# Time Series Analysis

**Topics:**Mean squared error, Time series, Mean absolute percentage error

**Pages:**7 (1874 words)

**Published:**April 9, 2012

Figure 1: Time series plot for raw Ford data.

Figure 1 shows a time series plot of the raw Ford stock prices against time. From this plot, a gradual but continuous upward trend can be observed. This trend was disrupted in 2005 when the stock prices experienced a huge rise moving from below 5 to above 25. This rise in stock price by Ford was not sustained as can be seen from the plot; the prices which reached a peak of above 25 fell to a about 10 by the end of 2005 and fell further in 2006 to a level below 5 fluctuations in the stock price existed and in 2007 the prices began to level out. Raw data is likely to be affected by non-stationarity and this can result in bias in the analysis. For the purpose of this analysis, it is required that the returns be continuously compounded. To achieve this I have taken the log and first difference of the raw data: this also achieves stationarity in the time series data. Figure 2 shows the plot of these new values where Ford is the raw data, l_Ford is the log of the raw data and d_l_Ford is the first difference of the log data. Figure 2

Using Box and Jenkins approach to estimate the best forecasting model requires 3 steps; identification, estimation and model checking. The aim is to choose a parsimonious model for forecasting. The table 1 below shows the values of the Akaike Criterion, Schwarz Criterion and Hannan-Quinn. IDENTIFICATION

In this paper the Information Criteria will be used instead of the ACF and PACF. The information criteria include a function of the RSS and some penalty for adding extra parameters. It is assumed that the lower the Information criteria the better the estimate. Fitting ARMA models (0, 0) to (5, 5) I have arrived at the values in table 1 for the Akaike Criterion, Schwarz Criterion and Hannan_Quinn. Table 1

ARIMA ModelAkaike CriterionSchwarz CriterionHannan-Quinn

(0,0)-10.87151-8.728378 -10.02861

(0,1)-7.178980-0.749576-4.650264

(0,2)-10.77113-2.198593-7.399510

(0,3)-8.8127281.902946-4.598201

(0,4)-8.4592824.399527-3.401849

(0,5)-8.8225456.179398-2.922208

(1,0)-7.379847-0.950442-4.851130

(1,1)-7.9474020.625137-4.575780

(1,2)-8.8761921.839481-4.661666

(1,3)-7.1574815.701327-2.100049

(1,4)-9.2276445.774299-3.327307

(1,5)-7.2885899.856489-0.545346

(2,0)-12.22176-3.649217-8.850134

(2,1)-12.33927-1.623594-8.124741

(2,2)-10.812302.046503-5.754873

(2,3)-8.9323036.069640-3.031965

(2,4)-7.2462269.898851-0.502983

(2,5)-5.29628013.991932.289868

(3,0) -10.666250.049424-6.451723

(3,1)-10.566172.292639-5.508737

(3,2)-8.8826036.119341-2.982265

(3,3)-11.498585.646496-4.755339

(3,4)-10.103549.184675-2.517389

(3,5)-7.56980113.861550.859252

(4,0)-10.046412.812398-4.988978

(4,1)-9.2167515.785192-3.316413

(4,2)-10.520396.624685-3.777150

(4,3)-9.5683759.719838-1.982226

(4,4)-10.7739610.65739-2.344906

(4,5)-4.86030418.714184.411655

(5,0)-8.8306596.171285-2.930321

(5,1)-7.3831429.761936-0.639899

(5,2)-9.09764410.19057-1.511496

(5,3)-8.60504612.82630-0.175992

(5,4)-4.03840019.536085.233559

(5,5)-2.29869123.418937.816173

ESTIMATION

The most significant model that minimises the value of the information criteria is the ARIMA (0, 0) model. I have identified this model using the information criteria. This model rejects the null hypothesis that Fordt = α + εt because the constant is not significant. The ARIMA (0, 0) Model has information criterion values as follows. Akaike Criterion:...

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