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Figure 1a: Ideal Filter Weights Figure 1b: p=12

Figure 1c: p=24 Figure 1d: p=36

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frequency, ω

frequency, ωfrequency, ω

frequency, ω frequency, ω

Figure 3: Comparing Filter Approximations - Monthly U.S. Inflation Case

0.6

0.7

0.8

0.9

1.0

0 60 120 180 240

0.0

0.2

0.4

0.6

0.8

1.0

0 60 120 180 240

0.0

0.2

0.4

0.6

0.8

1.0

0 60 120 180 240

0.6

0.7

0.8

0.9

1.0

1.1

0 60 120 180 240

0.6

0.7

0.8

0.9

1.0

1.1

0 60 120 180 240

0.6

0.7

0.8

0.9

1.0

1.1

0 60 120 180 240

++ + + +

| | | | |

-1.0

-0.5

0.0

0.5

1.0

+ +

+ + +

| | | | |0.0

0.2

0.4

0.6

0.8

1.0

+ + ++ +

| | | | |0.0

0.2

0.4

0.6

0.8

1.0

1.5 - 8 yrs

20 - 40 yrs

8 - 20 yrs Random Walk

Optimal Symmetric

Optimal Fixed

Random WalkOptimal

Optimal

k

k

k

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

corrt(yt,yt)^ corrt(yt, yt-k)

^

The panels in the first two columns contain curves differentiated according to the underlying procedure used to compute yt. See Table 2 for definitions of these procedures. While the sample size is T = 480, the statistics are

shown only for t = 1, 2, ..., 240 since the statistics are symmetric across the first and second halves of the sample. The third column displays five correlation functions, corrt(yt, yt-k), associated with the Random Walk filter for

t = 1, 31, 61, 121, 240 (1/12, 2 1/2, 5, 10, 20 years). Each correlation function is displayed for k = -24, -23, ..., 0, ..., 23, 24. The plus on the correlation function indicates the location of k = 0.

^

^

1/2[Vart(yt)/Var(yt)]

^

t t

t

t

t

t

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

1959 1964 1969 1974 1979 1984 1989 1994 1999

Figure 5a: M2 Growth and Inflation

Inflation

M2 Growth

M2 Growth Inflation

-6

-4

-2

0

2

4

6

-4.00

-2.75

-1.50

-0.25

1.00

2.25

3.50

4.75

6.00

1959 1964 1969 1974 1979 1984 1989 1994 1999

Figure 5b: Component, 1.5-8 years

Inflation

M2 Growth

M2 Growth Inflation

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-3

-2

-1

0

1

2

3

1959 1964 1969 1974 1979 1984 1989 1994 1999

Figure 5c: Component, 8-20 years

M2 Growth

M2 Growth Inflation

-2.50

-1.75

-1.00

-0.25

0.50

1.25

2.00

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1959 1964 1969 1974 1979 1984 1989 1994 1999

Figure 5d: Component, 20-40 years

Inflation

M2 Growth

M2 Growth Inflation

-25

-20

-15

-10

-5

0

5

10

15

20

25

-12

-9

-6

-3

0

3

6

9

12

15

18

1900 1910 1920 1930 1940 1950 1960

Figure 4a: M2 Growth and Inflation

Inflation

M2 Growth

M2 Growth Inflation

Inflation

-10

-8

-6

-4

-2

0

2

4

6

8

-15

-10

-5

0

5

10

1900 1910 1920 1930 1940 1950 1960

Figure 4b: Component, 2-8 years

Inflation

M2 Growth

M2 Growth

-8

-6

-4

-2

0

2

4

6

8

-7.5

-5.5

-3.5

-1.5

0.5

2.5

4.5

6.5

8.5

1900 1910 1920 1930 1940 1950 1960

Figure 4c: Component, 8-20 years

Inflation

M2 Growth

M2 Growth Inflation

-8

-6

-4

-2

0

2

4

6

8

-5.00

-3.75

-2.50

-1.25

0.00

1.25

2.50

3.75

5.00

1900 1910 1920 1930 1940 1950 1960

Figure 4d: Component, 20-40 years

Inflation

M2 Growth

M2 Growth Inflation

Inflation

Figure 6: Comparing Filter Approximations with the Hodrick-Prescott Filter

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80

0.7

0.8

0.9

1.0

1.1

0 20 40 60 80

0.6

0.7

0.8

0.9

1.0

1.1

0 20 40 60 80

0.6

0.7

0.8

0.9

1.0

1.1

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

Unemp

Inflation

GDP

Random Walk

H-P

Random WalkOptimal Fixed

Optimal

160 170 180 190 200

160 170 180 190 200

160 170 180 190 200

H-P

Random Walk

H-P

corrt(yt,yt)^ 1/2

[Vart(yt)/Var(yt)]^

tt

t

t

t

t T

T

T

The panels in each column contain curves differentiated according to the underlying procedure used to compute yt. The band pass filter approximations extract the frequency band between 2 quarters and 8 years. The Hodrick-

Prescott filter has the parameter lambda set to 1600. The time series representations are the quarterly models described in Table 1. While the sample size is T = 160, the statistics are shown only for t = 1, 2, ..., 80 since thestatistics are symmetric across the first and second halves of the sample. The third column contains the standard deviation of yt - yt for t = 160 and T equals 160, 161, ..., 200.

^

^

[VarT(y

160-y

160)]1/2

400x[VarT(y

160-y

160)]1/2

100x[VarT(y

160-y

160)]1/2

^

^

^

Figure 7: Comparing the Random Walk Filter with Trigonometric Regressions - Monthly U.S. Inflation Case

0.0

0.2

0.4

0.6

0.8

1.0

0 60 120 180 240

0.0

0.2

0.4

0.6

0.8

1.0

0 60 120 180 240

0.0

0.2

0.4

0.6

0.8

1.0

0 60 120 180 240

0

1

2

3

4

5

0 60 120 180 240

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 60 120 180 240

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 60 120 180 240

+

+ ++ +

| | | | |

-1.0

-0.5

0.0

0.5

1.0

+

+

+

++

| | | | |0.0

0.2

0.4

0.6

0.8

1.0

+ + ++

+

| | | | |0.0

0.2

0.4

0.6

0.8

1.0

1.5 - 8 yrs

20 - 40 yrs

8 - 20 yrs RandomWalk

Trigonometric Regression

Random Walk

Optimal

Optimal

k

k

k

0 0 0 0 0

0 0 0 0 0

Trigonometric Regression

corrt(yt,yt)^ 1/2

[Vart(yt)/Var(yt)]^

0 0 0 0 0

The panels in the first two columns contain curves differentiated according to the underlying procedure used to compute yt. While the sample size is T = 480, the statistics are shown only for t = 1, 2, ..., 240 since the

statistics are symmetric across the first and second halves of the sample. The third column displays five correlation functions, corrt(yt, yt-k), associated with the Trigonometric Regressions for t = 1, 31, 61, 121, 240 (1/12,

2 1/2, 5, 10, 20 years). Each correlation function is displayed for k = -24, -23, ..., 0, ..., 23, 24. The plus on the correlation function indicates the location of k = 0

^

^

corrt(yt, yt-k)^

t

t

t

t

t

t