ceps/instead, g.-d. luxembourg · .1.15.2 5 15 25 35 h = 1.11 (default) variable bandwidth...

Adaptive kernel density estimation in Stata

Philippe Van KermCEPS/INSTEAD, G.-D. Luxembourg

9th London Stata User Group Meeting, 19-20 May 2003

Kernel density function estimation

• Official command: kdensity

f̂f (x) =1n

n∑i=1

1h

K

(x− xi

h

)(1)

– ‘Point mass’ of sample data diffused around xi’s, and averaged at x

Adaptive kernel density estimation May 17, 2003

Kernel density function estimation

• Official command: kdensity

f̂f (x) =1n

n∑i=1

1h

K

(x− xi

h

)(1)

– ‘Point mass’ of sample data diffused around xi’s, and averaged at x

– Fixed/constant/global bandwidth h: ‘degree of diffusion’ constant for allxi’s


Adaptive kernel density function estimation

• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)

– Different bandwidths for different xi’s



• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)


– Degree of diffusion varies inversely with f(xi)



• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)



– Greater precision where data are abundant and ...

– ... greater smoothness where data are sparse



• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)





• Adaptive two-stage estimator (Abramson 1982):

– hi = h× λi;



• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)






– hi = h× λi; λi =(G/f̃(xi)

)0.5



• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)







)0.5

– First step: compute pilot estimate (fixed bandwidth h) to generate λi



• akdensity

f̂v(x) =1n

n∑i=1

1hi

K

(x− xi

hi

)(2)







)0.5

– First step: compute pilot estimate (fixed bandwidth h) to generate λi

– Second step: compute density estimate with hi local bandwidths


Variability bands as a bonus

• akdensity estimates variability bands:

– f̂v(x)± b× SE(x)




– f̂v(x)± b× SE(x)

– Pointwise variability bands




– f̂v(x)± b× SE(x)


– Not confidence intervals (accounts for sample variability but not bias!)




– f̂v(x)± b× SE(x)


– Not confidence intervals (accounts for sample variability but not bias!)

• Also for fixed bandwidth kernel estimates!


Syntax extract

• ‘High-level’ command (mimicks kdensity ):

akdensity varname ...[

, noadap tive width( #)[epan | gau ss

]stdb ands( #) ...

]


Syntax extract

• ‘High-level’ command (mimicks kdensity ):

akdensity varname ...[

, noadap tive width( #)[epan | gau ss

]stdb ands( #) ...

]

• ‘Low-level’ command (rarely used, but full control):

akdensity0 varname ... , width( # | varname) ...[

lambda( string) ...]


A simulated example

0.0

5.1

.15

dens

ity

0 10 20 30 40

True underlying density


0.0

5.1

.15

5 15 25 35

h = 1.11 (default)

FIXED bandwidth estimates


0.0

5.1

.15

5 15 25 35

h = 0.83



0.0

5.1

.15

5 15 25 35

h = 0.55



−.05

0.0

5.1

.15

.2

5 15 25 35

h = 1.11 (default)

VARIABLE bandwidth estimates


−.05

0.0

5.1

.15

.2

5 15 25 35

h = 0.83



−.05

0.0

5.1

.15

.2

5 15 25 35

h = 0.55



−.05

0.0

5.1

.15

.2

5 15 25 35

h = 1.11 (default) h = 0.83h = 0.55


−.05

0.0

5.1

.15

.2

5 15 25 35

h = 1.11 (default) h = 0.83h = 0.55



0.0

5.1

.15

.2

Variability−bias tradeoff−.

04−.

020

.02

.04

Poi

ntw

ise

std.

err

or.

5 15 25 35


A real example

KRTKPTMXGRJSPIR ITAFIBFISNONLUKSCAAUSDKNZLUS CH

1960

KRTK PTMX GR JSPIR ITAFIBFISNO NLUK SCAAUSDKNZ L US CH

1970

KRTK PTMX GR JSPIR ITAFIBF ISNONLUK S CAAUSDKNZ L USCH

1980

0 5000 10000 15000 20000 25000 30000

KRTK PTMX GR JSPIR ITAFIBF ISNONLUK S CAAUSDKNZ LUSCH

1990

0 5000 10000 15000 20000 25000 30000

Real GDP per capita − FIXED and VARIABLE bandwidths


KRTK PTMX GR JSPIR ITAFI BFISNO NLUK SCAAUS DKNZ L US CH

1970


KRTK PTMX GR JSPIR ITAFI BF ISNONLUK S CAAUSDKNZ L USCH

1980

0 5000 10000 15000 20000 25000 30000


Final remarks

• Fast implementation for large datasets (use of linear interpolation for thepilot estimate)


Final remarks


• Works under both version 7 and version 8:


Final remarks



– use new graphics engine if called from Stata 8, and former engine ifcalled from Stata 7


Final remarks




– graphics options therefore vary (in Stata 8, option plot(...)

especially useful!)


Final remarks





especially useful!)

– coded using if caller()<8 statements


Final remarks





especially useful!)

– coded using if caller()<8 statements

• Available in next Stata Journal issue (vol.3(2))


ceps/instead, g.-d. luxembourg · .1.15.2 5 15 25 35 h = 1.11 (default) variable bandwidth...

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