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Chapter-01

Introduction

1.1 Background of the study

Bangladesh is a developing country. As for developing purpose we have to know

about the marketing conditions. In developed countries overall developments

such as nancial development, economic development depends on dierent

marketing policy. To get the better result at rst we have to correctly specify the

model, because if we fail to correctly specify the model the analysis will give us

wrong answer.

Time series is an important topic not only in statistics but also in economics,geography, and also in government policy making. oreover, the developing

countries can!t develop their position if they have no appropriate time series

model for forecasting economic and other variables. "ften government

researchers are interested in determining the important economic variables or

indicators. If a developing country has a good model they can easily estimate

these economic variables and indicators like the ma#or import products in

Bangladesh, in$ation and other variables, to estimate the nancial or

agricultural trend of a country, to estimate the average the ma#or e%port and

import products in Bangladesh, important &uantities of a country and also the

improvement or decrease in production of the country, by showing the upwardor downward trend. Those are important for a country to take many valuable

decisions for future developments.

A time series analysis is a statistical methodology dealing with analysis of a

se&uence of observed data usually ordered in time. The time series analysis is a

movement of statistical literature that characteri'e a time series as stationary

and nonstationary in practical data may not be stationary to test the stationary

and non stationary of any statistical data the relative method is (raphical

analysis, )orrelogram, *+, A*+ test.

In time series problem very often the problem of spurious regression arises

when the data series are nonstationary. -uppose we have a data on the ma#or

import products in Bangladesh, which is related with the production of rice and

we assume that there is a linear relationship among the two variables. ow if we

obtain a very high / even though there is no meaningful relationship between

the variables, in this situation spurious regression arises. And the result will be

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misleading. (overnment use time series model to adopt some individual

research purpose. +or individual purpose researchers often use time series

model.

In this report we will see the data of the ma#or import products in Bangladesh for

relevant years.

1.2 Problems and Motiation

In our study, the problem is that we want to test the stationarity and non

stationarity and t the appropriate A/IA model. 0e have to prepare the report

regarding the topics provided our respective course teacher within a very short

time. -ince we have limited time, we have faced a lot of problems to carry out

our study. Also dierent physical and mental constraints are presented in our life

for our analysis.

There are several problems arises while dealing with test of stationarity and

nonstationarity and tting the appropriate A/IA model. The problems are

mentioned below

0hen the time series data are stationary, we cannot run our study. As a result,

we cannot precede our study. +or this reason, the time series must be non

stationary. 0e can say that testing the presence of white noise in the data,

making the time series stationary and developing the information!s for

forecasting may help a country to take many fruitful plans for their

development. -o we are motivated to testing white noise and developing thecorrect model.

1.! "b#ecties of the study

i. Investigate dierent properties from time series analyses

ii. Investigate dierent properties from correlogram

iii. To forecast import over ne%t ve 1234 years with appropriate model

iv. To test the stationarity and nonstationarity of the series

v. )orrectly specify the model

vi. To t the appropriate A/IA model by Bo%5enkins!s methodology

1.$ %ata Collection

*ata collection is the rst task for any research problem. In dealing with the real

life problem, it is often found data at hand are inade&uate, and hence becomes

necessary to collect data that are appropriate. +or this study, we collect

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information including the production of the ma#or import products in Bangladesh

from year 62271&84 to year 62981&64 from -tatistics department of Bangladesh

bank.

1.& "utline of the study

The entire report has been segmented into four chapters: The chapter29

discusses Background of the study, ;roblems and otivation, "b#ectives of the

study,


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Chapter-02

)iterature *eie+

2.1 Introduction

In the literature there are several investigations relating to Bangladesh from theviewpoint of time series econometrics such as Anam and /ahman 19CC94,Bhuiyan and /ashid 19CC84, Dossain 16222, 6229, 62284, Eilma' and Ferma19CC34, Dossain 16223, 622G4, @ddin et al 1622H4. The use of time seriesanalysis, searching for random walk and cointegration, is e%tensive inmacroeconomic literature. I outline some of those relating to trade andmacroeconomic indicators, as Dusted 19CC64, ;owell 19CC94, )uddington and

@r'a 19CHC4, *eaton and


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nonstationarity, methodology of time series, assumptions of time series, 0hitenoise, /andom 0alk model, correlogram, concept of A/1p4, A1&4, A/A1p,&4and A/IA1p,d,&4, methodology of Bo% 5enkins method, @nit /oot test, *ickey+uller test and Augmented *ickey+uller test. The terms function that are usedby those scholer that are mentioned in literature review are following orderly:

2.2 %escription of ,ime eries

2.2.1 %enition of ,ime eries

A time series is a set of observations ( )ntYt ,2,1 = each one being recordedat a specied time t . In other words, arrangement of statistical data in

accordance with occurrence of time is known as time series. A time series is a

set of data which recorded over a period of time. A Time -eries may be dened

as a collection of readings belonging to dierent time periods, of some economic

variable or composite of variables.

According to Ealun )hou, LA Time -eries may be dened as a collection of

readings belonging to dierent time periods, of some economic variable or

composite of variablesM.

athematically, a time series is dened by the functional relationship ( )tfYt= ,

where tY is the value of the phenomenon 1variable4 under consideration at time

t.

Thus, the values of a phenomenon or variable at times nttt ,,, 21 are

nYYY ,,, 21 respectively, then the series:

nt

n

YYYY

tttt

,,,:

,,,:

21

21

)onstitute a time series.

Thus a time series invariably gives a bivariate distribution, one of the two

variables being time ( )t and the other being the value ( )tY of the phenomenon

at dierent points of time.

any time series arise in economics, such as currency e%change rate on

successive days, average incomes in successive weeks, average incomes in

successive months, company prots in successive years etc.

2.2.2/ample of ,ime eries

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The e%amples of time series are the sales gure in successive weeks or months,

advertising e%penditure in successive time periods etc., the signal detection

problem and accidental death, the number of strikes per year, share prices on

successive days, average incomes in successive months, company prots in

successive years etc., rainfall in successive days and air temperature measured

in successive hours or months etc.

2.2.! ses of time series

The time series analysis is of great importance not only to businessman or an

economist but also to people working in various disciplines in natural social and

physical sciences.

The uses of time series are given below

i4 It enables us to determine the type and nature of the variations in the

data.ii4 It helps in planning and forecasting.iii4 It enables us to predict or estimate or forecast the behavior of the

phenomenon in future which is very essential for business planning.iv4 It helps us to compare the changes in the values of dierent phenomenon

at dierent times or places.v4 It is very essential in business and economics with the help of time series

we can make plans for the future.

vi4 It helps to compare the actual current performance of accomplishments

with the e%pected ones and analy'e the causes of such variation, if any.

2.2.$ "b#ecties of ,ime eries nalysis

"ur main ob#ectives of time series or main purpose of time series is to study

techni&ues for drawing inferences from such series. The e%amples of time series

are an e%tremely small sample from the multitude of time series encountered in

the elds of engineering, science, sociology, and economics. Dowever, it is

necessary to set up a hypothetical probability model to represent the data. After

an appropriate family of models has been chosen, it is then possible to estimate

parameters, check for goodness of t to the data, and possibly to use the ttedmodel to enhance our understanding of the mechanism generating the series.

"nce a satisfactory model has been developed, it may be used in a variety of

ways depending on the particular eld of application. The model may be used

simply to provide a compact description of the data. 0e may be able to

represent the accidental deaths data as the sum of a specied trend, and

seasonal and random terms. +or the interpretation of economic statistics such as

unemployment gures, it is important to recogni'e the presence of seasonal

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components and to remove them so as not to confuse them with longterm

trends. This process is known as seasonal ad#ustment. "ther applications of time

series models include separation 1or ltering4 of noise from signals, prediction of

future values of a series such as the red wine sales or the population data,

testing hypotheses such as global warming using recorded temperature data,

predicting one series from observations of another, e.g., predicting future salesusing advertising e%penditure data, and controlling future values of a series by

ad#usting parameters. Time series models are also useful in simulation studies.

+or e%ample, the performance of a reservoir depends heavily on the random

daily inputs of water to the system. If these are modeled as a time series, then

we can use the tted model to simulate a large number of independent

se&uences of daily inputs. nowing the si'e and mode of operation of the

reservoir, we can determine the fraction of the simulated input se&uences that

cause the reservoir to run out of water in a given time period. This fraction will

then be an estimate of the probability of emptiness of the reservoir at some

time in the given period.

2.2.& ,ime series analysis

The analysis of data which are arranged in accordance with occurrence of time

is known as time series analysis.

2.2.' ssumptions of a ,ime eries

The main assumptions of a time series are given below:

i. The time gap between various values must be as for as possible e&ual.ii. It must consist of a homogeneous set of values.

iii. *ata must be available for a long period

2.2.( %i3erent ,ypes of ,ime eries

There are two types of time series, such as:

i4 *iscrete time series,

ii4 )ontinuous time series.

%iscrete ,ime eries4 A time series is said to be discrete when observations

are taken only at a specic time.

Continuous ,ime eries4 A time series is said to be continuous when

observations are recorded continuously over some time interval.

2.! ,ypes of %ata

Three types of data may be available for empirical analysis

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i. ,ime series data4A time series is a set of observations on the values that a variable takes at

dierent times. -uch data may be collected at regular time intervals such

as daily weekly, monthly, &uarterly, annually, &uin&uennially, decennially

etc.

ii. Cross-ection data:)rosssection data are data on one or more variables collected at the

same point in time such as the census of population conducted by the

)ensus Bureau every 92 years.

iii. Pooled data4

)ombination of time series data and crosssection data are called pooled

data.

2.$ %i3erent of plot of time series

2.$.1 ,ime Plot

The rst step in analy'ing a time series is to plot the observations against time.

These plots are known as time plot. It is similar plot of scatter diagram. This will

show up important features such as trend, seasonality, discontinuities and

outliers etc.

The plot is vital, both describe the data and to help in formulating a seasonal

model, and several e%amples.

2.$.2 5istogram

"ne of the most common ways to portray a fre&uency distribution is histogram.

A histogram describes a fre&uency distribution using a series of ad#unct

rectangles where the height of each rectangle is proportional to the fre&uency

the class represents.

In histogram the classes are marked on the hori'ontal a%is and the class

fre&uencies are represented by the heights of the bars and drawn ad#acent toeach other. Thus it provides an easily interpreted visual representation of a

fre&uency distribution.

2.$.! Correlogram

The autocorrelation function at lag k , denoted by k is dened as

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( ) variance

lagatcovariance

0

kkk ==

0here ( ) ( )[ ] = +kttk YYE . -ince both covariance and variance are measured

in the same units of measurementk

against k , the graph we obtain in known

as population correlogram.

6.3Main Component of time series

The important components of time series are:

i4 Trend or secular trend or long term movement.

ii4 ;eriodic changes or short term movement

a. -easonal variations

b. )yclic variations

iii4 /andom or irregular variation.

,rend4

The general tendency of the time series data to increase or decrease during a

long period of time is called the secular trend or long term trend or simply trend.

The concept of trend does not include short range oscillations. This is true in

most of business and economic statistics. Trend may have either upward or

downward movement, such as production, prices, income are upward trend

while a downward trend is noticed in the time series relating to deaths,

epidemics etc. trend is the general, smooth, long term, average tendency.

There are dierent types of trend:

)inear or traight )ine ,rend4

If we get a straight line, when the values are plotted on a graph, then it is called

a linear trend.

6on-)inear ,rend4

If we get a curve after plotting the time series values then it is called nonlinear

or curvilinear trend.

Periodic Changes4It would be observed that in many social and economic phenomena there are

forces at work which prevent the smooth $ow of the series in a particular

direction and tend to repeat themselves over a period of time. These forces do

not act continuously but operate in a regular spasmodic manner. These changes

may be classied as:

a4 -easonal Fariations and

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b4 )yclic Fariations

easonal 7ariations4

A variation which occurs with some degree of regularity within a specied period

of one year or less than one year is called seasonal variation. There are hourly,

daily, weekly, monthly or half yearly variations also.

ost of the economic time series are in$uenced by seasonal swings, e.g., prices,

production and consumption of commodities sales and prots in a departmental

store bank clearings and bank deposits etc are all aected by seasonal

variations.

The seasonal variations may be attributed to the following two causes:

94 There resulting from natural forces, i.e., weather and seasons.

64 There resulting from nonmade conventions, i.e., ?id day, *urga ;oo#a,

)hristmas etc.

Cyclic 7ariation4

The oscillatory movements in a time series with period of oscillation more than

one year are termed as cyclic variations. "ne complete period is called a cycle.

The cyclic movements in a time series are generally attributed to the so called

LBusiness )ycleM which may also be referred to as the four phase cycle of

94 ;rosperity

64 /ecession

84 *epression

=4 /ecoveryAnd normally lasts from seven to eleven years. ost of the economic and

commercial series e.g. series relating to prices, production and wages etc are

aected by business cycles.

*andom or Irregular 7ariation4

Apart from the regular variations, almost all the series contain another factor

called the random or irregular or residual variations which are not accounted for

by secular trend, seasonal and cyclic variations. These variations are happened

due to uncertain and unusual cases. It is purely random and beyond the control

of human hand but a part of system. These are caused by earth&uakes, wars,$oods, famines, revolutions, epidemics etc.

2.' ,ime eries Model

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A time series model for the observed data { }tY is a specication of the #oint

distributions 1or possibly only the means and covariances4 of a se&uence of

random variables { }tY of which { }tx is postulated to be a reali'ation.

*ierent Types of Time -eries model:

The following are the models commonly used for the decomposition of a time

series into its components.

i4 *ecomposition by additive hypothesis

ii4 *ecomposition by multiplicative hypothesis

iii4 i%ed models.

%ecomposition by dditie 5ypothesis4

According to the additive model, a time series can be e%pressed as:

ttttt UCSTY +++=

where tY is the time series value at time t , tT is the trend value at time t , tS

is the seasonal variation at time t , tC is the cyclic variation at time t and tU

is the random variation at time t .

ssumptions4

i4 ttt UCS ,, "perate with e&ual absolute eect irrespective of the trend.

ii4 ( )tt SandC will have positive or negative values and the total of positive

and negative values for any cycle 1and any year4 will be 'ero.

iii4 tU will also have positive or negative values and in the long run tU

will be 'ero.

%ecomposition by Multiplicatie 5ypothesis4

According to the multiplicative model, a time series can be e%pressed as:

ttttt UCSTY =

The multiplicative decomposition of a time series is same as the additive

decomposition of the logarithmic vales of the original time series. i.e.,

ttttt UCSTY lnlnlnlnln +++=

ssumptions4

i4 All the values are positive.

ii4 The geometric mean of tS in a year, tC in a cycle and tU in a long

term period are unity.

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Mied Model4

The dierent combination of additive and multiplicative models are named as

mi%ed model dened as

ttttt

ttttt

ttttt

UCSTY

UCSTYUSCTY

++=

+=+=

2.( %i3erent Process of ,ime series

2.(.1 tochastic Process4

A stochastic process may be deed as a collection of random variables { }TtYt ,

, where T denotes the set of time points. 0e denote the random variable at

time t by tY

if T is discrete and by( )tY

if T is continuous. Thus a stochasticprocess is a collection of random variables which are ordered in time.

In time series analysis, it is often impossible to make more than one observation

at time t . -o that we may only have one observation on the random variable at

time t .evertheless we may regard the observed time series as #ust one

e%ample of the innite set of time series which might have been observed. This

innite set of time series is sometimes called the ensemble. ?very member of

the ensemble is a possible reali'ation of the stochastic process. The observed

time series can be thought of as one particular reali'ation and will be denoted

byt

Y for nt ,,2,1 = if time points are discrete.

2.(.2 II% 6oise4

The simplest model for a time series is one in which there is no trend or

seasonal component and in which the observations are simply independent and

identically distributed 1iid4 random variables with mean 'ero. 0e refer to such a

se&uence of random variables nYYY ,,, 21 as II* noise. By denition we

can write, for any positive integer n and real numbers nyyy ,,, 21 .

[ ] [ ] [ ] [ ]

( ) ( ) ( )n

nnnn

yFyFyF

yYPyYPyYPyYyYyYP

21

22112211 ,,,

=

=

where, ( )F is the cumulative distribution function, of each of the identically

distributed random variables nYYY ,,, 21 . In this model there is no

dependence between observations. In particular, for all 1h and all

nyyy ,,, 21

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[ ] [ ]yYPyYyYyYyYP hnnnhn = ++ ,,,| 2211

It shows that knowledge of nYYY ,,, 21 is of no value for predicting the

behavior of hnY + . (iven the values of nYYY ,,, 21 the function f that

minimi'es the mean s&uared error

( ){ } 221 ,,, nhn YYYfYE + is in fact identically 'ero.

Although, this means that II* noise is a rather uninteresting process for

forecasters, it plays an important role as a building block for more complicated

time series model.

2.(.! Binary Process


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If a time series is nonstationary, we can study its behavior only for the time

period under consideration. ?ach asset of time series data will therefore be for a

particular episode. As a conse&uence, it is not possible to generali'e it to other

time period. Therefore, for the purpose of forecasting, such 1nonstationary4 time

series may be of little practical value.

8eakly tationary

A time series { },2,1,0, =tYt is said to be weakly stationary if it!s mean

function and covariance function is independent of time t , i.e.,

i4 ( )tY is independent of time t and

ii4 ( )thtY ,+ Is independent of time t for each h .@sually weakly stationary is termed as simply stationary.

trictly tationary4

A time series { },2,1,0, =tYt is said to be strictly stationary if the #oint

distribution of nYYY ,,, 21 is the same as the #oint distribution of

hnhh YYY +++ ,,, 21 for all integers h and 0>n , i.e.

{ } { } Thnhh

dT

n

YYYYYY+++

= ,,,,,,2121

Properties of trictly tationary ,ime eries { }tY 4

The main properties of { }tY are given below:

i4 The random variables tY are identically distributed.

ii4 { } { } Thd

Thtt YYYY ++ = 11 ,, +or all integers

t and h .

iii4 { }tY is weakly stationary if { }


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+or e%ample stock prices or e%change rates, follow a random walk that is they

are nonstationary.

2.(.' 8hite noise process

A stochastic process if it has 'ero mean, constant variance 2 and is serially

uncorrelated then it is called purely random or white noise process.

+or e%ample, in classical linear regression model the error term t is

*istributed as ),0( 2 NIDt .

2.(.( *andom +alk process

The time series that they slowly wander upwards or downwards but with no

real pattern and the plot of these series is nonstationary then these time series

are called random walk.

-uppose that t is a white noise process or error term with mean 2 and

variance 2 .Then the series ty is said to be a random walk if

ttt yy += 1

*andom 8alk4

The random walk { },2,1,0, =tSt is obtained by cumulatively summing 1or

integrating4 iid random variables. Thus a random walk with 'ero mean is

obtained by dening 00 =S and

,2,1;21 =+++= tYYYS tt

0here { }tY is II* noise. If { }tY is the binary process then { },2,1,0, =tSt is

called a simple symmetric random walk.

There are two types of random walk such as

a. /andom walk without drift

b. /andom walk with drift

2.(.9 Integrated process

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-uppose a nonstationary series ty can be transformed to the stationary series

by dierencing once, then series is said to be integrated of ordered one and is

denoted by ty N I194.If the series needs to be dierent d times to be stationary

then the series is said to be integrated of order d and is denoted by ty N I1d4.

2.9 Basic Concept of :unction of ,ime series

2.9.1 Mean :unction of ,ime eries4


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( ) ( ) ( )ZYbZXaZCbYaX ,cov,cov,cov +=++

( ) ( ){ } ( ){ }[ ]

( ){ }{ }[ ] 0

,cov,

==

=

ccZEZE

cEcZEZEZcSinc

2.9.' ample uto-Correlation :unction4

0e know how to compute the autocorrelation function for a sample time series

model. But in practical problem we do not start with a model but with observed

data { }nyyy ,...,, 21 . To assess the degree of dependence in the data and to

select a model for the data, we use -ample Autocorrelation +unction 1sample

A)+4 of the data. If the data are stationary time series { }tY , then the sample A)+

will provide an estimate of the A)+ of { }tY . -ample Auto)orrelation +unction is

given by

ample mean4


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( )[ ]

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

=

= =

021

201

110

definitenegativenoniswhich 1,

nn

n

n

!i n

!in

2.9.9 ample Correlation Matri4

The sample correlation matri% is denoted by n" and dened by ( )[ ]n

!in !i" 1,

==

which is nonnegative. ?ach of its diagonal elements is e&ual to 9, since ( ) 10 = .

2.9.; Basic Properties of the uto-Coariance and uto-Correlation

The basic properties of autocovariance and autocorrelation function of a

stationary time series are ( ) 00

( ) ( ) hh 0

( ) is even function i.e. ( ) ( ) hhh =

2.; Conclusions

In this chapter, we have discussed about the time series, time series model,

types of data, graphical presentation of time series, component of time series,

choosing lag, stationary and nonstationary process, transformation of time

series, correlogram, @nit root test, *ickey+uller test, Augmented *ickey+uller

test, dierent process of time series and A/1p4, A1&4, A/A 1p, &4,A/IA1p,d,&4 which are the important topics related to the analysis.

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Chapter-0!

Methodology

!.1 Introduction

ethodology is the system of methods and principles used in a particular

discipline. In other words the branch of philosophy concerned with the science ofmethod and procedure.

This is very important chapter because in this chapter we will deliberate all

those, which are related to make this report. Dere we will discuss dierent

methods, how to write report, how to present report. 0e also discuss how we

make use of these steps in our problems and if not then e%plain clearly why we

did not use it.

The third chapter of this report is methodology. This is the vital chapter of a

report. "bviously in one sense this is the necessary food of a report. "therwise

he destroyed physically mentally gradually. As like as if we do not introduce

this chapter 1methodology4, the report would not be healthy. The ob#ective of

the report may deteriorate gradually.

In this chapter, we have described the methodology of @nit /oot test, *ickey

+uller test, and Augmented *ickey+uller test that we should need to use to

carry out our analysis. This chapter also represents the terms which are

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associated with the statistical techni&ues. In this chapter, we brie$y will describe

the concept of the data collection techni&ue and types of data. Dowever we try

our best to establish this chapter with gorgeously within short period of time.

!.2 ,est of tationarity

There are several tests of stationarity. -ome important tests of them are given

below

I. (raphical analysisII. The correlogram test or sample autocorrelation function.III. The unit root testIF. The Augmented *ickey+uller test

%escription of seeral tests of stationarity

-ome important tests of them are given below

!.2.1


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( ) ( )

( )0

hh =

Partial uto-Correlation :unction

The partial autocorrelation function of a A"#A process { }tY is the function( ) dened by the e&uations

( )

( ) 1,and

10

=

=

hh hh

0here, hh is the last component of

( )[ ]

( ) ( ) ( )[ ]=

=

=

=

h

!ih

h

h

h!i

hh

...,2,1and

1,

1

+or any set of observations { }nYYY ...,, 21 with !i YY for some i and ! the

sample partial autocorrelation function ( )h is given by

( )

( ) 1,and

10

=

=

hh hh

0here, hh is the last component of

hh h 1

=

!.2.! nit *oot ,est

A test of stationarity 1or nonstationarity4 that has become widely popular over

the past several years is the unit root test.


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-ubtract 1tY from both sides of ( )1 to obtain:

( ) ( )

1 1 1

11 ... ... ... 2

t t t t t

t t

Y Y Y Y $

Y $

= +

= +

This can be alternatively written as:

( )1 ... ... ... 3t t tY Y $ = +

0here ( )1 = and , as usual, is the rstdierence operator.

ow we estimate the model 184 and test the following hypothesis,

0:0 =% i.e. the time series is nonstationary.

0: a% i.e. the time series is stationary.

"r, e&uivalently

1:0 =%

1: a%

The appropriate test statistic is tau statistic or test is known as *ickey+uller test

1*+4 test is dened by,

=

s

Before we proceed to estimate ( )2 , it may be noted that if 0 = , ( )2 will become

( ) ( )1 ... ... ... 4t t t t Y Y Y $ = =

-ince t$ is a white noise error term, it is stationary, which means that the rst

dierences of a random walk time series are stationary.

ow let us turn to the rst dierences if tY and regress them on 1tY and if the

estimated slope coe>cient in this regression ( )= is 'ero or not.

If it is 'ero, we conclude that tY is nonstationary. But if it is negative use to

conclude that tY is stationary. The only &uestion is which test we use to nd out

if the estimated coe>cient of 1tY in ( )2 is 'ero or not. Dere we cannot use usual

t test, because under null hypothesis that 0= 1i.e. 1= 4, the t value of the

estimated coe>cient of 1tY does not follow the t distribution even in large

samples that is, it does not have an asymptotic normal distribution.

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!.2.$ %ickey-:uller =%:> test


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iY is a random walk with drift: ( )1 1 t t tY Y $ = + +

iY is a random walk with drift around a stochastic trend:

( )1 2 1 !t t tY t Y $ = + + +

0here tis the time or trend variable. In each case, the null hypothesis is that0 = that is there is a unit root the time series is nonstationary. The alternative

hypothesis is that is less than 'ero that is, the time series is stationary. If the

null hypothesis is re#ected, it means that tY is a stationary time series 'ero mean

in the case of ( )2 , that tY is stationary with a non'ero mean ( )1 1 = in the

case of ( ) , and that tY is stationary around a deterministic trend in ( ) .

It is e%tremely important to note that the critical values of the taw test to test

the hypothesis that 0 = , are dierent for each of the preceding threespecications of the *+ test.

The actual estimation procedure is as follows: estimate ( )3 by "cient of 1ty in each case by its standard error to compute the

( ) taw statistic and refer to the *+ tables 1or any statistical package4.

If computed absolute value of the taw statistic ( ) e%ceeds the *+ or

acinnon critical value, we re#ect the hypothesis that 0 = , in which case the

time series is stationary."n the other hand, if the computed does not e%ceed the critical taw value, we

do not re#ect the null hypothesis, in which case the time series is nonstationary.

!.2.& ugmented %ickey-:uller test

In conducting the *+ test as in ( ) ( ) ( )3 , !or , it was assumed that the error term

t$ is uncorrelated. But in case t$ is uncorrelated, *ickey and +uller have

developed a test, known as the augmented *ickey+uller test. This test is

conducted by adding the lagged values of the dependent variable tY

. To bespecic, suppose we use ( )! . The A*+ test here consists of estimating the

following regression:

( )1 2 11

... ... ... "&

t t i t i t

i

Y t Y Y =

= + + + +

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0here t is a pure white noise error term and where

( ) ( )1 1 2 2 2 3,t t t t t t Y Y Y Y Y Y = = , etc. The number of lagged dierence terms to

include is often determined empirically, the idea being to include enough terms

so that the error term in ( )" is serially uncorrelated. In A*+ we still test whether

0 = and the A*+ test follows the same asymptotic distribution as the *+statistic, so the same critical values can be used.

!.2.' Bo-?enkins Method

The in$uential work of Bo%5enkins ( )1#"0 shifted professional attention away

from the stationary serially correlated deviations from deterministic trend

paradigm toward the ( )'dpA"I#A ,, paradigm. It is popular because it can

handle any series, stationary or not with or without seasonal elements.

The basic steps in the Bo%5enkins methodology consist of the following vesteps

1. Identication of appropriate model4 "nce we have used the

dierencing procedure to get a stationary time series, we e%amine the

correlogram to decide on the appropriate orders of the A" and #A

components. The correlogram of a #A process is 'ero after a point, that

of an A" process declined geometrically. The correlogram of A"#A

process show dierent patterns 1but all dampers after a while4. Based on

these, one arrives at a tentative A"#A model. This step involves more of

a #udgment procedure than the use of any clearcut rules.2. /stimation of the model4 The ne%t step is the estimation of the

tentative A"#A model identied in step6. The estimation of A" model

is straight forward. 0e estimate then by "


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-chewat' Bayesian information criterion 1BI)4. If s is the total number ofparameters estimated

( )

( ) nsns(ICand

snsACI

loglog

2log

2

2

+=

+=

Dere n is the sample si'e. If =2 tZ"SS is the residual sum s&uares, then

pn

"SS

=2

If we are considering several A"#A models we choose the one with the

lowest AI) of BI). The two criteria can lead to dierent conclusions.

$. :orecasting4 -uppose that we have estimated the model with n

observations, we want to forecast knY + . This is called a kperiods ahead

forecast. +irst we need to write out the e%pression for knY

+ and thenreplace all future values ( )k!Y kn process

A series { }ty is said to be a autoregressive process of order p if,

tptpttt )yyyy ++++=

............

211

where, { }t) N 012, 2 4 and p ....,,........., 21 are constants. The series is

known as A/1p4 process.

!.2.9 Moing aerage process or M=@>

A series { }ty is said to be a moving average process of order & if,

't'tttt ))))y ++++= ............2211

where, { } )$%(0,& 2t) and ' ,,........., 21 are constants. This series is also

known as A1&4 process.

!.2.; *M =pA@> process

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The time series { }ty is an A/A1p,&4 process if it stationary and satises for

every t,

it

'

ii

p

iiti

't'tttptptt

))or

))))yyy

==

+=

++++=

1t

1t

221111

'(,

...................

where, { } )$%(0,& 2t) and the polynomials

( ) ( )''p

p )))) +++ ..........1and............1 11 have no common factors.

!.2.10 *IM =pA dA @> process

If the A/IA 1p, &4 process is integrated process than it is known as A/IA

process of order 1p, d, &4. Dere d represents the number of dierence the

needed to be stationary.

!.! Conclusions

In this chapter, we have discussed the dierent test procedures for testing the

stationarity and nonstationarity and tting the appropriate A/IA model. At last

of this chapter we can say that the time series estimation needs to consider

many things which are given above. ow from the discussion of the above

chapter we will be able to analy'e the data. "ur aim is to estimate a model and

then check whether the error term white noise is or not, if it is then the

estimated model is perfect.

Chapter-0$nalysis and *esults

$.1 Introduction

In this chapter, we will discuss the results and analysis for our study. Analysis is

very essential part in any statistical data analysis in our real life. 0e can take

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the decision by analy'ing the results. In this chapter, we present the result and

interpret on our analysis. +or our analysis, we used the secondary data. This

chapter will contain the results nding /console 1version 6.96.64. -o main part

of my report data analysis starts in this chapter. y data is the L;er Eear the

ma#or import products in BangladeshM. At rst we have to check the non

stationarity of the data. Then we have to t an appropriate model. At last themain work of my report starts, that is to check whether the errors are white

noise or not.

$.2 %ata analysis

0e use icrosoft ?%cel, -;--, IITAB, -A- and /)onsole etc. for the purpose

of data entry and analy'ing the data. "ne of the most important parts of a

research study is the plan for data analysis. The purpose of data analysis is to

provide answers to the research &uestion being studied. The analysis breaks the

data into dierent parts. The data are categori'ed, arranged, and summari'ed.

$.! %ata %escription

The task of data collection is so important after dening problem. In dealing with

the real life problem, it is often found data at hand are inade&uate, and hence, it

becomes necessary to collect data that are appropriate. There are several ways

of collecting the appropriate data, which dier considerably in conte%t of money

costs, time and other resources at the disposal of the researcher. +or this study,

we use secondary data. In our study, ; represents the period of time, and E

represents the ma#or import products of Bangladesh in every period.

+or our study, we are using the secondary data because we collect the data

based on the ma#or import products of Bangladesh from ?conomic Trends of

-tatistics *epartment of Bangladesh Bank.

$.$ ,est of stationary

There are dierent types of test in time series for testing stationarity in time

series data. -ome of them are given below:

i. (raphical Analysisii. Autocorrelation +unction and )orrelogramiii. The @nit /oot Test

a. *ickey+uller testb. Augmented *ickey+uller test

tationary test of the series

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In this part, we analy'e the data whether it is stationary or not which is the

initial and vital steps to determine the appropriate method by which we can

achieve our desired goal to fulll study. ?ventually most of the econometric data

series are nonstationary which results unusual forecasted values and give us

misleading decision. -o it should be taken seriously before any analysis that to

test the data series is stationary or not. To look for such property we will takethe following techni&ues: i4 (raphical Analysis and ii4 )orrelogram test.

$.& nalysis of Chemical Product4

$.&.1 ,ime series plot of Chemical Product4

Comment4+rom the line graph +ig29 1in Appendi%4 it is clear that the time

series data seems to be trending. -o from the time series plot we can see that

the time series data is nonstationary because it!s mean and variance depends

on time. It also visual that the mean and variance is not remains constant from

time to time, so we can say that the productions of )hemical products are not

stationary.

$.&.2 C: and PC: of the chemical Product4

Comment4It is evident from +ig26 and +ig8 1in Appendi%4 the A)+ and ;A)+ of

the ma#or import products of Bangladesh are show that autocorrelation

coe>cients at various lags are very high. The autocorrelation coe>cient starts

at a very high value and declines slowly towards 'ero. +rom the gure we

observe that the autocorrelation coe>cient starts at a very high value of A)+ at

lag 8 and of ;)+ at lag 6 declines slowly. There are single large spikes that arealso insignicant spikes at dierent lag. Thus we conclude that the time series

are nonstationary.

$.&.! :itted Model +ith %iagnostic Checking:+rom the +ig2= and +ig2=

1 in Appendi%4 graphs we conclude that the animal vegetable follows

model. The model can be e%press as:

-0.2&9(

0here, 2.63H7 and 2.629G are coe>cient and standard error of the model.

)all:

arima1% Q dt, order Q c19, 6, 244

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)oe>cients: ar9 2.63H7 s.e. 2.629G ,sigmaR6 estimated as =96G7H2GH:

log likelihood Q 6=C.=7, aic Q 326.C=

nit roots test4Bo-Pierce test

*ata: ftSresidual

)his&uared Q 9=.8HG6, df Q 3, pvalue Q 2.29888

%ecision: At 3 level of signicance, the pvalue is 2.29888, so we conclude

that the process follows unit root that means it follows -tationary time series

process.

$.&.$ :orecasting alues oer net e =0&> years of the chemical

Product4

The forecasting values of the animal vegetable are mentioned in the following

table 1294.

In the following table, the forecast value in 629= in the 9 st&uartile is G3C6G.7H

units.

Utr9 Utr6 Utr8 Utr=

6298 6289=.=H =27C9.36

629= G3C6G.7H C=G92.8C 96G382.96 9G98GC.9G

6293 9CHC23.C8 68HC3H.39 6H987H.=G 86G2=2.G6

629G 876H87.H3 =69G77.26 =76=7G.9C 3639G6.3H

6297 37CG72.C8

,able-014+orecasting values over ne%t ve 1234 years of chemical ;roduct

$.' nalysis of Machinery Mechanical product

$.'.1 ,ime series plot 0f Machinery Mechanical product4

Comment4 +rom the line graph +ig2G 1in Appendi%4 it is clear that the time




time to time, so we can say that the production of Animal vegetables is not

stationary.

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$.'.2 C: and PC: plot of plot Mechinary Mechanical product

Comment4It is evident from +ig27 and +ig2H 1in Appendi% the A)+ and ;A)+ of

the ma#or import products of Bangladesh are show that autocorrelation


at a very high value and declines slowly towards 'ero. +rom the gure weobserve that the autocorrelation coe>cient starts at a very high value of A)+ at

lag 3 and of ;)+ at lag 9 decline slowly. There are single large spikes that are

also insignicant spikes at dierent lag. Thus we conclude that the time series

are nonstationary.

$.'.! :itted Model +ith %iagnostic Checking: +rom the +ig2C and +ig92

1in Appendi%4 we conclude that the animal vegetable follows model.

The model can be e%press as:

0here, 2.39HG and 2.9HG9 are coe>cient and standard error of the model.

)all: arima1% Q dt, order Q c19, 8, 244

)oe>cients:

ar9 2.39HG , s.e. 2.9HG9, sigmaR6 estimated as 9GCHC7766: log likelihood Q

69H.2=, aic Q ==2.2H

nit roots test4

Bo-Pierce test

data: ftSresidual

)his&uared Q 9=.8HG6, df Q 3, pvalue Q 2.29888

%ecision: At 3 level of signicance, the pvalue is 2.29888, so we conclude

that the process follows unit root that means it follows -tationary time series

process.

I7.'.$ :orecasting alues oer net e =0&> years4The forecasting

values of the animal vegetable are mentioned in the following table

1264. In the following table, the forecast value in 629= in the 9st&uartile

is 72G=9.3= units.

Utr9 Utr6 Utr8 Utr=

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6298 13034.4) 34)"1.)"

629= "0!41.4 121202.13 1))).") 2"4014.2!

6293 3"#021.00 04)21.1! !2!41.2 )23!1".2

629G 101))4.13 123#3!".2! 14)!1)).03 1"!02"3.40

6297 20!2".12

Table 02: +orecasting values over ne%t ve 1234 years of echinary echanical

product

$.( nalysis of 7ehicles products

$.(.1 ,ime series plot of 7ehicles

Comment4 +rom the line graph +ig99 1in Appendi%4 it is clear that the time




time to time, so we can say that the production of Fehicles is not stationary.

$.(.2 C: plot and PC: plot of the 7ehicles

Comment4It is evident from +ig96 and +ig98 1in Appendi%4 the A)+ and ;A)+

of the ma#or import products in Bangladesh are show that autocorrelation


at a very high value and declines slowly towards 'ero. +rom the gure we

observe that the autocorrelation coe>cient starts at a very high value of A)+ at

lag 8 and of at lag 9 declines slowly. There are single large spikes that are also

insignicant spikes at dierent lag. Thus we conclude that the time series are

nonstationary.

$.(.! :itted Model +ith %iagnostic Checking:+rom the +ig9= and +ig93

1in Appendi%4 we conclude that the animal vegetable follows model.

The model can be e%press as:

0here, 2.G339 and 2.9G29 are coe>cient and standard error of the model.

)all:

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arima1% Q dt, order Q c19, 6, 244

)oe>cients:

ar9 2.G339

s.e. 2.9G29 sigmaR6 estimated as C=7G3G6H: log likelihood Q 666.C8, aic Q==C.HG

nit roots test4

Bo%;ierce test

data: ftSresidual

)his&uared Q G.39C=, df Q 3, pvalue Q 2.63HC

%ecision: At 3 level of signicance, the pvalue is 2.63C=, so we conclude that

the process does not follows unit root that means it follows non-tationary time

series process.

$.(.$ :orecasting alues oer net e =0&> years of 7ehicles

products4

The forecasting values of the animal vegetable are mentioned in the following

table 1284.

In the following table, the forecast value in 629= in the 9st&uartile is 6G6CC.CGH

units.

Utr9 Utr6 Utr8 Utr=

6298 C78=.7G= 9G893.996

629= 6G6CC.CGH 8G3H9.8H7 =HG9=.9H9 G9837.667

6293 736H6.HH2 HCCC2.H6= 923G86.8CG 96628G.=H9

629G 98C6=6.CC7 9379GC.GC7 973H97.H8G 9C39=6.238

6297 693989.2C9

,able-0!4+orecasting values over ne%t ve 1234 years of Fehicles

$.9 Concluding *emarks

+rom the above chapter we can see that the data which is the yearly production

the ma#or import products in Bangladesh are nonstationary and follows A/IA

19, 6, 24 model. By tting this model we nd from diagnostic checking that the

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model is correct. +rom the residual plot and from the Bo%;ierce test we have

seen that errors or residuals are 0hite oise, which was the main focus of this

study to test the white noise property of the tted time series model.

Chapter-0&

Conclusions

In our report, our main ob#ective is to investigate the stationary and nonstationary of the ma#or import products in Bangladesh and t the appropriate

A/IA model. It is very important and arduous task, because there are some

limitations to perform our task. Dowever we try our best to analy'e appropriate

the results and analysis.

In our analysis, we have been concerned with data consisting of the ma#or

import products in Bangladesh to forecast the amount of import over ne%t ve

years. +or our study, we have discussed the concept of time series and time

series model, component of time series, the concept of stationarity and non

stationarity, methodology of time series, assumptions of time series, 0hitenoise, /andom 0alk model, correlogram, concept of A/1p4, A1&4, A/A 1p,&4

and A/IA1p, d, &4, methodology of Bo% 5enkins method, @nit /oot test, *ickey

+uller test and Augmented *ickey+uller test.

0e have concerned about the methodology to complete the report. In chapter =,

we have concerned the results and analysis. +rom our analysis, we can see that

our data which is the &uarterly the ma#or import products in Bangladesh are

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nonstationary and follows A/IA model. By tting this model we nd from

diagnostic checking that the model is correct. +rom the residual plot and from

the Bo%;ierce test we have seen that errors or residuals are 0hite oise, which

was the main focus of this study to test the white noise property of the tted

time series model.

In this report we work on the white noise of the error term. +or this we had start

from the stationarity and nonstationarity test. Then we have tted a perfect

time series model for the data. Dence when we nd the model we have

performed diagnostic checking of the tted model. At last we have check the

error term. 0e used graphical and Bo%test to nd check that the errors are

white noise or not. +or our data LThe ma#or import products in BangladeshM we

nd that the model is an A/IA model and for this model the errors are 0hite

oise.


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final reportbody.doc

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