ECON3350/7350 VAR Models – I
Eric Eisenstat
The University of Queensland
Lecture 9
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 10
1 / 23
VAR Models
Recommended readings
Author
Title
Chapter
Call No
Enders Brooks
Verbeek
Applied Econometric Time Series, 4e Introductory Econo- metrics for Finance, 3e
A Guide to Modern Econometrics
5 7
9.1, 9.4
HB139 .E55 2015 HG173 .B76 2014
HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 2 / 23
Multivariate Time Series
Multiple equations models involve several variables as well as several equations.
Multiple equations models imply there are more than one endogenous variable.
Single equation models involve only one endogenous variables: the dependent variable.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10
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Multivariate Time Series
Endogenous Variables: variables that are explained by other variables in the system. Another name for endogenous variable is dependent variable.
Exogenous Variables: variables that are thought to be exogenous to the system, either by theoretical assumption or by model definition. Other names for exogenous variables include predictor variables and independent variables.
Multiple Equation Models (MEM) have as many equations as endogenous variables.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 4 / 23
Classification of MEMs
MEMs can be broadly classified in two main categories:
Simultaneous Equations Models (SEM): two or more endogenous variables are jointly determined and there is feedback (i.e., information flow in both directions) between them.
Examples: price and quantity demanded in a given time period are negatively related and there is feedback; Structural VAR Models (SVAR).
Non-simultaneous Equations Models: “reduced form” models derived from SEMs.
Examples: Seemingly Unrelated Regressions Models (SUR), Vector Autoregressive Models (VAR), Vector ARMA Models (VARMA), Vector Error Correction Models (VEC), VAR with exogenous variables (VARX).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 5 / 23
Vector Autoregressive Models (VARs)
VARs were popularized in economics by Chris Sims in 1980. These reduced form models are a generalisation of the univariate AR models.
VAR models are characterised by treating all variables as endogenous in the system.
VARs are special cases of VARMAs, which are generalizations of the univariate ARMA models.
VARs are extremely easy to work with—many extensions created in the past 30+ years:
VECM: An extension of the single equation ECM form of an ARDL model to the multivariate case.
VARX: One or more exogenous variables are included in the model.
Also, time-varying parameter VARs, regime-switching VARs, factor-augmented VARs, among many others.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10
6 / 23
Structural VAR (SVAR)
Recall the ARDL(1, 1) model:
yt = b10 − b12zt + γ11yt−1 + γ12zt−1 + εyt.
Suppose we can also formulate a similar model with zt as the dependent variable:
zt = b20 − b21yt + γ21yt−1 + γ22zt−1 + εzt.
Assume that εyt and εzt are uncorrelated white noise processes, i.e. εyt 0 εyt σy2 0
E ε
= 0 , Var ε = 0 σ2 . zt ztz
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 7 / 23
SVAR
Write the system in matrix form:
1 b12yt b10 γ11 γ12yt−1 εyt b1z=b+γγz+ε.
21 t 20 21 22 t−1 zt
B xt γ0 Γ1 xt−1 εt This is referred to as the structural VAR.
Now, let
xt = B−1γ0 + B−1Γ1xt−1 + B−1εt, xt = a0 + A1xt−1 + et.
This is referred to as the reduced form VAR. Eric Eisenstat (School of Economics) ECON3350/7350
Week 10
8 / 23
Reduced form VAR
For every SVAR, we can derive a reduced form VAR, where each equation only has one endogenous variable, but
eyt 1 b12−1 σy2 0 1 b21−1 ω12 ω12
Var e = b 1 0 σ2 b 1 = ω ω2 . zt21 z12 212
B−1 Σ (B−1)′ Ω
So, the reduced form errors eyt and ezt are correlated; often, these are also referred to as VAR residuals to distinguish them from the structural errors εyt and εzt.
However, reduced form errors are still white noise: E(eytey,t−s) = 0, E(eytez,t−s) = 0,
for s = 1,2,….
Eric Eisenstat (School of Economics) ECON3350/7350
Week 10 9 / 23
Relationship Between VARs and SVARs
For any reduced form VAR, there are infinitely many SVARs: there exist
′ other matrices B and Σ such that B−1Σ B−1
= Ω.
Example:
1 0.250
0.500−1 1.000 1 0
0 1 0.250−1 1.000 0.500 1
= =
1.6327 −0.9796
−0.9796 1.3878 ,
4.481−1 3.680
To derive the SVAR from a reduced form VAR, we need additional
information (return to this next week).
1
2.111 1 0
0 1 2.111−1 25.39 4.481 1
1.6327 −0.9796
−0.9796 1.3878 .
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 10 / 23
VAR in n Variables
VAR models have as many equations as endogenous variables, which we will denote by n.
The number of equations in a VAR is known as the dimension.
The number of variables in a VAR is identical to the dimension. A bivariate
VAR contains two endogenous variables and therefore two equations. In general, an n-variate VAR with with p lags is
p
xt =a0 +Apxt−p +et, E(et)=0,
j=1
where Ω is a symmetric and positive definite matrix.
Var(et)=Ω,
Eric Eisenstat (School of Economics) ECON3350/7350
Week 10
11 / 23
Features of VARs
An important feature of the VAR is that estimation is very easy: a VAR is just a system of Seamingly Unrelated Regressions (SUR).
Consistent estimates of a0 and A1, . . . , Ap can be obtained by running OLS on each equation (or more efficiently, GLS on the n equations).
However, the number of free parameters in a0 and A1, . . . , Ap grows quickly with n, i.e. k = n(1 + np).
npk
3 1 12
3 4 39 10 1 110 10 4 410
Eric Eisenstat
(School of Economics)
ECON3350/7350
Week 10
12 / 23
Properties of VARs
All relevant properties of AR processes generalize to VARs. To analyze the properties of VARs, define the lag operator:
A(L)=In −A1L−···−ApLp, and write the n-variate VAR(p) as
A(L)xt = et. The characteristic equation for the VAR(p) is
detA(z)=det(In −A1z−···−Apzp)=0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 13 / 23
Stationarity
Recall that the AR(1) model
xt = a0 + a1xt−1 + et
is stationary if the characteristic equation
1 − a1z = 0
has all roots greater than one in absolute value, which corresponds to |a1| < 1.
Similarly for the VAR(1), stationarity corresponds to det (In − A1z) = 0
having all roots greater than one in absolute value.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 14 / 23
Stationarity
Let z∗ be a root of det A(z). Then, by definition det (In − A1z∗) = 0,
which is equivalent to
1
det z∗In−A1 =0,
Therefore, z∗ is a root of det A(z) if and only if 1 is an eigenvalue of A . z∗ 1
This means, that stationarity for the VAR(1) corresponds to all eigenvalues of A1 being less than one in absolute value.
Note that when A1 has all eigenvalues less than one in absolute value Ah1 −→ 0 as h −→ ∞.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 15 / 23
Example of a Stationary VAR
Consider the VAR(1): yt
1.0846 −0.7385 yt−1 eyt
= 0.2462 −0.0846 Is this VAR stationary?
+ e . t−1 zt
z
t
z
Eric Eisenstat (School of Economics) ECON3350/7350
Week 10
16 / 23
Example of a Stationary VAR
Consider the VAR(1): yt
1.0846 −0.7385 yt−1 eyt
= 0.2462 −0.0846 z Is this VAR stationary?
+ e . t−1 zt
z
t
The 10-period ahead forecast is:
1.0846 −0.738510 0.4291 −0.3219 E(xt+10 | xt, xt−1, . . . ) = 0.2462 −0.0846 = 0.1073 −0.0805 .
The 50-period ahead forecast is:
1.0846 −0.738550 0.0063 −0.0048 E(xt+100 | xt, xt−1, . . . ) = 0.2462 −0.0846 = 0.0016 −0.0012 .
The eigenvalues of A1 are 0.9 and 0.1.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10
16 / 23
Example of a Stationary VAR
Consider the VAR(1):
yt 1.1462 −0.1846 yt−1 eyt
Is this VAR stationary?
z = 0.0615 0.8538 z + e . t t−1 zt
Eric Eisenstat (School of Economics) ECON3350/7350
Week 10
17 / 23
Example of a Stationary VAR
Consider the VAR(1):
yt 1.1462 −0.1846 yt−1 eyt
z = 0.0615 0.8538 z + e . t t−1 zt
Is this VAR stationary?
The 10-period ahead forecast is:
1.1462 −0.184610 3.1118 −2.0724 E(xt+10 | xt, xt−1, . . . ) = 0.0615 0.8538 = 0.6908 −0.1694
The 50-period ahead forecast is:
.
1.1462 −0.184650 144.4799 −108.3560 E(xt+100 | xt, xt−1, . . . ) = 0.0615 0.8538 = 36.1187 −27.0839 .
The eigenvalues of A1 are 1.1 and 0.9.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 17 / 23
VAR Companion Form
Consider the VAR(p):
xt =a0 +A1xt−1 +···+Apxt−p +et,
and add the auxiallary identities
xt−1 = xt−1, .
xt−p+1 = xt−p+1. Putting the system together yields
xt a0 A1 A2 ··· Apxt−1 et xt−1 0 In 0xt−2 0
. = . + .. . . +.. .. . ...
xt−p+1 0 In 0 xt−p 0
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10
18 / 23
VAR Companion Form
So any VAR(p) can be expressed as a VAR(1) with A1 A2 ··· Ap
In 0 A 1 = . . . . .
.. In 0
This is very useful for working with VARs. For example, to assess the stationarity of a VAR(p), we only need to check the eigenvalues of A1.
The companion form can also be used to analyze properties of a univariate AR(p).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 19 / 23
Determining the Lag Length p
Prior to estimation, we must determine the lag length p of the VAR. The general approach is the same as with univariate AR(p):
compute the multivariate generalisations of criteria such as the AIC and the BIC used in the univariate case;
compute a likelihood ratio (LR) which tests the number of lags against a shorter alternative under the null hypothesis.
LR Test Example: H0 : p = 4 against H1 : p = 8. Rejecting the null implies 4 lags is not enough to capture the dynamics of the model and including more lags will provide a better fit.
Aside from information criteria and hypothesis tests, it is also important to check for autocorrelation in the residuals.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 20 / 23
Example: The Returns of 5 Asian Stock Markets
Consider daily returns of the stock markets of: Thailand, Hong Kong, Singapore, Korea, Japan.
In a VAR model of these 5 stock markets we will estimate 5 equations (5 endogenous variables), one per country, with lagged values of all other stock markets appearing in each equation:
xt =a0 +A1xt−1 +···+At−pxt−p +et.
where
xt = (THAIRETt, HKRETt, SINGRETt, KOREARETt, JAPRETt)′.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 21 / 23
Example: The Returns of 5 Asian Stock Markets
Eric Eisenstat (School of Economics) ECON3350/7350 Week 10 22 / 23
VAR Lag Length Selection
VAR Lag Order Selection Criteria
Endogenous variables: HKRET JAPANRET KOREARET SINGRET THAIRET Sample: 1/01/1985 4/29/1999
Included observations: 3729
Lag LogL
0 51769.00
1 52053.11
2 52116.53
3 52191.60
4 52249.00
5 52274.33
6 52288.08
7 52308.92
8 52331.59
LR
NA 567.3073 126.4483 149.5024 114.1630 50.30679 27.27318 41.27481
FPE
6.03e-19 5.25e-19 5.14e-19 5.00e-19 4.92e-19 4.92e-19* 4.95e-19 4.96e-19
AIC
-27.76294 -27.90191 -27.92251 -27.94937 -27.96675 -27.96693* -27.96089 -27.95866 -27.95741
BIC
-27.75460 -27.85184* -27.83072 -27.81585 -27.79150 -27.74995 -27.70219 -27.65823 -27.61526
HQ
-27.75997 -27.88410 -27.88986 -27.90187 -27.90441* -27.88974 -27.86886 -27.85179 -27.83569
44.84274* 4.96e-19
* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level)
Eric Eisenstat (School of Economics) ECON3350/7350
Week 10 23 / 23