovations to the regressor. When the regressor is strictly stationary, such endogeneity has no impact on the properties of the estimator, but when the regressor is persistent in some manner, the properties of the estimator will be affected (e.g. Stambaugh, 1999). Return to text

3. The contemporaneous correlation between the innovations to the returns and the innovations to the regressor determines the endogeneity of a predictive regressor (see previous footnote). Campbell and Yogo (2006) show that for valuation ratios, such as the dividend-price ratio, this correlation is large and often greater than in absolute magnitude. For interest rate variables, however, the correlation is close to zero and, from the perspective of a predictive regression, these variables are thus nearly exogenous. Return to text

4. There are two primary reasons why the analysis of the long-run regression in equation (1) is of interest under the assumption that the true model is given by the short-run equation (2). First, there is the long standing popular belief that predictability is more evident in the long run, and that there may therefore be power gains to analyzing the long-horizon regression, even if the short-run specification given by equation (2) is correct; for instance, Campbell (2001) analyzes the power of long-run tests under the same specification that is used in this paper. Alternatively, since there appears to be no other data generating processes that are widely used for modeling return predictability, in the short- or long-run, the results derived under the data generating process given by equations (2) and (3) can be viewed as a benchmark against which to compare results from other specifications. Return to text

5. I use the '' sign here because in the framework of BRW, the asymptotic distribution of is not entirely pinned down by . Return to text

6. Apart from increasing slope coefficients, increasing have also been used as an argument in favor of long-run predictability. In Appendix C, I first replicate BRW's finding that increases with the forecast horizon under the null hypothesis. In addition, I show that the asymptotic properties of under the alternative hypothesis are not a function of the degree of endogeneity of the regressor and that still increases almost linearly with the forecasting horizon. Thus, unlike for , there is no systematic difference in the asymptotic properties of for exogenous and endogenous regressors under the alternative hypothesis. Return to text

7. The results in this paper are all based on the assumption that the regressor follows a near unit-root process. For a fixed autoregressive root strictly less than unity, the effects arising from endogeneity would not appear in the asymptotic analysis, although one could perhaps obtain some similar results using the finite sample bias derived in Stambaugh (1999). In practice, of course, the near unit-root construction is designed to asymptotically capture the finite sample bias that arises from highly persistent and endogenous regressors, which are not necessarily unit-root processes. For regressors with very low persistence, i.e. , there will be no effects from endogeneity and there should thus be no systematic difference between endogenous and exogenous regressors in the behavior of either the short-run or the long-run estimators. However, most relevant predictors of stock returns tend to be highly persistent. Return to text

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