Paris2012 session4

A collapsed dynamic factor analysis in STAMP

Siem Jan Koopman

Department of Econometrics, VU University AmsterdamTinbergen Institute Amsterdam

Univariate time series forecasting

In macroeconomic forecasting, time series methods are often used:

• Random walk : yt = yt−1 + εt ;

• Autoregression : yt = µ+ φ1yt−1 + . . .+ φpyt−p + εt ;

• Nonparametric methods;

• Unobserved components : . . .

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Trend and cycle decomposition

Many macroeconomic time series can be decomposed into trendand cyclical dynamic effects.

For example, we can consider the trend-cycle decomposition

yt = µt + ψt + εt , εt ∼ NID(0, σ2ε ),

where the unobserved components trend µt and cycle ψt arestochastically time-varying with possible dynamic specifications

µt = µt−1 + β + ηt , ηt ∼ NID(0, σ2η),

ψt = φ1ψt−1 + φ2ψt−2 + κt , κt ∼ NID(0, σ2κ),

for t = 1, . . . , n.

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Kalman filter methods

Time series models can be unified in the state space formulation

yt = Ztαt + εt , αt = Ttαt−1 + Rtηt ,

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.

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Limitations of univariate time series

Univariate 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 timeseries;

• 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.

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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.

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Multivariate time series with mixed frequencies

Define

zt =

(ytxt

), yt = target variable, xt = macroeconomic panel.

The 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).

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State space dynamic factor model

The state space dynamic factor model is given by

zt = µ+ Λft + εt , ft = Φ1ft−1 +Φ2ft2 + 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).

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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+ φ1yt−1 + . . .+ φpyt−p + β1gt−1 + βqgt−q + ξt ,

where ξt is a disturbance term.

• construction of PCs gt do not involve yt

• PCs gt can be noisy indicators

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Collapsed dynamic factor model

The 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 .

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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 ψt

may not be empirically identified; the Ft may be functional tocapture the cyclical properties in the time series.

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Collapsed state space dynamic factor model

Hence the model in state space form is given by

(ytgt

)=

(µ

0

)+

[1 λ′

0 Iq

](ψt

Ft

)+ εt ,

for 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 y ′ts since yt is a stationary process.

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Collapsed state space dynamic factor model

Hence the model in state space form is given by

(ytgt

)=

(µ

0

)+

[1 λ′

0 Iq

](ψt

Ft

)+ εt ,

for 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.

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PCs and their smoothed signals

1960 1965 1970 1975 1980 1985 1990 1995 2000

0.0

2.5

5.0

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

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Personal Income and its smoothed signal

1960 1965 1970 1975 1980 1985 1990 1995 2000

−5

−4

−3

−2

−1

0

1

2

3

4

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Forecasting set-upWe follow the forecasting approach of Stock and Watson (2002)using the data set ”sims.xls” of SW (2005). The target variable isyht as given by

yht =1200

h(logPt − logPt−h) ,

where Pt is typically an I(1) economic variable (eg Pt = IPI).

We generate forecasts of yht for horizons 1, 6, 12 and 24 monthsahead. The following models are considered

• Random walk yhT+j = yT

• AR(2) : yhT+j = γh1yT + γh2yT−1

• Stock and Watson : yhT+j = β′hgT + γh1yT + γh2yT−1

• MUC : reduced MUC for (y ′t , g′

t)′ : yhT+j from Kalman filter

for j = 1, 6, 12, 24, both γ and β are estimated by OLS.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 :

MSE = H−1j

Hj−1∑

i=0

(yhT+i+j − yhT+i+j)2,

MAE = H−1j

Hj−1∑

i=0

|yhT+i+j − yhT+i+j |,

with 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.

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Out-of-Sample Forecasting : Personal Income

1-month ahead 6-month aheadMSE MAE MSE MAE

RW 2.3939 1.1059 1.5626 0.9158AR(2) 0.9588 0.7099 1.0340 0.7674

SW(1) 0.9407 0.7041 1.1930 0.7894SW(2) 0.8873 0.6919 0.8956 0.7188SW(3) 0.8848 0.6926 0.8328 0.6876SW(4) 0.8893 0.6953 0.8162 0.6695SW(BaiNg) 0.8934 0.6942 0.8495 0.6974

MUC(1) 0.9317 0.7002 1.1261 0.7794MUC(2) 0.8756 0.6840 0.8815 0.7099MUC(3) 0.8784 0.6840 0.8947 0.7042MUC(4) 0.8636 0.6784 0.8161 0.6790MUC(BaiNg) 0.8761 0.6825 0.8882 0.6990

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Out-of-Sample Forecasting : Personal Income


RW 1.5946 0.9461 1.9735 1.1032AR(2) 0.9781 0.7504 0.7965 0.6750


MUC(1) 1.1528 0.7937 0.8698 0.7128

MUC(2) 0.8641 0.7151 0.7136 0.6731MUC(3) 0.9188 0.7252 0.7879 0.7025MUC(4) 0.8336 0.7061 0.7555 0.6911MUC(BaiNg) 0.9022 0.7284 0.7918 0.7088

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Out-of-Sample Forecasting : Industrial Production


RW 1.6046 0.9562 1.4264 0.8268AR(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 0.6347SW(4) 0.7740 0.6718 0.6972 0.6371

SW(BaiNg) 0.7865 0.6751 0.6961 0.6376


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Out-of-Sample Forecasting : Industrial Production


RW 1.6616 0.9657 2.4495 1.2178AR(2) 0.9176 0.7301 0.9044 0.7723


MUC(1) 0.8699 0.7165 0.8567 0.7483MUC(2) 0.8541 0.7369 0.8515 0.7291

MUC(3) 0.7740 0.7025 0.9450 0.7520MUC(4) 0.7034 0.6602 0.8772 0.7400

MUC(BaiNg) 0.7485 0.6854 0.9627 0.7725

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Out-of-Sample Forecasting : Quarterly GDP


RW 1.4659 0.9038 1.4659 0.9038AR(2) 1.3540 0.8609 1.3540 0.8609



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Out-of-Sample Forecasting : Quarterly GDP


RW 3.7226 1.4284 9.1548 2.3024AR(2) 3.2971 1.3690 8.1081 2.2408


MUC(1) 3.2195 1.3464 8.0067 2.2186MUC(2) 3.2696 1.3614 7.3850 2.1625MUC(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

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Conclusions

We 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 similarresults.

• Interpolation results (nowcasting) need to be analysed

• Inclusion of lagged factors

• Separate PCs for leading / lagging economic indicators

• Treatments for stock and flow variables

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Paris2012 session4

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