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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 11 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