Topics: Regression analysis, Time series, Numerical analysis Pages: 11 (3629 words) Published: September 3, 2012
Content
Introduction1
Part 1. Examine the data, looking for seasonal effects, trends and cycles2
Part2. Dummy Variables Model3
Linear trend model3
Cubic trend model7
Part 3. Decomposition and Box-Jenkins ARIMA approaches8
First difference:10
a. Create an ARIMA (4, 1, 0) model10
b. Create an ARIMA (0, 1, 4) model11
c. Create an ARIMA (4, 1, 4)11
d. Model overfitting12
Second difference13
Forecast based on ARIMA (0, 1, 4) model13
Return the seasonal factors for forecasting14
Part 4. Discussion of different methods and the results15
Comparison of different methods in terms of time series plot15
Comparison of different models in terms of error17
Assumptions and the discussion on the sensitivity of assumptions18
Conclusion18

Introduction
The data of this coursework were drawn from the UK national statistics. It is a quarterly series of total consumer credit gross lending in the UK from the second quarter 1993 to the second quarter 2009. In this coursework, the first 57data will be used to establish models and the latter 8 data will be used to test if the forecast is a good fit or not. Two forecasting methods will be used in this coursework, which are a regression with Dummy Variables method and a combination of the Decomposition and Box-Jenkins ARIMA approaches. In addition, further comparison will be made between models to select out the best fit one. Then the underlying assumptions of the chosen model and sensitivity of the model to these assumptions will be discussed. All the analyses are based on the outputs working out by SPSS software.

Part 1. Examine the data, looking for seasonal effects, trends and cycles It is the fundamental process that to find out trend-cycle and seasonality, in order to create a certain model for further forecasting. Two approaches can be used to examine the data: analysing the time series plot or ACF plot.

According to the charts given above, it can be identified that the data value has both trend-cycle and seasonal components. To begin with, it is obvious that the data shows a trend-cycle which maintains a steady upward trend for first few years then tend to decrease. This could be improved by acf the data, see the lower plot, the trend-cycle is clear for this dataset because the ACF comes down gently to zero. According to upper figure, however, the seasonal component is not fairly clear. In order to estimate the certain seasonal effect of the data, it is necessary to removing the trend-cycle effect by differencing the data. See lower chart at right side, The ACF of the first difference tells that there is a 4 point pattern repeat echoed at lags of 4,8,12 and 16, which indicates a quarterly seasonal component. Therefore, it is clear that the data has both trend-cycle and seasonal effects: a quarterly repeat pattern.

Part2. Dummy Variables Model
Based on the analysis given above, trend-cycle and seasonal effects are found out. Therefore in order to estimate both effects of the data in a certain category, dummy variable time series model is now used. Due to the fact that the data is quarterly repeat, dummy variables Q1, Q2, Q3, Q4 are created in which Q4 is removed for the reason of multicollinearity. Both linear trend-cycle and nonlinear trend-cycle component of the Dummy Variables method will be explored by approaching linear trend-cycle, quadratic trend-cycle and cubic trend-cycle model. Linear trend model

Firstly, new dummy variables TIME is introduced in to display the trend-cycle and Q1, Q2, Q3 are also contained in this model. Hence, the function of this multiple regression model with dummy variables is: Data = a + c time + b1Q1+b2Q2+b3Q3+ error

Via establishing regression model with SPSS, the output can be seen as follow: Model Summaryb|
Model| R| R Square| Adjusted R Square| Std. Error of the Estimate| 1| .970a| .940| .939| 3229.43240|...