How does a process react to a “shock” at a given time?
Definition: Given an ARMA(p,q) process {Xt}, the Impulse Response Function with origin x generated by a shock of size s at time t=k is x~t=x for t<k and x~t=x+∂ϵk∂Xt⋅s if t≥k
This represents giving a shock of size s to ϵk
Since a stable process has Wold representation Xt=∑j=0∞ψjϵt−j, we have ∂ϵk∂Xt=ψt−k
Example:
For AR(1), ψj=ϕj so the IRF is x~t={xx+ϕt−ksif t<kif t≥k
For more complicated time series, we can obtain a recursion for the partial derivatives from the recursive formulation of the model
Xt=ϕ1Xt−1+⋯+ϕpXt−p+ϵt+θ1ϵt−1+⋯+θqϵt−q
∂ϵk∂Xk=1 ∂ϵk∂Xk+1=ϕ1∂ϵk∂Xk+θ1=ϕ1+θ1 ∂ϵk∂Xk+2=ϕ1∂ϵk∂Xk+1+ϕ2∂ϵk∂Xk+θ2=ϕ12+ϕ1θ1+ϕ2+θ2
And so on…
Example:
Calculate the IRF of ARMA(1,1)
∂ϵk∂Xt=ϕ∂ϵk∂Xt−1+∂ϵk∂ϵt+θ∂ϵkϵt−1
∂ϵk∂Xk=1 ∂ϵk∂Xk+1=ϕ∂ϵk∂Xk+θ=ϕ+θ ∂ϵk∂Xk+2=ϕ∂ϵk∂Xk+1=ϕ(ϕ+θ)
… ∂ϵk∂Xk+h=ϕh−1(ϕ+θ) for h≥1
x~t=⎩⎨⎧xx+sx+ϕt−k−1(ϕ+θ)⋅sif t<kif t=kif t>k
In a triangular system (see ADL Models), we have {Yt} with a dependency on {Xt}, in which case the IRF effectively propagates
Example: ADL(1,0)+AR(1) system Yt=ϕYt−1+βXt+ϵt and Xt=γXt−1+ut
∂uk∂Xt=γt−k if t≥k and 0 otherwise ∂μk∂Yt=∂ukϕ∂Yt−1+β∂uk∂Xt