p1_ts.pdf

UNIVARIATE BAYESIAN MODELING

Spanish Treasury Bills 12m

Enrique M. Quilis

CONTENTS

Exploratory analysis

AR modeling

Litterman prior

Calibration

Estimation

Estimation

Analysis

MATLAB CODE: Commands

>> cstr=cal(1987,6,12)

cstr =

beg_yr: 1987

beg_yr: 1987

beg_per: 6

freq: 12

SERIES

10 12 14 16

Jan850 Jan90 Jan95 Jan00 Jan05

MATLAB CODE: Commands

SERIES: CORRELOGRAMS

0.6 0.8 1

Sample autocorrelation coefficients

s a c f v a lu e s 0.6 0.8 1

Sample partial autocorrelation coefficients

s p a c f v a lu e s

0 10 20 30 -0.2 0 0.2 0.4 k-values s a c f v a lu e s

MATLAB CODE: Commands

>> pmax=14

pmax =

14

14

ORDER SELECTION

0.45 0.5 0.55 0.6 0.65

AIC

0.55 0.6 0.65 0.7

BIC

0 5 10 15

0.2 0.25 0.3 0.35 0.4

AR order

0 5 10 15

0.35 0.4 0.45 0.5

LITTERMAN PRIOR

Underlying model: univariate

autoregression with exogenous variables

t

h

t

p

1

h

h

t

t

b

'

x

z

u

z

=

+

+

₋

+

=

LITTERMAN PRIOR

ARX model in compact notation:

1





[

]

_t

_t

_t

t

p

t

t

u

X

u

z

L

x

1

'

b

z

+

=

+





















LITTERMAN PRIOR

u

z

)

B

1

(

−

=

µ

+

Mean centered around a random walk with

drift:

t

t

u

z

)

B

1

LITTERMAN PRIOR: Variance







₌

=

)

(

)

(

v

)

V

(

diag

2

σ

θ

µ

_µ









∀

=

=

h

)

)

(

g

(

)

(

v

)

V

(

diag

2

2

h

1

h

θ

θ

σ

φ

LITTERMAN PRIOR: Variance

θ

_µ

>0

θ

_b

>0

θ

>0

σ

derived from BARX(p) model (unrestricted

LITTERMAN PRIOR: Variance

Temporal decay







_≤

_≤

=

1

h

1

0

1

g

θ

θ







∞

<

≤

=

−

1

h

0

h

g

1

θ

LITTERMAN PRIOR: Variance

Temporal decay

1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 Lag

LITTERMAN PRIOR: Variance

0.6 0.8 1 1.2

Prior on phi parameters: th1=0.1 th2=0.5

1 1.5 2

Prior on phi parameters: th1=1 th2=0.5

0 1 2 3 4 5 6 7 -0.2

0 0.2 0.4 0.6

0 1 2 3 4 5 6 7 -1

LITTERMAN PRIOR: Calibration

1.325 1.33 1.335

MSE

1.326 1.328 1.33 1.332 1.334

0.01

0.41

0.91

1.41 0.01

0.41

0.91

1.41 1.31

1.315 1.32

th2 th1

LITTERMAN PRIOR: Calibration

1.315 1.316 1.317

MSE

1.3145 1.315 1.3155 1.316

0.01

0.01

0.10 1.31

1.311 1.312 1.313 1.314

LITTERMAN PRIOR: Calibration

θ

_µ

θ

₁

θ

₂

σ

2

0.00

0.04

0.05

0.07

θ

_µ

θ

1

θ

2

σ

2

MATLAB CODE: Prior generation

close all; clc; clear all;

% Yield on Letras 12m: 1987.06 - 2007.03 z = load('y1.prn');

% Exogenous variables rex.X = [];

% Intercept: yes (1) or not (0) rex.opC = 1;

% Basic variable rex.z = z;

% Order of AR model rex.p = 6;

%---% Generating Litterman prior

%Type of decay function: harmonic (1) or geometric (2) rex.type = 1;

rex.th_mu = 1000; rex.th_b = 1000; rex.th = [0.10 0.05];

LITTERMAN PRIOR: Calibration

0.6 0.7 0.8 0.9 1

Decay function: harmonic

0.6 0.8 1 1.2

Prior on phi parameters

0 1 2 3 4 5 6 7 0

0.1 0.2 0.3 0.4 0.5 0.6

Lag

0 1 2 3 4 5 6 7 -0.2

0 0.2 0.4 0.6

MATLAB OUTPUT: Estimation

********************************************************************** ARX MODEL: BAYESIAN ESTIMATION

********************************************************************** LITTERMAN PRIOR

Hyperparameters: tightness Intercept 0.0000 Exogenous 1000.0000 Decay function: harmonic

Global 0.1000 Decay 0.0500

********************************************************************** Number of observations : 239

Number of effective observations : 233 Number of effective observations : 233 Number of series : 1 Number of lags : 6

********************************************************************** mu: Estimate, standard deviation and t-ratio (read columnwise)

---0.0000 ---0.0000 0.0001

---phi: Lag, estimate, standard deviation and t-ratio (read columnwise)

MATLAB CODE: Estimation

********************************************************************** ARX MODEL: BAYESIAN ESTIMATION

**********************************************************************

---RESIDUALS: standard deviation 0.2693

---Model performance (read columnwise):

- log-likelihood - log-likelihood - AIC and BIC - BIC

---24.9411 0.2742 0.3779

MATLAB CODE: Estimation

Sample autocorrelation coefficients

s a c f v a lu e s 0.2 0.4 0.6 0.8 1

Sample partial autocorrelation coefficients

MATLAB CODE: Outliers

0.5 1 1.5

Jul87 Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan05

MATLAB CODE: Analytics

0 10 20 30 40 0

0.5 1 1.5 2

PHI-WEIGHTS

0 2 4 6 8

-1.5 -1 -0.5 0 0.5

PI-WEIGHTS

lag lag

-0.5 0 0.5 1

6 ROOTS

2 4 6 8 10

MATLAB CODE: Forecasting

close all; clc; clear all;

% Yield on Letras 12m: 1987.06 - 2007.03 z = load('y1.prn');

% Exogenous variables: unit step 1999.01 - 2007.12 (EMU) rex.X = [];

% Intercept: yes (1) or not (0) rex.opC = 1;

% Basic variable % Basic variable rex.z = z;

% Order of AR model rex.p = 6;

%---% Generating Litterman prior

%Type of decay function: harmonic (1) or geometric (2) rex.type = 1;

rex.th_mu = 0.0000000001; rex.th_b = 1000;

rex.th = [0.10 0.05];

rex.plt = 0; %Plots or no plots

MATLAB CODE: Forecasting

---% Calling estimation function

res = barx(rex,prior);

%---% Number of forecasts

rex.npred = 9;

% Preparing forecasts of exogenous variables % rex.Xf = ones(rex.npred,1);

rex.Xf = [];

% Generating forecasts fore = upredict(rex,res);

% Joining forecasts and data

zz = [z fore.zf]; zli = [z

fore.zf - fore.sf]; zls = [z

MATLAB CODE: Forecasting

4 4.5 5 5.5

Forecasts and +/- s.e.

2004 2005 2006 2007 1.5

2 2.5 3 3.5