Local or Global Smoothing? A Bandwidth Selector for Dependent Data - - PowerPoint PPT Presentation

Local or Global Smoothing? A Bandwidth Selector for Dependent Data

Francesco Giordano – Maria Lucia Parrella

Department of Economics and Statistics University of Salerno

COMPSTAT 2010

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 1 / 115

Aims and prospects

This is part of a work in progress. The context and the goals of this research are:

1

analyzing the problem of bandwidth selection in local polynomial estimation (LPE) of dependent data;

2

proposing a locally adaptive bandwidth selector, based on the use of the neural network estimator (NNE);

3

evaluating the gain in using a local bandwidth instead of a global (fixed) bandwidth. Here we present some theoretical and computational results....

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 2 / 115

Motivations

Bandwidths are very crucial in LPE! One expects local bandwidths to perform better than global

• bandwidths. Is it really true when we use the estimated

bandwidths? The available selection procedures may present some drawbacks:

◮ often only for global bandwidths; ◮ often not for dependent data; ◮ sometimes computationally expensive; ◮ sometimes extremely biased and/or variable; ◮ sometimes not fully automatic.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 3 / 115

Recent research in bandwidth selection

Generally, bandwidth selection may be used as a tool to deal with different problems in several contexts. See, for example: Gao and Gijbels (JASA, 2008): size and power in nonparametric kernel testing. Gluhovsky and Gluhovsky (JASA, 2007): smooth conditions in kernel regression. Lafferty and Wassermann (AS, 2008): variable selection in high dimension problems. Prewitt and Lohr (JRSS, 2006): multicollinearity in local regression.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 4 / 115

Our framework

Our proposal applies to different setups. In this talk we consider, in particular, the following framework: DGP: real strictly stationary process {Xt} Model: Xt = m(Xt−1) + σ(Xt−1)ǫt, ǫt ∼ i.i.d.(0, 1) Context: estimation of the volatility function by LPE σ2(x) = Var [Xt|Xt−1 = x] , x ∈ R. Note: for simplicity, here we consider m(x) ≡ 0.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 5 / 115

The local polynomial estimator of volatility

The volatility function σ2(x) may be estimated by local polynomials: ˆ σ2(x; h) =

n

• t=1

X 2

t WK,h,p(x − Xt−1).

WK,h,p(·) are the weights, derived from a local approximation of σ2(x) through a polynomial of order p; K is the Kernel function; h is the bandwidth, which regulates the smoothness of ˆ σ2(x; h).

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 6 / 115

An illustrative example of LPE

Xt =

• 0.01 + 0.1X 2

t−1 + 0.35X 2 t−1I(Xt−1 < 0) ∗ εt

TIME PLOT

200 400 600 800 1000

• 0.4
• 0.2

0.0 0.2

VOLATILITY FUNCTION

X(-1) X^2

• 0.4
• 0.2

0.0 0.2 0.0 0.05 0.10 0.15

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 7 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.1 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 8 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.13 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 9 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.16 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 10 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.19 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 11 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.21 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 12 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.24 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 13 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.27 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 14 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.3 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 15 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.33 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 16 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.36 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 17 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.39 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 18 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.41 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 19 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.44 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 20 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.47 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 21 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.5 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 22 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.5 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 23 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.5 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 24 / 115

An illustrative example of LPE

σ2(x) = 0.01 + 0.1x2 + 0.35x2I(x < 0)

X_(t-1) volatility 0.02 0.04 0.06 0.08 0.10

• 0.4
• 0.2

0.0 0.2 0.4 0.1 0.3 0.5 bandwidth h True function Estimated (h= 0.5 )

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 25 / 115

The plug-in optimal local bandwidth

It is derived by minimizing the asymptotic mean square error hopt

L (x)

= arg min

h AMSE{ˆ

σ2(x; h)} =

• Cp,K × v(x)

n × d2(x) × fX(x) 1/(2p+3)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 26 / 115

The plug-in optimal local bandwidth

It is derived by minimizing the asymptotic mean square error hopt

L (x)

= arg min

h AMSE{ˆ

σ2(x; h)} =

• Cp,K × v(x)

n × d2(x) × fX(x) 1/(2p+3) The only unknown components (to estimate and plug-in) are

◮ v(x), the conditional variance of X 2

t ;

◮ d(x), the (p + 1)-th derivative of σ2(x); ◮ fX(x), the design density.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 26 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 27 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 28 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 29 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 30 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 31 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 32 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 33 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 34 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 35 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 36 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 37 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 38 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 39 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 40 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 41 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 42 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 43 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 44 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 45 / 115

An illustrative example: local (variable) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 46 / 115

The plug-in optimal global bandwidth

It is derived by minimizing the integrated AMSE hopt

G

= arg min

h

• AMSE{ˆ

σ2(x; h)}fX(x)dx = Cp,K × Rvar n × Rf,bias 1/(2p+3)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 47 / 115

The plug-in optimal global bandwidth

It is derived by minimizing the integrated AMSE hopt

G

= arg min

h

• AMSE{ˆ

σ2(x; h)}fX(x)dx = Cp,K × Rvar n × Rf,bias 1/(2p+3) The only unknown components (to estimate and plug-in) are:

◮ Rvar =

• v(x)dx;

◮ Rf,bias =

• d2(x)fX(x)dx.
• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 47 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 48 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 49 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 50 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 51 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 52 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 53 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 54 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 55 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 56 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 57 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 58 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 59 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 60 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 61 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 62 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 63 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x x+h true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 64 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 65 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 66 / 115

An illustrative example: global (fixed) bandwidth

X(-1) phi(X) x-h x true regression function LP estimate Kernel weights

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 67 / 115

Local vs global bandwidths

The following relation holds: AMSE

• x; hopt

L (x)

• ≤ AMSE
• x; hopt

G

• .
• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 68 / 115

Local vs global bandwidths

The following relation holds: AMSE

• x; hopt

L (x)

• ≤ AMSE
• x; hopt

G

• .

Anyway, the estimation of hopt

L (x) (=function) is less efficient than

the estimation of hopt

G

(=mean value)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 68 / 115

Local vs global bandwidths

The following relation holds: AMSE

• x; hopt

L (x)

• ≤ AMSE
• x; hopt

G

• .

Anyway, the estimation of hopt

L (x) (=function) is less efficient than

the estimation of hopt

G

(=mean value) Some questions follow:

◮ what is the effective gain in using the local bandwidth? ◮ Is it always convenient to perform a local smoothing instead of a

global smoothing?

◮ How to deal with pilot bandwidths?

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 68 / 115

Our proposal: a two stage procedure

Suppose we want to estimate the function σ2(x) on a support IX ⊂ R.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

Our proposal: a two stage procedure

Suppose we want to estimate the function σ2(x) on a support IX ⊂ R.

1

First stage: evaluating the “homogeneity” on the support Given the estimates of hopt

L (x) and hopt G , use some relative

indicator in order to evaluate the potential gain in using the local bandwidth instead of the global bandwidth on IX.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

Our proposal: a two stage procedure

Suppose we want to estimate the function σ2(x) on a support IX ⊂ R.

1

First stage: evaluating the “homogeneity” on the support Given the estimates of hopt

L (x) and hopt G , use some relative

indicator in order to evaluate the potential gain in using the local bandwidth instead of the global bandwidth on IX.

2

Second stage: deriving the locally global bandwidth Derive the optimal global bandwidths hopt

G

• n “homogeneous”

subsets of IX.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

Our proposal: a two stage procedure

Suppose we want to estimate the function σ2(x) on a support IX ⊂ R.

1

First stage: evaluating the “homogeneity” on the support Given the estimates of hopt

L (x) and hopt G , use some relative

indicator in order to evaluate the potential gain in using the local bandwidth instead of the global bandwidth on IX.

2

Second stage: deriving the locally global bandwidth Derive the optimal global bandwidths hopt

G

• n “homogeneous”

subsets of IX.

◮ How to estimate hopt

G

and hopt

L (x) on IX?

◮ Which relative indicator to use? ◮ How to smooth such bandwidths on IX?

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

First stage: deriving the relative indicator

Consider the relative increment of the AMSE, ∀x ∈ IX, ∆AMSE(x) = AMSE{ˆ σ2(x; hopt

G )} − AMSE{ˆ

σ2(x; hopt

L (x))}

AMSE{ˆ σ2(x; hopt

L (x))}

.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 70 / 115

First stage: deriving the relative indicator

Consider the relative increment of the AMSE, ∀x ∈ IX, ∆AMSE(x) = AMSE{ˆ σ2(x; hopt

G )} − AMSE{ˆ

σ2(x; hopt

L (x))}

AMSE{ˆ σ2(x; hopt

L (x))}

. We managed to express the ∆AMSE(x) of the estimator in a model free form: ∆AMSE(πh(x)) = 1 2p + 3 [πh(x)]−2(p+1) + 2p + 2 2p + 3 [πh(x)] − 1 where πh(x) = hopt

L (x)

hopt

G

≥ 0.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 70 / 115

First stage: deriving the relative indicator

∆AMSE(πh(x)) = 1 2p + 3 [πh(x)]−2(p+1) + 2p + 2 2p + 3 [πh(x)] − 1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 pi(x) Delta_AMSE

p=0 p=1 p=2 pi(x)

a 1 b

beta

Delta_AMSE

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 71 / 115

First stage: deriving the relative indicator

∆AMSE(πh(x)) = 1 2p + 3 [πh(x)]−2(p+1) + 2p + 2 2p + 3 [πh(x)] − 1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 pi(x) Delta_AMSE

p=0 p=1 p=2 pi(x)

a 1 b

beta

Delta_AMSE

Remark 1: as a function of πh(x), the relative indicator ∆AMSE(x) is completely model free. Remark 2 : ∆AMSE < β iif πh(x) ∈ [aβ, bβ], ∀β > 0.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 71 / 115

A global measure of ∆AMSE on the interval IX

A global measure of ∆AMSE would indicate, eventually, the need to use on the interval IX a local bandwidth. For example, the mean value of ∆AMSE on the interval IX could be considered...

• IX

∆AMSE(x)fX(x)dx. ... but we get some advantages if we consider the median value

• f ∆AMSE on the interval IX, or some other quantile.
• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 72 / 115

The median of ∆AMSE on the interval IX

pi(x)

a 1 b

beta

Delta_AMSE

• o
• *

* * *

For a fixed threshold β, consider the two set of points Sβ ={x : x ∈ IX, π(x) ∈ [aβ, bβ]}; Sβ ={x : x ∈ IX, π(x) / ∈ [aβ, bβ] Given the measure of the process µX, the median of ∆AMSE is the threshold value β∗ for which µX(Sβ∗) = µX(Sβ∗).

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 73 / 115

The algorithm of the first stage

1

Consider an estimation of both the global bandwidth hG and the local bandwidth function hL(Xt), such as

◮ the neural network method proposed in the second stage; ◮ some other local method proposed in the literature. 2

Estimate πh(Xt), for all Xt ∈ IX, then search for β∗ such that

• Xt∈IX

I {ˆ πh(Xt) ∈ [aβ∗, bβ∗]} ≈

• Xt∈IX

I {ˆ πh(Xt) / ∈ [aβ∗, bβ∗]} . where I(·) is the indicator function.

3

Split the subset IX if β∗ > τ, where τ represents the max relative increment tolerated for ∆AMSE when using a global bandwidth.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 74 / 115

Second stage: the locally global bandwidth

Given the “homogeneous” subset IX ⊂ R, we want to estimate hopt

G

=

• Cp,K × RIX

var

n × RIX

f,bias

1/(2p+3) where:

◮ RIX

var =

• IX v(x)dωIX ;

◮ RIX

f,bias =

• IX d2(x)fX(x)dωIX .

1

How to derive the measure ωIX ? (we used dωIX =

dx µ(IX ))

2

How to estimate v(x) and d2(x) on IX?

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 75 / 115

Estimating the functions v(x) and d2(x)

Note that: v(x) = Var{X 2

t |Xt−1 = x} ≡ m4(x) − m2 2(x)

d(x) = ∂p+1σ2(x) ∂xp+1 ≡ m(p+1)

2

(x)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 76 / 115

Estimating the functions v(x) and d2(x)

Note that: v(x) = Var{X 2

t |Xt−1 = x} ≡ m4(x) − m2 2(x)

d(x) = ∂p+1σ2(x) ∂xp+1 ≡ m(p+1)

2

(x) As usual, we should have to estimate separately the three functions m4(x), m2(x), m(p+1)

2

(x) (Fan & Yao, 1998; Franke & Diagne, 2006)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 76 / 115

Estimating the functions v(x) and d2(x)

Note that: v(x) = Var{X 2

t |Xt−1 = x} ≡ m4(x) − m2 2(x)

d(x) = ∂p+1σ2(x) ∂xp+1 ≡ m(p+1)

2

(x) As usual, we should have to estimate separately the three functions m4(x), m2(x), m(p+1)

2

(x) (Fan & Yao, 1998; Franke & Diagne, 2006) An interesting result here is that we estimate the three functions by only one estimator!

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 76 / 115

Estimating the functions v(x) and d2(x)

We use only the following neural network estimator ˆ η = arg min η

n

• t=2
• X 2

t − q(Xt−1; η)

2

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 77 / 115

Estimating the functions v(x) and d2(x)

We use only the following neural network estimator ˆ η = arg min η

n

• t=2
• X 2

t − q(Xt−1; η)

2 where

◮ q(Xt−1; η) = d

k=1 ckΓ (akXt−1 + bk) + c0 is the neural network

function;

◮ η = (c0, c1, . . . , cd, a1, . . . , ad, b1 . . . bd) is the vector of parameters

to be estimated;

◮ d is the number of nodes in the hidden layer; ◮ Γ (·) is the logistic activation function.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 77 / 115

Estimating the functions v(x) and d2(x)

By defining m4ε = E(ǫ4

t ), we reparameterize as follows:

v(x) = m4(x) − m2

2(x) = m2 2(x) [m4ε − 1]

d(x) = m(p+1)

2

(x) then we propose the following estimators ˆ m2(x) = q(x, ˆ η) ˆ m4ε = n

t=2 X 4 t

n

t=2[q(Xt, ˆ

η)]2 ˆ m(p+1)

2

(x) = q(p+1)(x; ˆ η)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 78 / 115

Estimating the locally global bandwidth

Finally, given the ergodicity of the process, the locally global bandwidth is estimated by: ˆ RIX

var =

n∗

i=1 ˆ

v(xi) n . ˆ RIX

f,bias =

• Xt∈IX
• ˆ

m(p+1)

2

(Xt) 2 n

t=1 I(Xt ∈ IX)

, ˆ hIX =

• Cp,K × ˆ

RIX

var

n × ˆ RIX

f,bias

1/(2p+3) The points {x1, x2, . . . , xn∗} in ˆ RIX

var are uniformly spaced in the

interval IX, and I(·) is the indicator function.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 79 / 115

Assumptions

(a1) the errors εt have continuous and positive density function and, for some δ > 4, E(ε2

t ) = 1,

E(εt) = E(ε3

t ) = 0,

E|εt|δ < ∞; (a2) the functions m (·) and σ(·) have continuous second derivative. Moreover, the function σ(·) is positive; (a3) there exist the constants M1 > 0 and M2 > 0 such that, for all y ∈ R, |m(y)| ≤ M1(1 + |y|), |σ(y)| ≤ M2(1 + |y|), M1 + M2

• E|εt|δ1/δ

< 1; (a4) the density function fX(·) of the (stationary) measure of the process µX exists, it is bounded, continuous and positive on every compact set in R. (b1) d

k=1 |ck| ≤ ∆n.

(b2) d ≡ dn, dn → ∞, ∆n → ∞ as n → ∞. (b3) Let K1(n) :=

∆2

ndn log(∆2 ndn)

√n

and K2(n) :=

∆4

n

n1−δ .

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 80 / 115

The consistency of the bandwidth selector

Using the framework in Franke and Diagne (2006), we can state the following results. Lemma 1 Under assumptions (a1)-(a4) and (b1)-(b3), the volatility function estimator is consistent in the sense that: If K1(n) → 0 as n → ∞, then E q (x; ˆ η) − σ2(x) 2 dµX(x) → 0 n → ∞ if, additionally, K2(n) → 0 for some δ > 0, then q (x; ˆ η) − σ2(x) 2 dµX(x)

a.s.

− → 0 n → ∞

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 81 / 115

The consistency of the bandwidth selector

Lemma 2 Under assumptions (a1)-(a4) and (b1)-(b3), the estimator of the second derivative of σ2(x) is consistent in the sense that: If K1(n) → 0 as n → ∞, then E q′′ (x; ˆ η) − [σ2(x)]′′2 dµX(x) → 0 n → ∞ if, additionally, K2(n) → 0 for some δ > 0, then q′′ (x; ˆ η) − [σ2(x)]′′2 dµX(x)

a.s.

− → 0 n → ∞

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 82 / 115

The consistency of the bandwidth selector

Theorem 1 Under the conditions (a1)-(a4) and (b1)-(b3), ˆ RIX

f,bias, with IX ⊆ R, is

consistent in the sense that: If K1(n) → 0 as n → ∞, then ˆ RIX

f,bias p

− → RIX

f,bias

n → ∞ if, additionally, K2(n) → 0 for some δ > 0, then ˆ RIX

f,bias a.s.

− → RIX

f,bias

n → ∞ Corollary Using the same conditions as in theorem 1, then ˆ m4ε is consistent in the sense that: If K1(n) → 0 as n → ∞, then ˆ m4ε

p

− → m4ε n → ∞ if, additionally, K2(n) → 0 for some δ > 0, then ˆ m4ε

a.s.

− → m4ε n → ∞

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 83 / 115

The consistency of the bandwidth selector

Theorem 2 Using the same conditions as in theorem 1, then ˆ RIX

var with IX ⊂ R and

n∗ = O(n), is consistent in the sense that: If K1(n) → 0 as n → ∞, then ˆ RIX

var p

− → RIX

var

n → ∞ if, additionally, K2(n) → 0 for some δ > 0, then ˆ RIX

var a.s.

− → RIX

var

n → ∞

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 84 / 115

The consistency of the bandwidth selector

For a fixed z ∈ R, let Iz be a non null measure set which contains the point z. Let nz be the number of observed values in Iz such that nz → ∞ when n → ∞. Theorem 3 Using the same conditions as in Lemma 2, if nz = o(n), for a set IX ⊂ R, then ˆ hIz is consistent in the sense that: If K1(n) → 0 as n → ∞, then sup

z∈IX

• ˆ

hIz − hopt

L (z)

• p

− → 0 n → ∞ if, additionally, K2(n) → 0 for some δ > 0, then sup

z∈IX

• ˆ

hIz − hopt

L (z)

• a.s.

− → 0 n → ∞

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 85 / 115

The consistency of the bandwidth selector

The previous results imply that:

1

ˆ hG

p

− → hopt

G

• r

ˆ hG

a.s.

− → hopt

G . 2

For all z ∈ IX ˆ hIz

p

− → hopt

L (z)

• r

ˆ hIz

a.s.

− → hopt

L (z), 3

Uniformly for each z ∈ IX ˆ πh(z)

p

− → πh(z)

• r

ˆ πh(z)

a.s.

− → πh(z)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 86 / 115

The simulation study: setting the models

Model 1: Xt = [ψ(Xt−1 + 1.2) + 1.5ψ(Xt−1 − 1.2)] ǫt ǫt ∼ N(0, 1) ψ(z)= d.f. of N(0, 1) Model 2: Xt =

• 0.1 + 0.3X 2

t−1ǫt

ǫt ∼ N(0, 1) Model 3: Xt =

• 0.01 + 0.1X 2

t−1 + 0.2X 2 t−1IXt−1<0ǫt

ǫt ∼ N(0, 1)

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 87 / 115

Model 1: from Härdle and Tsybakov (1997)

Xt = [ψ(Xt−1 + 1.2) + 1.5ψ(Xt−1 − 1.2)] ǫt; ǫt ∼ N(0, 1); ψ(z) = d.f. of N(0, 1)

TIME PLOT

200 400 600 800 1000

• 2
• 1

1 2

VOLATILITY FUNCTION

X(-1) volatility

• 1.0
• 0.5

0.0 0.5 1.0 0.20 0.25 0.30 0.35

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 88 / 115

Model 1: Estimated local and global bandwidths

n=500

x bandwidth

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 89 / 115

Model 1: Estimated local and global bandwidths

n=1000

x bandwidth

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 90 / 115

Model 1: Estimated local and global bandwidths

n=2000

x bandwidth

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 91 / 115

Model 1: πh(x) estimated functions

median of estimates

xx r(x)

• 1.0
• 0.5

0.0 0.5 1.0 2 4 6 8 10 12 14 true function n=500 n=1000 n=2000

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 92 / 115

Model 1: median of ∆(AMSE) estimates

0.0 0.1 0.2 0.3 0.4 0.5 n=500 n=1000 n=2000

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 93 / 115

Model 1: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=500

x

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 true function

Estimated volatility with GLOBAL bandwidth

x

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 94 / 115

Model 1: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=1000

x

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 true function

Estimated volatility with GLOBAL bandwidth

x

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 95 / 115

Model 1: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=2000

x

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 true function

Estimated volatility with GLOBAL bandwidth

x

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 96 / 115

Model 2: ARCH(1)

Xt =

• 0.1 + 0.3X 2

t−1ǫt;

ǫt ∼ N(0, 1)

TIME PLOT

200 400 600 800 1000

• 1

1

VOLATILITY FUNCTION

X(-1) volatility

• 0.5

0.0 0.5 0.10 0.15 0.20 0.25

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 97 / 115

Model 2: Estimated local and global bandwidths

n=500

x bandwidth

• 0.5

0.0 0.5 0.2 0.3 0.4 0.5 0.6 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 98 / 115

Model 2: Estimated local and global bandwidths

n=1000

x bandwidth

• 0.5

0.0 0.5 0.2 0.3 0.4 0.5 0.6 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 99 / 115

Model 2: Estimated local and global bandwidths

n=2000

x bandwidth

• 0.5

0.0 0.5 0.2 0.3 0.4 0.5 0.6 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 100 / 115

Model 2: πh(x) estimated functions

median of estimates

xx r(x)

• 0.5

0.0 0.5 0.8 1.0 1.2 1.4 1.6 true function n=500 n=1000 n=2000

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 101 / 115

Model 2: median of ∆(AMSE) estimates

0.0 0.1 0.2 0.3 0.4 0.5 0.6 n=500 n=1000 n=2000

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 102 / 115

Model 2: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=500

x

• 0.5

0.0 0.5 0.0 0.2 0.4 0.6 true function

Estimated volatility with GLOBAL bandwidth

x

• 0.5

0.0 0.5 0.0 0.2 0.4 0.6

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 103 / 115

Model 2: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=1000

x

• 0.5

0.0 0.5 0.0 0.2 0.4 0.6 true function

Estimated volatility with GLOBAL bandwidth

x

• 0.5

0.0 0.5 0.0 0.2 0.4 0.6

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 104 / 115

Model 2: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=2000

x

• 0.5

0.0 0.5 0.0 0.2 0.4 0.6 true function

Estimated volatility with GLOBAL bandwidth

x

• 0.5

0.0 0.5 0.0 0.2 0.4 0.6

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 105 / 115

Model 3: from Franke and Diagne (2006)

Xt =

• 0.01 + 0.1X 2

t−1 + 0.2X 2 t−1IXt−1<0ǫt;

ǫt ∼ N(0, 1)

TIME PLOT

200 400 600 800 1000

• 0.2

0.0 0.2

VOLATILITY FUNCTION

X(-1) volatility

• 0.2
• 0.1

0.0 0.1 0.2 0.010 0.015 0.020 0.025

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 106 / 115

Model 3: Estimated local and global bandwidths

n=500

x bandwidth

• 0.2
• 0.1

0.0 0.1 0.2 0.1 0.2 0.3 0.4 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 107 / 115

Model 3: Estimated local and global bandwidths

n=1000

x bandwidth

• 0.2
• 0.1

0.0 0.1 0.2 0.1 0.2 0.3 0.4 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 108 / 115

Model 3: Estimated local and global bandwidths

n=2000

x bandwidth

• 0.2
• 0.1

0.0 0.1 0.2 0.1 0.2 0.3 0.4 true estimated

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 109 / 115

Model 3: πh(x) estimated function

median of estimates

xx r(x)

• 0.2
• 0.1

0.0 0.1 0.2 0.8 1.0 1.2 1.4 1.6 1.8 true function n=500 n=1000 n=2000

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 110 / 115

Model 3: median of ∆(AMSE) estimates

0.0 0.1 0.2 0.3 0.4 0.5 0.6 n=500 n=1000 n=2000

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 111 / 115

Model 3: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=500

x

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.01 0.02 0.03 0.04 true function

Estimated volatility with GLOBAL bandwidth

x

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.01 0.02 0.03 0.04

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 112 / 115

Model 3: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=1000

x

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.01 0.02 0.03 0.04 true function

Estimated volatility with GLOBAL bandwidth

x

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.01 0.02 0.03 0.04

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 113 / 115

Model 3: LPE estimates of the volatility function

Estimated volatility with LOCAL bandwidth n=2000

x

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.01 0.02 0.03 0.04 true function

Estimated volatility with GLOBAL bandwidth

x

• 0.2
• 0.1

0.0 0.1 0.2 0.0 0.01 0.02 0.03 0.04

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 114 / 115

Conclusions

From the simulation study, it seems that the use of the local bandwidth sometimes does not produce better results. A global bandwidth derived on a suitable subset perform as well as the local bandwidth. Given a compact subset Iz, we derived a consistent estimator of the local bandwidth. Further research (under development):

1

improving the estimation of the derivative function m(p+1)

2

;

2

analysing the different orders of the two bandwidth estimators (local and global);

3

identifying some diagnostic tools useful for the choice of the most suitable type of smoothing;

4

extending the results to the multivariate framework.

• F. Giordano – M.L. Parrella (UNISA)

Local or Global Smoothing? COMPSTAT 2010 - Paris 115 / 115