paris2012 session3b

Time Series Forecasting

Siem Jan Koopmanhttp://personal.vu.nl/s.j.koopman

Department of EconometricsVU University Amsterdam

Tinbergen Institute2012

Unobserved components: decomposing time series

A basic model for representing a time series is the additive model

yt = µt + γt + εt , t = 1, . . . , n,

also known as the classical decomposition.

yt = observation,

µt = slowly changing component (trend),

γt = periodic component (seasonal),

εt = irregular component (disturbance).

In a Structural Time Series Model (STSM)or a Unobserved Components Model (UCM),the components are modelled explicitly as stochastic processes.

Basic example is the local level model.

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Illustrations

We present various illustrations of time series analysis andforecasting:

1. European business cycle

2. Bivariate analysis: decomposing and forecasting of Nordpooldaily (average) of spot prices and consumption.

3. Periodic dynamic factor analysis: joint modeling of 24 hoursin a daily panel of electricity loads.

4. Modelling house prices in Europe.

5. Modelling the U.S. Yield Curve.

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Illustration 1: European business cycle

Azevedo, Koopman and Rua (JBES, 2006) consider Europeanbusiness cycle based on

• a multivariate model consisting of generalised components fortrend and cycle with band-pass filter properties;

• data-set includes nine time series (quarterly, monthly) whereindividual series that may be leading/lagging GDP;

• a model where all equations have individual trends but shareone common “business cycle” component.

• a common cycle that is allowed to shift for individual timeseries using techniques developed by Runstler (2002).

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Shifted cycles

1980 1985 1990 1995

−0.4

−0.2

0.0

0.2

estimated cyclesgdp (red) versuscons confidence (blue)

1980 1985 1990 1995

−0.4

−0.2

0.0

0.2

estimated cyclesgdp (red) versusshifted cons confidence (blue)

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Shifted cycles

In standard case, cycle ψt is generated by

(ψt+1

ψ+t+1

)= φ

[cosλ sinλ− sinλ cos λ

](ψt

ψ+t

)+

(κtκ+t

)

The cyclecos(ξλ)ψt + sin(ξλ)ψ+

t ,

is shifted ξ time periods to the right (when ξ > 0) or to the left(when ξ < 0).

Here, −12π < ξ0λ <

12π (shift is wrt ψt).

More details in Runstler (2002) for idea of shifting cycles inmultivariate unobserved components time series model ofHarvey and Koopman (1997).

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The basic multivariate model

Panel of N economic time series, yit ,

yit = µ(k)it + λi

{cos(ξiλ)ψ

(m)t + sin(ξiλ)ψ

+(m)t

}+ εit ,

where

• time series have mixed frequencies: quarterly and monthly;

• generalised individual trend µ(k)it for each equation;

• generalised common cycle based on ψ(m)t and ψ

+(m)t ;

• irregular εit .

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Business cycle

Stock and Watson (1999) states that fluctuations in aggregateoutput are at the core of the business cycle so the cyclicalcomponent of real GDP is a useful proxy for the overall businesscycle and therefore we impose a unit common cycle loading andzero phase shift for Euro area real GDP.

Time series 1986 – 2002:quarterly GDPindustrial productionunemployment (countercyclical, lagging)industrial confidenceconstruction confidenceretail trade confidenceconsumer confidenceretail salesinterest rate spread (leading)

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Eurozone Economic Indicators

1990 1995 2000

13.90

13.95

14.00

14.05

14.10

14.15

14.20

14.25

14.30 GDP IPI Interest rate spread Construction confidence indicator Consumer confidence indicator

Retail sales unemployment Industrial confidence indicator Retail trade confidence indicator

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Details of model, estimation

• we have set m = 2 and k = 6 for generalised components

• leads to estimated trend/cycle estimates with band-passproperties, Baxter and King (1999).

• frequency cycle is fixed at λ = 0.06545 (96 months, 8 years),see Stock and Watson (1999) for the U.S. and ECB (2001)for the Euro area

• shifts ξi are estimated

• number of parameters for each equation is four (σ2i ,ζ , λi , ξi ,

σ2i ,ε) and for the common cycle is two (φ and σ2κ)

• total number is 4N = 4× 9 = 36

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Decomposition of real GDP

1990 1995 2000

13.9

14.0

14.1

14.2

GDP Euro Area Trend

1990 1995 2000

0.001

0.002

0.003

slope

1990 1995 2000

−0.01

0.00

0.01

Cycle

1990 1995 2000

−0.0050

−0.0025

0.0000

0.0025

0.0050

irregular

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The business cycle coincident indicator

Selected estimation results

series load shift R2d

gdp −− −− 0.31indutrial prod 1.18 6.85 0.67Unemployment −0.42 −15.9 0.78industriual c 2.46 7.84 0.47construction c 0.77 1.86 0.51retail sales c 0.26 −0.22 0.67consumer c 1.12 3.76 0.33retail sales 0.11 −4.70 0.86int rate spr 0.57 16.8 0.22

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Coincident indicator for Euro area business cycle

1990 1995 2000

−0.015

−0.010

−0.005

0.000

0.005

0.010

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Coincident indicator for growth

• tracking economic activity growth is done by growth indicator

• we compare it with EuroCOIN indicator

• EuroCOIN is based on generalised dynamic factor model ofForni, Hallin, Lippi and Reichlin (2000, 2004)

• it resorts to a dataset of almost thousand series referring tosix major Euro area countries

• we were able to get a quite similar outcome with a lessinvolved approach by any standard

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EuroCOIN and our growth indicator

1990 1995 2000

−0.0075

−0.0050

−0.0025

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

Coincident �Eurocoin

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Illustration 2: Nord Pool data

• we consider Norwegian electricity prices and consumptionfrom Nord Pool.

• mostly hydroelectric power stations; supply depends onweather.

• Norway’s yearly hydro power plant capacity is 115 Tw hours.

• Nord Pool is day ahead market: daily trades for next daydelivery.

• daily series of average of 24 hourly price and consumption.

• spot prices measured in Norwegian Kroner (8 NOK ≈ 1 Euro).

• sample: Jan 4, 1993 to April 10, 2005; 640 weeks or 4480days.

• data are subject to yearly cycles, weekly patterns, levelchanges, and jumps.

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Bivariate analysis: daily spot prices and consumptionOur unobserved components model is given by

yt = µt + γt + ψt + x ′tλ+ εt , εt ∼ NID(0, σ2ε ),

where

• yt is bivariate: electricity spot price and load consumption;

• µt is long term level;

• γt is seasonal effect with S = 7 (day of week effect);

• ψt is yearly cycle changes (summer/winter effects);

• x ′tλ has regression effects, mainly dummies for special days;

• εt is the irregular noise.

Parameter estimation and forecasting of observations have beencarried out by the STAMP 8 program of Koopman, Harvey,Doornik and Shephard (2008, stamp-software.com):user-friendly but still flexible, also for multivariate models.

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Daily spot electricity prices from the Nord Pool

100 200 300 400 500 600

3

4

5

6

(i)

100 200 300 400 500 600

−0.2

−0.1

0.0

0.1

(ii)

100 200 300 400 500 600

−1

0

1

(iii)

100 200 300 400 500 600

−0.25

0.00

0.25

0.50(iv)

Univariate decomposition of Nord Pool daily prices January 4, 1993 to April 10, 2005:

(i) data and estimated trend plus regression; (ii) seasonal component (S = 7, the day-of-week effect); (iii) yearly

cycle; (iv) irregular.

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Joint decomposition of electricity prices & consumption

450 500 550 600

5

6

7(i−a)

450 500 550 600

−0.25

0.00

0.25

0.50(i−b)

450 500 550 600

−0.05

0.05

(ii−a)

450 500 550 600

−0.005

0.000

0.005

0.010(ii−b)

450 500 550 600

0

1

(iii−a)

450 500 550 600

−0.25

0.00

0.25

0.50(iii−b)

450 500 550 600

−0.2

0.0

0.2

(iv−a)

450 500 550 600

−0.025

0.025

(iv−b)

Bivariate decomposition of prices and consumption: Feb 19, 2001 to April 10, 2005:

(ia,b) data and estimated trend plus regression; (iia,b) seasonal component (S = 7, the day-of-week effect); (iiia,b)

yearly cycle; (iva,b) irregular.

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Forecasting results

We present MAPE for forecasting of one- to seven-days ahead prices for both uni- and bivariate models. The one-

to seven-days ahead forecasts are for the next seven days. The first forecast is for Monday, March 14, 2005 in

Week 637. The last forecast is for Sunday, April 10, 2005 in Week 640. The weeks 638 and 639 contain calendar

effects for Maundy Thursday (March 24, 2005) and the days until Easter Monday (March 28, 2005).

week 637 week 638 week 639 week 640uni biv uni biv uni biv uni biv

horizon

1 M 0.83 1.11 0.15 0.07 0.83 1.01 0.92 0.272 T 0.86 0.94 0.51 0.53 1.20 1.36 0.74 0.203 W 1.43 1.55 0.67 0.79 1.40 1.52 0.62 0.164 T 1.94 2.09 0.64 0.88 1.71 1.75 0.60 0.145 F 1.69 1.93 0.65 0.72 2.01 2.00 0.60 0.306 S 1.62 1.95 0.58 0.69 2.26 2.17 0.67 0.437 S 1.61 2.05 0.68 0.90 2.44 2.27 0.79 0.56

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Illustration 3: periodic dynamic factor analysis

Aim: the joint modeling of 24 hours in a daily panel of electricityloads for EDF.Focus: modelling and short-term forecasting of hourly electricityloads, from one day ahead to one week ahead.

• EDF provides a long time series: 9 years of hourly loads

• We can establish a long-term trend component but also

• different levels of seasonality (yearly, weekly, daily)

• special day effects (EJP)

• weather dependence (temperature, cloud cover)

• We look at the intra-year as well as the long-run dynamics byusing these different components.

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Periodic dynamic factor model specificationThe adopted methodology builds on Dordonnat, et al (2008, IJF):

• Model is for high-frequency data, for hourly data);

• Hours are in the cross-section (yt is 24× 1 vector);

• The model dynamics are formulated for days: a multivariatedaily time series model;

• In effect, we adopt a periodic approach to time seriesmodelling;

• The right-hand side of the model is set-up as a multipleregression model;

• We let the regression parameters evolve over time (days);

• We have a time-varying regression model, written instate-space form;

• The 24-dimensional time-varying parameters are subject tocommon dynamics (random walks);

• Novelty: dynamic factors in the time-varying parameters.22 / 91

Daily National Electricity Load, 1995-2004

2000 2005

4000

060

000

8000

0

Year (a)

Nat

iona

l Loa

d (M

egaW

atts

)

2002 2003

4000

060

000

8000

0

Date (b)

Nat

iona

l Loa

d (M

egaW

atts

)

0 5 10 15 20

3000

040

000

5000

0

Days elapsed since August 8th,2004 (c)

Nat

iona

l Loa

d (M

egaW

atts

)

−5 0 5 10 15 20 25 30

4000

060

000

8000

0

National Temperature (°C) (d)

Nat

iona

l Loa

d (M

egaW

atts

)

Time series and temperature effects at 9 AM

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Daily National Electricity Load, 1995-2004

1 2 3 4 5 6 7 8 9 10 11 12

40

00

05

00

00

60

00

0M

ea

n L

oa

d (

Me

ga

Wa

tts)

4 8 12 16 20 24

40

00

05

00

00

60

00

0

Me

an

Lo

ad

(M

eg

aW

atts)

Hour (b)

4 8 12 16 20 24

40

00

05

00

00

60

00

0M

ea

n L

oa

d (

Me

ga

Wa

tts) Month (a)

Hour (c)4 8 12 16 20 24

40

00

05

00

00

60

00

0

Me

an

Lo

ad

(M

eg

aW

atts)

Hour (d)Average patterns

(c) Oct-Mar, (d) Apr-Sep

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Multivariate Time Series ModelA periodic approach: from a univariate hourly to a daily 24× 1vector:

yt = (y1,t . . . yS,t)′ , S = 24 hours per day, t = 1, . . . ,T days.

Our multivariate time-varying parameter regression model is givenby:

yt = µt +K∑

k=1

Bkt x

kt + εt , εt ∼ IIN (0,Σε) , t = 1, . . . ,T ,

• Trends: µt = (µ1,t . . . µS,t)′

• Daily vectors of explanatory variables xkt = (xk1,t . . . xkS,t)

′,k = 1, . . . ,K , depending only on the day or on the hour of theday.

• Regression coefficients βkt = (βk1,t . . . βkS,t)

′, k = 1, . . . ,K . In

matrix form: Bkt = diag(βkt ), k = 1, . . . ,K

• Irregular Gaussian white noise εt = (ε1,t . . . εS,t)′.

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Time-varying regressions and dynamic factors

yt = µt +K∑

k=1

Bkt x

kt + εt , εt ∼ IIN (0,Σε) , t = 1, . . . ,T ,

where the time-varying regression parameters are given by{µt = c0 + Λ0f 0t ,βkt = ck + Λk f kt , k = 1, . . . ,K , t = 1, . . . ,T ,

with constant c j = (c j1 . . . cjS )

′, S × R j factor loading matrices Λj

and R j dynamic factors f jt = (f j1,t . . . fj

R j ,t)′, for j = 0, . . . ,K and

0 ≤ R j ≤ S .

• Factor structure requires 0 < R j < S ;

• Constant parameter component for R j = 0;

• Model has unrestricted component when R j = S ;

• Identification restrictions apply.26 / 91

Dynamic factor specfications

Local linear trend model for factors in trends µt :

{f jt+1 = f jt + g j

t + v jt , v jt ∼ IIN(0,Σjv )

g jt+1 = g j

t + w jt , w jt ∼ IIN(0,Σjw )

• vector of dynamic factors f jt ,

• slope or gradient vector g jt ,

• level disturbance v jt and slope disturbance w jt .

Random walk model for factors in the regression coefficients:

f jt+1 = f jt + e jt , e jt ∼ IIN(0,Σje ), j = 1, . . . ,K , t = 1, . . . ,T ,

with regression coefficient disturbance e jt .

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Empirical application to French national hourly Loads• French national hourly electricity loads from Sept-95 untilAug-04

• Estimation of trivariate models for neighbouring hours• Smooth trends• Intentional missing values for special days (EJP) and turn ofthe year. No problem for state space models.

• Yearly pattern regressors: sine/cosine functions of time areused (2 frequencies)

• Day-of-the-week effects: day-type dummy regressors• Weather dependence: heating degrees, smoothed-heatingdegrees and cloud cover

• Heating degrees beneath treshold temperature of 15 C• Exponentially smoothed temperature• Cooling degrees above treshold temperature of 18 C• Implemention: SsfPack 3 of Koopman, Shephard andDoornik (2008, ssfpack.com) for Ox 6 (2008, doornik.com)

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Yearly pattern estimates per hour

µs,t +∑4

k=1

(

βks,tx

ks,t

)

for hours (a) s = 0, 1, 2 ; (b) s = 3, 4, 5 ; (c) s = 6, 7, 8 (with extra component), etc.

Estimation: Jan 1997 - Aug 2003, Graph: Jan 1998 - Aug 2003.

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Components 9 AM, factor model (blue) and univariate

2000 2002

0

500Factor Univariate

2000 2002

500

1000

1500

(b)

Factor Univariate

2000 2002

100

200

300Factor Univariate

2000 2002

−1500

−1000

−500(a)

(d)

Factor Univariate

2000 2002

−200

0

(e)

Factor Univariate

2000 2002

−10000

−8000

−6000(c)

(f)

Factor Univariate

2000 2002

−15000

−12500

−10000

(g)

Factor Univariate

2000 2002

50000

60000

(h)

Factor Univariate

(a) heating β99,t , (b) smoothed-heating, (c) cooling, (d) Monday, (e) Friday, (f) Saturday, (g) Sunday, (h) trend +

yearly pattern

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9 AM st. errors, factor model (blue) and univariate

2000 2002

100

200

300

(a)

Factor Univariate

2000 2002

250

500

(b)

Factor Univariate

2000 2002

25

50

75

(c)

Factor Univariate

2000 2002

100

200

300

(d)

Factor Univariate

2000 2002

50

60

70

80

(e)

Factor Univariate

2000 2002

300

500

(f)

Factor Univariate

2000 2002

250

500

750

(g)

Factor Univariate

2000 2005

1000

2000

(h)

Factor Univariate

(a) heating s.e. β99,t , (b) smoothed-heating, (c) cooling, (d) Monday, (e) Friday, (f) Saturday, (g) Sunday, (h)

trend + yearly pattern

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Sample ACFs of residuals (daily lags)

0 100 200 300

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0.2

0 100 200 300

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0 100 200 300

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0 100 200 300

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0 100 200 300

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0 100 200 300

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0 100 200 300

0.0

0.2

Selecting the right model requires experience and stamina!

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Conclusions

• A general, flexible and insightful methodology is developed.

• Many dynamic features of load and price data can becaptured.

• We can detect many interesting signals which are notdiscovered before.

• Decent forecasts.

• Decent diagnostics.

• Many possible extensions.

• Remaining challenge: a full multivariate unobservedcomponents model for all 24 hours to capture evolutions ofcomplete intradaily load pattern.

• More work is required !

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Short Bibliography

• “Multivariate structural time series models” by Harvey andKoopman (1997), Chapter in Heij et al. (1997) Wiley.

• “Time-series analysis by state-space methods” by Durbin andKoopman (Oxford, 2001)

• “Periodic Seasonal Reg-ARFIMA-GARCH Models for DailyElectricity Spot Prices” by Koopman, Ooms and Carnero(JASA, 2007).

• “An hourly periodic state-space for modelling French nationalelectricity load” by Dordonnat, et.al. (International Journal ofForecasting, 2008)

• “Forecasting economic time series using unobservedcomponents time series models” by Koopman and Ooms(2011), Chapter in Clements and Hendry, OUP Handbook ofForecasting.

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Illustration 4: The macroeconomy in the euro area

Quarterly time series, 1981 – 2008, GDP in volumes,for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain.

1980 1985 1990 1995 2000 2005 2010

12.4

12.6

12.8

(i)

1980 1985 1990 1995 2000 2005 2010

12.8

13.0

13.2

(ii)

1980 1985 1990 1995 2000 2005 2010

12.3

12.4

12.5

12.6

12.7 (iii)

1980 1985 1990 1995 2000 2005 2010

11.50

11.75

12.00

12.25 (iv)

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Illustration 4: The housing market in the euro area

Quarterly time series, 1981 – 2008, real house prices (HP),for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain.

1980 1985 1990 1995 2000 2005 2010

4.0

4.5

5.0

(i)

1980 1985 1990 1995 2000 2005 2010

0.0

0.1

0.2

0.3 (ii)

1980 1985 1990 1995 2000 2005 2010

−0.25

0.00

0.25

(iii)

1980 1985 1990 1995 2000 2005 2010

2.0

2.5

3.0(iv)

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Any (common) cyclical dynamics in the data ?

Autocorrelograms and sample spectra, based on first differences...

0 10 20

0

1(i)

GDP−Correlogram

0.0 0.5 1.0

0.2

0.4

GDP−Spectrum

0 10 20

0

1 HP−Correlogram

0.0 0.5 1.0

0.2

0.4

HP−Spectrum

0 10 20

0

1(ii)

0.0 0.5 1.0

0.1

0.2

0 10 20

0

1

0.0 0.5 1.0

0.25

0.50

0.75

0 10 20

0

1

0.0 0.5 1.0

0.2

0.4

0 10 20

0

1

0.0 0.5 1.0

0.25

0.50

0 10 20

0

1

(iii)

(iv)

0.0 0.5 1.0

0.2

0.4

0 10 20

0

1

0.0 0.5 1.0

0.5

1.0

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The basic multivariate modelMultiple set of M economic time series, yit , is collected inyt = (y1t , . . . , yMt)

′ and model is given by

yt = µt + ψ(1)t + ψ

(2)t + εt ,

where the disturbance driving each vector component is a vectortoo, with a variance matrix. The structure of the variance matrixdetermines the dynamic interrelationships between the M timeseries.

For example, if trend component µt follows the random walk,µt+1 = µt + ηt with disturbance vector ηt , with variance matrixΣη:

• diagonal Ση, independent trends;• rank(Ση) = p < M, common trends (cointegration);• rank(Ση) = 1, single underlying trend;• Ση is zero matrix, constant.

Similar considerations apply to other components.38 / 91

Dynamic factor representations

We can formulate the multivariate unobserved components modelalso by

yt = µ∗ + Aηµt + A(1)κ ψ

(1)t + A(2)

κ ψ(2)t + Aεεt ,

where, for the trend component, for example, the loading matrixAη is such that

Ση = AηA′

η,

and, similarly, loading matrices are defined for the other variancematrices of disturbances that drive the components.

Here the dynamic factors or unobserved components µt , ψ(1)t , ψ

(2)t

and εt are ”normalised”.

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STAMP

Model is effectively a state space model: Kalman filter methodscan be applied for maximum likelihood estimation of parameters(such as the loading matrices).

Kalman filter methods are employed for the evaluation of thelikelihood function and score vector.

Kalman filter and smoothing methods are employed for signalextraction or the estimation of the unobserved components.

User-friendly software is available for state space analysis.

We have used S T A M P for this research project: amulti-platform, user-friendly software: econometrics, time seriesand forecasting by clicking.

It can treat multivariate unobserved components time seriesmodels...

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Motivation of our study

• Evidence of any relationship between housing prices and GDPin the euro area.

• Focus on more recent developments...

• We prefer to model the time series jointly and establishinterrelationships between the time series

• Focus on cyclical dynamics, long-term and short-term

• Emphasis on real housing prices: relevant for the monetarypolicy

• We also like to discuss synchronisation of housing markets ineuro area

Empirical results are based on our data-set with two variables:GDP and real house prices (HP); and for four euro area countries:France, Germany, Italy and Spain.

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Relevant literature

• Unobserved components model: Harvey (1989)

• State space methods: Durbin and Koopman (2001)

• Multivariate unobserved components: Harvey and Koopman(1997), Azevedo, Koopman and Rua (2006);

• Parametric approaches for house prices:• Probit regressions: Borio and McGuire, 2004, van den Noord,

2006;• Dynamic Factor models: Terrones, 2004, DelNegro and Otrok,

2007, Stock and Watson, 2008;• VAR: Vargas-Silva, 2008, Goodheart and Hofmann, 2008.

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Univariate analysisObjectives:

• Verify the trend-cycle decomposition for each series

• Verify whether possible restrictions are realistic

Results for GDP:

• two short cycles in France and Italy are detected (¡6 years);

• Germany and Spain contain both a short cycle (5.42 and 3.62years, resp.) and a long cycle (13.5 and 9.11 years)

• Various cycles are deterministic (fixed sine-cosine wave)

Results for HP:

• Results are quite different for each series

• Two cycles for Germany (5.4 and 13.5 years)

• Two short cycles for Italy (3.0 and 5.5 years) and France (3.1and 5.8 years)

• For Spain a cycle reduces to an AR(1) process

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Univariate results for GDP

France Germany Italy Spain

GDP R R R R

Trend var 0.65 0.03 0.01 0.03 0.48 0.03 0.10 0.03Cycle 1 var 0.81 0.17 0.00 0.15 3.85 5.75 0.07 0.00Cycle 1 ρ 0.94 0.90 1.0 0.90 0.87 0.90 0.95 0.90Cycle 1 p 3.04 5 5.42 5 2.97 5 3.62 5Cycle 2 var 0.00 1 1.81 2.86 0.00 7.79 0.00 2.31Cycle 2 ρ 1.0 0.95 0.95 0.95 1.00 0.95 1.00 0.95Cycle 2 p 5.8 12 13.5 12 5.50 12 9.11 12Irreg var 1 0.0 1 1 1 1 1 1N 7.2 11.4 3.23 5.23 6.58 11.1 27.1 34.9Q 14.5 24.9 15.1 14.6 9.26 13.3 22.1 24.8R2 0.31 0.24 0.11 0.02 0.23 0.12 0.22 0.12

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Univariate results for HP


RHP R R R R

Trend var 0.59 0.03 0.34 0.03 0.00 0.03 0.39 0.03Cycle 1 var 0.00 0.01 0.31 1.51 0.04 0.02 1 0.01Cycle 1 ρ 1.0 0.90 0.97 0.90 0.96 0.90 0.34 0.90Cycle 1 p 6.34 5 4.48 5 1.11 5 – 5Cycle 2 var 0.00 2.19 1 19.9 1 49.4 0.00 39.5Cycle 2 ρ 1.0 0.95 0.61 0.95 0.99 0.95 0.99 0.95Cycle 2 p 8.37 12 2.82 12 13.3 12 – 12Irreg var 1 1 0 1 0 1 0 1N 23.8 0.59 5.89 9.95 7.03 8.32 36.1 11.9Q 10.6 187 55.5 111 13.7 68.4 29.3 127R2 0.61 0.25 0.35 0.15 0.56 0.22 0.47 0.28

45 / 91

Cycle correlations from univariate analysis

Correlations for combined cycles (ψ(1)t + ψ

(2)t ):

• Strong correlations between GDP of four countries(correlations range from 0.52 to 0.94)

• The correlations with German GDP are the lowest

• Correlations between HP of four countries range from 0.42 to0.94

• The highest correlation is between Spain and France HP’s

• Correlation on combined cycle are mostly due to long-termcycle, not to the short-term cycle

• Correlations between GDP and HP for each country rangefrom 0.06 for Germany to 0.76 for Spain

• Overall low cross-correlations between GDP of one countryand HP of another country

46 / 91

Correlations between combined cycles for eight series

Combined cycle (ψ(1)t + ψ

(2)t )

F GDP F HP G GDP G HP I GDP I HP S GDP S HP

F GDP 1.00 0.51 0.52 0.23 0.83 0.15 0.89 0.61

F HP 0.51 1.00 0.44 0.44 0.52 0.68 0.68 0.94

G GDP 0.52 0.44 1.00 0.50 0.54 0.47 0.61 0.44

G HP 0.23 0.44 0.50 1.00 0.08 0.80 0.22 0.42

I GDP 0.83 0.52 0.54 0.08 1.00 0.06 0.84 0.64

I HP 0.15 0.68 0.47 0.80 0.06 1.00 0.29 0.64

S GDP 0.89 0.68 0.61 0.22 0.84 0.29 1.00 0.76

S HP 0.61 0.94 0.44 0.42 0.64 0.64 0.76 1.00

47 / 91

Correlations between short cycle for eight series

Short cycle ψ(1)t


F GDP 1.00 0.46 0.40 0.24 0.64 -0.46 0.57 0.42

F HP 0.46 1.00 0.29 0.62 0.33 -0.51 0.35 0.39

G GDP 0.40 0.29 1.00 0.32 0.75 -0.16 0.67 0.58

G HP 0.24 0.62 0.32 1.00 0.18 -0.52 0.06 0.13

I GDP 0.64 0.33 0.75 0.18 1.00 -0.13 0.61 0.65

I HP -0.46 -0.51 -0.16 -0.52 -0.13 1.00 -0.25 -0.19

S GDP 0.57 0.35 0.67 0.06 0.61 -0.25 1.00 0.75

S HP 0.42 0.39 0.58 0.13 0.65 -0.19 0.75 1.00

48 / 91

Correlations between long cycle for eight series

Long cycle ψ(2)t


F GDP 1.00 0.51 0.53 0.23 0.89 0.16 0.90 0.63

F HP 0.51 1.00 0.46 0.44 0.58 0.68 0.68 0.94

G GDP 0.53 0.46 1.00 0.52 0.44 0.49 0.62 0.46

G HP 0.23 0.44 0.52 1.00 0.07 0.82 0.22 0.43

I GDP 0.89 0.58 0.44 0.07 1.00 0.08 0.90 0.72

I HP 0.16 0.68 0.49 0.82 0.08 1.00 0.29 0.64

S GDP 0.90 0.68 0.62 0.22 0.90 0.29 1.00 0.76

S HP 0.63 0.94 0.46 0.43 0.72 0.64 0.76 1.00

49 / 91

Bivariate analysis

For each country, we carry out a bivariate analysis between GDPand RHP:

yt = µt + ψ(1)t + ψ

(2)t + εt ,

where yt is a 2× 1 vector for two series: GDP and HP.

We can conclude that

• highest correlation is found for cycle components (exceptItaly)

• for France, high correlation for medium-term cycle (8 years)but no dependence for long-term cycle (15.6 years)

• for Spain, strong correlations for both medium-term (8.2years) and long-term (14.4 years)

• for Germany, correlations for both cycles, but with low periods(4.3 and 7 years)

50 / 91

Bivariate results for GDP and HP

GDP RHP corr per ρ diag GDP RHP

FRA

trend 0.0 0.0 0.0 – – N 3.25 13.4

cyc 1 3.0 3.3 0.88 8.0 0.98 Q 17.0 17.4

cyc 2 1.0 126 0.07 15.6 0.99 R2 0.38 0.63

irreg 0.6 1.6 -0.19 – –

GER

trend 0.0 0.003 0.0 – – N 8.52 1.08

cyc 1 2.5 5.4 -0.6 4.3 0.90 Q 6.86 42.1

cyc 2 3.1 0.5 1.0 7.0 0.98 R2 0.39 0.29

irreg 4.3 1.1 0.58 – –

51 / 91

Bivariate results for GDP and HP

GDP RHP corr per ρ diag GDP RHP

ITA

trend 0.1 0.9 -0.15 – – N 4.19 4.57

cyc 1 4.3 16.2 -0.08 6.0 0.92 Q 10.1 8.60

cyc 2 0.0 8.4 0.0 1.1 0.94 R2 0.14 0.47

irreg 0.8 1.2 0.96 – –

SPN

trend 0.0 0.0 0.0 – – N 9.05 21.7

cyc 1 3.3 11.9 0.95 8.2 0.98 Q 17.5 43.0

cyc 2 0.0 83.3 0.82 14.4 0.99 R2 0.45 0.73

irreg 3.9 7.7 -0.35 – –

52 / 91

Four-variate cross-country analysis of GDP and RHP

Now we incorporate earlier findings and impose a strict short- andlong-term cycle decomposition for our analysis.

In particular, we have

• an independent trend µt (i.e. diagonal variance matrix Ση fordisturbance vectors of µt+1 = µy + ηt)

• similarly, an independent irregular component εt (i.e. diagonalvariance matrix Σε)

• a two-cycle parametrization with restricted periods of 5 and12 years

• the rank of the 4× 4 cycle variance matrices Σκ is 2:common cyles ...

• we load the two ”independent” cycles on France and Germany,i.e. cyclical dynamics of Spain and Italy are obtained as linearfunctions of the two times two (short and long) cyclical factors

53 / 91

Four-variate decomposition for GDP, cross-country

1980 1990 2000 2010

12.50

12.75

13.00LFRA_GDP Level

1980 1990 2000 2010

12.75

13.00

13.25LGER_GDP Level

1980 1990 2000 2010

12.4

12.6LITA_GDP Level

1980 1990 2000 2010

11.50

11.75

12.00

12.25LSPA_GDP Level

1980 1990 2000 2010

−0.01

0.00

0.01LFRA_GDP−Cycle 1

1980 1990 2000 2010

−0.02

0.00

0.02LGER_GDP−Cycle 1

1980 1990 2000 2010

−0.01

0.00

0.01

0.02LITA_GDP−Cycle 1

1980 1990 2000 2010

−0.005

0.000

0.005

0.010LSPA_GDP−Cycle 1

1980 1990 2000 2010

−0.025

0.000

0.025LFRA_GDP−Cycle 2

1980 1990 2000 2010

−0.025

0.000

0.025LGER_GDP−Cycle 2

1980 1990 2000 2010

0.00

0.02LITA_GDP−Cycle 2

1980 1990 2000 2010

−0.025

0.000

0.025

0.050LSPA_GDP−Cycle 2

1980 1990 2000 2010

−0.001

0.000

0.001LFRA_GDP−Irregular

1980 1990 2000 2010

−0.01

0.00

0.01LGER_GDP−Irregular

1980 1990 2000 2010

−0.0025

0.0000

0.0025

0.0050LITA_GDP−Irregular

1980 1990 2000 2010

−0.01

0.00

0.01

0.02LSPA_GDP−Irregular

54 / 91

Four-variate results for cross-country: GDP

Fra Ger Ita Spn Fra Ger

Cycle short (cov ×10−6) factor loadingsFra 4.11 0.25∗ 0.77∗ -0.40∗ 1 0

Ger 1.77 11.8 0.81∗ 0.78∗ 0 1

Ita 5.65 10.1 13.1 0.27∗ 1.08 0.69

Spn -1.04 3.50 1.27 1.65 -0.41 0.35

Cycle long (cov ×10−6)Fra 8.08 0.79∗ 0.48∗ 0.98∗ 1 0

Ger 7.94 12.5 -0.16∗ 0.64∗ 0 1

Ita 3.43 -1.39 6.28 0.66∗ 1.42 -1.02

Spn 11.2 9.11 6.73 16.4 1.79 -0.41

55 / 91

Four-variate results for cross-country: GDP

• Diagnostic statistics are satisfactory

• Strong correlation France-Germany for long-term cycle

• Business cycles for Italy and Spain are closely connected withthe one for France (however, negative ??? marginalcorrelation Fra-Spa for short-term cycle)

• German cycles strongly affect business cycles in Italy andSpain (however, their marginal correlations for longer cycle arenegative)

56 / 91

Four-variate decomposition for HP, cross-country

1980 1990 2000 2010

4.0

4.5

5.0

5.5LFRA_RHprice Level

1980 1990 2000 2010

0.0

0.1

0.2

0.3LGER_RHprice Level

1980 1990 2000 2010

−0.25

0.00

0.25 LITA_RHprice Level

1980 1990 2000 2010

2.0

2.5

3.0 LSPA_RHprice Level

1980 1990 2000 2010

−0.02

0.00

0.02LFRA_RHprice−Cycle 1

1980 1990 2000 2010

−0.01

0.00

0.01

0.02LGER_RHprice−Cycle 1

1980 1990 2000 2010

0.00

0.05LITA_RHprice−Cycle 1

1980 1990 2000 2010

−0.02

0.00

0.02

0.04LSPA_RHprice−Cycle 1

1980 1990 2000 2010

−0.1

0.0

0.1LFRA_RHprice−Cycle 2

1980 1990 2000 2010

−0.025

0.000

0.025

0.050LGER_RHprice−Cycle 2

1980 1990 2000 2010

−0.1

0.0

0.1

0.2LITA_RHprice−Cycle 2

1980 1990 2000 2010

−0.2

0.0

0.2LSPA_RHprice−Cycle 2

1980 1990 2000 2010

−0.01

0.00

0.01LFRA_RHprice−Irregular

1980 1990 2000 2010

−5e−5

0

5e−5LGER_RHprice−Irregular

1980 1990 2000 2010

−0.001

0.000

0.001

0.002LITA_RHprice−Irregular

1980 1990 2000 2010

0.00

0.01LSPA_RHprice−Irregular

57 / 91

Four-variate results for cross-country: HP

Fra Ger Ita Spn Fra Ger

Cycle short (cov ×10−6) factor loadingsFra 15.5 0.37∗ -0.89∗ 0.05∗ 1 0

Ger 4.73 10.8 0.10∗ -0.91∗ 0 1

Ita -21.0 1.97 36.2 -0.50∗ -1.64 0.90

Spn 0.89 -14.6 -14.6 23.8 0.55 -1.60

Cycle long (cov ×10−6)Fra 44.5 0.38∗ 0.70∗ 0.93∗ 1 0

Ger 4.43 3.13 -0.40∗ 0.69∗ 0 1

Ita 66.9 -10.3 207.1 0.38∗ 2.13 -6.30

Spn 100.4 19.9 88.3 262.8 1.89 3.69

58 / 91

Four-variate results for cross-country: HP

• Overall, these results seem to indicate that there is lessevidence of common (cyclical) dynamics in HP series

• Low correlations between France and Germany

• Strong negative correlations for the 5-year cycle betweenFra-Ita and Ger-Spa

• However, more commonalities for the 12-year cycle (Fra-Spa,Fra-Ita, Ger-Spa)

• Similarities between correlation matrices for the 12-year HPand GDP cycles, except that relationship Fra-Ger is strongerfor GDP (0.79 against 0.38 for HP)

59 / 91

Eight-variate results: HP and GDP for four countries

Similar restrictions apply as in four-variate analyses.

We conclude that

• strong correlations among GDPs for short-term cycles but lessevidence for long-term cycles, especially for Germany

• low correlations among HP series.

• for short-term cycle, these correlations for HP Fra-Ger is 0.65and for HP Spa-Ger is -0.95.

• only a few positive correlations for the long-term cycle in HPhave been found: Fra-Spa (0.58) and Ger-Ita (0.57)

• correlations HP-GDP are only found for long-term cycle,especially for France and Spain.

60 / 91

Eight-variate results: short cycle correlations


GDP HP GDP HP GDP HP GDP HP

F-G 1 -0.33 0.67 0.10 0.81 -0.59 0.77 0.13

F-H 1 0.075 0.65 -0.35 -0.13 -0.12 -0.64

G-G 1 0.17 0.80 -0.27 0.88 -0.011

G-H 1 0.055 -0.26 -0.10 -0.95

I-G 1 -0.037 0.66 0.034

I-H 1 -0.55 -0.040

S-G 1 0.34

S-H 1

61 / 91

Eight-variate results: long cycle correlations


GDP HP GDP HP GDP HP GDP HP

F-G 1 0.95 0.19 0.043 0.72 0.41 0.54 0.50

F-H 1 0.44 0.24 0.63 0.43 0.57 0.58

G-G 1 0.41 -0.31 0.26 0.44 0.21

G-H 1 -0.005 0.57 0.036 0.29

I-G 1 0.045 0.12 0.37

I-H 1 0.13 0.099

S-G 1 0.61

S-H 1

62 / 91

Illustration 5 : Modelling U.S. Yield Curve

0 10 20 30 40 50 60 70 80 90 100 110 120

5.25

5.50

5.75

6.00

6.25

6.50

Maturity (in months)

Yield (in %)

63 / 91

Time Series of Four Maturities

1985 1990 1995 2000

4

6

8

Yield (in %)

Date

Time to maturity: 3 month

1985 1990 1995 2000

4

6

8

10 Yield (in %)

Date

Time to maturity: 1 year

1985 1990 1995 2000

6

8

10

Yield (in %)

Date


1985 1990 1995 2000

5.0

7.5

10.0

Yield (in %)

Date


64 / 91

Term Structure of Interest Rates over Time

Time

Maturity (Months)

Yie

ld (

Per

cent

)

1987.5 1990.0 1992.5 1995.0 1997.5 2000.0

25

50

75

100

125

5.0

7.5

10.0

65 / 91

Literature Review

Earlier analyses of this data:

• Affine Term Structure Models (ATSM):Vasicek (1977), Cox, Ingersoll, and Ross (1985), Duffie andKan (1996), Dai and Singleton (2000), and De Jong (2000)

• Nelson-Siegel Model (NS):Nelson and Siegel (1987), Diebold and Li (2006), Diebold,Rudebusch and Aruoba (2006), De Pooter (2007), andKoopman, Mallee, and Van der Wel (2009)

• Arbitrage-Free Nelson-Siegel Model (AFNS):Christensen, Diebold, and Rudebusch (2007)

• Functional Signal plus Noise (FSN):Bowsher and Meeks (2008)

In all cases: a dynamic factor model set-up !

66 / 91

Still Some Outstanding Issues...

• Which of these models provides an accurate description of thedata?

• Duffee (2002) and Bams and Schotman (2003) provideevidence against affine term structure models

• What are the dynamics of the underlying factors:Stationary or Nonstationary?

• Stationary: Affine Term Structure Models, Nelson-Siegel,Arbitrage-Free Nelson-Siegel

• Nonstationary: Campbell and Shiller (1987), Hall, Andersonand Granger (1992), Engsted and Tanggaard (1994) andBowsher and Meeks (2008)

• What are the dynamics of the underlying factors: #lags?• Almost all models take a VAR(1) for the factor dynamics• Exception: Bowsher and Meeks (2008)

67 / 91

Further Outline

• (General) Dynamic Factor Model (DFM)• General Set-Up• Stationary and Nonstationary

• Smooth Dynamic Factor Model (SDFM)• Specification• Knot Selection

• Other Restrictions of the DFM

• Results

• Conclusion

68 / 91

The Dynamic Factor Model (DFM)

• Time series panel of N monthly yield observationsyt = (yt(τ1), . . . , yt(τN))

′ with yt(τi ) the yield at time t withmaturity τi

• The general dynamic factor model is given by:

yt = µy + Λft + εt , εt ∼ N(0,H),

ft = Uαt

αt+1 = µα + Tαt + R ηt , ηt ∼ N(0,Q),

for t = 1, . . . , n

• In here ft is an r -dimensional stochastic process that isgenerated by the p-dimensional state vector αt and ηt is aq × 1 vector

• We take H diagonal and have for the p × 1 initial stateα1 ∼ N(a1,P1), and assume N >> r , r ≤ p and p ≥ q

69 / 91

The Dynamic Factor Model (DFM) – Cont’d

• Vectors µy and µα, and matrices Λ, H, U, T and Q aresystem matrices, R is selection matrix of ones and zeros

• Special case of state space model.

• All vector autoregressive moving average processes can beformulated in this framework (see, e.g., Box, Jenkins andReinsel (1994))

• In this paper: VAR and Cointegrated VAR (CVAR) for ft .Obtained by suitable specification of U, T and R

• Elements of system matrices µy , Λ, H, µα, T and Q generallycontain parameters that need to be estimated

70 / 91

The Dynamic Factor Model (DFM) – Cont’d

• Need to impose restrictions on loading and variance matricesto ensure identification, see Jungbacker and Koopman (2008):

• Vectors µy and µα not both estimated without restrictions=¿ Restrict µα = 0 to focus on loading matrix Λ

• Impose restrictions on Λ, T & Q that govern covariances=¿ Restrict r rows in Λ to be r × r identity matrix:

Λ =

1 0 00 1 00 0 1λ4,1 λ4,2 λ4,3...

......

λN,1 λN,2 λN,3

71 / 91

Dynamic Factor Model (DFM) – Stationary Case

• Take a VAR(k) model for the r × 1 vector ft :

ft+1 =

k−1∑

j=0

Γt−j ft−j + ζt , ζt ∼ NID(0,Qζ)

• Stationarity imposed by restriction that |Γ(z)| = 0 has allroots outside the unit circle

• Can write this ft in DFM. For example, for k = 1 have

αt = ft ,

U = Ir ,

R = Ir ,

T = Γ0,

Q = Qζ

72 / 91

Dynamic Factor Model (DFM) – Nonstationary Case

• Now the ft are generated by a cointegrated vectorautoregressive process:

∆ft+1 = βγ′ft +

k−1∑

j=0

Γj∆ft−j + ζt , ζt ∼ N(0,Qζ ),

• The r × s matrices β and γ have full column rank; in thenonstationary case s < r and all ft nonstationary

• We propose an alternative but observationally equivalentspecification for ft via factor rotation:

ft =

(f Ntf St

)=

[β⊥ γ

]′ft ,

also need to construct new loading matrix Λ =[ΛN ΛS

]

73 / 91

Dynamic Factor Model (DFM) – Estimation

• As noted earlier, the Dynamic Factor Model (DFM) can beseen as a special case of state space model

• Generally we can use likelihood-based methods: direct MLand/or EM methods

• However. . .• . . . the dimension of the observations vector is much larger

than the state vector• . . . there is a large number of parameters (DFM-VAR(1),

N = 17, r = 3: 91 parameters)

• We therefore estimate the models using the methodology ofJungbacker and Koopman (2008) and estimate parameters bydirect ML using analytical score expressions

74 / 91

Smooth Dynamic Factor Model (SDFM)

• For cross-sectional observation i we can write the DFM as:

yt(τi ) = µy ,i+

r∑

j=1

λij fjt+εit , t = 1, . . .T , i = 1, . . . ,N,

where λij is the loading of factor j on maturity i

• We propose to let the loading parameter be an unknownfunction gj (·) for each factor j , where the argument of thefunction relates to i

• Assume these functions g1(·), . . . , gr (·) smooth functions oftime to maturity:

λij = gj(τi )

• In practice: cubic spline for each gj(·)

• Call this Smooth Dynamic Factor Model (SDFM)

75 / 91

Smooth Dynamic Factor Model (SDFM) – Spline

• In a spline the location of the knots determines how the factorloadings behave for varying maturities

• Knot k for column j : s jk• Order the knots by time to maturity:

τ1 = s j1 < · · · < s jKj= τN , j = 1, . . . , r

• Get following loading function:

gj(τi ) = wijλj , λj =

gj (sj1)

...

gj(sjKj)

, j = 1, . . . , r ,

with wij a 1× Kj vector (only depends on the knot locations)and λj a Kj × 1 parameter vector

76 / 91

Smooth Dynamic Factor Model (SDFM) – Select Knots

• But how many knots Kj to select in the spline Wλ?• Small number of knots: Loadings lie on same polynomial for

considerable number of maturities• Large number of knots: Get closer to unrestricted DFM

• Propose using a Wald test procedure to determine the knots

• This is standard as we are testing linear restrictions

• Amounts to an iterative general-to-specific approach:

1. Start with all knots2. Calculate test statistic for all knots3. Remove knot with smallest non-significant statistic4. Continue with 2 and 3 until all knots are significant

77 / 91

Dynamic Factor Models for the Term Structure

• The general dynamic factor model is given by:

yt = µy + Λft + εt , εt ∼ N(0,H),

ft = Uαt

αt+1 = µα + Tαt + R ηt , ηt ∼ N(0,Q),

• It nests the term structure models mentioned earlier

• Functional Signal plus Noise – Bowsher and Meeks (2008)• Rather than a spline for the factor loadings they adopt the

Harvey and Koopman (1993) time-varying spline for the yieldcurve:

yt = µy +Wft + εt , εt ∼ NID(0,H),

with W as before and ft time-varying knot values• Take a CVAR(k) for ft and have restrictions on Λ

78 / 91

Dynamic Factor Models for the Term Structure – Cont’d

• Nelson-Siegel – Nelson and Siegel (1987), Diebold and Li(2006), Diebold, Rudebusch and Aruoba (2006)

• The yield curve is expressed as a linear combination of smoothfactors

gns(τ) = ξ1 +

(1− e−λτ

λτ

)· ξ2 +

(1− e−λτ

λτ− e−λτ

)· ξ3

which gives

yt = µy + Λns ft + εt , εt ∼ NID(0,H)

• Interpretation as level, slope and curvature for the factors• Typically: (restricted) VAR(1) for the state, µy = 0• Restrictions on Λ

79 / 91

Dynamic Factor Models for the Term Structure – Cont’d

• Arbitrage-Free NS – Christensen, Diebold and Rudebusch(2007)

• The NS model is not arbitrage free• CDR employ “reverse engineering” and obtain an

Arbitrage-Free NS model• Dynamics of latent factors now coming from solution of SDE,

plus ‘correction’ term for µy

• Restrictions on Λ, T and µy

• Affine Term Structure Models – Duffie and Kan (1996)• Term structure can be explained by dynamics of unobserved

short rate• Short rate depends on unobserved factors• Focus on Gaussian case• Restrictions on Λ, T and µy

80 / 91

Results

Strategy:

• Following, e.g., Litterman and Scheinkman (1991) we onlylook at 3-factor models

• Restrict ourselves to Gaussian models

• Use an existing dataset: unsmoothed Fama-Bliss

• For DFM, SDFM, FSN and NS: VAR and CVAR

• For CVAR case focus on 1 random walk

We show the following results:

• VAR and CVAR for DFM

• Results for SDFM

• Estimation results NS, FSN, AFNS and ATSM

• In-sample fit of all models

• Validity of restrictions

81 / 91

DFM Likelihoods and AIC

Below we show the value of the loglikelihood at the ML value(ℓ(ψ)) and AIC (AIC) for the Dynamic Factor Model (DFM):

Model ℓ(ψ) AIC Model ℓ(ψ) AIC

VAR(1) 3894.5 -7595 CVAR(1) 3899.0 -7606VAR(2) 3918.5 -7625 CVAR(2) 3923.7 -7637VAR(3) 3922.6 -7615 CVAR(3) 3927.7 -7627VAR(4) 3932.2 -7616 CVAR(4) 3937.3 -7628

Note: Similar results hold for the NS and FSN model

82 / 91

DFM-CVAR Influence of Factor Dynamics on Loadings

0 25 50 75 100 125

1.25

1.50

Panel B

0 25 50 75 100 125

0.0

0.5

1.0 Panel A CVAR(1) CVAR(3)

CVAR(2) CVAR(4)

0 25 50 75 100 125

0.0

0.5

1.0

0 25 50 75 100 1250.0

0.5

1.0

0 25 50 75 100 125

−2

−1

0

1

0 25 50 75 100 125

0.0

0.5

1.0

83 / 91

Smooth Dynamic Factor Model – Knot Selection

Maturity Unrestricted model Final result6 2.65 4.22∗ 6.08∗ 59.08∗∗ 6.50∗ 5.24∗

9 0.79 2.40 5.59∗ - 6.58∗ 8.92∗∗

12 0.23 1.35 4.29∗ - 16.25∗∗ 19.62∗∗

15 0.04 0.33 1.51 - 24.17∗∗ 26.83∗∗

18 0.00 0.02 0.28 - - -21 0.95 0.74 1.52 18.55∗∗ - -24 3.51 2.37 3.99∗ 23.13∗∗ - 7.35∗∗

36 1.14 1.50 6.68∗∗ - - 26.88∗∗

48 0.44 2.88 13.47∗∗ - 30.07∗∗ 52.87∗∗

60 1.19 4.99∗ 18.04∗∗ - 26.79∗∗ 54.39∗∗

72 2.59 5.74∗ 15.67∗∗ - 22.80∗∗ 43.00∗∗

84 2.59 4.57∗ 8.81∗∗ - - 17.85∗∗

96 0.76 1.68 1.79 7.68∗∗ - 5.10∗

108 0.01 0.05 0.00 - - -

Note: 3, 30 and 120 months not shown as these knots can not be removed

84 / 91

SDFM Factor Loadings – CVAR

10 20 30 40 50 60 70 80 90 100 110 120

0.0

0.5

1.0Loading 1 SDFM DFM

10 20 30 40 50 60 70 80 90 100 110 120

0.5

1.0Loading 2

10 20 30 40 50 60 70 80 90 100 110 120

0.0

0.5

1.0 Loading 3

85 / 91

VAR coefficient matrix estimates

Panel A: Stationary models

SDFM NS FSNreal img. real img. real img.

1 0.164 0.159 0.156 0.166 0.216 0.1432 0.164 -0.159 0.156 -0.166 0.216 -0.1433 0.607 0.134 0.593 0.056 0.642 0.2594 0.607 -0.134 0.593 -0.056 0.642 -0.2595 0.965 - 0.964 - 0.969 -6 0.992 - 0.992 - 0.993 -

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VAR coefficient matrix estimates (cont’)

Panel B: Nonstationary models

SDFM NS FSNreal img. real img. real img.

1 0.155 0.162 0.151 0.165 0.206 0.1432 0.155 -0.162 0.151 -0.165 0.206 -0.1433 0.601 0.123 0.594 0.099 0.649 0.2584 0.601 -0.123 0.594 -0.099 0.649 -0.2585 0.973 - 0.972 - 0.970 -6 1 - 1 - 1 -

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NS Influence of Factor Dynamics on Loadings

To get a feeling how the choice of factor dynamics affects thefactor loadings we estimate the factor loadings parameter λ in theNelson-Siegel model for different choices of factor dynamics:

Model p = 1 p = 2 p = 3 p = 4

VAR(p) 0.07303 0.07211 0.07216 0.07193CVAR(p) 0.07302 0.07210 0.07213 0.07191

Recall that for the Nelson-Siegel model we have

gns(τ) = ξ1 +

(1− e−λτ

λτ

)· ξ2 +

(1− e−λτ

λτ− e−λτ

)· ξ3

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All Models - Overview

Finally, we provide an overview of all models and test therestrictions imposed on the DFM by the various models

Model ℓ(ψ) nψ AIC

DFM-VAR(2) 3918.5 106 -7625DFM-CVAR(2) 3923.7 105 -7637SDFM-VAR(2) 3906.8 85 -7644SDFM-CVAR(2) 3913.6 85 -7657FSN-VAR(2) 3479.0 64 -6830FSN-CVAR(2) 3483.7 63 -6841NS-VAR(2) 3808.4 65 -7487NS-CVAR(2) 3813.5 64 -7499AFNS 3253.3 42 -6423ATSM 3393.0 30 -6726

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All Models - Ljung-Box Statistics

CVAR(2) factors VAR(1) factorsMaturity DFM NS FSN SDFM NS AFNS ATSM3 5.8 6.2 84.3∗∗ 6.0 11.6 10.5 17.96 7.1 7.4 11.2 7.4 12.6 11.8 33.1∗∗

9 19.2∗ 19.3∗ 11.6 18.7∗ 31.8∗∗ 39.9∗∗ 55.1∗∗

12 22.5∗∗ 29.2∗∗ 16.7 23.1∗∗ 36.2∗∗ 53.1∗∗ 52.6∗∗

18 12.8 13.0 13.2 12.7 22.2∗∗ 28.1∗∗ 22.2∗∗

21 12.2 12.0 13.8 12.2 18.8∗ 22.4∗∗ 19.1∗

24 10.2 11.2 15.3 10.6 18.9∗ 21.6∗∗ 22.0∗∗

30 9.3 9.4 10.5 9.1 17.2 15.8 16.060 5.9 5.7 9.2 5.6 11.5 10.3 11.284 8.4 9.4 10.4 9.1 15.3 12.9 17.4120 10.1 9.7 6.9 11.0 9.7 9.3 12.2

Note: To preserve space 15, 36, 48, 72, 96 and 108 months omitted here

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All Models - Tests of Restrictions

Stationary Models

Model LR k p-value

NS 220.2 41 0.000FSN 879.0 42 0.000SDFM 23.4 21 0.320

Nonstationary Models

LR k p-value

220.4 41 0.000879.8 42 0.00020.2 20 0.450

Panel C: Arbitrage-Free Models

Model LR k p-value

AFNS 1330.4 64 0.000ATSM 1051.0 76 0.000

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paris2012 session3b

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