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# VAR and VEC - PowerPoint PPT Presentation

VAR and VEC. Using Stata. VAR: Vector Autoregression. Assumptions: y t : Stationary K-variable vector v : K constant parameters vector A j : K by K parameters matrix, j=1,…,p u t : i.i.d.( 0 , S ) Exogenous variables X may be added. VAR and VEC.

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### VAR and VEC

Using Stata

• Assumptions:

• yt: Stationary K-variable vector

• v: K constant parameters vector

• Aj: K by K parameters matrix, j=1,…,p

• ut: i.i.d.(0,S)

• Exogenous variables X may be added

• If yt is not stationary, VAR or VEC can only be applied for cointegrated yt system:

• VAR (Vector Autoregression)

• VEC (Vector Error Correction)

• If there is no trend in yt, let P = ab’ (P is K by K, a is K by r, b is K by r, r is the rank of P, 0<r<K):

• No-constant or No-drift Model: g = 0, m = 0 (or v = 0)

• Restricted-constant Model: g = 0, m≠ 0 (or v = am)

• Constant or Drift Model: g≠ 0, m≠ 0

• If there is trend in yt

• No-drift No-trend Model: g = 0, m = 0, t = 0, r = 0 (or v = 0, d = 0)

• Restricted-constant Model: g = 0, m≠ 0, t = 0, r = 0 (or v = am, d = 0)

• Constant or Drift Model: g≠ 0, m≠ 0, t = 0, r = 0 (or d = 0)

• Restricted-trend Model:g≠ 0, m≠ 0, t = 0, r≠ 0 (or d = ar)

• Trend Model:g≠ 0, m≠ 0, t≠ 0, r≠ 0

• C: Personal Consumption Expenditure

• Y: Disposable Personal Income

• C ~ I(1), Y ~ I(1)

• Consumption-Income Relationship:Ct = a + bYt-1 + gCt-1 +…(+ dt) + e

• Cointegrated C and I: e ~ I(0)

• Johansen Test with constant model and 10 lags

• VAR:

• VEC:

• Rank 1 P = ab’

• Empirical Model: 1949q4 – 2006q4

• Constant, 10 lags

• Restricted-constant, 10 lags

• Parameters Restrictions

• Many of the parameters associated with augmented lags are 0

• Structural VAR

• Linking with macroeconomic theory

• Based on VMA representation, we can trace out the time path of the various shocks on the variables in the VAR system.

• Identification:

• A = -B-1G, ut = B-1et

• A = [A1,…,Ap,v], G = [G1,…, Gp,g0]

• B and G1,…,Gp are K by K parameters matrices

• Var(ut) = S = B-1Var(et)B-1’

• VAR as a reduced form of simultaneous equations model requires parameters restrictions so that the system model is identified or estimatable.

• A structural VAR corresponds to an identified simultaneous equations system.

• The restrictions are necessarily placed on the parameters matrix B and therefore the variance-covariance matrix S of VAR.

• 2-Equation System Model:

• Structural VAR:

• Structural VAR:

• Identification and Parameters Restrictions

Stata Programs

• us_dpi_pce.txt

• dpi_pce4.do

• dpi_pce5.do

• dpi_pce6.do

• dpi_pce7.do