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

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

The Dutch LFS • monthly estimates for the total numbers of the unemployed labour force; • five-wave rotating panel survey (from Oct 1999); • GREG estimator; • 1 st 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 𝒁 𝒖 with GREG estimates for the 5 waves: 𝐽 𝐽 𝑍 𝑓 𝑢 0 𝑢 1 𝐽𝐽 𝑆𝐻𝐶 𝑢 𝐽𝐽 𝐽𝐽 𝑍 𝑓 𝑢 1 𝑢 𝐽𝐽𝐽 𝑆𝐻𝐶 𝑢 𝐽𝐽𝐽 𝐽𝐽𝐽 1 𝒁 𝒖 = = ξ 𝑢 + + 𝑍 𝑓 𝑢 𝑢 𝐽𝑊 1 𝑆𝐻𝐶 𝑢 𝐽𝑊 𝐽𝑊 𝑍 𝑓 𝑢 𝑢 1 𝑊 𝑆𝐻𝐶 𝑢 𝑊 𝑊 𝑍 𝑓 𝑢 𝑢 rotation true population survey group parameter: ξ 𝑢 = 𝑀 𝑢 + 𝑇 𝑢 errors bias Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Stochastic components of the model 2 ; 𝑀 𝑢 - a stochastic trend with disturbances 𝜃 𝑢 ~𝑂 0, 𝜏 𝑀 𝑇 𝑢 - a trigonometric seasonal component with 2 ; disturbances 𝜕 𝑢 ~𝑂 0, 𝜏 𝑇 𝐽𝐽−𝑊 - random walk with disturbances 𝜘~𝑂 0, 𝜏 𝑆𝐻𝐶 2 𝑆𝐻𝐶 𝑢 ; 𝐽 = ν 𝑢 survey errors with 𝐽 ; 𝑓 𝑢 2 𝑥 ~𝑂 0, 𝜏 𝜉 𝑥 𝜉 𝑢 , w ={1, … 5}; 𝐽𝐽 = ρ𝑓 𝑢−3 𝐽 𝐽𝐽 , etc. 𝑓 𝑢 + ν 𝑢 waves II-V as AR(1) not Hyperparameter vector: known, 2 , 𝜏 𝜉 𝐽𝐽 2 , 𝜏 𝜉 𝐽𝐽𝐽 2 , 𝜏 𝜉 𝐽𝑊 2 , 𝜏 𝜉 𝑊 2 , ρ) 2 , 𝜏 𝑇 2 , 𝜏 𝑆𝐻𝐶 2 𝜾 = (𝜏 𝑀 , 𝜏 𝜉 𝐽 estimated Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

STS Model Estimation • the Kalman filter extracts signals (trends…) 𝛽 𝑢|𝑢 (𝜾) ; • MSE of 𝛽 𝑢|𝑢 (𝜾) at time t : used instead of 𝜾  but 𝜾 𝑁𝑇𝐹 𝑢|𝑢 is no longer the true MSE! 𝑢|𝑢 𝜾 − 𝛽 𝑢 ] 2 ; 𝑁𝑇𝐹 𝑢|𝑢 = 𝐹 𝑢 [𝛽 : • the true MSE that accounts for uncertainty around 𝜾 ) − 𝛽 𝑢|𝑢 𝜾 − 𝛽 𝑢 ] 2 +𝐹 𝑢 [𝛽 𝑢|𝑢 𝜾 ] 2 ; 𝑁𝑇𝐹 𝑢|𝑢 = 𝐹 𝑢 [𝛽 𝑢|𝑢 (𝜾 filter uncertainty filter uncertainty hyperparameter uncertainty Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Methods to Account for Hyperparameter Uncertainty • AA - asymptotic approximation (Hamilton (1986)); bootstraps: (Pfeffermann • PT1 – Pfeffermann-Tiller, parametric; and Tiller • PT2 - Pfeffermann-Tiller, non-parametric; (2005)) • RR1 – Rodriguez-Ruiz, parametric; Rodriguez and • RR2 - Rodriguez-Ruiz, non-parametric; Ruiz (2012) - PT : 𝐹 𝑢 taken unconditionally on the data; - RR : 𝐹 𝑢 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

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: 𝑈𝑆𝑉𝐹 = 1 − 𝛽 𝑛,𝑢 ] 2 50000 𝑁𝑇𝐹 𝑢 [𝛽 𝑛,𝑢 𝜾 ; 𝑛=50000 • sample lengths: T=80 , T=114 , T=200 months; • 4 versions of the DLFS model considered: Model 1 Model 2 Model 3 Model 4 2 =0 2 2 = 𝜏 𝑆𝐻𝐶 2 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 Original model =0 =0 Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Hyperparameter distribution under the DLFS model (Model 1) 2 ) 2 ) ln( 𝜏 𝑀 ln( 𝜏 𝑇 2 ) 2 ln( 𝜏 𝑆𝐻𝐶 ) ln( 𝜏 𝜉 𝐽 2 ) 2 ) ln( 𝜏 𝜉 𝐽𝐽𝐽 ln( 𝜏 𝜉 𝐽𝐽 2 ) 2 ) ln( 𝜏 𝜉 𝑊 ln( 𝜏 𝜉 𝐽𝑊

Hyperparameter distribution under Model 3 2 ) 2 ) ln( 𝜏 𝑇 ln( 𝜏 𝑀 2 ) 2 ) ln( 𝜏 𝜉 𝐽𝐽 ln( 𝜏 𝜉 𝐽 2 ) 2 ) ln( 𝜏 𝜉 𝐽𝑊 ln( 𝜏 𝜉 𝐽𝐽𝐽 2 ) ln( 𝜏 𝜉 𝑊

Signal MSE comparison for Model 3, T=114 months Naive Naive KF KF bias bias

Signal MSE relative bias, %, averaged over time T and simulations S T=80 T=114 T=200 Models M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 2 2 2 2 2 2 2 2 2 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 2 2 2 𝜏 𝑆𝐻𝐶 𝜏 𝑆𝐻𝐶 𝜏 𝑆𝐻𝐶 -3.0 -3.2 -2.1 -2.2 -2.1 -2.6 -2.4 -2.2 -1.3 -1.6 -1.3 -1.3 KF NA NA NA 14.9 NA NA NA 5.2 NA NA NA 5.9 AA 8.6 6.7 4.9 6.2 8.1 5.7 3.3 5.5 6.3 6.2 6.3 5.5 PT1 4.8 3.7 1.4 2.1 2.2 3.2 1.9 1.5 6.8 4.0 3.0 2.3 PT2 -7.2 -9.0 -7.3 -7.2 -8.3 -7.8 -6.4 -6.5 -8.0 -8.0 -4.9 -5.9 RR1 6.7 -3.5 -3.9 -3.7 -1.1 -6.0 -3.9 -3.5 -5.1 -5.6 -4.5 -5.0 RR2 Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Signal MSE relative bias, %, averaged over time T and simulations S T=80 T=114 T=200 Models M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 2 2 2 2 2 2 2 2 2 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 2 2 2 𝜏 𝑆𝐻𝐶 𝜏 𝑆𝐻𝐶 𝜏 𝑆𝐻𝐶 -3.0 -3.2 -2.1 -2.2 -2.1 -2.6 -2.4 -2.2 -1.3 -1.6 -1.3 -1.3 KF NA NA NA 14.9 NA NA NA 5.2 NA NA NA 5.9 AA 8.6 6.7 4.9 6.2 8.1 5.7 3.3 5.5 6.3 6.2 6.3 5.5 PT1 4.8 3.7 1.4 2.1 2.2 3.2 1.9 1.5 6.8 4.0 3.0 2.3 PT2 -7.2 -9.0 -7.3 -7.2 -8.3 -7.8 -6.4 -6.5 -8.0 -8.0 -4.9 -5.9 RR1 6.7 -3.5 -3.9 -3.7 -1.1 -6.0 -3.9 -3.5 -5.1 -5.6 -4.5 -5.0 RR2 Oksana Bollineni-Balabay, Jan van den Brakel, Franz Palm

Signal MSE relative bias, %, averaged over time T and simulations S T=80 T=114 T=200 Models M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 2 2 2 2 2 2 2 2 2 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 𝜏 𝑇 𝜏 𝑆𝐻𝐶 𝜏 𝑇 2 2 2 𝜏 𝑆𝐻𝐶 𝜏 𝑆𝐻𝐶 𝜏 𝑆𝐻𝐶 -3.0 -3.2 -2.1 -2.2 -2.1 -2.6 -2.4 -2.2 -1.3 -1.6 -1.3 -1.3 KF NA NA NA 14.9 NA NA NA 5.2 NA NA NA 5.9 AA 8.6 6.7 4.9 6.2 8.1 5.7 3.3 5.5 6.3 6.2 6.3 5.5 PT1 4.8 3.7 1.4 2.1 2.2 3.2 1.9 1.5 6.8 4.0 3.0 2.3 PT2 -7.2 -9.0 -7.3 -7.2 -8.3 -7.8 -6.4 -6.5 -8.0 -8.0 -4.9 -5.9 RR1 6.7 -3.5 -3.9 -3.7 -1.1 -6.0 -3.9 -3.5 -5.1 -5.6 -4.5 -5.0 RR2 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 ones; – 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