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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

VAR and VEC

Using Stata

var vector autoregression
VAR: Vector Autoregression
  • 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
var and vec1
VAR and VEC
  • If yt is not stationary, VAR or VEC can only be applied for cointegrated yt system:
    • VAR (Vector Autoregression)
    • VEC (Vector Error Correction)
vec vector error correction
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):
vec vector error correction1
VEC: Vector Error Correction
  • 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
vec vector error correction2
VEC: Vector Error Correction
  • If there is trend in yt
vec vector error correction3
VEC: Vector Error Correction
  • 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
example
Example
  • 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
example1
Example
  • VAR:
  • VEC:
  • Rank 1 P = ab’
example2
Example
  • Empirical Model: 1949q4 – 2006q4
    • Constant, 10 lags
    • Restricted-constant, 10 lags
restricted var
Restricted VAR
  • Parameters Restrictions
    • Many of the parameters associated with augmented lags are 0
  • Structural VAR
    • Linking with macroeconomic theory
impulse response function
Impulse Response Function
  • Based on VMA representation, we can trace out the time path of the various shocks on the variables in the VAR system.
var and simultaneous equations
VAR and Simultaneous Equations
  • 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’
structural var
Structural VAR
  • 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.
example3
Example
  • 2-Equation System Model:
  • Structural VAR:
example4
Example
  • Structural VAR:
  • Identification and Parameters Restrictions
stata programs
Stata Programs
  • us_dpi_pce.txt
  • dpi_pce4.do
  • dpi_pce5.do
  • dpi_pce6.do
  • dpi_pce7.do
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