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

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1 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

2 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 010(6) time span covered in this MSE study: 001(1)-010(6)

3 Numbers of unemployed in NL: design- and model-based estimates SE reduction: 4%

4 The DLFS model Vector Y t with GREG estimates for the 5 waves: Y t = Y t I Y t II Y t III Y t IV Y t V = ξ t + 0 RGB t II RGB t III RGB t IV RGB t V + e t I e t II e t III e t IV e t V true population parameter: ξ t = L t + S t rotation group bias survey errors

5 Stochastic components of the model L t - a stochastic trend with disturbances η t ~N 0, σ L ; S t - a trigonometric seasonal component with disturbances ω t ~N 0, σ S ; II V RGB t - random walk with disturbances θ~n 0, ; e I t = ν I t ; e II I t = ρe t 3 + ν t II, etc. survey errors with w ν t ~N 0, σ ν w, w={1, 5}; waves II-V as AR(1) Hyperparameter vector: θ = (σ L, σ S,, σ ν I, σ ν II, σ ν III, σ ν IV, σ ν V, ρ) not known, estimated

6 STS Model Estimation the Kalman filter extracts signals (trends ) α t t (θ); MSE of α t t (θ) at time t: MSE t t = E t [α t t θ α t ] ; but θ used instead of θ MSE t t is no longer the true MSE! the true MSE that accounts for uncertainty around θ: MSE t t = E t [α t t θ α t ] +E t [α t t (θ) α t t θ ] ; filter uncertainty hyperparameter uncertainty

7 Methods to Account for Hyperparameter Uncertainty AA - asymptotic approximation (Hamilton (1986)); bootstraps: PT1 Pfeffermann-Tiller, parametric; PT - Pfeffermann-Tiller, non-parametric; RR1 Rodriguez-Ruiz, parametric; RR - Rodriguez-Ruiz, non-parametric; -PT: E t taken unconditionally on the data; -RR: E t taken conditionally on the original data set; claimed to have better finite sample properties than PT. (Pfeffermann and Tiller (005)) Rodriguez and Ruiz (01)

8 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, PT, RR1, RR; true MSE obtained as: MSE TRUE t = [α m,t θ α m,t ] m=50000 ; sample lengths: T=80, T=114, T=00 months; 4 versions of the DLFS model considered: Model 1 Model Model 3 Model 4 Original model σ S =0 =0 σ S = =0

9 Hyperparameter distribution under the DLFS model (Model 1) ln(σ L ) ln( ) ln( ) ln(σ ν I) ln(σ ν II) ln(σ ν III) ln(σ ν IV) ln(σ ν V)

10 Hyperparameter distribution under Model 3 ln(σ L ) ln( ) ln(σ ν I) ln(σ ν II) ln(σ ν III) ln(σ ν IV) ln(σ ν V)

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

12 Signal MSE relative bias, %, averaged over time T and simulations S Models M1 M T=80 T=114 T=00 M1 M M1 M KF AA NA NA NA 14.9 NA NA NA 5. NA NA NA 5.9 PT PT RR RR

13 Signal MSE relative bias, %, averaged over time T and simulations S Models M1 M T=80 T=114 T=00 M1 M M1 M KF AA NA NA NA 14.9 NA NA NA 5. NA NA NA 5.9 PT PT RR RR

14 Signal MSE relative bias, %, averaged over time T and simulations S Models M1 M T=80 T=114 T=00 M1 M M1 M KF AA NA NA NA 14.9 NA NA NA 5. NA NA NA 5.9 PT PT RR RR

15 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.