Spence's function

Vector autoregression (VAR) is an econometric model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregression (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 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 models with simultaneous equations: The only prior knowledge required is a list of variables which can be hypothesized to affect each other intertemporally.

Specification

Definition

A VAR model describes the evolution of a set of k variables (called endogenous variables) over the same sample period (t = 1, ..., T) as a linear function of only their past values. The variables are collected in a k × 1 vector y_t, which has as the i ^th element, y_i,t, the time t observation of the i ^th variable. For example, if the i ^th variable is GDP, then y_i,t is the value of GDP at time t.

A p-th order VAR, denoted VAR(p), is

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+\cdots +A_p{y}_{t-p}+e_t,$

where the l-periods back observation y_t−l is called the l-th lag of y, c is a k × 1 vector of constants (intercepts), A_i is a time-invariant k × k matrix and e_t is a k × 1 vector of error terms satisfying

1. $\mathrm{E}(e_t)=0$ — every error term has mean zero;
2. $\mathrm{E}(e_t{e_t}^{\prime })=\mathrm{\Omega }$ — the contemporaneous covariance matrix of error terms is Ω (a k × k positive-semidefinite matrix);
3. $\mathrm{E}(e_t{e_{t^{\prime }-k}}^{\prime })=0$ 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.

A pth-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.Template:SfnTemplate:Sfn

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): one is in the standard case, i.e. a VAR in level
• All the variables are I(d) (non-stationary) with d > 0:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
  • The variables are 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 cointegrated: the variables have first 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) with a concise matrix notation:

$Y=BZ+U$

Details of the matrices are in a separate page.

Example

For a general example of a VAR(p) with k variables, see General matrix notation of a VAR(p).

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

$\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\end{array}\right]=\left[\begin{array}{c}{c}_1\\ c_2\end{array}\right]+\left[\begin{array}{cc}{A}_{1,1}& A_{1,2}\\ A_{2,1}& A_{2,2}\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ y_{2,t-1}\end{array}\right]+\left[\begin{array}{c}{e}_{1,t}\\ e_{2,t}\end{array}\right],$

(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

$y_{1,t}=c_1+A_{1,1}{y}_{1,t-1}+A_{1,2}{y}_{2,t-1}+e_{1,t}$

$y_{2,t}=c_2+A_{2,1}{y}_{1,t-1}+A_{2,2}{y}_{2,t-1}+e_{2,t}.$

Each variable in the model has one equation. The current (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 number of equations.

For example, the VAR(2) model

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+e_t$

can be recast as the VAR(1) model

$\left[\begin{array}{c}{y}_t\\ y_{t-1}\end{array}\right]=\left[\begin{array}{c}c\\ 0\end{array}\right]+\left[\begin{array}{cc}{A}_1& A_2\\ I& 0\end{array}\right]\left[\begin{array}{c}{y}_{t-1}\\ y_{t-2}\end{array}\right]+\left[\begin{array}{c}{e}_t\\ 0\end{array}\right],$

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

$B_0{y}_t=c_0+B_1{y}_{t-1}+B_2{y}_{t-2}+\cdots +B_p{y}_{t-p}+{ϵ}_t,$

where c₀ is a k × 1 vector of constants, B_i is a k × k matrix (for every i = 0, ..., p) and ε_t is a k × 1 vector of error terms. The main diagonal terms of the B₀ matrix (the coefficients on the i^th variable in the i^th equation) are scaled to 1.

The error terms ε_t (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 $\mathrm{E}({ϵ}_t{{ϵ}_t}^{\prime })=\mathrm{\Sigma }$ are zero. That is, the structural shocks are uncorrelated.

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

$\left[\begin{array}{cc}1& B_{0;1,2}\\ B_{0;2,1}& 1\end{array}\right]\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\end{array}\right]=\left[\begin{array}{c}{c}_{0;1}\\ c_{0;2}\end{array}\right]+\left[\begin{array}{cc}{B}_{1;1,1}& B_{1;1,2}\\ B_{1;2,1}& B_{1;2,2}\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ y_{2,t-1}\end{array}\right]+\left[\begin{array}{c}{ϵ}_{1,t}\\ {ϵ}_{2,t}\end{array}\right],$

where

$\mathrm{\Sigma }=\mathrm{E}({ϵ}_t{{ϵ}_t}^{\prime })=\left[\begin{array}{cc}{\sigma }_1^2& 0\\ 0& {\sigma }_2^2\end{array}\right];$

that is, the variances of the structural shocks are denoted $\mathrm{v}\mathrm{a}\mathrm{r}({ϵ}_i)={\sigma }_i^2$ (i = 1, 2) and the covariance is $\mathrm{c}\mathrm{o}\mathrm{v}({ϵ}_1,{ϵ}_2)=0$.

Writing the first equation explicitly and passing y_2,t to the right hand side one obtains

$y_{1,t}=c_{0;1}-B_{0;1,2}{y}_{2,t}+B_{1;1,1}{y}_{1,t-1}+B_{1;1,2}{y}_{2,t-1}+{ϵ}_{1,t}$

Note that y_2,t can have a contemporaneous effect on y_1,t if B_0;1,2 is not zero. This is different from the case when B₀ is the identity matrix (all off-diagonal elements are zero — the case in the initial definition), when y_2,t can impact directly y_1,t+1 and subsequent future values, but not y_1,t.

Because of the parameter identification problem, ordinary least squares estimation of the structural VAR would yield 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 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 B₀

$y_t=B_0^{-1}{c}_0+B_0^{-1}{B}_1{y}_{t-1}+B_0^{-1}{B}_2{y}_{t-2}+\cdots +B_0^{-1}{B}_p{y}_{t-p}+B_0^{-1}{ϵ}_t,$

and denoting

$B_0^{-1}{c}_0=c,B_0^{-1}{B}_i=A_i\text{ for }i=1,\dots ,p\text{ and }{B}_0^{-1}{ϵ}_t=e_t$

one obtains the pth order reduced VAR

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+\cdots +A_p{y}_{t-p}+e_t$

Note that in the reduced form all right hand side variables 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_t = B₀⁻¹ε_t. Thus, the occurrence of one structural shock ε_i,t can potentially lead to the occurrence of shocks in all error terms e_j,t, thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR

$\mathrm{\Omega }=\mathrm{E}(e_t{e_t}^{\prime })=\mathrm{E}(B_0^{-1}{ϵ}_t{{ϵ}_t}^{\prime }(B_0^{-1}{)}^{\prime })=B_0^{-1}\mathrm{\Sigma }(B_0^{-1}{)}^{\prime }$

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 (for details see this annex):

$Y=BZ+U$

• The multivariate least squares (MLS) for B yields:

$\hat{B}=Y{Z}^{\text{'}}(Z{Z}^{\text{'}}{)}^{-1}$

It can be written alternatively as:

$\mathrm{Vec}(\hat{B})=((Z{Z}^{\text{'}}{)}^{-1}Z\otimes I_k)\mathrm{Vec}(Y)$

Where $\otimes$ denotes the Kronecker product and Vec the vectorization of the matrix Y.

This estimator is consistent and asymptotically efficient. It is furthermore equal to the conditional maximum likelihood estimator.

• 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.

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:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. $\hat{\mathrm{\Sigma }}=\frac{1}{T}\sum _{t=1}^T{\widehat{ϵ}}_t{\widehat{ϵ}}_t^{\text{'}}$

OLS estimator:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. $\hat{\mathrm{\Sigma }}=\frac{1}{T-kp-1}\sum _{t=1}^T{\widehat{ϵ}}_t{\widehat{ϵ}}_t^{\text{'}}$ for a model with a constant, k variables and p lags.

In a matrix notation, this gives:

$\hat{\mathrm{\Sigma }}=\frac{1}{T-kp-1}(Y-\hat{B}Z)(Y-\hat{B}Z{)}^{\text{'}}.$

Estimation of the estimator's covariance matrix

The covariance matrix of the parameters can be estimated asPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

$\widehat{\text{Cov}}(\text{Vec}(\hat{B}))=(Z{Z}^{\prime }{)}^{-1}\otimes \hat{\mathrm{\Sigma }}.$

Interpretation of estimated model

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Properties of the VAR model are usually summarized using structural analysis using Granger causality, Impulse responses, and forecast error variance decompositions.

Forecasting using an estimated VAR model

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

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 Sims advocated VAR models, criticizing the claims and performance of earlier modeling in macroeconomic econometrics. 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.

Software

• R: there is a package called vars which deals with VAR models.
• SAS: VARMAX
• STATA: "var"
• EViews: "VAR"
• Gretl: "var"
• Regression analysis of time series
• Statistical Software Components

See also

• Bayesian vector autoregression
• Convergent cross mapping
• Granger causality
• Variance decomposition

Notes

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Further reading

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  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang