Monetary Policy & Asset Prices
David Backus, Mikhail Chernov,
and Stanley Zin
NYU Macro Lunch | March 8, 2012
This version: March 5, 2012
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
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The problem
Model: forward-looking difference equation
yt
= λEt yt+1 + x1t + x2t
Questions
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How do we identify λ?
Do we need to observe the shocks (x1 , x2 )?
Are forecasts and asset prices helpful? Cross-equation restrictions?
Connection to methods that identify shocks?
Answers
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We need “enough” observables and structure
We need to observe something, but one shock can be enough
Forecasts, asset prices, and economic structure all help
Identifying λ and x closely related
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
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Example 1: independent shocks
Example 1: independent shocks
Independent moving average shocks
Vary what we observe
Apply to model with Taylor rule
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Monetary policy
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Example 1: independent shocks
Example 1: independent shocks
Model
yt
= λEt yt+1 + xt
xt
= α(L)wt ,
wt ∼ NID(0, 1)
[|λ| < 1, α one-sided and square-summable, w fundamental for x]
Solution (agents observe everything)
yt
= β(L)wt , βj (α, λ) = αj + λαj+1 + λ2 αj+2 · · ·
What do we need to estimate λ?
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
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Example 1: independent shocks
Observable shock
(x, y )
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Joint process but degenerate (same innovations)
Cross-equation restrictions on parameters (β depends on λ, α)
Observe variable y (always)
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Estimate β
Observe shock x
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Estimate α
Use mapping β(α, λ) to infer λ
λ overdetermined if x is MA(q) and q > 1
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
4 / 24
Example 1: independent shocks
Hidden shock
(x, y ) still joint process, but we don’t observe x
Observe variable y
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Estimate β
But without x, we can’t distinguish roles of α and λ
Do forecasts help?
ftk = Et yt+k
=
[
]
β(L)/Lk wt
+
[Nope, depends only on β, can’t tell us about (λ, α)]
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
5 / 24
Example 1: independent shocks
Partly observable shock
Model and solution
⇒
yt
= λEt yt+1 + x1t + x2t
xit
= αi (L)wit ,
yt
= β1 (L)w1t + β2 (L)w2t
(w1t , w2t ) ∼ NID(0, I )
Observe variable y and shock x1 — but not x2
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Estimate α1 and β1
Use mapping β1 (α1 , λ) to infer λ
Do forecasts help?
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They help us identify the wi ’s
Then we can recover x2 from the difference equation
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
6 / 24
Example 1: independent shocks
Model with Taylor rule
Model (p = inflation, i = nominal interest rate)
it = r + Et pt+1 + x1t
(Euler equation)
it = r + τ pt + x2t
xit = αi (L)wit ,
(Taylor rule)
(w1t , w2t ) ∼ NID(0, I )
Solution
pt
= β1 (L)w1t + β2 (L)w2t
it
= r + [β1 (L)/L]+ w1t + [β2 (L)/L]+ w2t + x1t
What do we need to estimate λ = τ −1 ?
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
7 / 24
Example 1: independent shocks
Taylor rule: observable shocks
(x1 , x2 , p, i)
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Joint process but degenerate (two-dimensional innovation)
Cross-equation restrictions on parameters
Observe shocks (x1 , x2 )
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Estimate αi ’s
Observe variables (p, i)
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Estimate βi ’s
Use mapping βi (αi , λ) to infer λ = τ −1
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
8 / 24
Example 1: independent shocks
Taylor rule: hidden shock (“Cochrane version”)
Shocks: x1 = 0, x2 hidden
Observe variable p
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Estimate β2 (possible because shock is one dimensional)
Can’t distinguish roles of α2 and λ
Does extra variable i help?
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No, contains no information not in p
Do forecasts help?
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No, they add no information beyond p
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
9 / 24
Example 1: independent shocks
Taylor rule: partly observable shock (“Gertler version”)
Observe variables (p, i) and shock x1
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Estimate α1 and β1
Use mapping β1 (α1 , λ) to infer λ
Key insight
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i gives us an additional observable when x1 ̸= 0
How might we observe x1 ?
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Observe Et pt+1 directly — or derive from VAR in (p, i)
Then back x1 out of Euler equation
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Monetary policy
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Example 1: independent shocks
Example 1: summary
Identification may be possible with partial observability
Observability facilitated by forecasts
Independence needs further investigation
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Monetary policy
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Example 2: correlated shocks
Example 2: correlated shocks
Correlated shocks in state-space model
Vary what we observe
Apply to model with Taylor rule
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Monetary policy
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Example 2: correlated shocks
Example 2: correlated shocks
Model
yt
= λEt yt+1 + e ⊤ xt
xt+1 = Axt + Cwt+1 ,
{wt } ∼ NID(0, I )
[|λ| < 1, A stable & “regular”]
Solution
yt
= e ⊤ (I − λA)−1 xt = β ⊤ xt
What do we need to estimate λ?
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
13 / 24
Example 2: correlated shocks
Observable state and shock
Observe state x and vector e (hence shock e ⊤ x)
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Estimate A
Observe variable y
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Estimate β, use A to infer λ
λβ ⊤
= (β − e)⊤ A−1
[λ typically overdetermined]
Forecasts redundant
ftk = Et yt+k
Backus, Chernov, & Zin (NYU, LSE)
= β ⊤ Ak xt
Monetary policy
14 / 24
Example 2: correlated shocks
Digression on hidden shock & state
What’s hidden, state x or shock e ⊤ x?
One line of thought (Hansen-Sargent)
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Only part of state observed
→ may not be able to back out innovations [??]
We suggest: observe the state, not the shock
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Lots of econometric approaches
Or: build it from forecasts of y
ft
= [yt , ft1 , · · · , ftn−1 ]⊤
ft spans state x if A is regular and n = dim(x)
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
15 / 24
Example 2: correlated shocks
Hidden shock
Shock e ⊤ x not observed (e not known)
Observe variable y , state x
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From y : estimate β
From x: estimate A
But since we don’t know e, can’t infer λ
Do forecasts help?
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They span the state, not otherwise helpful
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
16 / 24
Example 2: correlated shocks
Partly observable shock
Model
⇒
yt
= λEt yt+1 + e1⊤ xt + e2⊤ xt
yt
= (e1 + e2 )⊤ (I − λA)−1 xt = β ⊤ xt
Observe variable y , state x
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From y : estimate β
From x: estimate A
Is it enough to observe shock e1⊤ x?
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Can’t easily disentangle effects of e1 and e2
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
17 / 24
Example 2: correlated shocks
Partly observable shock (continued)
Reminder: solution is
β ⊤ = (e1 + e2 )⊤ (I − λA)−1
What we know: β (n knowns)
What we don’t know: e1 , λ (n + 1 unknowns)
Needed: one or more restrictions
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Uncorrelated shocks
Other information that restricts the shocks
Tight economic structure
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
18 / 24
Example 2: correlated shocks
Model with Taylor rule
Model (p = inflation, i = nominal interest rate)
it = r + Et pt+1 + e1⊤ xt
it = r + τ pt +
(Euler equation)
e2⊤ xt
xt+1 = Axt + Cwt+1 ,
(Taylor rule)
{wt } ∼ NID(0, I )
Solution: pt = β ⊤ xt
β ⊤ = (e1 − e2 )⊤ (τ I − A)−1
What do we need to estimate λ = τ −1 ? One restriction on e2
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
19 / 24
Example 2: correlated shocks
Example 2: summary
Forecasts helpful in observing state
Identification then possible with
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Partial observability of shocks
Restriction(s) on hidden (monetary policy) shock
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
20 / 24
Example 3: bond pricing models
Representative agent model
Model
it = − log Et [mt+1 exp(−pt+1 )]
e2⊤ xt
it = r + τ pt +
log mt+1 = −δ − α log gt+1
(Euler equation)
(Taylor rule)
(pricing kernel)
log gt+1 = g + e1⊤ xt
xt+1 = Axt + Cwt+1 , {wt } ∼ NID(0, I )
Solution: pt = β ⊤ xt
β ⊤ = (αe1 − e2 )⊤ (τ I − A)−1
r
= δ + αg − (αe1 + β)⊤ CC ⊤ (αe1 + β)/2
Backus, Chernov, & Zin (NYU, LSE)
Monetary policy
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Example 3: bond pricing models
Representative agent model: identification
What do we need to estimate λ = τ −1 ?
Reminder: solution is
β ⊤ = (αe1 − e2 )⊤ (τ I − A)−1
r
= δ + αg − (αe1 + β)⊤ CC ⊤ (αe1 + β)/2
Identification strategies
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Observe log gt , e1
Long bond yields span state
Risk aversion α: match equity premium
One restriction on e2
Additional information from mean bond yields
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Monetary policy
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Summary and conclusion
Identification never a free lunch
We suggest
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Forecasts and bond yields aid observability of state
Additional restrictions can identify shock and “Taylor rule”
Representative agent bond pricing models can also help
The biggest challenge is not identification, but fit
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Monetary policy
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Related work (some of it)
Identifying monetary policy shocks
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Christiano-Eichenbaum-Evans, Cochrane, Hansen-Sargent,
Leeper-Sims-Zha
Factor models
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Sargent-Sims, Stock-Watson
Macro bond pricing models
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Gallmeyer-Hollifield-Zin, Piazzesi-Schneider, Rudebusch-Swanson,
Smith-Taylor ...
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Monetary policy
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