ECON 3350/7350
Single Equation Models of Multiple Time Series
Eric Eisenstat
The University of Queensland
Tutorial 4
Eric Eisenstat (School of Economics) ECON3350/7350 Week 1
1 / 6
ARMA(p,q) with deterministic trend
yt =a+a2t+a1yt−1+εt;|a1|<1
yt is trend stationary because if we take the trend out the new process is
stationary. We return to deterministic and stochastic trends next week. De-trending
yt −a2t=a0 +a1yt−1 +εt y=a+ay +ε
y is an ARMA(1,0) t
t 0 1t−1 t
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 1 2 / 6
ARDL(p, l, s) with trend
For ct, at, and yt we could have an ARDL(p, q, m): θ(L)ct = δ + γ(L)at + λ(L)yt + εt
Where,
p θ(L)=(1−θ1L−θ2L2 −...−θpLp)= θiLi
i=0 q
γ(L) = (γ0 + γ1L + γ2L2 − ... + γqLq) = γjLj j=0
m
λ(L) = (λ0 + λ1L + λ2L2 − ... + λmLm) = λjLj
j=0
Adding a deterministic trend
θ(L)ct = δ0 + δ1t + γ(L)at + λ(L)yt + εt
Eric Eisenstat (School of Economics) ECON3350/7350 Week 1 3 / 6
The ARDL Family of Models
Using ARDL(1, 1)
yt =δ+a1yt−1 +θ0xt +θ1xt−1 +εt
1. Static Regression:
yt = δ + θ0xt + εt; Restrictions: a1 = 0; θ1 = 0
2. First order autoregressive process:yt = δ + a1yt−1 + εt; Restrictions: θ0 = 0; θ1 = 0
3. Leading indicator equation: yt = δ + θ1xt−1 + εt; Restrictions: a1 = 0; θ0 = 0
4. Equation in first differences: ∆yt = δ + θ0∆xt + εt; Restrictions: a1 = 1, θ0 = −θ1
Eric Eisenstat (School of Economics) ECON3350/7350 Week 1 4 / 6
The ARDL Family of Models-II
yt =δ+a1yt−1 +θ0xt +θ1xt−1 +εt
5. First order distributed lag model:
yt =δ+θ0xt +θ1xt−1 +εt Restrictions: a1 = 0
6. Partial adjustment model: yt =δ+a1yt−1 +θ0xt +εt Restrictions: θ1 = 0
7. Dead Start model (lagged information only):
yt =δ+a1yt−1 +θ1xt−1 +εt Restrictions: θ0 = 0
8. Proportional Response Model:
yt =
δ+a1(yt−1 −xt−1)+θ0xt +εt Restrictions: θ1 = −a1
9. Error Correction Mechanism:
∆yt =δ+α(yt−1 −βxt−1)+θ0∆xt +εt
where, β = (θ1 + θ0); α = a1 − 1 (1−a1)
This is a re-arrangement of the ARDL equation.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 1 5 / 6
Multipliers
1 Immediate Response or Impact Multiplier ∂ct =γ0
∂at
2 The Effect after one period, two periods, ...
∂ct+1 = θ ∂ct + γ = θ γ + γ ∂a 1∂a 1 10 1
tt
∂ct+2 = θ ∂ct+1 = θ (θ γ + γ )
∂a 1∂a 110 1 tt
3 Long-run multiplier
LRM = γ(1) = (γ0 +γ1 +γ2 +...+γp)
θ(1) (1−θ1 −θ2 −...−θp)
Eric Eisenstat (School of Economics)
ECON3350/7350
Week 1
6 / 6