8. time series calculations

7/30/2019 8. Time Series Calculations

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1

2

3

4

Step 1

Step 2

5

Year QuarterImports

(X)Trends (T)

Deviation from trend

OR Detrended values

: ( X - T) = S + R

19-2 1 114

2 1423 155 137.0 +18.0

4 136 138.25 -2.25

19-3 1 116 139.0 -23.0

2 150 139.25 +10.75

3 153 141.25 +11.75

4 140 143.75 -3.75

19-4 1 128 146.75 -18.752 158 151.125 +6.875

3 169 154.625 +14.375

4 159 158.5 +0.5

19-5 1 137 164.125 -27.125

2 180 168.625 +11.375

Calculate the trend value for each observation

For each observation, calculate the detrended value

The calculation of seasonal variation (Additive model)

(Pg 139 OJ)

Seasonal variation is a swing around the trend line

Using the Additive model where, X = T + S + R

Variation in any particular quarter will be : X - T = S + R

Calculation of Detrended values

How to calculate Seasonal Variation


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3 192 171.25 +20.75

4 172 174.0 -2.00

19-6 1 145

2 194

Step 4: Calculate the total for each quarter

Step 5: Calculate the average for each quarter

Step 6: If sum of the average calculated for all 4 quarters is not equ

6

1 2 3 4

19-2 +18.0 -2.25

19-3 -23.0 +10.75 +11.75 -3.75

19-4 -18.75 +6.875 +14.375 +0.5

19-5 -27.125 +11.375 +20.75 -2.00

Total -68.875 +29.0 +64.875 -7.5

Average -22.958 +9.667 +16.219 -1.875

Adjust -0.263 -0.263 -0.263 -0.263

-23.221 +9.404 +15.956 -2.138

Seasonal variation -23 +9 +16 -2

7Assumptions

1

2 Each seasonal component ( ie each one of the four considere

3

8

What do these mean?

In a sufficiently long series, the residual influences tend to of

The seasonal variations are: Quarter 1 = -23, Quarter 2 = +9, Quarter 3 = +1

The sum of the 4 seasonals components for a given year is eq

Step 3: Arrange figures in the following order so as to ease the calc

YearQuarter


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Knowledge of Seasonal variation is useful for firms to make planning

for eg. -23 means that economic conditions are such that the index of import

will be 23 points below


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al to zero then we must adjust

=1.053

d in the example) is assumed to be constant over time

set each other

, Quarter 4= -2

ual to zero

lation of seasonal variation

=-68.96/3


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will tend to fall below the trend, and on average it


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1

2

3

Year Quarter Imports (X)Seasonal

variation

Series with seasonal

variation eliminated

19-2 1 114 -23 137

2 142 +9 133

3 155 +16 1394 136 -2 138

19-3 1 116 -23 139

2 150 +9 141

3 153 +16 137

4 140 -2 142

19-4 1 128 -23 1512 158 +9 149

3 169 +16 153

4 159 -2 161

19-5 1 137 -23 160

Data with Seasonal variation eliminated ( page 142 OJ)

How to Deseasonalise data (Additive model

Deseasonalised data = data with seasonal variation eliminated

Very easy to do: If seasonal variation is -23, meaning that our data is 23

below the trend line we must therefore increase it by 23 to eliminate se

Constant fluctuations tend to hide the underlying behaviour of the varia

comparisons ans assessment over time difficult to make hence the need

the data


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2 180 +9 171

3 192 +16 176

4 172 -2 174

19-6 1 145 -23 168

2 194 +9 185

Important To read page 142 and 143 OJ


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)

points on average

sonal influences

le making

o deseasonalised


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1

2

Year QuarterImports

(X)Trend (T)

Seasonal

variation

(S)

Residual = X - T - S

19-2 1 114 -23

2 142 +93 155 137.0 +16 2

4 136 138.25 -2 -0.25

19-3 1 116 139.0 -23 0

2 150 139.25 +9 1.75

3 153 141.25 +16 -4.25

4 140 143.75 -2 -1.75

19-4 1 128 146.75 -23 +4.25

2 158 151.125 +9 2.1253 169 154.625 +16 -1.625

4 159 158.5 -2 +2.5

19-5 1 137 164.125 -23 -4.125

2 180 168.625 +9 +2.375

3 192 171.25 +16 4.75

4 172 174.0 -2 0

19-6 1 145 -23

2 194 +9

To read OJ page 144

Calculation of Residual ( page 144 OJ)

Residual = Original data -trend data- S

Residual = X - T - S

Calculation of Residuals (Additive model) ( page 144 OJ)


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Year Quarter Trend (T)

19-2 19-2 1

19-2 2

19-2 3 137.019-2 4 138.25

19-3 19-3 1 139.0

19-3 2 139.25

19-3 3 141.25

19-3 4 143.75

19-4 19-4 1 146.75

19-4 2 151.125

19-4 3 154.62519-4 4 158.5

19-5 19-5 1 164.125

19-5 2 168.625

19-5 3 171.25

19-5 4 174.0

19-6 19-6 1

19-6 2

135136137138139140141142143144145146147148149150151152153154155156157158

159160161162163164165166167168169170171172173174175176177178

179180181182183184185

19-2

1

19-2

2

19-2

3

19-2

4

19-3

tr

end

value


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119-3 219-3 3 19-3 4 19-4

1

19-4

2

19-4

3

19-4

4

19-5

1

19-5

2

19-5

3

19-5

4

19-6 119-6

quarter

Trend (T)


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2

Trend (T)


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1

2

Year Quarter Imports Moving annual total Moving pair total

Col. 1 Col.2 Col.319-5 2

681

3 192 1370

689

4 172 1392

703

19-6 1 145

8x = 8*176.5

= 1412

703 + w = 8x =1412

w= 709

2 194

8y = 8*178.75

= 1430

709+v = 8y =1430

v =721

3 a =198

4 b =184

The basis of the forecasting lies in our knowledge of the behaviour of tren

The trend will not suddenly change directioon

Forecasting from time series using observations which are read direc

trend line graph (ie estimating the position of the trend by eye and w

backward) page 146 OJ

FORECASTING METHOD


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How to calculate a?

How to calculate b?

145 + 194 + 198 + b = 721

b = 721-145-194-198 = 184

172 + 145 + 194 + a = 709

a = 709 -172-145-194 =198


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Trend

Col.4

171.25

174

x = 176.5

y = 178.75

ly from

rk


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1

Year Quarter Trend (T)

19-2 3 137.0

4 138.25

19-3 1 139.0

2 139.25

3 141.25

4 143.75

19-4 1 146.75

2 151.125

3 154.625

4 158.5

19-5 1 164.125

2 168.625

3 171.25

4 174.0

2

3 Over the same period, the trend has changed 13 times

4 Average rate of change in trend is 37/13 = 2.846 units

5

Forecast trend for 19-6 Quarter 1 = 174.000 +2.846 = 176.846




Forecasting using average rate of change in tr

For the period over which the trend values are available, the trend has i

= 37 units

Future trend values can be predicted by adding the average rate of chavalues

The problem with projecting the trend by eye is that forecasts made usi

be consistent as opinions vary as to the future position of the trend


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6

6.1

Year Quarter Forecast trendSeasonal

variationForecast index

19-6 1 176.846 -23 153.846 =154

2 179.692 +9 188.692 = 189

3 182.538 +16 198.538 = 199

4 185.384 -2 183.384 = 183

7

8

To read page 148 and 149 OJ

Many statisticians would argue that finding the average rate of change

whole period for which data is available may not be a valid method of p

value of the trend

The most recent data may be the more valid and some statisticians thinvalues only should be used to estimate the future trend

Forecasts for these periods can now be made by adding the appropriate

variation or by 'working backward'.

Adding appropriate value for seasonal variation to forecast import


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nd

creased by : 174 - 137

ge to the previous

g this method will not


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f the trend over the

edicting the future

that the last 3 trend

value for seasonal


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1

1.1

2

3 Year Quarter Import Index Trend Actual/Trend

19-2 1 114

2 142

3 155 137.0 1.1314

4 136 138.25 0.9837

19-3 1 116 139.0 0.8345

2 150 139.25 1.0772

3 153 141.25 1.0832

4 140 143.75 0.9739

19-4 1 128 146.75 0.8722

2 158 151.125 1.04553 169 154.625 1.0930

4 159 158.5 1.0032

19-5 1 137 164.125 0.8347

2 180 168.625 1.0675

3 192 171.25 1.1212

4 172 174.0 0.9885

19-6 1 145

2 194

Year 1 2 3 4

19-2 1.1313 0.9837

4 Rearranging as below

MULTIPLICATIVE MODEL

The trend is calculated in exactly the same manner as for

additive model (using 4 Quarter , centred moving average)

X / T = S * R ( assumed to be insignificant)

X = T * S * R

Actual data = Trend * Seasonal * Residual


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19-3 0.8345 1.0771 1.0831 0.9739

19-4 0.8722 1.0454 1.0929 1.0031

19-5 0.8347 1.0674 1.1211 0.9885

Total 2.5414 3.1899 4.4284 3.9492

Average 0.8471 1.0633 1.1071 0.9873

Seasonal

ratios

0.846 1.062 1.106 0.986

Adjusting by multiplying each average for each quarter by

4/4.0048, which gives :


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1 To find a deseasonalised series, divide the actual data by the sea

Year QuarterImport

Index

Seasonal

ratio Deseasonalised index

19-2 1 114 0.846 135

2 142 1.062 134

3 155 1.106 140

4 136 0.986 138

19-3 1 116 0.846 137

2 150 1.062 141

3 153 1.106 1384 140 0.986 142

19-4 1 128 0.846 151

2 158 1.062 149

3 169 1.106 153

4 159 0.986 161

19-5 1 137 0.846 162

2 180 1.062 169

3 192 1.106 174

4 172 0.986 174

19-6 1 145 0.846 171

2 194 1.062 183

2 Using seasonal ratios for forecasting

3 Actual forecast = Forecast trend values are multiplied by the ses

Year QuarterForecast

trend

Seasonal

ratioForecast index

19-6 1 176.846 0.846 149.612 = 150

2 179.692 1.062 190.833 = 191

3 182.538 1.106 201.884 = 202

MULTIPLICATIVE MODEL ; DESEASONALISED SERIES pg 151 OJ


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4 185.384 0.986 182.789 = 183


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sonal variation

nal ratios

= 114 /0.846

=135

8. time series calculations

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