for m>0,and K is either upper or lower triangular for m=0, = 6x1 ∑=I, 0N-m)×1 0m×1 B= Imxm B股N-m 1(-m)x1 O(N-m)xm O(N-m)x(N-m) with the following parametric restrictions imposed: d≥0，m+1≤i≤N, K,0三∑K日，>0,1≤i≤m,K≤0,1≤j≤m,j≠i, j=1 Θ：≥0,1≤i≤m,Bi≥0,1≤i≤m,m+1≤j≤N. Then the sub-family Am(N)of affine term structure models is obtained by inclusion of all models that are invariant transformations of this canonical model or nested special cases of such transformed models.For the case of N risk factors,this gives N+1 non-nested sub-families of admissible affine mod- els.?Members of the families Am(N)include,among others,Vasicek [1977, Langetieg [1980],Cox,Ingersoll,and Ross [1985],Longstaff and Schwartz (1992],Chen and Scott [1993],Pearson and Sun [1994],Duffie and Singleton 1997,Balduzzi,Das,Foresi,and Sundaram 1996,Balduzzi,Das,and Foresi 1998,Duffie and Liu [2001],and Collin-Dufresne and Goldstein [2001al. Quadratic Gaussian Models If G(Yi,t;T)is quadratic in Yi,i.e., G(Yi,t;T)=A(T-t)+B(T-t)Yi+YC(T-t)Yi, then it must be the case that rt =a+B'Y:+YYi,where a =A'(0), B=B'(0),and y=C(0).Without loss of generality,we can assume that C(r)is symmetric.Thus y must also be symmetric and (14)becomes A+YB+Y:CY:=(a+YB+YyYi)+u(Yi,t)'[B(T)+2CY] (18) +Trace[a(,yCa,t训-号B+20Ya,a,y[B+20Y- 7Although the classification scheme was originally used by Dai and Singleton [2000] to characterize the state dynamics under the actual measure,it is equally applicable to the state dynamics under the Q-measure.Note that not all admissible affine DTSMs are subsumed by this classification scheme (i.e.,not all admissible models are invariant transformations of a canonical model). 15for m > 0, and K is either upper or lower triangular for m = 0, Θ =  ΘB m×1 0(N−m)×1  , Σ = I, α =  0m×1 1(N−m)×1  , B =  Im×m BBD m×(N−m) 0(N−m)×m 0(N−m)×(N−m)  , with the following parametric restrictions imposed: δi ≥ 0, m + 1 ≤ i ≤ N, KiΘ ≡ Xm j=1 KijΘj > 0, 1 ≤ i ≤ m, Kij ≤ 0, 1 ≤ j ≤ m, j 6= i, Θi ≥ 0, 1 ≤ i ≤ m, Bij ≥ 0, 1 ≤ i ≤ m, m + 1 ≤ j ≤ N. Then the sub-family Am(N) of affine term structure models is obtained by inclusion of all models that are invariant transformations of this canonical model or nested special cases of such transformed models. For the case of N risk factors, this gives N +1 non-nested sub-families of admissible affine mod￾els.7 Members of the families Am(N) include, among others, Vasicek [1977], Langetieg [1980], Cox, Ingersoll, and Ross [1985], Longstaff and Schwartz [1992], Chen and Scott [1993], Pearson and Sun [1994], Duffie and Singleton [1997], Balduzzi, Das, Foresi, and Sundaram [1996], Balduzzi, Das, and Foresi [1998], Duffie and Liu [2001], and Collin-Dufresne and Goldstein [2001a]. Quadratic Gaussian Models If G(Yt, t; T) is quadratic in Yt, i.e., G(Yt, t; T) = A(T − t) + B(T − t) 0 Yt + Y 0 t C(T − t)Yt, then it must be the case that rt = α + β0 Yt + Y 0 t γYt, where α = A0 (0), β = B0 (0), and γ = C0 (0). Without loss of generality, we can assume that C(τ ) is symmetric. Thus γ must also be symmetric and (14) becomes A˙ + Y 0 t B˙ + Y 0 t CY˙ t = (α + Y 0 t β + Y 0 t γYt) + µ(Yt, t) 0 [B(τ )+2CYt] (18) +Trace [σ(Yt, t) 0 Cσ(Yt, t)] − 1 2 [B + 2CYt] 0 σ(Yt, t)σ(Yt, t) 0 [B + 2CYt] . 7Although the classification scheme was originally used by Dai and Singleton [2000] to characterize the state dynamics under the actual measure, it is equally applicable to the state dynamics under the Q-measure. Note that not all admissible affine DTSMs are subsumed by this classification scheme (i.e., not all admissible models are invariant transformations of a canonical model). 15