1 we will now look at the properties of the ols regression estimators with the assumptions of model...
TRANSCRIPT
1
We will now look at the properties of the OLS regression estimators with the assumptions of Model B. We will do this within the context of the simple regression model. We will start by demonstrating unbiasedness.
MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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ii uaXX
YYXXb 222
iii uXY 21
We saw in Chapter 2 that the slope coefficient can be decomposed into the true value plus a weighted linear combination of the values of the disturbance term in the sample, where the weights depend on the observations on X.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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ii uaXX
YYXXb 222
iii uXY 21
2XX
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j
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ii uaXX
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iii uXY 21
2XX
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iiii uaEuaEbE 222
We now take expectations. b2 is just a constant, so it is unaffected.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
We have now used the first expectation rule to rewrite the expectation of the linear combination as the sum of the expectations of its components.
iinnnnii uaEuaEuaEuauaEuaE ...... 1111
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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iii uXY 21
2XX
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iiii uaEuaEbE 222
In Model A, the values of X were nonstochastic. This meant that the ai terms were also nonstochastic and could therefore be taken out of the expectations as factors. E(ui) = 0 for all i, and hence we proved unbiasedness.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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Model A
222 ii uEabE
0 iiii uEauaE
iii uXY 21
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j
ii
iiii uaEuaEbE 222
We cannot do this with Model B because we are assuming that the values of X are generated randomly (from a defined population).
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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Model A
222 ii uEabE
0 iiii uEauaE
iii uXY 21
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iiii uaEuaEbE 222
Instead we appeal to Assumption B.7. We saw in the Review chapter that if X and Y are two independent random variables, the expectation of the product of functions of them can be decomposed as the product of the expectations of the functions.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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Model B
iii uXY 21
2XX
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iiii uaEuaEbE 222
0 iiii uEaEuaE
ii uYgaXf )( ,)(
)()()()( YgEXfEYgXfE
Under Assumption B.7, ui is distributed independently of every value of X in the sample. It is therefore distributed independently of ai. So if X and u are independent, we can make use of the decomposition.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
0 iiii uEaEuaE
ii uYgaXf )( ,)(
)()()()( YgEXfEYgXfE
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ii uaXX
YYXXb 222
Model B
iii uXY 21
2XX
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j
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iiii uaEuaEbE 222
222 ii uEaEbE
Since E(ui) = 0 for all i, under Assumption B.4, we have proved unbiasedness, assuming E(ai) exists. For this to be the case, there must be some variation in X in the sample (Assumption B.3). Otherwise the denominator of the expression for ai would be zero.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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Model B
iii uXY 21
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iiii uaEuaEbE 222
0 iiii uEaEuaE
The next property, efficiency, we will take for granted. The Gauss–Markov theorem assures that the OLS estimators are BLUE (best linear unbiased estimators), provided that the regression model assumptions are valid.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
2222
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uuXX
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YYXXb
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iii uXY 21
We will prove consistency. We have decomposed the limiting value of the estimator of the slope coefficient into the true value and the limiting value of the error term.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
22
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222
var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
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ii
2222
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uuXX
XX
YYXXb
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iii uXY 21
We would now like to use the plim quotient rule. The plim of a quotient is the plim of the numerator divided by the plim of the denominator, provided that both of these limits exist.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
BA
BA
plim plim
plim
if A and B have probability limits
and plim B is not 0.
22
22
222
var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
i
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i
ii
22
22
222
var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
i
ii
i
ii
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
However, as the expression stands, the numerator and the denominator of the error term do not have limits. The denominator increases indefinitely and the numerator does not converge on a limit.
BA
BA
plim plim
plim
if A and B have probability limits
and plim B is not 0.
22
22
222
var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
i
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i
ii
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
To deal with this problem, we divide both the numerator and the denominator by n.
BA
BA
plim plim
plim
if A and B have probability limits
and plim B is not 0.
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22
222
var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
i
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ii
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
It can be shown that the limit of the numerator is the covariance of X and u and the limit of the denominator is the variance of X.
0,cov1
plim uXuuXXn ii
XXXn i var1
plim 2
Under Assumption B.7, X and u are independent. Hence the covariance of X and u is zero (see the Review chapter).
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
XXXn i var1
plim 2
0,cov1
plim uXuuXXn ii
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222
var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
i
ii
i
ii
Thus we demonstrate that b2 is a consistent estimator of b2, provided that the regression model assumptions are valid.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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var,cov
1
1
plim
plim plim
XuX
XXn
uuXXn
XX
uuXXb
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ii
2222
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uuXX
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YYXXb
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Finally, a note on Assumption B.8, that the disturbance term has a normal distribution. The justification is that it is reasonable to suppose that the disturbance term is jointly generated by a number of minor random factors.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
B.1 The model is linear in parameters and correctly specified.
Y = b1 + b2X2 + … + bkXk + u
B.2 The values of the regressors are drawn randomly from fixed populations.
B.3 There does not exist an exact linear relationship among the regressors.
B.4 The disturbance term has zero expectation.
B.5 The disturbance term is homoscedastic.
B.6 The values of the disturbance term have independent distributions.
B.7 The disturbance term is distributed independently of the regressors.
B.8 The disturbance term has a normal distribution.
A central limit theorem states that the combination of these factors should approximately have a normal distribution, even if the individual factors do not.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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B.8 The disturbance term has a normal distribution.
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If the disturbance term has a normal distribution, the regression coefficients also have normal distributions. This follows from the fact that a linear combination of normal distributions is also normal.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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B.8 The disturbance term has a normal distribution.
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What happens if we have reason to believe that the assumption is not valid? The central limit theorem comes into the frame a second time.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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B.8 The disturbance term has a normal distribution.
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The random component of a regression coefficient is a linear combination of the values of the disturbance term in the sample.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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B.8 The disturbance term has a normal distribution.
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By a central limit theorem, it follows that the combination will have an approximately normal distribution, even if the individual values of the disturbance term do not, provided that the sample is large enough.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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B.8 The disturbance term has a normal distribution.
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Hence asymptotically (in large samples) it ought to be safe to assume that the regression coefficients have normal distributions, even if Assumption B.8 is invalid, provided that the other regression model assumptions are satisfied.
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MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS
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B.8 The disturbance term has a normal distribution.
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2013.08.04
Copyright Christopher Dougherty 2013.
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