Common Trend Modelsfor

Time Series

Jee-Kwang Park

Post-Doctoral Fellow in QuaSSIDepartment of Political SciencePennsylvania State University

2

Factor Analysis

• The object of interest is not directly observable (measurable)

• But other seemingly related quantities are measurable

• Factor analysis explain the correlations between measurable variables in terms of underlying factors, which are not directly measurable.

3

Factor Analysis

• Ex) Mathematical abilities among students Geometry Algebra Calculus

1 68 76 70

2 90 94 94

3 86 80 78

4 88

subject

88 86

5 60 64 66

L

4

Common trend Models(Dynamic Factor Analysis)

• Estimate the underlying common trends among a group of time series.

• Do factor analysis with time series• Estimate the factor loadings and predict

factor scores• Business cycle, interest rates, stock prices• Three variants: DFA, TSFA, SSM

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Common Trend Model

Ex) Series 1 Series 2 Series 3

1 134.5 123.3 122.3

2 135.7 134.5 128.7

3 137.5 145.9 131.1

4 140.8 159.0

Time

153.5

5 141.9 165.6 153.7

L

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1) Dynamic Factor Analysis

• Geweke (1977) first proposes the dynamic factor analysis

• S-Plus/Finmetrics Program• DFA estimates the dynamics of the factors• Limit: valid with stationary series • most social science data are nonstationary

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Model Specification of DFA

( 1)( 1) ( 1) 1

2i

2 2

1

2

1

cov( , ) 0, for all t,s

( ) ( ) 0

var( ) ,var( ) ,

is a diagonal matrix with .

var( ) ,

the variance due to the common

t t tN KNN K N

t s

t t

t K t

K

it ij ij

K

ijj

Y f

f

E f E

f I D

Where D

Y

where

m b e

e

e

e

s

b s

b

´´´ ´ ´

=

=

= + +

=

= =

= =

= +

=

å

å

2

factors

=communality

the variance due to specific to each series = uniquenessis =

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Time Series Factor Analysis (TSFA)

• Gilbert and Meijer (2005)

• P-technique factor analysis (Cattell et al)

• works with nonstationary (weak bounded condition)

• MLE, estimate is unbiased and consistent• Errors are not assumed to be iid• R package (tsfa)

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State Space Model of Common Trends (SSM)

• Engle and Watson(1981, 1983), Molenaar (1985), Harvey (1989), Lütkepohl(1991), Zuur (2003)

• Works with nonstationary and short time series

• can include explanatory variables• most widely used• Ox/STAMP

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SSM Model Specification

'

' ' ' '1

'

time series & K common trends

, ~ (0, )

, ~ (0,D )

1,

of standardized factor loadings

D a diagonal matrix

is an 1 vector

in which the first element

t t t t

t t t t

t

N

y NID

NID

K

N K

N

N K

q e

q

m m e e

m m h h

h

m

-

= Q + +

= +

= ´

Q = ´

=

´

-

å

s are zeros

and the last elements are contained in a vector .K m

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Presidential Approval

• 6 series : Gallup, ABC/WP, CBS/NYT, Fox, Pew, Zogby

• Sources : www.pollingreport.com and roper center webpage

• Monthly Presidential Approval Series

• plural polls in a month, especially Gallup.• Averaging the plural polls is not so a good idea.• Cluster of polls, sample size

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2002 2003 2004 2005 2006

40

50

60

70

80

90 Gallup CBS/NYT Pew

ABC Fox Reuters/Zogby

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Presidential Approval

• Missing Data• all polls but Gallup have missing months• (Cubic) spline smooth interpolation (Zivot &

Wang 2002)• Spline ? nonparametric regression like L

oess• Gallup = 0, Fox = 1, CBS/NYT = 4, ABC/W

P = 9, Pew = 9, Zogby =12 missing observations

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Presidential Approval

• Previous Studies

• Beck, (2006), Franklin (2006), Chung(2006)…

• Franklin shows interesting findings on the house effects

• CBS polls tend to fall 3% below the overall trend in 2005-2006: 38.2% 38.6%

• Robert Chung’s finding : CBS polls have a small positive house effect during the pre-2005 period.

• Autocorrelation (serial correlation)

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Franklin (2006)

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Presidential Approval (DFA)• Using Finmetrics (S-Plus)• Wald test shows one common factor

> factor.fit <- factanal(Bush.multiple, factors=1, method="mle") > factor.fit Sums of squares of loadings: Factor1 5.816041 …

Test of the hypothesis that 1 factor is sufficientversus the alternative that more are required:The chi square statistic is 9.79 on 9 degrees of freedom.The p-value is 0.367

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Presidential Approval (DFA)• 97% of variance is explained by one common factor > summary(factor.fit)

Importance of factors: Factor1 SS loadings 5.8160408

Proportion Var 0.9693401Cumulative Var 0.9693401

• Variance unique to each series

The degrees of freedom for the model is 9.Uniquenesses: Gallup ABC/WP CBS Fox Pew Zogby 0.021 0.015 0.037 0.025 0.040 0.042

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Presidential Approval (DFA)

• Correlation among the polls

> fitted(factor.fit) Gallup ABC CBS Fox Pew ZogbyGallup 1.000 0.981 0.970 0.976 0.968 0.967ABC 0.981 0.999 0.973 0.979 0.971 0.970CBS 0.970 0.973 0.999 0.968 0.961 0.960Fox 0.976 0.979 0.968 0.999 0.966 0.965Pew 0.968 0.971 0.961 0.966 0.999 0.958Zogby 0.967 0.970 0.960 0.965 0.958 0.999

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Presidential Approval (SSM)

• The phenomenon conjectured by Franklin(2006) is supported by the residual plot

• The same phenomenon is also visible in other polls

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2005

50

75

100Gallup Trend_Gallup

2005

-5

0

5 Irr_Gallup

2005

50

75

100ABC Trend_ABC

2005

0

10 Irr_ABC

2005

50

75

100CBS/NYT Trend_CBS/NYT

2005

-10

0

10 Irr_CBS/NYT

2005

40

60

80Fox Trend_Fox

2005

0

10

20 Irr_Fox

2005

40

60

80Pew Trend_Pew

2005

0

10

20 Irr_Pew

2005

40

60

80Reuters/Zogby Trend_Reuters/Zogby

2005

-10

0

10 Irr_Reuters/Zogby

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Presidential Approval (SSM)

• Strength of SSM- We can compare a value in one series with one in another series

due to following properties of SSM

- 55% approval rate in Gallup is tantamount to 56.226% in ABC/WP poll.

- ABC = .775*Gallup + 13.601+ error

'1 1

'2 2 2 2

'

2 2 1 2

where is a univariate random walk. Thus we have

t t t

t t t

t

t t

y

y c

c

m e

bm e

m

m bm

= +

= + +

= +

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Tariff Rates (DFA)

• Tariff rates of 5 Latin American Countries

• Missing observations

• Two common factors• 80 % of variance is explained by 2

factors

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1980 1985 1990 1995 2000 2005

10

20

30

40

50

60

70

Venezuela Mexico Argentina

Peru Brazil

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Venezuela Mexico Brazil Argentina Peru

0.0

0.2

0.4

0.6

0.8

Factor1

Peru Brazil Argentina Venezuela Mexico

0.0

0.2

0.4

0.6

0.8

1.0

Factor2

25

10 20

-10

12

3

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Average Tariff Rates (SSM)

• Two Common trends

• Brazil seems to be best explained by the two common factors

• Argentina is the worst fit, which means its tariff rates seem to be more influenced by domestic variables compared to the others.

• The series co-moved more tightly in the second half of the period.

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1980 1990 2000

25

50

75 Venezuela Trend_Venezuela

1980 1990 2000

-5

0

5

10 Irr_Venezuela

1980 1990 2000

20

40Peru Trend_Peru

1980 1990 2000

0.0

2.5

5.0 Irr_Peru

1980 1990 2000

10

20

30Mexico Trend_Mexico

1980 1990 2000

-5

0

5 Irr_Mexico

1980 1990 2000

20

40

60Brazil Trend_Brazil

1980 1990 2000

-2

0

2 Irr_Brazil

1980 1990 2000

20

40

60Argentina Trend_Argentina

1980 1990 2000

-10

0

10

20 Irr_Argentina

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Conclusion

• Common trends model will be useful- We have reason to believe multiple time series are influenced by the

common factor(s)- The common factor cannot be directly measurable- We are interested in measuring the amount of the variances specific

to each series- Another Application: asset returns model