Modeling Nonlinearities in Interest Rate Setting

The United Kingdom 1970:2006 riedel@wiwi.hu-berlin.de Institut für Statistik and Ökonometrie, C2 Humboldt-Universität zu Berlin June 27, 2007

Aim Estimation of an augmented Taylor Rule for interest rates setting using Smooth Transition Regression (STR) which

Aim Estimation of an augmented Taylor Rule for interest rates setting using Smooth Transition Regression (STR) which Accounts for possible nonlinearities in the BoEs reaction on inflation and output

Aim Estimation of an augmented Taylor Rule for interest rates setting using Smooth Transition Regression (STR) which Accounts for possible nonlinearities in the BoEs reaction on inflation and output Accounts for possible asymmetries in the BoEs reaction on inflation and output

Forward Looking Taylor Rule Implementing Nonlinearities Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results

Forward Looking Taylor Rule Implementing Nonlinearities Taylor Rule and Interest Rate Smoothing I Taylor (1993): linear monetary policy rule for the US economy 1987 to 1992: r t = rr t + π t + β(π t π t ) + γy t where r t is the interest rate, rr t the optimal real interest rate, π t the inflation target, π t the inflation and y t the output gap at time t. (β = 1.5 and γ = 0.5) Clarida et al. (1998): Future inflation expectation are considered by the BoE for todays interest rate decisions. Therefore we consider one-quarter ahead inflation expectation. Introduce interest rate smoothing (empirical regularity)

Forward Looking Taylor Rule Implementing Nonlinearities Taylor Rule and Interest Rate Smoothing II Forward looking policy rule: r t = r + β(e[π t+1 Ω t ] π ) + γ(e[y t Ω t ]) rr t = rr + (β 1)(E[π t+1 Ω t ] π ) + γ(e[y t Ω t ]) Note: β > 1 (if not real rate not influenced) and γ > 0 Interest rate smoothing actual rate r t : r t = (1 ρ)r t + ρr t 1 + v t Obtain equation to estimate using realized values for expectations: r t = (1 ρ) [α + βπ t+1 + γy t ] + ρr t 1 + ε t where ε t = (1 ρ)(β(π t+1 E[π t+1 Ω t ]) + γ(y t E[y t Ω t ])) + v t

Forward Looking Taylor Rule Implementing Nonlinearities Taylor Rule and Interest Rate Smoothing III Estimates for π (target inflation) and rr (long run eq. real rate) cannot be obtained separately but: Since α r βπ and r = rr + π : rr = (β 1)π + α or π = rr α β 1 Finally, the reduced form model used in estimation has the following form r t = α + β π t+1 + γ y t + ρ 1 r t 1 + ρ 2 r t 2 + ε t with α = (1 2 j=1 ρ j)α, β = (1 2 j=1 ρ j)β and γ = (1 2 j=1 ρ j)γ

Forward Looking Taylor Rule Implementing Nonlinearities Augmented Taylor Type Rule I Benchmark linear model extended by a nonlinear part Literature: Assenmacher-Wesche(2006) (Markow-Switching); Kharel (2006) (UK:1992:2005); Kesriyeli et al. (2004) (US,UK,GER: 1984:2002, backward, interest rate differences as transvars); Expect no sharp changes STR modelling approach first proposed by Teräsvirta Logistic transition function to model monetary policy changes, allowing for two and three different regimes

Forward Looking Taylor Rule Implementing Nonlinearities Augmented Taylor Type Rule II r t = α 0 + β 0 π t+1 + γ 0 y t + ρ 01 r t 1 + ρ 02 r t 2 + [α 1 + β 1 π t+1 + γ 1 y t + ρ 11 r t 1 + ρ 12 r t 2 ] G(γ T, c, s t ) + ε t, G(γ, c, s t ) = where t = 1,..., T, ε t iid(0, σ 2 ) s t c γ ( 1 + exp { γ 1 K (s t c k )}), γ > 0 (ident.restr.) k=1 transition variable (econ. variable, trend or const.) (K 1) vector of location parameters slope parameter

Forward Looking Taylor Rule Implementing Nonlinearities Smooth Transition Regression Framework II Behaviour of the transition function for K = 1 (one regime switch) G(γ, c, s t ) = (1 + exp { γ(s t c)}) 1 γ determines speed of transition and c the location

Forward Looking Taylor Rule Implementing Nonlinearities Smooth Transition Regression Framework III Behaviour of the Transition Function for K = 2 (two regime switches) ( { G(γ, c, s t ) = 1 + exp γ }) 1 2 k=1 (s t d c k ) (a) Changes in gamma (b) Changes in c

Forward Looking Taylor Rule Implementing Nonlinearities Estimation Method: Maximum Likelihood Under certain regularity conditions we can use l(φ, θ, γ, c; y t x t, s t ) = α T 2 ln σ2 1 2σ 2 u 2 t l φ(γ, c)! = 0 l θ(γ, c)! = 0 Find starting values for γ and c and apply iterative methods like NR- or BFGS-Algorithm Find starting values using grid search: 1. Fix γ and c estimate φ(γ, c) and θ(γ, c) and calculate RSS 2. repeat 1.) N-times (for N different combis of γ and c) 3. choose the combination with minimum RSS To obtain scale invariant γ, divide γ by ˆσ s

Forward Looking Taylor Rule Implementing Nonlinearities Test of No Remaining Nonlinearity STR model with additive nonlinearity y t = φ z t + θ z t G(γ 1, c 1, s 1t ) + ψ z t H(γ 2, c 2, s 2t ) + u t Hypothesis H 0 : γ 2 = 0 H 1 : γ 2 0 Third order Taylor expansion around γ 2 = 0 yields y t = β 0z t + θ z t G(γ 1, c 1, s 1t ) + 3 β j( z t s j 2t ) + u t, with z t = (1, z t) j=1 H 0 : β 1 = β 2 = β 3 = 0 H 1 : β i 0 for at least one i

Forward Looking Taylor Rule Implementing Nonlinearities Test of Parameter Constancy Model with time dependent parameters (TV-STR) y t = φ(t) z t + θ(t)g(γ, c, s t ) z t + u t, u t N(0, σ 2 ) where φ(t) = φ + λ φ H φ (γ φ, c φ, t )and θ(t) = θ + λ θ H θ (γ θ, c θ, t ), t = t/t Hypothesis H 0 : γ φ = γ θ = 0 H 1 : γ φ > 0 or/and γ θ > 0

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results The Story 1973 end Bretton Woods + First Oil Crisis 1976 Thatcher: fight against inflation was announced 1979 Second Oil Crisis and 1979:M3 germany joins EMS/ EMS was founded 1980:1988 gulf war I 1990/91 gulf war II 1990:M10 UK joins EMS 1992 breakdown EMS I due to 1992M09 pound crisis 1992:M10 BoE starts targeting inflation (1-4 percent) 1997M5 UK target inflation (2.5 percent) (BoE operational autonomy) 1999 EURO+ECB (commitment to price stability, without explicit economic goals), EMS II

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Dataset I Short-term interest rate set by the BoE: 3-month Treasury Bill Rate (IMF IFS); TBR Inflation rate: year-on-year change in the s.a. RPI until 1992, 1992 onwards: exclude mortgage price (EcoWin Economics) Inflation gap: deviation of actual inflation from its target (target: 1970Q1:1991Q4 centered two year moving average of RPI, 1992Q1:2006:3 2.5 percent); RPIMIX Output: real GDP (OECD MEI) Output gap: 100*(real GDP-hptrend real GDP)/hptrend real GDP ; OUTDIFF Foreign interest rate: Federal funds rate (FFR) (IMF IFS) and the German overnight call money rate (CMR) (OECD MEI)

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Dataset II

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Unit Root Test Results ADF and KPSS tests suggests stationarity at least at a 10 percent for: TBR: (1970Q1:1992Q3, 1970Q1:98Q4) OUTDIFF: for each considered sample range RPIMIX: (1978Q1:2006Q2, 1992Q4:2006Q2) FFR: 1970Q1:1992Q3, 1970Q1:1998Q4 CMR: for each considered sample range But: assume that interest rates are stationary.

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Break Points Bootstrapped p-values, repl:1000 Break Point Test Sample Split Test 70Q1:06Q2 78Q1:06Q2 70Q1:06Q2 78Q1:06Q2 1977Q4 0.005-0.058-1992Q3 0.000 0.000 0.296 0.000 1998Q4 0.000 0.000 0.589 0.131 Samples: 1970Q1:2006Q2, 1978Q1:2006Q2, 1970Q1:1992Q3, 1970Q1:1998Q4, 1992Q4:2006Q2 supported by CUSUM analysis.

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Estimation Output: Linear Model I r t = ρ 1 r t 1 + ρ 2 r t 2 + (1 ρ 1 ρ 2 )(α + βπ t+1 + γy t ) + v t 70Q1:06Q2 78Q1:06Q2 70Q1:92Q3 92Q4:06Q2 α 5.409 3.872 8.986 4.209 ρ 1 1.065 0.867 1.035 1.363 ρ 2 0.158-0.179 0.449 β 0.409 0.857 0.146 0.419 γ 2.097 1.414 1.222 0.759 adj.r 2 0.923 0.953 0.827 0.921 Residual Tests JB 0.000 0.000 0.447 0.215 ARCH 0.000 0.004 0.002 0.344 AutoC yes no no no

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Estimation Output: Linear Model II Remarkably low coefficient for inflation and high coefficient for output gap Depend on estimation method and choice of instrument Cannot capture possible differences in CBs preferences No implementation of response to external shocks Coefficients in the linear model can be considered as simple averages over different regimes Thus, make the coefficients (systematically) change over time

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Linearity Tests 70Q1:06Q2 78Q1:06Q2 70Q1:92Q3 70Q1:98Q4 92Q4:06Q2 Trend LSTR1 LSTR1 LSTR1 LSTR1 LSTR1 OUTDIFF LSTR2 LSTR2 Linear Linear Linear RPIMIXf1 Linear LSTR1 Linear Linear Linear TBR(t-1) Linear LSTR1 Linear Linear Linear TBR(t-2) Linear - Linear Linear LSTR1 including FFR OUTDIFF LSTR1 LSTR2 Linear Linear Linear RPIMIXf1 Linear LSTR1 Linear Linear Linear/LSTR1 TBR(t-1) LSTR1 LSTR1 Linear Linear LSTR1/2 TBR(t-2) LSTR2 - LSTR2 LSTR1 LSTR1/- including CMR Trend LSTR2 LSTR1 LSTR1 LSTR2 LSTR1 OUTDIFF LSTR2 Linear Linear Linear Linear RPIMIXf1 Linear LSTR1 Linear Linear Linear TBR(t-1) LSTR1 LSTR1 Linear Linear Linear TBR(t-2) LSTR1 - Linear Linear LSTR1/-

Main Facts The Data, Unit Root Tests and Break-Point Analysis Linear Regression Results Choice of Transition Variable Based economic argumentation + supported by nonlinearity tests Future inflation π t+1 : High expected inflation BoE react stronger on changes in explanatories than in case of a low inflation regime (not necessarily increase in inflation coefficient). Output Gap y t+i : Higher gap causes stronger BoE reaction (maybe neg. more influencial) Lagged Interest Rates: Different reaction of the BoE with respect to past interest rate might be highly relevant, as the smoothing coefficient in the linear model is quite large. Trend: Transition over time to more restrictive reaction on inflationary pressure (change in the BoE preferences over time)

Estimation Output: Trend as Transition Variable 70Q1:06Q2 78Q1:06Q2 70Q1:92Q3 92Q4:06Q2 F linear part α 0 2.115 4.196 2.121 0.854 ρ 01 1.005 0.453 1.034 0.927 ρ 02 0.460-0.504-0.204 β 0 0.135 0.239 0.131 0.218 γ 0 0.186-0.149 0.171-0.156 ψ 0 - - - 0.289 nonlinear part α 1 1.778 4.283 0.788 1.670 ρ 11-0.204 0.444 0.547-0.134 ρ 12 0.460-0.525-0.130 β 1 0.165-0.022 0.317 - γ 1-0.080 0.582-0.249 0.447 ψ 1 - - - -0.156 γ T 0.237 0.100 0.167 17.631 c 1 40.534 44.311 41.535 22.96 adj.r 2 0.938 0.966 0.878 0.971

Remaining Nonlinearities, Parameter Constancy and Residual Analysis 70Q1:06Q2 78Q1:06Q2 70Q1:92Q3 92Q4:06Q2 F Residual Tests JB 0.000 0.050 0.135 0.911 ARCH 0.000 0.004 0.001 0.762 AutoC no no 10-14th - Remaining Nonlinearity: H 0 :no r t 1 0.037 0.745 0.587 0.686 r t 2 0.035-0.440 0.045 π t+1 0.000 0.206 0.020 0.141 y t 0.045 0.135 0.507 0.289 Parameter Constancy: H 0 :yes H1 0.006 0.727 0.749 0.005 H2 0.012 0.920 0.756 0.039 H3 0.100 0.686 0.860 0.461

Graphical Analysis trend as transvar, 7006

Grid Search Result

Coefficients over time: Output Gap

Coefficients over time: Inflation

Coefficients over time: Lagged Interest Rate

Estimation Output: OutDiff and Inflation as TransVars 70Q1:06Q2 O 78Q1:06Q2 O 78Q1:06Q2 I with CMR 92Q4:06Q2 I,F linear part α 0 0.645 7.454 0.448 1.244 2.398 ρ 01 0.971 0.487 0.915 0.772 0.322 ρ 02-0.119 - - - - β 0 0.041 0.128 - - 0.048 γ 0 0.648 0.631 0.145 0.245 0.251 ψ 0 - - - - 0.187 nonlinear part α 1-6.964 3.960-0.295 ρ 11 0.150 0.370 0.296-0.010 0.321 ρ 12-0.111 - - - - β 1-0.015 0.004 0.069 0.089 0.689 γ 1 0.663 0.809-0.049 - -0.218 ψ 1 - - - 0.112 0.062 γ T 418.2 4.582 80.85 1259.49 76.96 c 1-0.620 2.317 6.451 3.553 2.430 c 2 3.425 7.085 -Modeling Nonlinearities - in Interest Rate-Setting

Remaining Nonlinearities, Parameter Constancy and Residual Analysis 70Q1:06Q2 O 78Q1:06Q2 O 78Q1:06Q2 I with CMR 92Q4:06Q2 I,F Residual Tests JB 0.000 0.000 0.001 0.000 0.937 ARCH 0.001 0.663 0.717 0.805 0.586 AutoC no no 6-12th 12th 1-5th Remaining Nonlinearity: H 0 :no r t 1 0.841 0.005 0.008 0.112 0.377 r t 2 0.475 - - - - π t+1 0.952 0.552 0.435 0.003 0.100 y t 0.715 0.372 0.049 0.028 0.013 Parameter Constancy: H 0 :yes H1 0.095 0.032 0.011 0.001 0.027 H2 0.008 0.045 0.001 0.000 0.020 H3 0.067 NaN 0.000 0.000 0.020

Graphical Analysis

Grid Search Result

Coefficients over time: Inflation

Coefficients over time: Output Gap

Coefficients over time: Inflation

Coefficients over time: Lagged Interest Rate

Graphical Analysis 7806 infl as transition variable

Grid Search Result

Coefficients over time: Output Gap

Graphical Analysis 9206 infl as transition variable,ffr

Coefficients over time: Output Gap

Coefficients over time: Federal Funds Rate

Coefficients over time: Lagged Interest Rate

Estimation Output: Lagged Interest Rates as TransVar 70Q1:06Q2 F 78Q1:06Q2 F 78Q1:06Q2 92Q4:06Q2 F linear part α 0-0.011-0.091-0.269 2.487 ρ 01 1.096 0.778 0.869 0.917 ρ 02 0.242 - - 0.479 β 0 0.051 0.287 0.368-0.188 γ 0 0.019 - - 0.339 ψ 0 0.166 0.130-0.159 nonlinear part α 1-5.027-1.940-0.393 ρ 11 0.487 0.109-0.670 ρ 12-0.135 - - -0.592 β 1 0.204 0.256-0.251 0.227 γ 1 0.656 0.309 0.184 0.967 ψ 1 0.104-0.051 - - γ T 5.538 8.074 940.37 3.919 c 1 12.213 10.894 10.02 5.434 c 2-7.085 - - ad.r 2 0.945 Jana 0.964 Riedel Modeling 0.962 Nonlinearities in0.964 Interest Rate Setting

Remaining Nonlinearities, Parameter Constancy and Residual Analysis 70Q1:06Q2 F 78Q1:06Q2 F 78Q1:06Q2 92Q4:06Q2 F Residual Tests JB 0.000 0.029 0.002 0.761 ARCH 0.253 0.028 0.093 0.646 AutoC no no no no Remaining Nonlinearity: H 0 :no r t 1 0.638 0.604 0.536 0.147 r t 2 0.153 - - 0.137 π t+1 0.330 0.268 0.019 0.386 y t 0.057 0.042 0.025 0.963 Parameter Constancy: H 0 :yes H1 0.089 0.213 0.117 0.251 H2 0.168 0.174 0.115 0.200 H3 0.352 0.538 0.131 0.364

Graphical Analysis 7006 tbr as transition variable, ffr

Coefficients over time: Output Gap

Coefficients over time: Inflation

Coefficients over time: Lagged Interest Rate

Coefficients over time: Federal Funds Rate

Graphical Analysis 7806 tbr as transition variable

Coefficients over time: Output Gap

Nonlinear Using Monthly Data IPI as proxy for UK monthly GDP data Sometimes different results: definition of output gap + tendency to capture fluctuations in te nonlinear part

Transition variables Output Gap and Interest Rates perform better than Inflation In times of high inflation the BoE tends to smooth interest rates Former periods: Higher output coefficient before booming periods Recent periods: Higher output coefficient if fluctuations in the output gap are high and inflation is low Recent periods: Inflation coefficient stable and higher than former periods

Details on real interest rates Structure of the CB loss function/model Time-Varying STR Model Allowing additively for more than one transition function (num of obs.) GMM estimation (instrument choice) Forecasts (in general: LSTR vs. Linear +) Distinguish between different kinds of events that drive nonlinearity (e.g. oil shock versus BoE announcements) Problem: Sensitivity of results with respect to the construction of output gap, the choice of estimation method and additional regressors

Questions? Answers!

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