Multiple Time Series Models

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Lecture 2

• Multiple Time Series Models

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Vector ARMA Model

• VARMA
• Xt is a vector, Xt (X1t,.,Xgt)
• Xt F1Xt-1 .. FpXt-p ut Q1ut-1 ...
  Qput-q
• This model involves a nonlinear estimation.
• A simpler model is VAR model (without MA terms).
• Xt F1Xt-1 .. FpXt-p ut

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VAR Model

• Consider a VAR(1) Model, when g 2.
• Xt ? F1Xt-1 ut
• This implies two equations
• X1t ?1 ?111X1t-1 ?112X2t-1 u1t
• X2t ?2 ?121X1t-1 ?122X2t-1 u2t

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• Questions to understand VAR
• Q1. Why not include contemporaneous terms?
• Structural VAR
• X1t ?1 ?1 X2t ?11X1t-1 ?12X2t-1 v1t
• X2t ?2 ?2 X1t ?21X1t-1 ?22X2t-1 v2t
• Q2. Do we need GLS to estimate VAR models?
• OLS is just fine. Why?

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• Q3. How do we interpret those too many
  parameters?
• Whats usual practice? ? Impulse response
• Q4. We are interested in dynamic effects of
  shocks of the structural form model, not of the
  reduced form model. How do we do?
• ?Xi,ts/?vjt rather than ?Xi,ts/?ujt
• ? Orthogonalization

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• Q5. Should we difference non-stationary time
  series?
• VAR in differences (?Xt) or VAR in level (Xt)?
• or ECM (Error Correction Model)?
• Q6. Other issues
• Determining of VAR lags (p).
• Causality using VAR also Exogeneity
• Critics on VAR

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Impulse Response Analysis
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Innovation Accounting
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Orthogonalization

• Use a Cholesky decomposition.
• Xt ? FXt-1 ut, Cov(ut) W.
• We can find A, D such that
• W ADA AD1/2D1/2A PP with
• P AD1/2,
• A is lower triangular matrix
• D is diagonal matrix.

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• Let vt A-1ut .. structural form innovation
• Then, Evtvs dtt for t s, and zero
  otherwise.
• (orthogonal each other)
• Then,
• ?Xts/?vjt ?Xts/?ut?ut/ ?vjt Ysaj
• where Ys J F s J
• F is a big matrix in VAR(1) companion form
• aj j-th column of A

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• Interpretation
• Ykl,s (k, l) element of Ys
• reaction of the k-th series to one unit of
  orthogonalized shock of the l-th series occurring
  at s-period ahead.
• Alternatively, wt P-1ut
• Then, ?Xts/?wjt Ysaj djj1/2
• reaction to one standard error
  (orthogonalized) shock

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• The ordering matters!
• Due to lower-triangular decomposition, the order
  of Xt matters.
• When estimating VAR models, put relatively
  exogenous variables first.
• If X1t ? X2t, but not vice versa, put X1t
  first.

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Innovation Accounting

• Variance Decomposition
• Recall
• Xts EtXts forecast error
• MSE of the forecast error can be decomposed.
• wk,,l,s The proportion of the s-step forecast
  error variance of the k-th series accounted for
  the innovation of the l-th series.
• The plot of this is the innovation accounting.

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Determining of lags

• Use the Information Criteria
• AIC T log? 2N (also, BIC,.. )
• where N is of parameters.
• Select models with the minimum AIC.
• LR test
• LR (T-c)log?k - log?km
• where c is of parameters in each eq.
  (unrestricted).

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Granger Causality Test

• For the VAR(p) model,
• X1t ? ?1X1t-1 . .. ?pX1t-1 ?1X2t-1 ..
  ?pX2t-p u1t
• H0 ?1 .. ?p 0
• Use F-test.
• Ganger non-causality
• X2t does not Ganger-cause X1t, if past values
  of X2t help one to predict X1t in the presence of
  past values of X1t.

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Exogeneity Non-causality

• People often use the causality test to test
  whether indep. Variable(s) are exogenous. Its a
  mistake.
• Exogeneity implies non-casality but not vice
  versa.
• Why? Read Cooley Leroy (1985, JME).

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Critics on VAR

• The VAR is the best forecasting model. But, its
  an atheoretical model.
• The ordering matters for the impulse response
  analysis and the innovation accounting.
• Sensitive to Omitted variables.
• Sensitive to the lag selection.
• Dimensionality problem.