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Weekly UpdatesAbstract In this paper, we propose a new benchmarking procedure lying on cumulants for computing the factor loadings in financial models of returns. We apply this technique to the well-known augmented Fama and French (J Fin Econ 43 (2): 153-193, 1997) model and compare it with another technique of ours based on higher moments. Our new procedure confirms the fact that the alpha is supposed to decrease when we disaggregate HFR indices to the level of individual funds while correcting for specification errors. Our new technique is therefore useful for hedge funds selection or ranking based on the alpha of Jensen corrected for specification errors. This technique will also be useful for calibrating other financial models of returns like the simple market model or the conditional alpha and beta models.
Keywords Hedge funds returns * Alpha of Jensen * Financial models * Cumulants * Higher moments * Specification errors * Aggregation bias
JEL C10 . G10 . G20
Introduction
This paper deals with the identification and the correction of specification errors in the augmented Fama and French (1997) model. These specification errors may create a bias in the estimation of the alpha, which is especially important for stock-picking activities, and they may also give way to an overstatement or an understatement of the exposition of portfolios to risk factors. It is important to discard these biases in the model of F&F in order that the portfolio manager does not take wrong decisions based on the estimation of this model.
The financial literature has recorded many sources of specification errors in the F&F model. For many authors (Harvey and Siddique 2000; Lim 1989), these errors might be due to the neglect of higher order co-moments of factors in the F&F model. For others (Ferson and Schadt 1996; Christopherson et al. 1998), these specification errors might be related to the conditional character of alpha and beta or to the omission of an important factor, like the illiquidity premium (Chan and Faff 2005). In the context of hedge funds, which are studied in this article, one of the consequences of these specification errors in the F&F model would be an overstatement of the alpha of these funds, viewed as an absolute return.
In this paper, we pursue the studies on specification errors in the F&F model by using a new form of the Hausman (1978) test. The originality of this test lays in the choice of new instruments based on cumulants, which were introduced only recently in the literature of risk. We relate this financial literature to the econometric works of Dagenais and Dagenais (1997) and later generalized to financial applications by Racicot (2003) and Coen and Racicot (2007). These researchers propose cumulants of endogenous variables of a model as optimal instruments to discard specification errors. We therefore present two new Hausman estimations which are equivalent to a two-stage least squares regression. The advantage of our Hausman equations is that they inform directly on the overstatement or the understatement of the coefficients which are due to specification errors. We compare also our new method with GMM integrating classical instruments, e.g. predetermined variables of the model, and GMM run with our new instruments, which are based on higher moments and cumulants of the explanatory variables.
The organization of our paper is as follows. Firstly, we justify the choice of instrumental variables to estimate the F&F model. Then we develop two forms of the Hausman test based on artificial regressions. One resorts to higher moments and the other to cumulants. As we will see, these two forms constitute two new empirical versions of the F&F model which help to estimate the specification errors. The next section applies our two forms of the Hausman test to 22 indexes of hedge funds and 111 individuals' funds drawn from the database of HFR (Hedge Fund Research Inc.). These two samples will allow us to take care of the impact of aggregation of data on specification errors. We will compare the results of the two Hausman equations to iterated and non-iterated forms of GMM integrating classical instruments and our new instruments based on higher moments and cumulants.
The Choice of Instrumental Variables to Estimate the Augmented Fama and French (1997) Model (1)
The augmented F&F (1997) model is a purely empirical one which may be written as:
[R.sub.pt] - [R.sub.ft] = [alpha] + [[beta].sub.1]([R.sub.mt] - [R.sub.ft]) + [[beta].sub.2]SM[B.sub.t] + [[beta].sub.3]HM[L.sub.t] + [[beta].sub.4]UM[D.sub.t] + [[epsilon].sub.t] (1)
where [R.sub.pt] - [R.sub.ft] is the excess return of a portfolio, [R.sub.ft] being the risk-free return and [R.sub.mt] - [R.sub.ft], the market risk premium. SMB is a portfolio which mimics the "small firm anomaly", which is long in the returns of selected small firms and short in the returns of selected big firms. HML represents a portfolio which mimics the "value stock anomaly", which is long in returns of stocks of selected firms having a high (book value/market value) ratio (value stocks) and short in selected stocks having a low (book value/market value) ratio (growth stocks). Finally, UMD is a portfolio which mimics the "momentum anomaly", which is long in returns of selected stocks having a persistent upper trend and short in stocks having a persistent downward trend.
We postulate that Eq. 1 contains measurement errors. These errors might be due to many causes but the main plausible one may be that the F&F model is written in terms of the expected value of the explanatory variables. (2) Following these errors, the risk factors become endogenous and the condition of orthogonality between these factors of risk and the innovation term in Eq. 1 is violated: the estimators of the coefficients of Eq. 1 are no longer unbiased and consistent, To purge the coefficients from these biases, we must regress in a first pass the independent variables on instrumental variables.
As said before, it is difficult to find valuable instruments for the excess returns of the mimicking portfolios. Being long in some stocks and short in others, their cash flows are similar to those of hedge funds. Higher moments of returns, like asymmetry and kurtosis, might have a great influence on these returns. This suggests the use of higher moments of the variables on the RHS of Eq. 1 as instrumental variables. An econometric theory is indeed in construction on this subject. Following Durbin (1954) and Pal (1980), Dagenais and Dagenais (1997) showed that higher moments of independent variables of a regression might be valid instruments to remove errors in variables. But instead of defining higher moments as in these papers, we will adopt a method more akin to asset pricing theory which defines higher moments of returns by powers of these returns.
The method of asset pricing based on higher moments is not new. Samuelson (1970), Rubinstein (1973) and Kraus and Litzenberger (1976) put the foundations of the three-moment and four-moment CAPM. The three-moment CAPM integrated asymmetry of returns in the analysis while the four-moment CAPM added kurtosis. The n-moment CAPM can be written as follows:
[R.sub.i] - [R.sub.0] = [[alpha].sub.0] + [[alpha].sub.1]([R.sub.m] - [R.sub.0]) + [[alpha].sub.2][([R.sub.m] - [R.sub.0]).sup.2] + [[alpha].sub.3][([R.sub.m] - [R.sub.0]).sup.3] + ... + [[alpha].sub.n-1][([R.sub.m] - [R.sub.0]).sup.n-1]. (2)
A test on [[alpha].sub.2] is a test on skewness preferences in asset pricing and a test on [[alpha].sub.3], a test on kurtosis preferences, and so on. The higher moments are consequently powers of returns in this approach. We therefore use a financial theory, the n-moment CAPM, to give an object to the method of Dagenais and Dagenais (1997) for correcting errors in variables. Let us return to the variable SMB, which we want to correct for the problem of errors in variables. In the first pass of the regression, this variable will be regressed on:
S[^.M][B.sub.t] = f([F.sub.it-1], [F.sub.it-1.sup.2], [F.sub.it-1.sup.3],...,[F.sub.it-1.sup.5],...) (3)
where [F.sub.i] are the variables on the RHS of the equation of F&F (Eq. 1) including SMB. They stand for the higher moments of these variables. [F.sub.it-1.sup.2] stands for the skewness of factor [F.sub.i], [F.sub.it-1,sup.3] for its kurtosis, and so on. The variables appearing on the RHS of Eq. 3 will serve as instrumental variables in the first pass of the Hausman tests, as explained in the following section.
Hausman Specification Tests Based on Higher Moments and Cumulants
Test Based on Higher Moments
To detect specification in our sample of hedge funds, we could use the original Hausman h test (3). Hausman compares two sets of estimates of the parameters vector, say, [[beta].sub.OLS], the least-squares estimator (OLS), and [[beta].sub.A], an alternative estimator which can take a variety of forms but which for our purposes is the instrumental variable estimator which we designate by [[beta].sub.IV]. The hypotheses to test are [H.sub.0], being in our case the absence of errors in variables and [H.sub.1], being the presence of errors in variables. To do so, Hausman defines the following vector of contrasts or distances: [[beta].sub.IV] [[beta].sub.OLS]. The test statistic may be written as follows:
h = [([[^.[beta]].sub.IV] - [[^.[beta]].sub.OLS]).sup.T][[Var([[^.[beta]].sub.IV]) - Var([[^.[beta]].sub.OLS])].sup.-1]([[^.[beta]].sub.IV] - [[^.[beta]].sub.OLS]) ~ [[chi].sup.2](g) (4)
with Var([^.[beta]].sub.IV]) and Var ([^.[beta]].sub.OLS]) being consistent estimates of the covariance matrices of ([^.[beta]].sub.IV]) and [^.[beta]].sub.OLS]. g is the number of potentially endogenous regressors, that is the variables measured with errors in our case. [H.sub.0] will be rejected if the p-value of this test is less than [alpha], with [alpha] being the critical threshold of the test, say 5%. According to Mackinnon (1992), this test might run into difficulties if the matrix [Var ([^.[beta].sub.IV]) - Var ([^.[beta].sub.OLS])], which weights the vector of contrasts, is not positive definite. Fortunately, there is an alternative way to do the Hausman test which is much easier. This test goes as follows.
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