Understand Autocorrelation in 15 minutes - Dr. Tehseen Jawaid - YouTube

Bismillah-Hir-Rehman-Nir-Raheem

Today we are discuss the second assumption of OLS

So, What is the second assumption?

Error term observation are independent of each other

Or

They are not co-related with each other

Well, first we look at co-relation.

Like this we have the movement of a variable

This is the movement of the other variable

So see the movement of both is on the same time

Its increase is seeing its increase Its decrease is seeing its decrease

So if the moment is closer to each other and associate Co-relation will be so strong

And if the movement does not happen with each other, then the co-relation will be so weak

But this is the co-relation when two series are co-relating with each other

If I have series x and it has any values 10, 20, 30, 40, 50

And I want to know if x is correlating with its own values

Now I'm going to create another value here Xt this is current value

And I made a variable here t-1

t-1 means, each value with its previous value

40, 30, 20, 10.

If these two series are correlating with each other

So I can conclude this here

that x is correlating with its own value

So to correlate is called as correlation

And when any series starts correlating with its own value

So when it comes to its own values, so such correlation is called Auto corelation

Auto correlation can exist in every variable like we said GDP

Today's GDP will correlate with last year's GDP Absolutely logical

There is an investment, It is absolutely logical There is an saving, it is absolutely logical There is an consumption, it is absolutely logical

So every variable, most probably

Correlates with its old values

Now we look ahead. Let's assume that is a regression model.

Y we have dependent variable, x and z we have independent variable

And we have an error term.

We have discussed this

These error terms represent all variables that are not part of the model

And those that are part of the model appear as an independent variable in the model

Actually error, understand one thing about error

An error is called a mistake

And there is a basic characterstic of mistake

The main characteristic of a mistake is that the mistake or error is not intentionally do.

So when not done intentionally, there is no consistency

Mean's no sequence or trend exists in the error

So that's why it's an error term

Because this is also a mistake

So what is the most important characteristic of this error, the error should be raindom

The error should be random. The error should not be consistence.

There should not be exist any trend in this

Now what happened, A researcher developed this model

But due to lack of information Due to lack of literature review

It could not do the modeling correctly and the model stopped here.

Mean, Z was a important variable.

Which was part of this model but not considered in it.

Now you know that variables that are present in the model are present

and those that do not exist, that all exist in the error term.

In exactly the same way that Z was miss here

The Z that was missed went into the error term

It has many variables other then these variable are all in the error term so Z also goes there.

Z was an important variable that was very important for Y

And we have already seen this

Any variable and if it correlates with its own value

Develop a relationship or make a friend. It is not logical

Because, All variables in logical are auto-correlated

So it is not a problematic thing that the variables are correlated.

The same will happen here

Now we see that for example we have annual data, time series data.

So the value of z for each year is correlating

Correlating, showing consistency, creating a trend it is doing a Z

Now consider one thing here. There is a box in which I keep the remote control car

There is a big carton and the car is not visible to anyone

I moved the car and it came here

Actually, Who will be doing the move, this car but we will be seeing this box

We must have seen this box. In exactly the same way

Here Z is the consistence

Z forming a trend Z is correlating with its previous value

But Z's are not separate variables

It's added to the error term so we'll see it like this

This error term is consistency, the error term is correlating with its previous value

When the error term is correlated with its previous value

So we said that any series that correlates with its previous value is called autocorrelation

But then we also saw that there are no problems with auto-correlation

Each variable may be autocorrelated

The important point here is that the problem in autocorrelation will occur at this point

When autocorrelation is present in the error term

If the variable is autocorrelated. It is logical that it is acceptable and it should be

But when you talk about error, the error should not be autocorrelated

Because, as we have already seen above, errors should be random

If there is any consistency in the error term

A correlation exists by its own value

So this assumption will be valid

It is saying error term observation are independent of each other or they are not correlated with each other.

If this is not , it means that autocorrelation exists in error

So what was the problem that we saw? Autocorrelation

Autocorrelation

Now let's understand what is meant by consistency in autocorrelation

We know that the actual difference of Y-Y(cap) is called error

Assume this is a possibility

The firm invites any analyst every year Get your profitability predicted

So if the error is positive in one year, it means that the actual profit has increased

Estimated was given less

Next year, come again positive Next year, come again positive

When the error is coming with the same sign, it is coming with the consist sign

So this means that the analyst must be doing some kind of problem in forecasting that the error is repeating.

Consistency is happening

Or if it comes negative then negative

This means error consistency

A trend exists in error

that each incoming error carries the sign of the previous error

plus plus plus, this minu, minus, minus

So we will call it error consistency Error is autocorrelated

In exactly the same way, only the same sign does not show consistency

plus minus, plus minus, plus minus, plus minus

This is also autocorrelation

Because the opposite is being predicted in this too

Because consistency is also in the opposite sign plus minus plus minus

You can easily forecast

If the autocorrelation is such that the error is predicting the same sign. For your incoming error

So this is called positive correlation

And if opposite sign is predicting then it is called negative correlation

Now the question is if any autocorrelation occurs in the regression model

So what is illness, what problem comes in the model?

Well see this is a distribution

This is the distribution of a beta coefficient

So what happens now if positive correlation exists

So see, consistency is and how consistency is all on one side

mean are all plus or all minus

Most are on the same site

For that reason It is in the center

Because of that, if the autocorrelation with the same sign is mean positive.

So in them, because all the things are towards the same sign,

they are not going towards the opposite So the deviation is reduced

And in positive autocorrelation the distribution will look something like this

This means that the variable was previously insignificant

If there is a positive autocorrelation, it may be looking significant here

If the variable was previously insignificant, it may appear significant due to positive correlation

In exactly the same way as in negative correlation This is also consistency

But plus minus plus minus. Deviation increased

The distribution will look like this

Because of this, the variable that was previously significant will start to look insignificant.

This means that when autocorrelation exists in the model

The significance of the variable, the probe value, and its standard error are not reliable.

You cannot say this with confirmation

That variable is really significant or insignificant

So the issue that comes up in regression due to autocorrelation is that of significant values

So you can say here

Issues in significant which is illness.

What is illness, which issues come to the significant, the significant is not remain reliable

So the problem we discussed is Autocorrelation.

We talked about illness. Significant Issues. Means, the significant is not reliable

Now let's talk about Measure

The measure we use for this autocorrelation is Darwin Watson statics

Darwin Watson's range is zero to four

If its value is 2, it means no Autocorrelation

If the value is between zero and two, you would say a positive Autocorrelation

And between one and two four, then you will say negative Autocorrelation

So theoretically we will say that if the value is exist 2 then there is no autocorrelation

Even if the value is 1.9 or 2.1, the values will exist in autocorrelation

But now we also need to know

Is the autocorrelation coming in 1.9 so severe that we should move towards its removal

So let's do a test to check the severity

and this is Serial Correlation LM test

The hypothesis of serial correlation LM test is no autocorrelation

If its P value is greater than 0.5

If our level of significance is five percent

If it will be 10 percent, then we will see from 0.1

If it is more, then accept the hypothesis

If it is less, then reject the hypothesis

Accept means autocorrelation does not exist

If this hypothesis is rejected, it means that autocorrelation exists

So if value other than 2 is more or less than that

The serial correlation LM test should confirm that the hypothesis is being rejected

And the serial correlation severe is present in this model

So then which one will we move towards, we will move towards Removal

If this value is different from two, but the hypothesis is accepted, then no removal needs to be applied.

So now let's talk about removals, removals are very easy

And it is logical that as our regression model was

And in this the researcher forgot to add an important variable. which is Z

And Z had gone into error term. And the error consistence was visible

Actually the error was just showing consistence

But the consistence was z

So due to which variable the consistence error was visible.

Extract it and make it part of the model

Now the error term which was the consistency due to z

Which was trending because of Z

The autocorrelation due to Z was visible in this

Now that Z has come out of it

So now the error will go back to its old characterstic

Means error would be random now

Now the error has become random and as soon as the error has become random, the correlation between them has disappeared

Autocorrelation was eliminated from the model

So what is the most appropriate removal of it? Additional of relevant variable

Well it is not necessary to auto remove from one variable

Maybe you have to add one, maybe you have to add second,maybe you have to add a third variable

Well, the question arises how to know which variable to add

So there are two things to see in this, number one theory

The theoretical relationship of the dependent variable to which variable

Have you taken all those variables?

Second, what variables have been used in the literature

From there you will have the data to determine which is the important variable

Which could have been in the model and you didn't take it into the model

Number two, cochran orwatt and gernalise least square (gls)

Number three removal is AR1

GLS and AR1 is also a method to remove autocorrelation

These technically randomize the error term

If there is a trend and consistency in it, they eliminate it

But the problem in interpretation is that Y does not remain Y, Y becomes delta Y

With some technical explanation it becomes DeltaX

So, in effect, the autocorrelation is eliminated, the error term becomes random

But those that are variables change their form

So they don't have the same interpretation that we can do directly above

If a researcher is using a large sample in their data.The sample size is large

And he doesn't want to add any more variables, he thinks I have enough variables

So here is one more test, HAC test

Hetroscedasticity auto correlation consistence test

If a researcher performs the regression by selecting the HAC test

So the HAC test, due to autocorrelation, any changes in probe value and standard error

Adjusts them and after selecting the HAC test

Now when the regression comes to you, it is reliable

It can be significant and can be interpreted

This is the end of today's lecture.

If you like this video, please like, share, subscribe to our channel, and press the bell icon. Thank you.