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| {{more footnotes|date=February 2012}}
| | Hi there. Allow me start by introducing the author, her name is Sophia Boon but she by no means really liked that title. He is an info officer. Alaska is exactly where I've always been living. What me and my family adore is doing ballet but I've been using on new things recently.<br><br>Also visit my web blog :: real psychic readings ([http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ www.edmposts.com]) |
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| '''Vector autoregression''' ('''VAR''') is an [[econometric model]] used to capture the linear interdependencies among multiple [[time series]]. VAR models generalize the univariate ''[[autoregression]]'' [[AR model|(AR) models]] by allowing for more than one evolving variable. All variables in a VAR are treated symmetrically in a structural sense (although the estimated quantitative response coefficients will not in general be the same); each variable has an equation explaining its evolution based on its own [[Lag operator|lags]] and the lags of the other model variables. VAR modeling does 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 intertemporally.
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| ==Specification==
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| {{no footnotes|section|date=February 2012}}
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| ===Definition===
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| A VAR model describes the evolution of a set of ''k'' variables (called ''endogenous variables'') over the same [[sample (statistics)|sample]] period (''t'' = 1, ..., ''T'') as a [[linear]] function of only their past values. The variables are collected in a ''k'' × 1 [[vector space|vector]] ''y<sub>t</sub>'', which has as the ''i''<sup> th</sup> element, ''y''<sub>''i'',''t''</sub>, the time ''t'' observation of the ''i''<sup> th</sup> variable. For example, if the ''i''<sup> th</sup> variable is [[GDP]], then ''y''<sub>''i'',''t''</sub> is the value of GDP at time ''t''.
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| A ''p-th order VAR'', denoted '''VAR(''p'')''', is
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| :<math>y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + e_t, \, </math>
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| where the ''l''-periods back observation ''y''<sub>''t''−l</sub> is called the ''l''-th '''''lag''''' of ''y'', ''c'' is a ''k'' × 1 vector of constants ([[Y-intercept|intercepts]]), ''A<sub>i</sub>'' is a time-invariant ''k'' × ''k'' [[Matrix (mathematics)|matrix]] and ''e''<sub>''t''</sub> is a ''k'' × 1 vector of [[errors and residuals in statistics|error]] terms satisfying | |
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| #<math>\mathrm{E}(e_t) = 0\,</math> — every error term has [[Expected value|mean]] zero;
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| #<math>\mathrm{E}(e_t e_t') = \Omega\,</math> — the contemporaneous [[covariance matrix]] of error terms is Ω (a ''k'' × ''k'' [[positive-definite matrix|positive-semidefinite matrix]]);
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| #<math>\mathrm{E}(e_t e_{t-k}') = 0\,</math> for any non-zero ''k'' — there is no [[correlation]] across time; in particular, no [[serial correlation]] in individual error terms. See Hatemi-J (2004) for multivariate tests for autocorrelation in the VAR models.
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| A ''p''th-order VAR is also called a '''VAR with ''p'' lags'''. 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.{{sfn|Hacker|Hatemi-J|2008}}{{sfn|Hatemi-J|Hacker|2009}}
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| ===Order of integration of the variables===
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| Note that all variables have to be of the same [[order of integration]]. The following cases are distinct:
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| *All the variables are I(0) (stationary): one is in the standard case, i.e. a VAR in level
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| *All the variables are I(''d'') (non-stationary) with ''d'' > 0:{{Citation needed|date=April 2010}}
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| **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.
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| **The variables are not [[Cointegration|cointegrated]]: the variables have first to be differenced d times and one has a VAR in difference.
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| ===Concise matrix notation===
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| One can stack the vectors in order to write a VAR(''p'') with a concise matrix notation:
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| :<math> Y=BZ +U \, </math>
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| Details of the matrices are in a [[General matrix notation of a VAR(p)|separate page]].
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| ===Example===
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| For a general example of a VAR(''p'') with ''k'' variables, see [[General matrix notation of a VAR(p)]].
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| A VAR(1) in two variables can be written in matrix form (more compact notation) as
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| :<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>
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| (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
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| :<math>y_{1,t} = c_{1} + A_{1,1}y_{1,t-1} + A_{1,2}y_{2,t-1} + e_{1,t}\,</math>
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| :<math>y_{2,t} = c_{2} + A_{2,1}y_{1,t-1} + A_{2,2}y_{2,t-1} + e_{2,t}.\,</math>
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| 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.
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| ===Writing VAR(''p'') as VAR(1)===
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| 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 number of equations.
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| For example, the VAR(2) model
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| :<math>y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + e_t</math>
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| can be recast as the VAR(1) model
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| ::<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>
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| where ''I'' is the [[identity matrix]].
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| The equivalent VAR(1) form is more convenient for analytical derivations and allows more compact statements.
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| ==Structural vs. reduced form==
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| ===Structural VAR===
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| A '''''structural VAR with p lags''''' (sometimes abbreviated '''SVAR''') is
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| :<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>
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| where ''c''<sub>0</sub> is a ''k'' × 1 vector of constants, ''B<sub>i</sub>'' is a ''k'' × ''k'' matrix (for every ''i'' = 0, ..., ''p'') and ''ε''<sub>''t''</sub> is a ''k'' × 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.
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| 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 off the main diagonal of the covariance matrix <math>\mathrm{E}(\epsilon_t\epsilon_t') = \Sigma</math> are zero. That is, the structural shocks are uncorrelated.
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| For example, a two variable structural VAR(1) is:
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| :<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>
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| where
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| :<math>\Sigma = \mathrm{E}(\epsilon_t \epsilon_t') = \begin{bmatrix}\sigma_{1}^2&0 \\ 0&\sigma_{2}^2\end{bmatrix};</math>
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| 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>.
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| Writing the first equation explicitly and passing ''y<sub>2,t</sub>'' to the [[right hand side]] one obtains
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| :<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>
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| 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>.
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| 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.
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| 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:
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| :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.
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| :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.
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| ===Reduced-form VAR===
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| By premultiplying the structural VAR with the inverse of ''B''<sub>0</sub>
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| : <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>
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| and denoting
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| : <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> | |
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| one obtains the '''''p''th order reduced VAR'''
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| :<math>y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + e_t</math> | |
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| 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.
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| 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
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| :<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>
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| can have non-zero off-diagonal elements, thus allowing non-zero correlation between error terms.
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| ==Estimation==
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| ===Estimation of the regression parameters===
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| Starting from the concise matrix notation (for details see [[General matrix notation of a VAR(p)|this annex]]):
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| :<math> Y=BZ +U \, </math> | |
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| *The multivariate least squares (MLS) for B yields:
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| :<math> \hat B= YZ^{'}(ZZ^{'})^{-1} </math>
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| It can be written alternatively as:
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| :<math> \operatorname{Vec}(\hat B) = ((ZZ^{'})^{-1} Z \otimes I_{k})\ \operatorname{Vec}(Y) </math>
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| Where <math> \otimes </math> denotes the [[Kronecker product]] and Vec the [[Vectorization (mathematics)|vectorization]] of the matrix ''Y''.
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| This estimator is [[Estimator#Consistency|consistent]] and [[Estimator#Efficiency|asymptotically efficient]]. It is furthermore equal to the conditional [[Maximum likelihood|maximum likelihood estimator]].<ref>[[James D. Hamilton|Hamilton, James D.]] (1994) ''Time Series Analysis''. Princeton University Press. (p. 293)</ref>
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| * 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 | authorlink = Arnold Zellner | year = 1962 | title = An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias | url = | journal = [[Journal of the American Statistical Association]] | volume = 57 | issue = 298| pages = 348–368 }}</ref>
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| ===Estimation of the covariance matrix of the errors===
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| As in the standard case, the [[maximum likelihood estimator]] (MLE) of the covariance matrix differs from the ordinary least squares (OLS) estimator.
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| MLE estimator:{{citation needed|date=February 2012}} <math> \hat \Sigma = \frac{1}{T} \sum_{t=1}^T \hat \epsilon_t\hat \epsilon_t^{'}</math>
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| 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.
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| In a matrix notation, this gives:
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| : <math> \hat \Sigma = \frac{1}{T-kp-1} (Y-\hat{B}Z)(Y-\hat{B}Z)^'.</math>
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| ===Estimation of the estimator's covariance matrix===
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| The covariance matrix of the parameters can be estimated as{{citation needed|date=February 2012}}
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| : <math> \widehat \mbox{Cov} (\mbox{Vec}(\hat B)) =({ZZ'})^{-1} \otimes\hat \Sigma.\, </math>
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| ==Interpretation of estimated model==
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| {{Main|Variance decomposition of forecast errors}}
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| Properties of the VAR model are usually summarized using structural analysis using [[Granger causality]], [[Impulse response]]s, and [[Variance decomposition of forecast errors|forecast error variance decompositions]].
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| ==Forecasting using an estimated VAR model==
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| {{Main|Autoregressive model#n-step-ahead forecasting|Autoregressive model#Evaluating the quality of forecasts}}
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| 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.
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| ==Applications==
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| [[Christopher A. Sims|Christopher Sims]] 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|authorlink=Christopher A. Sims |last=Sims |first=Christopher |year=1980 |title=Macroeconomics and Reality |journal=[[Econometrica]] |volume=48 |issue=1 |pages=1–48 |jstor=1912017 }}</ref>
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| ==Software==
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| *[[R (programming language)|R]]: there is a package called vars which deals with VAR models.<ref>[http://cran.r-project.org/web/packages/vars/vignettes/vars.pdf Bernhard Pfaff VAR, SVAR and SVEC Models: Implementation Within R Package vars]</ref>
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| *SAS: VARMAX
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| *[[STATA]]: "var"
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| *[[EViews]]: "VAR"
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| *[[Gretl]]: "var"
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| *[[Regression analysis of time series]]
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| *[http://ideas.repec.org/s/boc/bocode.html Statistical Software Components]
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| ==See also==
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| *[[Bayesian vector autoregression]]
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| *[[Convergent cross mapping]]
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| *[[Granger causality]]
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| *[[Variance decomposition]]
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| ==Notes==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite book |last=Gujarati |first=Damodar N. |last2=Porter |first2=Dawn C. |title=Basic Econometrics |location=New York |publisher=McGraw-Hill |year=2009 |edition=Fifth international |isbn=978-007-127625-2 |chapter=Vector Autoregression (VAR) |pages=784–790 }}
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| *{{cite book |first=Walter |last=Enders |title=Applied Econometric Time Series |edition=2nd |publisher=John Wiley & Sons |year=2003 |isbn=0-471-23065-0 }}
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| *{{cite book |first=Helmut |last=Lütkepohl |title=New Introduction to Multiple Time Series Analysis |publisher=Springer |location=Berlin |year=2005 |isbn=3540401725 }}
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| *{{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=http://ideas.repec.org/a/taf/japsta/v35y2008i6p601-615.html |ref=harv}}
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| *{{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=http://ideas.repec.org/a/eee/ecmode/v21y2004i4p661-683.html |ref=harv}}
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| *{{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]] |volume=41 |issue=9 |pages=1489–1500 |url=http://ideas.repec.org/a/taf/applec/v41y2009i9p1121-1125.html |ref=harv}}
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| {{Statistics|analysis|state=expanded}}
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| {{Economics|state=collapsed}}
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| [[Category:Econometrics]]
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| [[Category:Time series models]]
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| [[Category:Multivariate time series analysis]]
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