A Time Series Analysis of Economical Phenomena in Japan’s Lost Decade

1. Try Test (S) on R
Analyzing Time-Series Data
2014.9.3

2. We introduce our R functions
You should be able to:
• Testify stationary of time series Z(n)
• Identify fracture points of stationary time series Z(n)
• Compute determinacy and causality values of Z1(n) and Z2(n)
• Visualize the results using R plot functions
• Apply these tests to various time series data
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3. Goals
• Using Statistics Programing language R
• Prototyping Each Test
• Stationary Test: Test(S)
• Abnormality Test: Test(ABN)
• Determinacy Test: Test(D)
• Causality Test: Test(ES)
• Merits
• Easy to handle
• Apply the tests to various data
• Spread the theory throughout the world

4. History of Prototyping
• These tests are developed as
– Basic Programs By Prof
• Test(S) , Test(ABN) 1988~
• Test(ES), Test(D) 2004~
– Java Programs (Converted from the Basic Programs)
• Test(S) , Test(ABN) 2009 ~ Test(ES), Test(D) 2011 ~
• Java programs have high performance.
However, require programming skills
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5. History: Prototyping Tests
• Developed by
– Basic Programs By
• 1988 ~
– Java Programs By
• 2009 ~
• R scripts + java class ( using Rjava ) By
• 2012~
(continued)
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6. Merits using R
1. Usability: Easy to handle
2. Data processing Performance
3. Powerful Visualization
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7. Demonstration of TestS-R
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8. 1 Easy to handle
$testABN(ts, term)
> testABN(M2CD, term=98)
Time Series Length = 186
Term = 98
1 1 0 1 0 0 3 1 0 4 14 19 15 16 18 9 9 12 12 9 7 4 1 2
1 1 1 4 4 3 2 1 3 4 8 16 12 11 5 6 7 5 5 3 3 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 6 35 103 84 81
88 89 77 66 77 59 60 50 55 87 84 96 88 51 43
• M2CD denotes a money supply data in Japan (1955q3 – 2001q4)
• Provided function $testABN calls Java class testS

9. 1 Easy to handle
$testD(ts1, ts2, term)
(continued)
> testD_result<-testD(GDP, M2CD, 183)
> testD_result$result[4,]
[1] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
…………………………………………………(skip)………………………………
[162] 0.0000000 0.0000000 0.0000000 0.7482668 0.8520689 0.8561230 0.8512788
[169] 0.8785611 0.8996980 0.9045383 0.8903662 0.8810984 0.8808611 0.8864593
[176] 0.9035437 0.8871590 0.8862733 0.8848502 0.8751460 0.8882548 0.8878465
[183] 0.8835506 0.8778081
• GDP denotes also Japan Nominal GDP data (1955q3 – 2001q4)
• testD_result$result[4,] denotes the Determinacy value of V3(n)

10. Data Accessing
• testD_result is a data frame of matrix of R
testD_result$result[4,]
Row Access
Column Access
testD_result$result[4,]
Dataframe_id$filed_id

11. 2 Data processing Performance
$scan, $log, $diff, $scale, $scale2
> M2CD <-scan("M2CD.DAT")
• Input M2CD data file
> M2CD_ld <- diff(log(M2CD))
Various data processing support: Log transformation, First difference
and Standardization. M2CD_ld is used for test(ABN).

12. 2 Data processing Performance
$plot, $ts
>M2CD <-scan("M2CD.DAT")
>plot(ts(M2CD, start=1955.5,,frequency=4))
(continued)

13. 2 Data processing Performance
$plot, $ts, $log
>plot(ts(log(M2CD), start=1955.5,,frequency=4))
(continued)

14. 2 Data processing Performance
$plot, $ts, $diff
>plot(ts(diff(log(M2CD)), start=1955.5,,frequency=4))
(continued)

15. 3 Visualization
$plotTestD(result, start)
> GDP <-scan("NGS-II.DAT")
> M2CD <-scan("M2CD.DAT")
>testD_result<-testD(GDP, M2CD, 183)
>plotTestD(testD_result, 4, 1955.5)
>abline(v=c(1981.25,1984,1996.5,1999),lty = 3)

16. Try Test(ABN), Test(D) with R
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17. Non linear transformation
$NLtransform(ts)
> NLtransform (diff(log(M2CD))
…………………..Return value…………………………………..
• Test(S) and Test(ABN) uses non–linear transformation up
to rank 6, which constructs 19 one-dimensional time
series ψi(Z(f)) from Z(n).
• $NLtransform returns 19 transformed time series as data
frame.

18. Stationary Test: $testS
$testS(ts, term)
> testS(diff(log(M2CD[1:99])), d=1)
Dimension d= 1
Time Series Length = 99
Test(S) Passed
[1] 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0
Time Series [2], [3], [5], [7], [18] is passed by TEST(S) is passed and considered
stationary.
• $TestS is applied to a subsequence of log- and diff-transformed
M2CD, whose data length is 99. ->M2CD[1:99]

19. Stationary Test
• Test the stationarity of the target time series
– Repeats Test(S) about subsequence TERM
– The length of TERM is set to 98: Economical Data
• $TestS returns an array of test(S) results about 19
transformed time series ψi(Z(f))
– one dimension: 19 results of 1d-test(S) for ψi(Z(f))
– Two dimension: results of 2d-test(S) for (ψi(Z(f)), ψj(Z(f)))
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20. Test(S) : Preliminary
• Time Series Data Z(m）
– Sample mean vector μZ ,
– Sample covariance matrix function RZ
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21. Test(S): Conditions
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• Three Test Values (M) (V) (O)
• Conditions
(M) mean
(V) variance
(O) Co-variance

22. Test(S): Conditions
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The Standard Rate of (M) i (V) i and (O)i
under which Test(S) is accepted.
Total Passed ／ Trials (N－M＋１)
•(M) i 80%
•(V) i 70%
•(O) i 80%
(continued)

23. Abnormality Test: $testABN
$testABN(ts, term)
>testABN_result <- testABN(diff(log(M2CD)), term=98)
Time Series Length = 186
Term = 98
1 1 0 1 0 0 3 1 0 4 14 19 15 16 18 9 9 12 12 9 7 4 1 2
1 1 1 4 4 3 2 1 3 4 8 16 12 11 5 6 7 5 5 3 3 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 6 35 103 84 81
88 89 77 66 77 59 60 50 55 87 84 96 88 51 43
• $TestABN uses log- and diff- transformed M2CD
• term is the length of subsequence tested by test(S) .

24. Abnormality Test
• Test the “degree of breaks in a time series”
• Repeats Test(S) about subsequence TERM
– The length of TERM is set to 98: Economical Data
• $TestABN returns a sequence of the Test(S) results
– Detect the points N -> 0 (N: natural number)
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25. Causality Test: $testCS
$testCS(ts, term)
>testCS_result<- testCS(M2CD, 100)
>testCS_result$shc
[1] NA NA NA NA 0.8317427 0.8493772 0.8471115
[8] 0.8476854 0.8432251 0.8474183 0.8943973 0.8768367 0.8864003 0.8975523
[15] 0.8943816 0.8876588 0.8841130 0.8554944 0.8761724 0.8838377 0.8857091
[22] 0.8928515 0.8600705 0.8339162 0.8276518 0.8275824 0.8666339 0.8456843
………………………………..(skipped)
• testCS_result$shc[4,] expresses the Causality value of V3(n)
• term is the length of subsequence

26. Causality Test: $testCS
• Causality Test checks the linear causality relation
from the time series X(n) to Y(n)
• X: e.g. GDP
• Y: e.g. M2CD
• $TestCS returns a data frame of 19 arrays of CR-values
CR(SH)i(y)
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27. Determinacy Test: $testD
$testD(ts1, ts2, term)
> testD_result<-testD(GDP, M2CD, 183)
> testD_result$result[4,]
[1] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
…………………………………………………(skip)………………………………
[162] 0.0000000 0.0000000 0.0000000 0.7482668 0.8520689 0.8561230 0.8512788
[169] 0.8785611 0.8996980 0.9045383 0.8903662 0.8810984 0.8808611 0.8864593
[176] 0.9035437 0.8871590 0.8862733 0.8848502 0.8751460 0.8882548 0.8878465
[183] 0.8835506 0.8778081
• The test(D) checks linear causality from M2CD to GDP in Japan
( 1955q3 – 2001q4)
• testD_result$result[4,] denotes the Determinacy value of V3(n)

28. Determinacy Test: $testD
• LN(6,2) determinacy property
- Time series Z(n) hold the property if D(Z) >= CR(SH)
• $TestD returns java object TestD including 19
determinacy function values as to V(n)k .
Fields of java class TestD
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$result 19 determinacy function values
$index1,2 Index pair of (ψi(Z(f)), ψj(Z(f))

29. Visualization Example(1)
$ts.plot, $abline
> full_result <-append (rep(NA,term), testABN_result) # fitting length to original data
> ts.plot(ts(full_result , 1955.5,,frequency=4),gpars=list(xlab="year",ylab="",yaxt = "n"))
> abline(v=c(1981.25),lty = 3)
• $abline depicts vertical lines at time points specified by “v=“

30. Visualization Example(2)
$plotTestD(result, start)
> plotTestD(testD_result, 4, 1955.5)
> abline(v=c(1981.25,1984,1996.5,1999),lty = 3)

31. Visualization Example(3)
$plotSample(result, result, num, start)
>plotSample(testCS_result$shc, testD_result, 4 ,1955.5)
>abline(v=c(1981.25,1984,1996.5,1999),lty = 3)

32. Advanced Visualization: ggplot2
$ggplot2
> abn<-read.table(file="plotABN.dat")
>ggplot(abn , aes(x, y, fill=range)) + geom_area(alpha=0.3)
+ ggtitle("Stationary Graph")+xlab("year")+ylab("M2CD")

33. Conclusion
• This is a provisional version
– TestD and TestCS requires further refinement
– TestS and TestABN are almost stable
– We try to elaborate Visualization skills
• We hope these Tests will be commonly used.
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34. References
1. Yuji Nakano, Yasunori Okabe : A Time Series Analysis of Economical
Phenomena in Japan’s Lost Decade (1): Determinacy Property of the
Velocity of Money and Equilibrium Solution ,Asia-Pacific Financial
Markets November 2012, Volume 19, Issue 4, pp 371-389
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