View source for Vector autoregression

{{Short description|Statistical model to calculate the value of multiple quantities as they change over time}}
{{other uses of|Var}}
{{more footnotes|date=February 2012}}
'''Vector autoregression''' ('''VAR''') is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of [[stochastic process]] model. VAR models generalize the single-variable (univariate) [[autoregressive model]] by allowing for multivariate [[time series]]. VAR models are often used in [[economics]] and the [[natural science]]s.

Like the autoregressive model, each variable has an equation modelling its evolution over time. This equation includes the variable's [[Lag operator|lagged]] (past) values, the lagged values of the other variables in the model, and an [[errors and residuals in statistics|error term]]. VAR models do not require as much knowledge about the forces influencing a variable as do [[structural equation modeling|structural models]] with [[simultaneous equations model|simultaneous equations]]. The only prior knowledge required is a list of variables which can be hypothesized to affect each other over time.

==Specification==
{{no footnotes|section|date=February 2012}}

===Definition===
A VAR model describes the evolution of a set of ''k'' variables, called ''[[Endogeneity (econometrics)|endogenous]] variables'', over time. Each period of time is numbered, ''t'' = 1, ..., ''T''. The variables are collected in a [[vector space|vector]], ''y<sub>t</sub>'', which is of length ''k.'' (Equivalently, this vector might be described as a (''k''&nbsp;×&nbsp;1)-[[Matrix (mathematics)|matrix.]]) The vector is modelled as a linear function of its previous value. The vector's components are referred to as ''y''<sub>''i'',''t''</sub>, meaning the observation at time ''t'' of the ''i'' th variable. For example, if the first variable in the model measures the price of wheat over time, then ''y''<sub>1,1998</sub>  would indicate the price of wheat in the year 1998.

VAR models are characterized by their ''order'', which refers to the number of earlier time periods the model will use. Continuing the above example, a 5th-order VAR would model each year's wheat price as a linear combination of the last five years of wheat prices. A ''lag'' is the value of a variable in a previous time period. So in general a ''p''th-order VAR refers to a VAR model which includes lags for the last ''p'' time periods. A ''p''th-order VAR is denoted "VAR(''p'')" and sometimes called "a VAR with ''p'' lags". A ''p''th-order VAR model is written as

:<math>y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + e_t, \, </math>

The variables of the form ''y''<sub>''t''−i</sub> indicate that variable's value ''i''  time periods earlier and are called the "i''th'' lag" of ''y''<sub>t</sub>. The variable ''c'' is a ''k''-vector of constants serving as the [[Y-intercept|intercept]] of the model. ''A<sub>i</sub>'' is a [[time-invariant]] (''k''&nbsp;×&nbsp;''k'')-matrix and ''e''<sub>''t''</sub> is a ''k''-vector of [[errors and residuals in statistics|error]] terms. The error terms must satisfy three conditions:

#<math>\mathrm{E}(e_t) = 0\,</math>. Every error term has a [[Expected value|mean]] of zero.
#<math>\mathrm{E}(e_t e_t') = \Omega\,</math>. The contemporaneous [[covariance matrix]] of error terms is a ''k''&nbsp;×&nbsp;''k'' [[positive-definite matrix|positive-semidefinite matrix]] denoted Ω.
#<math>\mathrm{E}(e_t e_{t-k}') = 0\,</math> for any non-zero ''k''. There is no [[correlation]] across time. In particular, there is no [[serial correlation]] in individual error terms.<ref>For multivariate tests for autocorrelation in the VAR models, see {{cite journal |last=Hatemi-J |first=A. |year=2004 |title=Multivariate tests for autocorrelation in the stable and unstable VAR models |journal=Economic Modelling |volume=21 |issue=4 |pages=661–683 |url=https://ideas.repec.org/a/eee/ecmode/v21y2004i4p661-683.html |doi=10.1016/j.econmod.2003.09.005}}</ref>

The process of choosing the maximum lag ''p'' in the VAR model requires special attention because [[inference]] is dependent on correctness of the selected lag order.<ref>{{cite journal |last1=Hacker |first1=R. S. |last2=Hatemi-J |first2=A. |year=2008 |title=Optimal lag-length choice in stable and unstable VAR models under situations of homoscedasticity and ARCH |journal=[[Journal of Applied Statistics]] |volume=35 |issue=6 |pages=601–615 |url=https://ideas.repec.org/a/taf/japsta/v35y2008i6p601-615.html |doi=10.1080/02664760801920473|bibcode=2008JApSt..35..601S }}</ref><ref>{{cite journal |last1=Hatemi-J |first1=A. |first2=R. S. |last2=Hacker |year=2009 |title=Can the LR test be helpful in choosing the optimal lag order in the VAR model when information criteria suggest different lag orders? |journal=[[Applied Economics (journal)|Applied Economics]] |volume=41 |issue=9 |pages=1489–1500 |doi=10.1080/00036840601019273 |url=https://ideas.repec.org/a/taf/applec/v41y2009i9p1121-1125.html }}</ref>

===Order of integration of the variables===
Note that all variables have to be of the same [[order of integration]]. The following cases are distinct:

*All the variables are I(0) (stationary): this is in the standard case, i.e. a VAR in level
*All the variables are I(''d'') (non-stationary) with ''d''&nbsp;>&nbsp;0:{{Citation needed|date=April 2010}}
**The variables are [[Cointegration|cointegrated]]: the error correction term has to be included in the VAR. The model becomes a Vector [[error correction model]] (VECM) which can be seen as a restricted VAR.
**The variables are not [[Cointegration|cointegrated]]: first, the variables have to be differenced d times and one has a VAR in difference.

===Concise matrix notation===

One can stack the vectors in order to write a VAR(''p'') as a [[stochastic]] [[matrix difference equation]], with a concise matrix notation:

:<math> Y=BZ +U \, </math>

===Example===
A VAR(1) in two variables can be written in matrix form (more compact notation) as

:<math>\begin{bmatrix}y_{1,t} \\ y_{2,t}\end{bmatrix} = \begin{bmatrix}c_{1} \\ c_{2}\end{bmatrix} + \begin{bmatrix}a_{1,1}&a_{1,2} \\ a_{2,1}&a_{2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\end{bmatrix} + \begin{bmatrix}e_{1,t} \\ e_{2,t}\end{bmatrix},</math>

(in which only a single ''A'' matrix appears because this example has a maximum lag ''p'' equal to 1), or, equivalently, as the following system of two equations

:<math>y_{1,t} = c_{1} + a_{1,1}y_{1,t-1} + a_{1,2}y_{2,t-1} + e_{1,t}\,</math>
:<math>y_{2,t} = c_{2} + a_{2,1}y_{1,t-1} + a_{2,2}y_{2,t-1} + e_{2,t}.\,</math>

Each variable in the model has one equation. The current (time ''t'') observation of each variable depends on its own lagged values as well as on the lagged values of each other variable in the VAR.

===Writing VAR(''p'') as VAR(1)===
A VAR with ''p'' lags can always be equivalently rewritten as a VAR with only one lag by appropriately redefining the dependent variable. The transformation amounts to stacking the lags of the VAR(''p'') variable in the new VAR(1) dependent variable and appending identities to complete the precise number of equations.

For example, the VAR(2) model

:<math>y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + e_t</math>

can be recast as the VAR(1) model

::<math>\begin{bmatrix}y_{t} \\ y_{t-1}\end{bmatrix} = \begin{bmatrix}c \\ 0\end{bmatrix} + \begin{bmatrix}A_{1}&A_{2} \\ I&0\end{bmatrix}\begin{bmatrix}y_{t-1} \\ y_{t-2}\end{bmatrix} + \begin{bmatrix}e_{t} \\ 0\end{bmatrix},</math>

where ''I'' is the [[identity matrix]].

The equivalent VAR(1) form is more convenient for analytical derivations and allows more compact statements.

==Structural vs. reduced form==

===Structural VAR===
A '''''structural VAR with p lags''''' (sometimes abbreviated '''SVAR''') is

:<math>B_0 y_t = c_0 + B_1 y_{t-1} + B_2 y_{t-2} + \cdots + B_p y_{t-p} + \epsilon_t,</math>

where ''c''<sub>0</sub> is a ''k''&nbsp;×&nbsp;1 vector of constants, ''B<sub>i</sub>'' is a ''k''&nbsp;×&nbsp;''k'' matrix (for every ''i'' = 0, ..., ''p'') and ''ε''<sub>''t''</sub> is a ''k''&nbsp;×&nbsp;1 vector of [[error]] terms. The [[main diagonal]] terms of the ''B''<sub>0</sub> matrix (the coefficients on the ''i''<sup>th</sup> variable in the ''i''<sup>th</sup> equation) are scaled to 1.

The error terms ε''<sub>t</sub>'' ('''''structural shocks''''') satisfy the conditions (1) - (3) in the definition above, with the particularity that all the elements in the off diagonal of the covariance matrix <math>\mathrm{E}(\epsilon_t\epsilon_t') = \Sigma</math> are zero. That is, the structural shocks are uncorrelated.

For example, a two variable structural VAR(1) is:

:<math>\begin{bmatrix}1&B_{0;1,2} \\ B_{0;2,1}&1\end{bmatrix}\begin{bmatrix}y_{1,t} \\ y_{2,t}\end{bmatrix} = \begin{bmatrix}c_{0;1} \\ c_{0;2}\end{bmatrix} + \begin{bmatrix}B_{1;1,1}&B_{1;1,2} \\ B_{1;2,1}&B_{1;2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\end{bmatrix} + \begin{bmatrix}\epsilon_{1,t} \\ \epsilon_{2,t}\end{bmatrix},</math>

where

:<math>\Sigma = \mathrm{E}(\epsilon_t \epsilon_t') = \begin{bmatrix}\sigma_{1}^2&0 \\ 0&\sigma_{2}^2\end{bmatrix};</math>

that is, the [[variance]]s of the structural shocks are denoted <math>\mathrm{var}(\epsilon_i) = \sigma_i^2</math> (''i'' = 1, 2) and the [[covariance]] is <math>\mathrm{cov}(\epsilon_1,\epsilon_2) = 0</math>.

Writing the first equation explicitly and passing ''y<sub>2,t</sub>'' to the [[right hand side]] one obtains

:<math>y_{1,t} = c_{0;1} - B_{0;1,2}y_{2,t} + B_{1;1,1}y_{1,t-1} + B_{1;1,2}y_{2,t-1} + \epsilon_{1,t}\,</math>

Note that ''y''<sub>2,''t''</sub> can have a contemporaneous effect on ''y<sub>1,t</sub>'' if ''B''<sub>0;1,2</sub> is not zero. This is different from the case when ''B''<sub>0</sub> is the [[identity matrix]] (all off-diagonal elements are zero — the case in the initial definition), when ''y''<sub>2,''t''</sub> can impact directly ''y''<sub>1,''t''+1</sub> and subsequent future values, but not ''y''<sub>1,''t''</sub>.

Because of the [[parameter identification problem]], [[ordinary least squares]] estimation of the structural VAR would yield [[Estimator#Consistency|inconsistent]] parameter estimates. This problem can be overcome by rewriting the VAR in reduced form.

From an economic point of view, if the joint dynamics of a set of variables can be represented by a VAR model, then the structural form is a depiction of the underlying, "structural", economic relationships. Two features of the structural form make it the preferred candidate to represent the underlying relations:

:1. ''Error terms are not correlated''. The structural, economic shocks which drive the dynamics of the economic variables are assumed to be [[Statistical independence|independent]], which implies zero correlation between error terms as a desired property. This is helpful for separating out the effects of economically unrelated influences in the VAR. For instance, there is no reason why an oil price shock (as an example of a [[supply shock]]) should be related to a shift in consumers' preferences towards a style of clothing (as an example of a [[demand shock]]); therefore one would expect these factors to be statistically independent.

:2. ''Variables can have a [[contemporaneous impact]] on other variables''. This is a desirable feature especially when using low frequency data. For example, an [[indirect tax]] rate increase would not affect [[tax revenues]] the day the decision is announced, but one could find an effect in that quarter's data.

===Reduced-form VAR===
By premultiplying the structural VAR with the inverse of ''B''<sub>0</sub>

: <math>y_t = B_0^{-1}c_0 + B_0^{-1} B_1 y_{t-1} + B_0^{-1} B_2 y_{t-2} + \cdots + B_0^{-1} B_p y_{t-p} + B_0^{-1}\epsilon_t,</math>

and denoting

: <math> B_{0}^{-1} c_0 = c,\quad B_{0}^{-1}B_i = A_{i}\text{ for }i = 1, \dots, p\text{ and }B_{0}^{-1}\epsilon_t = e_t</math>

one obtains the '''''p''th order reduced VAR'''

:<math>y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + e_t</math>

Note that in the reduced form all right hand side variables are predetermined at time ''t''. As there are no time ''t'' endogenous variables on the right hand side, no variable has a ''direct'' contemporaneous effect on other variables in the model.

However, the error terms in the reduced VAR are composites of the structural shocks ''e''<sub>''t''</sub> = ''B''<sub>0</sub><sup>−1</sup>''ε''<sub>''t''</sub>. Thus, the occurrence of one structural shock ''ε<sub>i,t</sub>'' can potentially lead to the occurrence of shocks in all error terms ''e<sub>j,t</sub>'', thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR

:<math>\Omega = \mathrm{E}(e_t e_t') = \mathrm{E} (B_0^{-1} \epsilon_t \epsilon_t' (B_0^{-1})') = B_0^{-1}\Sigma(B_0^{-1})'\,</math>

can have non-zero off-diagonal elements, thus allowing non-zero correlation between error terms.

==Estimation==

===Estimation of the regression parameters===
Starting from the concise matrix notation:

:<math> Y=BZ +U \, </math>

*The [[Multivariate regression|multivariate least squares]] (MLS) approach for estimating B yields:

:<math> \hat B= YZ'(ZZ')^{-1}. </math>

This can be written alternatively as:

:<math> \operatorname{Vec}(\hat B) = ((ZZ')^{-1} Z \otimes I_{k})\ \operatorname{Vec}(Y), </math>

where <math> \otimes </math> denotes the [[Kronecker product]] and Vec the [[Vectorization (mathematics)|vectorization]] of the indicated matrix.

This estimator is [[Estimator#Consistency|consistent]] and [[Estimator#Efficiency|asymptotically efficient]]. It is furthermore equal to the conditional [[Maximum likelihood|maximum likelihood estimator]].<ref>{{cite book |author-link=James D. Hamilton |last=Hamilton |first=James D. |year=1994 |title=Time Series Analysis |publisher=Princeton University Press |page=293 }}</ref>

* As the explanatory variables are the same in each equation, the multivariate least squares estimator is equivalent to the [[ordinary least squares]] estimator applied to each equation separately.<ref>{{cite journal | last1 = Zellner | first1 = Arnold | author-link = Arnold Zellner | year = 1962 | title = An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias | journal = [[Journal of the American Statistical Association]] | volume = 57 | issue = 298| pages = 348–368 | doi=10.1080/01621459.1962.10480664}}</ref>

===Estimation of the covariance matrix of the errors===

As in the standard case, the [[maximum likelihood estimator]] (MLE) of the covariance matrix differs from the ordinary least squares (OLS) estimator.

MLE estimator:{{citation needed|date=February 2012}} <math> \hat \Sigma = \frac{1}{T} \sum_{t=1}^T \hat \epsilon_t\hat \epsilon_t'</math>

OLS estimator:{{citation needed|date=February 2012}} <math> \hat \Sigma = \frac{1}{T-kp-1} \sum_{t=1}^T \hat \epsilon_t\hat \epsilon_t'</math> for a model with a constant, ''k'' variables and ''p'' lags.

In a matrix notation, this gives:

: <math> \hat \Sigma = \frac{1}{T-kp-1} (Y-\hat{B}Z)(Y-\hat{B}Z)'.</math>

===Estimation of the estimator's covariance matrix===

The covariance matrix of the parameters can be estimated as{{citation needed|date=February 2012}}

: <math> \widehat  \mbox{Cov} (\mbox{Vec}(\hat B)) =({ZZ'})^{-1} \otimes\hat \Sigma.\, </math>

===Degrees of freedom===
Vector autoregression models often involve the estimation of many parameters. For example, with seven variables and four lags, each matrix of coefficients for a given lag length is 7 by 7, and the vector of constants has 7 elements, so a total of 49×4 + 7 = 203 parameters are estimated, substantially lowering the [[degrees of freedom (statistics)|degrees of freedom]] of the regression (the number of data points minus the number of parameters to be estimated). This can hurt the accuracy of the parameter estimates and hence of the forecasts given by the model.

==Interpretation of estimated model==

===Impulse response===
Consider the first-order case (i.e., with only one lag), with equation of evolution
:<math>y_t=Ay_{t-1}+e_t,</math>

for evolving (state)  vector <math>y</math> and vector <math>e</math> of shocks. To find, say, the effect of the ''j''-th element of the vector of shocks upon the ''i''-th element of the state vector 2 periods later, which is a particular impulse response, first write the above equation of evolution one period lagged:

:<math>y_{t-1}=Ay_{t-2}+e_{t-1}.</math>

Use this in the original equation of evolution to obtain

:<math>y_t=A^2y_{t-2}+Ae_{t-1}+e_t;</math>

then repeat using the twice lagged equation of evolution, to obtain

:<math>y_t=A^3y_{t-3}+A^2e_{t-2}+Ae_{t-1}+e_t.</math>

From this, the effect of the ''j''-th component of <math>e_{t-2}</math> upon the ''i''-th component of <math>y_t</math> is the ''i, j'' element of the matrix <math>A^2.</math>

It can be seen from this [[mathematical induction|induction]] process that any shock will have an effect on the elements of ''y'' infinitely far forward in time, although the effect will become smaller and smaller over time assuming that the AR process is stable — that is, that all the [[eigenvalue#Eigenvalues and eigenvectors of matrices|eigenvalue]]s of the matrix ''A'' are less than 1 in [[absolute value]].

==Forecasting using an estimated VAR model==
{{Main article|Autoregressive model#n-step-ahead forecasting|Autoregressive model#Evaluating the quality of forecasts}}

An estimated VAR model can be used for [[forecasting]], and the quality of the forecasts can be judged, in ways that are completely analogous to the methods used in univariate autoregressive modelling.

==Applications==
[[Christopher A. Sims|Christopher Sims]] has advocated VAR models, criticizing the claims and performance of earlier modeling in [[macroeconomic]] [[econometrics]].<ref name=Sims/> He recommended VAR models, which had previously appeared in time series [[statistics]] and in [[system identification]], a statistical specialty in [[control theory]]. Sims advocated VAR models as providing a theory-free method to estimate economic relationships, thus being an alternative to the "incredible identification restrictions" in structural models.<ref name=Sims>{{cite journal|author-link=Christopher A. Sims |last=Sims |first=Christopher |year=1980 |title=Macroeconomics and Reality |journal=[[Econometrica]] |volume=48 |issue=1 |pages=1–48 |jstor=1912017 |doi=10.2307/1912017|citeseerx=10.1.1.163.5425 }}</ref> VAR models are also increasingly used in health research for automatic analyses of diary data<ref name= "Kr2016">{{cite journal |author= van der Krieke | display-authors=etal | year = 2016 | title = Temporal Dynamics of Health and Well-Being: A Crowdsourcing Approach to Momentary Assessments and Automated Generation of Personalized Feedback (2016) | journal = Psychosomatic Medicine | volume=79 | issue=2 | doi= 10.1097/PSY.0000000000000378 | pmid=27551988 | pages=213–223| url=https://pure.rug.nl/ws/files/40193705/00006842_201702000_00011.pdf }}</ref> or sensor data. Sio Iong Ao and R. E. Caraka found that the artificial neural network can improve its performance with the addition of the hybrid vector autoregression component. <ref name= "Ao2003">{{cite journal |author= Sio Iong Ao | display-authors= | year = 2003 | title = Analysis of the Interaction of Asian Pacific Indices and Forecasting Opening Prices by Hybrid VAR and Neural Network Procedures (2003) | journal = International Conf. On Computational Intelligence for Modelling, Control and Automation 2003}}</ref> <ref name= "Ca2021">{{cite journal |author= Caraka, R.E. | display-authors=etal | year = 2021 | title = Hybrid vector autoregression feedforward neural network with genetic algorithm model for forecasting space-time pollution data (2021) | journal = Indonesian Journal of Science and Technology | pages=243–266| doi=10.17509/ijost.v6i1.32732 | doi-access= free }}</ref>

==Software==
*[[R (programming language)|R]]: The package ''[https://cran.r-project.org/web/packages/vars/vars.pdf vars]'' includes functions for VAR models.<ref>{{Cite web|url=https://cran.r-project.org/web/packages/vars/vignettes/vars.pdf|title=Bernhard Pfaff VAR, SVAR and SVEC Models: Implementation Within R Package vars}}</ref><ref>{{Cite book|last1=Hyndman|first1=Rob J|url=https://otexts.com/fpp2/VAR.html|title=Forecasting: Principles and Practice|last2=Athanasopoulos|first2=George|publisher=OTexts|year=2018|isbn=978-0-9875071-1-2|pages=333–335|chapter=11.2: Vector Autoregressions}}</ref> Other R packages are listed in the CRAN Task View: Time Series Analysis.
*[[Python (programming language)|Python]]: The ''statsmodels'' package's tsa (time series analysis) module supports VARs. ''PyFlux'' has support for VARs and Bayesian VARs.
*[[SAS language|SAS]]: VARMAX
*[[Stata]]: "var"
*[[EViews]]: "VAR"
*[[Gretl]]: "var"
*[[Matlab]]: "varm"
*[[Regression analysis of time series]]: "SYSTEM"
*LDT

==See also==
*[[Bayesian vector autoregression]]
*[[Convergent cross mapping]]
*[[Granger causality]]
*[[Variance decomposition]]

==Notes==
{{Reflist}}

==Further reading==
* {{cite book |last1=Asteriou |first1=Dimitrios |last2=Hall |first2=Stephen G. |chapter=Vector Autoregressive (VAR) Models and Causality Tests |title=Applied Econometrics |location=London |publisher=Palgrave MacMillan |year=2011 |edition=Second |pages=319–333 }}
* {{cite book |first=Walter |last=Enders |title=Applied Econometric Time Series |edition=Third |location=New York |publisher=John Wiley & Sons |year=2010 |isbn=978-0-470-50539-7 |pages=272–355 }}
* {{cite book |first=Carlo A. |last=Favero |title=Applied Macroeconometrics |location=New York |publisher=Oxford University Press |year=2001 |isbn=0-19-829685-1 |pages=162–213 }}
* {{cite book |first=Helmut |last=Lütkepohl |author-link = Helmut Lütkepohl | title=New Introduction to Multiple Time Series Analysis |publisher=Springer |location=Berlin |year=2005 |isbn=3-540-40172-5 }}
* {{cite journal |first=Duo |last=Qin |title=Rise of VAR Modelling Approach |journal=[[Journal of Economic Surveys]] |volume=25 |issue=1 |year=2011 |pages=156–174 |doi=10.1111/j.1467-6419.2010.00637.x }}

{{Statistics|analysis|state=expanded}}
{{Economics|state=collapsed}}

[[Category:Time series models]]
[[Category:Multivariate time series]]