asymptotic and finite-sample distributions of the iv estimator 1 the asymptotic variance of the iv...

ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR

1

The asymptotic variance of the IV estimator is given by the expression shown. It is the expression for the variance of the OLS estimator, multiplied by the square of the reciprocal of the correlation between X and Z.

uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

2

What does this mean? We have seen that the distribution of the IV estimator degenerates to a spike. So how can it have an asymptotic variance?


uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

3

The contradiction has been caused by compressing several ideas together. We will have to unpick them, taking several small steps.


uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

4

The application of a central limit theorem (CLT) underlies the assertion. To use a CLT, we must first show that a variable has a nondegenerate limiting distribution. The CLT will then show that, under appropriate conditions, this limiting distribution is normal.


5

We cannot apply a CLT to b2IV directly, because it does not have a nondegenerate limiting

distribution. The expression for the variance may be rewritten as shown. MSD(X) is the mean square deviation of X.


uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

6

By a law of large numbers, the MSD tends to the population variance of X and so has a well-defined limit.


uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

7

2,

2

2,2

2

2,

2

22IV

2

1MSD

11

1var IV

2

ZX

u

ZXi

u

ZXi

ub

rXn

rXXn

n

rXXb

The variance of b2IV is inversely proportional to n, and so tends to zero. This is the reason

that the distribution of b2IV collapses to a spike.


uXY 21

XXZZ

YYZZb

ii

iiIV2 2

IV2 plim b

8

We can deal with the diminishing-variance problem by considering √n b2IV instead of b2

IV. This has the variance shown, which is stable.

2,

2IV2

1MSD

varZX

u

rXbn


2,

22 1

MSD IV2

ZX

ub rXn

9

However, √n b2IV still does not have a limiting distribution because its mean increases with

n.

2,

2IV2

1MSD

varZX

u

rXbn


2,

22 1

MSD IV2

ZX

ub rXn

nnbnE as 2IV2

2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

10

So instead, consider √n (b2IV – b2). Since b2

IV tends to b2 as the sample size becomes large, this does have a limiting distribution with zero mean and stable variance.


11

Under conditions that are usually satisfied in regressions using cross-sectional data, it can then be shown that we can apply a central limit theorem and demonstrate that √n (b2

IV – b2) has the limiting normal distribution shown.


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

12

The arrow with a d over it means ‘has limiting distribution’.


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

13

Having established this, we can now start working backwards and say that, for sufficiently large samples, as an approximation, (b2

IV – b2) has the distribution shown. (~ means ‘is distributed as’.)


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

2

2

2IV2

1MSD

,0~XZ

u

rXnNb

14

We can then say that, as an approximation, for sufficiently large samples, b2IV is distributed

as shown, and use this assertion as justification for performing the usual tests. This is what was intended by equation (8.50).


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

2

2

2IV2

1MSD

,0~XZ

u

rXnNb

2

,

2

2IV2

1MSD

,~ZX

u

rXnNb

15

Of course, we need to be more precise about what we mean by a ‘sufficiently large’ sample, and ‘as an approximation’.


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

2

2

2IV2

1MSD

,0~XZ

u

rXnNb

2

,

2

2IV2

1MSD

,~ZX

u

rXnNb

16

We cannot do this mathematically. This was why we resorted to asymptotic analysis in the first place.


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

2

2

2IV2

1MSD

,0~XZ

u

rXnNb

2

,

2

2IV2

1MSD

,~ZX

u

rXnNb

17

Instead, the usual procedure is to set up a Monte Carlo experiment using a model appropriate to the context. The answers will depend on the nature of the model, the correlation between X and u, and the correlation between X and Z.


2,

22 1

MSD IV2

ZX

ub rXn

22

2

2IV2

1,0

XZX

ud

rNbn

2

2

2IV2

1MSD

,0~XZ

u

rXnNb

2

,

2

2IV2

1MSD

,~ZX

u

rXnNb

18

Suppose that we have the model shown and the observations on Z, V, and u are drawn independently from a normal distribution with mean zero and unit variance. We will think of Z and V as variables and of u as a disturbance term in the model. l1 and l2 are constants.


uVZX 21 uXY 21

19

By construction, X is not independent of u and so Assumption B.7 is violated when we fit the regression of Y on X. OLS will yield inconsistent estimates and the standard errors and other diagnostics will be invalid.


uVZX 21 uXY 21

20

Z is correlated with X, but independent of u, and so can serve as an instrument. (V is included as a component of X in order to provide some variation in X not connected with either the instrument or the disturbance term.)


uVZX 21 uXY 21

21

We will set b1 = 10, b2 = 5, l1 = 0.5, and l2 = 2.0.


uVZX 21

uXY 510 uVZX 0.25.0

uXY 21

22

The diagram shows the distributions of the OLS and IV estimators of b2 for n = 25 and n = 100, for 10 million samples in both cases. Given the information above, it is easy to verify that plim b2

OLS = 5.19. Of course, plim b2IV = 5.00


0

5

10

4 5 6

OLS, n = 25

OLS, n = 100

IV, n = 100

IV, n = 25

uXY 510 uVZX 0.25.0

plim OLS = 5.19plim IV = 5.00

5.19

10 million samples

23

The IV estimator has a greater variance than the OLS estimator and for n = 25 one might prefer the latter. It is biased, but the smaller variance could make it superior, using some criterion such as the mean square error. For n = 100, the IV estimator looks better.


0

5

10

4 5 6

OLS, n = 25

OLS, n = 100

IV, n = 100

IV, n = 25

uXY 510 uVZX 0.25.0


5.19

10 million samples

24

This diagram adds the distribution for n = 3,200. Both estimators are tending to the predicted limits (the IV estimator more slowly than the OLS, because it has a larger variance). Here the IV estimator is definitely superior.

0

20

40

60

4 5 6

IV, n = 3,200

OLS, n = 3,200


uXY 510 uVZX 0.25.0


5.19

10 million samples

25

This diagram shows the distribution of √n (b2IV – b2) for n = 25, 100, and 3,200. It also

shows, as the dashed red line, the limiting normal distribution predicted by the central limit theorem.


0

0.1

0.2

-6 -4 -2 0 2 4 6

n = 25

n = 100

n = 3,200limiting normal distribution

uXY 510 uVZX 0.25.0

10 million samples

26

It can be seen that the distribution for n = 3,200 is very close to the limiting normal distribution and so inference would be safe with samples of this magnitude.


0

0.1

0.2

-6 -4 -2 0 2 4 6

n = 25

n = 100


uXY 510 uVZX 0.25.0

10 million samples

27

However, the distributions for n = 25 and n = 100 are distinctly non-normal. The distribution for n = 25 has fat tails. This means that if you performed a t test, the probability of suffering a Type I error will be much higher than the nominal significance level of the test


0

0.1

0.2

-6 -4 -2 0 2 4 6

n = 25

n = 100


uXY 510 uVZX 0.25.0

10 million samples

28

The distribution for n = 100 is better, in that the right tail is close to that of the normal distribution, but the left tail is much too fat and, as for n = 25, would give rise to excess instances of Type I error.


0

0.1

0.2

-6 -4 -2 0 2 4 6

n = 25

n = 100


uXY 510 uVZX 0.25.0

10 million samples

29

The distortion for small sample sizes is partly attributable to the low correlation between X and Z, 0.22.


0

0.1

0.2

-6 -4 -2 0 2 4 6

n = 25

n = 100


uXY 510 uVZX 0.25.0

10 million samples

30

Unfortunately, low correlations (‘weak instruments’) are common in IV estimation. It is difficult to find an instrument that is correlated with X but not the disturbance term. Indeed, it is often difficult to find any credible instrument at all.


0

0.1

0.2

-6 -4 -2 0 2 4 6

n = 25

n = 100


uXY 510 uVZX 0.25.0

10 million samples

Copyright Christopher Dougherty 2013.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2013.08.21

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asymptotic and finite-sample distributions of the iv estimator 1 the asymptotic variance of the iv...

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