Simple regression model

Question 1

SLR. 1

Answer 1

Population model is linear

Question 2

SLR. 2

Answer 2

Random sampling. X and Y are independent

Question 3

SLR. 3

Answer 3

Sample variation in x

Question 4

SLR. 4

Answer 4

Zero conditional mean E(u|x) That u is mean independent of x

Question 5

SLR. 5

Answer 5

Homoskedasticity - Var (u|x) = standard deviation squared

Question 6

SST

Answer 6

Squared difference of observation from the mean

Question 7

SSE

Answer 7

Squared difference of the fitted value from the mean

Question 8

SSR

Answer 8

Sum of all squared residuals - Squared difference between the observed value and the fitted value

Question 9

SST =

Answer 9

SSR + SSE

Question 10

Error

Answer 10

Deviation of the observed value from true value

Hence error term is unobservable

Question 11

Residual

Answer 11

Deviation between the observed value and the estimated value

Hence residual is observable

Question 12

Perfect collinearity

Answer 12

Explanatory variable lies exactly on the linear function

Question 13

ui hat =

completely broken down

Answer 13

yi − ybi = yi − β0 hat − β1xi hat

Question 14

F.O.C for B0

Answer 14

−2 Ε (yi − βc0 − βc1xi)= 0

Question 15

F.O.C for B1

Answer 15

−2 Ε xi (yi − βc0 − βc1xi) = 0

Question 16

What is the OLS estimator trying to do?

Answer 16

Minimise SSR by deriving it (first order condition)

Question 17

Finding Bo hat

Answer 17

Take -2 away
Eyi -nB0 -B1Exi = 0 
Divide everything by n
y bar - B0 - B1xbar =0 
B0= Y bar -B1 xbar
bar = mean

Question 18

Finding B1 hat

Answer 18

Take -2 away
We know B0 hat so plug that in
Left with E(Yi-Ybar) - Exi(BiXi -B1Xbar) = 0
EXi(Yi-Ybar) = B1E(xi(xi-xbar))

Question 19

Exi(yi-ybar) =

Answer 19

E(xi-xbar)(yi-ybar)

Question 20

Average value of x

Answer 20

1/nExi = Xbar

therefore Exi = nxbar

Question 21

E (Xi-Xbar)(Yi-Ybar) =

useful trick

Answer 21

= E Xi(Yi-Ybar) - Xbar E (Yi-Ybar)
= E Xi(Yi-Ybar) -XbarNYbar + XbarNYbar
= E Xi(Yi-Ybar)

Question 22

E(Xi-Xbar)^2 =

Answer 22

= E(Xi-Xbar)(Xi-Xbar)
= EXi(Xi-Xbar) - XbarE(Xi-Xbar)
= EXi(Xi-Xbar) - XbarNXbar + XbarNXbar
= EXi(Xi-Xbar)

Question 23

B1

Answer 23

E(Xi-Xbar)^2

Question 24

Only assumption needed to calculate the OLS estimator B1

Answer 24

E (Xi-Xbar)^2 > 0

Question 25

PRF

Answer 25

Population regression function - True relationship between x and y

Question 26

SRF

Answer 26

Sample regression function

Yhat = Bohat + B1xhat

Question 27

3 mathematical properties that hold in any sample of data. First two follow the two F.O.C

Answer 27

E Uihat = 0
E XiUihat = 0
(Xbar, Ybar) is always on the regression line

Question 28

Unbiasedness assumptions

Answer 28

E(B1hat) = B1 
E(B0hat) = B0

Question 29

Var(x)

Answer 29

1/1-n E (Xi-Xbar)^2

Question 30

ZCM assumptions in terms of Y
E(Y|X) =
Var(Y|X) =

Answer 30

B0 +B1x

σ^2

Question 31

y = B0 + B1x + u (Level - Level)

Answer 31

If you change x by one, we’d expect y to change by B1

Question 32

Ln(y) = B0 + B1x + u (Log - Level)

Answer 32

If we change x by one unit, we’d expect our y variable to change by 100 x B1 %

Question 33

Y= B0 + B1Ln(x) + u

Answer 33

If we increase x by 1% we would expect y to increase by B1/100 units of y

Question 34

Ln(y) = B0 + B1Ln(x) + u

Answer 34

If we change x by 1%, we would expect y to change by B1%