A generalization of generalized b eta distributions Michael B. Gordy Board of Governors of the Federal Reserve System April 8, 1998 Abstract This pap er intro duces the \comp ound con uent hyp ergeometric" (CCH) distribution. The CCH uni es and generalizes three recently intro duced generalizations of the b eta distribution: the Gauss hyp ergeometric (GH) distribution of Armero and Bayarri (1994), the generalized b eta (GB) distri- bution of McDonald and Xu (1995), and the con uenthyp ergeometric (CH) distribution of Gordy (forthcoming). In addition to greater exibility in tting data, the CCH o ers two useful prop erties. Unlike the b eta, GB and GH, the CCH allows for conditioning on explanatory variables in a natural and convenientway. The CCH family is conjugate for gamma distributed signals, and so may also prove useful in Bayesian analysis. Application of the CCH is demonstrated with two measures of household liquid assets. In each case, the CCH yields a statistically signi cant improvementin t over the more restrictive alternatives. JEL Co des: C10, D91, G11 The views expressed herein are my own and do not necessarily re ect those of the Board of Governors or its sta . I am grateful for the helpful comments of Karen Dynan, Arthur Kennickell, Whitney Newey, and Martha Starr-McCluer. Please address corresp ondence to the author at Division of Research and Statistics, Mail Stop 153, Federal Reserve Board, Washington, DC 20551, USA. Phone: (202)452-3705. Fax: (202)452-5295. Email: [email protected]. The b eta distribution is widely used in statistical mo deling of b ounded random variables. It is easily calculated, can take on a variety of shap es, and, p erhaps as imp ortantly, none of the other 1 commonly used distribution functions have compact supp ort. However, its application is limited in imp ortantways. First, as a two parameter distribution, it can provide only limited precision in tting data. It is desirable to have more parametrically exible versions of the b eta to allow a richer empirical description of data while still o ering more structure than a nonparametric estimator. Second, the b eta do es not o er a natural and convenient means of intro ducing explanatory variables. In a Beta(p; q ) distribution, the parameters (p; q ) jointly determine b oth the shap e and moments of the distribution. There is no satisfactory way of conditioning the mean by sp ecifying p and q as functions of explanatory variables and regression co ecients. Third, the b eta is inconvenient for use in Bayesian analysis. It is conjugate for binomial signals, but not for signals of any continuous distribution. Recent research has contributed three generalizations of the b eta which address one or more of these limitations. Armero and Bayarri (1994) de ne the Gauss hyp ergeometric (GH) distribution by the density function p1 q 1 r x (1 x) (1 + x) for 0 <x<1 (1) GH (x; p; q ; r;)= B(p; q ) F (r;p;p + q;) 2 1 with p>0;q>0 and where F denotes the Gauss hyp ergeometric function. The GH collapses to 2 1 the ordinary b eta if either r =0 or = 0, and to the b eta-prime if q = =1. Armero and Bayarri apply the GH to a Bayesian queuing theory problem. A related distribution is intro duced by McDonald and Xu (1995) as the \generalized b eta" 1 Except, of course, the uniform distribution, which is a sp ecial case of the b eta. 1 (GB) distribution. The GB is de ned by the p df ap1 a q 1 jajx (1 (1 c)(x=b) ) a a GB (x; a; b; c; p; q )= for 0 <x <b =(1 c) (2) ap a p+q b B(p; q )(1 + c(x=b) ) and zero otherwise with 0 c 1 and b; p and q p ositive. As in the ordinary b eta distribution, the parameters p and q control shap e and skewness. Parameters a and b control \p eakedness" and scale, resp ectively. Given a = b = 1, the parameter c shifts the GB from the ordinary b eta distribution (c = 0) to the b eta-prime distribution (c = 1). Gordy (forthcoming) generalizes the b eta in an unrelated direction. The \con uenthyp ergeo- metric" distribution CH(p; q ; s) is de ned by the p df p1 q 1 x (1 x) exp(sx) for 0 <x <1: (3) CH(x; p; q ; s)= B(p; q ) F (p; p + q; s) 1 1 where F is the con uenthyp ergeometric function de ned in Abramowitz and Stegun, eds (1968, 1 1 2 13.1.2) (hereafter cited as \AS"). Gordy (forthcoming) shows that a b eta prior and gamma signal gives rise to a CH p osterior and applies this prop erty to auction theory. In this pap er, I unify and further generalize these three distributions. The constant of prop or- tionality in the new p df is the pro duct of a b eta function and a comp ound con uenthyp ergeometric function. Therefore, I denote this distribution the \comp ound con uent hyp ergeometric" (CCH). The CCH is de ned and describ ed in Section 1. Sp ecial cases are discussed in Section 2. In partic- ular, I show that the b eta, GB, GH, CH and gamma distributions are all sp ecial cases of the CCH. Empirical applications to measures of household liquid assets are provided in Section 3. 2 This function is denoted there as M and referred to as the \degenerate" hyp ergeometric function in Gradshteyn and Ryzhik (1965, 9.210.1). In Mathematica, itistheHypergeometric1F1. 2 1 De nition of the CCH I de ne the CCH by the density function p1 q1 r x (1 x) ( +(1 )x) exp(sx) CCH(x; p; q ; r;s;;)= for 0 <x<1= (4) B(p; q )H (p; q ; r;s;;) 3 for 0 <p,0<q, r 2<, s2<,0 1, and 0 <. The function H is given by p H (p; q ; r;s;;)= exp(s= ) (q; r;p + q ; s= ; 1 ) (5) 1 where is the con uent hyp ergeometric function of two variables de ned in Gradshteyn and 1 Ryzhik (1965, 9.261.1) by 1 1 XX ( ) ( ) m+n n m n x y (6) ( ; ; ; x; y )= 1 ( ) m!n! m+n m=0 n=0 and where (a) is Pochhammer's notation, i.e., (a) =1; (a) =a; (a) =(a) (a + k 1). For 0 1 k k k1 convenience in exp osition, I refer to H as a \comp ound con uenthyp ergeometric" function rather than as a \con uenthyp ergeometric function of twovariables." In App endix A, I show that equation (4) integrates to one. To guarantee that the CCH distri- bution is well-de ned everywhere on the parameter space, I prove in App endix C the theorem Theorem 1 For all (p; q ; r;s;;) such that p>0, q>0, r2<, s2<, 0 < 1 and >0, H(p; q ; r;s;;) is a nite p ositive real numb er. It is straightforward to check that the moment generating function for the CCH is given by H(p; q ; r;s t; ; ) M (t)= H (p; q ; r;s;;) 3 = 0 is handled as a sp ecial case. See the UH distribution in Section 2. 3 and the k th order moments are given by (p) H(p+k; q; r;s;;) k k E (X )= : (7) (p+q) H(p; q ; r;s;;) k Theorem 1 is sucient to guarantee that all moments of X exist. Given the restrictions on the parameters, the function in equation (5) can always be ex- 1 pressed as an in nite series in which all terms are non-negative (see App endix B). Therefore, the H function can be calculated without numerical round-o problems. Despite its apparent com- 4 Computation time plexity, it is quickly calculated over most of the relevant parameter space. decreases with p; and and increases with q; jrj and jsj. It app ears that and jointly have the largest e ect. To take an easy example, computation of H (p =20;q=2;r=5;s=0;=1;=1) to 14 places accuracy takes 0.0002 seconds on a SparcStation10. To take a more dicult example, H (2; 20; 15; 10; 0:01 ; 0 :01 ) requires 0.6 seconds. To the extent that the CCH is employed to general- ize the b eta distribution, rather than the b eta-prime distribution, computationally easy cases will predominate in empirical applications. Figures 1a, 1b and 1c plot the CCH p df for a variety of parameter values. The gures show that the role of parameters p and q in the CCH is much the same as in the ordinary b eta distribution. Parameter rescales the distribution for longer or shorter supp ort. The remaining parameters r , s and \squeeze" the density function to the left or right. While the shap es p ortrayed in these gures are qualitatively familiar from the b eta distribution, the CCH can also take on a wide range of multi-mo dal or long-tailed shap es which the b eta cannot. Examples are presented in Figure 2. The parameter s allows for a convenient metho d of conditioning the CCH distribution on exogenous variables. The b ottom panel of Figure 1b shows that increasing (decreasing) s squeezes the distribution to the left (right). In App endix D it is proved that the mean changes monotonically 4 Software in C and MATLAB is available up on request.