Optimal Policy and Simple Rules: A Uni…ed Approach

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Optimal Policy and Simple Rules: A Uni…ed Approach

Øistein Røisland and Tommy Sveen

June 22, 2010

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Motivation

Two approaches:

Optimal policy (given a loss function) Simple instrument rules

Synthesis: Realistic Sensible Flexible

ØR&TS (NB) A uni…ed approach June 22, 2010 2 / 18

(3)

Motivation

Two approaches:

Optimal policy (given a loss function) Simple instrument rules

Synthesis:

Realistic

Sensible

Flexible

(4)

Motivation

Terminology

Model:

X _t ₊ ₁

Hx _t ₊ ₁ _j _t = A X t

x _t + Bi t + ^C

0 ε t + 1 , (1)

Optimal policy (targeting rules): L t = E t

∑ ∞ h = 0

β ^h Y _t ⁰ ₊ _h Λ Y _t + h . (2) where

Y _t = D 2 4

X t

x _t i _t

3 5 (3)

Instrument rule:

i t = f X _t

x t (4)

(5)

Motivation

Terminology

Model:

X _t ₊ ₁

Hx _t ₊ ₁ _j _t = A X t

x _t + Bi t + ^C

0 ε t + 1 , (1) Optimal policy (targeting rules):

L t = E t

∑ ∞ h = 0

β ^h Y _t ⁰ ₊ _h Λ Y _t + h . (2) where

Y _t = D 2 4

X t

x _t i _t

3 5 (3)

Instrument rule:

i t = f X _t

x t (4)

(6)

Motivation

Terminology

Model:

X _t ₊ ₁

Hx _t ₊ ₁ _j _t = A X t

x _t + Bi t + ^C

0 ε t + 1 , (1) Optimal policy (targeting rules):

L t = E t

∑ ∞ h = 0

β ^h Y _t ⁰ ₊ _h Λ Y _t + h . (2) where

Y _t = D 2 4

X t

x _t i _t

3 5 (3)

Instrument rule:

i t = f X _t

x t (4)

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Motivation

Terminology

Optimal instrument rule:

i _t = f X _t

x _t (5)

Simple instrument rule:

i _t ^s = f _s X _t x t

(6) i _t ^s = a _i i t 1 + ( 1 a _i ) [ a _π π t + a y y t + a y 1 y t 1 ]

Optimal simple rule:

i _t ^,s = f _s X _t

x _t (7)

(8)

Motivation

Terminology

Optimal instrument rule:

i _t = f X _t

x _t (5)

Simple instrument rule:

i _t ^s = f _s X _t x t

(6) i _t ^s = a _i i _t ₁ + ( 1 a _i ) [ a _π π t + a _y y _t + a _y ₁ y _t ₁ ]

Optimal simple rule:

i _t ^,s = f _s X _t

x _t (7)

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Motivation

Terminology

Optimal instrument rule:

i _t = f X _t

x _t (5)

Simple instrument rule:

i _t ^s = f _s X _t x t

(6) i _t ^s = a _i i _t ₁ + ( 1 a _i ) [ a _π π t + a _y y _t + a _y ₁ y _t ₁ ]

Optimal simple rule:

i _t ^,s = f _s X _t

x _t (7)

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Motivation

Simple vs optimal

Optimal policy approach

Minimizes loss (given a model)

Simple rules approach

Robustness (across models)

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Motivation

Simple vs optimal

Optimal policy approach

Minimizes loss (given a model) Simple rules approach

Robustness (across models)

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Motivation

Simple vs optimal

Taylor and Williams (2010):

”. . . Simple monetary policy rules are designed to take account of only the most basic principle of monetary policy of leaning against the wind of

in‡ation and output movements. Because they are not …ne tuned to speci…c assumptions, they are more robust to mistaken assumptions.”

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Motivation

Simple vs optimal

Simple rules approach Not realistic Svensson (2003)

Optimal policy approach

No role for simple rules

Asso, Kahn and Leeson (2009)

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Motivation

Simple vs optimal

Simple rules approach Not realistic Svensson (2003) Optimal policy approach

No role for simple rules Asso, Kahn and Leeson (2009)

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Motivation

Simple vs optimal

FOMC meeting in 1995 (Asso, Kahn and Leeson, 2009)

Janet Yellen:

"I do not disagree with the Greenbook strategy. But the Taylor rule and other rules. . . call for a rate in the 5 percent range, which is where we

already are. Therefore, I am not imagining another 150 basis points".

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A Uni…ed Approach

A proposal

Extended loss function:

L ˆ _t = ( 1 θ ) L _t + θ ( i _t i _t ^s ) ² (8) where

L _t = π ² _t + λy _t ² + δ ( i _t i _t ₁ ) ² (9) i _t ^s = a i i t 1 + ( 1 a i ) [ a _π π t + a y y t + a y 1 y t 1 ] (10)

Modi…ed loss functions:

Orphanides and Williams (2008)

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A Uni…ed Approach

A proposal

Extended loss function:

L ˆ _t = ( 1 θ ) L _t + θ ( i _t i _t ^s ) ² (8) where

L _t = π ² _t + λy _t ² + δ ( i _t i _t ₁ ) ² (9) i _t ^s = a i i t 1 + ( 1 a i ) [ a _π π t + a y y t + a y 1 y t 1 ] (10)

Modi…ed loss functions:

Orphanides and Williams (2008)

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A Uni…ed Approach

What do we do?

Three models of the US economy

Smets and Wouters (2007) Rudebusch and Svensson (1999) Taylor (1993)

Trade-o¤

Appropriate weight

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A Uni…ed Approach

What do we do?

Three models of the US economy Smets and Wouters (2007) Rudebusch and Svensson (1999) Taylor (1993)

Trade-o¤

Appropriate weight

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A Uni…ed Approach

What do we do?

Three models of the US economy Smets and Wouters (2007) Rudebusch and Svensson (1999) Taylor (1993)

Trade-o¤

Appropriate weight

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A Uni…ed Approach

What do we do?

Three models of the US economy Smets and Wouters (2007) Rudebusch and Svensson (1999) Taylor (1993)

Trade-o¤

Appropriate weight

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A Uni…ed Approach

Simulations

True loss function:

L _t = π ² _t + 0.5y _t ² + 0.1 ( i _t i _t ₁ ) ²

Optimal policy under commitment Approximated by:

i t =

∑ 4 j =1

b _i,j i _{t j} +

∑ 4 j=0

b _π,j π _{t j} +

∑ 4 j=0

b _y,j y _{t j}

Reference model: Smets and Wouters (2007)

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A Uni…ed Approach

Simulations

True loss function:

L _t = π ² _t + 0.5y _t ² + 0.1 ( i _t i _t ₁ ) ²

Optimal policy under commitment Approximated by:

i t =

∑ 4 j =1

b _i,j i _{t j} +

∑ 4 j=0

b _π,j π _{t j} +

∑ 4 j=0

b _y,j y _{t j}

Reference model: Smets and Wouters (2007)

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A Uni…ed Approach

Simulations

True loss function:

L _t = π ² _t + 0.5y _t ² + 0.1 ( i _t i _t ₁ ) ²

Optimal policy under commitment Approximated by:

i t =

∑ 4 j =1

b _i,j i _{t j} +

∑ 4 j=0

b _π,j π _{t j} +

∑ 4 j=0

b _y,j y _{t j}

Reference model: Smets and Wouters (2007)

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A Uni…ed Approach

Results – the classical Taylor rule

i _t = 1.5π _t + 0.5y _t

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A Uni…ed Approach

Results – the classical Taylor rule

i _t = 1.5π _t + 0.5y _t

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A Uni…ed Approach

Results – the classical Taylor rule

i _t = 1.5π _t + 0.5y _t

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A Uni…ed Approach

Results – the classical Taylor rule

i _t = 1.5π _t + 0.5y _t

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A Uni…ed Approach

Results – an optimal simple rule

i _t = 3.84π _t + 2.345y _t 0.008y _t ₁

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A Uni…ed Approach

Results – an optimal simple rule

i _t = 3.84π _t + 2.345y _t 0.008y _t ₁

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A Uni…ed Approach

Advantages

Realistic

Sensible Flexible

“The proposal to use simple instrument rules as mere guidelines is

incomplete and too vague to be operational” (Svensson, 2003)

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A Uni…ed Approach

Advantages

Realistic Sensible

Flexible

“The proposal to use simple instrument rules as mere guidelines is incomplete and too vague to be operational” (Svensson, 2003)

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A Uni…ed Approach

Advantages

Realistic Sensible Flexible

“The proposal to use simple instrument rules as mere guidelines is

incomplete and too vague to be operational” (Svensson, 2003)

(34)

A Uni…ed Approach

Advantages

Realistic Sensible Flexible

“The proposal to use simple instrument rules as mere guidelines is incomplete and too vague to be operational” (Svensson, 2003)

Optimal Policy and Simple Rules: A Uni…ed Approach

Optimal Policy and Simple Rules: A Uni…ed Approach

Øistein Røisland and Tommy Sveen

June 22, 2010

Motivation

Two approaches:

Optimal policy (given a loss function) Simple instrument rules

Synthesis: Realistic Sensible Flexible

Motivation

Two approaches:

Optimal policy (given a loss function) Simple instrument rules

Synthesis:

Realistic

Sensible

Flexible

Motivation

Terminology

Model:

X t + 1

Hx t + 1 j t = A X t

x t + Bi t + C

0 ε t + 1 , (1)

Optimal policy (targeting rules): L t = E t

∑ ∞ h = 0

β h Y t 0 + h Λ Y t + h . (2) where

Y t = D 2 4

X t

x t i t

3

5 (3)

Instrument rule:

i t = f X t

x t (4)

Motivation

Terminology

Model:

X t + 1

Hx t + 1 j t = A X t

x t + Bi t + C

0 ε t + 1 , (1) Optimal policy (targeting rules):

L t = E t

∑ ∞ h = 0

β h Y t 0 + h Λ Y t + h . (2) where

Y t = D 2 4

X t

x t i t

3

5 (3)

Instrument rule:

i t = f X t

x t (4)

Motivation

Terminology

Model:

X t + 1

Hx t + 1 j t = A X t

x t + Bi t + C

0 ε t + 1 , (1) Optimal policy (targeting rules):

L t = E t

∑ ∞ h = 0

β h Y t 0 + h Λ Y t + h . (2) where

Y t = D 2 4

X t

x t i t

3

5 (3)

Instrument rule:

i t = f X t

x t (4)

Motivation

Terminology

Optimal instrument rule:

i t = f X t

x t (5)

Simple instrument rule:

i t s = f s X t x t

(6) i t s = a i i t 1 + ( 1 a i ) [ a π π t + a y y t + a y 1 y t 1 ]

Optimal simple rule:

i t ,s = f s X t

x t (7)

X _t ₊ ₁

Hx _t ₊ ₁ _j _t = A X t

x _t + Bi t + ^C

β ^h Y _t ⁰ ₊ _h Λ Y _t + h . (2) where

Y _t = D 2 4

x _t i _t

i t = f X _t

X _t ₊ ₁

Hx _t ₊ ₁ _j _t = A X t

x _t + Bi t + ^C

β ^h Y _t ⁰ ₊ _h Λ Y _t + h . (2) where

Y _t = D 2 4

x _t i _t

i t = f X _t

X _t ₊ ₁

Hx _t ₊ ₁ _j _t = A X t

x _t + Bi t + ^C

β ^h Y _t ⁰ ₊ _h Λ Y _t + h . (2) where

Y _t = D 2 4

x _t i _t

i t = f X _t

i _t = f X _t

x _t (5)

i _t ^s = f _s X _t x t

(6) i _t ^s = a _i i t 1 + ( 1 a _i ) [ a _π π t + a y y t + a y 1 y t 1 ]

i _t ^,s = f _s X _t

x _t (7)

i _t = f X _t

x _t (5)

i _t ^s = f _s X _t x t

(6) i _t ^s = a _i i _t ₁ + ( 1 a _i ) [ a _π π t + a _y y _t + a _y ₁ y _t ₁ ]

i _t ^,s = f _s X _t

x _t (7)

i _t = f X _t

x _t (5)

i _t ^s = f _s X _t x t

(6) i _t ^s = a _i i _t ₁ + ( 1 a _i ) [ a _π π t + a _y y _t + a _y ₁ y _t ₁ ]

i _t ^,s = f _s X _t

x _t (7)

L ˆ _t = ( 1 θ ) L _t + θ ( i _t i _t ^s ) ² (8) where

L _t = π ² _t + λy _t ² + δ ( i _t i _t ₁ ) ² (9) i _t ^s = a i i t 1 + ( 1 a i ) [ a _π π t + a y y t + a y 1 y t 1 ] (10)

L ˆ _t = ( 1 θ ) L _t + θ ( i _t i _t ^s ) ² (8) where

L _t = π ² _t + λy _t ² + δ ( i _t i _t ₁ ) ² (9) i _t ^s = a i i t 1 + ( 1 a i ) [ a _π π t + a y y t + a y 1 y t 1 ] (10)