5.1 Component of time series | unit 5 analysis of time series

UNIT – 5

Analysis of time series

5.1 Component of time series

Introduction

A time series is set of data collected at successive point in a time or over successive period of time. A time series is a collection of observations made sequentially through time. The interval between observations can be any time interval (hours within days, weeks, months, years, etc.).

Some examples of time series are:

Malaria incidence or deaths over calendar years

Daily maximum temperatures

Hourly records of babies born at a maternity hospital

monthly unemployment,

weekly measures of money supply,

daily closing prices of stock indices, and so on

An analysis of a single sequence of data is called univariate time-series

analysis.

An analysis of several sets of data for the same sequence of time periods is called multivariate time-series analysis or, more simply, multiple time-series analysis.

COMPONENT OF TIME SERIES

Fluctuation in a time series is mainly due to four basic components.

1 Secular trend or trend (T)

2 Seasonal variation (S)

3 Cyclical variation or cyclic fluctuation (C)

4 Irregular or random moments (I)

Secular trend or trend (T)

Trend is the phenomenon of long term changed in a recorded data series, generally, in the same direction throughout the span of the series.

A sequence plot of time series (the time series value plotted vertically with respect to time itself on the horizontal axis) will usually reveal the presence of trend as a gentle upward or downward “drift” of the data path.

Upward sloping trend paths in a real- value time series may be indicative of growth phenomenon, a downward sloping path suggest contraction.

3. In a money-value time series an upward sloping path may represent some combination of real growth and inflation; a downward sloping trend path might indicate contraction with deflation.

4. Trend is usually the result of long-term factors such as changes in the population, demographics, technology, or consumer preferences.

Seasonal Variation

5. This is the pattern of variation within time series which repeat itself year to year.

6. Seasonality may be associated with agricultural functions, seasonal weather pattern, custom and convention, or religious or secular holidays.

It is important to remember that a seasonable pattern in one time series may or may not resemble that in another time series.

Fans and air-conditioned sales are high in the summer month, agricultural sales are high at harvest time, RAIN CAOTS, UMBERELLA SALES HIGH IN MONSOON

Cyclic Components:

Any regular pattern of sequences of values above and below the trend line lasting more than one year can be attributed to the cyclical component. Usually, this component is due to multiyear cyclical movements in the economy.

Cyclic variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year. Also, these variations are not regular as seasonal variation.

A business cycle showing these oscillatory movements has to pass through four phases-prosperity, recession, depression and recovery. In business, these four phases are completed by passing one to another in this order

Irregular Variation

Irregular variations are fluctuations in time series that are short in duration, erratic in nature and follow no regularity in the occurrence pattern. These variations are also referred to as residual variations since by after trend, cyclical and seasonal variations. Irregular fluctuations result due to the occurrence of unforeseen events like: FLOODs, EARTHQUAKES, WARS, and FAMINES etc.

How the component relates to the original series?

A model that expresses the time series variable Y in terms of the components T (trend), C (cycle), S (seasonal) and I (irregular).

Key Takeaways:

. A time series is a collection of observations made sequentially through time.

The interval between observations can be any time interval (hours within days, weeks, months, years, etc).

There are four components of time series Secular, Seasonal, Cyclical and Irregular.

5.2 Calculation of secular trend–moving average method and method of least squares.

Method of least Squares and curve fitting

Linear tend

It is a mathematical method and with it gives a fitted trend line for the set of data in such a manner that the following two conditions are satisfied.

The sum of the deviations of the actual values of Y and the computed values of Y is zero.
The sum of the squares of the deviations of the actual values and the computed values is least.

This method gives the line which is the line of best fit. This method is applicable to give results either to fit a straight-line trend or a parabolic trend.

The method of least squares as studied in time series analysis is used to find the trend line of best fit to a time series data.

Secular Trend Line

The secular trend line (Y) is defined by the following equation:

Y = a + b X

Where, Y = predicted value of the dependent variable

a = Y-axis intercept i.e. the height of the line above origin (when X = 0, Y = a)

b = slope of the line (the rate of change in Y for a given change in X)

When b is positive the slope is upwards, when b is negative, the slope is downwards

X = independent variable (in this case it is time)

To estimate the constants a and b, the following two equations have to be solved simultaneously:

ΣY = na + b ΣX

ΣXY = aΣX + bΣX2

To simplify the calculations, if the midpoint of the time series is taken as origin, then the negative values in the first half of the series balance out the positive values in the second half so that ΣX = 0. In this case, the above two normal equations will be as follows:

ΣY = na

ΣXY = bΣX2

Example: Below are given the figures of production (in thousand quintals) of a sugar factory:

Year	1992	1993	1994	1995	1996	1997	1998
Production	80	90	92	83	94	99	92

(a) Fit a straight-line trend to these figures.

(b) Plot these figures on a graph and show the trend line.

(a) Using normal equations and the sugar production data we can compute constants a and b as shown in Table 7.6:

Table : Calculations for Least Squares Equation

Year	Time period	Production			Trend values
1992 1993 1994 1995 1996 1997 1998	1 2 3 4 5 6 7	80 90 92 83 94 99 92	1 4 9 16 25 36 49	80 180 276 332 470 594 644	84 86 88 90 92 94 96
Total	28	630	140	2576

Therefore, linear trend component for the production of sugar is:

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Fig.7.4: Linear Trend for Production of Sugar

Plotting points on the graph paper, we get an actual graph representing production of sugar over the past 7 years. Join the point a = 82 and b = 2 (corresponds to 1993) on the graph we get a trend line as shown in Fig. 7.4.

The production of sugar for year 2001 will be thousand quintals

Parabolic Trend Model

The curvilinear relationship for estimating the value of a dependent variable y from an independent variable x might take the form

This trend line is called the parabola.

For a non-linear equation , the values of constants a, b, and c can be determined by solving three normal equations.

Example : The prices of a commodity during 1999-2004 are given below. Fit a parabola to these data. Estimate the price of the commodity for the year 2005.

Year	Price	Year	Price
1999 2000 2001	100 107 128	2002 2003 2004	140 181 192

Also plot the actual and trend values on a graph.

Exponential trend model

Logarithm y = aebx.

The equation is
y = aebx.
Taking log to the base e on both sides,
we get log y = log a + bx.
Which can be replaced as Y=A+BX,
where Y = logy, A = log a, B = b and X = x.

Key Takeaways:

Linear Trend is a mathematical method and with it gives a fitted trend line for the set of data in such a manner that the following two conditions are satisfied.
The secular trend line (Y) is defined by the following equation:

Y = a + b X

5.3 Utility of index numbers problems in the construction of index numbers simple and weighted index numbers

An alternative system of assigning weights lies in using value weights. The value weight for any single commodity is the product of its price and quantity, that is, If the index of weighted average of price relatives is defined as

…………….. (6-23)

then can be obtained either as

the product of the base period prices and the base period quantities denoted as

that is, or

the product of the base period prices and the given period quantities denoted

as that is,

When is , the index of weighted average of price relatives, is expressed as

…………….. (6-24)

It may be seen that is the same as the Laspeyre’s aggregative price index.

Similarly, When is , the index of weighted average of price relatives, is expressed as

……….. (6-25)

It may be seen that is the same as the Paasche’s aggregative price index.

If the index of weighted average of quantity relatives is defined as

…….. (6-26)

then v can be obtained either as

the product of the base period quantities and the base period prices denoted as

that is, , or

the product of the base period quantities and the given period prices denoted

as that is,

When is , the index of weighted average of quantity relatives, is expressed as

…….. (6-27)

It may be seen that is the same as the Laspeyre’s aggregative quantity index.

Similarly, When is , the index of weighted average of quantity relatives, is expressed as

…….. (6-28)

It may be seen that is the same as the Paasche’s aggregative quantity index.

Example 6-5

From the data in Example 6.2 find the:

Index of Weighted Average of Price Relatives, using

(i) as the value weights

(ii) as the value weights

Index of Weighted Average of Quantity Relatives, using

(i) as the value weights

(ii) as the value weights

Calculations for

Index of Weighted Average of Price Relatives

(Base Year = 1980)

Item
Fish Mutton Chicken	7500 10620 9900	9000 11520 11000	1000000 1357000 1080000	1200000 1472000 1200000
Sum 	28020	31520	3437000	3872000

Index of Weighted Average of Price Relatives, using

(i) as the value weights

(ii) as the value weights

Calculations for Index of Weighted Average of Quantity Relatives

(Base Year = 1980)

Item
Fish Mutton Chicken	7500 10620 9900	10000 13750 10800	9000000 1152000 1100000	1200000 1472000 1200000
Sum 	28020	34370	3152000	3872000

Index of Weighted Average of Quantity Relatives, using

(i) as the value weights

(ii) as the value weights

Although the indices of weighted average of price/quantity relatives yield the same results as the Laspeyre's or Paasche's price/quantity indices, we do construct these indices also in situations when it is necessary and advantageous to do so. Some such situations are as follows:

When a group of commodities is to be represented by a single commodity in the group, the price relative of the latter is weighted by the group as a whole.

Where the price/quantity relatives of individual commodities have been computed, these can be more conveniently utilised in constructing the index.

Price/quantity relatives serve a useful purpose in splicing two index series having different base periods.

Key Takeaways:

An alternative system of assigning weights lies in using value weights. The value weight
for any single commodity is the product of its price and quantity, that is,
If the index of weighted average of price relatives is defined as

5.4 Base shifting fishers ideal index numbers and tests of reversibility

Reversal Test

There are two types of test which the different methods of index numbers can be put to check: Time Reversal Test and Factor Reversal Test

A test that may be used under the axiomatic approach which requires that if the prices and quantities in the two periods being compared are interchanged the resulting price index is the reciprocal of the original price index.

When an index satisfies this test, the same result is obtained whether the direction of change is measured forwards in time from the first to the second period or backwards from the second to the first period.

The time reversal test requires that the index for the later period based on the earlier period should be the reciprocal of that for the earlier period based on the later period; one of the desirable features of the “Fisher Ideal” price and volume indexes is that they satisfy this test.

Time Reversal Test

It is an important test for testing the consistency of a good index number. This test maintains time consistency by working both forward and backward with respect to time (here time refers to base year and current year). Symbolically the following relationship should be satisfied, P01 × P10 =1

Fisher’s index number formula satisfies the above relationship

when the base year and current year are interchanged, we get

Factor Reversal Test

This is another test for testing the consistency of a good index number. The product of price index number and quantity index number from the base year to the current year should be equal to the true value ratio. That is, the ratio between the total value of current period and total value of the base period is known as true value ratio. Factor Reversal Test is given by,