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A Theory of Data-Oriented Identification with a SVAR Application

Nikolay Arefiev Higher School of Economics August 20, 2015 The 11th World Congress of the Econometric Society, Montr´ eal, Canada and August 26, 2015 The 30th Annual Congress of the European Economic Association, Mannheim, Germany

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Acknowledgments

◮ This research was inspired by online course at Coursera

“Probabilistic Graphical Models” lectured by Daphne Koller.

◮ I thank Alina Arefeva, Antoine d’Autume, Svetlana

Bryzgalova, Jean-Bernard Chatelain, Boris Demeshev, Jean-Pierre Drugeon, Jean-Marie Dufour, Bulat Gafarov, James Hamilton, Oleg Itskhoki, Maarten Janssen, Grigory Kantorovich, Ramis Khabibulin, Sergey Kusnetzov, Jessie Li, Judea Pearl, Anatoly Peresetsky, Christopher Sims, Alain Trognon, Ilya Voskoboynikov, and Mark Watson for helpful comments and suggestions.

◮ I also thank participants of groups 7inR and Macroteam, and

• ther colleagues from University Paris-1 Panth´

eon-Sorbonne and from The Higher School of Economics for fruitful discussions.

◮ This study (research grant No 14-01-0088) was supported by

The National Research University Higher School of Economics’ Academic Fund Program in 2014/2015

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Research question

◮ Correlation does not imply causation;

corr(x, y) = 0 may be because: x y x y x u y

◮ How to measure causal effects?

◮ Use theory to formulate identification assumptions ◮ Competing theories may produce contradictory conclusions 3 / 31

Research question

◮ Correlation does not imply causation;

corr(x, y) = 0 may be because: x y x y x u y

◮ How to measure causal effects?

◮ Use theory to formulate identification assumptions ◮ Competing theories may produce contradictory conclusions

◮ Research question: Is this possible to find a set of inclusion

and exclusion restrictions, such that:

◮ Each restriction be not only theoretically justified but also

empirically verified

◮ Taken together, the restrictions be sufficient for the full

identification the structural model?

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Result

Linear Gaussian SEM with orthogonal structural shocks: AY = BZ + C + E Y and Z: vectors of endogenous and exogenous variables A, B, C : matrices and vector of parameters E : vector of structural shocks E

• EET

is diagonal The central result: For linear Gaussian models with orthogonal structural shocks I show how to use over-identifying restrictions:

• 1. To formulate many testable inclusions and exclusions (may be

more than the number of parameters in A, B and C)

• 2. To verify whether they suffice for the full testable identification

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Contributions to the theory of graphical identification

• 1. Graphical identification of cyclical models

◮ Instead of relying on recursiveness assumption, I use graphical

interpretations of:

◮ Rank condition ◮ Rubio-Ram´

ırez, Waggoner and Zha (2010)

◮ Partial identification

• 2. More testable identifying restrictions

AY = BZ + C + E

◮ Not only ATA, but also ATB, BTB, ATIR∞ and many others

• 3. Application example: SVAR monetary model for the US

economy

◮ Lucas’ Islands economy assumption fully identifies the model

using only testable restrictions, producing no IRF anomalities

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Plan

• 1. The central idea in a simple example
• 2. Potential pitfalls and recipes
• 3. Application: SVAR monetary model for the US Economy

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• 1. Example:

How the method works

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Example: Demand and Supply

q + αp = c1 + γzd + εd q − βp = c2 + δzs + εs q and p are the quantity and the price zd and zs are some demand and supply determinants

Table : Identifying restrictions

Instrument Exclusions Inclusions zd zd → supply zd → demand zs zs → demand zs → supply My result: E(εdεs) = 0 ⇒ each identifying restriction is testable

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The key tool

◮ The (generalized) moral graph:

◮ can be estimated from the data in almost all parameter points ◮ may define a unique structural model 9 / 31

Moral graph

q + αp = c1 + γzd + εd q − βp = c2 + δzs + εs

◮ The nodes: X =

• p, q, zd, zs

zd q p zs

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Moral graph

q + αp = c1 + γzd + εd q − βp = c2 + δzs + εs

◮ The nodes in the example: X =

• p, q, zd, zs

◮ Each structural equation produces a clique

zd q p zs

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Moral graph

q + αp = c1 + γzd + εd q − βp = c2 + δzs + εs

◮ The nodes in the example: X =

• p, q, zd, zs

◮ Each structural equation produces a clique

zd q p zs

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Moral graph

q + αp = c1 + γzd + εd q − βp = c2 + δzs + εs

◮ The nodes in the example: X =

• p, q, zd, zs

◮ Each structural equation produces a clique

zd q p zs

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This MG defines a unique structural model

zd q p zs

◮ Clique Cover Problem (CCP): find as few cliques as possible

to cover the entire graph

◮ In this example the CCP has a unique solution with 2 cliques.

◮ Each cliques includes the variable from one structural equation. ◮ MG uniquely defines the structural equations 11 / 31

Testing individual edges of the moral graph

AY = BZ + C + E

◮ Let X = {Y , Z} ◮ For two variables {xi, xj}, where at least one is endogenous:

xixj ∈ MG ⇔ corr(xi, xj|X−i,j) = 0 almost everywhere

◮ More general restrictions (for larger moral graphs):

◮ ATA, ATB, BTB, ATIR∞ are the same for all observationally

equivalent models

◮ Each entry of these matrices is associated with an edge in MG 12 / 31

How the method works

The moral graph:

◮ can be estimated from the data almost everywhere ◮ may define a unique structural model

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• 2. Pitfalls and recipes

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Is the Generic Assumption crucial?

zd q p zs

◮ May be unsatisfied where 2 or more cliques overlap ◮ If theory available: do not test qp ◮ No theory: may be a problem

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Is sequential hypothesis testing (SHT) a problem?

zd q p zs

◮ If theory available: do not remove insignificant blue edges ◮ No theory: may be a problem

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Correlated structural shocks

◮ E(EEd) is not diagonal ◮ Every edge is significant ⇒ identification rejected ◮ Insignificant edges present ⇒ misidentification

◮ Limitation of the scientific approach 17 / 31

The theory is wrong

zd q p zs

◮ Edge zdzs is present and significant ⇒ identification rejected ◮ Edge zdzs is present but insignificant ⇒ misidentification

◮ Limitation of the scientific approach 18 / 31

Potential pitfalls and recipes

◮ No prior theory ⇒ unreliable results ◮ With a theory:

◮ The method inherits all limitation of the scientific approach ◮ Do not rely on the tests associated with the edges for which

the generic assumption may be not satisfied

◮ Do not remove insignificant edges if the theory predicts that

they may be present

◮ With these reservations, this is a reliable approach of

data-oriented identification

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Application example

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SVAR monetary model for the US economy

◮ Objective: to estimate the Taylor rule and the IRFs for the

monetary policy shocks

◮ US data 1967Q1:2007Q4; no unconventional policy ◮ Plan for this section:

• 1. Choosing variables to include
• 2. Choosing identification restrictions
• 3. Estimated IRFs

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• 1. Which variables to include? Models with three variables

◮ r is the federal interest rate

g is the GDP growth rate π is the GDP deflator inflation rate

◮ Problem: no direct significant effect of r on π

r g π “−” may be “±” estimated “−”

◮ If {r → π mediated by g} ⇒ {biased IRF, price puzzle} ◮ Weak or irrelevant? Ill identification in either case

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Considered models with 3 and 4 variables

r Federal interest rate + + + + c The capacity utilization rate

• +

g The GDP growth rate + + + + π The GDP deflator inflation rate +

• +

+ πc The commodity price inflation rate

• +

+

• Testable restrictions on A, B, and IR∞

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Model with r, c, g, π

◮ r is the federal interest rate

c is the capacity utilization g is the GDP growth rate π is the GDP deflator inflation r g c π − +

◮ The mediators can distinguish between the AD and AS shocks

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From 4 to 6 variables

◮ New problem in 4-vars model: In response to a restrictive MP

shock, c jumps up and only then starts to decrease

◮ May be because of confounding

◮ Inclusion of the unemployment rate u makes the jump

insignificant;

◮ Inclusion of u and πc completely eliminates the jump.

◮ The final model:

r is the federal interest rate; u is the unemployment rate; c is the capacity utilization rate; g is the GDP growth rate; π is the GDP deflator inflation rate; πc is the commodity price inflation rate.

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Theory for identification: Lucas’ Islands economy

Example:

◮ Full information new Keynesian Phillips curve (NKPC):

πt = βEtπt+1 + . . .

◮ πt adjacent to each contemporaneous variable

◮ NKPC augmented with the Lucas’s Island assumption justified

by rational inattention: πt = βE (πt+1|ct, gt, πt, πc) + . . .

◮ πt adjacent to ct, gt, πc ◮ May be not adjacent to rt, ut 26 / 31

Estimated Moral Graph

r u c g π πc Lr Lu Lc Lg Lπ Lπc L2r L2u L2c L2g L2π L2πc 0.5 < q and p ≤ 0.5 0.1 < p and q ≤ 0.5 0.1 < q and p ≤ 0.1 0.05 < q and q ≤ 0.1 0.01 < q and q ≤ 0.05 0.005 < q and q ≤ 0.01 q ≤ 0.005

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Rejecting the full information hypothesis

◮ Arefiev, Kusnetzov (2015) estimate MG for the real data and

for the Smets Wouters’ (2007) model generated data

◮ Much more dense MG for DSGE data ◮ Consider edge consumption - investment: ◮ Absent for real data, present for DSGE-generated data ◮ Disappears when Lucas’ assumption introduced

◮ The full information hypothesis is rejected in the Smets and

Wouters’ model

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Estimated response to restrictive monetary policy shock

5 10 15 20 −0.4 0.0 0.4 0.8

Interest rate

5 10 15 20 −0.1 0.0 0.1 0.2

Unemployment

5 10 15 20 −0.8 −0.4 0.0 0.2

Capacity utilization

5 10 15 20 −0.6 −0.4 −0.2 0.0 0.2

GDP Growth

5 10 15 20 −0.4 −0.3 −0.2 −0.1 0.0

Inflation

5 10 15 20 −0.8 −0.6 −0.4 −0.2 0.0

Commodity price inflation

Quarters

◮ Gray zone: outside 90% bootstrap confidence intervals ◮ Dashed lines: ±1σ

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All IRFs

5 10 15 20 −0.4 0.2 0.8

Interest rate Restrictive monetary policy shock

5 10 15 20 −0.1 0.1

Unemployment

5 10 15 20 −1.0 −0.4 0.2

Capacity utilization

5 10 15 20 −0.6 −0.2 0.2

GDP Growth

5 10 15 20 −0.4 −0.2 0.0

Inflation

5 10 15 20 −0.8 −0.4 0.0

Commodity price inflation

5 10 15 20 −3 −1 1

Unidentified unemployment shock

5 10 15 20 −0.5 0.5 1.5 5 10 15 20 −1 1 3 5 5 10 15 20 −3 −1 1 5 10 15 20 0.0 1.0 2.0 5 10 15 20 −2 2 4 5 10 15 20 0.0 0.4 0.8

Positive AD shock

5 10 15 20 −0.3 0.0 0.2 5 10 15 20 −0.5 0.5 1.5 5 10 15 20 −0.5 0.5 5 10 15 20 −0.1 0.2 0.5 5 10 15 20 0.0 1.0 5 10 15 20 −0.05 0.10

Positive AS shock

5 10 15 20 −0.08 0.00 5 10 15 20 −0.2 0.0 0.2 5 10 15 20 −0.2 0.4 0.8 5 10 15 20 −0.05 0.05 5 10 15 20 0.0 0.4 5 10 15 20 0.0 0.4

GDP deflator Inflation

5 10 15 20 0.0 0.2 5 10 15 20 −0.6 0.0 0.4 5 10 15 20 −0.4 0.0 0.4 5 10 15 20 −0.2 0.4 0.8 5 10 15 20 0.0 1.0 5 10 15 20 −0.10 0.00

Stagflation shock

5 10 15 20 −0.04 0.02 5 10 15 20 −0.15 0.00 5 10 15 20 −0.3 −0.1 5 10 15 20 −0.04 0.02 5 10 15 20 −0.2 0.4 0.8

Response Impulse

Quarters

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Conclusions

◮ A theory of data-oriented identification for cyclical models ◮ A reliable approach when prior theory is available ◮ Application example: full testable identification produces IRF

consistent with the macroeconomic theory

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