Vector auto regressive models

Vector autoregressive models ( VAR models for short ) are very popular econometric models for estimating multiple equations simultaneously. They are the multidimensional analogue of the autoregressive model . They belong to the top model class of the VARMA models. In this type of time series model , the endogenous variables are determined both by their own past values and by the past values ​​of the other endogenous variables. The variables are therefore also referred to as delayed exogenous . So there is feedback between the variables when the covariance matrix is non-diagonal.

motivation

A simple two-dimensional VAR model contains two time series and , which are explained, and another time series , which is used for explanation. The model equations are then ${\displaystyle x_{t}^{(1)}}$ ${\displaystyle x_{t}^{(2)}}$ ${\displaystyle z_{t}}$

${\displaystyle x_{t}^{(1)}=\nu _{1}+a_{11}x_{t-1}^{(1)}+a_{12}x_{t-1}^{(2)}+b_{1}z_{t}+\epsilon _{1}}$,

${\displaystyle x_{t}^{(2)}=\nu _{2}+a_{21}x_{t-1}^{(1)}+a_{22}x_{t-1}^{(2)}+b_{2}z_{t}+\epsilon _{2}}$.

or in matrix notation:

${\displaystyle X_{t}=\nu +A_{1}\cdot X_{t-1}+B\cdot Z_{t}+\epsilon _{t}}$

The values ​​of the two time series and currently depend on ${\displaystyle x_{t}^{(1)}}$ ${\displaystyle x_{t}^{(2)}}$ ${\displaystyle t}$

• the past of both time series,
  • in the VAR (1) model only from the values ​​of the previous period, and , ${\displaystyle x_{t-1}^{(1)}}$ ${\displaystyle x_{t-1}^{(2)}}$
  • In the VAR (p) model, further lags (from English "delays") can be included,
• further explanatory time series, here: currently as well as ${\displaystyle z_{t}}$ ${\displaystyle t}$
• the error terms . ${\displaystyle \epsilon _{i}}$

In the model, the model parameters

• ${\displaystyle \nu ={\begin{pmatrix}\nu _{1}\\\nu _{2}\end{pmatrix}}}$,
• ${\displaystyle A_{1}={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}}$ and
• ${\displaystyle B={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}$

iteratively estimated from the data.

Differentiation from transfer function models

There are similarities between the VAR models and the transfer function models . However, a VAR (1) model must not be viewed as a causal transfer function model. The reason is the respective contemporary correlation of the shock variables . By orthogonalizing the shock variables ( diagonalizing the covariance matrix ), a VAR (1) model can nevertheless be converted into a causal transfer function model.

Specification and estimation of VAR models

In vector-matrix form , a -dimensional VAR (p) model can be written as ${\displaystyle n}$

${\displaystyle X_{t}=\nu +A_{1}\cdot X_{t-1}+A_{2}\cdot X_{t-2}+\ldots +A_{p}\cdot X_{t-p}+B\cdot Z_{t}+\epsilon _{t}}$,

where denotes the vector of the endogenous variable, the vector of the exogenous variable and the error term. The vectors and the matrices are to be estimated. This is a linear model, so Gauss-Markov's theorem applies and the model can be estimated efficiently using the least squares method . For the derivation of the estimator, the covariance matrix etc. see z. B. Lütkepohl (1991) ${\displaystyle X_{t}}$ ${\displaystyle Z_{t}}$ ${\displaystyle \epsilon _{t}}$ ${\displaystyle \nu ,B\in \mathbb {R} ^{n}}$ ${\displaystyle A_{1},\dotsc ,A_{p}\in \mathbb {R} ^{n\times n}}$

To choose the optimal number of delays (optimal lag order) z. B. the Akaike or the Schwarz information criterion can be used. Even if the least squares estimate itself has no prerequisites in terms of expected value or variance of , constant moments are required for quality statements and the specification of the lags via the information criteria . For this reason, time series are usually trend-adjusted (for example with a Hodrick-Prescott filter ) and thus made stationary before VAR models are estimated . An alternative approach is to estimate vector error correction models. ${\displaystyle p}$ ${\displaystyle X_{t}}$

Advantages and problems of VAR

Benefits

VAR models owe their current popularity in economics to a large extent to the profound criticism by Christopher Sims (1980) of theory-based multi-equation models that were used in the 1960s and 1970s for economic forecasts. B. reference to the FRB-MIT model. These models consisted of up to several dozen equations that described individual sectors of the economy and estimated separately. The model equations put i. d. Usually only some of the endogenous variables of the overall model in relation. Sims criticizes that this is synonymous with a large number of more or less arbitrary restrictions in the model that can massively distort model results. Furthermore, such models are significantly affected by the Lucas criticism and potentially suffer from additional statistical problems.

VAR models are theory-free and therefore do not set any more or less arbitrary restrictions. Therefore, despite their relative simplicity, they are prognostically superior to classic multi-equation models and are still used today as a benchmark to evaluate the prognosis quality of modern theory-based models (especially DSGE models ).

Furthermore, with Structural VARs, i.e. VARs with restrictions in the covariance matrix or parameters, there are possibilities to use prior information specifically to improve the forecast quality or to derive structural dynamic statements from VARs.

Problems

• Due to the large number of parameters to be estimated, VAR models are relatively imprecise for common economic data set sizes, as Sims (1980) already points out.
• VAR models are not necessarily unique; H. Even with the help of restrictions, it can be impossible to derive economic statements from an SVAR.
• VAR models are not free from the Lucas criticism; H. they cannot easily be used for policy simulations.

literature

• Damodar N. Gujarati, Dawn C. Porter: Basic Econometrics . 5th edition. McGraw-Hill, New York 2009, ISBN 978-0-07-127625-2 , pp. 784-790 .

Individual evidence

1. ^ Helmut Lütkepohl: Introduction to Multiple Time Series Analysis . 1991, p. 63 ff ., doi : 10.1007 / 978-3-662-02691-5 ( springer.com [accessed on July 23, 2018]).
2. ↑ Christopher A. Sims: Macroeconomics and Reality . In: Econometrica . tape 48 , no. 1 , 1980, p. 1-48 , doi : 10.2307 / 1912017 .
3. ↑ Ando, ​​Albert; Modigliani, Franco; Rasche, Robert: Appendix To Part 1: Equations and Definitions af Variables for the FRB-MIT-Penn Econometric Model . In: Hickman, Bert G. (Ed.): Economic Models of Cyclical Behavior . NBER, November 1969, p. 543-598 .
4. ^ Negro, Marco Del: On the Fit of New Keynesian Models . In: Journal of Business and Economic Statistics . tape 2 , April 2007, p. 123-143 .
5. ^ Helmut Lütkepohl: New Introduction to Multiple Time Series Analysis . 2005, p. 359 , doi : 10.1007 / 978-3-540-27752-1 ( springer.com [accessed July 23, 2018]).
6. ^ VV Chari, Patrick J. Kehoe, Ellen R. McGrattan: A critique of structural VARs using business cycle theory . 2005 ( psu.edu [accessed July 23, 2018]).
7. ↑ Handbook of Applied Econometrics. Volume I: Macroeconomics . In: Handbook of Applied Econometrics. Volume I: Macroeconomics . Blackwell Publishing Ltd, Oxford, UK 1999, ISBN 978-0-631-21558-5 , pp. 105 ff ., doi : 10.1111 / b.9780631215585.1999.00003.x ( wiley.com [accessed July 24, 2018]).