Model: the Dutch Labour Force Survey Oksana Bollineni-Balabay, Jan - - PowerPoint PPT Presentation

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm Accounting for Hyperparameter Uncertainty in SAE Based on a State-Space Model: the Dutch Labour Force Survey

The Dutch LFS

• monthly estimates for the total numbers of the

unemployed labour force;

• five-wave rotating panel survey (from Oct 1999);
• GREG estimator;
• 1st wave net sample size ≈ 6500 persons;
• a structural time series model in production since 2010(6)
• time span covered in this MSE study: 2001(1)-2010(6)

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Numbers of unemployed in NL: design- and model-based estimates

SE reduction: 24%

The DLFS model

Vector 𝒁𝒖 with GREG estimates for the 5 waves: 𝒁𝒖 = 𝑍

𝑢 𝐽

𝑍

𝑢 𝐽𝐽

𝑍

𝑢 𝐽𝐽𝐽

𝑍

𝑢 𝐽𝑊

𝑍

𝑢 𝑊

= 1 1 1 1 1 ξ𝑢 + 𝑆𝐻𝐶𝑢

𝐽𝐽

𝑆𝐻𝐶𝑢

𝐽𝐽𝐽

𝑆𝐻𝐶𝑢

𝐽𝑊

𝑆𝐻𝐶𝑢

𝑊

+ 𝑓𝑢

𝐽

𝑓𝑢

𝐽𝐽

𝑓𝑢

𝐽𝐽𝐽

𝑓𝑢

𝐽𝑊

𝑓𝑢

𝑊

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

true population parameter: ξ𝑢 = 𝑀𝑢 + 𝑇𝑢 survey errors rotation group bias

Stochastic components of the model

𝑀𝑢 - a stochastic trend with disturbances 𝜃𝑢~𝑂 0, 𝜏𝑀

2 ;

𝑇𝑢 - a trigonometric seasonal component with disturbances 𝜕𝑢~𝑂 0, 𝜏𝑇

2 ;

𝑆𝐻𝐶𝑢

𝐽𝐽−𝑊 - random walk with disturbances 𝜘~𝑂 0, 𝜏𝑆𝐻𝐶 2

; 𝑓𝑢

𝐽 = ν𝑢 𝐽;

𝑓𝑢

𝐽𝐽 = ρ𝑓𝑢−3 𝐽

+ ν𝑢

𝐽𝐽, etc.

Hyperparameter vector: 𝜾 = (𝜏𝑀

2, 𝜏𝑇 2, 𝜏𝑆𝐻𝐶 2

, 𝜏𝜉𝐽

2 , 𝜏𝜉𝐽𝐽 2 , 𝜏𝜉𝐽𝐽𝐽 2 , 𝜏𝜉𝐽𝑊 2 , 𝜏𝜉𝑊 2 , ρ)

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

survey errors with 𝜉𝑢

𝑥 ~𝑂 0, 𝜏𝜉𝑥 2

, w={1, … 5}; waves II-V as AR(1) not known, estimated

STS Model Estimation

• the Kalman filter extracts signals (trends…) 𝛽

𝑢|𝑢(𝜾);

• MSE of 𝛽

𝑢|𝑢(𝜾) at time t: 𝑁𝑇𝐹𝑢|𝑢 = 𝐹𝑢[𝛽 𝑢|𝑢 𝜾 −𝛽𝑢]2;

• the true MSE that accounts for uncertainty around 𝜾

: 𝑁𝑇𝐹𝑢|𝑢 = 𝐹𝑢[𝛽 𝑢|𝑢 𝜾 −𝛽𝑢]2+𝐹𝑢[𝛽 𝑢|𝑢(𝜾 )−𝛽 𝑢|𝑢 𝜾 ]2;

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

but 𝜾 used instead of 𝜾  𝑁𝑇𝐹𝑢|𝑢 is no longer the true MSE!

filter uncertainty hyperparameter uncertainty filter uncertainty

Methods to Account for Hyperparameter Uncertainty

• AA - asymptotic approximation (Hamilton (1986));

bootstraps:

• PT1 – Pfeffermann-Tiller, parametric;
• PT2 - Pfeffermann-Tiller, non-parametric;
• RR1 – Rodriguez-Ruiz, parametric;
• RR2 - Rodriguez-Ruiz, non-parametric;
• PT: 𝐹𝑢 taken unconditionally on the data;
• RR: 𝐹𝑢 taken conditionally on the original data set; claimed to

have better finite sample properties than PT.

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

(Pfeffermann and Tiller (2005)) Rodriguez and Ruiz (2012)

Monte-Carlo Study of MSE Approximation Approaches

• S=1000 series generated from the DLFS model;
• B=500 draws per series s made for AA;
• B=300 bootstrap series generated per series s for

PT1, PT2, RR1, RR2;

• true MSE obtained as:

𝑁𝑇𝐹𝑢

𝑈𝑆𝑉𝐹 = 1 50000

[𝛽 𝑛,𝑢 𝜾 − 𝛽𝑛,𝑢]2

𝑛=50000

;

• sample lengths: T=80, T=114, T=200 months;
• 4 versions of the DLFS model considered:

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Model 1 Model 2 Model 3 Model 4 Original model 𝜏𝑇

2=0

𝜏𝑆𝐻𝐶

2

=0 𝜏𝑇

2=𝜏𝑆𝐻𝐶 2

=0

Hyperparameter distribution under the DLFS model (Model 1)

ln(𝜏𝑀

2)

ln(𝜏𝑇

2)

ln(𝜏𝑆𝐻𝐶

2

) ln(𝜏𝜉𝐽

2 )

ln(𝜏𝜉𝐽𝐽

2 )

ln(𝜏𝜉𝐽𝐽𝐽

2 )

ln(𝜏𝜉𝐽𝑊

2 )

ln(𝜏𝜉𝑊

2 )

Hyperparameter distribution under Model 3

ln(𝜏𝑀

2)

ln(𝜏𝑇

2)

ln(𝜏𝜉𝐽

2 )

ln(𝜏𝜉𝐽𝐽

2 )

ln(𝜏𝜉𝐽𝐽𝐽

2 )

ln(𝜏𝜉𝐽𝑊

2 )

ln(𝜏𝜉𝑊

2 )

Signal MSE comparison for Model 3, T=114 months

Naive KF bias Naive KF bias

Signal MSE relative bias, %, averaged

• ver time T and simulations S

T=80 T=114 T=200

Models M1 M2 𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

M1 M2

𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

M1 M2

𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

KF

• 3.0
• 3.2
• 2.1
• 2.2
• 2.1 -2.6 -2.4 -2.2 -1.3 -1.6 -1.3 -1.3

AA

NA NA NA 14.9 NA NA NA 5.2 NA NA NA 5.9

PT1

8.6 6.7 4.9 6.2 8.1 5.7 3.3 5.5 6.3 6.2 6.3 5.5

PT2

4.8 3.7 1.4 2.1 2.2 3.2 1.9 1.5 6.8 4.0 3.0 2.3

RR1

• 7.2
• 9.0
• 7.3
• 7.2
• 8.3 -7.8 -6.4 -6.5 -8.0 -8.0 -4.9 -5.9

RR2

6.7

• 3.5
• 3.9
• 3.7
• 1.1 -6.0 -3.9 -3.5 -5.1 -5.6 -4.5 -5.0

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Signal MSE relative bias, %, averaged

• ver time T and simulations S

T=80 T=114 T=200

Models M1 M2 𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

M1 M2

𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

M1 M2

𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

KF

• 3.0
• 3.2
• 2.1
• 2.2
• 2.1 -2.6 -2.4 -2.2 -1.3 -1.6 -1.3 -1.3

AA

NA NA NA 14.9 NA NA NA 5.2 NA NA NA 5.9

PT1

8.6 6.7 4.9 6.2 8.1 5.7 3.3 5.5 6.3 6.2 6.3 5.5

PT2

4.8 3.7 1.4 2.1 2.2 3.2 1.9 1.5 6.8 4.0 3.0 2.3

RR1

• 7.2
• 9.0
• 7.3
• 7.2
• 8.3 -7.8 -6.4 -6.5 -8.0 -8.0 -4.9 -5.9

RR2

6.7

• 3.5
• 3.9
• 3.7
• 1.1 -6.0 -3.9 -3.5 -5.1 -5.6 -4.5 -5.0

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Signal MSE relative bias, %, averaged

• ver time T and simulations S

T=80 T=114 T=200

Models M1 M2 𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

M1 M2

𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

M1 M2

𝜏𝑇

2

M3

𝜏𝑆𝐻𝐶

2

M4

𝜏𝑇

2

𝜏𝑆𝐻𝐶

2

KF

• 3.0
• 3.2
• 2.1
• 2.2
• 2.1 -2.6 -2.4 -2.2 -1.3 -1.6 -1.3 -1.3

AA

NA NA NA 14.9 NA NA NA 5.2 NA NA NA 5.9

PT1

8.6 6.7 4.9 6.2 8.1 5.7 3.3 5.5 6.3 6.2 6.3 5.5

PT2

4.8 3.7 1.4 2.1 2.2 3.2 1.9 1.5 6.8 4.0 3.0 2.3

RR1

• 7.2
• 9.0
• 7.3
• 7.2
• 8.3 -7.8 -6.4 -6.5 -8.0 -8.0 -4.9 -5.9

RR2

6.7

• 3.5
• 3.9
• 3.7
• 1.1 -6.0 -3.9 -3.5 -5.1 -5.6 -4.5 -5.0

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Conclusions

– the naive KF MSE does not have huge biases in the DLFS model ; – MSE biases become smaller with the series length; – AA may fail in models with small hyperparameters; – non-parametric bootstraps overperform the parametric

• nes;

– RR perform consistently worse than PT-bootstraps, with negative biases larger than those of the naive Kalman filter.

Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm