Lesson 15: Building ARMA models. Examples

1 Lesson 15: Building ARMA models. Examples Umberto Triacca Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica Universita dell'Aquila, Umberto Triacca Lesson 15: Building ARMA models. Examples Examples In this Lesson , in order to illustrate the time series modelling methodology we have presented so far, we analyze some time series. Umberto Triacca Lesson 15: Building ARMA models. Examples example 1. By using a computer program we have generated a time series graph of the series is presented in the following figure Figure : A simulated time series Umberto Triacca Lesson 15: Building ARMA models.

2 Examples example 1. The objective is to build an ARMA model this time series. The first step in developing a model is to determine if the series is stationary. Umberto Triacca Lesson 15: Building ARMA models. Examples example 1. Our time series seems the realization of a stationary process with zero mean, thus we can look at sample autocorrelation and partial autocorrelation function to establish the orders p and q of the ARMA model . Figure : Sample autocorrelation and sample partial autocorrelation Umberto Triacca Lesson 15: Building ARMA models.

3 Examples example 1. Since the SACF cuts off after lag 2 and the SPCAF follows a damped cycle, an MA(2) model xt = ut + 1 ut 1 + 2 ut 2 , ut WN(0, 2 ). seems appropriate for the sample data. Umberto Triacca Lesson 15: Building ARMA models. Examples example 1. Table reports the result of the ML estimation. 300 observations, Dependent Variable x Variabile Coefficient St. error t statistic p-value 1 1,68559 0,0456203 36,9481 0,0000. 2 0,883683 0,0492842 17,9303 0,0000. Variance of innovations Umberto Triacca Lesson 15: Building ARMA models.

4 Examples example 1. Now, we consider the graph of the residuals Figure : Residuals from MA(2) model Umberto Triacca Lesson 15: Building ARMA models. Examples example 1. Figure : SACF and SPACF of residuals from MA(2) model Umberto Triacca Lesson 15: Building ARMA models. Examples example 1. By analysing the SACF and SPACF of residuals presented in figure, we note that any term isn't significant and Q25 = do not indicate any autocorrelation in the residuals. They can be assimilate to a white noise process. Umberto Triacca Lesson 15: Building ARMA models.

5 Examples example 1. We conclude that the MA(2) model defined by xt = ut + 1 + 2 , ut WN(0, ). appear to fit the data very well. Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. Consider the montly series of the foreign exchange rate Lira per US Dollar from Jannuary 1973 until October 1989 (202. observations). Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. It can be observed that the series displays a nonstationary pattern with an upward trending behavior. Figure : Foreign exchange rate Lira per $ from Jannuary 19t3.

6 Until October 1989. Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. The first difference of the series seems to have a constant mean, although inspection of the graph (see Figure ) suggests thye variance is an increasing function of time. Figure : First difference of the foreign exchange rate Lira per US. dollar Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. As we can see in figure, the first difference of the logarithm is the most likely candidate to be covariance stationary. Figure : First difference of the logarithm of the foreign exchange rate Lira per US Dollar.

7 Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. Now, we examine the autocorrelation and partial autocorrelation functions of the logarithmic change in the foreign exchange rate Lira per US Dollar. Figure : SACF and SPACF for the logarithmic change in the foreign exchange rate Lira per US Dollar. Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. An AR(1) model is fitted by using the exact maximum likelihood estimation. The parameter estimates are summarized in the following table Umberto Triacca Lesson 15: Building ARMA models.

8 Examples example 2. Sample 1973:02 1989:10. Dependent Variable: First difference of log of the foreign exchange rate Lira per US Dollar Coefficient Std. error t statistic p-value Variance innov. 0,381412 0,0652368 5,8466 0,0000 0,000543700. Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. The AR(1) model fit indicates a highly significant parameter 1 with estimate 1 = 0, 381. Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. The Q statistic (Q20 = 16, 5014) and the graphs of the SACF. and SPACF of residuals (see Figure )indicate that the autocorrelations of the residuals are not statistically significant.

9 Figure : SACF and SACFP of residuals from the model AR(1). Umberto Triacca Lesson 15: Building ARMA models. Examples example 2. Thus we conclude that the AR(1) model xt = 1 + ut where ut WN(0, ) and xt is the first difference of log of the foreign exchange rate Lira per US Dollar, fits the data well. Umberto Triacca Lesson 15: Building ARMA models. Examples example 3. We consider the log of GNP deflator series in USA observed on the period 1955:1- 2000:4. The objective is to build an ARMA model for this time series. Umberto Triacca Lesson 15: Building ARMA models.

10 Examples example 3. The time serie graph is shown in figure. We note that the mean is changing over time. The variable exhibts a strong trend. Thus the series cannot be considered a realization of a stationary process. Figure : Log of GNP deflator series in USA observed on the period Umberto Triacca Lesson 15: Building ARMA models. Examples example 3. In order to make stationary the series, we consider the first differeces. Figure : Graphical plot of the first difference of log of GNP. deflator series Umberto Triacca Lesson 15: Building ARMA models.