stationarity assumption,now 71 ur= e(Xr) a sizeable reduction in the number of unknown parameters from T+T(T+1)/2 to(T+1). It is important, however, to note that even in the case of stationarity the number of parameters increase with the size of the subset(t1, . tr) although the parameters do not depend on t E T. This is because time-homogeneity does not restrict the 'memoryof the process. In the next section we are going to consider 'memory' restrictions in an obvious attempt to solve the problem of the parameters increasing with the size of the subset(t1, t2, .,tr) of T 2.2 Restricting the memory of a stochastic process n the case of a typical economic times series, viewed as a particular realization of a stochastic process (Xt, teT one would expect that the dependence between Xt, and X,, would tend to weaken as the distance(t-ti) increase. Formally, this dependence can be described in terms of the joint distribution F(at, Ita,. atr) as follows. Definition A stochastic process (Xt, tET is said to be asymptotically independent if for any subset(t1, t2,. tr) of and anv 7 B(r) defined by F(xt1,x2…,xr,xt1+,…,xxr+,)-F(xt1,c2…,x)F(x1+1…,xxr+ ≤B(7) goes to zero as T→∞. That is if B(7)→0asr→0 the two subsets(Xt1,X2,…,Xn)and(X1+,…,Xtr+) become independent a particular case of asymptotic independence is that of m-dependence which restricts B()to be zero for all T>m. That is, Xt and Xt, are independent for t1-t2|>mstationarity assumption, now µT = E(XT ) =         µ µ . . . µ         VT =         γ0 γ1 . . . γT −1 γ1 γ0 . . . γT −2 . . . . . . . . . . . . . . . . . . γT −1 . . . . γ0         , a sizeable reduction in the number of unknown parameters from T +[T(T +1)/2] to (T + 1). It is important, however, to note that even in the case of stationarity the number of parameters increase with the size of the subset (t1, ...,tT ) although the parameters do not depend on t ∈ T . This is because time-homogeneity does not restrict the ’memory’ of the process. In the next section we are going to consider ’memory’ restrictions in an obvious attempt to ’solve’ the problem of the parameters increasing with the size of the subset (t1,t2, ...,tT ) of T . 2.2 Restricting the memory of a stochastic process In the case of a typical economic times series, viewed as a particular realization of a stochastic process {Xt , t ∈ T } one would expect that the dependence between Xti and Xtj would tend to weaken as the distance (tj −ti) increase. Formally, this dependence can be described in terms of the joint distribution F(xt1 , xt2 , ..., xtT ) as follows: Definition: A stochastic process {Xt , t ∈ T } is said to be asymptotically independent if for any subset (t1,t2, ...,tT ) of T and any τ , β(τ ) defined by |F(xt1 , xt2 , ..., xtT , xt1+τ , ..., xtT +τ ) − F(xt1 , xt2 , ..., xtT )F(xt1+τ , ..., xtT +τ )| ≤ β(τ ) goes to zero as τ → ∞. That is if β(τ ) → 0 as τ → ∞ the two subsets (Xt1 , Xt2 , ..., XtT ) and (Xt1+τ , ..., XtT +τ ) become independent. A particular case of asymptotic independence is that of m−dependence which restricts β(τ ) to be zero for all τ > m. That is, Xt1 and Xt2 are independent for |t1 − t2| > m. 6