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# State Space Model

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### State Space Model

1. 1. A collapsed dynamic factor analysis in STAMP Siem Jan Koopman Department of Econometrics, VU University Amsterdam Tinbergen Institute Amsterdam
2. 2. Univariate time series forecastingIn macroeconomic forecasting, time series methods are often used: • Random walk : yt = yt−1 + εt ; • Autoregression : yt = µ + φ1 yt−1 + . . . + φp yt−p + εt ; • Nonparametric methods; • Unobserved components : . . . 2 / 24
3. 3. Trend and cycle decompositionMany macroeconomic time series can be decomposed into trendand cyclical dynamic eﬀects.For example, we can consider the trend-cycle decomposition 2 yt = µt + ψt + εt , εt ∼ NID(0, σε ),where the unobserved components trend µt and cycle ψt arestochastically time-varying with possible dynamic speciﬁcations 2 µt = µt−1 + β + ηt , ηt ∼ NID(0, ση ), 2 ψt = φ1 ψt−1 + φ2 ψt−2 + κt , κt ∼ NID(0, σκ ),for t = 1, . . . , n. 3 / 24
4. 4. Kalman ﬁlter methodsTime series models can be uniﬁed in the state space formulation yt = Zt αt + εt , αt = Tt αt−1 + Rt ηt ,with state vector αt and disturbance vectors εt and ηt ; matricesZt , Tt and Rt (together with the disturbance variance matrices)determine the dynamic properties of yt .Kalman ﬁlter and related methods facilitate parameter estimation(by exact MLE), signal extraction (tracking the dynamics) andforecasting. 4 / 24
5. 5. Limitations of univariate time seriesUnivariate time series is a good starting point for analysis.It draws attention on the dynamic properties of a time series.Limitations : • Information in related time series may be used in the analysis; • Established relations between time series should be explored; • Interesting to understand dynamic relations between time series; • Economic theory can be veriﬁed; • Simultaneous eﬀects to variables when events occur; • Forecasting should be more precise, does it ?Hence, the many diﬀerent discussions in economic time seriesmodelling and economic forecasting. 5 / 24
6. 6. Features of Large Economic Databases• Quarterly and Monthly time series• Unbalanced panels : many series may be incomplete• Hence many missing observations• Series are transformed in growth terms (stationary)• Series are ”seasonally adjusted”, ”detrended”, etc. 6 / 24
7. 7. Multivariate time series with mixed frequenciesDeﬁne ytzt = , yt = target variable, xt = macroeconomic panel. xtThe time index t is typically in months.Quarterly frequency variables have missing entries for the monthsJan, Feb, April, May, July, Aug, Oct and Nov.Stocks and ﬂows should be treated diﬀerently;this requires further work as in Proietti (2008). 7 / 24
8. 8. State space dynamic factor modelThe state space dynamic factor model is given by zt = µ + Λft + εt , ft = Φ1 ft−1 + Φ2 ft2 + ηt ,where µ is a constant vector, Λ is matrix of factor loadings, ft isdynamic factor modelled as a VAR(2) and εt is a disturbance term.The panel size N can be relatively large while the time seriesdimension can be relatively short.The coeﬃcients in the loading matrix Λ, the VAR and variancematrices need to be estimated; see Watson and Engle (1983),Shumway and Stoﬀer (1982), Jungbacker and Koopman (2008).We can reduce the dimension of zt by replacing xt for a limitednumber of principal components which we denote by gt ; see thesuggestions in Stock and Watson (2002). 8 / 24
9. 9. Stock and Watson (2002)Consider the macroeconomic panel xt and apply principalcomponent analysis. Missing values can be treated via an EMmethod.The q extracted principal components (PCs) vector time series arelabelled as gt .The PCs are then used in autoregressive model for yt , yt = µ + φ1 yt−1 + . . . + φp yt−p + β1 gt−1 + βq gt−q + ξt ,where ξt is a disturbance term. • construction of PCs gt do not involve yt • PCs gt can be noisy indicators 9 / 24
10. 10. Collapsed dynamic factor modelThe collapsed dynamic factor model is given by yt = µy + ψt + λ′ Ft + εy ,t , gt = Ft + εg ,t ,where ψt ∼ AR(2), Ft ∼ VAR(2). Since Var(gt ) = I byconstruction, we can treat the elements of Ft as independentAR(2)s.The model is reduced to a parsimonious dynamic factor model.Realistic model for yt : own dynamics in ψt whereas parameters inλ determine what additional information from Ft is needed.We do not insert gt directly in equation for yt : not interested inthe noise of gt , only in the signal Ft . 10 / 24
11. 11. Collapsed dynamic factor modelThe collapsed dynamic factor model is given by yt = µy + ψt + λ′ Ft + εy ,t , gt = Ft + εg ,t ,where ψt ∼ AR(2), Ft ∼ VAR(2). Since Var(gt ) = I byconstruction, we can treat the elements of Ft as independentAR(2)s.It relates to recent work by Doz, Giannone and Reichlin (2011, J ofEct) in which they show that an ad-hoc dynamic factor approachwhere the loadings are set equal to the eigenvectors of theprincipal components lead to consistent estimates of the factors.The model can also be useful for univariate trend-cycledecompositions when the time series span is short. The cycle ψtmay not be empirically identiﬁed; the Ft may be functional tocapture the cyclical properties in the time series. 11 / 24
12. 12. Collapsed state space dynamic factor modelHence the model in state space form is given by yt µ 1 λ′ ψt = + + εt , gt 0 0 Iq Ftfor t = 1, . . . , n, where ψt ∼ AR(2), Ft ∼ VAR(2), Var(ǫt ) = Dε .The time series of yt can be quarterly and of gt is monthly.We can simplify the model further by approximating ψt as aweighted sum of lagged yt′ s since yt is a stationary process. 12 / 24
13. 13. Collapsed state space dynamic factor modelHence the model in state space form is given by yt µ 1 λ′ ψt = + + εt , gt 0 0 Iq Ftfor t = 1, . . . , T , where ψt ∼ AR(2), Ft ∼ VAR(2), Var(ǫt ) = Dε .Here, VAR(2) consists of q cross-independent AR(2)’s. Weconsider diﬀerent q’s. 13 / 24
14. 14. PCs and their smoothed signals 5.0 2.5 0.0 1960 1965 1970 1975 1980 1985 1990 1995 2000 2.5 0.0−2.5 1960 1965 1970 1975 1980 1985 1990 1995 2000 2.5 0.0−2.5 1960 1965 1970 1975 1980 1985 1990 1995 2000 14 / 24
15. 15. Personal Income and its smoothed signal 4 3 2 1 0−1−2−3−4−5 1960 1965 1970 1975 1980 1985 1990 1995 2000 15 / 24
16. 16. Forecasting set-upWe follow the forecasting approach of Stock and Watson (2002)using the data set ”sims.xls” of SW (2005). The target variable isyth as given by 1200 yth = (log Pt − log Pt−h ) , hwhere Pt is typically an I(1) economic variable (eg Pt = IPI).We generate forecasts of yth for horizons 1, 6, 12 and 24 monthsahead. The following models are considered ˆh • Random walk yT +j = yT ˆh • AR(2) : yT +j = γh1 yT + γh2 yT −1 ˆ ˆ ˆh ˆ′ ˆ • Stock and Watson : yT +j = βh gT + γh1 yT + γh2 yT −1 ˆ ˆ ˆh • MUC : reduced MUC for (yt′ , gt′ )′ : yT +j from Kalman ﬁlter ˆfor j = 1, 6, 12, 24, both γ and β are estimated by OLS. ˆ 16 / 24
17. 17. Out-of-Sample Forecasting : designOur forecasting results are based on a rolling-sample starting atJanuary 1970 and ending at December 2003 (nr.forecasts is391 − h).Depending on forecasting horizon, we have, say, 400 forecasts.We compute the following forecast error statistics : Hj −1 MSE = Hj−1 (yT +i +j − yT +i +j )2 , h h i =0 Hj −1 h h MAE = Hj−1 |yT +i +j − yT +i +j |, i =0with number of forecasts Hj and forcast horizon j.The signiﬁcance of the gain in forecasting precision against abenchmark model is measured using the Superior Predictive Ability(SPA) test of Hansen. 17 / 24
18. 18. Out-of-Sample Forecasting : Personal Income 1-month ahead 6-month ahead MSE MAE MSE MAE RW 2.3939 1.1059 1.5626 0.9158 AR(2) 0.9588 0.7099 1.0340 0.7674 SW(1) 0.9407 0.7041 1.1930 0.7894 SW(2) 0.8873 0.6919 0.8956 0.7188 SW(3) 0.8848 0.6926 0.8328 0.6876 SW(4) 0.8893 0.6953 0.8162 0.6695 SW(BaiNg) 0.8934 0.6942 0.8495 0.6974 MUC(1) 0.9317 0.7002 1.1261 0.7794 MUC(2) 0.8756 0.6840 0.8815 0.7099 MUC(3) 0.8784 0.6840 0.8947 0.7042 MUC(4) 0.8636 0.6784 0.8161 0.6790 MUC(BaiNg) 0.8761 0.6825 0.8882 0.6990 18 / 24
19. 19. Out-of-Sample Forecasting : Personal Income 12-month ahead 24-month ahead MSE MAE MSE MAE RW 1.5946 0.9461 1.9735 1.1032 AR(2) 0.9781 0.7504 0.7965 0.6750 SW(1) 1.2361 0.8072 0.9125 0.7248 SW(2) 0.8865 0.7210 0.7559 0.6861 SW(3) 0.8656 0.7095 0.8082 0.7078 SW(4) 0.8537 0.7030 0.8023 0.6964 SW(BaiNg) 0.8738 0.7224 0.8339 0.7314 MUC(1) 1.1528 0.7937 0.8698 0.7128 MUC(2) 0.8641 0.7151 0.7136 0.6731 MUC(3) 0.9188 0.7252 0.7879 0.7025 MUC(4) 0.8336 0.7061 0.7555 0.6911 MUC(BaiNg) 0.9022 0.7284 0.7918 0.7088 19 / 24
20. 20. Out-of-Sample Forecasting : Industrial Production 1-month ahead 6-month ahead MSE MAE MSE MAE RW 1.6046 0.9562 1.4264 0.8268 AR(2) 0.9249 0.7217 0.9280 0.6943 SW(1) 0.8057 0.6773 0.8160 0.6675 SW(2) 0.8028 0.6813 0.7933 0.6769 SW(3) 0.7881 0.6738 0.6837 0.6347 SW(4) 0.7740 0.6718 0.6972 0.6371 SW(BaiNg) 0.7865 0.6751 0.6961 0.6376 MUC(1) 0.8485 0.6899 0.9398 0.7242 MUC(2) 0.8470 0.6888 0.9371 0.7291 MUC(3) 0.8428 0.6941 0.8807 0.7211 MUC(4) 0.8323 0.6998 0.7458 0.6746 MUC(BaiNg) 0.8323 0.6928 0.8355 0.7103 20 / 24
21. 21. Out-of-Sample Forecasting : Industrial Production 12-month ahead 24-month ahead MSE MAE MSE MAE RW 1.6616 0.9657 2.4495 1.2178 AR(2) 0.9176 0.7301 0.9044 0.7723 SW(1) 0.8419 0.6825 0.8876 0.7482 SW(2) 0.8432 0.7325 0.9098 0.7460 SW(3) 0.7130 0.6721 0.9945 0.7735 SW(4) 0.7341 0.6689 0.9654 0.7750 SW(BaiNg) 0.7308 0.6760 1.0337 0.8026 MUC(1) 0.8699 0.7165 0.8567 0.7483 MUC(2) 0.8541 0.7369 0.8515 0.7291 MUC(3) 0.7740 0.7025 0.9450 0.7520 MUC(4) 0.7034 0.6602 0.8772 0.7400 MUC(BaiNg) 0.7485 0.6854 0.9627 0.7725 21 / 24
22. 22. Out-of-Sample Forecasting : Quarterly GDP 1-month ahead 2-month ahead MSE MAE MSE MAE RW 1.4659 0.9038 1.4659 0.9038 AR(2) 1.3540 0.8609 1.3540 0.8609 SW(1) 1.3715 0.8814 1.3715 0.8814 SW(2) 1.3512 0.8617 1.3512 0.8617 SW(3) 1.3330 0.8515 1.3330 0.8515 SW(4) 1.3327 0.8529 1.3327 0.8529 SW(BaiNg) 1.3304 0.8497 1.3304 0.8497 MUC(1) 1.3605 0.8711 1.2848 0.8259 MUC(2) 1.3848 0.8680 1.3020 0.8288 MUC(3) 1.3523 0.9014 1.2102 0.8286 MUC(4) 1.5355 0.9705 1.2303 0.8309 MUC(BaiNg) 1.4213 0.8837 1.2717 0.8235 22 / 24
23. 23. Out-of-Sample Forecasting : Quarterly GDP 6-month ahead 12-month ahead MSE MAE MSE MAE RW 3.7226 1.4284 9.1548 2.3024 AR(2) 3.2971 1.3690 8.1081 2.2408 SW(1) 3.6037 1.4231 10.150 2.3537 SW(2) 3.3012 1.3879 8.3149 2.2384 SW(3) 3.2910 1.3686 8.4193 2.2692 SW(4) 3.3125 1.3738 8.2524 2.2433 SW(BaiNg) 3.2825 1.3698 8.3608 2.2379 MUC(1) 3.2195 1.3464 8.0067 2.2186 MUC(2) 3.2696 1.3614 7.3850 2.1625 MUC(3) 3.2455 1.3395 6.4904 2.0133 MUC(4) 2.9741 1.3040 6.8064 2.0657 MUC(BaiNg) 3.2871 1.3674 7.3965 2.1562 23 / 24
24. 24. ConclusionsWe have presented a basic DFM framework for incorporating amacroeconomic panel for the forecasting of key economic variables.This methodology will be implemented for STAMP 9.Possible extensions: • Forecasting results are promising, specially for long-term • Short-term forecasting : diﬀerent approaches produce similar results. • Interpolation results (nowcasting) need to be analysed • Inclusion of lagged factors • Separate PCs for leading / lagging economic indicators • Treatments for stock and ﬂow variables 24 / 24