TimeSeriesBook.pdf

Yt policy reaction curve x 2 t m t a 2 y t γ 21 y t

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yt policy reaction curve: X 2 t = m t = a 2 y t + γ 21 y t - 1 + γ 22 m t - 1 + v mt Note the contemporaneous dependence of y t on m t in the AD-curve and a corresponding dependence of m t on y t in the policy reaction curve. These equations are typically derived from economic reasoning and may characterize a model explicitly derived from economic theory. In statistical terms the simultaneous equation system is called the structural form . The error terms { v yt } and { v mt } are interpreted as demand shocks and money supply shocks, respectively. They are called structural shocks and are assumed to follow a multivariate white noise process: V t = v yt v mt WN (0 , Ω) with Ω = ω 2 y 0 0 ω 2 m . Note that the two structural shocks are assumed to be contemporaneously uncorrelated which is reflected in the assumption that Ω is a diagonal matrix. This assumption in the literature is uncontroversial. Otherwise, there would remain some unexplained relationship between them. The structural shocks can be interpreted as the statistical analog of an experiment in the natural sciences. The experiment corresponds in this case to a shift of the AD- curve due to, for example, a temporary non-anticipated change in government expenditures or money supply. The goal of the analysis is then to trace the reaction of the economy, in our case represented by the two variables { y t } and { m t } , to these isolated and autonomous changes in aggregate demand and money supply. The structural equations imply that the reaction is not restricted to contemporaneous effects, but is spread out over time. We thus represent this reaction by the impulse response function. We can write the system more compactly in matrix notation: 1 - a 1 - a 2 1 y t m t = γ 11 γ 12 γ 21 γ 22 y t - 1 m t - 1 + 1 0 0 1 v yt v mt or AX t = Γ X t - 1 + BV t where A = 1 - a 1 - a 2 1 , Γ = γ 11 γ 12 γ 21 γ 22 and B = 1 0 0 1 .
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280 CHAPTER 15. INTERPRETATION OF VAR MODELS Assuming that a 1 a 2 6 = 1, we can solve the above simultaneous equation system for the two endogenous variables y t and m t to get the reduced form of the model: X 1 t = y t = γ 11 + a 1 γ 21 1 - a 1 a 2 y t - 1 + γ 12 + a 1 γ 22 1 - a 1 a 2 m t - 1 + v yt 1 - a 1 a 2 + a 1 v mt 1 - a 1 a 2 = φ 11 y t - 1 + φ 12 m t - 1 + Z 1 t X 2 t = m t = γ 21 + a 2 γ 11 1 - a 1 a 2 y t - 1 + γ 22 + a 2 γ 12 1 - a 1 a 2 m t - 1 + a 2 v yt 1 - a 1 a 2 + v mt 1 - a 1 a 2 = φ 21 y t - 1 + φ 22 m t - 1 + Z 2 t . Thus, the reduced form is a VAR(1) model with error term { Z t } = { ( Z 1 t , Z 2 t ) 0 } . The reduced form can also be expressed in matrix notation as: X t = A - 1 Γ X t - 1 + A - 1 BV t = Φ X t - 1 + Z t where Z t WN(0 , Σ) with Σ = A - 1 B B 0 A 0- 1 . Whereas the structural form represents the inner economic relations between the variables (economic model), the reduced form given by the VAR model summarizes their outer directly observable characteristics. As there is no un- ambiguous relation between the reduced and structural form, it is impossible to infer the inner economic relationships from the observations alone. This is known in statistics as the identification problem . Typically, a whole family of structural models is compatible with a particular reduced form. The models
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