seasonal modeling in time series with r

Presentation onProject:

Time series Analysis & Forecasting

Presented By

Mahmuda Mohammad Reg no:2013134029 MSc 2nd Semester Department of StatisticsShahjalal University of Science & Technology, Sylhet

2Shahjalal University of Science & Technology,Sylhet

Coordinated By

Dr. Mohammad Shahidul Islam MS, PhD & Postdoc (Canada), MSc (SUST)

ProfessorDepartment of Statistics

Shahjalal University of Science & Technology, Sylhet


Index


Monthly Yields on Treasury Securities

• Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec• 1953 2.83 3.05 3.11 2.93 2.95 2.87 2.66 2.68 2.59• 1954 2.48 2.47 2.37 2.29 2.37 2.38 2.30 2.36 2.38 2.43 2.48 2.51• 1955 2.61 2.65 2.68 2.75 2.76 2.78 2.90 2.97 2.97 2.88 2.89 2.96• 1956 2.90 2.84 2.96 3.18 3.07 3.00 3.11 3.33 3.38 3.34 3.49 3.59• 1957 3.46 3.34 3.41 3.48 3.60 3.80 3.93 3.93 3.92 3.97 3.72 3.21• 1958 3.09 3.05 2.98 2.88 2.92 2.97 3.20 3.54 3.76 3.80 3.74 3.86• 1959 4.02 3.96 3.99 4.12 4.31 4.34 4.40 4.43 4.68 4.53 4.53 4.69• 1960 4.72 4.49 4.25 4.28 4.35 4.15 3.90 3.80 3.80 3.89 3.93 3.84• 1961 3.84 3.78 3.74 3.78 3.71 3.88 3.92 4.04 3.98 3.92 3.94 4.06• 1962 4.08 4.04 3.93 3.84 3.87 3.91 4.01 3.98 3.98 3.93 3.92 3.86• 1963 3.83 3.92 3.93 3.97 3.93 3.99 4.02 4.00 4.08 4.11 4.12 4.13• 1964 4.17 4.15 4.22 4.23 4.20 4.17 4.19 4.19 4.20 4.19 4.15 4.18• 1965 4.19 4.21 4.21 4.20 4.21 4.21 4.20 4.25 4.29 4.35 4.45 4.62• 1966 4.61 4.83 4.87 4.75 4.78 4.81 5.02 5.22 5.18 5.01 5.16 4.84• 1967 4.58 4.63 4.54 4.59 4.85 5.02 5.16 5.28 5.30 5.48 5.75 5.70• 1968 5.53 5.56 5.74 5.64 5.87 5.72 5.50 5.42 5.46 5.58 5.70 6.03• 1969 6.04 6.19 6.30 6.17 6.32 6.57 6.72 6.69 7.16 7.10 7.14 7.65• 1970 7.79 7.24 7.07 7.39 7.91 7.84 7.46 7.53 7.39 7.33 6.84 6.39• 1971 6.24 6.11 5.70 5.83 6.39 6.52 6.73 6.58 6.14 5.93 5.81 5.93• 1972 5.95 6.08 6.07 6.19 6.13 6.11 6.11 6.21 6.55 6.48 6.28 6.36• 1973 6.46 6.64 6.71 6.67 6.85 6.90 7.13 7.40 7.09 6.79 6.73 6.74• 1974 6.99 6.96 7.21 7.51 7.58 7.54 7.81 8.04 8.04 7.90 7.68 7.43• 1975 7.50 7.39 7.73 8.23 8.06 7.86 8.06 8.40 8.43 8.14 8.05 8.00• 1976 7.74 7.79 7.73 7.56 7.90 7.86 7.83 7.77 7.59 7.41 7.29 6.87•

• To b continued

library(tseries)data(package="tseries")data(tcm) TS<-tcm10yTS


Monthly Yields on Treasury Securities

• 1977 7.21 7.39 7.46 7.37 7.46 7.28 7.33 7.40 7.34 7.52 7.58 7.69• 1978 7.96 8.03 8.04 8.15 8.35 8.46 8.64 8.41 8.42 8.64 8.81 9.01• 1979 9.10 9.10 9.12 9.18 9.25 8.91 8.95 9.03 9.33 10.30 10.65 10.39• 1980 10.80 12.41 12.75 11.47 10.18 9.78 10.25 11.10 11.51 11.75 12.68 12.84• 1981 12.57 13.19 13.12 13.68 14.10 13.47 14.28 14.94 15.32 15.15 13.39 13.72• 1982 14.59 14.43 13.86 13.87 13.62 14.30 13.95 13.06 12.34 10.91 10.55 10.54• 1983 10.46 10.72 10.51 10.40 10.38 10.85 11.38 11.85 11.65 11.54 11.69 11.83• 1984 11.67 11.84 12.32 12.63 13.41 13.56 13.36 12.72 12.52 12.16 11.57 11.50• 1985 11.38 11.51 11.86 11.43 10.85 10.16 10.31 10.33 10.37 10.24 9.78 9.26• 1986 9.19 8.70 7.78 7.30 7.71 7.80 7.30 7.17 7.45 7.43 7.25 7.11• 1987 7.08 7.25 7.25 8.02 8.61 8.40 8.45 8.76 9.42 9.52 8.86 8.99• 1988 8.67 8.21 8.37 8.72 9.09 8.92 9.06 9.26 8.98 8.80 8.96 9.11• 1989 9.09 9.17 9.36 9.18 8.86 8.28 8.02 8.11 8.19 8.01 7.87 7.84• 1990 8.21 8.47 8.59 8.79 8.76 8.48 8.47 8.75 8.89 8.72 8.39 8.08• 1991 8.09 7.85 8.11 8.04 8.07 8.28 8.27 7.90 7.65 7.53 7.42 7.09• 1992 7.03 7.34 7.54 7.48 7.39 7.26 6.84 6.59 6.42 6.59 6.87 6.77• 1993 6.60 6.26 5.98 5.97 6.04 5.96 5.81 5.68 5.36 5.33 5.72 5.77• 1994 5.75 5.97 6.48 6.97 7.18 7.10 7.30 7.24 7.46 7.74 7.96 7.81• 1995 7.78 7.47 7.20 7.06 6.63 6.17 6.28 6.49 6.20 6.04 5.93 5.71• 1996 5.65 5.81 6.27 6.51 6.74 6.91 6.87 6.64 6.83 6.53 6.20 6.30• 1997 6.58 6.42 6.69 6.89 6.71 6.49 6.22 6.30 6.21 6.03 5.88 5.81• 1998 5.54 5.57 5.65 5.64 5.65 5.50 5.46 5.34 4.81 4.53 4.83 4.65• 1999 4.72 5.00 5.23 5.18 5.54 5.90 5.79 5.94 5.92


Abstruct


The aim of the project is to conduct a time seriesanalysis and forecasting the monthly yields of treasury securities. Among the most effective approches for analyzing time series data ARIMA is employed in the data. In this project appropriate model is adaptivelyformed based on the given data.

Introduction

The treasury yield is the return on investment, expressed as percentage, on the us govt’s debt obligations (bonds, notes and bills).

Looked at the another way the treasury yield is the interest rate the US govt pays to borrow money for different lengths of time. The higher the yields on 10-,20-,30- years treasuries, the better the economic outlook.

Shahjalal University of Science & Technology,Sylhet

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Methods


• BoxCox transformation• Adf test• Kpss test• ARIMA, ACF, PACF• AIC,BIC• Ljung-Box test • McLeodLi test

Material

R software Version 3.2.2


Analysis

Start the analysis with R


11

Plot of the Data

plot(TS)

There is a stong trend and seasonality pattern

Time

TS

1960 1970 1980 1990 2000

24

68

1012

14


Box test for transformationlibrary(MASS) boxcox(TS~1)

To stabilize the fluctuation we use the Boxcox; the family of transformation

-2 -1 0 1 2

-155

0-1

500

-145

0-1

400

-135

0-1

300

log-

Like

lihoo

d

95%


Lambda value a<- boxcox(TS~1) lam<- a$x[which.max (a$y)]lam[1] 0.2222222 TSV<- ((TS^lam)-1)/lamplot(TSV)

We get the value of the lambda which is .22222 Which leads us to the decision that no transformation is needed.


Time

TSV

1960 1970 1980 1990 2000

1.0

1.5

2.0

2.5

3.0

3.5

Adf test adf.test(TSV)

Augmented Dickey-Fuller Test

data: TSVDickey-Fuller = -1.4095, Lag order = 8, p-value = 0.8282alternative hypothesis: stationary So we have come to conclusion that the series is nonstationary


KPSS test kpss.test(TSV)

KPSS Test for Level Stationarity

data: TSVKPSS Level = 5.375, Truncation lag parameter = 5, p-value = 0.01

Warning message:In kpss.test(TSV) : p-value smaller than printed p-value

So we may accept the alternative hypothesis that it is difference stationary. Which leads us to take the difference of the series to produce a stationary series.


Decompositiondec<-decompose(TS)plot(dec)

24

68

12

obse

rved

24

68

12

trend

-0.1

00.

00

seas

onal

-1.5

-0.5

0.5

1.5

1960 1970 1980 1990 2000

rand

om

Time

Decomposition of additive time series


difference to de- seasonalize the data diff<-diff(TSV)plot(diff)

After taking the seasonal difference the seasonality has gone but there is still the evidence of trend


Time

diff

1960 1970 1980 1990 2000

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

To remove the trend we take the first difference diff1<-diff(diff)plot(diff1)

So after taking both the seasonal difference and non seasonal 1st difference stationary has gone now


19

Time

diff1

1960 1970 1980 1990 2000

-0.3

-0.2

-0.1

0.0

0.1

0.2

Adf test

Augmented Dickey-Fuller Test

data: diff1Dickey-Fuller = -6.7294, Lag order = 8, p-value = 0.01alternative hypothesis: stationary

Warning message:In adf.test(diff1) : p-value smaller than printed p-value

The series is now stationary as we may accept the alternative hypothesis with the p value less than 0.05


To choose the order of the seasonal ARIMA we investigate the ACF and PACF:

index<-1:60> col<-rep("white",60)> col[c(1,12,24,36,48,60)]<-"black"> col<-rep("white",60)> col[c(1,12,24,36,48,6)]<-"black" par(mfrow=c(2,1))> acf(diff1,60,col=col)> pacf(diff1,60,col=col)


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0 1 2 3 4 5

-0.5

0.5

Lag

AC

F

Series diff1

0 1 2 3 4 5

-0.4

0.0

Lag

Par

tial A

CF

Series diff1

library(forecast)tsdisplay(diff1)


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diff1

1960 1970 1980 1990 2000

-0.3

-0.1

0.1

0 5 10 20 30

-0.4

-0.2

0.0

0.2

Lag

AC

F

0 5 10 20 30

-0.4

-0.2

0.0

0.2

Lag

PA

CF

Seasonal: Looking at lags that are multiples of 12 (we have monthly data). Not much is going on there, although there is a (barely) significant spike in the ACF at lag 1. Nothing significant is happening at the higher lags. Maybe a seasonal MA(1) or MA(2) might work.

Non-seasonal: Looking at just the first 2 or 3 lags, it seems possible that a MA(1) might work based on the single spike in the ACF and the PACF tapering to 0. With S=12 the nonseasonal aspect is sometimes difficult to interpret in such a narrow window.

On the basis of the ACF and PACF of the 12th differences, we identified an

ARIMA(0,1,1)×(0,1,1)12 model as a possibility.


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Possible Combination Arima(TS,order=c(0,1,1),seasonal=list(order=c(0,1,1),12),lambda=lam)$bic-1662.896Arima(TS,order=c(0,1,2),seasonal=list(order=c(0,1,1),12),lambda=lam)$bic-1661.526Arima(TS,order=c(0,1,1),seasonal=list(order=c(0,1,2),12),lambda=lam)$bic -1657.353Arima(TS,order=c(0,1,2),seasonal=list(order=c(0,1,2),12),lambda=lam)$bic-1655.843 Arima(TS,order=c(0,1,3),seasonal=list(order=c(0,1,2),12),lambda=lam)$bic -1649.544Arima(TS,order=c(0,1,2),seasonal=list(order=c(0,1,3),12),lambda=lam)$bic-1649.876Arima(TS,order=c(0,1,3),seasonal=list(order=c(0,1,3),12),lambda=lam)$bic-1643.578Arima(TS,order=c(1,1,1),seasonal=list(order=c(1,1,1),12),lambda=lam)$bic -1655.57


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BIC values

( p,d,q) (P,D,Q) (BIC)(0,1,1) (0,1,1) -1662.896

(0,1,2) (0,1,1) -1661.526

(0,1,1)

(0,1,2) -1657.353

(0,1,2) (0,1,2) -1655.843

(0,1,3) (0,1,2) -1649.544

(0,1,2) (0,1,3) -1649.876

(0,1,3) (0,1,3) -1643.578

(1,1,1) (1,1,1) -1655.57Shahjalal University of Science & Technology,Sylhet

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Residual AnalysisFit<-Arima(TS,order=c(0,1,1),seasonal=list(order=c(0,1,1),12),lambda=lam)e<-resid(Fit) tsdisplay(e)

Except for significance at lag 11 and 21 the model seems to have captured the essence of dependence in the series .


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e

1960 1970 1980 1990 2000

-0.2

0.0

0.1

0.2

0 5 10 20 30

-0.1

00.

000.

10

Lag

AC

F

0 5 10 20 30

-0.1

00.

000.

10

Lag

PA

CF

The Ljung-Box testBox.test(e,type="Ljung")

Box-Ljung testdata: e• X-squared = 1.3268, df = 1, p-value = 0.2494

The test leading to p value .2494 a further indication that the model captured the dependence in the time series .


27

For formally testing whether the squared residuals are autocorrelated :

library(TSA)McLeod.Li.test(y=e)

The figure shows that the McLeod .Li tests are all significant at the 5% significance level.This formally shows strong evidence for ARCH in the data.


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0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

Lag

P-v

alue

summary(Fit)

Series: TS ARIMA(0,1,1)(0,1,1)[12] Box Cox transformation: lambda= 0.2222222

Coefficients: ma1 sma1 0.5047 -1.0000s.e. 0.0410 0.0439

sigma^2 estimated as 0.002457: log likelihood=840.9AIC=-1675.8 AICc=-1675.75 BIC=-1662.9

Training set error measures: ME RMSE MAE MPE MAPE MASETraining set -0.01004224 0.2580049 0.1730594 -0.1069817 2.423545 0.1951395 ACF1Training set -0.08384966


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Forecast plot(forecast(Fit),h=10)

Forecasting or predicting future as yet unobserved values is one of the main reasons for developing time series model. We showed how to do this with ARIMA model.


30

Forecasts from ARIMA(0,1,1)(0,1,1)[12]

1990 2000

24

68

1012

14

Discussion

From the visual pattern of the McLeodLi test we get the strong evidence for ARCH in the data .So we should conduct the further modeling .But in our analysis we have stopped here .

The Seasonal ARIMA model we have got is of order (0,1,1)* (0,1,1),12


Conclusion


The main aim of time series modeling is to carefully collect and regorously study the past observations of a time series to develop an appropriate model which describes the inherent structure of the series .This model is then used to generate future values for the series .Time series forecasting thus can be termed as the act of predicting the future by understanding the past.

In this project among the most effective approches for analyzing time series data ,ARIMA is employed and appropriate

model is adaptively formed based on the given data.

Thank You


seasonal modeling in time series with r

Economy & Finance