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A collapsed dynamic factor analysis in STAMP Siem Jan Koopman Department of Econometrics, VU University Amsterdam Tinbergen Institute Amsterdam
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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