Regression

start with the regression and then I

don't know how much time will it take if

time then we will do either sources of

capital or one of

that okay regression

uh so first the equation Y = C + MX

which is now called as yal a + BX or

called as y equal B 0 + b1x okay that c

or uh the a or that b0 is called as

starting point which is also called as

intercept okay uh so we call for today

as Y A Plus BX but in the exam you could

say y b 0 b1x C

MX for today say Y is equal to a plus BX

so the a is called as starting point

which is also called as

intercept if that is positive then the

graph will start from above and if that

a is negative the graph will Start From

Below so that's the importance of the

intercept where does the line start from

a positive that means the line will

start from above above the origin and a

negative line will start below the

origin second is the B which is called

as slope coefficient calculate formula

I'm taking after some time if the B

value is positive the line would be

upward sloping if the B value is zero

line would be flat and if the B value is

negative line would be downward sloping

if the B value is negative line be

downward sloping if B value is zero line

would be flat if the B value is positive

line would be upward sloping and if

value is more positive line would be

more upward sloping okay

yeah so a is the starting point if a is

positive line will start from above

origin if a is negative line will start

from below origin okay if B is negative

line will be going down if B is zero

line will be horizontal if B is positive

line would be upward sloping and this

more positive line would be more steeply

upward sloping

uh this x which is the horizontal axis

is called as independent variable and

this Y which is the vertical axis is

called as dependent variable so the

equation y equal to a plus BX Y is

dependent variable X is independent

variable a is intercept and B is your

slope all right so a starting point B

the steepness X independent variable y

dependent

variable this line is called as the

regression line okay this line is called

as regression line and this model is

called as linear linear model this model

is linear linear

model Y A Plus BX you have Ln of Y = A

Plus BX then that's called as log linear

model instead of yal a plus BX you have

y = a + b into log of x that is Ln of X

is called as linear log model and again

it's Ln for y also Ln for X also then

called as log log model so yal a plus b

x x Ln

y linear model

y l linear model

y log

model l l

model you should know the formula for

the B although a and the B can be

calculated directly from that second

seven and second 8 so if you press

second seven put all data second 8 and

linear other second enter

and then scroll down down down down down

you'll get a b and then R so a that's

the a intercept and B slope coefficient

but obviously in the exam they might ask

you the formula what is the

formula B

formula between Y and

xid variance of x y and x variance of X

that's the formula for the B Co between

Y and X that is dependent and

independent variable by variance of X

that is variance of independent variable

I think example S&P 5 and some ABC stock

So Co ABC stock was the Y and S&P 500

was the X so cence between the X and the

Y would be cence between S&P and ABC

stock and divid by variance of X and

that X wasp finded so variance ofp find

in the

denominator so general form is co

between Y and X variance of X and once

you get the value of then only you can

get the value of a so over there instead

of Y you put the Y bar that is average

of Y instead of X you put the xar which

is average of X so Y is equal to a plus

BX in of y y bar instead of x xar b we

already calculated so only thing which

is unknown is a value okay so y y bar

average of Y is known to you X xar is

already known to you B value calculated

so the only thing which is unknown is a

value so that's the way you're going to

get a value

then that coefficient B whether that is

statistically different from zero or it

is statistically zero find out we have

three ways but why it is important to

find out what is statistically zero

statistically different from zero

statistically zero zero useless then

that case that X will become useless so

X coefficient B if that b is

statistically zero then if it is zero

then it's useless if it is statistically

different from zero then it is useful so

once you calculate the value of B I

think what 64 value I think so B value

which was 64 if that is statistically

zero or statistically non zero

is important because if the B value is

statistically zero then that's useless

and if B value is statistically non zero

then that's useful so how do we go for

that testing so there is three ways one

is the easiest way which is called as P

value if P value is given to be3

percentage then Max

99.7% confidence level then the B would

be non zero that means H1 would be

selected and H1 would have B non zero so

B will be different from zero or B will

not be zero that would be your H1 and

that H1 would be selected Max up to

99.7% if the given P value is3

percentage so the P value is.3 100 -3

99.7% so up to 99.7% confidence we can

say that the H1 would be selected which

means we can say that the B is non zero

so take an example P value is 1

percentage and the confidence sorry

Alpha is 1 percentage P value is 1

percentage and Alpha is. 1 percentage

whether B is statistically zero

statistically non

zero I I repeat P value 1 percentage

Alpha

.1% whether the B is statistically zero

statistically non

zero what is P value 1 percentage so

what's Max confidence up to 99

percentage H1 would be selected up to 99

percentage B will be different from zero

H1 means not equal to that is different

from zero so up to 99 percentage the B

would be non zero but the question had

said Alpha of

Point

99.9 that means it was beyond the 99 so

up to 99 H1 would be selected Beyond 99

H1 would be rejected h0 would be

selected and h0 would stay that b is

equal to Z H1 was B not equal to Z h0

was b equal to Z so up to 99 percentage

based on P value of 1 percentage up to

99 percentage H1 would be selected

Beyond 99 percentage H1 would be

rejected h0 would be selected and h0 was

b equal to z b equal to Z get selected

would mean that the coefficient is

statistically zero and hence useful yeah

useless useless because statistically

zero

useless second method is you do the

testing and for that you have to first

write the null and the H1 null would be

opposite of your belief so generally n

would be b equal to Z and H1 would be B

not equal to

Z is the coefficient greater

than2 so greater than to less than equal

toer less

equal equal to sign can never come in H1

so greater than 020 would be there in H1

and less than or equal to 020 will be in

h0 because equal to that will be h0 I

say greater than that means less than

equal to would be h0 and greater than

would be H1 so greater than would mean

right tail okay so you'll be focusing on

only the upper critical value okay so

now we have to look at the T Test

compulsorily over here we don't have

the standard Dev of Pop is known to Z

standard of population is not known to

use use test so in this example when I

said the H1 would be B greater than 020

then greater than would mean upper tail

so there's only one tail so we'll be

looking at one tail Alpha would be given

to you let's say 5% alpha is given to

you so one 5 percentage we'll be looking

that in that column and degree of

Freedom would

be good n minus 2 for correlation and

regression the degree of freedom is

going to be n minus 2 so in that column

degree of Freedom n minus 2 one tail and

Alpha

5 previous example h b equal to Z and H1

B not equal to Z so not equal to left

tail and right tail both so 2.5 over

here 2.5 over here so you will be one

tail 2.5 and two tail five so 2 till

five degree of Freedom nus 2 will be

looking at the lower and upper critical

second example B is greater than 020 so

greater than upper tail only so only

entire 5 percentage will be over here so

one tail 5 percentage degree of Freedom

n

minus so you'll get the area of H1 and

the h0 and then you have to apply the

formula remember write down that formula

given value okay I'll not speak out the

formula you write down the formula and

then check it on the

board how many have written like this

raise

hand this is wrong denominator

root it's only this formula okay the

value of B that you calculated minus the

value the

H1

sorry so H1 h0 was like

this and H1

was H1 Zer so that will put over here

and just second

H1

h120 so that H1 be putting

020 all right so whatever is the value

in the H1 H1 zero value so put that H1

over here zero value and h120 you put

that 020 over here okay so the

calculated value of B which I believe

was 64 right

64 and then B1 will be given to you

that's called as standard

error of slope

coefficient how theable will look

like uh you'll be given some y value

over here which is like let's say stock

return

return then you will be given over here

intercept and then the x value which is

let's say Market

return return

coefficient and then standard

error so let's say these values are like

this two three and let's say uh 2.5 and

4.5 so how to read this the equation

will be stock return return that will be

your Y is equal to a +

BX that a intercept that intercept a is

this

two plus the slope is three and that X

is Market

return all right so the Y will be

written over here or it will be

given y value that is y meaning stock

return this is your X which is your

Market return intercept is a

value and this is your coefficient value

B

value okay so this is the B value three

which you'll be putting it over here in

the formula B value

three value in the H1 is over here or

over here whatever the H1 they will give

it to you and the standard error of B1

is this standard error of B1 is this

this

4.5 So based on that you'll get your

calculated value calculated value h0

area h0 would be selected calculated

value H1 area H1 would

be okay so

P hypothesis

testing confidence interval so what you

do over here you get the B value which

is 64 in example

that was

three so you add as well as you

subtract something called as s into

T you get B Plus St and over here you

get B minus that s is standard ER which

given 4.5 standard ER so the standard

error s is 4.5 and T value will have to

be calculated using the T table degree

of Freedom n minus 2 and it's two tail

because left left and right right

inter okay so it's two tail and then

Alpha will be given to you yeah this

interval will be given to you so this is

95 percentage so

2.5

2.5 so 5 and then get the T value let's

say t value is 2 so 4.5 into 2 is 9 so

here it will be 3 +

9 and here it will be 3 - 9 so 3 - 9 is

-

6 and 3 + 9 is 12 and now the important

thing is interval minus 6 or 12 be zero

is there because lower limit is negative

- 5 -4 -3 -2 -1 zero and then 1 2 3 4 5

6 7 8 9 10 11 12 zero has come zero has

come 0 which we

say b equal to Z is going to be selected

and H1 B not equal to Zer will be

rejected so that's the third method I

repeat you'll be given B values let's

say B value is two you have to do B plus

s and then the right tail and then you

have to do B minus St and then the left

tail s Val will be given to you let's

say s value is

1.5 and to get the T value you have to

look at the T table degree of Freedom n

minus 2 that's one tail that second tail

so look for two tail and Alpha will be

given to you so based on that you get

the T value let's say t value is let's

say two or let's say t value is yeah

let's say two

then you do s into T So 1.5 into two

that would be three so two this B value

two 2 + 3 and over here B Val is two and

2 -

3 so s into T that will be 1.5 into 2

that will be 3 so 2 + 3 and 2 - 3 so

again 2 - 3 is

-1 and 2 + 3 is 5 -1 or 5 be zero will

be there zero would be

there at zero b equal to Z would be

selected and if it is zero it is

statistically

useless okay so these are the three

suppose interval minus 5 minus two

suppose interval came like this then is

the zero there over here

Z so Z is not there then H1 B not equal

to Z is going to be

selected so if it is not equal to zero

it's meaningful it's useful

similarly

range7 to

7.5 in this range is zero

there so Z is not in this range so H1 b

equal to Z is going to be

rejected so B is not I'm sorry will be

selected sorry H1 B not equal to Z sorry

H1 is B not equal to Z so that not equal

to Z is going to be selected

so it's not equal to zero so it's

useful so three methods one is P value

which I believe is the easiest one

hypothesis testing and then the

confidence

inter you calculate the value of y a and

the B value will be given to you either

in the table

like intercept is given and Market

return was given so what was the of

intercept I gave you

Che coefficient and standard error which

was I think

4.5 so y equal to intercept is

two and uh the market return return the

coefficient is three so that's

three x is your Market return so that's

your X Market

return and Y is your stock

return so you'll be given Market return

put that market return so let's say

Market return is let's say 5 percentage

you put that over there and you get the

value of stock return okay 5 into 3 15

15 you be 2 percentage so 2 percentage

so 15 percentage plus 2% will be 17

percentage that's called as estimated

value this is not the actual value this

is the estimated value actual value

could be higher or actual value could be

lower

and that will give us an error and that

error is called as actual value minus

predicted value this 17 percentage is

called as a predicted value actual value

could be higher or lower if actual value

is higher then the error would be

positive or

negative good the formula is actual

minus expected actual minus predicted

predicted is 17 if actual is higher than

that something like 18 so 18 - 17 error

would be plus one and if actual is

lesser than that something like 16 then

the error would be actual minus

estimated 16 minus 17 so that would be

negative 1 so if actual is higher error

would be positive if actual is lesser

error would be

negative so what we do we again

construct a confidence interval around

this so again confidence interval we put

the the calculated value in the middle

that is 17 percentage in the

middle what do we add and what do we

subtract St 17 plus St and 17 minus St s

this will be S for the forecasted

value which you don't have to that

formula I don't expect that formula to

be tested so this s for the forecast

will be given to you but there's a

formula given remember in the text

s e s into bracket 1 + 1 upon X so that

big formula is that the standard error

for forecast is not required to be

by so this would be that s standard

error of the forecast and T value again

you have to calculate it's a two tail

left tail right tail so two tail and

degree of freom would be n minus 2 so

from that you'll get the T value then

multiply the St add it and then subtract

ract it and you'll get your confidence

inter so once you calculate the

estimated value you could get two types

of variations over there one variation

is going to

be calculate the error so they give you

actual value and I have seen students

making a silly mistake they do estimated

minus actual as error which is a wrong

actual minus estimated will be the error

so basically you calculate the y value

using a plus BX formula yal a plus BX

and that would be the estimated value of

y stock return 17 that was the estimated

value actual value will be given to you

and actual minus estimated would be the

error value not the other way around

that could be one type of question and

second would be the confidence interval

put the estimated value at the center 17

then add the St subtract the S 17 plus

St 17 minus St you'll get a confidence

interval s will be given to you T you

have to look for two tail degree of

Freedom n minus 2 and then you'll get

the T value St add St subtract from that

calculated value of

y all

right then Anova full form analysis of

variance so over here let's write that

table down and from that only we'll be

able to reconnect everything

so first would be over here regression

sum of

square and then error sum of square and

then that becomes total sum of

square sometimes it is a sum of square

of regression sum of square of error sum

of square of total sometimes called as

regression sum of square error sum of

square and total sum of

square so these values will be given to

you it's coming

and then coefficient of determination R

square is to be calculated and that

would be RSS upon

TSS regression sum of square upon total

sum of

square what is co efficient of

determination

second y degree of Freedom will be

required to be known the degree of is K

over here which in level one is one and

error sum of square it's N - K - 1 so N

- 1 - 1 so in level 1 it's N -

2 actual formula is K and nus Kus

one in level one the K is

one so that becomes nus 1 - 1 so that

becomes nus

2 yes silent please then mean sum of

regation so that will be RSS upon K that

is RSS upon one or SSR upon one and this

is m e which is

SS upon n minus 2 or actually n minus K

minus

one all right so this is SSR or RSS

divided by one that would be MSR and

then SS upon n minus 2 would be

MSE and then finally the ratio of MSR

upon MSE would be your F value MSR upon

MSC would be F calculated

value F critical value mostly will be

given to you f critical value will be

mostly given to you this will be given

to you if not you have to look at the F

table

and degree of freedom for the numerator

and degree of freedom for the

denominator so numerator degree of

freedom is one and denominator degree of

freedom is n minus 2 okay so degree of

freedom for the numerator and degree of

freedom for the denominator is 1 andus

respectively f

tablee

forat for the denominator intersection

point will give you the critical

value and if the calculated value is

greater if the calculated Valu is coming

out to be greater then the model is

better model or model is good model

problem r s r value higher the model is

good but what is the cut off we don't

know over here so if I have R square of

82 percentage we don't know whether it's

good or bad so if his R square is 85

percentage then my R square of 82

percentage is bad and if his R square is

65 percentage then my R square of 82

percentage is good so R square cut off

and look at the F test which is MSR upon

MSC that's the calculated value and

compare that with the critical value if

the calculated value is higher than the

model is

good it's all

which is standard deviation of

error which is called as s standard

error of

estimate or also called as standard

deviation of

error which is

m m square

root okay so the formulas are like this

RSS plus SS is

SST that's one formula then RSS upon TSS

is R

squ then RSS upon K which is one is

MSR ssse upon n minus K minus one that's

M and in the ratio is f calculated value

and then finally square root of M is

called as standard deviation of error

also called as S E standard error of

estimate yal a + BX and just formula for

the B I'll write it one more time b and

a formula B is equal to coari between y

and

x divided by

variance of

X all right once we get the value of y

sorry once we get the value of B then y

y bar x xar b is known so only unknown

would be the a so you calculate B into

xar you put it on the other side with a

negative

sign this is called as linear linear

model Ln

so that's called as log linear

model L it's called as linear log

model and finally L it's called as log

log

model just recapping B test three

methods P

value which will will give you Max

confidence level till the time you can

say H1 would be

selected which says that a b is not

equal to zero will be selected second

one is uh the confidence interval where

you have B minus St

and B is not equal to Zer in the B minus

HD to B plus h z is there that mean b is

equal to Z Z is not there then the B is

not equal to Z and the third one is

hypothesis testing where you write h0 b

equal to Z H1 B not equal to zero and

then you write the make the normal

distribution with both tailes because

not equal to then get the T values using

degree of Freedom which is n minus 2 and

the two tail 5 percentage and then apply

the formula which is value of B minus

the value in the H1 divided standard

error upon root of

only standard er only standard not

divided root so that's the third

method then you'll be applied you'll be

asked to use this yal to a plus BX so

values of A and B will be given to

you something like this and they will

not be given to you like this they will

given to you in the table

table so using that they will give you x

value and they will ask you to calculate

y value and then you'll be asked either

for the error or the confidence interval

error predicted value so actual value

minus the predicted value will be your

error so using this formula the value

that you get is predicted value and then

actual value will be given to you so

difference between them actual value and

predicted value is the error that's one

and second would be once you get the

predicted

value subtract

St and add St and you'll be asked to do

a confidence interal for that

s will be given to you that formula for

the S will not be asked to be byhe

hearted and the T value would be like

degree of Freedom n minus two and it's

right tail and left tail so it's a two

tail Alpha will be given to

you and then

that analysis of

variance it's only there for the theor

it's not there that's no numerical

everything or miss out something any

theory that you missed

out please have a look at it I'm just

scrolling if I out any

Theory that's linear log log log model

so no numericals on this

this we covered confidence in terms ofed

as I said this formula is not required

to be

byed we did

itting RSS upon

TSS this one RSS upon TSS

this one if they give you correlation

coefficient then R square will be

coration coefficient

Square so if they give you correlation

between the Y and the X as let's say

80 then R squ which is called as

coefficient of determination would be 80

Square

this is the point I missed out

correlation between the dependent and

independent variable R square here so

you get the R square coefficient of

determination typ

of hypothesis testing type of er if you

want me to revise I do that but types

of all right

so point let me highlight the

point because

act or

line is is

positive

or and line is so this will be called as

negative eror line will give you

predicted values so predicted value

value so this will be a negative error

and over here actual values over here

predicted value somewhere over here so

this will be a positive error and this

will be a negative ER so Point actual

points are so actual minus predicted

will be a negative error and points

actual points are above predicted points

are below so that's

positive I formula let me write that

formula what we are going to do is we

have the Y over here and the Y Bar over

here that's Yus Y Bar which we call it

as a

deviation all right y- Y Bar so we'll do

Yus Y Bar Square which is called as TSS

total sum of

squ then we introduce that y predicted

value

YP okay and then we do YP minus y bar

which is called as the regression sum of

square and then Yus YP is called as

error actual minus is called as error so

that becomes error sum of stent so one

more time this is y and this is average

value of y and the difference between

them is called as deviation and square

it's total sum of

square and then predicted value in the

middle somewhere so YP minus y bar

that's the regression sum of square and

then this is error actual minus VOR

that's called as error sum of

square so why minus YP square and over

here YP minus y

s and total sum of square is like Yus Y

Bar

Square I

repeat y minus y bar that's called as

total sum of

square then YP minus y bar square is

called as regression sum of square

and lastly y minus YP that's called as

error sum of

square we didn't cover this errors

so I'll talk about this

errors okay apart from okay so let let's

cover this

errors or assumptions or errors or this

are assumptions and if this assumptions

are violated that becomes

error first one the name of the chapter

what's the name of the

chapter linear regression that means

there has to be a linear relationship

between Y and X the dependent variable Y

and the independent variable X be

linear relationship so there is no

linear relationship or there's a

nonlinear relationship nonlinear y = x²

or Y = root of x or y = x Cub this is

called as nonlinear y = a + BX that X is

like X to one so when it is X to one not

X squ or X Cube X to one that's called

as linear relationship so assumption

over here is that Y and X have a linear

line relationship not a curve

relationship Square Cube square roots if

you plot them in the Excel sheet you'll

get a curve you will not get a line so

assumption is the Y and the X have a

linear relationship error would be when

they have a nonlinear

relationship second variance variance

should be constant if the variance is

not constant then that is going to have

error which is called as

heteroscedasticity it's mentioned over

there variance not being constant is

called as

can someone number I

here 20 so page number 2011 so variance

constant to homoscedastic variance not

constant to

heteroscedastic now let me explain what

do you mean by variance not constant and

variance

constant stock market returns okay stock

market is it on every day same volatile

or on some days is more volatile some

days more volatile some days less

volatile that is stock market is heteros

stas if the volatility would have been

constant every day then it would have

been homoscedastic so if the volatility

that is Sigma is going to be constant

across the time then the stock market

would be homos gastic but it is not some

days the stock market is stable some

days like very very volatile so the

volatility is not constant hence stock

market is

hetas if the standard deviation or the

volatility is constant homoscedastic if

the standard deviation or the uh the

volatility is not constant then it's

called as

hetas the first one is linear

relationship obviously because the name

of the chapter is linear regression so

assumption is there is a linear

relationship between the Y and the

X if there is not a linear relationship

that is there is a nonlinear

relationship then that would be an

error second one if if there is the

error sorry the if the volatility is

constant then that's called as

homoscedastic if the volatility is not

constant that's called as

hetas listen to me carefully the

error first error and

second

Rel what does that mean

act second value

be chances are higher that's called as

positive dependency between the errors

okay this should not happen the

assumption is the error of one period

and the error of Next Period should not

have any dependency which is in simple

English called as errors should be

random the error

not happen like

that in every time the chance of six

is 1 upon six all right every time the

chance of six is 1 upon six so every

time whether six will be there or not

there that will be randomly determined

so basically similarly eror positive

second error be

posi that means errors are

dependent that means errors are not

random and

violation have they a name is there in

level

two called as serial correlation in

level two anyways but I think so

basically they require you that the

errors should be inde of each that

means chance of the second error coming

out to be positive or negative whether

first error is positive or negative

won't have any dependency on the second

error so three assumptions first Y and X

linear relationship not Square not Cube

not square root that is it's not a

nonlinear relationship second the

variance or the volatility is going to

be constant which is homoskedastic

variance not constant that error is

called as heteroskedastic and third

errors should be independent of each

other that means error should not have

any dependency

each so I repeat it one more time y = a

+ b x² y = a + b s root of x y = a + b x

CU not allowed y = a + b x is linear

ination which is allowed all

right then the standard deviation or the

variance of the volatility should be

constant which is called as

hosas not constant that's error hetas

and errors no dependency if there's a

dependency that's error which is called

as serial correlation in level

two and normally distribution

noral

distrib all right so that completes the

regression