Do Idiosyncratic Risks in Multi-Factor Asset Pricing Models Really Contain a Hidden Non-Diversifiable Factor? A Diagnostic Testing Approach ()
1. Introduction
Empirical studies on asset pricing models generally require the application of some ad hoc information sets of proxies or reference variables. Since the genuine factor structures are unobservable for the specification of risk premia, the set of proxies or reference variables only represents the incomplete information set in a presumed factor structure for risk premium. In empirical studies, it is possible that the disturbance term of a presumed factor structure includes omitted factors in the process of projecting excess returns on an ad hoc information set. Numerous empirical studies, such as Goyal and Santa-Clara [4], Mayers [5], and Malkiel and Xu [6], show that the idiosyncratic risk volatility on a presumed factor structure or asset-pricing model may contain useful information in forecasting the stock returns. In a follow-up study to Goyal and Santa-Clara [4], Guo and Savickas [7] argue that the predictive power of idiosyncratic volatility goes down if consumption-wealth ratio is controlled for in the forecasting equation. T. Bali, N. Cakici, X. Yan and Z. Zhang [8] also shows that the finding of Goyal and Santa-Clara [4] is partially due to liquidity premium and small stocks traded in NASDAQ. With extended samples of stock returns, Balie [8] does not find any significant relation between equally weighted average stock volatileity and the value-weighted portfolio returns. A. Ang, R. J. Hodrick, Y. Xing, and X. Zhang [9], however, find that stocks with high idiosyncratic volatility (relative to the Fama and French [3] model) have low average returns pointing to a negative relation between idiosyncratic risks and stock returns. Fu [10] argues that idiosyncratic volatilities are time-varying and the results in A. Ang, R. J. Hodrick, Y. Xing, and X. Zhang [9] are largely driven by the return reversal of a subset of small stocks with high idiosyncratic volatilities. He instead finds a significantly positive relation between the estimated conditional idiosyncratic volatilities and expected returns using EGARCH models. Guo and Savickas [11] show that idiosyncratic volatility has statistically significant predictive power for aggregate stock market returns, and that idiosyncratic volatility performs just as well as the book-to-market factor in explaining the cross section of stock returns. However, all these time series/cross-sectional empirical studies focus mainly on the finding of explanatory variables for the stock returns. An issue that needs further attention is the diversifiability of the identified factor(s). Given that the systematic risk/factor(s) is non-diversifiable in asset pricing models, verification of diversifiability on these presumed factor(s) should be provided in addition to the identification of explanatory variables. In brief, the detection of possible explanatory factor(s) for stock returns although essential, should be followed by a study on the diversifiability of the identified factor(s).
Hence, discussions on additional hidden factor(s) in the idiosyncratic risks from presumed asset pricing models raise the following questions: Do idiosyncratic risks on a presumed asset pricing model really contain an essential hidden non-diversifiable factor that should be applied to asset pricing models? If so how can we test it?
Chamberlain and Rothschild [12] show that if asset returns have a k-factor approximate factor structure where k < n, and n is the number of assets then, the first k eigenvalues of the variance-covariance matrix of asset returns will be unbounded while the k + 1 eigenvalue of the matrix will be finite, as n grows to infinity. However, in empirical studies of asset returns, for any finite sample collected, eigenvalues of the variance-covariance matrix of asset returns under presumed factor structure are all finite. Verification on the growth of eigenvalues of these matrices across increasing sample sizes of asset returns is needed in empirical studies since the claim of approximate factor structure of Chamberlain and Rothschild [12] is based on an infinite dimensional setting. This paper provides an alternative test based on the fact that if the factor loadings of a hidden factor (in idiosyncratic risks) are non-diversifiable with respect to all well-diversified portfolios, a persistent cross-sectional memory among idiosyncratic risks will result. Specifically, if the asset returns follow the approximate factor structure with k + 1 factors where the fitted model is only equipped with k identified factors, then the (k + 1)-th eigenvalue of the variance-covariance matrix for asset returns (due to the hidden non-diversifiable factor loadings) will be unbounded, according to Chamberlain and Rothschild [12]. However, this property of unbounded non-diversifiable factor loadings can be shown to lead to a persistent crosssectional memory among the presumed idiosyncratic risks. Instead of applying eigenvalues or statistical procedures such as principal component analysis to examine the factor structure, the verification of the persistence of cross-sectional dependence among idiosyncratic risks becomes a diagnostic tool for asset pricing models and/or hidden factor(s). Many empirical studies show that idiosyncratic risk for asset pricing models is essential in pricing the asset returns. However, the verification for idiosyncratic risks as systematic/non-diversifiable factor(s) is still required for further studies. Specifically, if idiosyncratic risks or proxies of their volatility are indeed the common non-diversifiable factors among all asset returns, then they should appear as consistent components with persistent cross-sectional dependence in presumed asset pricing models. Although it may seem optimistic for empiricists to consider idiosyncratic risks as hidden factors when cross-sectional dependence or shortrun time-series predictability is discovered, these findings may not necessarily guarantee a non-diversifiable factor required for asset pricing models, unless further study is provided to verify that the identified factor or proxy is non-diversifiable.
The method developed in this study can verify the existence of a non-diversifiable hidden factor by testing if the error terms of asset pricing models are subject to a cross-sectional long dependence. Intuitively, under linear factor structure of excess returns, if there is a hidden non-diversifiable factor in idiosyncratic risks, then none of the weighted combinations of asset returns in the well-diversified portfolios will eliminate this factor. Specifically, the weighted sums of factor loadings for this hidden factor will not converge to zero asymptotically when the number of assets expands. Similar to time series setting, this property is closely related to the definition of cross-sectional long memory as the sums of cross-sectional covariances of these idiosyncratic risks will expand to infinity asymptotically. Therefore, the cross-sectional memory condition of idiosyncratic risk becomes a necessary condition, and thus an alternative test for the existence of non-diversifiable hidden factor.
Our diagnostic test provides an effective way to determine whether the factors found in these studies are transitory or persistent in cross-sectional memory, and thus to ascertain the validity of their findings. In other words, the verification of factor structure is reduced to the verification of persistence of cross-sectional dependence (for idiosyncratic risks) among these asset returns. Specifically, a forecast-error-based extended Rescaled Variance test is introduced here as an alternative statisticcal inference for idiosyncratic risk. Although long memory or long dependence has been discussed in many time series studies, the extension can be provided in random fields or set-indexed data such as Lavancier [13]. On the other hand, re-stacking and exchanging the time series horizon and cross-sectional indices as in White [14] for panel data, the asymptotic arguments on long memory time series can also be applied to the cross-sectional long dependence. The rest of this paper is organized as follows. Section 2 shows the model and a theorem for the long-memory properties of idiosyncratic risks where their partial sums of cross-sectional co-variances grow to infinity asymptotically if the omitted factor is non-diversifiable. The proofs of these arguments are shown in the appendix. Section 3 shows the modified T-sample Rescaled Variance Test when given with finite-timehorizon panel data. Section 4 shows applications and empirical findings of the test in detecting the omitted non-diversifiable factor on idiosyncratic risks of FamaFrench [3] model for 1391 excess returns, followed by the conclusion.
2. The Model
To derive the test, we first introduce a few prerequisites. Definition D1 describes a Hilbert space of real squaredintegrable random variables defined on the probability space1. The excess returns of assets forms a subset of this Hilbert space. Assumption A1 shows the conditions of factors or pre-specified reference variables applied as factors. The assumption allows different choice sets of instrumental or reference variables applied to specify the risk premium. These choice sets may be evolving over time or different sample sizes. In particular, the reference variables may be generated by the innovations from the multivariate time series models of pre-specified economic variables in a conditional expectation approach. Assumption A2 is to describe the presumed multi-factor model of the asset returns. Assumption A3 shows the possible missing factor and idiosyncratic risk in the excess return after projecting on the presumed explanatory variables. Assumption A4 provides a condition for diversification of the genuine noises in pricing models.
Definition D1: Let 
be a Hilbert space of squared-integrable (with respect to probability measure P) real-valued random variables on complete probability space 
where H is endowed with the 
-norm
|, where 
for
.
Let the inner product of H be denoted as 
for all
. In addition, 
represents a sequence of all assets’ excess returns contained in H.
| 3As this studyfocuses on the identification of hidden non-diversifiable factors, the possibly time-varying risk premium is left for future studies. |
Assumption A1: Let 
be a vector of p orthogonal perceived factors2 (defined in H) at time t for the multi-factor pricing model, 
is the j-th factor at time t. Or, in particular, 
s are orthogonal reference variables from the available information set for the asset-pricingfactors. In addition, 
, where
for all
.
Assumption A2: Let the excess return 
for each asset i at time t be expressed in the fitted factor structure as
, (1)
where 
as randomly assigned subindices for the asset returns, and
, where 
represents the information filtration (including lagged dependent variables) up to time t, 
stands for the conditional expected excess return 
3, 
stands for the projection error (or so-called “presumed” idiosyncratic risk) for asset i and time t with the assumed multifactor pricing model.
Assumption A3: Let the projection error or presumed idiosyncratic risk (if contains a hidden factor) be expressed as, 
, where 
represents a stationary stochastic hidden factor with a non-degenerated distribution, 
, 
, such that 
is orthogonal to all pre-selected factors as
, and 
is cross-sectional stationary for all assets and inter-temporal independent over time. The 
represents the non-stochastic unobservable factor loading for asset i on the hidden factor 
for all i’s, 
The 
is a cross-sectional mean-zero stationary random noise with finite moments and is independent of 
,
.
Also, let
such that all sequences of 
belong to the factor-loading space 
with 
-norm.
Assumption A3 imposes an asymptotic condition for the absolute factor loading(s) on the hidden factor. This is to ensure that as the size of portfolio expands, the factor loadings won’t be exploding—provided that the excess returns are in the Hilbert space H. To apply the ideas of diversification, a few definitions on the feasible weights in the factor pricing models are introduced in the followings. Definition D2 and D3 are to formulate the diversification in an infinite dimensional quadratic functional to define the opportunity set for sequence of feasible weights applied to each asset. The definition of non-diversifiable hidden factor is provided in D4. The notation 
represents the numbers of assets n grows sufficiently large asymptotically. Specifically, the limit of weighted sums of hidden factor loadings
is denoted as 
and the limit of weights 
as
, respectively.
Definition D2: Let W be a compact sub-space of 
space endowed with the 
-norm that for any
where W consists of all real bounded sequences of nondegenerated diversifying feasible weights 
such that 
and
, where 
contains sequences of 
with only finitely many numbers of non-zero weights that for each
and W* is the closed subset of W with 
4, 
and
,
where 
stands for the mean return for asset i, 
is the mean rate of return for the portfolio5. Let 
=
, 
, 
is an infinite dimensional closed sub-space of 
-space for bounded sequences of factor-loading
endowed with the 
-norm as
.
Definition D3: Let 
be a weakly lower semi-continuous diversification functional, where
for a given 
in
. The diversification problem of the portfolio is defined as
where W* is the closed subset of W as defined in Definition D2.
Given the above definition, the idea of diversifiable factor rests on the opportunity set of the infinite dimensional optimization problem. For the non-diversifiable factor, its hidden factor loadings can’t be eliminated by any possible sequence of feasible diversifying weights in W* as
.
Definition D4: The hidden factor 
for all the factor-loading sequences such that
where
, is denoted as W*-diversifiable with increasing numbers of assets if and only if
for all
, and the hidden factor is denoted as nondiversifiable in W* when 
for all
as
. That is, the hidden factor is W*-non-diversifiable, if and only if all sequences of diversifying weights will not drive the weighted sums of factor loadings to zero6.
Assumption A4 (Diversification of random noises):
the weighted sum as 
almost surely for all
as
.
With the above framework, we can now show that diversification of the hidden factor depends on the crosssectional memory conditions (or intensity of memory) of hidden factor. This in turns provides us with statistical hypotheses to test the existence of non-diversifiable hidden factor.
Theorem 1.1: If for any date t, there exists a non-diversifiable stationary hidden factor in idiosyncratic risk 
and let the distance function between indices of asset returns with arbitrary orders (denoted as s and i respectively) be defined as 
for
, and the cross-sectional covariance function be defined as
the presumed idiosyncratic risk 
is a crosssectional perturbed long-dependent series for any date t, such that
for all the presumed idiosyncratic risk 
as numbers of assets n grow. Conversely, if 
is covariance-stationary and 
for all i’s and t’s, where L(j) is a slowly-varying function of j,
, 
(i.e. the 
is a crosssectional long dependent series for any t), and for some sufficiently small
, and for all
7, (2)
is a slowly-varying function of i that
for 
then the hidden factor is non-diversifiable in W*.
Moreover, according to Zhou and Taqqu [15], a completely random re-ordering of samples will not impact the data’s long dependence. Consequently, when the cross-sectional sub-indices to asset returns are assigned randomly, like in many empirical studies, the cross-sectional long memory will not be influenced either. Hence, the diagnostic test on the memory condition of projection errors (or idiosyncratic risks) becomes an alternative method for identifying the non-diversifiable hidden factor.
3. Hypothesis Testing
In the following, we extend the framework of the Rescaled-Variance test of L. Giraitis, P. Kokoszaka, R. Leipus, and G. Teyssiere [1] with simultaneous inferences across finite forecast horizon to test the memory of idiosyncratic risks in fitted multifactor pricing models. The test statistics converge asymptotically to the T-sample Watson’s test statistic on unit circle where the asymptotic distribution is shown in Maag [2]. For the framework of analysis, we will assume that the time horizon is fixed. That is, the time horizon T for performing the test is finite while the numbers of cross-sectional observations n can be extended asymptotically as
. A few definitions and assumptions are needed to establish the pre-requisites.
Definition D5: A random process 
is called “self-similar”, if and only if
, (3)
where 
is the Hurst exponent, the notation 
stands for the equivalence of distributions. In addition, without loss of generality, we set the initial condition 
almost surely. According to Embrechts and Maejima [16], for any self-similar process, 
if and only if 
almost surely. When
, it implies that all autocorrelations of 
are equal to one.
Definition D6: Let 
A mean-zero Gaussian process 
is called “fractional Brownian motion” if its auto-covariance function can be shown as
(4)
In particular, when
, the process will have long-range dependence, when
, 
will become the usual Brownian motion with independent identically distributed increments, if
, the process is called “anti-persistent” with sum of auto-covariances being finite.
Assumption A5: Given Theorem 1.1 that for any date t, 
in the pre-determined finite time horizon k of hypothesis testing, and for numbers of assets
, where
, and the hidden factor
is W*-non-diversifiable, and if 
is covariance-stationary such that
and the long run variance 
of partial sum
follows that
for 
is a slowly-varying function of n such that 
for all
, will converge in distribution as
(5)
where
, 
, 
is the largest integer less than or equal to
, 
is a fractional Brownian motion with Hurst exponent
.
Assumption A6: Let the idiosyncratic risk 
for any date t in the pre-determined finite time horizon T that may have a diversifiable hidden factor, contain only cross-sectional short memory as
.
The sequence 
satisfies the strong mixing condition such that the mixing coefficient 
as
, where the mixing coefficient
where the 
represents the 
-field generated by the set of random variables 
and 
follows the invariance principle that as 
(6)
We hereby extend the Rescaled Variance statistic of L. Giraitis, P. Kokoszaka, R. Leipus, and G. Teyssiere [1] to a T-sample setting that analyzes the memory of the idiosyncratic risk with finite time horizon for panel data. Notice that the assumption A6 can be extended to include the (cross-sectional) heteroscedasticity in 
and the following 
statistic converges to the same functional of Brownian bridges under the null hypothesis even though 
is not covariance-stationary. The conditions still holds under the non-stationary case as indicated by L. Giraitis, P. Kokoszaka, R. Leipus, and G. Teyssiere [1]. However, due to additional complexity that needs to be introduced, this is left for further studies.
Theorem 1.2: Given the above setting, for any date t, 
, T is the finite time horizon for hypothesis testing, where
, and
is the fourth order cumulant of cross-sectional covariance-stationary 
at date t then, under the null hypothesis that there is no non-diversifiable hidden factor in the idiosyncratic risk, for hypothesis testing with finite time horizon
, the modified Rescaled Variance test is given as
(7)
(8)
are T independent Brownian bridges where by definition,
and the weights 
are the Bartlett weights such that
, 
as 
and the sample cross-sectional covariance function such as
is the cross-sectional average, 
are the error terms from the presumed asset pricing model for a given date t, 
stands for the largest integer that is less than or equal to the real number
, where the asymptotic distribution of 
(denoted as
) is shown by Maag [2] as
(9)
and 
is the Hermite polynomial of order 
for argument z. Under the alternative hypothesis 
where 
is cross-sectional covariance-stationary and there is a non-diversifiable factor in the idiosyncratic risk of presumed asset pricing model. In addition, for simplicity, let
, 
then,
(10)
where 
is the fractional Brownian Bridge with scaling parameter H. The test is also consistent that the statistic 
under the alternative hypothesis.
In other words, we reject the null hypothesis of short memory when 
is large relative to the critical value chosen according to the significance level. When T is equal to one, the test is reduced to the Rescaled Variance test distributed as Watson’s test statistic. The reason that the test is referred to 
is that the test statistic converges to the same functional as Maag’s [2] statistic in testing identical distribution for all T groups without knowledge of the underlying cumulative distribution function for the data. The asymptotic distribution of such statistic is in turns, identical to the test statistic of 
groups under assumed cumulative distribution function according to Maag [2].
Notice that the above result is based on the genuine idiosyncratic risk
. Yet, in the empirical studies, we can only observe either residuals or (out-ofsample) forecast errors from the fitted asset pricing models. Hence, we need to obtain the consistency of the sample statistics for any given (out-of-sample) forecast errors with cross-sectional observations 
Theorem 1.3: Given that 
for any date t in the finite time horizon T are the (cross-sectional) forecast errors for the presumed asset pricing model where the parameters for
, are consistently estimated, and under the null hypothesis of no hidden non-diversifiable factor in
, such that
when
, the forecast error-based Rescaled Variance statistic is shown as
, (11)
where
(12)
(13)
is the estimate for 
by using
, the forecast errors from the fitted asset pricing model.
4. Empirical Applications and Data
To test whether the idiosyncratic risks in Fama-French [3] model contain a hidden non-diversifiable factor or not, we collect the excess returns of 1391 randomly selected firms with different capitalizations and from different industries including agricultural products, chemical productions, financial, transportation, consumer goods, retail, entertainment, etc. These excess returns are collected from the firms in CRSP that have continuous monthly data in our sample period of January 1987 to December 2006. If there is a hidden non-diversifiable factor in idiosyncratic risks from Fama-French [3] model, finding of persistent memory in these idiosyncratic risks may provide the empirical evidence to verify the claim for its existence.
With regards to the possibility of survivorship bias in the CRSP data source, Li and Xu [17] argue that such survivorship bias is unlikely significant for the U.S. stock returns. In addition, Barber and Lyon [18] also show that the selection bias does not significantly affect the estimated factor premiums in equity returns of Fama-French [3] model, which implies that sample selection does not affect the empirical results of Fama-French [3] model. Hence, we select the explanatory variables for the regressions as those in the Fama-French [3] model as the asset pricing models of interest. And for comparison, we also apply our diagnostic test to the idiosyncratic risks from the CAPM. Furthermore, following the argument of Li and Mayer [19] that correction of dynamic selection bias does not improve the accuracy of forecasts of the model, the application with forecast errors from the fitted model seems more appropriate since our study is diagnostic for specifications in regressions. Therefore, we consider the out-of-sample forecast errors of both Fama-French [3] model and CAPM to test the null hypothesis that the hidden factor is diversifiable in the idiosyncratic risks.
Based on the tables and asymptotic cumulative distribution function provided by Maag’s [2], we select 24 months out-of-sample forecast errors (that is, T = 24) for the test period in verifying cross-sectional memory of idiosyncratic risks. The forecast errors for all these excess returns are obtained with the parameters estimated in the previous months of the in-sample period from January 1987 to December 2004, and are conditional on the updated explanatory variables in the asset pricing models applied here. The test period is from January 2005 to December 2006. The test statistics for different forecast horizons are calculated from T = 1 to T = 24 using the asymptotic cumulative distribution functions from Maag [2]. Notice that when T = 2, the test statistic actually conforms with Watson’s [19] two-sample test where its asymptotic distribution is identical to the statistic as in L. Giraitis, P. Kokoszaka, R. Leipus, and G. Teyssiere [1]. Hence, the asymptotic distributions applied to T = 2 and T = 1 are identical according to the asymptotic arguments in Maag [2] and Watson [20] as far as the two-sample tests are of concern, with an empirical application of Magg’s [2] tests shown in Brown [21].
Our extended Rescaled Variance 
tests are performed under the null hypothesis that the hidden factor is diversifiable. The p-values of the statistics are calculated from the asymptotic cumulative distribution function provided by Maag [2]. Since the distribution function is not conventional, and that Maag’s [2] tabulation of critical values is limited to the case when maximum forecast horizon is T = 5, the 
statistics are considered as rejecting the null if the p-values are relatively smaller than the given 5% or 1% significance levels.
Table 1 presents the 
test statistics and the corresponding p-values for testing the cross-sectional memory of idiosyncratic risks in the Fama-French [3] model. These test statistics are provided with different selections of q, which are denoted as the lags for possible crossNotes: The time index T stands for each month in the forecast horizon. For instance, T = 1 stands for January 2005, T = 2 stands for February 2005, and so forth. The table reports the test statistic with corresponding p-values denoted in parenthesis. If the p-value is less than the significance levels such as 5% or 1%, the null hypothesis that there is no non-diversifiable hidden factor is rejected.
Table 1. T-sample rescaled variance tests on forecast errors of Fama-French (1993) Model.
sectional short-run memory in estimating long run variances of idiosyncratic risk under the null. For simplicity of exposition, we only report cases when q = 5, 10, 15, respectively.
For comparison purpose, we also perform the same diagnostic tests on the idiosyncratic risks from alternative models such as the CAPM, and the over-simplified case when excess returns are regressed only with a constant term. The test statistics are shown in Tables 2 and 3, respectively. All test statistics from these two alternative models reported in Tables 2 and 3 show that their p-values are all well below the 1% confidence level.
Table 2. T-sample rescaled variance tests on forecast errors of the CAPM.