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Economics 210c/236a Christina Romer Fall 2018 David

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LECTURE 3

The Effects of Monetary Changes: Statistical Identification September 5, 2018

Economics 210c/236a Christina Romer Fall 2018 David Romer

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  • I. SOME BACKGROUND ON VARS
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A Two-Variable VAR

Suppose the true model is: where Ξ΅1t and Ξ΅2t are uncorrelated with one another, with the contemporaneous and lagged values of the right-hand side variables, and over time.

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Rewrite this as:

  • r

where

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This implies where The assumptions of the model imply that the elements of 𝑉𝑒 are uncorrelated with π‘Œπ‘’βˆ’1. We can therefore estimate the elements of 𝛲 by estimating each equation of π‘Œπ‘’ = π›²π‘Œπ‘’βˆ’1 + 𝑉𝑒 by OLS.

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Extending to K Variables and N Lags

The β€œtrue model” takes the form: where C is K x K, X is K x 1, B is K x K, and E is K x 1. This leads to: where

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Consider where The elements of B and C are not identified. Thus, to make progress we need to make additional assumptions.

An Obvious (at Least in Retrospect) Insight

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  • II. CHRISTIANO, EICHENBAUM, AND EVANS: β€œTHE

EFFECTS OF MONETARY POLICY SHOCKS: EVIDENCE FROM

THE FLOW OF FUNDS”

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  • The most common approach to identifying assumptions in

VARs is to impose zero restrictions that give the matrix of contemporaneous effects, C, a recursive structure.

  • Specifically, one chooses an ordering of the variables, 1, 2, …,

n, and assumes that variable 1 is not affected by any of the

  • ther variables within the period and potentially affects all

the others within the period; variable 2 is not affected by any variables other than variable 1 within the period and potentially affects all the others except variable 1; …; variable n is potentially affected by all the other variables within the period and does not affect any of the others.

  • Such assumptions are known as timing or ordering

assumptions, or Cholesky identification.

Timing Assumptions

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  • Two variables, one lag:
  • The reduced form is:

Simplified Version of Christiano, Eichenbaum, and Evans

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From: Christiano, Eichenbaum, and Evans

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From: Christiano, Eichenbaum, and Evans

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From: Christiano, Eichenbaum, and Evans

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From: Christiano, Eichenbaum, and Evans

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Two Other Ways of Estimating the Effects of Monetary Policy Shocks under CEE’s Assumptions

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Other Types of Restrictions to Make VARs Identified

  • Zero restrictions other than ordering assumptions.
  • Long-run restrictions.
  • Imposing coefficient restrictions motivated by

theory.

  • …
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  • III. ROMER AND ROMER: β€œA NEW MEASURE OF

MONETARY SHOCKS: DERIVATION AND IMPLICATIONS”

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Deriving Our New Measure

  • Derive the change in the intended funds rate around

FOMC meetings using narrative and other sources.

  • Regress on Federal Reserve forecasts of inflation and
  • utput growth.
  • Take residuals as new measure of monetary policy

shocks.

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Regression Summarizing Usual Fed Behavior

ff is the federal funds rate y is output; Ο€ is inflation; u is the unemployment rate ~ over a variable indicates a Greenbook forecast

From: Romer and Romer, β€œA New Measure of Monetary Shocks”

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What kinds of thing are in the new shock series?

  • Unusual movements in funds rate because the Fed

was also targeting other measures.

  • Mistakes based on a bad model of economy.
  • Change in tastes.
  • Political factors.
  • Pursuit of other objectives.
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From: Romer and Romer, β€œA New Measure of Monetary Shocks”

Burns Volcker Miller

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Evaluation of the New Measure

  • Key issue – is there useful information used in setting

policy not contained in the Greenbook forecasts?

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Digression: Kuttner’s Alternative Measure of Monetary Shocks

  • Get a measure of unexpected changes in the federal

funds rate by (roughly) comparing the implied change indicated by fed funds futures and the actual change.

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From: Kenneth Kuttner, β€œMonetary Policy Surprises.”

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Single-Equation Regression for Output

y is the log of industrial production S is the new measure of monetary policy shocks D’s are monthly dummies

From: Romer and Romer, β€œA New Measure of Monetary Shocks”

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Fitting this Specification into the Earlier Framework

Suppose the true model is: 𝑧𝑒 = 𝑏1π‘§π‘’βˆ’1 + 𝑐1π‘—π‘’βˆ’1 + πœπ‘§π‘’, (1) 𝑗𝑒 = 𝑏2𝑧 𝑒 + 𝑐2𝜌 𝑒 + πœπ‘—π‘’, (2) where 𝑧 and 𝜌 are the forecasts as of period t, and Ξ΅it is uncorrelated with all the

  • ther things on the right-hand side of (1) and (2) (π‘§π‘’βˆ’1, π‘—π‘’βˆ’1, 𝑧

𝑒, 𝜌 𝑒, and πœπ‘§π‘’). (2) implies: π‘—π‘’βˆ’1 = 𝑏2𝑧 π‘’βˆ’1 + 𝑐2𝜌 π‘’βˆ’1 + πœπ‘—,π‘’βˆ’1. Substituting this in to (1) gives us: 𝑧𝑒 = 𝑐1πœπ‘—,π‘’βˆ’1 + πœ€π‘’, where πœ€π‘’ = 𝑏1π‘§π‘’βˆ’1 + 𝑐1 𝑏2𝑧 π‘’βˆ’1 + 𝑐2𝜌 π‘’βˆ’1 + πœπ‘§π‘’. Under the assumptions of the model, Ξ΄t is uncorrelated with Ξ΅i,t-1, and so we can estimate this equation by OLS and recover the effect of i on y (b1).

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From: Romer and Romer, β€œA New Measure of Monetary Shocks”

Single-Equation Regression for Output

Using the New Measure of Monetary Shocks Using the Change in the Actual Funds Rate

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From: Romer and Romer, β€œA New Measure of Monetary Shocks”

Single-Equation Regression for Prices

Using the New Measure of Monetary Shocks Using the Change in the Actual Funds Rate

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VAR Specification

  • Three variables: log of IP, log of PPI for finished

goods, measure of monetary policy.

  • Monetary policy is assumed to respond to, but not to

affect other variables contemporaneously.

  • We include 3 years of lags, rather than 1 as

Christiano, Eichenbaum, and Evans do.

  • Cumulate shock to be like the level of the funds rate.
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VAR Results

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  • 3.5
  • 3.0
  • 2.5
  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 4 8 12 16 20 24 28 32 36 40 44 48 Percent Months after the Shock

Comparison of VAR Results: Impulse Response Function for Output

Funds Rate Romer and Romer Shock

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  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 4 8 12 16 20 24 28 32 36 40 44 48 Percent Months after the Shock

Comparison of VAR Results: Impulse Response Function for Prices

Funds Rate Romer and Romer Shock

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  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0 2.5 4 8 12 16 20 24 28 32 36 40 44 48 Percentage Points Months after the Shock

Impulse Response Function

  • f DFF to Shock
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From: Coibion, β€œAre the Effects of Monetary Policy Shocks Big or Small?”

Solid black line includes the early Volcker period, dashed blue line excludes it.

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  • IV. GERTLER AND KARADI: β€œMONETARY POLICY

SURPRISES, CREDIT COSTS, AND ECONOMIC ACTIVITY”

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Key Features of Gertler and Karadi’s Approach

  • A different strategy for trying to isolate useful

identifying variation in monetary policy: surprise changes in measures of monetary policy around the times of FOMC decisions.

  • An IV approach to VARs: β€œexternal instruments.”
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Surprise Changes in Measures of Monetary Policy around the Times of FOMC Decisions

  • The idea that changes in financial market variables in

a very short window around an FOMC announcement are almost entirely responses to the announcement seems very reasonable.

  • Concerns?
  • Potentially leaves out a lot of useful variation.
  • Is a surprise change in monetary policy necessarily

the same as a pure monetary policy shock?

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  • Suppose we want to estimate

𝑧𝑒 = π‘π‘™π‘—π‘’βˆ’π‘™

𝐿 𝑙=0

+ 𝑓𝑒, and that we have a variable zt that we think is correlated with it and uncorrelated with the e’s.

  • It might be tempting to estimate the equation by IV, with

instrument list zt, ztβˆ’1, …, ztβˆ’K.

  • Concerns:
  • We think itβˆ’k is affected by ztβˆ’k and not the other z’s. So, at

the very least, this approach creates a lot of inefficiency.

  • Conjecture: This approach could magnify the bias caused

by small misspecification.

  • How do we extend this to a VAR?

Background on External Instruments: A NaΓ―ve Approach to IV

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  • Suppose the true model is:

𝑧𝑒 = πœ„π‘—π‘’ + 𝑐11π‘§π‘’βˆ’1 + 𝑐12π‘—π‘’βˆ’1 + πœπ‘§π‘’, 𝑗𝑒 = 𝛿𝑧𝑒 + 𝑐21π‘§π‘’βˆ’1 + 𝑐22π‘—π‘’βˆ’1 + πœπ‘—π‘’.

  • The reduced form is:

π‘Œπ‘’ = π›²π‘Œπ‘’βˆ’1 + 𝑉𝑒, (where: π‘Œπ‘’ ≑ 𝑧𝑒 𝑗𝑒 , Ξ  ≑ C-1B, ≑ 1 βˆ’πœ„ βˆ’π›Ώ 1 , 𝐢 ≑ 𝑐11 𝑐12 𝑐21 𝑐22 , Ut ≑ C-1Et, 𝐹𝑒 ≑ πœπ‘§π‘’ πœπ‘—π‘’ ).

External Instruments in a Simple 2-Variable Modelβ€”Set-Up

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  • Suppose we have a variable zt that is correlated with

πœπ‘—π‘’ and not systematically correlated with πœπ‘§π‘’.

  • Let 𝑣𝑧𝑒 and 𝑣𝑗𝑒 be the two elements of Utβ€”that is,

the reduced form innovations in y and i.

  • One can show that 𝑣𝑧𝑒 can be written in the form:

𝑣𝑧𝑒 = πœ„π‘£π‘—π‘’ + πœπ‘§π‘’.

  • So, a regression of 𝑣𝑧𝑒 on 𝑣𝑗𝑒, using zt as an

instrument allows us to estimate πœ„. External Instruments in a Simple 2-Variable Modelβ€” Using an Instrument

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  • Once we know πœ„, we can find Ξ³.
  • And once we know Ξ³, we know all the elements of C (the

matrix of contemporaneous coefficients).

  • This allows us to go from estimates of Ξ  (the reduced form

relationship between (yt,it) and (ytβˆ’1,itβˆ’1), which we can estimate by OLS) to estimates of B (the causal effect of (ytβˆ’1,itβˆ’1) on (yt,it)), using B = C Ξ .

  • Notice that we use only the variation in i associated with

variation in z to estimate the contemporaneous impact of i on y, but all the variation in i to estimate the dynamics.

External Instruments in a Simple 2-Variable Modelβ€” Using an Instrument (continued)

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  • We haven’t discussed how to compute standard errors.

(Addressed briefly in n. 13 of Gertler-Karadi.)

  • Gertler and Karadi have multiple instruments, and they

consider various candidate measures of i.

  • Instruments: surprise in the current federal funds

futures rate; surprise in the 3-month ahead federal futures rate; surprises in in the 6-month, 9-month and 12-month ahead futures on 3-month Eurodollar deposits.

  • Candidate measures of i: the federal funds rate; the
  • ne-year government bond rate; the two-year

government bond rate.

External Instrumentsβ€”Complications

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  • β€œThe monetary shocks … have a standard deviation of
  • nly about 5 basis points. This β€˜power problem’

precludes … directly estimating their effect on future

  • utput” (Nakamura and Steinsson, 2018).
  • The way Gertler and Karadi avoid this problem is by

not directly estimating the effects of the shocks: they use the high frequency identification to find the contemporaneous effects, but the full variation to find the dynamics.

  • Concern: what if the b’s aren’t structural parameters?

Where Is Our Ability to Estimate the Impulse Response Functions Coming from?

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  • Suppose the IV estimation of 𝑣𝑧𝑒 = πœ„π‘£π‘—π‘’ +

πœπ‘§π‘’ leads to an estimate of πœ„ of zero.

  • In that case, the external instruments approach is

identical to the Christiano-Eichenbaum-Evans approach.

  • But the possibility of πœ„ β‰  0 was not our only (or even
  • ur main) concern about CEE. Most notably, the

possibility of forward-looking monetary policy was a larger one.

A Concrete Example of These Concerns

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From: Gertler and Karadi

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Gertler and Karadi’s Baseline VAR

  • Four variables: Log industrial production, log CPI,

β€œGilchrist-ZakrajΕ‘ek excess bond premium,” 1-year government bond rate.

  • Use one instrument: surprise in the 3-month ahead

federal futures rate;

  • Monthly data, 1979:7-2012:6.
  • Only partially identified: the external IV approach

allows Gertler and Karadi to identify the effects of monetary policy shocks (shocks to the 1-year rate), but not the effects of shocks to the other variables).

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From: Gertler and Karadi

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Gertler and Karadi’s Extended VARs

  • Add the mortgage spread, the commercial paper

spread, and (one at a time): the federal funds rate, the 2-year, 5-year, or 10-year government bond rate, market-based measures of expected inflation, various private sector interest rates or spreads, ….