Topic: Time Series Analysis

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• Session 6
• Topic Time Series Analysis
• Faculty Ms Prathima Bhat K
• Department of Management Studies
• Acharya Institute of Technology
• Bangalore 90
• Contact prathimabhatk_at_gmail.com
• 9242187131

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• Fit an equation of the form y a b x c x2
  to the data given below.

ANSWER
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Fitting of a Second Degree (Parabolic) Trend ?y
na b?x c?x2 171 5a0b10c ..(i) ?xy
a?x b?x2 c?x3 530a10b0c ..(ii) ?x2y
a ?x2 b?x3 c?x4 35110a0b34c ..(iii) B
y (ii) b 5.3 Solving (i) and (iii) Multiply
(i) by 2 and deduct that from (iii) we get c
o.64 (14c 9) and a 32.92 (171-100.645a) Ther
efore the equation is y 32.92 5.3 x 0.64
x2
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• Fit an equation of the form y A. Bx to the
  data given below.

ANSWER
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Fitting of a Exponential Curve y A. Bx
..(i) Taking Logarithm we get log y log A
x log B Y a bx ..(ii) Y log y a log
A b log B ..(iii) Equation (ii) can be
written as ?Y na b?x 5.6983 5a
15b ..(iv) ?xY a?x b?x2 21.8315
15a55b ..(v) By solving (iv) (v) we get b
0.4737 a -0.2814 Take Antilog we get A
0.5231 B 2.977 Therefore the trend equation
is y 0.5231(2.977)x
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METHOD OF MOVING AVERAGES
This is the simple and flexible method of
measuring trend. Moving Average is a averaging
process that smoothens out the fluctuations and
ups downs in the given data. The Moving Average
of period m is a series of successive averages
of m overlapping values at a time, starting with
1st, 2nd, 3rd value and so on.
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SEASONAL VARIATIONS

• The variations due to such forces which operate
  in a regular periodic manner with period less
  than one year. The objectives of studying this is
  as follows
• To isolate seasonal variations To determine
  the effect of seasonal swings on the values of a
  given phenomenon.
• To eliminate them To determine the value of
  the phenomenon if there were no seasonal ups
  downs.

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METHODS

• Method of Simple Averages
• Ratio to Trend Method
• Ratio to Moving Averages Method
• Link Relative Method

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SIMPLE AVERAGES

• This is the simplest method of measuring the
  seasonal variations in a time series and involves
  the following steps
• Arrange the data by years months
• Compute the average for the months
• Compute the overall average
• Obtain seasonal Indices for different months

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Compute the seasonal index from the data
given
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• X 4.06 (3.84.253.854.35)/4
• (3.8/4.06)10093.6
• (4.25/4.06)100104.7
• (3.85/4.06)10094.8
• (4.35/4.06)100107.1

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RATIO TO TREND

• This is a method which is an improvement over the
  previous method. This is on the assumption that
  seasonal fluctuations for any season are a
  constant factor of the trend. This involves the
  following steps
• Compute the trend values by the appropriate
  method
• Assuming multiplicative model, trend is
  eliminated.
• Arrange values according to the years, months
  or quarters
• These seasonal indices are adjusted to the
  total of 1200 for monthly data or 400 for
  quarterly data.

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Using Ration to Trend method, determine
seasonal index.
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Fitting of Linear Trend y a b x To find a
b ?y n a b? x 307 a5 b0 a
61.4 ?xy a ?x b ?x2 -14.5 a0
b10 b -1.45 Therefore the equation will
be given by y 61.4 -1.45x Quarterly values
will be increment of (-1.45/2 -0.36) Between
II III quarter - 0.36/2 -0.18
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Sum of the averages 106.8 95.84 95.33
101.9 399.90 Trend Eliminated Values are
Given Value for that Quarter 100 Trend
Value for that Quarter Therefore the Correction
Factor is 400 399.90
1.00025
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RATIO TO MOVING AVERAGES

• This is a method which is an improvement over the
  previous method. This is a widely used measure
  which involves the following steps
• Obtain 12-month (4-quarter) moving average
  values.
• Express the original values as a percentage of
  centered moving average.
• Arrange these according to the
  years/months/quarter
• These indices should be 1200 or 400.

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• Calculate the seasonal indices.

Answer Ratio to Moving Averages (61/63.125)100
96.63 (63/62.250)100 101.20 .. and so on.
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LINK RELATIVES

• This is the value of the given phenomenon in any
  season expressed as a percentage of its value in
  the preceding season. This involves the following
  steps
• Convert the original data into link relatives.
• Average these link relatives for each month.
• Convert Link Relatives into Chain relatives.
• Obtain CR for the first month
• Obtain Corrected Chain relatives.

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Link Relatives for any month (Current Months
Value / Previous Months Value) 100 Chain
Relative for any month (Link Relative of that
month Chain Relative of the preceding month) /
100 New CR for the First Quarter (LR of I Qtr.
CR of last Qtr.)/100 (123.303
89.81) / 100 112.54 d ¼(New CR of first Qtr.
-100) ¼(112.54 100) 3.135 Adjusted CR
78.395 3.135 75.26 72.69 6.27
66.42 89.81 9.405 80.41
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CYCLICAL VARIATIONS
This is an approximate or crude method of
measuring cyclical variations, which consists of
estimating trend, seasonal components and then
eliminating their effect from the given Time
Series.
RANDOM VARIATIONS
These can not be estimated accurately, we can not
obtain an estimate the variance of random
components.