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1 Gauss-Markov Theorem

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### Transcript of ECONF241 GaussMarkov Theorem

• 1

Gauss-Markov Theorem

• 2

1. Yi = 1 + 2Xi + ui 2. E(u i) = 0 E(Yi) = 1 +2Xi 3. cov(ui, Xi) =0

4. cov(u i, u j) = cov(Yi,Yj) = 0

5. var(ui) = u2 < == > var(Yi) = y

2

6. XK Xm, Xi not constant for every observation

7. ui~N(0, 2) Yi~N(1+2Xi,

2)

Properties of the Linear Regression Model

• 3 The Sampling Distributions of the

Regression Estimators

The population parameters 1 and 2are

unknown population constants.

The formulas that produce the sample estimates

of 1 and 2 are called the estimators of 1 and

2.

Theoretical: Yi = 1 + 2Xi + ui

ii XY 2

1 Estimated:

• 4

Estimators are Random Variables

If the least squares estimators 1 and 2 are random variables, then what are their means, variances, covariances and probability distributions?

^ ^

The estimators, 1 and 2 are random variables because they are different from sample to sample

^ ^

• 5

The Expected Values of 0 and 1

The least squares estimators in the simple regression

2 = nXiYi - XiYi

nXi -(Xi) 2 2

= xiyi

xi2

^

^ ^

where y = yi / n and x = xi / n

1 = Y - 2X ^ ^

• 6 Substitute Yi = 1 + 2Xi + u i Into 1 formula to get:

^

22 )(

)21()21(2

ii

iiiiii

XXn

uXXuXXn

22

22

)(

))(21()2(2

ii

iiiiiiii

XXn

uXXXnuXnXnXn

22

22

)(

))(222

ii

iiiiii

XXn

uXXuXnXn

22

22

)(

)(])([22

ii

iiiiii

XXn

uXuXnXXn

• 7

2 = 2 + nxi ui - xi ui

nxi -(xi) 2 2

^

The mean of 2 is:

E(2) = 2 + nxiE(ui)- xi E(ui)

nxi -(xi) 2 2

^

^

Since E(ui) = 0, then E(2) = 2 . ^

=0

• 8 An Unbiased Estimator

Unbiasedness: The mean of the distribution of sample

estimates is equal to the parameter to be estimated.

The result E(2) = 2 means that

the distribution of 2 is centered at 2.

^

^

Since the distribution of 2

is centered at the true 2 ,we say that

2 is an unbiased estimator of 2.

^

^

• 9

The unbiasedness result on the

previous slide assumes that we

are using the correct model

Wrong Model Specification

For example:

Y = 1 + 2X1 + (3X2 +u) = 3X2 +u

E(ui) 0

If the model is of the wrong form

or is missing important variables,

then E(ui)= 0, then E(2) = 2 ^

• 10

Unbiased Estimator of the Intercept

In a similar manner, the estimator 1

of the intercept or constant term can be

shown to be an unbiased estimator of 1 when the model is correctly specified.

^

E(1) = 1 ^

• 11

^

u2 ^

Var(i) ^

• 12

2 = (Xi X )Yi Y )

xi x ) 2

= xiyi

xi2

= 590.12

92.5 = 6.38

1 = Y - 1X = 169.4 (6.38*10.35) = 103.4

Y X y x x2 xy X2

140 5 -29.40 -5.35 28.62 157.29 25

157 9 -12.40 -1.35 1.82 16.74 81

205 13 35.60 2.65 7.02 94.34 169

162 10 -7.40 -0.35 0.12 2.59 100

174 11 4.6 0.65 0.42 2.99 121

165 9 -4.4 -1.35 1.82 5.94 81

Sum 3388 207 0 0 92.5 590.2 2235

mean 169.4 10.35

• 13

Variance of 2

Se(2)= (8.50)2/92.55 = 0.7809 = 0.8836

^

2 is a function of the Yi values, but

var(2) does not involve Yi directly.

^

^

^

Given that both Yi and ui have variance 2,

the variance of the estimator 1 is:

xi x

2

2 var(2) = =

xi 2

2

^ ^ ^

^

• 14

Variance of 1 ^

Given 1 = Y 2X ^ ^

the variance of the estimator 1 is:

var(1)= 2

n x i x 2

x i 2

nxi 2

x i 2

2

^

^

Se(1)= (8.50)2(2235/20(92.55)) = 87.238 = 9.34 ^

• 15

If x = 0, slope can change without affecting

the variance.

Covariance of 1 and 2 ^ ^

cov(1,2)= 2

xi 2

x

x t x 2

x = 2

^ ^

• 16 Estimating the variance

of the error term, 2

ui ^

i =1

T 2

n 2 = ^

is an unbiased estimator of 2 ^

Smaller variance higher efficiency

N variances

Var(ui|Xi) variances of betas

ui = Yi 1 2Xi ^ ^ ^

• 17

1. 2: When uncertainty about Yt values

uncertainty about 1, 2 and their relationship.

2. The more spread out the Xi values,

the more confidence in 1, 2

3. The larger the sample size, N,

the smaller the variances and covariances.

4. When the (squared) Xt values are far from zero (in

either +/- direction), The larger variance of i.

5. Changing the slope, 2, has no effect on the intercept,

1, when the sample mean is zero.

But if sample mean is positive, the covariance between

1and 2 will be negative, and vice versa.

What factors determine variance and covariance ?

^ ^

^ ^

^

^

^

^ ^

• 18

Gauss-Markov Theorem

Note: Gauss-Markov Theorem doesnt

apply to non-linear estimators

Given the assumptions of classical linear

regression model, the ordinary least

squares (OLS) estimators 0 and 1

are the best linear unbiased estimators

(BLUE) of 1 and 2. This means that 1 and

2 have the smallest variance of all linear

unbiased estimator of 1 and 2.

^ ^

^ ^

• 19

B: Best (minimum variance, efficiency)

L: Linear (linear in Yis)

U: Unbiased ( E( ) = k )

E: Estimator (a formula, a functional form)

k

OLS estimators are BLUE.

Efficiency (B) and unbiasedness (U) are

desirable properties of estimators.

3. The Gauss-Markov Theorem

• 20

Prob.

(1)

True value

of 2

^

Unbiased

E(2)=

E(2)

^

^

E(2)>

Biased overestimate

E(2)

^

^

E(2)

• 21 Probability Distribution

of Least Squares Estimators

1~ N 1 , nx i2

u2 x i

2

^

2 ~ N 2 , x i2 u

2 ^

• 22

: An estimator is more efficient

than another if it has a smaller variance.

Efficiency

Prob.

(b1)

True value of 2 2

1 2

1 is more efficient

than 2.

E(2) ^

• 23 Consistency:

k is a consistent estimator of k means that as the

sample size gets larger, the k will become more

accurately.

^

^

Prob.

(1)

True value of 2

N=5

0

N=100

N=500

E(1) ^

^

• 24 Summary of BLUE estimators

Mean

E(1)=1 E(2)=2 and ^ ^

Variance

nxi 2

x i 2

Var(1)= 2 and

xi 2

2 Var(2)=

^ ^

Standard error or standard deviation of estimator K

Se(k) = var(k) ^ ^

• 25

Estimated Error Variance

ui ^

i =1

T 2

n-K-1 u

= ^

Standard Error of Regression (Estimate), SEE

ui ^

i =1

T 2

n K-1

u = ^ u =

^ K # of independent

excludes

the Constant term

^

E(u2) = u

2