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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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