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Vector autoregression

From Wikipedia, the free encyclopedia
Statistical model to calculate the value of multiple quantities as they change over time
For other uses of "Var", seeVar (disambiguation).
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Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type ofstochastic process model. VAR models generalize the single-variable (univariate)autoregressive model by allowing for multivariatetime series. VAR models are often used ineconomics and thenatural sciences.

Like the autoregressive model, each variable has an equation modelling its evolution over time. This equation includes the variable'slagged (past) values, the lagged values of the other variables in the model, and anerror term. VAR models do not require as much knowledge about the forces influencing a variable as dostructural models withsimultaneous equations. The only prior knowledge required is a list of variables which can be hypothesized to affect each other over time.

Specification

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Definition

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A VAR model describes the evolution of a set ofk variables, calledendogenous variables, over time. Each period of time is numbered,t = 1, ...,T. The variables are collected in avector,yt, which is of lengthk. (Equivalently, this vector might be described as a (k × 1)-matrix.) The vector is modelled as a linear function of its previous value. The vector's components are referred to asyi,t, meaning the observation at timet of thei th variable. For example, if the first variable in the model measures the price of wheat over time, theny1,1998 would indicate the price of wheat in the year 1998.

VAR models are characterized by theirorder, 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. Alag is the value of a variable in a previous time period. So in general apth-order VAR refers to a VAR model which includes lags for the lastp time periods. Apth-order VAR is denoted "VAR(p)" and sometimes called "a VAR withp lags". Apth-order VAR model is written as

yt=c+A1yt−1+A2yt−2+⋯+Apyt−p+et,{\displaystyle y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots +A_{p}y_{t-p}+e_{t},\,}

The variables of the formyt−i indicate that variable's valuei time periods earlier and are called the "ith lag" ofyt. The variablec is ak-vector of constants serving as theintercept of the model.Ai is atime-invariant (k × k)-matrix andet is ak-vector oferror terms. The error terms must satisfy three conditions:

  1. E(et)=0{\displaystyle \mathrm {E} (e_{t})=0\,}. Every error term has amean of zero.
  2. E(etet′)=Ω{\displaystyle \mathrm {E} (e_{t}e_{t}')=\Omega \,}. The contemporaneouscovariance matrix of error terms is ak × kpositive-semidefinite matrix denoted Ω.
  3. E(etet−k′)=0{\displaystyle \mathrm {E} (e_{t}e_{t-k}')=0\,} for any non-zerok. There is nocorrelation across time. In particular, there is noserial correlation in individual error terms.[1]

The process of choosing the maximum lagp in the VAR model requires special attention becauseinference is dependent on correctness of the selected lag order.[2][3]

Order of integration of the variables

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Note that all variables have to be of the sameorder 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) withd > 0:[citation needed]
    • The variables arecointegrated: the error correction term has to be included in the VAR. The model becomes a Vectorerror correction model (VECM) which can be seen as a restricted VAR.
    • The variables are notcointegrated: first, the variables have to be differenced d times and one has a VAR in difference.

Concise matrix notation

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

Y=BZ+U{\displaystyle Y=BZ+U\,}

Example

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

[y1,ty2,t]=[c1c2]+[a1,1a1,2a2,1a2,2][y1,t−1y2,t−1]+[e1,te2,t],{\displaystyle {\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}},}

(in which only a singleA matrix appears because this example has a maximum lagp equal to 1), or, equivalently, as the following system of two equations

y1,t=c1+a1,1y1,t−1+a1,2y2,t−1+e1,t{\displaystyle y_{1,t}=c_{1}+a_{1,1}y_{1,t-1}+a_{1,2}y_{2,t-1}+e_{1,t}\,}
y2,t=c2+a2,1y1,t−1+a2,2y2,t−1+e2,t.{\displaystyle y_{2,t}=c_{2}+a_{2,1}y_{1,t-1}+a_{2,2}y_{2,t-1}+e_{2,t}.\,}

Each variable in the model has one equation. The current (timet) 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)

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A VAR withp 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

yt=c+A1yt−1+A2yt−2+et{\displaystyle y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+e_{t}}

can be recast as the VAR(1) model

[ytyt−1]=[c0]+[A1A2I0][yt−1yt−2]+[et0],{\displaystyle {\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}},}

whereI is theidentity matrix.

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

Structural vs. reduced form

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Structural VAR

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Astructural VAR with p lags (sometimes abbreviatedSVAR) is

B0yt=c0+B1yt−1+B2yt−2+⋯+Bpyt−p+ϵt,{\displaystyle B_{0}y_{t}=c_{0}+B_{1}y_{t-1}+B_{2}y_{t-2}+\cdots +B_{p}y_{t-p}+\epsilon _{t},}

wherec0 is ak × 1 vector of constants,Bi is ak × k matrix (for everyi = 0, ...,p) andεt is ak × 1 vector oferror terms. Themain diagonal terms of theB0 matrix (the coefficients on theith variable in theith equation) are scaled to 1.

The error terms εt (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 matrixE(ϵtϵt′)=Σ{\displaystyle \mathrm {E} (\epsilon _{t}\epsilon _{t}')=\Sigma } are zero. That is, the structural shocks are uncorrelated.

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

[1B0;1,2B0;2,11][y1,ty2,t]=[c0;1c0;2]+[B1;1,1B1;1,2B1;2,1B1;2,2][y1,t−1y2,t−1]+[ϵ1,tϵ2,t],{\displaystyle {\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}},}

where

Σ=E(ϵtϵt′)=[σ1200σ22];{\displaystyle \Sigma =\mathrm {E} (\epsilon _{t}\epsilon _{t}')={\begin{bmatrix}\sigma _{1}^{2}&0\\0&\sigma _{2}^{2}\end{bmatrix}};}

that is, thevariances of the structural shocks are denotedvar(ϵi)=σi2{\displaystyle \mathrm {var} (\epsilon _{i})=\sigma _{i}^{2}} (i = 1, 2) and thecovariance iscov(ϵ1,ϵ2)=0{\displaystyle \mathrm {cov} (\epsilon _{1},\epsilon _{2})=0}.

Writing the first equation explicitly and passingy2,t to theright hand side one obtains

y1,t=c0;1−B0;1,2y2,t+B1;1,1y1,t−1+B1;1,2y2,t−1+ϵ1,t{\displaystyle 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}\,}

Note thaty2,t can have a contemporaneous effect ony1,t ifB0;1,2 is not zero. This is different from the case whenB0 is theidentity matrix (all off-diagonal elements are zero — the case in the initial definition), wheny2,t can impact directlyy1,t+1 and subsequent future values, but noty1,t.

Because of theparameter identification problem,ordinary least squares estimation of the structural VAR would yieldinconsistent 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 beindependent, 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 asupply shock) should be related to a shift in consumers' preferences towards a style of clothing (as an example of ademand shock); therefore one would expect these factors to be statistically independent.
2.Variables can have acontemporaneous impact on other variables. This is a desirable feature especially when using low frequency data. For example, anindirect tax rate increase would not affecttax revenues the day the decision is announced, but one could find an effect in that quarter's data.

Reduced-form VAR

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By premultiplying the structural VAR with the inverse ofB0

yt=B0−1c0+B0−1B1yt−1+B0−1B2yt−2+⋯+B0−1Bpyt−p+B0−1ϵt,{\displaystyle 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},}

and denoting

B0−1c0=c,B0−1Bi=Ai for i=1,…,p and B0−1ϵt=et{\displaystyle 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}}

one obtains thepth order reduced VAR

yt=c+A1yt−1+A2yt−2+⋯+Apyt−p+et{\displaystyle y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots +A_{p}y_{t-p}+e_{t}}

Note that in the reduced form all right hand side variables arepredetermined at timet. As there are no timet endogenous variables on the right hand side, no variable has adirect contemporaneous effect on other variables in the model.

However, the error terms in the reduced VAR are composites of the structural shockset =B0−1εt. Thus, the occurrence of one structural shockεi,t can potentially lead to the occurrence of shocks in all error termsej,t, thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR

Ω=E(etet′)=E(B0−1ϵtϵt′(B0−1)′)=B0−1Σ(B0−1)′{\displaystyle \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})'\,}

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

Estimation

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Estimation of the regression parameters

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Starting from the concise matrix notation:

Y=BZ+U{\displaystyle Y=BZ+U\,}
B^=YZ′(ZZ′)−1.{\displaystyle {\hat {B}}=YZ'(ZZ')^{-1}.}

This can be written alternatively as:

Vec⁡(B^)=((ZZ′)−1Z⊗Ik) Vec⁡(Y),{\displaystyle \operatorname {Vec} ({\hat {B}})=((ZZ')^{-1}Z\otimes I_{k})\ \operatorname {Vec} (Y),}

where⊗{\displaystyle \otimes } denotes theKronecker product and Vec thevectorization of the indicated matrix.

This estimator isconsistent andasymptotically efficient. It is furthermore equal to the conditionalmaximum likelihood estimator.[4]

  • As the explanatory variables are the same in each equation, the multivariate least squares estimator is equivalent to theordinary least squares estimator applied to each equation separately.[5]

Estimation of the covariance matrix of the errors

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

MLE estimator:[citation needed]Σ^=1T∑t=1Tϵ^tϵ^t′{\displaystyle {\hat {\Sigma }}={\frac {1}{T}}\sum _{t=1}^{T}{\hat {\epsilon }}_{t}{\hat {\epsilon }}_{t}'}

OLS estimator:[citation needed]Σ^=1T−kp−1∑t=1Tϵ^tϵ^t′{\displaystyle {\hat {\Sigma }}={\frac {1}{T-kp-1}}\sum _{t=1}^{T}{\hat {\epsilon }}_{t}{\hat {\epsilon }}_{t}'} for a model with a constant,k variables andp lags.

In a matrix notation, this gives:

Σ^=1T−kp−1(Y−B^Z)(Y−B^Z)′.{\displaystyle {\hat {\Sigma }}={\frac {1}{T-kp-1}}(Y-{\hat {B}}Z)(Y-{\hat {B}}Z)'.}

Estimation of the estimator's covariance matrix

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The covariance matrix of the parameters can be estimated as[citation needed]

Cov^(Vec(B^))=(ZZ′)−1⊗Σ^.{\displaystyle {\widehat {\mbox{Cov}}}({\mbox{Vec}}({\hat {B}}))=({ZZ'})^{-1}\otimes {\hat {\Sigma }}.\,}

Degrees of freedom

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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 thedegrees 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

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Impulse response

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Consider the first-order case (i.e., with only one lag), with equation of evolution

yt=Ayt−1+et,{\displaystyle y_{t}=Ay_{t-1}+e_{t},}

for evolving (state) vectory{\displaystyle y} and vectore{\displaystyle e} of shocks. To find, say, the effect of thej-th element of the vector of shocks upon thei-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:

yt−1=Ayt−2+et−1.{\displaystyle y_{t-1}=Ay_{t-2}+e_{t-1}.}

Use this in the original equation of evolution to obtain

yt=A2yt−2+Aet−1+et;{\displaystyle y_{t}=A^{2}y_{t-2}+Ae_{t-1}+e_{t};}

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

yt=A3yt−3+A2et−2+Aet−1+et.{\displaystyle y_{t}=A^{3}y_{t-3}+A^{2}e_{t-2}+Ae_{t-1}+e_{t}.}

From this, the effect of thej-th component ofet−2{\displaystyle e_{t-2}} upon thei-th component ofyt{\displaystyle y_{t}} is thei, j element of the matrixA2.{\displaystyle A^{2}.}

It can be seen from thisinduction process that any shock will have an effect on the elements ofy 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 theeigenvalues of the matrixA are less than 1 inabsolute value.

Forecasting using an estimated VAR model

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Main articles:Autoregressive model § n-step-ahead forecasting, andAutoregressive model § Evaluating the quality of forecasts

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

Applications

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Christopher Sims has advocated VAR models, criticizing the claims and performance of earlier modeling inmacroeconomiceconometrics.[6] He recommended VAR models, which had previously appeared in time seriesstatistics and insystem identification, a statistical specialty incontrol 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.[6] VAR models are also increasingly used in health research for automatic analyses of diary data[7] 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.[8][9]

Software

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See also

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Notes

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  1. ^For multivariate tests for autocorrelation in the VAR models, seeHatemi-J, A. (2004)."Multivariate tests for autocorrelation in the stable and unstable VAR models".Economic Modelling.21 (4):661–683.doi:10.1016/j.econmod.2003.09.005.
  2. ^Hacker, R. S.; Hatemi-J, A. (2008)."Optimal lag-length choice in stable and unstable VAR models under situations of homoscedasticity and ARCH".Journal of Applied Statistics.35 (6):601–615.Bibcode:2008JApSt..35..601S.doi:10.1080/02664760801920473.
  3. ^Hatemi-J, A.; Hacker, R. S. (2009)."Can the LR test be helpful in choosing the optimal lag order in the VAR model when information criteria suggest different lag orders?".Applied Economics.41 (9):1489–1500.doi:10.1080/00036840601019273.
  4. ^Hamilton, James D. (1994).Time Series Analysis. Princeton University Press. p. 293.
  5. ^Zellner, Arnold (1962). "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias".Journal of the American Statistical Association.57 (298):348–368.doi:10.1080/01621459.1962.10480664.
  6. ^abSims, Christopher (1980). "Macroeconomics and Reality".Econometrica.48 (1):1–48.CiteSeerX 10.1.1.163.5425.doi:10.2307/1912017.JSTOR 1912017.
  7. ^van der Krieke; et al. (2016)."Temporal Dynamics of Health and Well-Being: A Crowdsourcing Approach to Momentary Assessments and Automated Generation of Personalized Feedback (2016)"(PDF).Psychosomatic Medicine.79 (2):213–223.doi:10.1097/PSY.0000000000000378.PMID 27551988.
  8. ^Sio Iong Ao (2003). "Analysis of the Interaction of Asian Pacific Indices and Forecasting Opening Prices by Hybrid VAR and Neural Network Procedures (2003)".International Conf. On Computational Intelligence for Modelling, Control and Automation 2003.
  9. ^Caraka, R.E.; et al. (2021)."Hybrid vector autoregression feedforward neural network with genetic algorithm model for forecasting space-time pollution data (2021)".Indonesian Journal of Science and Technology:243–266.doi:10.17509/ijost.v6i1.32732.
  10. ^"Bernhard Pfaff VAR, SVAR and SVEC Models: Implementation Within R Package vars"(PDF). Archived fromthe original(PDF) on 2016-08-18. Retrieved2016-07-19.
  11. ^Hyndman, Rob J; Athanasopoulos, George (2018). "11.2: Vector Autoregressions".Forecasting: Principles and Practice. OTexts. pp. 333–335.ISBN 978-0-9875071-1-2.

Further reading

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  • Asteriou, Dimitrios; Hall, Stephen G. (2011). "Vector Autoregressive (VAR) Models and Causality Tests".Applied Econometrics (Second ed.). London: Palgrave MacMillan. pp. 319–333.
  • Enders, Walter (2010).Applied Econometric Time Series (Third ed.). New York: John Wiley & Sons. pp. 272–355.ISBN 978-0-470-50539-7.
  • Favero, Carlo A. (2001).Applied Macroeconometrics. New York: Oxford University Press. pp. 162–213.ISBN 0-19-829685-1.
  • Lütkepohl, Helmut (2005).New Introduction to Multiple Time Series Analysis. Berlin: Springer.ISBN 3-540-40172-5.
  • Qin, Duo (2011). "Rise of VAR Modelling Approach".Journal of Economic Surveys.25 (1):156–174.doi:10.1111/j.1467-6419.2010.00637.x.
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