Download Yield Rate of 10-Year Government Bond - Neas

Yield Rate of 10-Year
Government Bond
Time Series Project
xxxxxxx xxxxxxx
VEE Time Series - Fall 2012
1/18/2012
Content
1.
Introduction .......................................................................................................................................... 2
2.
Descriptive Statistics Analysis ............................................................................................................... 3
3.
4.
2.1.
Trend ............................................................................................................................................. 3
2.2.
Seasonality .................................................................................................................................... 4
2.3.
Volatility ........................................................................................................................................ 5
2.4.
Unusual Values.............................................................................................................................. 6
2.5.
Changes in Pattern ........................................................................................................................ 7
Statistical Test for Trend and Seasonality ............................................................................................. 8
3.1.
Test for the Presence of Trend ..................................................................................................... 8
3.2.
Test for the Presence of Seasonality ............................................................................................. 9
Modeling ............................................................................................................................................. 10
4.1.
ARIMA (Auto Regressive Integrated Moving Average) Model ................................................... 10
4.1.1.
Preliminary Stage ................................................................................................................ 10
4.1.2.
Stage 1: Identification ......................................................................................................... 11
4.1.3.
Stage 2: Estimation and Test of Significance ...................................................................... 12
4.1.4.
Stage 3: Diagnostic Checking .............................................................................................. 13
4.2.
Intervention Model ..................................................................................................................... 17
4.3.
Regression with Other Time Series Data .................................................................................... 19
4.4.
ARCH/GARCH Model ................................................................................................................... 21
5.
Summary ............................................................................................................................................. 23
6.
Attachments........................................................................................................................................ 24
1
1. Introduction
The yield rate of 10-year government bond is commonly use as the interest rate in
computing the accrued liability1 of a company. This long term liability is funded by the company
by investing on different securities. To simplify computation, it is assumed that the investment
return yields the same as the 10-year government bonds.
The primary goal of this paper is to create a good model of interest rate assumptions that
aims to explain the behavior of the series; forecast expected value of the series and analyzes
other factors affecting the series.
The first part of the paper discusses the descriptive statistics that extracts information from
the series and describe the features in time series such as trend, seasonality, volatility, unusual
values and changes in pattern.
The second part of the paper discusses some statistical test that justifies the descriptive
analysis done on the first part of the paper. This test includes the test for presence of trend and
the test for presence of seasonality.
The third part of the paper discusses the modeling part of the series such as ARIMA/ARMA
model, intervention model, regression with other time series data and ARCH/GARCH volatility
model of the series.
The fourth part discusses the conclusions and recommendations that summarize the results
from part 3.
1
Accrued liability is an actuarial term which is the expected retirement liability of a company at the date of
valuation.
2
2. Descriptive Statistics Analysis
Figure 1 Descriptive statistics computations
Descriptive analysis characterizes or describes the features of a time series data such as
trend, seasonality, volatility, unusual observation and changes in pattern. Figure 1 shows the
computation done in analyzing the features of the yield rate of the 10-year government bonds.
2.1.
Trend
Figure 2 Time plot of the yield rate of 10-year government bonds
3
Trend describes the long term tendency of the value of the series to go up or to go
down over a period of time. Figure 2 depicts the time plot of the yield rate of 10-year
government bonds, which shows a decreasing trend.
2.2.
Seasonality
Figure 3 Multiple line chart of the yield rate of 10-year government bonds
Seasonality is the repeated patterns or fluctuations observe from year to year.
Figure 3 shows the multiple line chart of the yield rate of 10-year government bond for
each year, which suggest that there is no seasonality on the series.
4
2.3.
Volatility
Figure 4 Coefficient of variation of the yield rate of 10-year government bonds
Volatility is measured by the squared deviation of the observation from its mean
(variance) or the positive square root of the variance (standard deviation). Figure 4
depicts the bar graph of the coefficient of variation on the yield rate of the 10-year
government bonds, which shows that the volatility of the series is not constant
(heteroscedastic). Moreover, Figure 2 also shows that the size of fluctuation of the
series is not constant.
5
2.4.
Unusual Values
Figure 5 Analysis on the Time Plot
Figure 5 shows the point on the time plot that shows large increase on the value of
the observation. This point is on at the middle of the year 2006.
6
2.5.
Changes in Pattern
Figure 6 Analysis on trend
Figure 6 depicts the change in trend such as from year 2003 to year 2004, which
shows increasing trend on the series, also, from year 2007 to year 2008, which shows
increasing trend on the series.
7
3. Statistical Test for Trend and Seasonality
3.1.
Test for the Presence of Trend
Using regression model to test for the presence of trend, the regression model is
given as:
The null and alternative hypothesis is:
The series has no linear trend if
trend.
, otherwise, the test suggest that it has a linear
Table 1 Linear Trend Model
Eviews output suggests that there exist a linear trend on the yield rate of the 10year government bonds since p-value of the coefficient of Time is 0. The negative sign
on the coefficient of Time means that the series has decreasing trend over time.
8
3.2.
Test for the Presence of Seasonality
Using Kruskal Wallis Nonparametric Test to test the presence of seasonality, the null
and alternative hypothesis is:
H0: The time series is not seasonal
Ha: The time series is seasonal
The H statistic is computed as
where L is the length of seasonality
ni is the number of observations in the ith season
Reject H0 at α level of significance if the computed value of H >χ2(α,L-1), otherwise,
conclude H0.
Table 2 Computation of Kruskal Wallis H-Statistics
The value of test statistic is 15.66 < 19.68 = χ2(0.05,11), therefore, we conclude that
the series is not seasonal.
9
4. Modeling
4.1.
ARIMA (Auto Regressive Integrated Moving Average) Model
The ARIMA model is type of analysis where the past observations of the variable of
interest are used to describe the behavior of the time series data and forecast (short
term forecasting) future values.
4.1.1. Preliminary Stage
ARIMA/ARMA model requires stationary series where the series has constant
mean and variance and its autocorrelation function is not function of time. This
stage determines if the series is stationary and make necessary transformation to
make the series stationary. In reference to 2.1 and 3.1, the series has decreasing
trend over time hence the mean of the series is not constant and make necessary
transformation to make the mean constant. One such transformation is
differencing2. Another method is use in determining if the series has constant mean
is the graph of ACF (Auto Correlation Function) of the series. The mean of the series
is stationary if the estimated ACF drops off rapidly to zero.
Letting Yt = dlog(yield rate of 10-year government bonds) be the yield rate of 10year government bonds after transformation.
Figure 7 Correlogram of yield_rate
2
Non seasonal differencing is appropriate to the series since in reference to 2.2 and 3.2, the series is non
seasonal.
10
Figure 7 shows the correlogram of the series and the graph of ACF suggests that
the mean of the series is not stationary.
Figure 8 Correlogram of d(yield_rate)
Figure 8 shows the correlogram of the series after first differencing and the
graph of ACF suggest that the mean of the series after first differencing is stationary.
The variance of the series is not constant in reference to 2.3. Log transformation
is necessary to make variance of the series constant. Finally, the stationary series is
Yt = dlog(yield rate of 10-year government bonds). All the discussion below will refer
to this series.
4.1.2. Stage 1: Identification
In this stage, preliminary model is determined using the characteristics of ACF
and PACF of the series. As a guide in including the AR or MA terms in the model,
compare ACF and PACF of the series to theoretical ACF and PACF of the model. A
purely AR process exhibits ACF that decays fast to zero and PACF that cut off sharply
to zero after the few spikes while a purely MA process exhibits ACF that cut off
sharply to zero after the few spikes and PACF that decays fast to zero. The lag
length of last spike on the PACF of AR process is its order and the lag length of the
last spike of the ACF of MA process is its order. ARMA process exhibits ACF that
decays fast to zero and stays after q lags and PACF that decays fast to zero and stays
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after p lags. Box and Jenkins suggest that the maximum number of useful estimated
auto correlations is roughly T/4 where T is the number of observations or 72/4 = 18
lags in this paper.
Figure 9 Correlogram of dlog(yield_rate)
Figure 9 shows the ACF and PACF of the stationary series and a candidate to
include to the model is MA(8).
4.1.3. Stage 2: Estimation and Test of Significance
Table 3 Testing the Coefficients of the Preliminary Model
12
Stage 1 suggests a model of the series
. Testing the
coefficients are summarize on Eviews results as shown on Table 1 suggesting that
is not significant while is significant in the model.
4.1.4. Stage 3: Diagnostic Checking
Table 4 MA(8) Model
Dropping the intercept, the model is
as the result
shown on Table 6 and the next step to determine if the residual (at) is white noise
with mean 0 and constant variance. Ljung-Box test is use to test at is white noise
with null and alternative hypothesis
and concluding H0 means that at is white noise. The Q-stat and the p-value is shown
on Figure 10 suggesting that at is white noise.
13
Figure 10 Ljung-Box Test for Residual at
ARCH (Auto-Regressive Conditional Heteroskedasticity) Test is use to test for
constancy of variance and Table 5 shows the Eviews result on the ARCH test
suggesting that at has constant variance.
Table 5 ARCH Test Result
Finally, Jarque-Bera test is use to test for normality of at where the null and
alternative hypothesis is
H0: The distribution is normal
Ha: The distribution is not normal
14
Figure 11 Jarque-Bera Test for Residual
Figure 11 shows the Eviews result for Jarque-Bera test of the residual (at)
suggesting that the distribution is normal since the p-value is 0.1397. Therefore, one
model to consider is
Repeating the process from stage 1 to stage 3, Table 6 shows another model
that can be consider which is
The residual of the model at is white noise by Ljung-Box test; it satisfies the
normality assumption by Jarque-Bera test and the p-value of the test statistic is
equal to 0.2099 > 0.05; and it also satisfies the assumption of constancy of variance
by the ARCH test using 12 lags and the p-value of Obs*R-squared is equal to 0.9435
> 0.05.
15
Table 6 Result of ARMA(8,16) Model
There are two ARIMAARMA models that are candidate for the yield rate of the 10-Year
10-Year government bonds. The first model is a MA(8) model, which suggests that the series is only
the series is only related to the past error terms of the series while the second model is an
model is an ARMA(8,16) model, which suggests that the series is related to both the past
past observation and past error terms of the series. Table 7 and
Table 8 show the forecast errors for the two models. Comparing the mean absolute
percentage error (MAPE), there is no major difference on the performance of
forecast between the two models. Hence, it is enough to use the MA(8) model on
short term forecasting of the series.
12
Forecast: FORECAST_IMA
Actual: YIELD_RATE_UPDATE
Forecast sample: 2009M01 2012M12
Included observations: 48
11
10
9
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
8
7
6
5
0.812174
0.578688
8.791973
0.058400
0.010814
0.096211
0.892975
4
3
2009
2010
FORECAST_IMA
2011
2012
± 2 S.E.
Table 7 Out-of-Sample Forecast Errors of the MA(8) Model of the Series
16
12
Forecast: FORECAST_ARIMA
Actual: YIELD_RATE_UPDATE
Forecast sample: 2009M01 2012M12
Included observations: 48
11
10
9
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
8
7
6
5
0.788023
0.553759
8.483287
0.056534
0.024651
0.075831
0.899518
4
3
2009
2010
2011
FORECAST_ARIMA
2012
± 2 S.E.
Table 8 Out-of-Sample Forecast Errors of the ARMA(8,16) Model of the Series
4.2.
Intervention Model
Figure 12 Plot of dlog(yield_rate) to Determine Intervention
An intervention to the series is spotted to the time plot of dlog(yield_rate) as shown
in Figure 12 and identified to be in June 2006 observation. This observation is related to
the news that the Banko Sentral ng Pilipinas (BSP) might increase the interest rate
because of high inflation rate due to continuous increase on the price of crude oil. Such
17
effect of known intervention (event) can be incorporated into the model via a pulse
function
Table 9 is the Eviews result of the regression model with intervention and it happens
that the intervention is significant to the model. Therefore, the model with
incorporation of intervention is
The residual of this model is white noise by Ljung-Box test; satisfies normality by
Jarque-Bera test with p-value = 0.6605 > 0.05 and has constant variance by ARCH test
using lag of 12 and has p-value = 0.2877 > 0.05.
Table 9 Regression Model with Intervention
This model suggests that the yield rate of 10-year government bond can be
explained by the past lag error of the series and the intervention that happen on June
2006.
Table 10 shows the forecasting performance of this model and suggests that there is
much larger deviation from the forecast value as compared with the first two models
which has MAPE equal to 8.99%.
18
12
Forecast: FORECAST_WINTERVENTION
Actual: YIELD_RATE_UPDATE
Forecast sample: 2009M01 2012M12
Included observations: 48
11
10
9
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
8
7
6
5
0.811132
0.596234
8.986491
0.058299
0.012498
0.096329
0.891173
4
3
2009
2010
2011
FORECAST_WINTERVENTION
2012
± 2 S.E.
Table 10 Out-of-Sample Forecast IMA(8) with Intervention Model
4.3.
Regression with Other Time Series Data
Let Xt be the yield rate of the 20-year government bond at time t; and Yt be the yield
rate of the 10 year government bond at time t. By unit root test,
and
.
Table 11 Unit Root Test of Residual in Regressing Y to X
19
Eviews output as shown on Table 11 suggests that Yt and Xt are co-integrated since
the p-value of Augmented Dickey-Fuller test statistic t is 0.0025 < 0.05. Hence, the error
term (at) in regressing Yt to Xt is stationary.
Table 12 Regression Model with AR(1)
A possible regression model is
and
be rewritten as
since
where
as shown on Table 12. Dropping the constant term, this model can
.
The residual (vt) of this model is white noise by using the Ljung-Box test; it satisfies
the normality assumption by Jarque-Bera test with p-value of its test statistic equal to
0.4767 > 0.05; and also it satisfies the constancy variance by ARCH test using lag 12 and
the p-value of Obs*R-squared equal to 0.4978 > 0.05.
The model suggests that there exist statistically relationship between the yield rate
of 10-year government bond and the 20-year government bond. This relationship shows
that behavior of Yt can be explained by its observation 1 month ago (Yt-1) and the series
Xt and the value of observation on X 1 month ago (Xt-1).
20
9
Forecast: FORECAST_WREGRESSION
Actual: YIELD_RATE_UPDATE
Forecast sample: 2009M01 2012M12
Included observations: 48
8
7
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
6
5
0.224046
0.145349
2.050875
0.016354
0.100688
0.063385
0.835927
4
3
2009
2010
2011
FORECAST_WREGRESSION
2012
± 2 S.E.
Table 13 Out-of-Sample Forecast Errors with Regression on Other Time Series
Table 13 shows the out-of-sample forecast of the model with regression on the time
series, which is much accurate than the ARIMA models since the MAPE of this model is
equal only to 2.05%.
4.4.
ARCH/GARCH Model
Table 14 shows the regression model with General Auto Regressive Conditional
Heteroskedasticity (GARCH) process. Including the a model on volatility (ARCH/GARCH)
to the regression model on 4.3, the regression model becomes
where
and
21
Table 14 Regression with ARCH/GARCH
The variance of the model is determined to be white noise by Ljung-Box test; it does
not satisfy normality assumption by using Jarque-Bera test with test statistic equal to 0
<0.05; but it satisfies the constancy of variance by ARCH test using 12 lag with test
statistic Obs*R-squared equal to 0.5720.
This regression model relating the yield rate of 10-year government bond (Yt) is can
be explained by the observations from its immediate past (Yt-1) and the observation
from the yield rate of the 20-year government bond (Xt) and its immediate past (Xt-1).
The volatility model describes how the volatility (ht) is related to the volatility of other
periods. Above results shows that ht is explained by the news from lag 1 and lag 2.
Table 15 shows the out-of-sample forecast of the model with GARCH which has
MAPE of 2.12%, which is much accurate than the ARIMA models and with intervention
model but less accurate than with Regression on Other Time Series.
22
10
Forecast: FORECAST_GARCH
Actual: YIELD_RATE_UPDATE
Forecast sample: 2009M03 2012M12
Included observations: 46
9
8
7
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
6
5
4
3
2009
2010
2011
FORECAST_GARCH
0.211003
0.149691
2.120053
0.015545
0.487870
0.060779
0.451352
2012
± 2 S.E.
.7
.6
.5
.4
.3
.2
.1
.0
2009
2010
2011
2012
Forecast of Variance
Table 15 Out-of-Sample Forecast Errors with GARCH Model
5. Summary
The ARIMA model is one of the models considered on this paper which relates the
series only to its past values. The form of stationary is Yt =dlog(yield rate of 10-year
government bonds). The ARIMA model selected on this paper is
which relates the series to its errors on 8 months ago. Forecasting errors such as MAPE
is analyzed between the two ARIMA models and suggesting that both the two models
predicts very well with MAPE equal to 8.79% and 8.48%. The model with shorter lag is
considered since this model is simple but it can completely describe the behavior of the
series.
Further improvement is added to the above model by incorporating the
intervention on the June 2006 and the model becomes
However, the addition of this intervention (event) makes the forecast errors such as the
MAPE suffer, which increase to 8.99%.
23
A model that includes regression of other time series with ARIMA on error terms is
also considered. This model relates the series to another time series and the error terms
is a function of the past error terms. The series considered are the original series and
letting Yt as the yield rate of 10-year government bonds; and Xt as the yield rate of 20year government bonds. This two series are found to be cointegrated and the model is
where
This model relates the series to the current observation on X and to the observation
on X on month ago. Also it relates the series to its past value one month ago. This model
is more accurate for short term forecasting, which has MAPE of 2.05%.
Extending the model to GARCH(3,1) by including the model on volatility, which has
MAPE of 2.12%. The model is
where
and
Problem exists in forecasting the variance since the coefficient of
is negative and
may be negative. Moreover, this model is not really a good model since one of the
assumptions on
is violated, which is the normality assumption. One recommendation
for future study on this series is to relate the series to other time series such as the
Philippine Composite Index which is the benchmark of all investor.
As a conclusion, the best model to consider in this paper is the model with
regression with other time series with ARIMA terms.
6. Attachments
Below are time series data and other computations used in this paper:
Data.xls
Data and Descriptive YieldRatesTestForSe
Stats.xls
asonality.xls
24