Convert VAR to VMA(infinite)

Convert VAR process to infinite vector MA process

Usage

VARtoVMA(object, lag_max)

Arguments

object

A varlse object

lag_max

Maximum lag for VMA

Value

VMA coefficient of k(lag-max + 1) x k dimension

Details

Let VAR(p) be stable. $$Y_t = c + \sum_{j = 0} W_j Z_{t - j}$$ For VAR coefficient \(B_1, B_2, \ldots, B_p\), $$I = (W_0 + W_1 L + W_2 L^2 + \cdots + ) (I - B_1 L - B_2 L^2 - \cdots - B_p L^p)$$ Recursively, $$W_0 = I$$ $$W_1 = W_0 B_1 (W_1^T = B_1^T W_0^T)$$ $$W_2 = W_1 B_1 + W_0 B_2 (W_2^T = B_1^T W_1^T + B_2^T W_0^T)$$ $$W_j = \sum_{j = 1}^k W_{k - j} B_j (W_j^T = \sum_{j = 1}^k B_j^T W_{k - j}^T)$$

References

Lütkepohl, H. (2007). New Introduction to Multiple Time Series Analysis. Springer Publishing.