1. A collapsed dynamic factor analysis in STAMP
Siem Jan Koopman
Department of Econometrics, VU University Amsterdam
Tinbergen Institute Amsterdam
2. Univariate time series forecasting
In 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 : . . .
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3. Trend and cycle decomposition
Many macroeconomic time series can be decomposed into trend
and 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 are
stochastically 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.
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4. Kalman filter methods
Time 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 ; matrices
Zt , 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) and
forecasting.
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5. 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 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 series
modelling and economic forecasting.
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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.
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7. Multivariate time series with mixed frequencies
Define
yt
zt = , yt = target variable, xt = macroeconomic panel.
xt
The time index t is typically in months.
Quarterly frequency variables have missing entries for the months
Jan, 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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8. State space dynamic factor model
The 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 is
dynamic factor modelled as a VAR(2) and εt is a disturbance term.
The panel size N can be relatively large while the time series
dimension can be relatively short.
The coefficients in the loading matrix Λ, the VAR and variance
matrices 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 limited
number of principal components which we denote by gt ; see the
suggestions in Stock and Watson (2002).
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9. Stock and Watson (2002)
Consider the macroeconomic panel xt and apply principal
component analysis. Missing values can be treated via an EM
method.
The q extracted principal components (PCs) vector time series are
labelled 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
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10. 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 by
construction, we can treat the elements of Ft as independent
AR(2)s.
The model is reduced to a parsimonious dynamic factor model.
Realistic model for yt : own dynamics in ψt whereas parameters in
λ determine what additional information from Ft is needed.
We do not insert gt directly in equation for yt : not interested in
the noise of gt , only in the signal Ft .
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11. 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 by
construction, we can treat the elements of Ft as independent
AR(2)s.
It relates to recent work by Doz, Giannone and Reichlin (2011, J of
Ect) in which they show that an ad-hoc dynamic factor approach
where the loadings are set equal to the eigenvectors of the
principal components lead to consistent estimates of the factors.
The model can also be useful for univariate trend-cycle
decompositions when the time series span is short. The cycle ψt
may not be empirically identified; the Ft may be functional to
capture the cyclical properties in the time series.
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12. Collapsed state space dynamic factor model
Hence the model in state space form is given by
yt µ 1 λ′ ψt
= + + εt ,
gt 0 0 Iq Ft
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 a
weighted sum of lagged yt′ s since yt is a stationary process.
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13. Collapsed state space dynamic factor model
Hence the model in state space form is given by
yt µ 1 λ′ ψt
= + + εt ,
gt 0 0 Iq Ft
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. We
consider different q’s.
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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
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16. Forecasting set-up
We follow the forecasting approach of Stock and Watson (2002)
using the data set ”sims.xls” of SW (2005). The target variable is
yth as given by
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yth = (log Pt − log Pt−h ) ,
h
where Pt is typically an I(1) economic variable (eg Pt = IPI).
We generate forecasts of yth for horizons 1, 6, 12 and 24 months
ahead. 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
ˆ
for j = 1, 6, 12, 24, both γ and β are estimated by OLS.
ˆ
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17. Out-of-Sample Forecasting : design
Our forecasting results are based on a rolling-sample starting at
January 1970 and ending at December 2003 (nr.forecasts is
391 − 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 =0
with number of forecasts Hj and forcast horizon j.
The significance of the gain in forecasting precision against a
benchmark model is measured using the Superior Predictive Ability
(SPA) test of Hansen.
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24. Conclusions
We have presented a basic DFM framework for incorporating a
macroeconomic 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
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