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Department Of Economics, University Of Ghana Level 300, Econs 305: Applied Statistics for Economist Lecturer: Dr. D. K. Twerefou Problem Set 3: Time series analysis

1. The table below provides the components of a time series from 1985 to 1997. Using the additive model complete the table.

Year Trend Cycle Seasonal Random Actual values1985 3 1 1 162

1986 159 4 2 1 1987 160 5 2 167 1989 162 6 2 169 1990 5 1 2 173 1991 166 6 -1 2 1992 168 7 2 180 1993 171 8 1 3 1994 174 2 3 185 1995 177 7 1 184 1996 180 2 2 192 1997 184 8 -1 192

2. The table below shows some of the values of the components and actual time series from 1996 to 2005. Using the multiplicative model complete the table.

Years Trend Cycle Seasonal Random Actual Values 1996 0.60 1.04 0.40 18335.37 1997 88150.80 0.80 0.50 35965.53 1998 96965.88 1.10 0.40 25598.99 1999 99874.86 0.70 0.97 0.50 2000 104868.60 0.70 0.95 41842.57 2001 111160.72 0.90 0.40 30013.39 2002 124500.00 0.65 0.90 0.50 2003 151890.00 0.70 0.40 40402.74 2004 162522.30 0.85 0.60 58020.46 2005 0.80 0.88 0.50 63615.13 2006 196990.03 0.80 0.90 0.50

3. The table below shows the values of some of the components and actual values of a time series for 1996 to 2005. Using the additive model complete the table.

Year Trend Cycle Seasonal Random Actual 1995 91823.75 734.59 36.73 92962.36 1996 110188.50 440.75 44.08 111334.46 1997 121207.35 727.24 48.48 122467.91 1998 499.37 749.06 49.94 126141.94 1999 131085.75 524.34 52.43 132449.04 2000 138950.89 833.71 55.58 140395.98 2001 622.50 933.75 62.25 157243.50 2002 189862.50 7 1139.18 75.95 191837.07 2003 812.61 1218.92 81.26 205265.67 2004 225906.00 903.62 1355.44 228255.42 2005 246237.54 984.95 1477.43 98.50 2006 253624.66 1003.00 1521.75 256250.86

4. The table below shows the production of cocoa by a farmer in Ghana from 2000 to 2009.

a. Fit the trend line using the method of semi-averages.

b. Find the equation of the straight line

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Years 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Tons of Cocoa 300 400 500 100 300 400 600 200 400 500

5. The table below provides some of the components and actual values of a time series from 1990 to 2000. Using the multiplicative model complete the table.

Years Trend Cycle Seasonal Random Actual 1990 1251 1.04 1.2 1405.12 1991 1241 0.8 1.02 1.2 1992 1231 0.7 1.2 1127.10 1993 1222 0.7 0.97 1.1 912.71 1994 0.8 0.95 1.1 1019.92 1995 1200 0.9 0.9 972.00 1996 1195 0.9 1 860.40 1997 1195 0.7 1 752.85 1998 1194 0.8 1.1 935.14 1999 1192 0.8 0.88 923.08 2000 0.9 0.9 0.9

6. The table below shows the production of rice (in thousands of tons) from 2000 to 2008. Find the trend equation using the method of semi-averages.

Years 2000 2001 2002 2003 2004 2005 2006 2007 2008 Prod. (000’ tons) 30 40 50 10 30 40 60 20 40

7. The table below shows the production of timber from 1973 to 1987. Find the trend equation using the 3-yearly moving averages.

Year Production Year Production 1973 15 1980 56 1974 21 1981 63 1975 30 1982 70 1976 36 1983 74 1977 41 1984 82 1978 46 1985 93 1979 50 1986 95 1987 102

8.The table below shows the nominal GDP of a country from 2000 to 2008. Fit the trend equation using 3-yearly moving averages.

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 GDP (trillion cedis) 235 242 255 258 285 291 292 297 300

9. The table below shows the number of people who attend morning, afternoon and evening school for a week.

Monday Tuesday Wednesday Thursday Friday Morning 122 128 129 123 133 Afternoon 111 112 143 109 103 Evening 116 140 141 125 125

a. Find the trend equation using 3-weekly moving averages

b. Draw the trend line.

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10. The data below shows the output of a food processing company. Assume a 4-yearly cycle and find the trend line by the method of moving averages. Using the trend line estimate the output for 1990

Year 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 Output (tons) 464 515 518 467 502 540 557 571 586 612

11. The table below shows the quarterly sales of a supermarket.

Year I II III IV 2000 - - 80 40 2001 120 144 172 102 2003 50 75 124 46 2004 88 36 180 -

i. Using the method of 4-quaterly moving average find the trend values and the trend equation

ii. What will be the sales in the 3rd quarter of 2004

12. The table below shows the unemployment data of graduates from 2004-2009 in a certain country.

Year 2004 2005 2006 2007 2008 2009

Unemployment 1005 1120 980 1030 1040 1000

i. Determine the least squares trend equation

ii. Estimate unemployment for 2015.

13. The following data shows the number of road accidents from 1996 to 2008.

Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 No. of accidents 194 215 220 224 235 242 255 258 285 291 292 297 300

i. Obtain the trend equation using the method of least squares

ii. Estimate the number of road accidents in 2016

14. The table below shows the production of rice in a country in metric tons.

i. Fit the trend line using the method of least squares.

ii. Estimate rice production for the year 2020

Year 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 776 860 880 896 940 968 1020 1033 1139 1163 1167

Year 2003 2004 2005 2006 2007 2008 1190 1202 1248 1280 1300 1360

15. The table below shows the number of people who attend morning, afternoon and evening school for a week.

Monday Tuesday Wednesday Thursday Friday Morning 122 128 129 123 133 Afternoon 111 112 143 109 103 Evening 116 140 141 125 125

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a. Using the method of moving averages and the multiplicative model, Determine the seasonal indices for morning, afternoon and evening

b. Deseasonalize the data.

16. The table below shows the actual (Y) and Trend (T) estimate of the production of aluminum ingots in a country. Calculate the cyclical relative and interpret your result

Year 1994 1995 1996 1997 1998 1999 2000 2001 2002

Y 700 850 900 1100 1225 1324 1500 1675 1700

T 728 799 972 1078 1225 1347 1485 1682 1564

17. The table below shows the sales of a supermarket in Ghana in billion cedis.

Quarter Year First Second Third Fourth

2003 18 80 30 522 2004 24 105 54 527 2005 33 141 48 538 2006 40 150 75 545 2007 35 180 55 550 2008 48 205 70 557

a. Calculate the trend using the method of moving averages b. Calculate the average seasonal Index under the multiplicative model c. Deseasonalize the data.

18. The table below shows the actual (Y), Trend (T) estimate and the cyclical relative (C) of wheat production in a country. Calculate the missing values. Interpret your results for 1998, 1999 and 2005.

Year Actual Value(Y) Trend value (T) Cyclical relative (%) 1998 1938 88 1999 1850 100 2000 1900 2242 2001 11100 10978 2002 11225 83 2003 12844 103 2004 15001 86 2005 11208 104 2006 17001 17239

19. The data below shows the quarterly sales from a shopping mall from 2000 – 2005 in billion cedis.

I II III IV 2000 40 78 92 250 460 2001 55 82 101 260 498 2002 68 90 121 265 544 2003 72 95 120 270 557 2004 78 101 125 275 579 2005 85 112 130 280 607

a. Plot the quarterly data for all the years b. Which component of time series is visibly present c. Calculate the seasonal indices

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d. Estimate the trend line using the method of least square e. Forecast the sales value for the year 2006 f. Forecast the sales value for the first and fourth quarters of year 2006

20. The data below shows quarterly export of a cash crop from a country from 1995 – 2000 in million metric tons.

I II III IV 1995 5 8 9.5 26 48.5 1996 6.5 8.7 10.5 27 52.7 1997 6.8 9 12.1 26.5 54.4 1998 7.5 9.5 12 28 57 1999 7.8 10.5 12.5 29.5 60.3 2000 9.5 11.5 13.5 30 64.5

a. Plot the data for all the years b. Which component of time series is visibly present c. Calculate the seasonal indices d. Estimate the annual trend line using the method of least square e. Forecast the sales value for the year 2008 f. Forecast the sales value for the first and fourth quarters of year 2008

21. The table below shows the monthly export of cocoa for the first five months of the year in thousand metric tons.

January February March April May 15.2 30.9 41 18.3 36.2

If the forecast for January is 20,000 metric tons, determine forecasts for February through June using exponential smoothing with a smoothing constant of α=0.17

22. The table below shows the sales of a shopping mall

Year I II III IV Total 2003 70 300 90 150 610 2004 80 350 100 165 695 2005 85 380 110 170 745 2006 90 400 120 180 790 2007 95 410 130 190 825 2008 100 430 135 195 860

a. Plot the quarterly data for all the years b. Which component of time series is visibly present c. Calculate the annual trend equation using the method of semi averages d. Calculate the annual trend equation using the 3 yearly moving averages e. Calculate the annual trend equation using the method of least squares f. Calculate the seasonal index g. Forecast the sales value for the year 2010 h. Forecast the sales value for the first and fourth quarters of year 2010 based on the

least square trend equation and the seasonal index.

23. A well known Casino made the following annual sales, in millions of cedis.

2004 2006 2007 2008 123.3 172.3 230.3 315.5

If the forecast for 2004 was 125, forecast the sales for 2009 using the method of exponential smoothing with a smoothing constant of α=0.15