Forecasts for the Year 2011 for Hughes Travel PLC - Coursework Example

 Hughes Travel PLC Forecasts for the year 2011 Table of Contents Table of Contents 1 Analysis of the Time Series Data and its Structure 2 Forecasting Methods 3 Forecasts using the ARIMA Model 6 Error Measures 7 References 9 Appendix 10 Time-Series Analysis 10 Exponential Smoothing 15 ARIMA Model 18 Analysis of the Time Series Data and its Structure A time series is the combination of observations recorded sequentially over time. Usually, the observations are taken at equal time intervals. If there is a single variable being observed then it is known as a univariate time series. The main speculation behind any time series is that at least some of the patterns observed in the past would continue in the future (Ramasubramanian, n.d.). The data available is of Hughes Travel PLC monthly travel data collected over the span of January 1986 – December 2010. It consists of two variables, namely: number of overseas visitors travelling to the UK and the number of UK residents travelling abroad. The time interval of data collection for both the variables is a month. As both the variable data is independent of each other, hence, we have two univariate time series. The data does not depict a particular trend. Analysis of UK Residents Time Series Figure 1 of appendix A shows the month wise distribution of UK residents travelling abroad. The graph shows that highest number of UK residents travel abroad during the months of August, September, and July. The graph shows that August has had highest number of UK residents travel abroad and it has happened consistently for the past 25 years. Similarly, figure 2 of appendix A shows the cumulative data on UK residents travelling abroad on a yearly basis. The data shows a steady rise in the number of UK residents travelling abroad with the highest being year 2010. Figure 1 in appendix A also depicts that UK residents travel least during the months of December, January, February. Analysis of UK Residents Time Series Figure 3 of appendix A shows the cumulative sum of overseas travellers on a monthly basis. The data shows that overseas travellers travel most during the months of July and August, which is the most busy season for travelling. The months of May, June, September and October show similar travelling patterns and more or less the same number of overseas travellers travel during those months. Figure 4 of the appendix shows that the rate of Overseas UK travels has risen considerably from the previous years and it was the highest in 2010 from the past 25 years whereas the years 2007-2009 saw the lowest travelling statistics. This probably has to do with the recession and the credit crunch during these years. Figure 5 displays the overseas travelling statistics distributed over the 25 years. The graph clearly shows that not once in the period of 25 years, the months of July and August have never seen a decline in the number of overseas travellers as compared to other months. Moreover, the graph also shows that as the years 2007-2009 were an all time low for overseas travellers, the number of travellers declined to their lowest during the July-August of these years as well. Forecasting Methods Several forecasting methods have been developed over the years and each of them have their advantages and accuracy. We have chosen the two most basic and common forecasting models: exponential smoothing model, and ARIMA Model. Exponential Smoothing – This method is most common forecasting method for different types of time series data. It was developed by Brown and Holt. A basic approach towards time series modelling is to look at each observation as the combination of a constant and an error term. The value of constant would vary with time but is constant in a short interval of time. One way of modelling this is to assign greater weight to the most recent values of the constant as compared to the older observations also termed as moving averages, which is the basis of simple exponential smoothing. Following is the formula of simple exponential smoothing: St = *Xt + (1-)*St-1 Overseas Visitors travelling to UK In order to forecast the number of overseas visitors travelling to UK in 2011, we use the exponential smoothing model that results in Maximum Absolute Percentage Error (MAPE) of 5.138 provided in table 3 and figure 6, which tells us the mean uncertainty of the model that is 5.14%. This means that in the worst case scenario, there is a 5.14% chance that the prediction would be incorrect. The stationary R square of the model is 54.3%, which tells us that the 54% of the variation in the series is explained by model. Moreover, if we look at the series chart of observed versus the forecasted values in figure 7; we see that the model fits perfectly. The figure tells us how well the model fits the observed values as both the red and the blue line overlap each other and it perfectly captures the seasonal peaks and upward trend in the data. Table 3 provides a summary of the model information as well as the goodness-of-fit statistics of the model. The significance value of the model for the Ljung-Box statistic is .403 telling us that the model accounts all the structural details of the data. UK Residents travelling abroad Forecasting the number of UK residents travelling abroad with the same exponential smoothing model results in similar results as for the earlier variable. Table 4 depicts the MAPE statistic for this model that is 4.948%, which is an acceptable level of uncertainty. Similarly, the stationary R square of the model is 51.9% and the Ljung-Box statistic showing the specification of the model has a significance value of less than 0.05 telling us that there is structure in the observed series that the model has not taken into account. The series chart in figure 8 showing the observed and the fit values show that the model captures the seasonality and upward trend of the data perfectly. However, due to the inability of the model to take the structure of the data into account, we cannot use the model for forecasting the number of UK residents travelling abroad during the year 2011. ARIMA Model – ARIMA stands for Autoregressive Integrated Moving Average. Real-life research as in our case consists of complex patterns of data that are difficult to indentify. Moreover, the observations consist of errors and we need to identify patterns to help generate forecasts. The ARIMA model helps us in doing all the above-mentioned tasks. Box and Jenkins developed the model in 1976 and it has become very popular because of its power and flexibility. It is complex to use and requires experience. Overseas Visitors travelling to UK The model summary in table 5 provides the statistics of the model that help us in analyzing the accuracy of the model. MAPE of the model is 5.3%, which is more than the MAPE of the model with exponential smoothing. Similarly, the stationary R square is 53.1%, which is also less than the exponential smoothing model. However, the Ljuang Significance statistic of the model is higher than exponential smoothing and the series chart displaying the fit values shows a better fit of the observed values towards seasonal peaks and the overall upward trend. UK Residents Travelling abroad The model summary in table 6 displays the model statistics. It shows that the MAPE of the model is 5.14%, which is again more than the exponential smoothing for the same variable. Similarly, the stationary R sq. statistic is 53.8% that is more than exponential smoothing and the Ljuang Significance Statistic here is not significant as the value is .279 whereas in the case of exponential smoothing it was significant showing an error in that model. The series chart also shows a perfect fit of the values again reaffirming the goodness-of-fit of the model. Although, the exponential smoothing model does show some better statistical values, it was unable to capture the structure of the data for one of the variables whereas the ARIMA model captures the structural details of both the variables perfectly well. Hence, the ARIMA Model would be used for forecasting the number of overseas visitors travelling to UK and the number of UK residents travelling abroad in 2011. Forecasts using the ARIMA Model Forecast of Overseas visitors to UK in 2011 Model Jan 2011 Feb 2011 Mar 2011 Apr 2011 May 2011 Jun 2011 Jul 2011 Aug 2011 Sep 2011 Oct 2011 Nov 2011 Dec 2011 Overseas visitors to UK from Jan 1985-Model_1 Forecast 2058 1886 2191 2481 2503 2690 3129 3267 2589 2474 2274 2345 UCL 2267 2109 2426 2728 2762 2959 3408 3556 2889 2783 2592 2672 LCL 1849 1664 1956 2234 2245 2421 2849 2977 2290 2165 1956 2018 For each model, forecasts start after the last non-missing in the range of the requested estimation period, and end at the last period for which non-missing values of all the predictors are available or at the end date of the requested forecast period, whichever is earlier. Forecast of UK residents going abroad in 2011 Model Jan 2011 Feb 2011 Mar 2011 Apr 2011 May 2011 Jun 2011 Jul 2011 Aug 2011 Sep 2011 Oct 2011 Nov 2011 Dec 2011 UK residents going abroad from Jan 1985-Model_1 Forecast 4034 4063 4429 5319 5423 6611 6561 8256 6864 6299 4506 3897 UCL 4390 4435 4817 5722 5841 7034 6988 8687 7300 6740 4951 4354 LCL 3678 3690 4040 4915 5004 6188 6134 7824 6428 5859 4061 3441 For each model, forecasts start after the last non-missing in the range of the requested estimation period, and end at the last period for which non-missing values of all the predictors are available or at the end date of the requested forecast period, whichever is earlier. Analyzing the forecasts by the model, we see that the model has followed the seasonality and upward trend of the data. The forecasts for the months of July to September are higher than any other months of the years as it is the travelling season due to summer breaks and other factors. Hence, the number of people travelling to or out of UK increases tremendously during these months. Error Measures There are various types of error statistics that can be used to test the accuracy of any model. However, all the error measures have some positives and negatives. Mean Error (ME) – The mean error and mean percentage error tells us if the forecasts that have been made are biased and their level of biasness. However, ME is not useful for measuring accuracy as it does not include the errors in forecasts. A ME score of 0 does not necessarily does not exclude the cancellation effect of large errors with opposite signs. Moreover, a non-zero ME does not necessarily mean that the ME is independent of the forecast value (The Mean Error, 2007). Mean Absolute Deviations (MAD) & Mean Absolute Error (MAE) – Mean absolute deviation calculates the divergence from the mean. This error statistic measures average absolute deviation of observations from their forecasts where as Mean Absolute error is a measure of the errors in forecast in a time series analysis. The mean absolute error measures the closeness of a forecast to its actual outcomes and it is an average of the absolute errors (Hyndman, & Koehler, 2005). Mean Square Error (MSE) & Root Mean Square Error (RMSE) – The Mean squared error measures the accuracy of an estimator by quantifying the difference in the values estimated by the estimator and the actual values of that quantity. It is a measure of the average of the squares of the error. The difference in the estimated and actual value exists because of random values or due to the inability of the estimator to take all information into account. Taking a square root of Mean square error produces the root mean square error (RMSE), which is the square root of variance known as standard error. This error statistic is useful in selection of estimators as well as in selecting models, it helps in finding the difference between the actual observations and the forecast predicted by the model helping us in determining if the model fits the data or not (Lehmann, & Casella, 1998). Mean Percentage Error (MPE) – It is the average of the percentage errors by which the forecasts differ from the actual values of a particular variable. It can tell us the error in the estimation of a variable. However, it is not very useful for comparing forecasting models (Khan, & Hildreth, 2003). Mean Absolute Percentage Error (MAPE) – It measures the accuracy of a fit in a particular time-series depicting trends. It expresses the accuracy as a percentage. One major advantage of using the statistic is that it corrects the ‘cancelling out’ effect and keep the different scales into account. Hence, it has the ability of comparing different models (iPredict, 2010). References Khan, Aman U.; Hildreth, W. Bartley (2003). Case studies in public budgeting and financial management. New York, N.Y: Marcel Dekker. Ramasubramanian, V. (n.d.). Time Series Analysis. I.A.S.R.I. Library Avenue, New Delhi. Retrieved on April 20, 2011 from < http://www.iasri.res.in/ebook/EBADAT/5-Modeling%20and%20Forecasting%20Techniques%20in%20Agriculture/2-time_series_analysis_22-02-07_revised.pdf> iPredict, (2010). Time-series Forecasting Error Statistics. Retrieved on April 29, 2011 from < http://www.ipredict.it/ErrorStatistics.aspx> The Mean Error, (2007). Extreme Forecast Index. © ECMWF. Retrieved on April 29, 2011 from < http://www.ecmwf.int/products/forecasts/guide/The_mean_error.html> Hyndman, R. and Koehler A. (2005). Another look at measures of forecast accuracy. Retrieved on April 29, 2011 from < http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.154.9771&rep=rep1&type=pdf> Lehmann, E. L., Casella, George (1998). Theory of Point Estimation (2nd ed.). New York: Springer. Retrieved on April 29, 2011 from < http://www.ams.org/mathscinet-getitem?mr=1639875> Appendix Time-Series Analysis Table 1 - Year and Month wise Statistics of total UK residents travelling abroad Year 1 2 3 4 5 6 7 8 9 10 11 12 Total 1986 823 780 1058 1397 1438 1821 1950 2777 2258 1509 1004 690 17505 1987 833 818 1067 1521 1559 1828 2177 3143 2547 1812 1006 737 19048 1988 962 872 1106 1671 1524 2261 2247 3268 2741 2018 1087 854 20611 1989 947 875 1156 1616 1646 2142 2677 3187 2711 2027 1098 912 20994 1990 1035 885 1336 1717 1828 2436 2480 3150 2968 2054 1235 950 22074 1991 1043 872 1365 1644 1651 2288 2277 3150 2830 2050 1424 1014 21608 1992 1129 1004 1572 1607 2116 2620 2865 3740 3318 2457 1465 1055 24948 1993 1305 1292 1644 2074 2391 2849 3148 4041 3462 2540 1603 1098 27447 1994 1406 1384 1679 2080 2133 3130 3326 3967 3729 3077 1695 1224 28830 1995 1724 1627 2055 2211 2478 3263 3353 4392 3878 3006 1647 1398 31032 1996 1808 1532 1925 2535 2468 3216 3392 4285 3799 3091 1884 1215 31150 1997 1695 1429 2024 2703 2317 2896 3323 4301 3734 3072 1929 1386 30809 1998 1879 1800 2391 2912 2993 3098 3645 4479 3937 3146 2075 1477 33832 1999 2046 1995 2572 2762 2968 3692 3845 4825 4215 3793 2240 1767 36720 2000 2035 2117 2618 2994 3329 4119 4161 5113 4620 4037 2388 2100 39631 2001 2267 2207 2711 3360 3486 4147 4356 5055 4617 4241 2537 2362 41346 2002 2253 2314 2965 3325 3299 4118 4142 5178 4569 4405 2975 2508 42051 2003 2541 2582 3030 3359 3911 4619 4580 5736 4865 4556 3459 2719 45957 2004 2817 3035 3327 3818 4604 4820 5139 6188 5648 4910 3661 2906 50873 2005 3183 3052 3644 4244 4605 5726 5391 6525 5821 5423 3465 2802 53881 2006 3020 3340 3782 4568 4693 5532 5931 7031 6275 5706 3747 3213 56838 2007 3533 3362 3947 4995 4589 6077 5982 7472 6198 5271 3919 2935 58280 2008 3326 3503 4114 4848 4695 6068 5877 7520 6332 5827 3908 3359 59377 2009 3564 3663 4279 4830 5401 6066 6219 7855 6256 5717 4080 3494 61424 2010 3878 3986 4020 5244 5075 6573 6357 8142 6704 6205 4260 3693 64137 Grand Total 51052 50326 61387 74035 77197 95405 98840 124520 108032 91950 59791 47868 940403 Figure 1 – Month wise travelling of UK residents over the 25 year period Figure 2 – Year wise travelling of UK residents over the 25 year period Table 2 - Sum of Overseas UK Travellers’ Statistics per year and per month Year 1 2 3 4 5 6 7 8 9 10 11 12 Total 1986 739 602 740 1028 1088 1124 1699 1839 1200 963 755 642 12419 1987 695 540 685 962 1007 1039 1430 1650 1181 954 689 619 11451 1988 789 578 646 1071 1034 1069 1471 1629 1155 943 661 592 11638 1989 723 550 740 1023 988 1189 1697 1748 1269 1051 796 690 12464 1990 745 580 828 1170 1096 1316 1763 2011 1406 1093 882 752 13642 1991 817 652 867 1208 1282 1468 1819 2139 1447 1140 802 809 14450 1992 926 732 921 1028 1125 1167 1680 2048 1337 1195 910 828 13897 1993 1038 678 925 1315 1305 1427 1883 2228 1507 1348 948 963 15565 1994 1021 792 964 1324 1191 1498 1930 2084 1535 1366 1073 1021 15799 1995 1132 869 1336 1302 1388 1575 2071 2257 1633 1448 1183 1145 17339 1996 1184 968 1171 1411 1482 1644 2206 2309 1791 1524 1204 1123 18017 1997 1015 786 1035 1321 1474 1502 2007 2281 1724 1484 1299 1197 17125 1998 1200 966 1179 1648 1594 1655 2037 2430 1721 1483 1235 1385 18533 1999 1164 1153 1294 1724 1801 1697 2290 2614 1794 1582 1410 1340 19863 2000 1215 1058 1568 1667 1725 1876 2339 2555 1998 1799 1531 1461 20792 2001 1482 1167 1568 2073 1924 1954 2624 2896 2226 2077 1842 1704 23537 2002 1526 1366 1827 2118 2239 2323 2826 2991 2314 2123 1653 1856 25162 2003 1664 1377 1899 2022 2194 2230 3000 2940 2228 2082 1820 2059 25515 2004 1631 1406 1767 2227 2431 2176 2785 3002 2241 2255 1773 2052 25746 2005 1680 1511 1855 2160 2295 2344 2712 3014 2187 2061 1836 1739 25394 2006 1621 1553 1819 2187 2304 2242 2819 2909 2214 2028 1682 1830 25208 2007 1589 1455 1820 2008 2024 2247 2546 2589 1965 1631 1582 1381 22837 2008 1527 1261 1737 2181 1966 2228 2782 2694 2079 2061 1774 1889 24179 2009 1792 1446 1705 1833 1823 2417 2588 2766 2180 2170 2011 1983 24714 2010 1781 1771 1943 2267 2344 2403 2912 3137 2385 2259 2149 2359 27710 Grand Total 30696 25817 32839 40278 41124 43810 55916 60760 44717 40120 33500 33419 482996 Figure 3 - Sum of Overseas UK travelers/month Fi Figure 4 - Sum of overseas UK travelers/Year Figure 5 - Month wise overseas travelling statistics Exponential Smoothing Figure 6 – MAPE Plot for Exponential Smoothing Model for the Overseas Visitors to UK Table 3 – Model Statistics of Exponential Smoothing for Overseas visitors to UK Model Statistics Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared MAPE Statistics DF Sig. Overseas visitors to UK from Jan 1985-Model_1 0 .543 5.138 16.732 16 .403 0 Figure 7 – Series Chart of Overseas Visitors to UK Figure 8 – Histogram of MAPE for Residents of UK residents going abroad Table 4 – Model Statistics of UK residents going abroad for Exponential Smoothing Model Model Statistics Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared MAPE Statistics DF Sig. UK residents going abroad from Jan 1985-Model_1 0 .519 4.948 26.416 15 .034 0 Figure 8 – Series Chart of UK Residents going abroad ARIMA Model Table 5 – Model Statistics of ARIMA for Overseas visitors to UK Model Statistics Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared MAPE Statistics DF Sig. Overseas visitors to UK from Jan 1985-Model_1 0 .531 5.303 14.108 16 .591 0 Figure 9 – MAPE Plot for ARIMA Model for the Overseas Visitors to UK Figure 10 – Series Chart Table 6 – Model Statistics of UK residents going abroad for ARIMA Model Model Statistics Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared MAPE Statistics DF Sig. UK residents going abroad from Jan 1985-Model_1 0 .538 5.140 16.586 14 .279 1 Figure 11 –Histogram of MAPE for Residents of UK residents going abroad Figure 12 – Series Chart Read More