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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 895, May 2007--- Screen Reader
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Abstract:

Since the seminal work of Mandelbrot (1963), -stable distributions with infinite variance have been regarded as a more realistic distributional assumption than the normal distribution for some economic variables, especially financial data. After providing a brief survey of theoretical results on estimation and hypothesis testing in regression models with infinite-variance variables, we examine the statistical properties of the coefficient of determination in models with -stable variables. If the regressor and error term share the same index of stability , the coefficient of determination has a nondegenerate asymptotic distribution on the entire [0, 1] interval, and the density of this distribution is unbounded at 0 and 1. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable. In contrast, if the indices of stability of the regressor and error term are unequal, the coefficient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability. In an empirical application, we revisit the Fama-MacBeth two-stage regression and show that in the infinite-variance case the coefficient of determination of the second-stage regression converges to zero in probability even if the slope coefficient is nonzero.

Keywords: Regression models, -stable distributions, infinite variance, coefficient of determination, Fama-MacBeth regression, Monte Carlo simulation, signal-to-noise ratio, density transformation theorem.

JEL classification: C12, C13, C21, G12

Granger and Orr (1972) begin their article, " 'Infinite variance' and research strategy in time series analysis," by questioning the uncritical use of the normal distribution assumption in economic modelling and estimation:

Due in part to the influential seminal work of Mandelbrot
(1963), -stable distributions are often
considered to provide the basis for more realistic distributional
assumptions for some economic data, especially for high-frequency
financial time series such as those of exchange rate fluctuations
and stock returns. Financial time series are typically fat-tailed
and excessively peaked around their mean--phenomena that can be
better captured by -stable distributions
with
rather than by the normal
distribution, for which .^{4} The -stable distributional assumption with
is thus a generalization of rather
than an alternative to the Gaussian distributional assumption. If
an economic series fluctuates according to an -stable distribution with , it
is known that many of the standard methods of statistical analysis,
which often rest on the asymptotic properties of sample second
moments, do not apply in the conventional way. In particular, as we
demonstrate in this paper, the coefficient of determination--a
standard criterion for judging goodness of fit in a regression
model--has several nonstandard statistical properties if
.

The linear regression model is one of the most commonly used and
basic econometric tools, not only for the analysis of macroeconomic
relationships but also for the study of financial market data.
Typical examples for the latter case are estimation of the
*ex-post* version of the capital asset pricing model (CAPM)
and the two-stage modelling approach of Fama and MacBeth (1973).
Because of the prevalence of heavy-tailed distributions in
financial time series, it is of interest to study how regression
models perform when the data are heavy-tailed rather normally
distributed.

The first purpose of the present paper is to survey theoretical results of estimation and hypothesis testing in regression models with infinite-variance distributions, and the second is to establish that infinite variance of the regression variables has important consequences for the statistical properties of the coefficient of determination and tests of the hypothesis that this coefficient is equal to zero. Third, we revisit the Fama-MacBeth two-stage regression approach and demonstrate that infinite variance of the regression variables can affect decisively the interpretation of the empirical results.

The rest of our paper is structured as follows. In Section 2 we provide a brief summary of the properties of -stable distributions and of aspects of estimation, hypothesis testing, and model diagnostic checking in regression models with infinite-variance regressors and disturbance terms. Section 3 provides a detailed analysis of the asymptotic properties of the coefficient of determination in regression models with infinite-variance variables. In our empirical application, presented in Section 4, we revisit the data used in Fama and French (1992), and we show that the statistical and/or economic interpretation of their findings can be quite different under the maintained assumption of -stable distributions from an interpretation based on the assumption of normal distributions. Section 5 summarizes the paper and offers some concluding remarks.

A random variable is said to have a stable
distribution if, for any positive integer ,
there exist constants and
such that
, where
are independent copies
of and
signifies equality in
distribution. The coefficient above is
necessarily of the form
for some
(see Feller, 1971,
Section VI). The parameter is called the
index of stability of the distribution, and a random
variable with index is
called -stable. An -stable
distribution is described by four parameters and will be denoted by
. Closed-form
expressions for the probability density functions of -stable distributions are known to exist only for
three special cases.^{5} However, closed-form expressions for
the characteristic functions of -stable
distributions are readily available. One parameterization of the
logarithm of the characteristic function of
is

where for , for , and for ; and for and for .

The tail shape of an -stable distribution is determined by its index of stability . Skewness is governed by ; the distribution is symmetric about if and only if . The scale and location parameters of -stable distributions are denoted by and , respectively. When , the log characteristic function given by equation (1) reduces to , which is that of a Gaussian random variable with mean and variance . For and , the tail properties of an -stable random variable satisfy

i.e., both tails of the probability density function (pdf)
of are asymptotically Paretian. For
and
(), the distribution is maximally
right-skewed (left-skewed) and only the right (left) tail is
asymptotically Paretian.^{6} The term in
equations (2)
and (3) is
given by

and for ; see, e.g., Samorodnitsky and Taqqu (1994), p. 17. The function , which is shown in Figure 1, is continuous and strictly decreasing in , with and .

Because
, it follows that
for
and
for
if is
-stable with
.^{8} Only moments of order
up to but not including are finite if
, and a non-Gaussian stable
distribution's index of stability is also equal to its maximal
moment exponent.^{9} In particular, if
, the variance is infinite but
the mean exists. For , it follows that
; in addition, for , is equal to the
distribution's mode and median irrespective of the value
of , justifying the use of the term
``central location parameter'' for in the
finite-mean or symmetric cases. In addition, for
, one can show that
.^{10} We make use of this property below
in the derivations of Theorem 1 and
Remark 3.

The class of -stable distributions is an interesting distributional candidate for disturbances in regression models because (i) it is able to capture the relative frequencies of extreme vs.observations in the economic variables, (ii) it has the convenient statistical property of closure under convolution, and (iii) only -stable distributions can serve as limiting distributions of sums of independent and identically distributed (iid) random variables, as proven in Zolotarev (1986). The latter two properties are appealing for regression analysis, given that disturbances can be viewed as random variables which represent the sum of all external effects not captured by the regressors. For more details on the properties of -stable distributions, we refer to Gnedenko and Kolmogorov (1954), Feller (1971), Zolotarev (1986), and Samorodnitsky and Taqqu (1994). The role of the -stable distribution in financial market and econometric modelling is surveyed in McCulloch (1996) and Rachev et al.1999).

Let and be two jointly
symmetric -stable (henceforth, ) random variables with ,
i.e., we require and to have
finite means. Our main reason for concentrating on the case
lies in its empirical relevance.
Estimated maximal moment exponents for most empirical financial
data, such as exchange rates and stock prices, are generally
greater than 1.5; see, for example, de Vries (1991) and
Loretan and Phillips (1994). An econometric (purposeful) reason for
studying the case is that, for
-stable distributions with , regression analysis that is based on sample
second moments, such as least squares, is still asymptotically
consistent for the regression coefficients, even though the limit
distributions of these regression coefficients are
nonstandard.^{11} Suppose that the regression of a
random variable on a random
variable is linear, i.e., there exists a
constant such that

with

where is the scale parameter of the variable and in the numerator is covariation (covariance in the Gaussian case), which can be calculated as , for all with .

For estimation and diagnostics, the relation (5) can be written as a regression model with a constant term,

where the maintained hypothesis is that is iid , with . The econometric issues of interest are to estimate properly, to test the hypothesis of significance for the estimated parameter, usually based on the -statistic, as well as to compute model diagnostics, such as the coefficient of determination, the Durbin-Watson statistic, and the -test of parameter constancy across subsamples.

The effects of infinite variance in the regressor and disturbance term can be substantial. If the variables share the same index of stability , the ordinary least squares (OLS) estimate of is still consistent, but its asymptotic distribution is -stable with the same as the underlying variables. Furthermore, the convergence rate to the true parameter is , smaller than the rate which applies in the finite-variance case. If , OLS loses its best linear unbiased estimator (BLUE) property, i.e., it is no longer the minimum-dispersion estimator in the class of linear estimators of . In addition, the asymptotic efficiency of the OLS estimator converges to zero as the index of stability declines to . Blattberg and Sargent (1971) (henceforth, BS) derived the BLUE for in (6) if the value of is known. The BS estimator is given by

(7) |

which coincides with the OLS estimator if . Kim and Rachev (1999) prove that the asymptotic distribution of the BS estimator is also -stable. Samorodnitsky et al.2007) consider an optimal power estimate based on the BS estimator for unknown , and they also provide an optimal linear estimator of the regression coefficients for various configurations of the indices of stability of and . Other efficient estimators of the regression coefficients have been studied as well; Kanter and Steiger (1974) propose an unbiased -estimator, which excludes very large shocks in its estimation to avoid excess sensitivity due to outliers. Using a weighting function, McCulloch (1998) considers a maximum-likelihood estimator which is based on an approximation to a symmetric stable density.

Hypothesis testing is also affected considerably when the regressors and disturbance terms have infinite-variance stable distributions. For example, the -statistic, commonly used to test the null hypothesis of parameter significance, no longer has a conventional Student- distribution if . Rather, as established by Logan et al.1973), its pdf has modes at and ; for these modes are infinite. Kim (2003) provides empirical distributions of the -statistic for finite degrees of freedom and various values of by simulation. The usual applied goodness-of-fit test statistics, such as the likelihood ratio, Lagrange multiplier, and Wald statistics, also no longer have the conventional asymptotic distribution, but have a stable distribution, a term that was introduced by Mittnik et al.1998).

In time series regressions with infinite-variance innovations, Phillips (1990) shows that the limit distribution of the augmented Dickey-Fuller tests for a unit root are functionals of Lévy processes, whereas they are functionals of Brownian motion processes in the finite-variance case. The -test statistic for parameter constancy that is based on the residuals from a sample split test has an -distribution in the conventional, finite-variance case. Kurz-Kim et al.2005) obtain the limiting distribution of the -test if the random variables have infinite variance. As shown by the authors, as well as by Runde (1993), the limiting distribution of the -statistic for behaves completely differently from the Gaussian case: whereas in the latter case the statistic converges to 1 under the null as the degrees of freedom for both numerator and denominator of the statistic approach infinity, in the former case the statistic converges to a ratio of two independent, positive, and maximally right-skewed -stable distributions. This result is used below to derive closed-form expressions for the pdf and cumulative distribution function (cdf) of the limiting distribution of the statistic if the regressor and disturbance term share the same index of stability .

Moreover, commonly used criteria for judging the validity of
some of the maintained hypotheses of a regression model, such as
the Durbin-Watson statistic and the Box-Pierce -statistic, would be inappropriate if one were to rely on
conventional critical values. Phillips and Loretan (1991) study the
properties of the Durbin-Watson statistic for regression residuals
with infinite variance, and Runde (1997) examines the properties of
the Box-Pierce -statistic for random variables
with infinite variance. Loretan and Phillips (1994) and Phillips
and Loretan (1994) establish that both the size of tests of
covariance stationarity under the null and their rate of divergence
of these tests under the alternative are strongly affected by
failure of standard moment conditions; indeed, standard tests of
covariance stationarity are *inconsistent* if population
second moments do not exist.

For the general asymptotic theory of stochastic processes with stable random variables, we refer to Resnick (1986) and Davis and Resnick (1985a, 1985b, 1986). Our results in this section are, in large part, an application of their work to the regression diagnostic context.

The maintained assumptions are:

- The relationship between the dependent and independent variable
conforms to the classical bivariate linear regression model,
- is iid , with .
- is exogenous and is also iid , with .
- The regressor and the error term have the same index of stability, i.e., .
- The coefficients
and are consistently estimated
by and
.
^{12}

The fourth assumption, that the regressor and the error term have the same index of stability, is rather strong, and its validity may be difficult to ascertain in empirical applications. In Corollary 2 below, we examine the consequences of having unequal values for the indices of stability for and for the asymptotic properties of the coefficient of determination.

The coefficient of determination measures the proportion of the total squared variation in the dependent variable that is explained by the regression:

Because and , where and are the respective sample averages of and , and because =0 by construction, the coefficient of determination may be written as

Since and are in the normal domain of attraction of a stable distribution with index of stability , norming by rather than by is required to obtain non-degenerate limits for the sums of the squared variables. Because by the assumption of consistent estimation, an application of the law of large numbers to , the continuous mapping theorem, and the results of Davis and Resnick (1985b) yield the following expression for the joint limiting distribution of the elements in equation (9):

For , the random variables
and are
independent, maximally right-skewed, and positive stable random
variables with index of stability
, ,
,^{13} , and log characteristic function

We therefore conclude that, under the five maintained assumptions of this section, the statistic of the regression model (8) has the following asymptotic distribution.

Thus, for and , the coefficient of determination does *not*
converge to a constant but has a nondegenerate asymptotic
distribution on the interval . This
contrasts starkly with the standard, finite-variance result, which
is stated here for completeness.

In the finite-variance case, the model's asymptotic signal-to-noise ratio, , is constant, as is therefore the limit of the coefficient of determination. In contrast, in the infinite-variance case the model's limiting signal-to-noise ratio is given by , where and , and is therefore a random variable even asymptotically; it is this feature that causes the randomness of . We postpone a fuller discussion of the intuition that underlies this result to the end of this section, after we provide a detailed analysis of the statistical properties of .

Before doing so, however, we note that the fourth maintained assumption, i.e., that the indices of stability of the regressor and error term in (8) be the same, is crucial for obtaining the result that the asymptotic distribution of is nondegenerate. Indeed, if the two indices of stability differ, the asymptotic properties of the statistic are as follows.

**Corollary 2**
*Suppose that the maintained assumptions of
**Theorem 1 **apply
except that
, i.e., suppose that
the indices of stability of the regressor and error term are
unequal. Let
to rule out the trivial case from
further consideration. Then,*

- if , ; and
- if , .

*Thus, converges to
in probability if
, and it converges
to 0 in probability if
.*

Similarly, if , , and .

Heuristically, if and , the limiting distribution of the statistic is degenerate at 0 or 1 because the model's asymptotic signal-to-noise ratio is either zero (if ) or infinite (if ). From an examination of the proof of this corollary, we can also deduce that if , the fifth maintained assumption--that the regression coefficients are estimated consistently--could be relaxed, to require merely that an estimation method be employed that guarantees ; the result that converges either to 0 or 1 would continue to hold in this case.

Returning to the main case of , we note that the random variable is defined for all values of , even though in a regression context one would typically assume that . We now establish some important qualitative properties of .

Because and are iid and have continuous cdfs, by an application of Fubini's Theorem.

Thus, is equal to the non-random limit
of in the finite-variance case. Since
and are positive
*a.s.*, we also have
,
i.e., the median of is equal to 1,
regardless of the value of . As we will
demonstrate rigorously later in this paper, the probability mass
of is highly concentrated around 1 for
values of close to 2. Conversely, for
small values of ,
is *unlikely* to be close to 1; instead, it is very
likely that one will obtain a draw of that is
either very small, i.e., close to 0, or very large. A small or
large draw of has a crucial effect on the
model's signal-to-noise ratio, , and
therefore also on . This suggests that an
informal measure of the effect of infinite variance in the
regression variables on the value of in
a given sample may be based on the *difference* between the
model's coefficient of determination and a consistent estimate of
its median , say
, where
. The larger the difference between
and , the more important the effect is of
having obtained a small (or large) value of .

The following remark shows that a finite-variance property
of
for ,
*viz.*,
, carries over in
a natural way to
.

The symmetry of about for follows immediately from this result and the fact that the distribution's support is the interval [0, 1].

Next, as the following remark shows, the pdf of
has *infinite* modes
at 0 and , i.e., at the
*endpoints* of its support.

where the joint pdf is nonzero on . The case can occur only if ; if , however, the random variables and are perfectly dependent, their joint pdf is nonzero only on the positive -halfline, and the joint pdf reduces to , . Hence, for we find

By Remark 2, we have as well. The continuity of the cdf of on for follows from the continuity of the cdfs of and on and the fact that their pdfs are equal to zero at the origin. For example, one finds that ; the result then follows from Remark 2.

The fact that the probability density function of
has infinite singularities
may seem unusual. However, the presence of singularities is a
regular feature of pdfs that are based on *ratios* of stable
random variables. For example, Logan et al.1973) and Phillips
and Hajivassiliou (1987) showed that if ,
the density of the -statistic has infinite
modes at and ; similarly,
Phillips and Loretan (1991) demonstrated that if , this feature is also present in the asymptotic
distributions of the von Neuman ratio and the normalized
Durbin-Watson test statistic.

The remarks in the preceding subsection provide important
qualitative information about some of the distributional properties
of
. However, they do not
address issues such as whether the distribution has modes beyond
those at 0 and 1, whether the discontinuity of the pdf at
the endpoints is simple or if
diverges--and, if so,
at which rate--as
or
, or how much of the
distribution's mass is concentrated near the endpoints of the
support. To examine these issues, we provide expressions for the
cdf and pdf of
in this subsection.
It is possible to do so because
is a continuously
differentiable and invertible function of the *ratio* of two
independent, maximally right-skewed, and positive -stable random variables, and because closed-form
expressions for the cdf and pdf of this ratio are known. The latter
expressions are provided in the following proposition.

The cdf of the random variable is shown in
Figure 2 for various values of
between 1.98 and 0.25.^{18}The random variable
has several interesting properties. First,
note that
and that
the rate of divergence to infinity of as
is given by
; thus, the pdf
of has a one-sided infinite singularity
at 0. Second, as
,
for a suitable
constant . This result, along with
, implies that lies in the normal domain of attraction of a positive
stable distribution, say
, with index of stability
and , the
same parameters as that of the variables and .^{19} Hence, the mean
of is infinite for all values of . Third, in the special case of , and are
each distributed as a Lévy -stable
random variable, which is well known to be equivalent to the
inverse of a
random variable. For , then, the pdf of reduces to
, which is
also the pdf of an distribution; see
Runde (1993).

As was noted earlier, the median of is
equal to 1 for all values of
. The regression model's
signal-to-noise ratio is given by the random
variable if ,
whereas it is given by the constant in
the standard, i.e., finite-variance case. The fact that the random
variable which multiplies has a median
of 1 helps to develop further the intuition that underlies the
result of Remark 1, *viz.*,the
median of
,
, is the same in both the
finite-variance and the infinite-variance cases. Finally, an
inspection of equation (13) reveals that
and
; put
differently,
. The
probability mass of therefore becomes perfectly
concentrated at 1 as
, even though, of course, its
mean remains infinite as long as .

From Theorem 1, we have
, say.
Note that
satisfies the conditions
of Proposition 1 and that the
function
is continuously differentiable and strictly increasing in the
interior of its domain. We are therefore able to provide the
following expressions for the cdf and pdf of
by an application of the
density transformation theorem.^{20}

*The pdf of
for
is given by*

The probability density functions and cumulative distribution functions of for values of between 0.25 and 1.98 are graphed in Figures 3 and 4. (In all cases, we have set .) The pdfs in Figure 3 are shown with a logarithmic scale on the ordinate. Since we know that , we graph the functions only for . The graphs show that

- If is close to but less than 2, e.g., if or , the pdf has an interior mode, and most of the probability mass of is concentrated near its median. Conversely, only very little mass is located near 0 and 1, and the pdfs register only mild increases as approaches either edge of the distribution's support.
- For and , the distribution of continues to have an interior mode (as well as, of course, the two unbounded modes at 0 and 1). However, the distribution is noticeably less concentrated around the interior mode than if is closer to 2.
- By , the interior mode has disappeared and the distribution is nearly uniform over the entire interval .
- If takes on even smaller values, less and less of the probability mass of is located near the median, and more and more of it is concentrated close to 0 and 1.
- If , about 75 percent of the probability mass lies within 0.001 of the two endpoints of the distribution, while the probability of observing a realization of for is less than 5 percent.

A heuristic summary of these properties of is straightforward. We begin by recalling that the multiplicative term , shown in equation (4) and Figure 1, affects the probability of tail-region values of the random variables in question, and that the rate of decline in the tail areas of density of -stable random variables increases as . Suppose first that is very close to 2; then, is close to 0, and the fraction of observations of and that fall into the respective Paretian-tail regions is therefore very low; moreover, given the fairly rapid decay of the density's tails for close to 2, the likelihood of obtaining a very large draw, conditional on obtaining a draw from the Paretian tail area, is also low. As a result, the probability of observing large observations of and is quite low. This, in turn, makes it unlikely to observe a very large draw of either or and thus of observing a value of that is either close to 0 or very large. Therefore, if is very close to 2, is likely close to its median of 1, and most of the mass of is concentrated near its median, . Next, as moves down and away from 2, say to around 1.5, increases rapidly, leading to a higher frequency of observing tail-region draws for and . In addition, as the density in the tail region declines more slowly for smaller values of , it is much more likely of obtaining very large draws of the regressor and error term than if is close to 2. In consequence, if is around 1.5, it is quite likely to obtain draws of that are either very close to zero or very large, and thus more of the probability mass of is located near the edges of its support. Conversely the interior mode of is considerably less pronounced than if is close to 2. Finally, as decreases further, rises further, and both the frequency of tail observations and the likelihood that any draws from the tail areas will be very large increase. Therefore, it is very likely that the largest few observations of or will dominate the realization of and therefore the realization of . As a result, if is small the central mode of vanishes entirely and almost all of its probability mass is located very close to the endpoints of the distribution's support. In the limit, as , converges to a Bernoulli random variable, for which all of the probability mass is located at 0 and 1.

Fama and MacBeth (1973) proposed the so-called Fama-MacBeth
regression to test the hypothesis of a linear relationship between
risk and risk premium in stock returns in a cross-sectional
setting. Let be the return on market
portfolio at time ,
where
and
; denote the average return of
portfolio as
;
denote the average portfolio return at time
as
; and
denote the average portfolio return across all time periods by
. The
first-stage Fama-MacBeth regression is an *ex post* CAPM,

where , , and is iid the same index as . We may assume that the distribution of has a finite mean and variance, say, and . Denote the OLS estimates of the regression coefficients in equation (17) by and . The second-stage Fama-MacBeth regression is given by

where is iid the same index as , , and .

The statistic of the second-stage Fama-MacBeth regression is given by

This statistic has the following asymptotic properties.

- If , , where ; and
- If , .

*Thus, if ,
, at a rate that is
proportional to
.*

This result does not conflict with the one provided in Theorem 1, as the present case is one of an unbalanced regression design: the regressor has an asymptotically finite variance, whereas the error term has infinite variance, implying that the asymptotic signal-to-noise ratio is zero. Instead, this result is closely related to the one provided in Corollary 2, which examined the asymptotic limit of if . We note that even if is fixed (as is generally taken to be the case in Fama-MacBeth regressions), the dispersion of will likely be quite a bit smaller than that of , indicating that the model's signal-to-noise ratio, , and hence the median of , in the second-stage regression will be quite small unless is sufficiently large in absolute value.

These qualitative observations are confirmed by a small-scale
Monte Carlo simulation, shown in Table 1, in which we
report the median value of as a function
of two values of and selected values
of , ,
and .^{21} It is evident for
both
and
that the median value
of declines as
increases if is fixed, that this effect
is particularly strong if is large, and that
this effect is more pronounced for
than it is for
. The final result is as one
would expect, given that Theorem 3 states that the
rate of convergence of to zero
increases as moves down further
from 2.

On the basis of the small value of coefficient of determination from the Fama-MacBeth regression, Jagannathan and Wang (1996) confirm the finding of Fama and Macbeth (1973) of a ``flat'' relation between average return and market beta. They report a very low coefficient of determination of 1.35%=0.0135 for the Sharpe-Lintner-Black (SLB) static CAPM. Regarding ``thick-tailed'' phenomena in empirical data, Fama and French (1992) conjectured that neglecting the heavy-tails phenomenon of the data does not lead to serious errors in the interpretation of empirical results. In the following, we use the same CRSP dataset as was used by Jagannathan and Wang (1996); the data are very similar to those that were used in the study of Fama and French (1992). The data consist of stock returns of nonfinancial firms listed on the NYSE and AMEX from July 1963 until December 1990 covered by CRSP alone; the frequency of observation is monthly. In the preceding notation, we have and . Figure 5 displays the time series of these monthly returns.

For our analysis we need to obtain point estimates the index of
stability of the stock returns and determine whether the estimates
are less than 2. Under the assumption of symmetry, which
implies that the left and right tails of the returns distribution
possess the same maximal moment exponent and dispersion
coefficient, the point estimate of for
monthly stock returns in the CRSP dataset using the Hill method
(Hill, 1975) is 1.77, with a standard deviation
of 0.15.^{22} On the basis of these estimates,
normality () can be excluded only at a
confidence level of approximately 87.5 percent. However,
inference about the width of the confidence interval for the Hill
estimator is valid only asymptotically; in finite samples, the
Hill-method estimates are known to be quite sensitive to even minor
departures from exactly Paretian tail behavior.^{23} In contrast, the
method of Dufour and Kurz-Kim (2007) provides exact confidence
intervals for finite samples. By their method, the point estimate
of for the monthly stock returns data
is 1.78, and the exact finite-sample 90interval for this point
estimate is [1.64, 1.99]. This result also does not offer very
strong evidence against the hypothesis .
Nevertheless, because of estimation uncertainty in small samples,
and because this uncertainty is especially severe if is close to 2, the data can be regarded as being
in the domain of attraction of a stable distribution with .^{24} We therefore proceed
to investigate the consequences of this finding for the proper
interpretation of the low statistic
reported by Jegannathan and Wang (1996).

We designed Monte Carlo simulations to obtain the cdf of for our empirical data, first under the assumption that the returns data are in the domain of attraction of an -stable distribution with , and second under the assumption of normality (). The simulation was calibrated to the main characteristics the empirical data; we set , , , and we set the expected return equal to the average annual return in the full sample, i.e., . The number of replications of the first-stage and second-stage Fama-MacBeth regressions is 100,000, for the both values of . The simulated cdfs of the -statistic are shown in Figure 6, where a vertical line is drawn at to indicate the in-sample value of the coefficient of determination. The shapes of the two curves are rather different, with the one for rising much more quickly for small values of .

The simulated median of the second-stage Fama-MacBeth regression is 0.384 for , but it is only 0.072 for . The simulated probability of obtaining is a minuscule 1.55 percent for , but it is a much more sizable 21.88 percent for ; thus, if the event is about 14 times more probable than if . On the basis of these findings, we conclude that the inference drawn from the low value of by Fama and French (1992)--that the empirical usefulness of the SLB CAPM is refuted--does not seem to be robust once proper allowance is made for the distributional properties of the data that give rise to this statistic.

After providing a brief overview of some of the properties of -stable distributions, this paper surveys the literature on the estimation of linear regression models with infinite-variance variables and associated methods of conducting hypothesis and specification tests. Our paper adds to the already-wide body of knowledge that there are substantial differences between regression models with infinite-variance and finite-variance regressors and error terms by examining the properties of the coefficient of determination. In the infinite-variance case with iid regressors and error terms that share the same index of stability , we find that the statistic does not converge to a constant but instead that it has a nondegenerate asymptotic distribution on the interval, with a pdf that has infinite singularities at 0 and 1. We provide closed-form expressions for the cdf and pdf of this limit random variable. If the regressors and error term do not have the same index of stability, we show that the coefficient of determination collapses either to 0 or to 1, depending on whether the model's signal-to-noise ratio converges asymptotically to zero or infinity. Finally, we provide an empirical application of our methods to the Fama-MacBeth two-stage regression setup, and we show that the coefficient of determination asymptotically converges to 0 in probability if the regression variables have infinite variance. This, in turn, strongly suggests that low values of the statistic should not, by themselves, be taken as proof of a ``flat'' relationship between the dependent variable and the regressor.

In view of the random nature of the limit law
if the regressors and error
terms share the same index of stability, and given our related
finding that the coefficient of determination converges to zero in
probability if the tail index of the disturbance term is smaller
than that of the regressor, a case that may be difficult to rule
out in empirical practice unless the sample size is very large, we
view our results as establishing that one should *not* rely
on as a measure of the goodness of fit of
a regression model whenever the regressors and disturbance terms
are sufficiently heavy-tailed to call into question the existence
of second (population) moments. At the very least, if one chooses
to report the coefficient of determination in regressions with
infinite-variance variables at all, one should also report a point
estimate of the median of
,
,
where is as in Theorem 1. In
addition, one should indicate whether the error terms and
regressors may reasonably be assumed to share the same index of
stability. If the validity of that assumption is in doubt, the
authors should also indicate which of the two parameters is likely
to be smaller and how far apart the two parameters may plausibly
be.

It is widely known, and it is certainly stressed in all
introductory econometrics textbooks, that a *high* value
of does not provide a sufficient basis
for concluding that an empirical regression model is a ``good''
explanation of the dependent variable, or even that the regression
is correctly specified. Nevertheless, one suspects, researchers may
view *low* values of in an
empirical regression as an indication that the (linear)
relationship is either weak or unreliable. A direct implication of
the work presented in this paper is that whenever the data are
characterized by significant outlier activity, a low value
of should not, by itself, be used to
disqualify the model from further consideration.

Several extensions to the work presented here are possible. First, the regression -statistic is a simple function of the coefficient of determination; e.g., in the bivariate regression case. Given the close connection between the two statistics, it seems useful to study if and how the distributional properties of the regression -statistic are affected by the presence of -stableand error terms under both the null hypothesis, , and the alternative hypothesis, . It would also be useful to elaborate on our idea, offered after Remark 1 in subsection 3.2, that the difference between the estimate of and a consistent estimate of its median may serve as a diagnostic check of the size of the effect of infinite variance on . For example, it may be feasible to develop an asymptotic theory of the distributional properties of this difference.

It also seems desirable to study how well the distribution of approximates the empirical distribution of in finite samples, for various types of heavy-tailed distributions that are in the domain of attraction of distributions, and for various types of estimators (such as OLS, Blattberg-Sargent's BLUE, and the least-absolute deviation estimator). In addition, an extension to a multiple-regression framework may produce additional insights into the properties of the coefficient of determination in the infinite-variance case. Finally, the theoretical results presented in our paper depend crucially on the assumption that the random variables are iid. Relaxing this assumption would seem to be useful, as many economic and financial time series--especially if they are sampled at very high frequencies--are characterized by interesting dependence and heterogeneity features. Introducing serial dependence and heterogeneity, especially conditional heterogeneity, would serve the purpose of studying how the properties of may be affected by such departures from the basic case of iid variables. The authors are considering conducting research to extend the work presented in this paper along these lines.

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Table 1. Median Value of as a Function of , , , and

=0.0 | =0.1 | =0.3 | =0.5 | =1.0 | |||
---|---|---|---|---|---|---|---|

1.50 | 100 | 30 | 0.0404 | 0.0425 | 0.0576 | 0.1009 | 0.2963 |

1.50 | 100 | 100 | 0.0162 | 0.0172 | 0.0292 | 0.0596 | 0.2019 |

1.50 | 100 | 500 | 0.0068 | 0.0075 | 0.0147 | 0.0325 | 0.1206 |

1.50 | 100 | 1000 | 0.0046 | 0.0055 | 0.0114 | 0.0264 | 0.0973 |

1.50 | 250 | 30 | 0.0402 | 0.0426 | 0.0779 | 0.1598 | 0.4417 |

1.50 | 250 | 100 | 0.0161 | 0.019 | 0.0448 | 0.1058 | 0.3304 |

1.50 | 250 | 500 | 0.0064 | 0.0075 | 0.022 | 0.0565 | 0.202 |

1.50 | 250 | 1000 | 0.0047 | 0.0058 | 0.0172 | 0.0452 | 0.1667 |

1.50 | 1000 | 30 | 0.0387 | 0.0484 | 0.1499 | 0.332 | 0.6748 |

1.50 | 1000 | 100 | 0.0162 | 0.0223 | 0.094 | 0.2272 | 0.5558 |

1.50 | 1000 | 500 | 0.0065 | 0.0104 | 0.0521 | 0.1341 | 0.3994 |

1.50 | 1000 | 1000 | 0.0046 | 0.0072 | 0.0399 | 0.1079 | 0.3443 |

1.50 | 2500 | 30 | 0.0403 | 0.058 | 0.2478 | 0.4806 | 0.7962 |

1.50 | 2500 | 100 | 0.0155 | 0.0294 | 0.1621 | 0.3581 | 0.6973 |

1.50 | 2500 | 500 | 0.0066 | 0.013 | 0.0883 | 0.2243 | 0.5507 |

1.50 | 2500 | 1000 | 0.0047 | 0.0103 | 0.0737 | 0.1896 | 0.497 |

1.75 | 100 | 30 | 0.0488 | 0.0543 | 0.1332 | 0.2944 | 0.641 |

1.75 | 100 | 100 | 0.026 | 0.0328 | 0.1032 | 0.2413 | 0.5756 |

1.75 | 100 | 500 | 0.0177 | 0.0222 | 0.0779 | 0.1941 | 0.5055 |

1.75 | 100 | 1000 | 0.0149 | 0.0199 | 0.072 | 0.1778 | 0.4792 |

1.75 | 250 | 30 | 0.0474 | 0.0642 | 0.2509 | 0.4899 | 0.7993 |

1.75 | 250 | 100 | 0.0265 | 0.043 | 0.2066 | 0.4264 | 0.756 |

1.75 | 250 | 500 | 0.0169 | 0.029 | 0.1571 | 0.35 | 0.695 |

1.75 | 250 | 1000 | 0.0143 | 0.0251 | 0.144 | 0.3273 | 0.673 |

1.75 | 1000 | 30 | 0.047 | 0.1193 | 0.5351 | 0.7665 | 0.9309 |

1.75 | 1000 | 100 | 0.0265 | 0.0871 | 0.4612 | 0.7124 | 0.9115 |

1.75 | 1000 | 500 | 0.0169 | 0.0635 | 0.391 | 0.6507 | 0.8865 |

1.75 | 1000 | 1000 | 0.0144 | 0.0579 | 0.3663 | 0.6257 | 0.8744 |

1.75 | 2500 | 30 | 0.048 | 0.2185 | 0.7214 | 0.8804 | 0.9677 |

1.75 | 2500 | 100 | 0.0255 | 0.1704 | 0.6599 | 0.8474 | 0.9578 |

1.75 | 2500 | 500 | 0.0169 | 0.1251 | 0.59 | 0.8066 | 0.9452 |

1.75 | 2500 | 1000 | 0.0149 | 0.1202 | 0.5674 | 0.7902 | 0.9394 |

The numbers in the body of the table are the medians from simulated distributions with 100,000 replications.

Figure 1. The Function , 0 <

Figure 2. Cumulative Distribution Functions of

Figure 3. Probability Density Functions of ,

Figure 4. Cumulative Distribution Functions of ,

Figure 5. CRSP Returns, July 1963 to December 1992

Figure 6. Simulated cdf of , Second-Stage Fama-MacBeth Regressions

1. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the staff of the Deutsche Bundesbank, the Board of Governors of the Federal Reserve System, or of any other person associated with the Federal Reserve System. We are grateful to Jean-Marie Dufour, Neil R. Ericsson, Peter C.B. Phillips, Werner Ploberger, Jonathan H. Wright, and participants of a workshop at the Federal Reserve Board for valuable comments, and to Zhenyu Wang for the data used in the empirical section of this paper. Return to text

2. Corresponding author. Research support from the Alexander von Humboldt Foundation is gratefully acknowledged. Deutsche Bundesbank, Wilhelm-Epstein-Strasse 14, 60431 Frankfurt am Main, Germany; Email: jeong-ryeol.kurz-kim@bundesbank.de; Tel: +49-69-9566-4576, Fax: +49-69-9566-2982. Return to text

3. Mailstop 18, Board of Governors of the Federal Reserve System, Washington DC 20551, USA; Email: mico.loretan@frb.gov; Tel: +1-202-452-2219; Fax: +1-202-263-4850. Return to text

4. The normal distribution is the only member of the family of -stable distributions that has finite second (and higher-order) moments; all other members of this family have infinite variance. Return to text

5. The three special cases are: (i) the Gaussian distribution , (ii) the symmetric Cauchy distribution , and (iii) the Lévy distribution ; see Zolotarev (1986), Section 2, and Rachev et al. (2005), Section 7. Return to text

6. For and , , i.e., the distribution's support is bounded below by . Zolotarev (1986, Theorem 2.5.3) and Samorodnitsky and Taqqu (1994, pp. 17-18) provide expressions for the rate of decline of the non-Paretian tail if and . Return to text

7. The function is smooth on the entire interval . The numerator and the second term in the denominator of equation (4) both converge to 0 as ; therefore follows from an application of L'Hôpital's Rule. Return to text

8. Ibragimov and Linnik (1971,
Theorem 2.6.4) show that this result holds not only for
-stable distributions, but that it
pertains to *all* distributions that are in the domain of
attraction of an -stable distribution.
Ibragimov and Linnik (1971, Theorem 2.6.1) provide necessary and
sufficient conditions for a probability distribution to lie in the
domain of attraction of an -stable
law. Return to text

9. The maximal moment exponent of a distribution is either a finite positive number, or it is infinite if a distribution has finite moments of all orders. For a Student- distribution, the degrees of freedom parameter is equal to its maximal moment exponent. Return to text

10. This result also holds for the case
*and* . Return to text

11. Another reason for this restriction comes from the viewpoint of statistical modelling. The conditional expectation of the bivariate symmetric stable distribution in (5) is, as in the Gaussian case, linear in only if . The regression function is in general nonlinear, or rather only asymptotically linear, under other conditions. For more on bivariate linearity, see Samorodnitsky and Taqqu (1994, Sections 4 and 5). Return to text

12. If , OLS is known to generate consistent estimates of and . See Samorodnitsky et al. (2007) for an overview and discussion of estimation methods that are consistent for various combinations of and . Return to text

13. To prove that , see equation (13.3.14) on p. 529 of Brockwell and Davis (1991). In that equation, put , where is given by equation (4), and employ the recursive relationship . Return to text

14. Observe that if and only if , as the dispersion parameters and are necessarily positive. Return to text

15. Recall that in the finite-variance case, ; therefore, norming by and in equation (10) produces a constant of . Return to text

16. See, e.g., Resnick (1999, p. 155). Return to text

17. See, e.g., Mood, Graybill, and Boes (1974), p. 187. Return to text

18. Runde (1993) graphs pdfs of for values of between and . Return to text

19. See Mittnik et al. (1998) for a discussion of some of the properties of the stable law . Return to text

20. See, e.g., Mood, Graybill, and Boes (1974, p. 200). Return to text

21. The design of the simulation and the choices of values for , , and were influenced by a desire to maximize the empirical relevance of the simulation exercise. We chose and because for most empirical economic data. We study the cases of , , , and because corresponds approximately to the number of business days in a calendar year. The values of , , , and correspond to the numbers of stocks contained in certain well-known stock price indices, such as the U.S. Dow-Jones Industial and German DAX indices, the U.K. FTSE-100 index, the U.S. S&P-500 index, etc. The choice of provides a reference to contrast the cases of and ; is particularly relevant for the empirical study provided below. Return to text

22. In this estimation, we used 0.0031 as the centering offset for the empirical data; this adjustment is necessary because the Hill estimator is not location-invariant. The offset is equal to the estimated location parameter obtained by the quantile estimation method of McCulloch (1986). The choice of the number of order statistics to include in the Hill method used was determined by the Monte Carlo method of Dufour and Kurz-Kim (2007). For the present dataset, this method indicated the use of 43% of all observations.

The Hill estimator uses extreme observations from both tails of the empirical distribution under the assumption of symmetry, but it uses only observations from the right (left) tail under the assumption of right-skewed (left-skewed) asymmetry. In the case of the monthly stock returns, the distribution is clearly left-skewed, i.e., the largest negative returns are larger in the sample than the largest positive returns; see Figure 5. Under the assumption of left-skewed asymmetry, the point estimate of for the left tail using the Hill method is 1.47, with one standard deviation of 0.18. Return to text

23. Stable distributions have tails that
are *asymptotically* Paretian. In finite samples, and
especially if the index of stability is not far below 2, it is
known that the tails of stable distributions are not approximated
particularly well by Pareto distributions with the same value
of . See Resnick (2006, pp. 86-9) for
a discussion of the consequences of these finite-sample features
for the reliability of the Hill estimator. Return to text

24. For a broader discussion of how to decide if , see McCulloch (1997). 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