MFE Macroeconomics Week 3 Exercise

MFE Macroeconomics Week 3 Exercise The first row in the figure below shows monthly data for the Federal Funds Rate and CPI inflation for the period 199m1-18m8. 1 FFR CPI inflation 8 1 6 4 1 199 1995 5 1 15 199 1995 5 1 15 1 FFR HP trend.5 CPI inflation HP trend 8.4 6.3 4..1 199 1995 5 1 15 199 1995 5 1 15 FFR HP detrended CPI inflation HP detrended 1 1 1 1 199 1995 5 1 15 199 1995 5 1 15 The second row shows the Hodrick-Prescott trend with smoothing parameter λ 144, the standard value for monthly data. 1. Comment on the trends identified in the data and what they imply for the detrended data in the third row of the figure. We are interested in the effects of interest rate monetary policy shocks on inflation. To begin, we estimate an unrestricted bivariate VAR with 4 lags in R t HP detrended and π t demeaned but not detrended. 1

Rt 1.3. Rt 1 π t.31.45.5.3 π t 1 Rt +.3.18.. π t Rt 3 +.9..1.7 π t 3 Rt 4 +.6.3 π t 4 + e t with [.147.57 ] e t N ;.57.511 Assume that interest rates react to shocks to inflation with a lag this could be justified if the central bank does not have access to the current inflation data when it decides on policy, the Bernanke-Blinder 199 identification, in which case the residuals of the bivariate VAR are related to the fundamental disturbances by θ1 u1t e t. Calculate the coeffi cients θ 1, θ 3, θ 4 implied by the estimation. The Excel spreadsheet MFE_week3_class.xlsx available from the course webpage is already part-programmed to help you produce impulse response functions from the estimate parameters of the VAR and different identification schemes. 3. Use the spreadsheet to calculate the impulse response functions to interest rate shocks and inflation shocks, under the assumption that shocks to inflation do not affect the interest rate within the period. To do this you simply need to transfer the values of θ 1, θ 3, θ 4 you just calculated into cells C3, C4, D3 and D4 of the spreadsheet hint, D3 will be zero when inflation shocks do not contemporaneously affect interest rates. The spreadsheet checks whether your values of θ 1, θ 3, θ 4 are consistent with the variancecovariance matrix of residuals of the VAR in cells A13 and A14 i.e. have you done your calculation correctly. If they are then you should see some sensible impulse responses. What is the response of the economy to an interest rate shock? And to an inflation shock? Do these make intuitive sense? We now look at sign restrictions and suppose that we know that interest rate shocks have a positive effect on interest rates on impact this is incontrovertible and a negative effect on inflation on impact we could argue u t

about this - it s basically saying that tightening monetary policy is contractionary for inflation. Consider the set of rotational matrices indexed by λ [ π, π] that characterise all possible identification schemes. e t θ1 cos λ sin λ sin λ cos λ u1t u t 4. Write down two inequalities that must be satisfied such that interest rate shocks have a positive contemporaneous impact on interest rates and a negative contemporaneous impact on inflation. What is the range of λ for which both inequalities are satisfied? 5. Choose a value of λ comfortably within the admissible range, calculate what that value implies for the rotation matrix θ 1 θ θ 3 θ 4 θ1 cos λ sin λ sin λ cos λ and substitute these values of θ 1, θ, θ 3, θ 4 into cells C3, C4, D3 and D4 of the spreadsheet hint, D3 will no longer be zero to calculate the sign-restricted impulse response functions. If you have rotated correctly then the consistency check in cells A13 and A14 should not be a problem. 6. Compare and contrast what you find with your earlier results. We now replace the sign restrictions with an external instrument, the surprise movement in the interest rate on monetary policy decision dates, as measured by data from the fed funds futures market. This is the proxy for monetary policy shocks in Gertler and Karadi 15. It is shown below for 199m1-1m6..1 z.5.5.1.15..5.3 199 1995 5 1 15 Notice the big surprise interest rate cuts in April and May 1, which occurred even outside the usual 3

scheduled meetings. Here s CNN Money at the time. The covariance of the proxy with the VAR residuals is Ee 1t z t.4378 1 3 Ee t z t 6.73 1 4 Identification is achieved by assuming that the proxy co-varies with unexpected interest rate shocks u 1t but not with inflation shocks u 1t, as expected if the proxy captures monetary policy shocks. 7. Use the covariances of the proxy with the residuals of the VAR to derive a relationship between two of the parameters in the general form. e t θ1 θ u1t u t 8. Use the three restriction implied by the variance-covariance matrix of the VAR residuals to calculate all of θ 1, θ, θ 3, θ 4. Derive new impulse response functions with the spreadsheet, comparing and contrasting with those derived so far. 4

1. There is a clear downward trend in interest rates but not in inflation. This means that detrending matters for interest rates but not so much for inflation. Some would argue that the fall in interest rates is caused by other factors demographics and secular stagnation so removing the trend is valid.. From the lecture notes we have σ 1 θ 1.147, σ 1 θ 1 θ 3.57, σ θ 3 + θ 4.511. It follows that θ1 θ.11.469.11 3. The interest rate shock increases inflation, which is counterintuitive and we have a price puzzle. In addition, interest rates barely react to inflation shocks, if anything falling slightly. This is again counterintuitive as we would expect monetary policy to raise the interest rate after an inflation shock. 4. Directly from the lecture notes we need θ 1 cos λ > and θ 3 cos λ θ 4 sin λ <. From the previous calculations we know θ 1, θ 3, θ 4 so these become.11 cos λ > and.469 cos λ.11 sin λ <. The first inequality simply requires cos λ > so λ [ π/, π/]. A plot of the second inequality 5

.469 cos λ.11 sin λ is shown below, from which the relevant range is then λ [ tan 1.469.11, π/ ]. 5. We select λ π/4 as comfortably in the admissible range, in which case θ 1 θ.11 cos π/4 sin π/4 θ 3 θ 4.469.11 sin π/4 cos π/4.857.857.131.1895 6. The impulse response functions are 6

These look much better. A shock to the interest rate reduces inflation on impact and in the next month, whereas a positive inflation shock leads to an increase in the interest rate. 7. Lifting directly from the lecture notes, we have that z t is correlated only with u 1t and the covariances can be written E [ e t z t ] E [ θ1 θ u1t The ratio of the covariances pins down the ratio of θ 1 to θ 3. θ 1 Ee 1tz t θ 3 Ee t z t u t φu 1t + ν t.4378 1 3 4.141 6.73 1 4 8. From the variance-covariance matrix of the VAR residuals we have 3 restrictions θ1 θ θ1 θ 3 σ1 σ 1 θ θ 4 σ 1 σ θ 1 + θ σ 1 θ 1 θ 3 + θ θ 4 σ 1 θ 3 + θ 4 σ ] θ1 φ The restriction from the proxy is θ 1 γθ 3 where γ 4.141 so the three equations become θ 3 φ γ θ 3 + θ σ 1 γθ 3 + θ θ 4 σ 1 θ 3 + θ 4 σ Substituting in for θ 3 σ θ 4 γ σ θ 4 + θ σ 1 γ σ θ 4 + θ θ 4 σ 1 Substituting in for θ 1 θ 4 σ1 γ σ θ 4 means that θ4 satisfies γ 1 σ θ 4 + σ1 γ σ θ 4 σ 1 θ 4 which after some tedious algebra solves for θ 4 θ 4 σ 1 σ 1 γσ + γ σ γ σ σ 1 + σ 1 γ γ σ In our case the numeric value is θ 4.5. Without loss of generality we select the positive root θ 4.4 so that positive inflation shocks cause an increase in inflation selecting the negative root is equally valid, but then the impulse response functions would be drawn for an unexpected fall in the interest rate. From the remaining formulae we get θ 1 θ.1145.399 θ 3 θ 4.85.4 7

The resulting impulse response functions are We get some partial success on the price puzzle. Inflation does fall on impact, although only for 1 period. The advantage of the proxy method over the sign restrictions is that we are not creating nice impulse response functions by construction. The response of interest rates to an inflation shock is positive, as under sign restrictions but not with the causal ordering. 8