Comments on Invertibility and
the Virtues of Differencing
by Albert Marcet
George Tauchen, un Minnesoto de Sims
Duke University
Macroeconomics and Reality, 25 years Later
Los Minnesotos de Sims, 2005
Outline
1. Overview
2. Comments
George Tauchen un Minnesoto de Sims
2
Macroeconomics and Reality, 25 years Later
Los Minnesotos de Sims, 2005
Overview: Radical Approach
• Difference all of the seemingly non-stationary series
• Run the VAR
• Estimate the MA representation by inverting and
accumulating differences
• Appears to work better than running everything in
levels or error-correction form
George Tauchen un Minnesoto de Sims
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Macroeconomics and Reality, 25 years Later
Los Minnesotos de Sims, 2005
Idea: project
yt = α +
p
X
Γk yt−k + ²t
k=1
and then invert
Γ̂p(L) = I − Γ̂p1L − ... − Γ̂ppLp
by doing the usual matrix polynomial long division.
George Tauchen un Minnesoto de Sims
4
Macroeconomics and Reality, 25 years Later
Los Minnesotos de Sims, 2005
After forming
h
i−1
Ĉp(L) = Γ̂p(L)
accumulate differences as appropriate and then
display the cumulated elements of Ĉp(L) as the
elements of the MA representation of the noninvertible levels variables.
George Tauchen un Minnesoto de Sims
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Macroeconomics and Reality, 25 years Later
Los Minnesotos de Sims, 2005
Comments
• How to impose stability on Γ̂p(L)? Include the
likelihood of the first p observations. Hard to do
in practice.
• Econometric theory: Iterated limits, first let T →
∞, then p → ∞, is a suspect paradigm. Explore
local to unity asymptotics.
• Something is amiss with the simulations.
George Tauchen un Minnesoto de Sims
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Macroeconomics and Reality, 25 years Later
Los Minnesotos de Sims, 2005
In the levels autoregression, say,
∆yt = α + ρyt−1 + ψ∆yt−1 + ²t
it makes a huge difference if the intercept α = 0 or
not. In the former case the OLS estimate satisfies
D
T (ρ̂ − 1) → DF, while in the latter case (α 6= 0)
the OLS estimates (α̂ ρ̂ ψ̂) are jointly asymptotically
√
T -Gaussian and the usual theory applies. See
Hamilton (1994, p. 497, 528).
George Tauchen un Minnesoto de Sims
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