Linear regression

[Javad05]

Hello
in a simple linear regression we have
Y=b0 +b1X +ε where ε is a random variable
but when we use a sample
we have
Yi = b0 + b1Xi + εi, i = 1, …, n where εi are random variable too
I’m a little bit lost ε is a random variable and its occurrence εi are random variable too?
and why E(ε)=0 would imply E(εi) for each i?

 

[S2000magician]

Suppose that you have three data points:

• (1, 1)
• (2, 6)
• (3, 5)

The regression equation is:
y = 2.0x + 0.0 + ε
In this case:

• ε1 = −1.0
• ε2 = 2.0
• ε3 = −1.0

 

[Harrogath]

Yes, as S2000 says, those ” i ” means the data observations, they can be temporal data or cross-sectional data. For example, set X variable as consumption. So Xi is the ” i-th ” observation of the sample you got about consumption:
X1 = 50 dollar consumption of Harrogath on August, 2015
X2 = 75 dollar consumption of S2000Magician on August, 2015
…
Xn = 62 dollar consumption of Javad05 on August, 2015
This is cross-sectional data of X variable.
Time-series data of variable X would be:
X1 = 62 dollar consumption of Javad05 on August, 2015
X2 = 70 dollar consumption of Javad05 on September, 2015
…
Xn = 128 dollar consumption of Javad05 on January, 2038
Regards

 

[JZA]

S2000magician wrote:
Suppose that you have three data points:

• (1, 1)
• (2, 6)
• (3, 5)

The regression equation is:
y = 2.0x + 0.0 + ε
In this case:

• ε1 = 1.0
• ε2 = −2.0
• ε3 = 1.0

Based on the equation, shouldn’t the signs be flipped on the above ε’s?
(1) = 2.0(1) + 0.0 + ε1
ε1=-1
(6) = 2.0(2) + 0.0 + ε2
ε2=+2
…

 

[S2000magician]

JZA wrote:

S2000magician wrote:
Suppose that you have three data points:

• (1, 1)
• (2, 6)
• (3, 5)

The regression equation is:
y = 2.0x + 0.0 + ε
In this case:

• ε1 = 1.0
• ε2 = −2.0
• ε3 = 1.0

Based on the equation, shouldn’t the signs be flipped on the above ε’s?
(1) = 2.0(1) + 0.0 + ε1
ε1=-1
(6) = 2.0(2) + 0.0 + ε2
ε2=+2
…

Absolutely correct.
Mea culpa.
I’ve corrected it.

 

[Javad05]

Thank you guys
but I was not asking about that !
if you look to the assumption of linear regression (6)
-The variance of the error term is the same for all observations: E(ε2i)=σ2ε , i = 1, …, n.
-The error term, ε, is uncorrelated across observations. Consequently, E(εiεj) = 0 for all i not equal to j

you can see that εi are not only value but random variable on their own
so do I have to dwell on that or move on
thank you

 

[S2000magician]

The outcome of the roll of a die is a random variable, but once you roll the die, it takes on a specific value.
So it goes with linear regression: the error term is a random variable, but once you run the regression with a specific data set, each random error term then takes on a specific value.