STA457 – Spring 2008 - Assignment 1 - Solution 1) The sample autocorrelation is calculated using Minitab. Here is the output

Autocorrelation Function: TSE Lag ACF T LBQ 1 0.981067 13.52 185.78 2 0.959454 7.73 364.40 3 0.937084 5.92 535.71 4 0.913822 4.93 699.49 5 0.882800 4.25 853.17 6 0.849663 3.75 996.30 7 0.815207 3.36 1128.77 8 0.780353 3.04 1250.83 9 0.742335 2.76 1361.90 10 0.703116 2.52 1462.09 11 0.662253 2.29 1551.47 12 0.619858 2.09 1630.21 13 0.576165 1.90 1698.63 14 0.534927 1.73 1757.94 15 0.492420 1.57 1808.49 16 0.451462 1.42 1851.22 17 0.409361 1.27 1886.56 18 0.368263 1.14 1915.32 19 0.325701 1.00 1937.95 20 0.284496 0.87 1955.32 21 0.242093 0.74 1967.97 22 0.199368 0.60 1976.60 23 0.157225 0.48 1982.00 24 0.121037 0.37 1985.22 25 0.085369 0.26 1986.83 Here is the correlogram

Lag

Aut

ocor

rela

tion

24222018161412108642

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

Autocorrelation Function for TSE(with 5% significance limits for the autocorrelations)

b) Here is the correlogram of the first difference

Lag

Aut

ocor

rela

tion

24222018161412108642

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

Autocorrelation Function for Diff1_TSE(with 5% significance limits for the autocorrelations)

c) The correlogram of the data in part (a) decays to 0 very slowly, which means that this is a long memory time series. The correlogram of the first difference, decays very quickly to 0 and most of the data are within 2.0± which implies that the first difference is approximately white noise. Taking the first difference has removed the trend in the data. 2) a) Here is the time series plot of the data

Index

Sale

s

9988776655443322111

25000

20000

15000

10000

5000

Time Series Plot of Sales

As we can see, the sales increase over time in general, i.e. car sales in the late 60th are higher than those earlier in that decade. Also, it looks like there is a trend within each year, increasing in the beginning of the year and decreasing later towards the end of the year. This is in fact a seasonal effect.

b) Here is the plot of the smoothed and the original data

Index

Sale

s

9988776655443322111

25000

20000

15000

10000

5000

Moving AverageLength 13

Accuracy MeasuresMAPE 21MAD 3039MSD 13934682

VariableActualSmoothed

Moving Average Plot for Sales

Smoothing the data helped remove the random variation. However, the seasonal pattern is no longer apparent in the smoothed data; all we can see is the long-term increasing trend. c) Here is the plot of the differenced data

Index

Diff

_12

9988776655443322111

5000

4000

3000

2000

1000

0

-1000

-2000

-3000

Time Series Plot of Diff_12

As we can see, differencing has removed the overall increasing trend in the data, but did not eliminate the seasonal trend.