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{{MSC|60G10|60G15,60G25,60G35}}


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'''Vector Autoregressive Models'''


by

Helmut Lütkepohl 

Department of Economics, European University Institute, Via della
Piazzuola 43, I-50133 Firenze, ITALY, email:
helmut.luetkepohl@eui.eu

[1ex]
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Vector autoregressive (VAR) processes are popular in economics and
other sciences because they are flexible and simple models for
multivariate time series data. In econometrics they became standard
tools when \can{sims:80} questioned the way classical simultaneous
equations models were specified and identified and advocated VAR
models as alternatives. A textbook treatment of these models with
details on the issues mentioned in the following introductory
exposition is available in \can{lue:05}.

=== The model setup ===
The basic form of a VAR process is
$$
y_t=Dd_t+ A_1y_{t-1}+\cdots + A_py_{t-p}+u_t,
$$
where $y_t=(y_{1t},\dots,y_{Kt})'$ (the prime denotes the transpose)
is a vector of $K$ observed time series variables, $d_t$ is a vector
of deterministic terms such as a constant, a linear trend and/or
seasonal dummy variables, $D$ is the associated parameter matrix,
the $A_i$'s are $(K\times K)$ parameter matrices attached to the
lagged values of $y_t$, $p$ is the lag order or VAR order and $u_t$
is an error process which is assumed to be white noise with zero
mean, that is, $E(u_t)=0$, the covariance matrix,
$E(u_tu_t')=\Sigma_u$, is time invariant and the $u_t$'s are
serially uncorrelated or independent.

VAR models are useful tools for forecasting. If the $u_t$'s are
independent white noise, the minimum mean squared error (MSE)
$h$-step forecast of $y_{t+h}$ at time $t$ is the conditional
expectation given $y_s$, $s\le t$,
$$
y_{t+h|t} = E(y_{t+h} | y_t, y_{t-1}, \dots)= Dd_{t+h}+A_1
y_{t+h-1|t} + \cdots + A_p y_{t+h-p|t},
$$
where $y_{t+j|t} = y_{t+j}$ for $j \le 0$. Using this formula, the
forecasts can be computed recursively for $h=1,2,\dots$. The
forecasts are unbiased, that is, the forecast error $y_{t+h} -
y_{t+h|t}$ has mean zero and the forecast error covariance is equal
to the MSE matrix. The 1-step ahead forecast errors are the $u_t$'s.

VAR models can also be used for analyzing the relation between the
variables involved. For example, \can{gra:69} defined a concept of
causality which specifies that a variable $y_{1t}$ is causal for a
variable $y_{2t}$ if the information in $y_{1t}$ is helpful for
improving the forecasts of $y_{2t}$. If the two variables are
jointly generated by a VAR process, it turns out that $y_{1t}$ is
not Granger-causal for $y_{2t}$ if a simple set of zero
restrictions for the coefficients of the VAR process are
satisfied. Hence, Granger-causality is easy to check in VAR
processes.

Impulse responses offer another possibility for analyzing the
relation between the variables of a VAR process by tracing the
responses of the variables to impulses hitting the system. If the
VAR process is stable and stationary, it has a moving average
representation of the form
$$
y_t = D^*d_t+\sum_{j=0}^\infty \Phi_j u_{t-j},
$$
where the $\Phi_j$'s are $(K\times K)$ coefficient matrices which
can be computed from the VAR coefficient matrices $A_i$ with
$\Phi_0=I_K$, the $(K\times K)$ identity matrix. This
representation can be used for tracing the effect of a specific
forecast error through the system. For example, if
$u_t=(1,0,\dots,0)'$, the coefficients of the first columns of the
$\Phi_j$ matrices represent the marginal reactions of the $y_t$'s.
Unfortunately, these so-called ''forecast error impulse
responses'' are often not of interest for economists because they
may not reflect properly what actually happens in a system of
variables. Given that the components of $u_t$ are typically
instantaneously correlated, such shocks or impulses are not likely
to appear in isolation. Impulses or shocks of interest for
economists are usually instantaneously uncorrelated. They are
obtained from the forecast errors, the $u_t$'s, by some
transformation, for example, $\varepsilon_t=Bu_t$ may be a vector
of shocks of interest if the $(K\times K)$ matrix $B$ is such that
$\varepsilon_t\sim(0,\Sigma_\varepsilon)$ has a diagonal
covariance matrix $\Sigma_\varepsilon$. The corresponding moving
average representation in terms of the $\varepsilon_t$'s becomes
$$
y_t = D^*d_t+\sum_{j=0}^\infty \Theta_j \varepsilon_{t-j},
$$
where $\Theta_j=\Phi_jB^{-1}$.

There are many $B$ matrices with the property that $Bu_t$ is a
random vector with diagonal covariance matrix. Hence, there are
many shocks $\varepsilon_t$ of potential interest. Finding those
which are interesting from an economic point of view is the
subject of ''structural VAR analysis''.

=== Estimation and model specification ===
In practice the process which has generated the time series under
investigation is usually unknown. In that case, if VAR models are
regarded as suitable, the lag order has to be specified and the
parameters have to be estimated. For a given VAR order $p$,
estimation can be conveniently done by equationwise ordinary least
squares (OLS). For a sample of size $T$, $y_1,\dots, y_T$, and
assuming that in addition presample values $y_{-p+1},\dots,y_0$ are
also available, the OLS estimator of the parameters
$B=[D,A_1,\dots,A_p]$ can be written as
$$
\hat B=
\left(\sum_{t=1}^Ty_tZ_{t-1}'\right)\left(\sum_{t=1}^TZ_{t-1}Z_{t-1}'\right)^{-1},
$$
where $Z_{t-1}'=(d_t',y_{t-1}',\dots,y_{t-p}')$. Under standard
assumptions the estimator is consistent and asymptotically normally
distributed. In fact, if the residuals and, hence, the $y_t$'s are
normally distributed, that is, $u_t\sim$ i.i.d.$\n(0,\Sigma_u)$, the
OLS estimator is equal to the maximum likelihood (ML) estimator with
the usual asymptotic optimality properties. If the dimension $K$ of
the process is large, then the number of parameters is also large
and estimation precision may be low if a sample of typical size in
macroeconomic studies is available for estimation. In that case it
may be useful to exclude redundant lags of some of the variables
from some of the equations and fit so-called subset VAR models. In
general, if zero or other restrictions are imposed on the parameter
matrices, other estimation methods may be more efficient.

VAR order selection is usually done by sequential tests or model
selection criteria. ''Akaike's information criterion'' (AIC) is,
for instance, a popular model selection criterion (\can{aik:73}).
It has the form
$$
\mbox{AIC}(m) = \log \det(\hat \Sigma_m) + 2mK^2/T,
$$
where $\hat \Sigma_m=T^{-1}\sum_{t=1}^T\hat u_t\hat u_t'$ is the
residual covariance matrix of a VAR($m$) model estimated by OLS. The
criterion consists of the determinant of the residual covariance
matrix which tends to decline with increasing VAR order whereas the
penalty term $2mK^2/T$, which involves the number of parameters,
grows with $m$. The VAR order is chosen which optimally balances
both terms. In other words, models of orders $m=0,\dots,p_{\max}$
are estimated and the order $p$ is chosen such that it minimizes the
value of AIC.

Once a model is estimated it should be checked that it represents
the data features adequately. For this purpose a rich toolkit is
available. For example, descriptive tools such as plotting the
residuals and residual autocorrelations may help to detect model
deficiencies. In addition, more formal methods such as tests for
residual autocorrelation, conditional heteroskedasticity,
nonnormality and structural stability or tests for parameter
redundancy may be applied.

=== Extensions ===
If some of the time series variables to be modelled with a VAR
have stochastic trends, that is, they behave similarly to a random
walk, then another model setup may be more useful for analyzing
especially the trending properties of the variables. Stochastic
trends in some of the variables are generated by models with unit
roots in the VAR operator, that is, $\det (I_K-A_1z-\cdots
-A_pz^p)=0$ for $z=1$. Variables with such trends are
nonstationary and not stable. They are often called integrated.
They can be made stationary by differencing. Moreover, they are
called cointegrated if stationary linear combinations exist or, in
other words, if some variables are driven by the same stochastic
trend. Cointegration relations are often of particular interest in
economic studies. In that case, reparameterizing the standard VAR
model such that the cointegration relations appear directly may be
useful. The so-called ''vector error correction model'' (VECM)
of the form
$$
\Delta y_t = Dd_t+\Pi y_{t-1}+\Gamma_1 \Delta y_{t-1} + \cdots + \Gamma_{p-1} \Delta y_{t-p+1}
+u_t
$$
is a simple example of such a reparametrization, where $\Delta$
denotes the differencing operator defined such that $\Delta
y_t=y_t-y_{t-1}$, $\Pi= -(I_K-A_1-\cdots -A_p)$ and $\Gamma_i=
-(A_{i+1}+\cdots +A_p)$ for $i=1,\dots,p-1$. This parametrization
is obtained by subtracting $y_{t-1}$ from both sides of the
standard VAR representation and rearranging terms. Its advantage
is that $\Pi$ can be decomposed such that the cointegration
relations are directly present in the model. More precisely, if
all variables are stationary after differencing once, and there
are $K-r$ common trends, then the matrix $\Pi$ has rank $r$ and
can be decomposed as $\Pi=\alpha\beta'$, where $\alpha$ and
$\beta$ are $(K\times r)$ matrices of rank $r$ and $\beta$
contains the cointegration relations. A detailed statistical
analysis of this model is presented in \can{joh:951} (see also
Part II of \can{lue:05}).

There are also other extensions of the basic VAR model which are
often useful and have been discussed extensively in the associated
literature. For instance, in the standard model all observed
variables are treated as endogenous, that is, they are jointly
generated. This setup often leads to heavily parameterized models,
imprecise estimates and poor forecasts. Depending on the context,
it may be possible to classify some of the variables as exogenous
and consider partial models which condition on some of the
variables. The latter variables remain unmodelled.

One may also question the focus on finite order VAR models and allow
for an infinite order. This can be done by either augmenting a
finite order VAR by a finite order MA term or by accounting
explicitly for the fact that the finite order VAR approximates some
more general model. Details on these and other extensions are
provided, e.g., by \can{hade:88} and \can{lue:05}.


====References====
{|
|-
|valign="top"|{{Ref|Akaike}}||valign="top"| {1973}{aik:73} Akaike, H. (1973).  Information theory and an extension of the maximum likelihood principle, ''in'' B. Petrov & F. Csáki (eds), ''2nd International Symposium on Information Theory'', Académiai Kiadó, Budapest, pp. 267--281. 
|-
|valign="top"|{{Ref|Granger}}||valign="top"| {1969}{gra:69} Granger, C. W. J. (1969).  Investigating causal relations by econometric models and cross-spectral methods, ''Econometrica'' '''37''': 424--438. 
|-
|valign="top"|{{Ref|Hannan & Deistler}}||valign="top"| {1988}{hade:88} Hannan, E. J. & Deistler, M. (1988).  ''The Statistical Theory of Linear Systems'', Wiley, New York. 
|-
|valign="top"|{{Ref|Johansen}}||valign="top"| {1995}{joh:951} Johansen, S. (1995).  ''Likelihood-based Inference in Cointegrated '''Vector Autoregressive Models''''', Oxford University Press, Oxford. 
|-
|valign="top"|{{Ref|Lütkepohl}}||valign="top"| {2005}{lue:05} Lütkepohl, H. (2005).  ''New Introduction to Multiple Time Series Analysis'', Springer-Verlag, Berlin. 
|-
|valign="top"|{{Ref|Sims}}||valign="top"| {1980}{sims:80} Sims, C. A. (1980).  Macroeconomics and reality, ''Econometrica'' '''48''': 1--48.
|-
|}


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