Time Series Theory.2

7/31/2019 Time Series Theory.2

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WEEK 9:TIME SERIES

This is a graphical representation of observations

taken for along time, at specific times, and usuallyat equal intervals.

Examples: the daily closing price of a share onSE., total monthly sales, hourly temperatures

announced by a weather station, population,un/employment figures, the turnover of a firm,prices, quantities etc.

It is a function of time. I.e. Y = F (t).

What are the factors that affect the comparabilityof values used in constructing a time series?Calendar, price & population variations.

See overleaf slide for details.


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ATTRIBUTES TO VARIATIONS IN DETAILS

Calendar days variation affect time seriesexpressed on monthly basis. E.g. monthlywages that vary with working days. To adjust forthis we divide monthly figurewith respectivedays of the month.

Price changes due to change in value of money.

Price index could be used in adjusting thevalues.

Large population changes affect the unit basisor per capita cases. Adjustment is C/N.

A change in units, definition, classification ortype of product.


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TIME SERIES CHARACTERISTICS


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CHARACTERISTICS OF A TIME SERIES

Long term or secular movements or variations:

the general or overall trend of the data, drawn in adotted line. This trend line can be used to predictthe future values of the series.

Seasonal variation: they are fluctuations attributedto seasonal factors. E.g. demand for certainproducts like toys.

Other elements: cyclicalfluctuations (repeat

themselves), and randomor irregular fluctuations. See a typical time series on the provided hand out.


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METHODS OF ESTIMATING THE TREND

The least squares method where,

D = d21+ d22+ d23++ d2n is a minimum. The method of semi-averages: This entails the

separation of data into two equal parts which areaveraged to give two points that are used to draw

the trend. E.g. 1970 1981(12 years) 1970 -75,then 1976 81. For odd-n the middle period isomitted. E.g. 1970- 1982 (13years) 1976 isomitted.

The free hand method by looking at the graph. The moving average method. This has some

disadvantages. List two of them. See example in the next slide.


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MOVING AVERAGE EXAMPLES

Computation of the 3-

unit moving average of:

23, 21, 22, 23, 27, 25,

and 25.

Try: 170, 231, 261,267,278,302,299,

298, 340, 273, 210,and 158.

Value MovingTotal

Moving

Aver.

23 - -

21 66 22

22 66 22

23 72 24

27 75 25

25 77 25 2/3

25 - -


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SAMPLING THEORY

The theory has to do with the study of the

relationship existing between a population andsamples drawn from the population. Its importance includes:

(i) Estimation of parameters from known sample

statistics.(ii) Determining the significanceof the difference

between two samples, or it may be due tochance variation. The answers to such are

obtained through the use oftests ofsignificance and hypothesis.


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SAMPLING DISTRIBUTION

This is a distribution where the variable is a

statisticobtained from samples of size Ndrawn from their parent population.

The statistic computed will vary from sample to

sample giving a distribution of the statistic. Examples: Sampling distribution of means, of

medians and of standard deviations.


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SAMPLING DISTRIBUTION FOR MEANS

For an infinite population of size N where all

possible samples of size n are drawn withoutreplacement, the sampling distribution mean is xand that of the population is p, are such that

x = p. Thus, sample mean is an estimator ofthat of the population.

Estimates of SE and SD are given as, SE x = s /nand SD x = /n, where s is the SD of one of thesamples and I the population SD.

If the population is finite we have, x = p and

SE = x = ( / n) [(N n)/(N - 1)], Find x for N = 15,000, = 20 and n =1,000.

( 0.611) What is a finite population correction factor (FPC)?


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PRE-REQUISITES FOR THEVERIFICATION OF THE RELATIONS.1

Use of random sampling: This provides every

member of the population with an equal chance ofbeing selected into the sample.Combination of samples: For a finite population ofsize N P the number of combinations of samples ofsize N, drawn from it is given as a combinatory of,(n) = N p C N = N P! / [N!(N P N)! ] N!/ [n! (N n)!].Chance of a sample being selected: Each of the

possible samples is chosen with the sameprobability, p = 1/ (N p C N) 1/ (N C n).

Self- check: Compute the number of distinctsamples of size, n = 2, that can be drawn from a

population of size, N = 5. (10)


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PRE-REQUISITES FOR THEVERIFICATION OF THE RELATIONS.2

Probability based sample statistics The mean value, or expected value, E (x)is given by, = xi pi, where pi is therespective probability of occurrence of a

value, xi, in this case it remains the same forall samples.

The variance value, is given by,

2 = (xi - )2pi = pi x2i - (pixi)2.

Use these concepts to draw a sampledistribution of the means for random samplesof size, n = 2 from a population of values: 1,3, 5, 7 and 9.


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DETAILED TABULATION

pi pi (xi-) (xi-)2 (xi-)2pi

2 1/10 1/5 -3 9 9/103 1/10 3/10 -2 4 4/10

4 2/10 8/10 -1 1 2/10

5 2/10 1 0 0 0

6 2/10 11/5 1 1 2/10

7 1/10 7/10 2 4 4/10

8 1/10 8/10 3 9 9/10

pi=1 pi=5 (xi-)2pi=3


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FINITE POPULATION CORRECTION FACTOR

It is the factor [(N n)/(N - 1)]

The FPC is used to covert the SE formula from

infinite populations case to finite one.

FPC is usually ignored when the sample

constitutes 5% or less, for when the sample sizeis much smaller than the population,the latter isconsidered to be effectively infinite in size.

Find the value of the finite population correction

factor for n = 100 and N = 10,000. (0.995)

Also find SE when n = 20, N = 1,200 and

= 346.55 (76.874)


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STANDARD ERRORS SUMMARY


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STANDARD ERRORS SUMMARY

Means: = s z(S.E), S.E = s / n .

Difference of means: dp

= ds

z(S.E),

S.E = (12/ n1 + 22/n2). See p. 12.-12.5 Gupta.

Variance: 2 = 2 z(S.E), S.E = s2 (2/n).

Standard deviations: = s z(S.E), S.E = (s2/2n).

Difference of s d: 1 - 2 = d s z(S.E),

S.E= (12/2n1 + 22/2n2). The best is of d f = n -1.

Proportion: = p z(S.E), S.E= (p q/n) (Bernoulli

process). Difference of proportion: 1 - 2=d s z(S.E),

S.E = [(p1 q1/n1) +(p2 q2/n2)].

See estimators properties (estimation theory) overleaf.


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WEEK 10: ESTIMATOR CONSISTENCY ANDSUFFICIENCY

Consistency is attributed to the absolute differencebetween the statistics used to estimate the

parameter tending to zero (CLT) as the size of the

sample increases (precision). E.g. | s - | 0, as

n N. This is true for proportion, the mean and s d.

Give any other suggestions.

Sufficiency has to do with an estimators formula

using all the available data. E.g. the mean, E(x), andthe s d; unlike the mode and median whose

formulae use only part of the data or scores.


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UNBIASED ESTIMATES

Is attributed to the sample distribution of a

statistic being equal to the correspondingpopulation parameter estimate.

E.g. the mean and median of the means ofinfinite sets of samples equals those of the

population. The rx y has this property when rx y for the

population is zero.

The sample variance, s for d f = n -1 does so.

The conversion formula of sample variances is,s2 = n/(n-1) 2. Check this with 1,2,8,and 9.

(7.5, 6 2/3)


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EXAMPLES .1

A poll of 400 randomly selected registered votersin a given constituency show that 186 intend tovote for candidate X. Construct a 95% C I for theportion of all the voters who will vote forcandidate X. (0.416 - 0.514)

During a sale by firm X,150 units of givenproduct were sold at a mean price of $ 1,400with a s d of 120. 200 units of the same productwere sold by firm Y at a mean price of 1,200 witha s d of 80. Construct a 95% C I of the limits ofthe difference of the means of the two sales.(200 22.16, 177.8 - 222.16)


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Time Series Theory.2

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