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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 932, June 2008--- Screen Reader
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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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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 http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

We show that the general bias reducing technique of jackknifing can be successfully applied to stock return predictability regressions. Compared to standard OLS estimation, the jackknifing procedure delivers virtually unbiased estimates with mean squared errors that generally dominate those of the OLS estimates. The jackknifing method is very general, as well as simple to implement, and can be applied to models with multiple predictors and overlapping observations. Unlike most previous work on inference in predictive regressions, no specific assumptions regarding the data generating process for the predictors are required. A set of Monte Carlo experiments show that the method works well in finite samples and the empirical section finds that out-of-sample forecasts based on the jackknifed estimates tend to outperform those based on the plain OLS estimates. The improved forecast ability also translates into economically relevant welfare gains for an investor who uses the predictive regression, with jackknifed estimates, to time the market.

Keywords: Bias correction, jackknifing, predictive regression, stock return predictability

JEL classification: C22, G1

Ordinary Least Squares (OLS) estimation of predictive
regressions for stock returns generally results in biased
estimates. This is true in particular when valuation ratios, such
as the dividend- and earnings-price ratios, are used as predictor
variables. The bias has been analyzed and discussed in numerous
articles and a number of potential solutions have been suggested
(e.g., Mankiw and Shapiro, 1986, Stambaugh, 1999, and Jansson and
Moreira, 2006). However, most of the attention in the literature
has been directed at constructing valid *tests* in the case of
a single regressor that follows an auto-regressive process, and
much less attention has been given to the problem of obtaining
better *estimators*, both in the case of single or multiple
predictor variables.^{1}

Although the testing problem is arguably the more fundamental issue from a strictly statistical point of view, the estimation problem is of great interest from an economic and practical perspective. The statistical tests answer the question whether there is predictability, but the coefficient estimate speaks more directly to the economic magnitude of the relationship. Since there is an emerging consensus in the finance profession that stock returns are to some extent predictable, it is of vital interest to determine the economic importance of this predictability. In addition, if forecasting regressions are to be used for out-of-sample forecasts, which is often their ultimate purpose, the point estimate obviously takes on the main role.

In this paper, we propose the application of a general bias reduction technique, the jackknife, to obtain better point estimates in predictive regressions. Unlike most other methods that have been proposed, this procedure does not assume a particular data generating process for the regressor and allows for multiple predictor variables. The jackknifed estimator, which is based on a combination of OLS estimates for a small number of subsamples, is also trivial to implement and could easily be used with common statistical packages. In relation to previous work, the current paper contributes to both the emerging literature on bias-reducing techniques in predictive regressions, such as Amihud and Hurvich (2004) and Eliasz (2005), as well as the ongoing debate on out-of-sample predictability in stock-returns, as exemplified by Goyal and Welch (2003, 2007) and Campbell and Thompson (2007).

In a series of Monte Carlo experiments, we show that the jackknifed estimator can reduce the bias in the estimates of the slope coefficients in predictive regressions. This applies both to the standard one-regressor, one-period regression as well as to the case of multiple regressors and longer forecasting horizons. Although the jackknifed estimates have a larger variance than the OLS estimates, the jackknifed estimates still often outperform the OLS ones in a mean squared error sense. Thus, to the extent that it is desirable to have as small a bias as possible, for a given mean squared error, the jackknifed estimator tends to dominate the OLS estimator.

In the empirical section of the paper, we consider forecasting of aggregate U.S. stock returns, using five different predictor variables: the dividend- and earnings-price ratios, the smoothed earnings-price ratio suggested by Campbell and Shiller (1988), the book-to-market ratio, and the short interest rate. Although many other stock return predictors have been proposed (see, for instance, Goyal and Welch, 2007), the above valuation ratios are of most interest here, since they tend to result in the largest biases in the OLS estimates. The short interest rate is also analyzed since some recent work by Ang and Bekaert (2007) suggests that it works well as a predictor together with the dividend-price ratio, which thus provides an opportunity to study the performance of the jackknifed estimator with multiple regressors.

The in-sample results show that the jackknifed estimates, in some cases, deviate substantially from the OLS estimates. For instance, the magnitude of the coefficient for the book-to-market ratio is often drastically smaller when using the jackknife procedure. On average, the OLS estimates often overstate the magnitude of predictability compared to the jackknife estimates.

In order to evaluate whether these discrepancies in the
full-sample estimates actually translate into better real time
forecasting ability, we perform two different out-of-sample
exercises. First, we calculate the out-of-sample *R*^{2}*s* for the different predictor variables, and find that
the forecasts based on the jackknifed estimates typically dominate
those based on the OLS estimates; this is true also if one imposes
some of the forecast restrictions proposed by Campbell and Thompson
(2007). In a second out-of-sample exercise, we estimate the welfare
gains to a mean-variance investor who uses either the OLS estimates
or the jackknifed estimates to form his portfolio weights in order
to time the market. In this case, the jackknifed estimates produce
even clearer gains, dominating both the portfolio choices based on
the OLS estimates as well as the baseline choice based on the
historical average returns. Overall, the promising results seen in
the Monte Carlo simulations carry over to the real data.

The rest of the paper is organized as follows. Section 2 outlines the jackknife procedure and provides an explicit example of how it works in a predictive regression. Section 3 presents the results from the Monte Carlo exercises. The empirical analysis is performed in Section 4 and Section 5 concludes.

Let *T* be the sample size available for the
estimation of some parameter . Decompose the
sample into *m* consecutive subsamples, each with
*l* observations, so that *T* = *m* × *l*. The jackknife estimator, which
was introduced by Quenoille (1956), is given by

(1) |

where and are the estimates of based on the full sample and the th subsample, respectively, using some given estimation method such as OLS or maximum likelihood. In the current paper, we rely only on OLS for obtaining . Under fairly general conditions, which ensure that the bias of and can be expanded in powers of , it can be shown that the bias of will be of an order instead of ; Phillips and Yu (2005) provide a longer discussion on this.

Note that may be a single parameter, or a vector of parameters, estimated from some model using any feasible estimation method. Furthermore, may also represent a combination, or complicated function, of estimated parameters. For instance, Phillips and Yu (2005) show how jackknifing bond option prices directly, rather than the estimated parameters that enters the bond option formula, can help reduce the bias in the estimated option prices. The jackknife is thus a very generally applicable method. Within the context of estimating models for stock return predictability, we consider, in addition to the standard single regressor case, also a case with multiple regressors, as well as overlapping observations. Whereas the bias in the single regressor case is well analyzed, less is understood about the biases in the case of multiple regressors or the case of long-run forecasting regressions with overlapping observations. Again, in all three cases the analysis of the bias is usually restricted to the case where the regressors follow an auto-regressive process; see, for instance, Amihud and Hurvich (2004) for a discussion on some bias reduction methods for the case of multiple regressors.

A simple example helps to illustrate how the jackknifing procedure reduces the bias in estimates. Consider the traditional predictive regression with a single regressor which follows an process:

(2) | ||

(3) |

Suppose and are bivariate normally distributed with mean zero and covariance matrix ; the correlation between and is denoted by in the simulations below. As shown in Stambaugh (1999), the bias in the OLS estimator of is given by

(4) |

The jackknife estimator of for , based on OLS estimation, is equal to

(5) |

and

(6) |

Taking expectations on both sides and using the expression in (4), it follows that

(7) |

Thus, the bias is reduced from to .

This result would hold for any *m*, which raises
the question of what value *m* should be set to in
practice. As shown by the simulations in the following section,
setting *m* = 2 works very well and usually eliminates
almost all of the bias. However, the simulations also show that an
increase in *m* (to 3 or 4)
can reduce the variance of the jackknife estimate without any
substantial increase in the bias. In general, the root mean squared
error is smallest for *m* = 4 in the simulations
presented below. Phillips and Yu (2005) present results along
similar lines and provide some brief theoretical arguments that
support these findings. In a given context, an optimal choice of
*m* may therefore exist, although there appears
to be no studies on how to choose this optimal *m*.
The empirical section, which presents results for *m* = 2, 3, and 4, *m* = 3
may be the best choice on average, although the differences are
generally not great between the three alternatives, and there
appears to be no choice of *m* that strictly
dominates empirically.

We analyze the finite sample performance of the jackknife method
by simulating data from the model defined by equations (2) and (3). The assumption
that the predictor variable follows an *AR*(1) process is probably the
most common one in the analysis of stock-return predictability.
This stems primarily from the relative ease with which the
properties of estimators of *β* can be analyzed
in this setup, and because the model captures the most salient
features of typical forecasting variables such as valuation ratios
and interest rates. The results from the *AR*(1) specification should also
be qualitatively similar to those from a more general *AR*(*p*) model. In general, the
jackknife procedure should help reduce bias in other setups as
well, but we focus on its properties for this familiar model which
is easy to parametrize in a realistic manner, such that the OLS
estimator will be biased in finite samples. In addition to
considering the case with a single regressor, we also simulate from
a model with two forecasting variables, where each of these follows
an *AR*(1) process as specified in
detail below. Finally, we also consider the case when forecasts are
formed at a horizon different from that at which the data were
sampled.

Equations (2) and
(3) are simulated
for the case when is a scalar. The
innovation terms and are drawn
from a multivariate normal distribution with unit variances. The
correlation between and , denoted , takes on three
different values: -0.9, -0.95, and -0.99. The
auto-regressive root is set equal to either
0.9, 0.95, or 0.999. The
sample size, *T*, is equal to 100 or
500 observations. The parameters
and are all
set to zero, although an intercept is still estimated in the
predictive regression; since the bias in the OLS estimator is not a
function of the values of these parameters (e.g. Stambaugh, 1999),
this standardization does not affect the results. Campbell and Yogo
(2006) show that values such as these for and
are often encountered empirically, when
using valuation ratios as predictors.

Note that, if , so that the error terms and are uncorrelated, the OLS estimator is unbiased and equal to the full information maximum likelihood estimator. Furthermore, for close to zero, the OLS estimator will also be unbiased, even when . In general, the bias for the OLS estimator is thus greater as gets closer to unity, and the closer the absolute value of is to one. We therefore restrict the analysis to the part of the parameter space where there actually is a bias to correct in the OLS estimator. Results for are shown since this is the empirically most relevant case and the case of is completely analogous.

The Monte Carlo simulation is conducted by generating
10,000 sample paths from equations (2) and (3), for each
combination of parameter values. From each set of generated returns
and regressors, the OLS estimate of and the
jackknife estimates for *m* = 2, 3 and 4,
are calculated. The average bias and root-mean-squared errors
(RMSE) for these estimators are then calculated across the
10,000 samples. The results are reported in
Table 1, which shows
the bias and the RMSE in parentheses below, for each parameter
combination.

An inspection of the results in Table 1 quickly reveals
three distinct findings: (i) the OLS estimates are upward biased
for all of the parameter combinations under consideration, (ii) the
jackknife estimates are virtually unbiased in all cases, and (iii)
the RMSEs for the jackknife estimates are always less than or equal
to the RMSE for the OLS estimates for *m* = 3 and 4, and fairly similar to the RMSE for the OLS
estimates for *m* = 2. These simulation results thus
suggest that the jackknifing procedure reduces the bias without
inducing enough variance to inflate the RMSE.

Figure 1
provides some additional insights into the workings of the
jackknifed estimator. It shows density plots for the OLS estimator
as well as the jackknife estimators for* m* = 2, 3 and 4. The densities are estimated with kernel
methods from 100,000 samples, with *T* = 100,
and
. The density of the OLS
estimate is almost completely to the right of the true value for
, and is also highly skewed towards the
right. The jackknifed estimates are both more centered around the
true value as well as more symmetric. For *m* = 2, the
jackknife estimator has a distribution that is centered almost
exactly at the true value and is also fairly symmetric. For
*m* = 3 and 4, the densities are
more peaked, reflecting the lower RMSEs shown in Table 1, but also
slightly less centered at the true value; these densities are also
somewhat more skewed. As mentioned in the previous section, these
results indicate that there is a trade off between bias and
variance in the choice of *m*, and an optimal
choice of *m* in terms of RMSE may therefore exist.
However, no formal results along these lines appear to be
available.

In order to understand the magnitude of the bias in the OLS estimator, and the importance of the bias reduction achieved with the jackknife estimators, it is useful to consider typical values of the estimates of in actual data. The results in Campbell and Yogo (2006) are particularly convenient for such a comparison since they present their estimates in a standardized manner conforming with the model simulated here; that is, they scale the estimate of to correspond to a model with unit variance in and . Campbell and Yogo (2006) consider stock return predictability for aggregate U.S. stock returns. They show that OLS estimates of are typically in the range of 0.1 to 0.2 in annual data and most often in the range of 0.01 to 0.02 in monthly data. Thus, if one uses 100 years of annual data, the bias in the OLS estimate may be between 20 and 50 percent of the actual parameter value, as seen from the results in Table 1. If one relies on a shorter (in years covered) monthly series with 500 observations, the bias could easily be as large as the parameter value itself. In proportion to the size of the parameter value, the bias reduction in the jackknifing procedure is therefore at least substantial and potentially huge.

Although the simple forecasting regression with just one predictor is by far the most studied and commonly used in the literature, there are instances when the use of several forecasting variables may be advantageous. For instance, Ang and Beekaert (2007) argue that the dividend-price ratio works much better as a predictor when used jointly with the short rate, rather than on its own.

In order to evaluate the properties of the jackknife estimator
in the multiple regressor case, we restrict the attention to the
case with two forecasting variables and follow a similar setup to
the one used in the single regressor case. In particular, it is
assumed that the data is generated by a multivariate version of the
model described by equations (2) and (3). The
auto-regressive matrix for the two predictor variables is set to
and the innovations and
are
again normally distributed with unit variance. The correlation
vector between and is
labeled
and the correlation between
and is
labeled , such that the variance-covariance
matrix for is equal to
. Table 2 shows
the results for the estimates of the two coefficients,
and ,
that correspond to the first and second predictor variable, for
various values of *A* and different correlations
between the innovations. Results for *T* = 100 and
*T* = 500 are presented and the results are
based on 10,000 repetitions.

The first two columns of results in Table 2 represent
perhaps the most empirically interesting case. For these results,
,
,
and . That is, the first predictor is
the most persistent one and is also highly endogenous, whereas the
second predictor is exogenous and less persistent. This setup
corresponds fairly well to the case with the dividend-price ratio
and the short interest rate as predictors, since the dividend-price
ratio is highly endogenous whereas the short rate is nearly
exogenous, and usually somewhat less persistent than the
dividend-price ratio (Campbell and Yogo, 2006). The correlation of
0.4 between the *innovations* to the
two regressors results in an average correlation of around
0.25 between the levels of the regressors,
which is similar to the empirical correlation between the
dividend-price ratio and the short interest rate observed in the
data used in this paper.^{2}

Intuitively, given the results for the single regressor case,
one would expect the OLS estimate for the coefficient for the first
regressor
to be highly biased
whereas the second estimate
should perform
better, since it is only indirectly affected by the endogeneity
bias through the correlation of the two regressors. This intuition
is bourne out to some extent by the simulation results, which show
a large bias for the first predictor but a smaller, although still
substantial, bias for the second. The jackknife works very well for
, resulting in almost unbiased
estimates with only a small increase in RMSE for *m* = 2 and a significant reduction in RMSE for *m* = 3 and 4. Jackknifing the estimates for
also results in unbiased
estimates, but with a slight increase in RMSE, particularly for
*T* = 100.

The following two sets of results in Table 2 represent cases where both regressors are endogenous. In order for the overall covariance matrix between and to be well defined, this forces and to be fairly highly correlated as well. In particular, and in the first case and and in the second case. The persistence parameters are set to and in the first specification, and to in the second one. Thus, both variables are highly endogenous and highly co-linear. In the first specification, the first predictor is more persistent than the other, whereas in the latter specification both have the same persistence. These parametrizations could correspond to, for instance, various combinations of valuation ratios, which may have somewhat different degrees of persistence and endogeneity, and may also have different correlations with each other.

As seen in Table 2, both of these
parametrizations result in OLS estimates that are biased, both for
and . The
jackknifing reduces the bias substantially, although there is a
little bias left in the jackknifed estimates when using the first
parameter specification for *T* = 100, as seen in
the middle columns of Table 2. For *m* = 3 and 4, the RMSE for the jackknifed estimates
are similar to the OLS ones. The second specification, shown in the
last two columns of Table 2, is symmetric
for the two regressors and the OLS biases for the two coefficients
are virtually identical. For *T* = 100 the
jackknifed estimates are now virtually unbiased, with only a small
increase in the RMSE relative to the OLS estimates. For *T* = 500, the bias is completely removed by the jackknifing.

In summary, the jackknife appears to work well in the multivariate case and generally results in virtually unbiased estimates. Given the multitude of possible parameter combinations that arise as soon as one leaves the simplicity of the single regressor case, the results presented in this section are far from exhaustive but hopefully capture some of the more salient features of biases in multivariate regressions.

Finally, we consider the performance of the jackknife estimator
for predictive regression with overlapping observations. Inference
with overlapping observations is a topic that has a long history in
the finance literature, but most of the effort has been directed at
constructing valid test statistics rather than reducing the bias in
OLS estimates.^{3} The jackknife procedure provides a
simple but flexible way of addressing the estimation problem.

To keep things tractable, the single regressor case is analyzed.
The data is generated in exactly the same manner as described in
Section 3.1,
generating sample paths from equations (2) and (3). However,
instead of estimating equation (2), the sums of
future *q*-period returns are now regressed on the
value of . The forecasting horizon *q* is set equal to 10 for *T* = 100 and equal to 12 for *T* = 500. These two cases capture common applications of
long-run forecasts using a century of annual data and annual
forecasts based on monthly data.

The results are shown in Table 3. The bias in the OLS estimates is of an order of magnitude larger than the ones shown in Table 1. This is entirely in line with the analytical results of Bodoukh et al. (2006), who show that one should expect the estimate, and hence the bias, to increase almost linearly with the forecasting horizon. The jackknifing reduces the bias substantially in all cases, although not always completely. The RMSEs for the jackknifed estimates is slightly larger than those for the OLS estimates in some cases, although there are also substantial reductions for some parameter combinations.

It is evident that the jackknife is also applicable in
long-horizon regressions. From the results presented here, it
appears to be most useful when the overlap is not too large
relative to the number of observations; the results for *q* = 12 and *T* = 500 are generally stronger than
those for *q* = 10 and *T* = 100.
Overall, however, the results are very promising and the jackknife
clearly presents a simple way of alleviating estimation biases in
long-horizon regressions, an issue which is often ignored in
applied work.

We next apply the jackknife method to real stock market data. Since the purpose of the jackknife method is to obtain better point estimates, we primarily evaluate its usefulness by an out-of-sample (OOS) forecasting exercise. However, it is also of interest to analyze the full sample point estimates, since they directly show the differences between the plain OLS estimates and the bias corrected jackknifed estimates.

As the dependent variable, we use monthly total excess returns on the S&P 500 index, starting in February 1872 and ending December 2005; after 1920, the T-Bill rate is used to form excess returns and before that, commercial paper rates. Five separate forecasting variables are used: the dividend- and earnings-price ratios (D/P and E/P), the smoothed earnings-price ratio of Campbell and Shiller (1988), the book-to-market ratio (B/M), and the short term interest rate as measured by the three-month T-Bill rate. The smoothed earnings-price ratio is defined as the ratio of the 10-year moving average of real earnings to the current real price. Although many other stock return predictors have been proposed (see, for instance, Goyal and Welch, 2007), the above valuation ratios are of most interest here since they tend to result in the largest biases in the OLS estimates (e.g. Campbell and Yogo, 2006). The short interest rate is also analyzed, since recent work by Ang and Bekaert (2007) suggests that it works well as a predictor together with the dividend-price ratio, which provides an opportunity to study the performance of the jackknifed estimator with multiple regressors; the short interest rate is generally negatively related to future stock returns, and we therefore flip the sign on this predictor variable in all regressions so that the expected sign is always positive. All data are recorded on a monthly basis and regressions are run either at this monthly frequency or at an annual frequency, using overlapping observations based on the original monthly data. The annual results thus provide an illustration of the jackknifed procedures applied to regressions with overlapping observations. In all cases, excess stock returns are regressed on the lagged predictor variable(s) and an intercept, following the basic structure of equation (2).

These are a subset of the same data as those used by Campbell
and Thompson (2007) in their study of out-of-sample return
predictability.^{4} The jackknifed OOS predictions can
thus be directly compared to their results. In line with Campbell
and Thompson, we use the level, and not logs, of the predictor
variables as well as simple rather than log-returns.

The first set of empirical results is given in Table 4 and shows the
full sample OLS estimates, *t*-statistics and
*R*^{2}, along with the jackknifed estimates;
the *t*-statistics for the annual data with
overlapping observations are formed using Newey and West (1987)
standard errors. Results for the monthly and annual frequencies are
displayed, and two different sample periods are considered: the
longest available sample for each predictor variable, as well as
the forecast period used in the out-of-sample forecasts below.

As is well established, predictive regressions like these tend
to generate significant *t*-statistics but fairly small
*R*^{2}, which increase with the horizon.
Inference based on the *t*-statistics is generally
subject to pitfalls, as documented in, for instance, Stambaugh
(1999) and Campbell and Yogo (2006), and they are primarily shown
here for completeness. The focus in this paper is on the point
estimates in the predictive regression, which are also shown in
Table 4. Four sets
of estimates are shown: the standard OLS ones and the jackknifed
ones using *m* = 2, 3, and 4 subsamples.
Within the standard stock return predictability model, where the
regressors follow an auto-regressive process, the OLS estimates for
the valuation ratios are generally upward biased, whereas for the
short interest rate the OLS estimator should be nearly unbiased.
This suggests that the jackknifed estimates, which attempt to
correct the OLS bias, should generally be smaller than the OLS
estimates. Overall, this is the case, especially when using
*m* > 2. This is particularly true for the
book-to-market ratio in the shorter sample, and for the coefficient
on the dividend-price ratio in the regressions that include the
dividend-price ratio and the short rate jointly. The jackknifed
estimates using *m* = 2 are often close to the OLS
estimates, although they sometimes deviate substantially as well.
Qualitatively, the results are similar for the monthly and annual
data.

The results in Table 4 suggests that standard OLS estimates are likely to exaggerate the size of the slope coefficient in these predictive regressions. However, from these full sample estimates alone, it is difficult to tell whether the jackknifed estimates are actually more accurate than the OLS estimates and we therefore turn to out-of-sample exercises to evaluate this question.

In order to evaluate the OOS performance of the jackknifed estimates, we calculate an OOS , defined as

(8) |

where
is the fitted value from a
predictive regression estimated using data up till time *t* - 1 and
is the historical average return
estimated using all available data up till time *t* - 1. The out-of-sample forecasts begin in 1927, at which
point high quality monthly CRSP data becomes available, or 20 years
after the first available observation for a given predictor
variable, whichever comes later. Thus, *s*, in equation
(8), represents the
length of this initial 'training-sample', which is used to obtain
the estimates on which the first round of forecasts is based. Note
that the historical average forecast,
, is always based on all the data
back to 1872, which preserves its real world advantage. The
statistic is positive when the
conditional forecast based on the predictive regression outperforms
the historical mean. Thus, the out-of-sample *R*^{2} is positive when the root mean squared error of the
conditional forecast is less than that of the historical mean
forecast. Given that the out-of-sample *R*^{2} and a
comparison of the root mean squared errors yield identical
qualitative results, we focus on the out-of-sample *R*^{2} since it is measured in comparable units to the
in-sample *R*^{2} and thus allows for more direct
comparison.

In addition to the standard forecasts based on the predictive
regression and the historical mean, we also analyze the effects of
imposing some of the forecast restrictions proposed by Campbell and
Thompson (2007). That is, Campbell and Thompson argue that rather
than mechanically forecasting stock returns based on the estimated
forecasting equation, it is reasonable to impose the following
restrictions: if an estimated coefficient does not have the
expected sign, it is set equal to zero, and if the forecast of the
equity premium is negative, the forecast is set equal to zero.
These restrictions rule out some of the perverse results that can
otherwise occur in the rolling regressions that are used in the
out-of-sample forecasts.^{5}

Table 5
shows the OOS *R*^{2}s for the OLS estimator and the
jackknifed estimator with *m* = 2, 3 and 4, for both the restricted forecasts, which
impose the Campbell and Thompson restrictions, and the unrestricted
ones. For each predictor, the highest OOS *R*^{2} is
shown in bold type. In general, the results show that the forecasts
based on the jackknifed estimates tend to outperform the ones based
on the plain OLS estimates, although there is no given value of
*m* that consistently produces the highest OOS
*R*^{2}. The jackknifing procedure appears to
be somewhat more useful on the monthly, rather than the annual
data, in line with the simulation results above, although the
results are somewhat mixed. Qualitatively, the results are similar
for both the unrestricted and restricted forecasts. As might be
expected from the full-sample coefficient estimates in Table
4, where the
full-sample jackknifed estimates were drastically different from
the OLS estimate, the advantages of the jackknifing are
particularly clear for the book-to-market ratio.

With regards to the choice of *m*, there is no value
that clearly produces the best results. However, using *m* = 3 in the restricted forecasts consistently dominates the
OLS forecasts in the monthly data and is close to, or better, in
the annual data; only for the smoothed earnings price ratio in the
annual data is there a material difference in favor of the OLS
forecasts. In the unrestricted case, there is no *m*
for which the jackknifed estimates consistently dominate the OLS
ones for all predictor variables. This is clearly a drawback,
since, as mentioned before, there are no clear guidelines for
choosing *m*. However, as shown in the section below,
the results become clearer when one considers the implementation of
actual portfolio strategies.

In summary, the jackknifed estimator often improves upon the OLS
estimator in out-of-sample forecasts. This seems to be particularly
true when one also imposes the forecast restrictions proposed by
Campbell and Thompson (2007), in which case the jackknifed
estimator with *m* = 3 almost completely dominates the
OLS estimator.

Campbell and Thompson (2007) discuss how the OOS *R*^{2} can be translated into gains in economic terms for an
investor that attempts to time the market using these predictor
variables. However, practical considerations such as short selling
constraints may render such theoretical relationships less
accurate; a more reliable approach to gauging the economic
importance of the improvement in out-of-sample forecasts is to
directly simulate a portfolio choice strategy. To keep the
calculations tractable, consider an investor with a single-period
investment horizon and mean-variance preferences; that is, in each
period the investor myopically chooses the optimal portfolio based
on his quadratic preferences. The investor's utility function is
the expected excess return minus
times the portfolio
variance, where can be viewed as the
coefficient of relative risk aversion. The weight on the risky
asset for this investor is given by

(9) |

where
and
represents the
expected value and variance of the excess returns over the next
period, conditional on the information at time *t*.
If the investor does not use the predictive regression (2), it follows
that

(10) |

where and . If the investor does use regression (2),

(11) |

The out-of-sample economic gains of the predictive ability of equation (2) are evaluated by comparing the utilities from an investor who uses the weights in (11) to one who disregards the predictability in returns and uses the weights in (10).

The weights
are calculated using only
information available at time *t*. When the
predictive regression is not used, the weights at each time
*t* are estimated by

(12) |

where
is the historical average return
estimated using all available data up till time *t*
and
is the variance of
returns estimated using a five year rolling window of data; i.e.
is estimated using the
last five years of data before time *t*. The weights
based on the predictive regression are given by

(13) |

where and
are the estimates of the
intercept and slope coefficient in the predictive regression, using
the data up till time *t*, and
is the variance of the
residuals, again estimated using a five year rolling window of
data.^{6} In order for the portfolio weights to
be compatible with real world constraints, we impose a no short
selling restriction and a maximum of 50% leverage,
so that the portfolio weights are restricted to lie between 0 and
150%. Finally, the risk aversion parameter
is set equal to three.

Table 6 reports the welfare benefits from using the weights , using either the OLS estimator or the jackknifed estimators, instead of the weights . The utility differences are expressed in terms of expected annualized returns and can thus be interpreted as the (maximum) management fee that an investor would be willing to pay a portfolio manager that exploits the predictive ability of equation (2). As in Table 5, we consider both the forecasts that impose the Campbell and Thompson restrictions and those that do not. Qualitatively, the results in Table 6 tell the same story as those in Table 5. The portfolio strategies based on the jackknifed estimates tend to outperform those based on the OLS estimates, and, importantly, offer welfare gains over the strategies based purely on historical average returns. Again, the jackknifed estimator appears to work best for the monthly data.

The portfolio results in Table 6 provides even
stronger support of the benefit of the jackknifed estimates than
the OOS *R*^{2}s reported in Table 5. In the monthly
data, the results for the OLS portfolio weights are dominated by
the jackknife weights, for any *m*, in almost all
cases. This is true both for the restricted and unrestricted
forecasts. If one were to choose a single *m* for all
predictors, *m* = 3 would appear to be the best choice;
in the monthly data, it dominates the OLS results in all cases.

Compared to the OLS weights, the utility gains from using the
jackknife procedure are relatively large, often between 50 and 100
basis points. Although this may not sound that large in absolute
terms, the gains from using the predictive regression (with OLS
estimates) in the first place, instead of the historical average
return, are typically no larger than 50-60 basis points. In fact,
the welfare gains from the OLS weights are quite often negative,
whereas the jackknife weights, especially for *m* = 3, are almost always positive. The welfare gains from the
jackknife weights are also similar to those reported by Campbell
and Thompson (2007) based on their completely restricted forecasts
where the coefficient in the predictive regression is totally
pinned down by theoretical arguments and not estimated at all. The
results here thus suggest that improving the estimation procedures
can lead to at least as big an improvement as the imposition of
theoretical constraints. These result also add further evidence to
the case that returns are predictable out-of-sample, in contrast to
the conclusions of Goyal and Welch (2003, 2007).

A simple bias reducing method, the jackknife, is proposed for predictive regressions of stock returns. Unlike most previous work on inference in stock return predictability regressions, this paper puts the focus on obtaining good point estimates rather than correctly sized tests, a task which has become increasingly more important as the focus in the literature has shifted towards out-of-sample forecasts and practical portfolio choice based on return forecasts. In addition, the jackknife is a general method that does not rely on specific assumptions on the data generating process.

Monte Carlo simulations show that the jackknife method works
well in finite samples and virtually eliminates the bias in OLS
estimates of predictive regressions. Most importantly, it also
works well on actual stock returns data, and leads to substantial
improvements in out-of-sample forecasts. This is illustrated not
only by purely statistical measures such as out-of-sample
*R*^{2}, but also through simulated portfolio
strategies, which often perform significantly better when the
forecasts are based on the jackknifed estimates rather than the OLS
ones.

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Table 1: Monte Carlo results for the Single Regressor Case - Panel A: *T* = 100

The table shows the mean bias and root mean squared error (in parentheses) for the OLS estimator and the jackknifed estimators with *m* = 2, 3, and 4 subsamples. The differing values of *δ*, the correlation between the innovations to the returns and the regressor, are given in the top row. The sample size (*T*) and the value of the auto-regressive root (*ρ*) are given above each set of results. All results are based on 10,000 repetitions.

Estimator | T=100ρ=0.9δ=-0.90 | T=100ρ=0.9δ=-0.95 | T=100ρ=0.9δ=-0.99 | T=100ρ=0.95δ=-0.90 | T=100ρ=0.95δ=-0.95 | T=100ρ=0.95δ=-0.99 | T=100ρ=0.999δ=-0.90 | T=100ρ=0.999δ=-0.95 | T=100ρ=0.999δ=-0.99 |
---|---|---|---|---|---|---|---|---|---|

OLS: Mean | 0.038 | 0.040 | 0.041 | 0.042 | 0.044 | 0.046 | 0.048 | 0.051 | 0.053 |

OLS: RMSE | (0.069) | (0.070) | (0.072) | (0.066) | (0.068) | (0.069) | (0.065) | (0.067) | (0.068) |

m = 2: Mean | -0.001 | -0.002 | -0.002 | -0.002 | -0.002 | -0.002 | 0.003 | 0.003 | 0.002 |

m = 2: RMSE | (0.074) | (0.075) | (0.076) | (0.071) | (0.073) | (0.073) | (0.066) | (0.067) | (0.069) |

m = 3: Mean | -0.002 | -0.003 | -0.002 | -0.001 | -0.002 | -0.002 | 0.003 | 0.004 | 0.003 |

m = 3: RMSE | (0.068) | (0.069) | (0.070) | (0.061) | (0.064) | (0.064) | (0.056) | (0.056) | (0.057) |

m = 4: Mean | -0.002 | -0.002 | -0.002 | 0.000 | -0.001 | -0.001 | 0.004 | 0.004 | 0.004 |

m = 4: RMSE | (0.065) | (0.066) | (0.067) | (0.058) | (0.060) | (0.061) | (0.052) | (0.052) | (0.053) |

Table 1: Monte Carlo results for the Single Regressor Case - Panel B: *T* = 500

Estimator | T=500ρ=0.9δ=-0.90 | T=500ρ=0.9δ=-0.95 | T=500ρ=0.9δ=-0.99 | T=500ρ=0.95δ=-0.90 | T=500ρ=0.95δ=-0.95 | T=500ρ=0.95δ=-0.99 | T=500ρ=0.999δ=-0.90 | T=500ρ=0.999δ=-0.95 | T=500ρ=0.999δ=-0.99 |
---|---|---|---|---|---|---|---|---|---|

OLS: Mean | 0.007 | 0.007 | 0.008 | 0.008 | 0.008 | 0.008 | 0.010 | 0.010 | 0.011 |

OLS: RMSE | (0.022) | (0.022) | (0.022) | (0.018) | (0.018) | (0.018) | (0.013) | (0.014) | (0.014) |

m = 2: Mean | 0.000 | 0.000 | 0.000 | 0.000 | -0.001 | -0.001 | 0.000 | 0.000 | 0.000 |

m = 2: RMSE | (0.022) | (0.022) | (0.023) | (0.018) | (0.018) | (0.018) | (0.014) | (0.014) | (0.014) |

m = 3: Mean | 0.000 | 0.000 | 0.000 | 0.000 | -0.001 | -0.001 | 0.000 | 0.000 | 0.000 |

m = 3: RMSE | (0.021) | (0.022) | (0.022) | (0.017) | (0.018) | (0.017) | (0.011) | (0.012) | (0.012) |

m = 4: Mean | 0.000 | 0.000 | 0.000 | 0.000 | -0.001 | -0.001 | 0.000 | 0.000 | 0.000 |

m = 4: RMSE | (0.021) | (0.022) | (0.022) | (0.017) | (0.017) | (0.017) | (0.011) | (0.011) | (0.011) |

Table 2: Monte Carlo results for the Multiple Regressor Case - Panel A: *T* = 100

The table shows the mean bias and root mean squared error (in parentheses) for the OLS estimator and the jackknifed estimators with *m* = 2, 3, and 4 subsamples, for the two slope coefficients in a predictive
regression with two predictor variables. The top row indicates the value of the auto-regressive roots for the two
regressors, with the auto-regressive matrix given by *A* = [(*a*_{11}, 0), (0, *a*_{22})]´. The second row indicates the correlation vector, *ω _{uv}* , between
the innovations to the returns and the two regressors. The third row gives the variance-covariance matrix Ω

Estimator | a_{11}=0.999a_{22}=0.95ω=(-0.9,0)´_{uv}Ω _{uv}=[(1,0.4),(0.4,1)]´β_{1} | a_{11}=0.999a_{22}=0.95ω=(-0.9,0)´_{uv}Ω _{uv}=[(1,0.4),(0.4,1)]´β_{2} | a_{11}=0.999a_{22}=0.95ω=(-0.7,-0.7)´_{uv}Ω _{uv}=[(1,0.5),(0.5,1)]´β_{1} | a_{11}=0.999a_{22}=0.95ω=(-0.7,-0.7)´_{uv}Ω _{uv}=[(1,0.5),(0.5,1)]´β_{2} | a_{11}=0.999a_{22}=0.999ω=(-0.9,-0.9)´_{uv}Ω _{uv}=[(1,0.8),(0.8,1)]´β_{1} | a_{11}=0.999a_{22}=0.999ω=(-0.9,-0.9)´_{uv}Ω _{uv}=[(1,0.8),(0.8,1)]´β_{2} |
---|---|---|---|---|---|---|

OLS: Mean | 0.064 | -0.028 | 0.038 | 0.026 | 0.035 | 0.036 |

OLS: RMSE | (0.084) | (0.068) | (0.062) | (0.068) | (0.093) | (0.093) |

m = 2: Mean | -0.004 | -0.001 | 0.005 | -0.003 | 0.002 | 0.002 |

m = 2: RMSE | (0.085) | (0.085) | (0.075) | (0.085) | (0.123) | (0.123) |

m = 3: Mean | -0.004 | -0.001 | 0.005 | -0.003 | 0.002 | 0.003 |

m = 3: RMSE | (0.072) | (0.075) | (0.064) | (0.076) | (0.106) | (0.105) |

m = 4: Mean | -0.002 | -0.002 | 0.007 | -0.002 | 0.003 | 0.004 |

m = 4: RMSE | (0.066) | (0.071) | (0.059) | (0.071) | (0.100) | (0.100) |

Table 2: Monte Carlo results for the Multiple Regressor Case - Panel B: *T* = 500

Estimator | a_{11}=0.999a_{22}=0.95ω=(-0.9,0)´_{uv}Ω _{uv}=[(1,0.4),(0.4,1)]´β_{1} | a_{11}=0.999a_{22}=0.95ω=(-0.9,0)´_{uv}Ω _{uv}=[(1,0.4),(0.4,1)]´β_{2} | a_{11}=0.999a_{22}=0.95ω=(-0.7,-0.7)´_{uv}Ω _{uv}=[(1,0.5),(0.5,1)]´β_{1} | a_{11}=0.999a_{22}=0.95ω=(-0.7,-0.7)´_{uv}Ω _{uv}=[(1,0.5),(0.5,1)]´β_{2} | a_{11}=0.999a_{22}=0.999ω=(-0.9,-0.9)´_{uv}Ω _{uv}=[(1,0.8),(0.8,1)]´β_{1} | a_{11}=0.999a_{22}=0.999ω=(-0.9,-0.9)´_{uv}Ω _{uv}=[(1,0.8),(0.8,1)]´β_{2} |
---|---|---|---|---|---|---|

OLS: Mean | 0.011 | -0.005 | 0.008 | 0.004 | 0.007 | 0.007 |

OLS: RMSE | (0.015) | (0.018) | (0.012) | (0.018) | (0.019) | (0.019) |

m = 2: Mean | -0.001 | 0.000 | 0.000 | -0.001 | 0.000 | 0.000 |

m = 2: RMSE | (0.015) | (0.020) | (0.013) | (0.020) | (0.025) | (0.025) |

m = 3: Mean | -0.002 | 0.000 | 0.000 | -0.001 | 0.000 | 0.000 |

m = 3: RMSE | (0.013) | (0.019) | (0.011) | (0.019) | (0.022) | (0.022) |

m = 4: Mean | -0.002 | 0.000 | 0.000 | -0.001 | 0.000 | 0.000 |

m = 4: RMSE | (0.012) | (0.019) | (0.011) | (0.018) | (0.021) | (0.020) |

Table 3: Monte Carlo results for long-horizon regressions with overlapping observations - Panel A: *T* = 100 and *q* = 10

The table shows the mean bias and root mean squared error (in parentheses) for the OLS estimator and
the jackknifed estimators with *m* = 2, 3, and 4 subsamples, for the slope coefficient in a long-horizon
predictive regression with overlapping observations and forecast horizon *q*. A single regressor is used in
the regression. The differing values of *δ*, the correlation between the innovations to the returns
and the regressor, are given in the top row. The sample size *T*, the forecast horizon *q*, and the value of the
auto-regressive root *ρ*, are given above each set of results. All results are based on 10,000 repetitions.

Estimator | T=100q=10ρ=0.9δ=-0.90 | T=100q=10ρ=0.9δ=-0.95 | T=100q=10ρ=0.9δ=-0.99 | T=100q=10ρ=0.95δ=-0.90 | T=100q=10ρ=0.95δ=-0.95 | T=100q=10ρ=0.95δ=-0.99 | T=100q=10ρ=0.999δ=-0.90 | T=100q=10ρ=0.999δ=-0.95 | T=100q=10ρ=0.999δ=-0.99 |
---|---|---|---|---|---|---|---|---|---|

OLS: Mean | 0.284 | 0.307 | 0.316 | 0.339 | 0.361 | 0.368 | 0.401 | 0.419 | 0.435 |

OLS: RMSE | (0.469) | (0.481) | (0.482) | (0.480) | (0.492) | (0.495) | (0.496) | (0.513) | (0.523) |

m = 2: Mean | 0.043 | 0.052 | 0.050 | 0.086 | 0.098 | 0.084 | 0.166 | 0.175 | 0.171 |

m = 2: RMSE | (0.573) | (0.577) | (0.577) | (0.549) | (0.550) | (0.554) | (0.506) | (0.521) | (0.515) |

m = 3: Mean | 0.076 | 0.086 | 0.087 | 0.133 | 0.146 | 0.139 | 0.223 | 0.229 | 0.238 |

m = 3: RMSE | (0.507) | (0.510) | (0.506) | (0.480) | (0.480) | (0.476) | (0.454) | (0.464) | (0.462) |

m = 4: Mean | 0.106 | 0.120 | 0.121 | 0.173 | 0.188 | 0.186 | 0.267 | 0.277 | 0.286 |

m = 4: RMSE | (0.480) | (0.482) | (0.475) | (0.460) | (0.461) | (0.457) | (0.454) | (0.465) | (0.463) |

Table 3: Monte Carlo results for long-horizon regressions with overlapping observations - Panel B: *T* = 500 and *q* = 12

Estimator | T=500q=12ρ=0.9δ=-0.90 | T=500q=12ρ=0.9δ=-0.95 | T=500q=12ρ=0.9δ=-0.99 | T=500q=12ρ=0.95δ=-0.90 | T=500q=12ρ=0.95δ=-0.95 | T=500q=12ρ=0.95δ=-0.99 | T=500q=12ρ=0.999δ=-0.90 | T=500q=12ρ=0.999δ=-0.95 | T=500q=12ρ=0.999δ=-0.99 |
---|---|---|---|---|---|---|---|---|---|

OLS: Mean | 0.063 | 0.067 | 0.073 | 0.078 | 0.080 | 0.084 | 0.113 | 0.121 | 0.123 |

OLS: RMSE | (0.206) | (0.208) | (0.208) | (0.180) | (0.181) | (0.181) | (0.147) | (0.155) | (0.157) |

m = 2: Mean | -0.001 | -0.001 | 0.001 | 0.002 | -0.002 | -0.002 | 0.013 | 0.016 | 0.013 |

m = 2: RMSE | (0.226) | (0.228) | (0.226) | (0.196) | (0.199) | (0.198) | (0.149) | (0.152) | (0.154) |

m = 3: Mean | -0.001 | -0.002 | 0.002 | 0.002 | -0.001 | 0.000 | 0.018 | 0.022 | 0.019 |

m = 3: RMSE | (0.218) | (0.221) | (0.218) | (0.187) | (0.188) | (0.188) | (0.125) | (0.130) | (0.130) |

m = 4: Mean | -0.001 | -0.001 | 0.002 | 0.004 | 0.000 | 0.001 | 0.023 | 0.027 | 0.024 |

m = 4: RMSE | (0.215) | (0.218) | (0.215) | (0.182) | (0.184) | (0.183) | (0.118) | (0.122) | (0.121) |

Table 4: In-Sample Empirical Results - Panel A: Monthly, Full Sample

The table shows the OLS and jackknifed point estimates of the slope
coefficients in predictive regressions of excess stock returns, using the predictor variables indicated in the
first column. In addition, the OLS *t*-statistics and *R*^{2} (expressed in percent) are shown. Four sets of
results are shown, using either monthly or annual overlapping data, based on the original monthly observations, and
either the longest available full sample for each predictor variable or the forecast sample used in the
subsequent out-of-sample exercises. The first column in the table indicates the predictor
variable(s) used in the predictive regression, and the second column shows the start date of the sample; all samples
end in December 2005. The next four columns show the OLS and jackknifed point estimates, with *m* = 2, 3, and 4
subsamples, for the slope coefficient of the first (and typically only) predictor in the forecasting regression. The next
four columns show the estimates of the slope coefficient for the second regressor; this is only applicable in
the regression with both the dividend-price ratio and the T-Bill rate included jointly. The final three columns show the
OLS $t-$statistics for the two slope coefficients and the OLS *R*^{2} in percent. The *t*-statistics for the
annual data with overlapping observations are calculated using Newey and West (1987) standard errors.

Predictor(s) | Sample Begins |
_{1,OLS} |
_{1,m=2} |
_{1,m=3} |
_{1,m=4} |
_{2,OLS} |
_{2,m=2} |
_{2,m=3} |
_{2,m=4} |
t_{1,OLS} |
t_{2,OLS} |
R²_{OLS} (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

D/P | 1872m2 |
1.99 |
0.93 |
2.03 |
1.84 |
- |
- |
- |
- |
1.02 |
- |
0.37 |

E/P | 1872m2 |
1.05 |
1.04 |
1.32 |
1.13 |
- |
- |
- |
- |
1.73 |
- |
0.24 |

Smoothed E/P | 1881m2 |
1.49 |
1.34 |
1.23 |
1.38 |
- |
- |
- |
- |
1.77 |
- |
0.56 |

B/M | 1926m6 |
0.21 |
0.24 |
0.15 |
0.12 |
- |
- |
- |
- |
1.28 |
- |
1.19 |

T-Bill rate | 1920m1 |
1.37 |
1.01 |
0.74 |
1.22 |
- |
- |
- |
- |
1.88 |
- |
0.38 |

D/P and T-Bill rate | 1920m1 |
1.87 |
0.53 |
1.36 |
1.13 |
1.65 |
-0.57 |
0.38 |
1.01 |
1.79 |
2.08 |
1.21 |

Table 4: In-Sample Empirical Results - Panel B: Monthly, Forecast Sample

Predictor(s) | Sample Begins | _{1,OLS} | _{1,m=2} | _{1,m=3} | _{1,m=4} | _{2,OLS} | _{2,m=2} | _{2,m=3} | _{2,m=4} | t_{1,OLS} | t_{2,OLS} | R²_{OLS} (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

D/P | 1927m1 | 3.93 | 4.22 | 2.68 | 3.00 | - | - | - | - | 1.25 | - | 1.12 |

E/P | 1927m1 | 2.06 | 2.06 | 2.01 | 1.62 | - | - | - | - | 2.28 | - | 0.71 |

Smoothed E/P | 1927m1 | 3.02 | 2.57 | 2.76 | 2.62 | - | - | - | - | 1.85 | - | 1.35 |

B/M | 1946m6 | 0.18 | 0.01 | 0.01 | 0.05 | - | - | - | - | 1.96 | - | 0.61 |

T-Bill rate | 1940m1 | 1.53 | 0.88 | 1.50 | 1.73 | - | - | - | - | 2.46 | - | 0.87 |

D/P and T-Bill rate | 1940m1 | 2.91 | 0.10 | -1.74 | 0.21 | 1.35 | -0.22 | 0.28 | 1.23 | 2.33 | 2.13 | 1.56 |

Table 4: In-Sample Empirical Results - Panel C: Annual, Full Sample

Predictor(s) | Sample Begins | _{1,OLS} | _{1,m=2} | _{1,m=3} | _{1,m=4} | _{2,OLS} | _{2,m=2} | _{2,m=3} | _{2,m=4} | t_{1,OLS} | t_{2,OLS} | R²_{OLS} (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

D/P | 1872m2 | 2.55 | 1.41 | 2.53 | 2.32 | - | - | - | - | 2.41 | - | 5.14 |

E/P | 1872m2 | 1.52 | 1.49 | 1.70 | 1.51 | - | - | - | - | 2.76 | - | 4.30 |

Smoothed E/P | 1881m2 | 1.77 | 1.65 | 1.48 | 1.46 | - | - | - | - | 2.35 | - | 6.89 |

B/M | 1926m6 | 0.23 | 0.25 | 0.15 | 0.15 | - | - | - | - | 4.05 | - | 13.71 |

T-Bill rate | 1920m1 | 0.99 | 1.33 | 0.63 | 0.98 | - | - | - | - | 1.75 | - | 1.91 |

D/P and T-Bill rate | 1920m1 | 2.66 | 2.38 | 2.29 | 2.16 | 1.32 | 0.78 | 0.50 | 1.35 | 3.75 | 2.32 | 16.01 |

Table 4: In-Sample Empirical Results - Panel D: Annual, Forecast Sample

Predictor(s) | Sample Begins | _{1,OLS} | _{1,m=2} | _{1,m=3} | _{1,m=4} | _{2,OLS} | _{2,m=2} | _{2,m=3} | _{2,m=4} | t_{1,OLS} | t_{2,OLS} | R²_{OLS} (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

D/P | 1927m1 | 3.97 | 4.33 | 2.47 | 3.11 | - | - | - | - | 3.24 | - | 10.89 |

E/P | 1927m1 | 2.05 | 2.07 | 1.97 | 1.76 | - | - | - | - | 3.12 | - | 6.78 |

Smoothed E/P | 1927m1 | 3.09 | 2.66 | 2.85 | 2.74 | - | - | - | - | 3.25 | - | 13.57 |

B/M | 1946m6 | 0.22 | 0.04 | 0.05 | 0.10 | - | - | - | - | 2.09 | - | 8.26 |

T-Bill rate | 1940m1 | 1.12 | 0.53 | 1.16 | 1.34 | - | - | - | - | 2.18 | - | 4.26 |

D/P and T-Bill rate | 1940m1 | 3.70 | 2.16 | 0.50 | 1.18 | 0.87 | 0.57 | 0.55 | 0.94 | 2.87 | 1.82 | 14.28 |

Table 5: Out-of-Sample Results - Panel A: Monthly

The table shows the out-of-sample *R*^{2} (expressed in percent)
that result from the forecasts of excess stock returns using the predictor variables indicated in the first column. The
forecasts are formed using either the OLS estimates or the jackknifed estimates, with *m* = 2, 3, and 4, and with or
without imposing the restrictions on the forecasts recommended by Campbell and Thompson (2007). Results for
both the monthly and annual data are shown. For each row, and for both the unrestricted and restricted
sets of forecasts, the highest out-of-sample *R*^{2} is shown in bold type. The first column indicates the predictor
variable(s) that the forecasts are based on, and the following two columns show the date at which the sample begins
and the date at which the out-of-sample forecasts begin, respectively. The difference between columns two and three
represents the 'training-sample' that is used to form the initial estimates for the first forecast. The following four columns
show the out-of-sample *R*^{2} for the unrestricted forecasts that do not impose the Campbell and Thompson
restrictions, and the last four columns show the correpsonding results with the Campbell and Thompson
restrictions in place.

Predictor(s) | Sample Begins | Forecast Begins | Unrestricted: R²_{OLS} | Unrestricted: R²_{m=2} | Unrestricted: R²_{m=3} | Unrestricted: R²_{m=4} | Restricted: R²_{OLS} | Restricted: R²_{m=2} | Restricted: R²_{m=3} | Restricted: R²_{m=4} |
---|---|---|---|---|---|---|---|---|---|---|

D/P | 1872m2 | 1927m1 | -0.66 | -0.31 | -0.62 | -0.54 | 0.16 | 0.44 | 0.33 | 0.36 |

E/P | 1872m2 | 1927m1 | 0.12 | 0.39 | 0.31 | 0.29 | 0.24 | 0.46 | 0.37 | 0.38 |

Smoothed E/P | 1881m2 | 1927m1 | 0.32 | 0.67 | 0.17 | 0.10 | 0.44 | 0.74 | 0.56 | 0.41 |

B/M | 1926m6 | 1946m6 | -0.44 | -0.38 | 0.72 | 0.42 | -0.01 | 0.13 | 0.78 | 0.47 |

T-Bill rate | 1920m1 | 1940m1 | 0.54 |
-13.74 | -0.29 | -4.30 | 0.58 | -13.75 | 0.84 | -0.10 |

D/P and T-Bill rate | 1920m1 | 1940m1 | 0.12 |
-10.50 | -1.22 | -3.00 | 0.17 | -9.21 | 1.09 | 0.22 |

Table 5: Out-of-Sample Results - Panel B: Annual

Predictor(s) | Sample Begins | Forecast Begins | Unrestricted: R²_{OLS} | Unrestricted: R²_{m=2} | Unrestricted: R²_{m=3} | Unrestricted: R²_{m=4} | Restricted: R²_{OLS} | Restricted: R²_{m=2} | Restricted: R²_{m=3} | Restricted: R²_{m=4} |
---|---|---|---|---|---|---|---|---|---|---|

D/P | 1872m2 | 1927m1 | 5.53 | 7.69 |
4.72 | 5.29 | 5.63 | 7.73 | 4.79 | 5.27 |

E/P | 1872m2 | 1927m1 | 4.93 | 5.23 |
4.65 | 3.92 | 4.94 | 5.24 | 4.72 | 3.97 |

Smoothed E/P | 1881m2 | 1927m1 | 7.89 |
5.67 | 3.15 | 2.01 | 7.85 | 5.70 | 4.23 | 2.61 |

B/M | 1926m6 | 1946m6 | -3.38 | -10.80 | 4.47 | 2.94 | 1.39 | -3.61 | 5.83 | 3.81 |

T-Bill rate | 1920m1 | 1940m1 | 5.54 | -2.24 | 8.20 | -0.98 | 7.47 | 0.34 | 9.45 | 7.40 |

D/P and T-Bill rate | 1920m1 | 1940m1 | 8.84 | 1.95 | 9.24 | 11.31 | 7.87 | 12.94 | 10.40 | 9.46 |

Table 6: Portfolio Choice Results - Panel A: Monthly

The table shows the utility gains, expressed in percent annualized expected returns, for an investor who uses the predictor variables indicated in the first column, instead of the historical mean, to time the market; the investor has mean-variance preferences with relative risk aversion equal to three. The portfolio weights are based on forecasts of the excess stock returns, formed using either the OLS estimates or the jackknifed estimates, with *m* = 2, 3, and 4, and with or without imposing the restrictions on the forecasts recommended by Campbell and Thompson (2007). Results for both the monthly and annual data are shown. For each row, and for both the unrestricted and restricted sets of forecasts, the highest utility gain is shown in bold type. The first column indicates the predictor variable(s) that the forecasts are based on, and the following two columns show the date at which the sample begins and the date at which the out-of-sample forecasts begin respectively. The difference between columns two and three represents the 'training-sample' that is used to form the initial estimates for the first forecast. The following four columns show the utility gains from the portfolio decisions based on the unrestricted forecats that do not impose the Campbell and Thompson restrictions, and the last four columns show the corresponding results with the Campbell and Thompson restrictions in place.

Predictor(s) | Sample Begins | Forecast Begins | Unrestricted: OLS | Unrestricted: m=2 | Unrestricted: m=3 | Unrestricted: m=4 | Restricted: OLS | Restricted: m=2 | Restricted: m=3 | Restricted: m=4 |
---|---|---|---|---|---|---|---|---|---|---|

D/P | 1872m2 | 1927m1 | -0.52 | 0.41 |
-0.06 | -0.09 | -0.43 | 0.50 | 0.08 | 0.03 |

E/P | 1872m2 | 1927m1 | 0.23 | 0.53 | 0.51 | 0.56 | 0.37 | 0.65 | 0.63 | 0.71 |

Smoothed E/P | 1881m2 | 1927m1 | -0.30 | -0.11 | 0.08 |
-0.09 | -0.26 | -0.07 | 0.38 | 0.08 |

B/M | 1926m6 | 1946m6 | -0.70 | -0.64 | 0.39 |
0.20 | -0.71 | -0.64 | 0.38 | 0.20 |

T-Bill rate | 1920m1 | 1940m1 | 1.68 | 2.12 | 2.15 |
1.55 | 1.67 | 2.11 | 2.22 | 1.75 |

D/P and T-Bill rate | 1920m1 | 1940m1 | -0.65 | 0.92 | 0.77 | 1.01 | -0.65 | 0.50 | 0.89 | 2.47 |

Table 6: Portfolio Choice Results - Panel B: Annual

Predictor(s) | Sample Begins | Forecast Begins | Unrestricted: OLS | Unrestricted: m=2 | Unrestricted: m=3 | Unrestricted: m=4 | Restricted: OLS | Restricted: m=2 | Restricted: m=3 | Restricted: m=4 |
---|---|---|---|---|---|---|---|---|---|---|

D/P | 1872m2 | 1927m1 | -0.54 | 0.30 | -0.30 | -0.35 | -0.55 | 0.28 | -0.30 | -0.35 |

E/P | 1872m2 | 1927m1 | 0.62 | 0.58 | 0.46 | 0.42 | 0.62 | 0.58 | 0.46 | 0.42 |

Smoothed E/P | 1881m2 | 1927m1 | 0.52 | 0.14 | -0.26 | 0.16 | 0.52 | 0.14 | 0.03 | 0.33 |

B/M | 1926m6 | 1946m6 | -0.57 | -1.64 | -0.02 | -0.45 | -0.62 | -1.63 | -0.03 | -0.46 |

T-Bill rate | 1920m1 | 1940m1 | 1.55 | 1.42 | 1.95 | 1.52 | 1.53 | 1.41 | 1.89 | 1.56 |

D/P and T-Bill rate | 1920m1 | 1940m1 | 0.00 | 1.33 | 0.95 | 2.07 | -0.06 | -0.51 | -0.31 | 0.23 |

Figure 1: Density Plots

Density plots for the OLS and jackknife estimates, based on 100,000 simulations for *T* = 100, *ρ* = 0.999 and *δ* = -0.99. The graphs shows the kernel density estimates of the bias in the OLS and jackknifed estimates, with *m* = 2, 3, and 4. The vertical solid line indicates a zero bias.

* Helpful comments have been provided by Daniel Beltran, Lennart Hjalmarsson, Randi Hjalmarsson, and Mike McCracken. Corresponding author: Erik Hjalmarsson. 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 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. Return to text

1. The only bias corrections, in predictive
regressions, that have been used to any great extent are ad hoc
corrections for the bias dervied by Stambaugh (1999), for the case
of a single regressor that follows an *AR*(1) process. Amihud and Hurvich
(2004) provide justifications for similar corrections in the case
of multiple regressors. Lewellen (2004) provides a 'conservative'
bias correction, also based on a single *AR*(1) regressor, which is
primarily useful as a tool for obtaining conservative test
statistics, since in general the corrected estimate will not be
unbiased but, rather, underestimate the true parameter value. In
fact, one of the main reasons that testing, rather than estimation,
has been the main focus is that most studies on inference in
predictive regressions resort to some conservative test, which does
not deliver a unique estimation analogue; e.g., Cavanagh et al.
(1995) and Campbell and Yogo (2006). Return to text

2. The case when the regressors are completely orthogonal to each other is obviously most desirable in empirical specifications, although it is not often likely to hold. In that case, however, there will be little to no difference between the individual coefficient estimates from separate regressions on each regressor and the estimates obtained from a multiple regression. Thus, there is little point in analyzing this case, since it would merely confirm the results obtained in the previous section. Return to text

3. See, for instance, Hansen and Hodrick (1980), Richardson and Stock (1989), Richardson and Smith (1991), Goetzman and Jorion (1993), Campbell (2001), Valkanov (2003), Torous et al. (2004), and Boudoukh et al. (2006). Nelson and Kim (1993) briefly discuss the magnitude of the Stambaugh (1999) bias in regressions with overlapping observations. Return to text

4. The data were obtained from Professor John Campbell's website and are described in more detail in Campbell and Thompson (2007). Return to text

5. One could consider various ways of implementing the restrictions on the jackknife estimates. Here we take the simplest approach and set the parameter estimate equal to zero if it has the wrong sign. Alternatively, one could restrict the individual sub-sample estimates in the jackknife estimator to have the right sign. Since the first approach immediately generalizes to the case of multiple regressors, unlike the second one which would become complicated to implement for more than one regressor, we use the first approach. In all cases, the intercepts are calculated to line up with the, potentially restricted, slope coefficient such that the residuals have mean zero. In the regressions with two predictor variables, each coefficient is restricted separately and the intercept is again estimated to produce zero mean residuals. Return to text

6. The use of a five-year window to estimate the variance of the (unexpected) returns conforms with the approach taken by Campbell and Thompson (2007). It can be justified by the fact that it is easier to calculate the variance of returns, as opposed to the expected value, over shorter time horizons, and there is a large literature that shows that variances change over time. Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text