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Good afternoon.This is Dr.Pradhan here. Welcome
to NPTEL project on Econometric Modeling.Today
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we will discuss the issue of Bivariate Econometric
Modeling.In the lastcouple of lectures we
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have discussed various aspects of Econometric
Modeling, various structures of data analysis,Univariate
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analysis, Bivariateanalysis and Multivariate
analysis.We have discussed various issues
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under Univariate analysis, various issues
under Bivariate analysis and little bit idea
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about the Multivariate analysis. So, today
westructure is mostly on the analysis of econometric
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modeling.
So, let us start with what is all about the
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structure of Bivariate Econometric Modeling.
So, it it consist oftwoaspects Bivariate setup
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and modeling. So, we have discussed what is
the Bivariate data structure and we have also
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discussed the modeling issues. So, let me
first highlight what is theEconometric issue
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behind the Bivariate Modeling. Econometric
is the product of statistics. Basically it
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is an extension ofregression modeling.
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So, the basic idea of regression analysis
is to start with cause and effect relationship
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betweentwovariables that is the dependent
and independent variables.
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So, we like to know how the econometric modeling
is very close or you can say somewhat different
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from the basic statistical modeling. So, the
basic idea behind regression analysis is to
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start the cause and effect relationship. It
is alsoa similar way of econometric modeling.
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So, here this issue is something different
when you will go for basic regression analysis.
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So, we are not bothering about the various
typical issues or typical problems behind
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this analysis.We have discussed the simple
structure of regression analysis.
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But, if you go in deep higher version of modelingthen,
the regression analysis is very complex and
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very complicated. So far as econometric modeling
is concerned,it is the root or beginning from
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this basic regression analysis. So, now, when
will you talk about the Bivariate Econometric
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Modelingobviously, the basic idea behind this
issue is to study the cause and effect relationship
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betweentwovariableslike we have discussed
this issue.That means,it is cause and effect
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relationship betweentwo variables.
So, what is the cause and what is the effect?For
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instance, if we havetwo variables say X and
Y then if we will write Y is a function of
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X then X is represented as a cause and Y represented
as a effect. That means, this is effect side
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and this is cause side alright. So, nowfor
as econometric modeling is concerned, it is
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the extension of the basic regression modeling.
Econometric Modeling the issue is on the structure
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of research. So, there are many ways the structure
can be analyzed.For instance, we are discussing
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here cause and effect relationship betweentwo
variables.There are many names we can discuss
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regarding the cause and effect relationship.
For instance, it can be the a relationship
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between independent variables and dependent
variables.Otherwise it is alsotheissue between
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explanatory variables explanatory variable
and explained variables. So, the structure
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is like this independent to dependent explain
to explanatory.Then similarly, it can be also
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represented as this is predictor and it will
stand to predictant. So, similarly, it is
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otherwise known as exogenous variablesand
this is otherwise known as endogenous variables.
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So, similarly, it is otherwise known as stimulus
and this is otherwise called as a response.This
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is otherwise called as a regressor this is
otherwise called as a regressant. So, likewise
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there are various waysit can be represented.
So, its cause and effect relationship the
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issue between or the nexus between dependent
variable independent variable, explained variable
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and explanatory variables,covariance and covariate,
then similarly, exogenous and endogenous variable,
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stimulus and responseandlast, but, not the
least is called as a regressor and regressant.
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Because Econometrics is a statistical technique
which has many application in many areas.There
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are variousyou know ways it can be represented.Means
the paper is must learn application oriented.
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So, as a result the same words it can be represented
in a very ways it means many ways. So, it
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is just like oldwine in a new bottle. So,
the pictures aremore or less same, but, the
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representation is the somewhat different.
So, before we start with this BivariateModeling,you
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know Econometric Modeling.Let me highlight
what is the issue behind thisBivariate Econometric
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Modeling.
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The Bivariate Econometric Modeling, since
the issue starts with the bivariateobviously,
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there aretwosets of variables. One we call
it X and another is you can say Y. So, before
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we enter to Bivariateeconometric modeling
we must haveessential requirements otherwise
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we cannot do anything for this econometric
modeling. So, now, the issue is what are this
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essential requirement before we proceed for
econometric modeling that too Bivariate analysis.
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So, 1stthing is numberoneconditions or constraints
of Bivariate econometric modeling.
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So, what are the conditions we need to have
to build a Bivariate econometric modeling.
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So, number one condition is that first there
must be two variablesin the system. So, let
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us say X and Y or X 1 X 2 like this. So, this
is first and foremost condition of thisparticular
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problemssecond is whichthere must be classification
of dependent and independent. So, dependent
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variables and independent variable classifications
because it is the cause and effect relationship
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issue.So, obviously, there is a dependent
structure and there isa independent structure.
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So, when we talk about cause it is usually
represented as an independent structure and
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when we call it effect it is called as a dependent
structure. So, the issue iswhen we talk about
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Bivariate econometric modeling then, there
must betwo variables in the system this is
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the first condition and second condition you
have to classify which 1 is dependent and
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which 1 is independent.Becausethe model isbased
on that systemof course, there is a Bivariate,you
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know bidirectional causality issue.So, obviously
when there aretwo variables in a particular
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systemthen either X causes Y or Y causes X
or both can be go simultaneously.
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So, that particular structure is called as
a time series issue. So, we are not in the
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process of discussing the detail about time
series modeling. So, we are just in the process
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of beginning. So, thisrestriction must be
here is that to the bidirectional issue or
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reverse causality. So, we are not considering
here the reverse causality.That means, if
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X influence yso, we are assuming that Y does
not depends upon X. Y may depend upon X, but,
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Y is cannot be treated as a again independent
variables. That means, a dependent and independent
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classification must be very essentialand very
you can say accurate or that is you know right
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choice of this particular modeling.
So, now third is that should be that means,
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in other words it called as a n greater than
equal to two. So, n stands for n stands for
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number of number of variables in the system.Just
it is extension of the first condition.Fourk
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must be greater than equal to two. So, that
means, in this particularprogram k is treated
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as a independent variables setup. So, when
will we go for bivariate issue then, obviously,
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k may be or k is exactly equal to 1.So, when
we will go for Trivariatemodeling then k equal
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to 2.When we will go for multivariate modeling
then obviously, k greater than equal to two.
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But for the Bivariate analysis or Bivariate
econometric modelingif k is represented as
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the number of independents variables then,
obviously, in this particular setup k must
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be exactly equal to 1. That means, in this
particular Bivariatemodeling your k must be
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exactly equal to 1. So, k is treated as a
number of independent variables in this particular
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system. So, then finally, N greater than n
represents here the capital N represents total
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number of observations in the systems and
n represents total number of variablesin the
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systems.
That means, sinceit is a Bivariate setup then
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there aretwo variables. So, your sample size
should be more thantwoat least.If your sample
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size is less than 2 then the system itself
is inconsistent we cannot proceedfurther.So,
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when there is a issue of multivariate similarly,
the representation of you knowcapital N small
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n will be very serious issue. So, for instance
for trivariate analysis obviously, there arethreevariables
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in the systems.That means, n represents 3
and capital N must be greater than equal to
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three.
But, even if it is equal tothreethe system
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cannot be operated properly. So, to operate
the system properly you must have sufficientsample
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size or samples; that means,you must have
some sufficient data points.Until and unless
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you have sufficient data point you cannot
able to you can say build the econometric
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models.
So, your model building or the consistency
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of the model or the feasibility of the model
depends upon the sample setup.Higher is the
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sample size better is the accuracy of the
model or better is the feasibility of the
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models.If the sample size is very less or
you can say very minimum then, obviously,
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it will affect the system and the model by
default it will be inconsistent oneand you
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cannot use this particular model for any forecasting
or any policy use.For you knowfor the objective
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of policy use or forecasting your model must
be perfectly ok.
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So, what we otherwise called as a best fitted
models, to get the best fitted models you
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must havehigher and higher sample size. So,
that is you know toughest issue in this modeling
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setup that too econometric modeling. So, your
sample size will be substantially or absolutely
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very high than the number of variable in the
particular systems. So, there is a trick how
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to determine the a minimum number of you can
say a minimum sample size in a particular
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system.
We will discuss in details when we will go
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in the different version of the econometric
modeling. In thea in the very beginningyou
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must have knowledge that whatever variables
you are using in a particular systems your
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samplessample size should be absolutely greater
than to number of variables in the systems.
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So, this is the another conditionof Bivariateyou
can say econometric modeling.Thenwe are talking
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about sixth condition.We are talking abouttwo
variablesin the systems X and y, but, we remember
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thatthere must be some variability in X and
Y for instanceif I will take some observation
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on X and some observation on Y.If the observations
are not perfectlyok as per the modeling rules
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and you can say modeling formalities then
of course, again the model will be you can
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say inconsistent or infeasible one.
For instance,let us take a case.Here is if
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I have a X variables and I have Y variable.If
I will take only X consist of somany observations
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like X 1 X 2 X 3 like up to X n similarly,
Y consist of Y 1 Y 2 Y 3 up to Y n. So, now,
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X 1 X 1 X 2 X 3 X n these are all you know
implicitly from other means we do not know
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it is X 1 what is X 2 what is X 3 what is
X n and we also do not know what is Y 1 Y
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2 Y 3 Y n. What we can say represent here
that X 1 X 2 X n are the sample points of
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X and Y 1 Y 2 Y n are the sample points of
Y.
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But what is X 1 what is X 2 or what is Y 1
or what is Y 2 we have no idea.Now I will
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give the structure.Let us say X 1 equal to
2, X 2 equal to 2, X 3 equal to 2,and X n
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equal to 2 then in this particular setup there
is no variation on X samples.That means, every
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point it is 2.If there is a no such variation
and obviously, by default the model will be
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inconsistent. So, there must be a some kind
of variability in the sample observations
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it should not be highly distance or it should
not be or you can say very equal. So, you
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have to find out the optimumone.So that means,
it is the midpoint or you can say somewhat
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middle between.You can say exclusive equality
and exclusive inequality. So, there should
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be some optimum one.
For instance, if like the sample observation
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like thisinstead of 2,2,2 if I will put X
1equal to 1, X 2 equal to 2, X 3equal to 4,
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X 4 is another sample say is equal to seven
then X n is another sample say eight then
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there is a some kind of variability. So, this
particular setup is very consistent for the
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a model building.Of course, by initial look
this data points are somewhat, but, still
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there is a statistical test whether this particular
variables observations are definitelyor not
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we have to verify it. So, there is a statistical
techniques that means, we have to check the
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normal distributionstructures before you go
for a any econometric modeling.
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Similarly, in the case of Y there should not
be any problem like a 2, 2, 2case. So, there
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should be some variability in Y also for instance
Y 1 equal to 2, Y 2 equal to five, Y 3 equal
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to seven and Y five equal to eight. That means,if
the setup is a some kind ofvariations then,
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obviously, there is a way to build a model.
If all the data points are equal then we sometimes
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very handicap to handle the particular situation.
So, we need some variability in the data points
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and that variability should not be so high.If
it is high then again it will turn to inconsistent.
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So, to make the consistent it should not be
absolutely equal and it should not be absolutely
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unequal. So, it has to be in between the two.
So, that is what we call as a optimum ones
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all right.
Then seventh or last, but, not the least condition
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is that X and Y should be random in natures
so that means, somewhatit is attach with the
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issue of probability.Means there is a some
kind of chance factor which can you can say
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involve in this modeling scenario that too
Bivariate analysis with this particular setupof
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you can saycondition of Bivariate econometric
modeling.
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We have to proceed furtherto what is the structure
of this particularissue let us the basic structure
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ofBivariatemodelingBivariate Econometric Modeling.
So, Bivariatemodeling the initial starting
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point we must havetwo variables Y and X ok.
So, here we will assume that X is represented
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as a independent variable clusterand Y is
represented as a dependent variable clusters.
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So, now, the extension is like this Y equal
to function of X this is just like a mathematics.
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Sonow the extension is the how you have to
set this particular Bivariate econometric
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modeling. So, lets Y equal to the structure
of Y 1 Y 2 up to Y n and X represents X 1
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X 2 up to X n all right. So, now, when we
have Y and X then we cantrans the functional
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relationship between the 2.
When we trans the functional relationship
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between the two then there are 2 different
techniques.We usually use either correlation
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that is the extension of covariance and there
is another technique called as a regression
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that tooextensions are vacant correlations.What
is upon as econometric modeling is concerned
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there is something more thanthat it is not
just to correlate and just to regress. So,
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in between there is a lots of hidden issues.
So, that hidden issue is the crucial point
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of you know higher version ofresults or you
can say higher complex problem. So, now, we
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have to discuss all these issue and the beginning
of econometric is that from this particular
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basic levels.
So, when you have a particular relationship
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that too correlation and regression then there
are various measuresyou can say bistructure.We
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can represent the relationship and that to
be in we perfect ones to get the perfect 1
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we have lots of by you can say structures
techniques tools to get the better picture.
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So, now, we have to see how quicklywe can
have that particular best fitted models and
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what are the problems we have to face or we
have to find out to get this best fitted models.
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So, this is you know very complex issue and
very you can say typical issue we have to
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discuss step by step. So, let us start with
this particular relationship. So, when we
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havetwo variables in this particular system
having the observation Y 1 Y 2 up to Y n and
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a the observation X 1 or X 2 up to X n then;
obviously, the first and foremoststep you
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can say is that we have to build a mathematical
form of the model that is nothing, but, Y
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equal to function of X that is we called as
a mathematical models.It is simply mathematical
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model. So, now, we know we have a variable
in the systems.
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So, first you transfer this particular theoreticalinformation
to mathematical informationthat too the individual
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variable into some functional forms that functional
form is treated as a mathematical form of
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the models. So, now, we have to transfer this
mathematical form of the model into statistical
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form of the models then the econometric issue
will be coming in to the pictures. So, now,
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before we go to statistical form of the model
let usjust represent the a explicit format
192
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of this particular problem you can say model
or relationship. Y equal to function of X
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means there are many ways this Y and X can
be you can say worked out. So, what you have
194
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to do here is..
We have to see in anexplicitly format; that
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means, with the relationship is linearoneor
the linearshipis a non-linear or not because
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this is very strong issue for this modeling
behaviors. So, let us assume that there is
197
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a relationship and that too linear relationships.
198
00:24:00,270 --> 00:24:07,210
So, now, we have to represents Y equal to
alpha plus beta X.Just like it is a straight
199
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line equation Y equal to m X plus c where
m is the slope and c is the constant. Here
200
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instead of putting Y equal to m X plus c we
are putting beta X plus alpha, beta is the
201
00:24:18,850 --> 00:24:25,309
slope and alpha is represented as a supporting
factor, constant factor and X is independent
202
00:24:25,309 --> 00:24:31,450
variable Y is the dependent variable. So,
this is what we called as a explicit format
203
00:24:31,450 --> 00:24:37,929
of mathematical models.
So, now this is this second step of this particular
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process and this is this step 1 process of
this bivariate setup. So, step 1 is slope.Bivariate
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00:24:47,260 --> 00:24:54,260
econometric modeling is that we have to bringtwo
variables and you have to build its relationship
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00:24:56,409 --> 00:25:03,169
and that too in a functional form and that
functionalform again has to be in a explicitly
207
00:25:03,169 --> 00:25:09,460
format. So, now, when you have a explicitly
mathematical model then we have to transfer
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in to statistical form of the models. So,
what is the statistical form of the models
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that too you have to move in to step three.
So, Y equal to alpha plus beta X plus another
210
00:25:21,799 --> 00:25:28,799
term called as a U where U is represented
as a error term and this particular model
211
00:25:32,240 --> 00:25:39,240
is called as a statistical form of the model.
So, we have simple Y and X then we have to
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00:25:42,820 --> 00:25:48,760
transfer in to mathematical form of the models
that too in explicitly format Y equal to alpha
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00:25:48,760 --> 00:25:54,370
plus beta X.Then again this particular model
has to be transferred in to statistical format
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00:25:54,370 --> 00:25:58,179
that is Y equal to alpha plus beta X plus
0.
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00:25:58,179 --> 00:26:05,179
For instance, I will put it in differentway.
So, Y equal to alpha plus beta x. So, that
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00:26:06,039 --> 00:26:13,039
is mathematical models. So, then we are transferring
Y equal to alpha plus beta X plus U it is
217
00:26:13,970 --> 00:26:20,289
statistical models.
So, now the difference between these 2 models
218
00:26:20,289 --> 00:26:26,320
is with respect to this particular U term.
So, U is represented as a error term. So,
219
00:26:26,320 --> 00:26:33,320
now, the issue is what is this component U
why there is a U in this particular systemand
220
00:26:34,019 --> 00:26:41,019
where you have brought this U?Initially we
have our beginning isour journey is from Y
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00:26:42,019 --> 00:26:48,630
and x. So, now, in between U is introduced
in the system. So, now, the question is why
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00:26:48,630 --> 00:26:55,630
because there is a debate about this issue.
So, basically there is a always fighting between
223
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the mathematics and statistics mathematicians
are alwaysin the believe that everything is
224
00:27:03,750 --> 00:27:10,110
in exact; that means, what is in the right
side it should be exactly equal to in the
225
00:27:10,110 --> 00:27:17,110
left side.But statistician does not you can
say like all these issues they are in the
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00:27:18,419 --> 00:27:25,390
view that there is this something who is hidden
in natures.So that means,nothing can exact
227
00:27:25,390 --> 00:27:31,899
in the society. So, there is a always in exact
process.So that means,something which we cannot
228
00:27:31,899 --> 00:27:38,899
exactly explore or we cannot exactly represent
in the particular system. So, if you do not
229
00:27:39,940 --> 00:27:46,070
exactlyrepresent in that particular system
then there is a something gap. So, that gap
230
00:27:46,070 --> 00:27:50,970
can be fill through the term U that is nothing,
but, error components.
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00:27:50,970 --> 00:27:56,019
That means, if I will put this particular
equationslet us say this is equation number
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00:27:56,019 --> 00:28:03,019
1 and this is equation number 2 this is statistical
form of the model.So that means,this is cause
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00:28:03,389 --> 00:28:07,590
sight and this is effect sight this is independent,
structure this is dependent structure. So,
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00:28:07,590 --> 00:28:13,669
now when there is a question of effect.So,
now, there are 2 dimension here this 1 dimension
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00:28:13,669 --> 00:28:18,720
is alpha plus beta X and [another] another
dimension is called as a U so; that means,
236
00:28:18,720 --> 00:28:25,720
alpha plus beta X is called as a explained
factorsthis U error term is called as a unexplained
237
00:28:29,440 --> 00:28:36,440
factors and this is called as a explained
factors and this is somewhat it is called
238
00:28:39,490 --> 00:28:45,200
as a total factors.
So, that means, the total effect depends upon
239
00:28:45,200 --> 00:28:52,200
from explained issue that is your that is
derived from the X issue and which is not
240
00:28:52,870 --> 00:28:59,870
derived through X or through independent variables
then it will go to you can U component errorcomponent
241
00:29:00,850 --> 00:29:07,850
that is not in your hand.So, that means, all
explained items are known to us why U is unexplained
242
00:29:12,419 --> 00:29:18,399
in nature because it is in not in your concludes
and we do not have any idea about that particular
243
00:29:18,399 --> 00:29:25,179
item. So, our target is to find out what is
lacking in the system.So, that means, how
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00:29:25,179 --> 00:29:31,779
much we could not represent or we could not
explain in this particular system. So, that
245
00:29:31,779 --> 00:29:37,590
is the main issue or main agenda of this econometric
modeling.
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00:29:37,590 --> 00:29:44,590
So, we like to know what is the error component
which we cannotyou can say have in the beginning.
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00:29:45,919 --> 00:29:52,580
So, that has to be adjusted continuously so
that means, our objective is always we have
248
00:29:52,580 --> 00:29:59,580
to build a model in such a way that the error
components would be at the minimum levels.So
249
00:30:00,500 --> 00:30:06,289
that means, we have to build or the model
can beyou can say model can be represented
250
00:30:06,289 --> 00:30:13,080
as a best fitted.If everything can be explained
nothing can be unexplained, but, it is very
251
00:30:13,080 --> 00:30:17,389
difficult to say something is a total explained
and nothing isunexplained. So, there is a
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00:30:17,389 --> 00:30:24,010
little even if it is 1 percent then also that
1 percent has also weight age sometimes.
253
00:30:24,010 --> 00:30:30,269
So, we have to see or we like to know why
there is a error component in the particular
254
00:30:30,269 --> 00:30:37,269
system of statistical modeling. So, now, the
issue is why error component in the issueof
255
00:30:39,929 --> 00:30:46,929
statistical modeling. So, our ideais to see
why you like to use error component in the
256
00:30:48,980 --> 00:30:55,980
systems. So, now, there are many ways we can
represent this error issue. So, you see we
257
00:30:58,990 --> 00:31:05,059
are in a Bivariatesystems even if in the case
of multivariate system error component is
258
00:31:05,059 --> 00:31:11,850
must. So, now, the issue is your our justification
is why we like to introduce error component.There
259
00:31:11,850 --> 00:31:16,450
are many reasons for that.
Why error components in the systems?Number
260
00:31:16,450 --> 00:31:23,450
one.First is there are certain variables which
can explain the dependent variables, but,
261
00:31:24,539 --> 00:31:31,539
we are not in a position to include that variables
there may be many reasons for that may be
262
00:31:32,659 --> 00:31:39,179
not available in our head with respect to
information wise or with respect to structure
263
00:31:39,179 --> 00:31:46,179
wise or sometimes what happens even sometimes
the idea is there, but, we are not able to
264
00:31:47,039 --> 00:31:53,029
represent in a particular format or sometimes
we have no idea at all.Some variables may
265
00:31:53,029 --> 00:31:58,309
be effecting, but, for the time being we are
not in a position to represent with that particular
266
00:31:58,309 --> 00:32:03,730
variable which can also influence the Y component.
That means some of the relevant variables
267
00:32:03,730 --> 00:32:10,730
or you can say useful variables are not included
in the systems. So, since some variables are
268
00:32:22,330 --> 00:32:29,330
not using the systems so, obviously,there
is this some percentage which cannot be explained
269
00:32:31,299 --> 00:32:38,299
so, that means, there must be some error component.
So, useful variables not included in the system
270
00:32:39,460 --> 00:32:46,460
secondsome ofthe you know unnecessary variables
means or not relevant variables are included
271
00:32:57,429 --> 00:33:04,429
in the system are includedin the systems.
So, this may be also because of this you know
272
00:33:06,490 --> 00:33:13,019
unexplanations. So, there may be error tone
because some of the variables which may not
273
00:33:13,019 --> 00:33:18,460
have any contribution, but, it will affect
the system. So, as a result we have to introduce
274
00:33:18,460 --> 00:33:25,460
the error problem.That means, what is our
commenting factors for this effect sight.Third
275
00:33:27,100 --> 00:33:34,100
is there is a sometimes mathematicalimperfection
of the models.For instance so, we are sayings
276
00:33:51,500 --> 00:33:58,500
Y and X and we are just putting Y equal to
function of X all right. So, that too Y equal
277
00:33:59,159 --> 00:34:05,500
to alpha plusbeta X that is our issue.
But there are many ways alpha I mean Y and
278
00:34:05,500 --> 00:34:11,000
X can be represented for instance Y can be
alpha into beta to the power X or Y equal
279
00:34:11,000 --> 00:34:18,000
to alpha by beta to the power X or you can
say alpha beta to the power X or you can say
280
00:34:18,340 --> 00:34:24,889
alpha log X log beta like many ways we have
to represent the relationship.Since for the
281
00:34:24,889 --> 00:34:30,149
time being we are assuming Y equal to alpha
plus beta X then there may be some problems
282
00:34:30,149 --> 00:34:36,819
technical problem or mathematical problem.At
a particular point of time we have to use
283
00:34:36,819 --> 00:34:42,700
only 1 relationship so, that means, at a time
we cannot take Y equal to alpha plus beta
284
00:34:42,700 --> 00:34:45,210
X or Y equal to alpha into beta to the power
x.
285
00:34:45,210 --> 00:34:51,020
So, yes of course, what we can do we have
to test the model with a different function
286
00:34:51,020 --> 00:34:55,800
alpha.For instance Y equal to alpha plus beta
X in oneextent and another extent we will
287
00:34:55,800 --> 00:35:00,010
take Y equal to alpha into beta to the power
X and we have to setup different forms of
288
00:35:00,010 --> 00:35:04,970
themodel.Test the model to get the best fitted
model between these 2 which 1 is the best.We
289
00:35:04,970 --> 00:35:11,970
have to consider finally, and we have to say
that this is the best fitted model which we
290
00:35:12,839 --> 00:35:19,250
have derived on the basis of some decision
making process. So, now,likewise there are
291
00:35:19,250 --> 00:35:26,250
many ways the functional form can be established.
So, now feel there are many ways the functionalforms
292
00:35:26,400 --> 00:35:32,660
are represented then the model building structure
will be completely different and or also the
293
00:35:32,660 --> 00:35:39,170
result will be completely different.But, there
is way for particular mathematical form has
294
00:35:39,170 --> 00:35:46,170
to be use. So, basically we will start with
the simple oneand by chance will we get the
295
00:35:47,500 --> 00:35:53,220
best fitted model with the simple one.Then
we are in the right track.If the model accuracy
296
00:35:53,220 --> 00:35:59,690
is not on the basis of the above information
or above functional form then; obviously,
297
00:35:59,690 --> 00:36:06,690
we have togo onebyonewith or we have to proceedoneafter
another process to get the best fitted model.
298
00:36:07,520 --> 00:36:14,520
So, mathematical imperfection of the modelalsooneof
the committingfactor which came you can sayinvolving
299
00:36:17,000 --> 00:36:24,000
in the eve issuenext fourth there is a misspecification
of the random terms.There is a question on
300
00:36:24,470 --> 00:36:30,369
misspecification of the random terms.We are
always talking about X and Y are random in
301
00:36:30,369 --> 00:36:35,589
nature.So that means,there must be some level
of or some environmental probability in the
302
00:36:35,589 --> 00:36:42,589
particular system.The term probability itself
represent this chance of occurrence.That means
303
00:36:42,940 --> 00:36:48,630
something which is not in your control. So,
now, which is something not in your control
304
00:36:48,630 --> 00:36:54,440
means obviously, that control may be in many
ways.It can be at a higher level it can be
305
00:36:54,440 --> 00:37:01,440
at lower level it can be at the medium levels.
So, now what you have to do sincewe have no
306
00:37:01,560 --> 00:37:08,560
idea whether it is higher onelower oneor medium
one. So, we have to assume at leastonethen
307
00:37:09,270 --> 00:37:15,440
accordingly that error involvement must be
incorporate in the systems.Since we are not
308
00:37:15,440 --> 00:37:22,440
sure whether the impact is higheroneloweroneor
medium one. So, we have to do that. So, this
309
00:37:23,650 --> 00:37:30,650
is how the error is involve in the systems.Then
fifth,there may be some question of luck in
310
00:37:31,230 --> 00:37:38,230
the systemstake a case of you know social
problems.For instance you like to know what
311
00:37:38,810 --> 00:37:45,810
is the implement of expenditure on a particulars
you can say sales revenue. So, expenditure
312
00:37:48,079 --> 00:37:53,220
that too lead to you can say advertising expenditures.
So, now the theoretical knowledge is that
313
00:37:53,220 --> 00:37:59,960
it will put more and moreinvestment on advertising.Then
obviously, there is a strong impacton you
314
00:37:59,960 --> 00:38:06,960
knowsales revenue. So, now, the issue may
be in something different because we are discussing
315
00:38:08,510 --> 00:38:14,450
about one problem.So, that means, we are discussing
a particular issue.Let us say thisis a pen.
316
00:38:14,450 --> 00:38:21,450
So, now, I like to know if there is a some
kind of investment on this pen advertisement.Then
317
00:38:21,940 --> 00:38:28,550
obviously, the growthor sales of this particular
pen will bein a increasing sequence.
318
00:38:28,550 --> 00:38:35,550
But you are you are not in, you can say monopoly
situation.There are many competitors in this
319
00:38:35,770 --> 00:38:42,770
particular business environment. So, everybody
is doing like this way. So, behind there is
320
00:38:42,950 --> 00:38:47,819
a competitive issue.Then obviously, the formula
issomewhat you know direct one. So, you are
321
00:38:47,819 --> 00:38:54,180
involving other peoples are also involving.
So, by the ways there are certain factors
322
00:38:54,180 --> 00:39:01,180
means which is again you can say third variable
in natures.It can also you can sayincorporate
323
00:39:01,400 --> 00:39:08,400
or you can saygive the accuracy of the models.
So, as a result there isyou can say luck which
324
00:39:09,359 --> 00:39:14,460
can involve in the issue.Everybody is you
can say objective that if you will put more
325
00:39:14,460 --> 00:39:19,200
and more investment on advertising our sales
will be go increasing.
326
00:39:19,200 --> 00:39:26,040
So, just we are believing thatone.That means,we
are assuming that if you will put more on
327
00:39:26,040 --> 00:39:30,940
advertising then obviously, sales is good.So
that means,we are not sure it is just not
328
00:39:30,940 --> 00:39:35,650
like your mathematical way, you will put 2
plus 3 you will get five.You will put 4 you
329
00:39:35,650 --> 00:39:40,050
can say 3 plus 3 you will get six.It is not
like that way because we are putting something
330
00:39:40,050 --> 00:39:46,819
then the effect will be somewhat in other
way.That means, in between cause and effect
331
00:39:46,819 --> 00:39:52,930
there are certain variables which can be also
effect the system. So, as a result sometimes
332
00:39:52,930 --> 00:39:58,010
that factor may be considered as a luck factors.
So, that luck because of you know you are
333
00:39:58,010 --> 00:40:05,010
not sure and sometimes luck is notsupporting
you for this particular issue.Then obviously,
334
00:40:05,109 --> 00:40:11,200
you can say moral cannot be accurateoneor
cannot be perfectly explained one. So, some
335
00:40:11,200 --> 00:40:18,200
part of unexplained is there. So, as a result
error must be in the systems.Then last, but,
336
00:40:19,099 --> 00:40:26,099
not the least is called as a external factors.Besides
luck there are certain factors which is not
337
00:40:33,359 --> 00:40:40,359
in your control.For instance either you are
not aware of it or it may be coming in certain
338
00:40:46,170 --> 00:40:53,170
you can say at the particular situation. So,
in that contest since you are not sure or
339
00:40:53,319 --> 00:40:57,700
you are not certain then, obviously, there
is a error issue.
340
00:40:57,700 --> 00:41:04,700
For instance take a case ofterrorist impact.
So, everything is a planned in a proper way
341
00:41:06,119 --> 00:41:11,490
we are we are very serious and we know all
these variables are explained which is used
342
00:41:11,490 --> 00:41:18,490
in a particular system.We are in the process
to design that build that, but, unfortunately
343
00:41:19,170 --> 00:41:26,170
there is in between there is a terrorist activity.
So, as a result there should be some inconsistent.Take
344
00:41:26,800 --> 00:41:33,450
a case of same thing in say you know in between
advertising and sales. If you know putting
345
00:41:33,450 --> 00:41:38,280
more advertising and increasing the earns
among the people. So, that your sales of that
346
00:41:38,280 --> 00:41:45,280
particular item can go on increasing.
But by any chance terrorist attack on the
347
00:41:45,359 --> 00:41:51,150
your plant say then obviously, your plant
will be get damaged totally and whatever investment
348
00:41:51,150 --> 00:41:55,579
you have done on advertising on that particular
product and thatproduct cannot be also available
349
00:41:55,579 --> 00:42:00,800
for the market.That means, since production
is not there. So, whateveramount you have
350
00:42:00,800 --> 00:42:05,770
put on advertising it is no meaning at all.
So, it is no impact at all. So, as a result
351
00:42:05,770 --> 00:42:11,930
some of the external factor which are not
in your control as a result we have to put
352
00:42:11,930 --> 00:42:18,930
it in new component that is error component.So
that means, your erroris always there in a
353
00:42:19,280 --> 00:42:22,900
in a particular system.
When we will talk about statistical form of
354
00:42:22,900 --> 00:42:29,780
the model,now we have to see what are the
variableswhich are particularly explained
355
00:42:29,780 --> 00:42:34,109
in nature and what are the variable which
are not explained that we will represent in
356
00:42:34,109 --> 00:42:41,109
the form of a U. U is treated as a proxy for
unexplained variables which is a not known
357
00:42:44,450 --> 00:42:51,450
to us or which is not exactly identified.
So, since we have no idea about it. So, we
358
00:42:52,059 --> 00:42:58,569
are assuming that it is in U only. So, error
will incorporate all this defects which is
359
00:42:58,569 --> 00:43:00,059
not in our controls.
360
00:43:00,059 --> 00:43:05,470
So, that means,.So, for a Bivariate econometric
modeling is concerned then system will be
361
00:43:05,470 --> 00:43:11,809
like this our starting point will be Y then
X in between U is introduced in the system.
362
00:43:11,809 --> 00:43:18,809
So, now, like this Y 1 Y 2 Y 3 Y 4 up to Y
n then X 1 X 2 X 3 X 4 and up to X n similarly,
363
00:43:25,410 --> 00:43:32,410
U 1 U 2 U 3 U 4 and U n. So, now if will we
go by simply mathematics.Then obviously, Y
364
00:43:39,109 --> 00:43:46,109
1 equal to X 1 plus U 1, Y 2 equal to X 2
plus U 2, Y 3 equal to X 3 plus U 3, Y 4 equal
365
00:43:54,480 --> 00:44:01,480
to X 4 into U four. So, similarly, Y n equalto
X n plus U n so that means,all X are inonegroup
366
00:44:07,500 --> 00:44:14,500
and all U are in another group and the total
effect will be on Y. So, this is explained
367
00:44:17,349 --> 00:44:24,349
effect, this is unexplained effect.
So, now our objective is to minimize this
368
00:44:28,829 --> 00:44:34,220
particular activities sothat means, the way
we have to minimize you need to have a best
369
00:44:34,220 --> 00:44:41,220
fitted models. So, you need to have a best
fitted models for instance like this. So,
370
00:44:41,740 --> 00:44:48,740
now, if will we go by simple framework then
let us assume that for this particular variable
371
00:44:49,040 --> 00:44:54,380
your graphic structure will be like thisside
X measurement and this side Y measurement.
372
00:44:54,380 --> 00:45:00,390
Since the functional form is Y equal to alpha
plus beta X then alpha is a constant. So,
373
00:45:00,390 --> 00:45:07,390
thiswill be just supporting factor like this.
So, now, for every X 1 every X 2 every X 3
374
00:45:07,550 --> 00:45:14,550
every X 4 X 5like this. So, X 1 there may
be someyou can say Y 1 for X 2 there may be
375
00:45:15,740 --> 00:45:22,740
Y 2, X 3 there may be Y 3, X 4 there may be
you can say Y 4, X 5 then there will be Y
376
00:45:26,280 --> 00:45:33,280
5then X 6 then obviously, Y 6 like this.
So, now if will we join all such points then
377
00:45:34,950 --> 00:45:41,349
the picture will be coming like this and this
is the true picture of this particular setup.So,
378
00:45:41,349 --> 00:45:48,349
that means, we have Y and Y information and
X information and our idea is how Y and X
379
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are related to each other.This is first objective
and if Y and X are related to each other how
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best can they beyou can say related to each
other.This is the basic objective behind econometric
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modeling.Obviously, when you will go for investing
the relationship between Y and X there will
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be certain relationship.Eitheryou assume it
or by theory you have to bring these variable
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in such a way there is a somewhat relationship.
So, that means, that relationship is there,
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00:46:24,550 --> 00:46:31,550
but, we have to predict or we have to forecasthow
best they can be you can say related to each
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other. So, that the effect will be very positive
and you know very accurate. So, that is how
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00:46:39,660 --> 00:46:45,130
that is that should be our main agenda. So,
as a result within the particular setup we
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have to build a you can still line which is
the bestfor you.
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00:46:49,910 --> 00:46:56,680
That means, if you will consider this is a
path and this path is very much uneven in
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nature.It is not at all straight forwarditjust
like a non-linearone.So, that means, we have
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00:47:03,530 --> 00:47:10,240
to being in to a linear path what difference
you have to bring in a best part so that the
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00:47:10,240 --> 00:47:14,000
model accuracy or forecasting can be very
perfect one.
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00:47:14,000 --> 00:47:19,819
So, as a result let us assume that this particular
line is the best fitted one. So, if will you
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00:47:19,819 --> 00:47:26,819
say that best fittedonein statistic we call
it as the Y headestimated lines. So, this
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00:47:26,890 --> 00:47:33,890
is what we call it as a estimated lineor otherwise
called as a expected line.Yhead equal to alpha
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00:47:36,450 --> 00:47:43,450
head plus beta head X.Yhead equal to alpha
head plus beta headX.That means, in other
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00:47:43,970 --> 00:47:50,970
waywe havethreeforms of functions. Y equal
to alpha plus beta X this is mathematical
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00:47:51,079 --> 00:47:58,079
form of the model.
Then we have alpha plus beta X plus U.This
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00:47:58,290 --> 00:48:05,290
is input statistical form of the model.Then
we have Y head equal to alpha head plus beta
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00:48:05,520 --> 00:48:10,829
head X.When will you call it Y head equal
to alpha plus beta head X,this is estimated
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line.So, we will be say that this is the estimated
line this is the best line this is the perfect
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00:48:16,180 --> 00:48:23,099
line then of course, the error issue will
not be there.So, that means, the way we will
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choose the model is the best one.Obviously,
as per your knowledge it should be veryexplained
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00:48:29,240 --> 00:48:36,240
one.
So, if it is explained 1 then, obviously,
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00:48:36,579 --> 00:48:43,579
theerror will be not there in the system.So,
that means, we firststart with the exact model
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00:48:45,950 --> 00:48:51,750
then in between you have to assume that the
model is not exact.So, that means, we have
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00:48:51,750 --> 00:48:57,359
to bring the inexact of that particular system
then again we have to verify or you have to
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00:48:57,359 --> 00:49:04,359
come to this stage again.It will be transferredto
the same exact model that is ina way of mathematical
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00:49:04,380 --> 00:49:08,599
form of the model.
But the mathematical form of the model in
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00:49:08,599 --> 00:49:14,270
a initial setting in the estimated model of
the model the latter setting may not be exactly
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equal. So, here the issue of this mathematical
form of the structureform of the model is
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that we know this should be the relationship
that is the exact relationships. Now why there
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is a statistic because statistic always object
this particular mathematics. So, if there
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00:49:31,250 --> 00:49:38,160
is a question of objection then there is a
need of information to verify that one. So,
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00:49:38,160 --> 00:49:43,630
statisticsstatisticians you can say assignment
is to the way they will be verify to that
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00:49:43,630 --> 00:49:50,220
particular mathematical problem mathematical
form of the model. So, that the judgment can
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00:49:50,220 --> 00:49:56,020
be accurate onein same way the structure isall
about the econometric modeling.
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00:49:56,020 --> 00:50:02,440
So, now this is called as the best fitted
model. So, now,if we will integrate all this
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things then we will have Y equal to Y head
plus e. So, this is the another form of the
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00:50:09,030 --> 00:50:16,030
equation.So that means,yourtrue value Y which
depends upon estimated value and error terms.
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00:50:19,369 --> 00:50:25,880
So, that estimated value may be perfectlystill
there is a question of error term. So, we
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00:50:25,880 --> 00:50:32,819
have to see how much error is committed in
the system and whatever you know error is
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00:50:32,819 --> 00:50:39,819
that in the systemthat should be in least.
So, that is the more you knowaccuracy of this
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00:50:42,160 --> 00:50:45,790
particular system.
So, altogether..
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00:50:45,790 --> 00:50:52,790
So, what we have discuss is that Y equal to
alpha plus beta X.This is mathematical form
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00:50:53,380 --> 00:51:00,380
of the models.Then Y equal to alpha head plus
beta head X.This is estimated models.In between
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00:51:05,690 --> 00:51:12,119
from the mathematical models we assume that
Y equal to alpha plus beta X plus U is the
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00:51:12,119 --> 00:51:17,710
statistical form of the model. So, this is
what mathematical form of the model, this
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00:51:17,710 --> 00:51:24,710
is what statistical form of the model and
this is the estimated model.Now if will integrate
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00:51:28,010 --> 00:51:31,609
all these things then Y equal to Y head plus
e.
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00:51:31,609 --> 00:51:38,609
So, now what is e?e is the other way or other
way representation of error terms. So, because
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00:51:41,210 --> 00:51:47,059
we have already in the estimation process.
Obviously, we are putting the error components
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00:51:47,059 --> 00:51:52,869
in different names, but, it is more or less
same. So, e equal to basically a difference
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00:51:52,869 --> 00:51:59,869
betweenYminus Y head.That means, whatis the
true value and what is theestimated value,that
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00:52:00,980 --> 00:52:07,980
is the committee of errors. So, error meanswhat
is the actual and what is the predicted or
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00:52:10,010 --> 00:52:16,630
estimatedor assumed value. So, we are we are
assuming that this is should be the perfect
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00:52:16,630 --> 00:52:23,240
one. So, there should be actual. So, we like
to know what is the difference between the
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00:52:23,240 --> 00:52:29,339
actual and estimated. So, that difference
is called as an error issue like this. So,
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00:52:29,339 --> 00:52:33,500
these here these are the true value ok.
So, we are assuming this is estimated value.
439
00:52:33,500 --> 00:52:40,500
So, now, the differenceis all about this error
issue. So, these here are called as a error
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00:52:42,290 --> 00:52:47,240
issue.Like you know for e it called as e 1
this is e 2 this is e 3 like this. So, this
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00:52:47,240 --> 00:52:53,930
side is the X measurement and this side Y
measurement.This is estimated models.Yheadin
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00:52:53,930 --> 00:53:00,240
between it shaped to with the integration
between true lines or true points actual points
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00:53:00,240 --> 00:53:07,240
and theestimated point. So, now, this is the
typical issue of this or this is the basic
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00:53:07,480 --> 00:53:13,630
statistic point of econometric modeling. So,
now, what is our next objective?Our next objective
445
00:53:13,630 --> 00:53:19,200
is to minimize this error terms.
So, far as minimization is concerned or of
446
00:53:19,200 --> 00:53:25,569
course, when there is a question of optimization,
we cannot optimize this single one. So, we
447
00:53:25,569 --> 00:53:32,569
have to optimize the minimum sum of the squares.
So, sum of the sumof the errors, square errorshas
448
00:53:37,150 --> 00:53:44,119
to be minimized. So, that process is a more
complex and very interesting and very useful
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00:53:44,119 --> 00:53:50,650
very systematic. So, that we will discuss
in next class.So, it is not possible to start
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00:53:50,650 --> 00:53:55,809
here now. So, the detail structure we have
to discuss in the next.Thank you very much
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00:53:55,809 --> 00:53:56,690
have a nice day.