The University of Queensland
(School of Economics) Applied Econometrics for Macro and Finance Week 10 1 / 24
ECON7350 Multivariate Processes – I
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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.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 2 / 24
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.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 3 / 24
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).
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 Regression Models (SUR), Vector Autoregressive Models (VAR), Vector ARMA Models (VARMA), Vector Error Correction Models (VEC), VAR with exogenous variables (VARX).
(School of Economics) Applied Econometrics for Macro and Finance Week 10 4 / 24
Vector Autoregressive Models (VARs)
VARs were popularised in economics by 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 generalisations of the univariate ARMAs.
VARs are extremely easy to work with—many extensions 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.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 5 / 24
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
(School of Economics) Applied Econometrics for Macro and Finance Week 10 6 / 24
= 0 , Var ε = 0 σ2 . zt ztz
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.
xt = B−1γ0 + B−1Γ1xt−1 + B−1εt, xt = a0 + A1xt−1 + et.
This is referred to as the reduced form VAR.
(School of Economics) Applied Econometrics for Macro and Finance
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,
(School of Economics) Applied Econometrics for Macro and Finance Week 10 8 / 24
for s = 1,2,….
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:
0.500−1 1.000 4.481−1 3.680
0 1 0.250−1 1.000 0.500 1 =
0 1 2.111−1 25.39 4.481 1 =
1.6327 −0.9796
1.6327 −0.9796
−0.9796 1.3878 ,
−0.9796 1.3878 .
To derive the SVAR from a reduced form VAR, we need additional information (return to this next week).
(School of Economics) Applied Econometrics for Macro and Finance Week 10 9 / 24
VAR in n Variables
VAR models have as many equations as endogenous variables, which is denoted 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
xt =a0 +Ajxt−j +et, E(et|xt−1,…)=0,
where Ω is a symmetric and positive definite matrix.
Var(et|xt−1,…)=Ω,
(School of Economics) Applied Econometrics for Macro and Finance
Features of VARs
An important feature of the VAR is that estimation is very easy: a VAR is just a system of Seemingly 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).
3 4 39 10 1 110 10 4 410
(School of Economics)
Applied Econometrics for Macro and Finance
Properties of VARs
All relevant properties of AR processes generalize to VARs.
To analyse the properties of VARs, define the polynomial matrix in 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.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 12 / 24
Recall that the AR(1) model
is stable 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), stability corresponds to
det (In − A1z) = 0
having all roots greater than one in absolute value.
xt = a0 + a1xt−1 + et
(School of Economics) Applied Econometrics for Macro and Finance Week 10 13 / 24
Let z∗ be a root of det A(z). Then, by definition
det (In − A1z∗) = 0,
which is equivalent to
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 stability 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 −→ ∞. (School of Economics) Applied Econometrics for Macro and Finance Week 10 14 / 24
Example of a Stable VAR
Consider the VAR(1):
Is this VAR stable?
1.0846 −0.7385 yt−1 eyt
= 0.2462 −0.0846 z
+ e . t−1 zt
(School of Economics)
Applied Econometrics for Macro and Finance
Example of a Stable VAR
Consider the VAR(1):
E(xt+10 | xt, xt−1, . . . ) = The 50-period ahead forecast is:
E(xt+100 | xt, xt−1, . . . ) =
1.0846 −0.738510 0.4291 −0.3219 0.2462 −0.0846 = 0.1073 −0.0805 .
1.0846 −0.738550 0.0063 −0.0048 0.2462 −0.0846 = 0.0016 −0.0012 .
1.0846 −0.7385 yt−1 eyt
= 0.2462 −0.0846 z The 10-period ahead forecast is:
+ e . t−1 zt
Is this VAR stable?
The eigenvalues of A1 are 0.9 and 0.1.
(School of Economics) Applied Econometrics for Macro and Finance Week 10
Example of a Stable VAR
Consider the VAR(1):
Is this VAR stable?
yt 1.1462 −0.1846 yt−1 eyt z = 0.0615 0.8538 z + e .
(School of Economics)
Applied Econometrics for Macro and Finance
Example of a Stable VAR
Consider the VAR(1):
Is this VAR stable?
The 10-period ahead forecast is:
E(xt+10 | xt, xt−1, . . . ) = The 50-period ahead forecast is:
1.1462 −0.184610 3.1118 −2.0724 0.0615 0.8538 = 0.6908 −0.1694
yt 1.1462 −0.1846 yt−1 eyt z = 0.0615 0.8538 z + e .
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.
(School of Economics) Applied Econometrics for Macro and Finance Week 10
VAR Companion Form
Consider the VAR(p):
xt =a0 +A1xt−1 +···+Apxt−p +et, and add the auxiliary 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
(School of Economics) Applied Econometrics for Macro and Finance
VAR Companion Form
So any VAR(p) can be expressed as a VAR(1) with
A1 A2 ··· Ap
In 0 A 1 = . . . . .
This is very useful for working with VARs. For example, to assess the stability of a VAR(p), we only need to check the eigenvalues of A1.
The companion form can also be used to analyse properties of a univariate AR(p).
(School of Economics) Applied Econometrics for Macro and Finance Week 10 18 / 24
Adequate Set of VAR(p) Models
VARs are specified by the lag length p along with parameter restrictions.
The general approach to constructing an adequate set of VAR(p) models is the same as with univariate AR(p) models; we use the following basic tools:
Compute the multivariate generalisations of criteria such as the AIC and the BIC used in the univariate case;
Look for evidence of significant autocorrelation in the estimated residuals.
The estimated VAR residuals will be an n-variate system: we can use the Ljung-Box tests on individual residuals series, but this is not optimal.
A more common way to test for autocorrelations in estimate VAR residuals is the Lagrange-Multiplier test.
It is a multi-step procedure that involves augmenting the original VAR with lags of estimated residuals.
The null is H0 : first K cross-auto-correlations are zero, for a pre-specified K.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 19 / 24
Example: The Returns of 5 Asian Stock Markets
Consider daily returns of the stock markets of: Thailand, , 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.
xt = (THAIRETt, HKRETt, SINGRETt, KOREARETt, JAPRETt)′.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 20 / 24
Example: The Returns of 5 Asian Stock Markets
(School of Economics) Applied Econometrics for Macro and Finance Week 10 21 / 24
VARs and Information Criteria
VAR Lag Order Selection Criteria
Endogenous variables: HKRET JAPANRET KOREARET SINGRET THAIRET Sample: 1/01/1985 4/29/1999
Included observations: 3729
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
NA 567.3073 126.4483 149.5024 114.1630 50.30679 27.27318 41.27481
6.03e-19 5.25e-19 5.14e-19 5.00e-19 4.92e-19 4.92e-19* 4.95e-19 4.96e-19
-27.76294 -27.90191 -27.92251 -27.94937 -27.96675 -27.96693* -27.96089 -27.95866 -27.95741
-27.75460 -27.85184* -27.83072 -27.81585 -27.79150 -27.74995 -27.70219 -27.65823 -27.61526
-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)
(School of Economics) Applied Econometrics for Macro and Finance
Forecasting with VARs
VARs are commonly used for forecasting, and they have been used to forecast macroeconomic variables with huge success.
The procedure is very similar to forecasting with univariate ARs. Letting xt+h ≡ E(xt+h | It), forecasts are computed recursively:
xt+h = a0 + A1xt+h−1 + · · · + Apxt+h−p.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 23 / 24
Quantifying Estimation Uncertainty
To account for estimation uncertainty, a standard approach is to construct confidence intervals through bootstrapping and simulation. Basic idea:
1 Sample residuals from a distribution (e.g. Normal), then construct new simulated data using estimated coefficients.
2 Re-estimate the VAR using the newly simulated sample and re-compute forecasts.
3 Repeat Steps 1-2 above for some number of iterations (e.g., 1000).
4 Choose a significance level α, then discard the largest (100 × α/2)% and smallest (100 × α/2)% of all forecasts generated at horizon h.
5 The largest and smallest remaining forecasts for a given horizon h represent the (100 × (1 − α))% predictive interval for that horizon.
(School of Economics) Applied Econometrics for Macro and Finance Week 10 24 / 24
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