ECON3350/7350
Single Equation Models of Multiple Time Series
Eric Eisenstat
The University of Queensland
Lecture 4
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4
1 / 22
Multiple Time Series Models
Recommended readings
Author
Title
Chapter
Call No
Patterson
Enders Hill et al.
An Introduction to Applied Economet- rics: A Time Series Approach
Applied Econometric Time Series, 4e Principles of Econo- metrics
2 (except 2.6)
5.1-5.3 9
HB139 .P373 2000
HB139 .E55 2015 HB139 .H548 2008
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 2 / 22
Autoregressive Distributed Lag
In this lecture, we explore the autoregressive distributed lag (ARDL) model as a way to jointly model two or more variables (e.g. yt, xt, wt).
Today’s lecture is an introduction to several topics coming in the following weeks:
stochastic trends (next week) equilibrium error correction multiple equation models
The ARDL encompasses many other models.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4
3 / 22
The ARDL Model
The ARDL model uses lags of both yt and xt to permit richer modelling of dynamics.
The ARDL(p, l) general form:
yt =a0 +a1yt−1 +···+apyt−p +θ0xt +θ1xt−1 +···+θlxt−l +εt, pl
=a0 +aiyt−i +θjxt−j +εt. i=1 j=0
Examples:
ARDL(2, 2) : yt = a0 + a1yt−1 + a2yt−2 + θ0xt + θ1xt−1 + θ2xt−2 + εt,
ARDL(4,1): yt =a0 +a1yt−1 +···+a4yt−4 +θ0xt +θ1xt−1 +εt.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 4 / 22
The ARDL with Several Variables
Using lag operator notation, the ARDL(p, l) can be written: a(L)yt = a0 + θ(L)xt + εt.
This is useful to represent ARDLs with several variables–e.g., an ARDL(p, l, s) is given by:
where
a(L)yt = a0 + θ(L)xt + γ(L)wt + εt,
a(L) = 1 − a1L − · · · − apLp, θ(L)=θ0 +θ1L+···+θlLl, γ(L)=γ0 +γ1L+···+γsLs.
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 4 5 / 22
Dynamic Effects – Multipliers in the ARDL
ARDL allows us to construct IRFs of yt to shocks in either xt or yt. Example: ARDL(1, 1) given by yt = a0 + a1yt−1 + θ0xt + θ1xt−1 + εt.
1 Immediate response or impact multiplier: ∂yt =θ0.
∂yt+1 = a ∂yt + θ = a θ + θ , ∂x 1∂x 1 10 1
∂xt
2 The response after one period, two periods, etc.:
tt
∂yt+2 = a ∂yt+1 = a (a θ + θ ),
∂x 1∂x 110 1 tt
.
Eric Eisenstat
(School of Economics)
ECON3350/7350
Week 4
6 / 22
Dynamic Effects – Multipliers in the ARDL
The effect on yt+h is decreasing as long as |a1| < 1 (stability condition). The long run multiplier (LRM) is the sum of effects over time:
Given the stability condition is satisfied, LRM in the ARDL(1, 1) is
LRM = θ0 + a1θ0 + θ1 + a1(a1θ0 + θ1) + · · · ,
=(1+a1 +a21 +···)(θ0 +θ1),
= θ0 +θ1. 1−a1
We return to the long-run effect later in the lecture.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 7 / 22
Multipliers ARDL(1, 1) - Summary
RECAP
1) Immediate response
2) The effect after 1-period 3) The effect after 2-periods Sum 1)+2)+3)+···
Time (t) t = 0
t + 1 t + 2 LRM
Multiplier
θ0 a1θ0 + θ1
a1(a1θ0 + θ1)
θ0 +θ1 1−a1
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 4
8 / 22
The ARDL Family of Models
ARDL(1, 1): yt = a0 + a1yt−1 + θ0xt + θ1xt−1 + εt
1. Static Regression:
yt = a0 + θ0xt + εt; Restrictions: a1 = 0, θ1 = 0.
2. First-order autoregression:
yt = a0 + a1yt−1 + εt; Restrictions: θ0 = 0, θ1 = 0.
3. Leading indicator equation:
yt = a0 + θ1xt−1 + εt; Restrictions: a1 = 0, θ0 = 0.
4. Equation in first differences:
∆yt = a0 + θ0∆xt + εt; Restrictions: a1 = 1, θ1 = −θ0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 9 / 22
The ARDL family of Models
ARDL(1, 1): yt = a0 + a1yt−1 + θ0xt + θ1xt−1 + εt
5. First-order distributed lag:
yt =a0 +θ0xt +θ1xt−1 +εt; Restrictions: a1 = 0.
6. Partial adjustment model:
yt =a0 +a1yt−1 +θ0xt +εt; Restrictions: θ1 = 0.
7. Dead start model:
yt =a0 +a1yt−1 +θ1xt−1 +εt; Restrictions: θ0 = 0.
8. Proportional response model:
yt = a0+a1(yt−1−xt−1)+θ0 xt+εt; Restrictions: a1 = 1, θ1 = −a1.
9. Error correction model:
∆yt = a0 + α(yt−1 − βxt−1) + θ0∆xt + εt; This is a re-arrangement of the ARDL equation.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 10 / 22
The Error Correction Model (ECM)
Start with the ARDL(1, 1):
yt =a0 +a1yt−1 +θ0xt +θ1xt−1 +εt.
Add and subtract yt−1 and θ0xt−1:
yt =a0 +a1yt−1 +yt−1 −yt−1
+ θ0xt + θ0xt−1 − θ0xt−1 + θ1xt−1 + εt, then re-arrange:
yt −yt−1 =a0 +(a1 −1)yt−1 +(θ0 +θ1)xt−1 +θ0(xt −xt−1)+εt, ∆yt =a0 +(a1 −1)yt−1 +(θ0 +θ1)xt−1 +θ0∆xt +εt,
∆yt = a0 + α(yt−1 − βxt−1) + θ0∆xt + εt, this is the ECM.
where α = −(1 − a1) and β = θ0+θ1 . 1−a1
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 11 / 22
The ECM and Long-Run Equilibrium
The ECM has a particularly important interpretation in terms of the long-run equilibrium.
Two non-stationary variables yt and xt are said to co-exist in long-run equilibrium if zt = yt − βxt is stationary (with E(zt) = 0).
In the short-run shocks to yt−1 and/or xt−1 cause the variables to deviate from the long-run equilibrium, such that yt−1 − βxt−1 ̸= 0.
Since the long-run relationship is stationary, yt must adjust to restore equilibrium; the adjustment ∆yt is proportional to the magnitude of the deviation, i.e. α(yt−1 − βxt−1).
α determines the speed of adjustment.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 12 / 22
The ECM and Long-Run Equilibrium
The equilibrium solution of the ARDL is the mean path of (stationary) yt when xt is constant.
The equilibrium between yt and xt can be found by setting them all to their long-run values and εt = 0:
yt =yt−1 =y∗ xt =xt−1 =x∗. Substituting into the ARDL(1, 1), we obtain:
y∗ = a0 +a1y∗ +θ0x∗ +θ1x∗, y∗ = a0 + θ0 + θ1 x∗.
1−a1 1−a1
The mean path response is therefore y∗ = μ + βx∗, where β is the LRM.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 13 / 22
Illustration
Consider the problem of estimating the relationship between quarterly capital expenditures and appropriations for manufacturing firms.
The data are from Hill et al. (2001), Chapter 15. The model is ARDL(p, l):
where
pl
yt =a0 +aiyt−i +θjxt−j +εt,
i=1 j=0
yt : quarterly capital expenditures, xt : appropriations,
T = 88 quarters.
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 4 14 / 22
Output for ARDL(1, 1)
Dependent Variable: Y
Method: Least Squares
Sample (adjusted): 2 88
Included observations: 87 after adjustments
Variable Coefficient C −2.403615
Std. Error
28.95567 0.018967 0.023504 0.017010
t-Statistic
−0.083010 1.833658 6.645260 47.07956
Prob.
0.9340 0.0703 0.0000 0.0000
4331.851 2196.238 12.31555 12.42892 12.36120 1.549251
X X(-1) Y(-1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 11041.90 Prob(F-statistic) 0.000000
0.034779 0.156188 0.800800
0.997501 0.997410 111.7642 1036772.
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
−531.7262
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 4
15 / 22
Output for ARDL(1, 2)
Dependent Variable: Y
Method: Least Squares
Sample (adjusted): 3 88
Included observations: 86 after adjustments
Variable Coefficient C −0.031351
Std. Error
27.90879 0.018318 0.036297 0.021178 0.038246
t-Statistic
−0.001123 2.339324 1.883555 35.82410 3.103414
Prob.
0.9991 0.0218 0.0632 0.0000 0.0026
4358.070 2195.381 12.23387 12.37656 12.29129 1.545836
X X(-1) Y(-1) X(-2)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 8979.669 Prob(F-statistic) 0.000000
0.042851 0.068367 0.758669 0.118692
0.997750 0.997639 106.6768 921775.5
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
−521.0562
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 4
16 / 22
Output for ARDL(2, 1)
Dependent Variable: Y
Method: Least Squares
Sample (adjusted): 3 88
Included observations: 86 after adjustments
Variable Coefficient C 5.707282
X 0.037896 X(-1) 0.108816 Y(-1) 1.128740 Y(-2) −0.284722
R-squared 0.997808
Std. Error
27.67370 0.017908 0.026142 0.095923 0.082128
t-Statistic
0.206235 2.116172 4.162494 11.76720
Prob.
0.8371 0.0374 0.0001 0.0000 0.0008
4358.070 2195.381 12.20786 12.35056 12.26529 2.265866
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 9216.766 Prob(F-statistic) 0.000000
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
0.997699 105.2988 898115.2
−3.466813 Mean dependent var
−519.9381
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 4
17 / 22
Output for ARDL(2, 2)
Dependent Variable: Y
Method: Least Squares
Sample (adjusted): 3 88
Included observations: 86 after adjustments
Variable Coefficient C 7.520561
X 0.043773 X(-1) 0.049477 Y(-1) 1.037766 X(-2) 0.091098 Y(-2) −0.233847
R-squared 0.997955
Std. Error
26.90555 0.017575 0.035453 0.100637 0.037961 0.082584
t-Statistic
0.279517 2.490608 1.395584 10.31201 2.399744
Prob.
0.7806 0.0148 0.1667 0.0000 0.0187 0.0059
4358.070 2195.381 12.16161 12.33284 12.23052 2.143801
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 7807.755 Prob(F-statistic) 0.000000
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
0.997827 102.3356 837806.1
−2.831613 Mean dependent var
−516.9491
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 4
18 / 22
Illustration - Model Comparison
Model Information Criteria Computed Values:
ARDL(1,1) AIC
SC 12.4289
12.3155
ARDL(2,1) AIC
SC 12.3505
12.2978
ARDL(1,2) AIC
SC 12.3765
ARDL(2,2) AIC
SC 12.3328*
12.2338
12.1616*
Eric Eisenstat
(School of Economics)
ECON3350/7350
Week 4
19 / 22
Estimated Marginal Effects from ARDL(2, 2)
.20
.16
.12
.08
.04
.00
0 2 4 6 8 10 12 14 16
Periods After Impact
Total multiplier: 0.9402 =⇒ for each dollar in appropriation money, 0.94 cents are spent on capital expenditures.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 4 20 / 22
Capital Expenditure