Economics 210c/236a Christina Romer Fall 2018 David Romer L ECTURE 3 The Effects of Monetary Changes: Statistical Identification September 5, 2018
I. S OME B ACKGROUND ON VAR S
A Two-Variable VAR Suppose the true model is: where Ξ΅ 1 t and Ξ΅ 2 t are uncorrelated with one another, with the contemporaneous and lagged values of the right-hand side variables, and over time.
Rewrite this as: or where
This implies where The assumptions of the model imply that the elements of π π’ are uncorrelated with π π’β1 . We can therefore estimate the elements of π² by estimating each equation of π π’ = π²π π’β1 + π π’ by OLS.
Extending to K Variables and N Lags The βtrue modelβ 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
An Obvious (at Least in Retrospect) Insight Consider where The elements of B and C are not identified. Thus, to make progress we need to make additional assumptions.
II. C HRISTIANO , E ICHENBAUM , AND E VANS : βT HE E FFECTS OF M ONETARY P OLICY S HOCKS : E VIDENCE FROM THE F LOW OF F UNDS β
Timing Assumptions The most common approach to identifying assumptions in β’ 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, β¦, β’ n , and assumes that variable 1 is not affected by any of the other 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; β¦; 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 β’ assumptions, or Cholesky identification .
Simplified Version of Christiano, Eichenbaum, and Evans Two variables, one lag: β’ The reduced form is: β’
From: Christiano, Eichenbaum, and Evans
From: Christiano, Eichenbaum, and Evans
From: Christiano, Eichenbaum, and Evans
From: Christiano, Eichenbaum, and Evans
Two Other Ways of Estimating the Effects of Monetary Policy Shocks under CEEβs Assumptions
Other Types of Restrictions to Make VARs Identified β’ Zero restrictions other than ordering assumptions. β’ Long-run restrictions. β’ Imposing coefficient restrictions motivated by theory. β’ β¦
III. R OMER AND R OMER : βA N EW M EASURE OF M ONETARY S HOCKS : D ERIVATION AND I MPLICATIONS β
Deriving Our New Measure β’ Derive the change in the intended funds rate around FOMC meetings using narrative and other sources. β’ Regress on Federal Reserve forecasts of inflation and output growth. β’ Take residuals as new measure of monetary policy shocks.
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, βA New Measure of Monetary Shocksβ
What kinds of thing are in the new shock series? β’ Unusual movements in funds rate because the Fed was also targeting other measures. β’ Mistakes based on a bad model of economy. β’ Change in tastes. β’ Political factors. β’ Pursuit of other objectives.
Miller Burns Volcker From: Romer and Romer, βA New Measure of Monetary Shocksβ
Evaluation of the New Measure β’ Key issue β is there useful information used in setting policy not contained in the Greenbook forecasts?
Digression: Kuttnerβs Alternative Measure of Monetary Shocks β’ 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.
From: Kenneth Kuttner, βMonetary Policy Surprises.β
Single-Equation Regression for Output y is the log of industrial production S is the new measure of monetary policy shocks D βs are monthly dummies From: Romer and Romer, βA New Measure of Monetary Shocksβ
Fitting this Specification into the Earlier Framework Suppose the true model is: π§ π’ = π 1 π§ π’β1 + π 1 π π’β1 + π π§π’ , (1) π π’ = π 2 π§ οΏ½ π’ + π 2 π οΏ½ π’ + π ππ’ , (2) where π§ οΏ½ and π οΏ½ are the forecasts as of period t , and Ξ΅ it is uncorrelated with all the other things on the right-hand side of (1) and (2) ( π§ π’β1 , π π’β1 , π§ οΏ½ π’ , π οΏ½ π’ , and π π§π’ ) . (2) implies: π π’β1 = π 2 π§ οΏ½ π’β1 + π 2 π οΏ½ π’β1 + π π , π’β1 . Substituting this in to (1) gives us: π§ π’ = π 1 π π , π’β1 + π π’ , where π π’ = π 1 π§ π’β1 + π 1 π 2 π§ οΏ½ π’β1 + π 2 π οΏ½ π’β1 + π 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 ( b 1 ).
Single-Equation Regression for Output Using the New Measure of Monetary Shocks Using the Change in the Actual Funds Rate From: Romer and Romer, βA New Measure of Monetary Shocksβ
Single-Equation Regression for Prices Using the New Measure of Monetary Shocks Using the Change in the Actual Funds Rate From: Romer and Romer, βA New Measure of Monetary Shocksβ
VAR Specification β’ Three variables: log of IP, log of PPI for finished goods, measure of monetary policy. β’ Monetary policy is assumed to respond to, but not to affect other variables contemporaneously. β’ We include 3 years of lags, rather than 1 as Christiano, Eichenbaum, and Evans do. β’ Cumulate shock to be like the level of the funds rate.
VAR Results
Comparison of VAR Results: Impulse Response Function for Output 1.0 0.5 0.0 -0.5 Funds Rate Percent -1.0 -1.5 -2.0 -2.5 Romer and Romer Shock -3.0 -3.5 0 4 8 12 16 20 24 28 32 36 40 44 48 Months after the Shock
Comparison of VAR Results: Impulse Response Function for Prices 1 Funds Rate 0 -1 Percent -2 -3 Romer and Romer Shock -4 -5 -6 0 4 8 12 16 20 24 28 32 36 40 44 48 Months after the Shock
Impulse Response Function of DFF to Shock 2.5 2.0 1.5 Percentage Points 1.0 0.5 0.0 -0.5 -1.0 0 4 8 12 16 20 24 28 32 36 40 44 48 Months after the Shock
Solid black line includes the early Volcker period, dashed blue line excludes it. From: Coibion, βAre the Effects of Monetary Policy Shocks Big or Small?β
IV. G ERTLER AND K ARADI : βM ONETARY P OLICY S URPRISES , C REDIT C OSTS , AND E CONOMIC A CTIVITY β
Key Features of Gertler and Karadiβs Approach β’ 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. β’ An IV approach to VARs: βexternal instruments.β
Surprise Changes in Measures of Monetary Policy around the Times of FOMC Decisions β’ 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. β’ Concerns? β’ Potentially leaves out a lot of useful variation. β’ Is a surprise change in monetary policy necessarily the same as a pure monetary policy shock?
Background on External Instruments: A NaΓ―ve Approach to IV Suppose we want to estimate β’ πΏ π§ π’ = οΏ½ π π π π’βπ + π π’ , π=0 and that we have a variable z t that we think is correlated with i t and uncorrelated with the e βs. It might be tempting to estimate the equation by IV, with β’ instrument list z t , z t β1 , β¦, z t β K . Concerns: β’ We think i t β k is affected by z t β k and not the other z βs. So, at β’ the very least, this approach creates a lot of inefficiency. Conjecture: This approach could magnify the bias caused β’ by small misspecification. How do we extend this to a VAR? β’
External Instruments in a Simple 2-Variable ModelβSet-Up Suppose the true model is: β’ π§ π’ = ππ π’ + π 11 π§ π’β1 + π 12 π π’β1 + π π§π’ , π π’ = πΏπ§ π’ + π 21 π§ π’β1 + π 22 π π’β1 + π ππ’ . The reduced form is: β’ π π’ = π²π π’β1 + π π’ , (where: π π’ β‘ π§ π’ 1 βπ , πΆ β‘ π 11 π 12 π π’ , Ξ β‘ C -1 B, β‘ π 22 , βπΏ 1 π 21 U t β‘ C -1 E t , πΉ π’ β‘ π π§π’ π ππ’ ).
External Instruments in a Simple 2-Variable Modelβ Using an Instrument β’ Suppose we have a variable z t that is correlated with π ππ’ and not systematically correlated with π π§π’ . β’ Let π£ π§π’ and π£ ππ’ be the two elements of U t βthat is, the reduced form innovations in y and i . β’ One can show that π£ π§π’ can be written in the form: π£ π§π’ = ππ£ ππ’ + π π§π’ . β’ So, a regression of π£ π§π’ on π£ ππ’ , using z t as an instrument allows us to estimate π .
Recommend
More recommend
