2 Vector Autoregressive Process, VAR A pth order vector autoregression, denoted V AR(p) is written as yt=C+重yt-1+重yt-2+…+中pyt-p+Et where c denotes an(k×1) vector of constants and重;an(k×k) matrix of au toregressive coefficients for j=1, 2, . p and Et is a vector white noise process 2.1 Population Characteristics Let ci denotes the ith element of the vector c and let oia denote the row i, column j element of the matrix s, then the first row of the vector system in(1) specifies that y,t-2+12y2:-2+…+01k9kt y/1,t- 9k,t- Thus, a vector autoregression is a system in which each variable is regressed n a constant and p of its own lags as well as on p lags of each of the other(k-1) variables in the V AR. Note that each regression has the same explanatory vari- Using lag operator notation, (1)can be written in this form k-重1L-重2L 更LP 更(L)y Here p(L) indicate an k x k matrix polynomial in the lag operator L. The row i, column j elements of p(L)is a scalar polynomial in L where Si; is unity if i=j and zero otherwise2 Vector Autoregressive Process, V AR A pth order vector autoregression, denoted V AR(p) is written as; yt = c + Φ1yt−1 + Φ2yt−2 + ... + Φpyt−p + εt , (1) where c denotes an (k × 1) vector of constants and Φj an (k × k) matrix of au￾toregressive coefficients for j = 1, 2, ..., p and εt is a vector white noise process. 2.1 Population Characteristics Let ci denotes the ith element of the vector c and let φ (s) ij denote the row i, column j element of the matrix Φs, then the first row of the vector system in (1) specifies that y1t = c1 + φ (1) 11 y1,t−1 + φ (1) 12 y2,t−1 + ... + φ (1) 1k yk,t−1 +φ (2) 11 y1,t−2 + φ (2) 12 y2,t−2 + .... + φ (2) 1k yk,t−2 +.... + φ (p) 11 y1,t−p + φ (p) 12 y2,t−p + ... + φ (p) 1k yk,t−p + ε1t . Thus, a vector autoregression is a system in which each variable is regressed on a constant and p of its own lags as well as on p lags of each of the other (k −1) variables in the V AR. Note that each regression has the same explanatory vari￾ables. Using lag operator notation, (1) can be written in this form [Ik − Φ1L − Φ2L 2 − ... − ΦpL p ]yt = c + εt or Φ(L)yt = c + εt . (2) Here Φ(L) indicate an k × k matrix polynomial in the lag operator L. The row i, column j elements of Φ(L) is a scalar polynomial in L: Φ(L)ij = [δij − φ (1) ij L 1 − φ (2) ij L 2 − ... − φ (p) ij L p ], where δij is unity if i = j and zero otherwise. 5