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A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

A nonlinear cointegration model for US bond

Emanuele Armillotta

UNIVERSIT` A POLITECNICA DELLE MARCHE Department of Economics and Social Sciences Ph.D. in Economics XV cycle

October 2015

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 1 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

1

Introduction

2

Aim of the thesis

3

Literature review Linear Model Alternative Models Nonlinear Model

4

Preliminary analysis Data set Out-of-sample analysis

5

Work in progress

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 2 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Nonlinear model

Recently a stand of literature has been interested in the use

• f nonlinear model in different macroeconomic areas

Power purchasing parity: Lo e Zivot(2001), Taylor e Siklos(2001), Kapetanios e Shin(2002), Lundberg e Ter¨ asvirta(2006), Nam(2011), Beckmann(2012) Employment rate: Skalin e Ter¨ asvirta(2002), Caner e Hansen(2003), Akram(2005), P´ erez-Alonso e Di Sanzo(2010)

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 3 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Aim of the thesis

1 Could nonlinear adjustment mechanism capture

macroeconomic dynamics in the term of structure?

2 Could the nonlinear framework represent in a stylised way

the American monetary policy?

3 Which approximation function is best suited for this

purpose?

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 4 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Linear Model

Campbell e Shiller(1987) were the first who used a cointegration model (VECM) in which the spread represents a linear stationary combination of two processes I(1)

Problems

1 Spreads aren’t stationary because of:

time-varying risk premium failure rational expectations hypothesis correlation with macro factors

2 Tests reject the hypothesis that there is only one common

trend

3 Restrictions imposed by linear model Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 5 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Alternative Models

Literature has tried to verify empirically the expectations hypothesis through different methods:

1 By including exogenous macroeconomic variables (inflation

and real activity indicator) Ang e Piazzesi (2003), Carriero et al.(2004), Valente et al.(2004)

2 By empirical analysis on yields with shorter maturity (daily

• r weekly maturity)

Longstaff(2000), Della Corte et al.(2007)

3 By using nonlinear models

Blake e Fomby(1992), Enders e Granger(1998), Hansen e Seo(2002)

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 6 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Alternative Models

Literature has tried to verify empirically the expectations hypothesis through different methods:

1 By including exogenous macroeconomic variables (inflation

and real activity indicator) Ang e Piazzesi (2003), Carriero et al.(2004), Valente et al.(2004)

2 By empirical analysis on yields with shorter maturity (daily

• r weekly maturity)

Longstaff(2000), Della Corte et al.(2007)

3 By using nonlinear models

Blake e Fomby(1992), Enders e Granger(1998), Hansen e Seo(2002)

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 6 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Alternative Models

Literature has tried to verify empirically the expectations hypothesis through different methods:

1 By including exogenous macroeconomic variables (inflation

and real activity indicator) Ang e Piazzesi (2003), Carriero et al.(2004), Valente et al.(2004)

2 By empirical analysis on yields with shorter maturity (daily

• r weekly maturity)

Longstaff(2000), Della Corte et al.(2007)

3 By using nonlinear models

Blake e Fomby(1992), Enders e Granger(1998), Hansen e Seo(2002)

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 6 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Alternative Models

Literature has tried to verify empirically the expectations hypothesis through different methods:

1 By including exogenous macroeconomic variables (inflation

and real activity indicator) Ang e Piazzesi (2003), Carriero et al.(2004), Valente et al.(2004)

2 By empirical analysis on yields with shorter maturity (daily

• r weekly maturity)

Longstaff(2000), Della Corte et al.(2007)

3 By using nonlinear models

Blake e Fomby(1992), Enders e Granger(1998), Hansen e Seo(2002)

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 6 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Threshold Model

Balke e Fomby(1997) argue the hypothesis of the existence of two regimes in which only one presents adjustment mechanism

1 Presence of transaction costs 2 Monetary authority’s “discrete” interventions

Adjustment mechanism of Band-TAR

g(st) =      φ1(1 − ρ1) + ρ1st se st > φ1 se φ2 < st < φ1 φ2(1 − ρ2) + ρ2st se st < φ2

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 7 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Threshold Model

Balke e Fomby(1997) argue the hypothesis of the existence of two regimes in which only one presents adjustment mechanism

1 Presence of transaction costs 2 Monetary authority’s “discrete” interventions

Adjustment mechanism of Band-TAR

g(st) =      φ1(1 − ρ1) + ρ1st se st > φ1 se φ2 < st < φ1 φ2(1 − ρ2) + ρ2st se st < φ2

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 7 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Threshold Model

Results

1 Empirical evidence of threshold adjustment 2 Consistency with theory

Problems

1 Computational issues 2 Drawback to generalise when the number of processes is

greater than two

3 Issues with diagnostic tests Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 8 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Threshold Model

Results

1 Empirical evidence of threshold adjustment 2 Consistency with theory

Problems

1 Computational issues 2 Drawback to generalise when the number of processes is

greater than two

3 Issues with diagnostic tests Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 8 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

NEC model

Escribano(2004) proposes to approximate a nonlinear adjustment mechanism by using Pad´ e polynomials for money demand

Advantages

1 Cubic polynomial function is more flexible and useful to

approximate unknown parametric functions

It detects asymmetries It detects threshold points (unique or multiple equilibria)

2 It satisfies stability condition

Disadvantages

1 High number of parameters 2 Difficulties in a multivariate setting Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 9 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

NEC model

Escribano(2004) proposes to approximate a nonlinear adjustment mechanism by using Pad´ e polynomials for money demand

Advantages

1 Cubic polynomial function is more flexible and useful to

approximate unknown parametric functions

It detects asymmetries It detects threshold points (unique or multiple equilibria)

2 It satisfies stability condition

Disadvantages

1 High number of parameters 2 Difficulties in a multivariate setting Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 9 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

NEC model

Escribano(2004) proposes to approximate a nonlinear adjustment mechanism by using Pad´ e polynomials for money demand

Advantages

1 Cubic polynomial function is more flexible and useful to

approximate unknown parametric functions

It detects asymmetries It detects threshold points (unique or multiple equilibria)

2 It satisfies stability condition

Disadvantages

1 High number of parameters 2 Difficulties in a multivariate setting Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 9 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Lucchetti e Palomba(2009) Model

Lucchetti e Palomba(2009) approximate the threshold function with third-order Taylor expansion

Adjustment mechanism

g(st) = µ + α′st + θ′(st ⊗ st)+ + λ′(st ⊗ st ⊗ st)+ Where

1 st = [smt

slt]′

2 (st ⊗ st)+ = [sm2

t

smt · slt sl2

t ]′

3 (st ⊗ st ⊗ st)+ = [sm3

t

sm2

t · slt

smt · sl2

t

sl3

t ]′

(·)+ is the Moore-Penrose inverse of duplication matrix

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 10 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Data Set

Univariate analysis on US zero coupon bonds with 3 months, 2 years and 10 years maturity

Data set

Frequency Monthly Weekly Period Oct.1982 - May.2015 04 Oct.1982 - 12 May.2015

• Tot. obs

392 1702 Obs out-of-sample 12 52

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 11 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Series Analysis

Identification tests

1 Unit root tests for interest rate reject the hypothesis of

stationarity

2 Stationarity tests for spreads confirm the absence of unit

roots

3 Results from Johansen test do not confirm the presence of

cointegration

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 12 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Diebold-Mariano Test

Monthly interest rates Model / Series r 3month

t

r 2year

t

r 10year

t

Random Walk

• 0,1526
• 1,6533*

0,0403 D VAR

• 0,3106
• 1,2314

0,0553 VECM1

• 1,6647*
• 1,6979*

0,5358 VECM2

• 2,3817**
• 1,6682*

0,2865 L VAR

• 0,0216
• 0,6087

1,0039 Weekly interest rates Model / Series r 3month

t

r 2year

t

r 10year

t

Random Walk 0,2289

• 0,3698

0,0258 D VAR 0,4690

• 0,3473

0,0440 VECM1 0,1611

• 0,3050

0,2141 VECM2 0,1880

• 0,2975

0,0904 L VAR 0,1432 0,1185 0,3525

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 13 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

RMSE

Monthly interest rates Model / Series r 3month

t

r 2year

t

r 10year

t

NEC 0,053980 0,079971 0,694814 Random Walk 0,094392 0,316874 0,277539 D VAR 0,078926 0,409292 0,290484 VECM1 0,216619 0,230057 0,388958 VECM2 0,167628 0,207296 0,321528 L VAR 0,084367 0,108481 0,541723 Weekly interest rate Model / Series r 3month

t

r 2year

t

r 10year

t

NEC 0,40106 0,07893 0,682532 Random Walk 0,145992 0,365118 0,272367 D VAR 0,16374 0,369555 0,269511 VECM1 0,304065 0,301471 0,340994 VECM2 0,296793 0,308972 0,301293 L VAR 0,057428 0,068776 0,480826

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 14 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Conclusion

Achieved Results

1 Diagnostic tests confirm the presence of nonlinear

adjustment

2 Adjustment function is a stylised representation of

• ccasional events, like monetary policy interventions

3 An improvement in the monthly data previsions

Problems

1 Presence of heteroskedasticity 2 Equal predictive accuracy of weekly data with linear

models

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 15 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Conclusion

Achieved Results

1 Diagnostic tests confirm the presence of nonlinear

adjustment

2 Adjustment function is a stylised representation of

• ccasional events, like monetary policy interventions

3 An improvement in the monthly data previsions

Problems

1 Presence of heteroskedasticity 2 Equal predictive accuracy of weekly data with linear

models

Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 15 / 16

A nonlinear cointegration model for US bond Armillotta E. Introduction Aim of the thesis Literature review

Linear Model Alternative Models Nonlinear Model

Preliminary analysis

Data set Out-of-sample analysis

Work in progress

Work in progress

Future aims

1 Use of alternative approximative functions (Pad´

e polynomials or splines)

2 Nonlinearity check with nonparemetric or semiparametric

estimations

3 Comparison with a larger variety of models 4 Conditional variance modelling Armillotta E. (UNIVPM) A nonlinear cointegration model for US bond October 2015 16 / 16