The University of Queensland
(School of Economics) Applied Econometrics for Macro and Finance Week 4 1 / 36
ECON7350 Dynamic Relationships
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Dynamic Relationships
In addition to forecasting, time series data is often used to infer dynamic relationships between economic or financial variables.
We regard economic or financial variables as representing stochastic processes.
The fundamental assumption is that variables are driven by unanticipated shocks.
By modelling the underlying stochastic process, we can quantify how variables respond to shocks at different points in time.
Dynamic relationships are central to many policy decisions.
How does inflation respond to a monetary policy shock in the short term?
What is the long run increase in consumption following a decrease in the tax rate? How do sales respond over time to an increase in marketing expenditures?
How much does news about future productivity affect business cycles?
(School of Economics) Applied Econometrics for Macro and Finance Week 4 2 / 36
Impulse Response Functions
If the dynamics of an economic or financial variable are characterized by a stochastic process, then we can quantify dynamic effects using impulse response functions (IRFs).
IRFs are derived based on the idea that a shock to the stochastic process represents new information.
If a shocks occurs at time t, there is new information relative to the past (time t−1,t−2,…).
The new information changes the expected path of the process from time t onward (t,t+1,… .).
IRFs quantify the difference in the expected paths after the shock occurs (including the new information at time t) and before the shock occurs.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 3 / 36
Impulse Response Functions
Let It denote the information set at time t.
The impulse response of yt at horizon h to a shock εt occurring at time t of magnitude δ is
generally computed as
IRy,h = E(yt+h | It−1, εt = δ) − E(yt+h | It−1). The IRF for yt (IRFy) is the plot of IRy,h against h = 0,1,….
By convention, δ is typically set to either 1 or the standard deviation of εt.
The definition of It and formulae for IRy,h are made more precise by specifying a particular
model for the stochastic process {yt} in terms of the shock process {εt}.
Because IRFs are defined through conditional expectations, they are not subject to uncertainty
due to an unknown future.
However, specification and estimation uncertainty is generally present in inference about IRFs
and must be accounted for in practice.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 4 / 36
IRFs From the ARMA(p, q)
Suppose yt is an ARMA(p, q) process: a(L)yt = a0 + b(L)εt.
Using polynomial expansion and division, θ(L) = b(L)/a(L) can be decomposed as: θ(L)= a(L) =1+θ1L+···+θhLh +β(L)Lh+1 +α(L)b(L)Lh+1,
h is an integer greater than or equal to 0;
θ1, . . . , θh are obtained recursively using the method of undetermined coefficients; β(L) is a polynomial of degree q + h;
α(L) is a polynomial of degree (p − 1)(h + 1);
dividing (1 − α(L))/a(L) produces a polynomial of degree ph.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 5 / 36
IRFs From the ARMA(p, q)
Using this decomposition, and setting γ(L) ≡ (1 − α(L))/a(L), we can write a(L)yt+h = a0 + a(L) b(L) εt+h
h b(L)
=a0 +a(L) (1+θ1L+···+θhL )εt+h +β(L)εt−1 +α(L)a(L)εt−1 =a +a(L)(1+θ L+···+θ Lh)ε +β(L)ε +α(L)b(L)ε
0 1 h t+h t−1 t−1
=a +a(L)(1+θ L+···+θ Lh)ε +β(L)ε +α(L)(a(L)y −a )
0 1 h t+h t−1 t−1 0 =a(L)(1+θ L+···+θ Lh)ε +β(L)ε +α(L)y +(1−α(L))a ,
1ht+ht−1t−1 0 yt+h = εt+h + θ1εt+h−1 + · · · + θhεt + β(L)εt−1 + α(L)yt−1 + γ(1)a0.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 6 / 36
IRFs From the ARMA(p, q)
In an ARMA(p, q) the information set It contains all the realised variables and errors up to
time t: It = {yt,yt−1,…,εt,εt−1,…}.
The additional information in It relative to It−1 is εt. To derive the IR at horizon h, we
assume that εt = δ, which leads to:
E(yt+h | It−1) = β(L)εt−1 − α(L)yt−1 + γ(1)a0,
E(yt+h | It−1, εt = δ) = θhδ + β(L)εt−1 − α(L)yt−1 + γ(1)a0. Hence, for an ARMA(p, q) process, IRy,h = θhδ.
IRFs depend on the mean-independence of errors, i.e. E(εt+h | It−1) = 0, h ≥ 0; deterministic terms (such as a0) do not matter—we can ignore them;
if the ARMA(p, q) is stable, then IRFs decay to zero;
if the ARMA(p, q) is not stable then IRFs either explode to ±∞ as h −→ ∞ or, in a special case that will be discussed soon, converge to a non-zero constant.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 7 / 36
Multiple Time Series Variables
Dynamic relationships are typically analysed for two or more variables.
Suppose in addition to the process {yt}, we also have another process {xt}. A relationship
between the two processes can be modelled as:
a(L)yt = a0 + b(L)xt + ut,
a(L) = 1 − a1L − · · · − apLp, b(L)=b0 +b1L+···+blLl.
Hence, a(L) is a polynomial of degree p and b(L) is a polynomial of degree l.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 8 / 36
The Autoregressive Distributed Lag Model
The time-t information set is It = {yt,yt−1,…,xt,xt−1,…,ut,ut−1,…}.
In general ut is correlated and depends on innovations in both processes {xt},{yt}.
For example, it is sensible to specify:
ut = ζ(L)εx,t + ξ(L)εy,t,
where εx,t is the mean-independent innovation to xt and εy,t is the mean-independent
innovation to yt.
The autoregressive distributed lag model ARDL(p, l) assumes ζ(L) = 0 and ξ(L) = 1, so that
ut ≡ εy,t, and εy,t satisfies: Mean-independence: E(εy,t | It−1, xt) = 0. Homoscedasticity: Var(εy,t | It−1, xt) = σε2.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 9 / 36
The ARDL Model
The ARDL(p, l) can be viewed as a standard regression:
ARDL(2, 2) : ARDL(4,1): ARDL(1, 0) : ARDL(0, 1) :
yt = a0 + a1yt−1 + a2yt−2 + b0xt + b1xt−1 + b2xt−2 + εy,t, yt =a0 +a1yt−1 +···+a4yt−4 +b0xt +b1xt−1 +εy,t,
yt = a0 + a1yt−1 + b0xt + εy,t,
yt = a0 + b0xt + b1xt−1 + εy,t.
yt =a0 +a1yt−1 +···+apyt−p +b0xt +b1xt−1 +···+blxt−l +εt, pl
=a0 +aiyt−i +bjxt−j +εy,t. i=1 j=0
ARDL(p, l) parameters can be estimated with OLS.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 10 / 36
The ARDL with Several Variables
ARDLs can be specified with any number of variables–e.g., an ARDL(p, l, s) is given by: a(L)yt = a0 + b(L)xt + c(L)wt + εy,t,
a(L) = 1 − a1L − · · · − apLp, b(L)=b0 +b1L+···+blLl, c(L)=c0 +c1L+···+csLs,
and εy,t satisfies E(εy,t | It−1, xt, wt) = 0, Var(εy,t | It−1, xt, wt) = σε2.
We will discuss methods and intuition for the three-variable ARDL(p, l, s); they extend in a
straightforward way to ARDLs with an arbitrary number of variables.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 11 / 36
IRFs in the ARDL(p, l, s)
Given an ARDL(p, l, s), the time-t information set is
It = {yt,yt−1,…,xt,xt−1,…,wt,wt−1,…,εy,t,εy,t−1,…}. Let μx,t+j ≡ E(xt+j | It−1) and μw,t+j ≡ E(wt+j | It−1) for all j ≥ 0.
Suppose we are interested in the response of yt at horizon h to a shock in xt of magnitude δ, which is given by
IRy,x,h = E(yt+h | It−1, xt = μx,t + δ) − IRy,x,h = E(yt+h | It−1).
To derive IRFs, we need to also specify: E(wt+j |It−1,xt) for j = 0,…,h; E(xt+j |It−1,xt) for j = 1,…,h.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 12 / 36
IRFs in the ARDL(p, l, s)
Standard Assumption 1: {wt} and {xt} are mean-independent; i.e.,
E(wt+j |It−1,xt) = μw,t+j for all j = 0,…,h.
Standard Assumption 2: {xt} is a process with either
1 E(xt+j |It−1,xt) = μx,t+j for all j = 1,…,h, or
2 E(xt+j | It−1, xt) = μx,t+j + δ for all j = 1, . . . , h.
IRFs are obtained by decomposing the ratio of polynomials
θx(L) = b(L) = θx,0 + θx,1L + · · · + θx,hLh + βx(L)Lh+1 + αx(L) b(L) Lh+1 a(L) a(L)
and applying the assumptions above.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 13 / 36
IRFs in the ARDL(p, l, s)
Let vt = a0 + c(L)wt + εy,t so that the ARDL(p, l, s) can be expressed as a(L)yt = b(L)xt + vt.
Using mean-independence of εy,t and Standard Assumption 1:
E(vt+j |It−1,xt) = E(vt+j |It−1) for all j = 0,…,h.
Using the decomposition of θx(L) = b(L)/a(L),
yt+h = θx,0xt+h + θx,1xt+h−1 + · · · + θx,hxt + γx(L)vt+h,
θx,0, . . . , θx,h are obtained from a(L) and b(L) using the method of undetermined coefficients, and γx(L) is a polynomial of degree ph.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 14 / 36
IRFs in the ARDL(p, l, s)
Under Standard Assumption 2.1:
E(yt+h | It−1, xt) = θx,0μx,t+h + θx,1μx,t+h−1 + · · · + θx,h(μx,t + δ) + E(vt+h | It−1, xt) + γx,1E(vt+h−1 | It−1, xt)
+ · · · + γx,phE(vt−(p−1)h | It−1, xt) = θx,hδ + E(yt+h | It−1).
Under Standard Assumption 2.2:
E(yt+h | It−1, xt) = θx,0(μx,t+h + δ) + θx,1(μx,t+h−1 + δ) + · · · + θx,h(μx,t + δ) + E(vt+h | It−1, xt) + γx,1E(vt+h−1 | It−1, xt)
+ · · · + γx,phE(vt−(p−1)h | It−1, xt) =(θx,0 +···+θx,h)δ+E(yt+h|It−1).
(School of Economics) Applied Econometrics for Macro and Finance Week 4 15 / 36
Dynamic Effects and Multipliers in the ARDL
What is the effect on yt, yt+1, . . . of an unanticipated shock to xt?
If the unanticipated shock at time t is:
1 one-off (Standard Assumption 2.1), then the effect on yt+h is IRy,x,h = θx,hδ;
2 permanent (Standard Assumption 2.2) then the effect on yt+h is IRy,x,h = (θx,0 + · · · + θx,h)δ.
Horizon h = 0 is called the impact horizon; for both one-off and permanent shocks, the response on impact or impact multiplier is IRy,x,0 = θx,0δ = b0δ.
Horizon h −→ ∞ is called the long-run horizon. The long-run response for 1 a one-off shocks is IRy,x,∞ = limh→∞ θx,hδ, if it exists.
2 apermanentshockisIR =∞ θ δ=θ (1)δ= b(1)δ,ifitexists;thisisalso y,x,∞ j=0 x,j x a(1)
called the long-run multiplier (LRM).
(School of Economics) Applied Econometrics for Macro and Finance
Week 4 16 / 36
Stability in an ARDL
The ARDL is stable if and only a(z) ̸= 0 for all |z| ≤ 1. If the ARDL is stable, the long-run response exists.
If the ARDL is unstable, the long-run response to a one-off shock exists only in special cases, but the LRM does not exist.
The stability of the ARDL is unrelated to the stationarity of {xt} and/or {wt}. As a consequence, the ARDL could be stable but {yt} non-stationary.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 17 / 36
Dynamic Effects and Multipliers in the ARDL(1, 1)
Example: ARDL(1, 1) given by yt = a0 + a1yt−1 + b0xt + b1xt−1 + εy,t. If δ = 1, the impact multiplier is:
IRy,x,0 = b0.
If the shock is one-off, the response after one period, two periods, etc.:
IRy,x,1 = a1IRy,x,0 + b1 = a1b0 + b1, IRy,x,2 = a1IRy,x,1 = a1(a1b0 + b1),
IRy,x,h = a1IRy,x,h−1 = ah−1(a1b0 + b1),
(School of Economics)
Applied Econometrics for Macro and Finance
Dynamic Effects and Multipliers in the ARDL(1, 1)
The ARDL(1, 1) is stable if and only if |a1| < 1.
If the ARDL(1, 1) is stable, the effect on yt+h of a one-off shock is decreasing. This holds even for non-stationary {xt} (and, consequently, {yt}).
If the ARDL(1, 1) is stable, the LRM in the ARDL(1, 1) is
LRM = b0 + a1b0 + b1 + a1(a1b0 + b1) + · · · ,
=(1+a1 +a21 +···)(b0 +b1), =b0+b1 =θx(1).
If a1 = 1, then the long-run response to a one-off shock is IRy,x,∞ = b0 + b1, but the LRM
does not exist; if a1 = −1 neither effect exists unless b1 = b0.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 19 / 36
Inference on Dynamic Effects with an ARDL
In practice, IRFs are computed from a realisation of an ARDL(p, l, s) model.
Working with ARDLs is similar to working with ARMAs: a realisation is obtained through
model specification and model estimation.
ARDL specification consists of:
setting lag orders p, l and s;
setting restrictions on parameters (e.g. stability).
ARDL model estimation consists of determining parameter values from a sample {yt, xt, wt}Tt=1 given lag orders and subject to restrictions.
If no restrictions are present, estimation can be carried out with OLS.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 20 / 36
Estimating Dynamic Effects with a Specified ARDL
Given a specified ARDL(p, l, s) model, estimation uncertainty is characterised by
2 Var(a0,...,ap,b0,...,bp,c0,...,cp,σε).
The joint variance can be used to derive confidence sets for the impulse responses. For example, if the responses are to a shock in x, then the confidence set is for IRy,x,0, . . . , IRy,x,∞.
Estimated IRFs are typically presented with confidence intervals.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 21 / 36
Specification of an ARDL
The specification of an ARDL is implemented by constructing an adequate set of models, in a similar fashion to ARMAs.
Use information criteria (e.g., AIC and BIC) to assess fit and parsimony.
Analyse the SACF and SPACF of estimated residuals to diagnose possible violations of the mean-independence assumption.
The AIC and BIC are computed in the same way but adjusted for the total number of estimated coefficients.
For an ARDL(p,l,s), the formulas use p+l+s+1 instead of p+q+1.
The Ljung-Box test requires the additional assumption that all independent variables (e.g., xt and wt in an ARDL(p, l, s)) are strictly exogenous.
Strict exogeneity: corr(xt±j,εy,t) = 0 for all j ≥ 1; same for wt. This is a difficult assumption to justify in most settings.
If the assumption is satisfied, then QK ∼ χ2K−p.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 22 / 36
General Strategy to Working with an ARDL
Overall, the general strategy to working with ARDLs is the same as with ARMAs.
1 Construct a large set of potential models with alternative lag orders and/or parameter restrictions.
2 Reduce the set by iteratively comparing information criteria (AIC/BIC) and analysing autocorrelations in the estimated residuals.
3 Compare results across the final set of models; interpret accordingly.
Because ARDLs can generally be estimated with OLS, large sets of models can be processed
quickly: computation is not an issue!
(School of Economics) Applied Econometrics for Macro and Finance Week 4 23 / 36
Error Correction Decomposition
Letα(L)beapolynomialinLofdegreer: α(L)=α0+α1L+···+αrLr. For every α(L), there exists a polynomial α(L) of degree r − 1, such that:
α(L)(1 − L) = α(L) − α(1)L.
The same decomposition can be applied to α(L) to obtain a polynomial of degree r − 2, etc.
This property of polynomials is used to transform the ARDL(p, l, s) into another useful representation called the error correction model (ECM) form.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 24 / 36
The Error Correction Model
and substitute these into
a(L) = a(L)(1 − L) + a(1)L, b(L) = b(L)(1 − L) + b(1)L,
c(L) = c(L)(1 − L) + c(1)L,
a(L)yt = a0 + b(L)xt + c(L)wt + εy,t.
Define the difference operator ∆ = 1 − L such that ∆yt = yt − yt−1 and a(L)yt = a(L)∆yt + a(1)yt−1,
b(L)xt = b(L)∆xt + b(1)xt−1, c(L)wt = c(L)∆wt + c(1)wt−1.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 25 / 36
The Error Correction Model
We obtain:
a(L)∆yt + a(1)yt−1 = a0 − γ + γ + b(L)∆xt + b(1)xt−1 + c(L)∆wt + c(1)wt−1 + εy,t, a(L)∆yt = γ−a(1)yt−1 + (a0 − γ) + b(1)xt−1 + c(1)wt−1
+ b(L)∆xt + c(L)∆wt + εy,t.
a0 − γ b(1) c(1)
If a(1) ̸= 0, this yields:
a(L)∆yt = γ−a(1) yt−1 − a(1) − a(1)xt−1 − a(1)wt−1
+ b(L)∆xt + c(L)∆wt + εy,t,
= γ + α(yt−1 − μ − βxxt−1 − βwwt−1) + b(L)∆xt + c(L)∆wt + εy,t,
where α = −a(1), μ = a0−γ , βx = θx(1) and βw = θw(1). a(1)
(School of Economics) Applied Econometrics for Macro and Finance Week 4 26 / 36
The ECM Form of an ARDL(1, 1)
Start with the stable ARDL(1, 1):
yt =a0 +a1yt−1 +b0xt +b1xt−1 +εy,t, |a1|<1.
Add and subtract yt−1, b0xt−1 and γ:
yt =a0−γ+γ+a1yt−1 +yt−1 −yt−1
+ b0xt + b0xt−1 − b0xt−1 + b1xt−1 + εy,t,
then re-arrange:
yt −yt−1 =γ+(a0 −γ)+(a1 −1)yt−1
+ (b0 + b1)xt−1 + b0(xt − xt−1) + εy,t,
∆yt =γ+(a1 −1)yt−1 +(a0 −γ)+(b0 +b1)xt−1 +b0∆xt +εy,t, ∆yt =γ+α(yt−1−μ−βxxt−1)+b0∆xt+εy,t, thisistheECM.
whereα=−(1−a1),μ=a0−γ andβx=b0+b1. 1−a1 1−a1
(School of Economics)
Applied Econometrics for Macro and Finance Week 4 27 / 36
Interpretation of the Error Correction Model
The ECM form of the ARDL exists as long as a(1) ̸= 0; otherwise, there exists an ARDL with dependent variable ∆yt and an ECM form as long as a(1) ̸= 0.
If the ARDL is stable:
ECM coefficients βx and βw may be interpreted as long-run multipliers.
More generally, the ECM represents theoretically meaningful relationships between {yt}, {xt} and {wt}.
If it is unstable, the ECM exists but cannot be interpreted in a meaningful way.
Neither the existence of the ECM form nor the interpretation of βx and βw as LRMs depends
on the properties of {xt} and/or {wt}.
In particular, {xt} and/or {wt} can be stationary or non-stationary processes.
However, the type of long-run relationship between {yt}, {xt} and {wt} that is implied by the
LRMs does depend on whether or not these processes are stationary.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 28 / 36
Equilibrium in Dynamic Systems
An important concept underlying much of economics is equilibrium between two or more variables.
For dynamic systems comprised of times series variables such as yt and xt, equilibrium is represented by a function f(·) satisfying
E(f(yt, xt)) = 0.
In linear models, we can write f(yt, xt) = yt − μ − βxxt, so that equilibrium is characterised by
E(yt −μ−βxxt)=0.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 29 / 36
The ECM form of an ARDL(1, 1) for Stationary Processes
Suppose {yt} and {xt} are stationary. Then they have finite unconditional means y∗ = E(yt) and x∗ = E(xt), also called steady states (SS).
The equilibrium condition then implies
y∗ − μ − βxx∗ = 0.
Hence, it is a relationship between the steady states of two time series variables.
We call this relationship the steady-state equilibrium (SSE) for stationary processes. It can be derived from the ARDL(1, 1) by taking expectations:
E(yt) = a0 + a1E(yt−1) + b0E(xt) + b1E(xt−1) + E(εy,t), y∗ =a0 +a1y∗ +b0x∗ +b1x∗,
y∗ − a0 − b0 + b1 x∗ = 0. 1−a1 1−a1
(School of Economics) Applied Econometrics for Macro and Finance Week 4 30 / 36
The ECM Form of an ARDL(1, 1) for Stationary Processes This is precisely what appears in the ECM with γ = 0.
IntheECM,μ= a0 andβx=b0+b1. 1−a1 1−a1
Note: for stationary processes, we must have γ = 0!
The ECM provides a transparent view of the mechanism underlying SS dynamics.
An intuitive overview is as follows.
Suppose the two variables are in SSE at a given point in time.
A shock occurs that moves the system out of SSE.
In the following period, the dependent variable must adjust.
The adjustment is towards SSE and proportional to the deviation from the SSE.
The amount of adjustment in each period determines how long it takes to restore SSE: this is governed by the coefficient α.
(School of Economics) Applied Econometrics for Macro and Finance Week 4 31 / 36
The ECM Form of an ARDL(1, 1) and Long-Run Equilibrium
Suppose {yt} and {xt} are non-stationary such that they do not have unconditional means.
Nevertheless, zt = yt − βxt may be stationary with E(zt) = μ. Then, E(yt − μ −
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