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VAR and VEC

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 ) Trend may be included: d t, where d is K by 1 Exogenous variables X may be added.

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VAR and VEC

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  1. VAR and VEC Using Stata

  2. 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) • Trend may be included: dt, where d is K by 1 • Exogenous variables X may be added

  3. 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)

  4. 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):

  5. VEC: Vector Error Correction • No-constant or No-drift Model: v = 0 g = 0, m = 0 • Restricted-constant Model: v = amg = 0, m≠ 0 • Constant or Drift Model: g≠ 0, m≠ 0

  6. VEC: Vector Error Correction • If there is trend in yt

  7. VEC: Vector Error Correction • No-drift No-trend Model: v = 0, d = 0 g = 0, m = 0, t = 0, r = 0 • Restricted-constant Model: v = am, d = 0 g = 0, m≠ 0, t = 0, r = 0 • Constant or Drift Model: d = 0 g≠ 0, m≠ 0, t = 0, r = 0 • Restricted-trend Model: d = arg≠ 0, m≠ 0, t = 0, r≠ 0 • Trend Model:g≠ 0, m≠ 0, t≠ 0, r≠ 0

  8. Example • C: Personal Consumption Expenditure • Y: Disposable Personal Income • C ~ I(1), Y ~ I(1) • Consumption-Income Relationship:Ct = vc + acc1Ct-1 + acy1Yt-1 +…(+ dct) + ecYt = vy + ayc1Ct-1 + ayy1Yt-1 +…(+ dyt) + ey • Cointegrated C and I: e ~ I(0)? • Which model? (trend and/or drift) • How many lags?

  9. ExampleJohansen Test with trend models and 6 lags • VAR: • VEC: • Rank 1: P = ab’

  10. ExampleJohansen Test with constant models and 6 lags • VAR: • VEC: • Rank 1: P = ab’

  11. ExampleJohansen Test with constant models and 6 lags • Constant model: v = am + g (a 0,g0) • Restricted-constant model: v = am (a 0) • No-constant model: v = 0

  12. ExampleJohansen Test with trend models and 6 lags • Trend model:d = ar+t g≠ 0, m≠ 0, t≠ 0, r≠ 0 • Restricted trend model: d = ar g≠ 0, m≠ 0, t = 0, r≠ 0 • No trend or constant model: d = 0 g≠ 0, m≠ 0, t = 0, r= 0

  13. Example VEC estimated models with 6 lags • Empirical Model: 1948q3 – 2012q3 (N=257) • No-constant: LL=1766.15 (K=23) • Restricted-constant: LL=1771.47 (K=24) • Restricted-trend: LL=1774.15 (K=27)

  14. VEC Forecasting Estimated Models: 1948q3 – 2009q4 • No-constant model, 6 Lags (LL=1684.37, 23 parameters, N=246) • Restricted-constant model, 6 lags (LL=1689.23, 24 parameters, N=246) • Restricted-trend model, 6 lags (LL = 1691.9, 27 parameters, N=246)

  15. VEC ForecastingRestricted-trend Model, 6 Lags Y = ln(PCE), X = ln(DPI), 2010q1 -

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