christopher dougherty ec220 - introduction to econometrics (chapter 2) slideshow: testing a...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 2)Slideshow: testing a hypothesis relating to a regression coefficient

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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 2). [Teaching Resource]

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Model: Y = 1 + 2X + u

Null hypothesis:

Alternative hypothesis

TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT

0220 : H0221 : H

This sequence describes the testing of a hypothesis at the 5% and 1% significance levels. It also defines what is meant by a Type I error.

Model: Y = 1 + 2X + u

Null hypothesis:



0220 : H

We will suppose that we have the standard simple regression model and that we wish to

test the hypothesis H0 that the slope coefficient is equal to some value 20.

2

0221 : H

Model: Y = 1 + 2X + u

Null hypothesis:



0220 : H

The hypothesis being tested is described as the null hypothesis. We test it against the

alternative hypothesis H1, which is simply that 2 is not equal to 20.

3

0221 : H

Model: Y = 1 + 2X + u

Null hypothesis:


Example model: p = 1 + 2w + u

Null hypothesis:

Alternative hypothesis:

4


As an illustration, we will consider a model relating price inflation to wage inflation. p is the rate of growth of prices and w is the rate of growth of wages.

0220 : H

0.1: 20 H

0.1: 21 H

0221 : H

Model: Y = 1 + 2X + u

Null hypothesis:


Example model: p = 1 + 2w + u

Null hypothesis:

Alternative hypothesis:


0220 : H

0.1: 20 H

We will test the hypothesis that the rate of price inflation is equal to the rate of wage

inflation. The null hypothesis is therefore H0: 2 = 1.0. (We should also test 1 = 0.)

5

0221 : H

0.1: 21 H

6


probability densityfunction of b2

Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)

b21.0 1.10.90.80.70.6 1.2 1.3 1.4

If this null hypothesis is true, the regression coefficient b2 will have a distribution with mean 1.0. To draw the distribution, we must know its standard deviation.

We will assume that we know the standard deviation and that it is equal to 0.1. This is a very unrealistic assumption. In practice you have to estimate it.

7


1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


8


Here is the distribution of b2 for the general case. Again, for the time being we are assuming that we know its standard deviation (sd).

Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)


b2

0

2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0

9


Suppose that we have a sample of data for the price inflation/wage inflation model and the estimate of the slope coefficient, b2, is 0.9. Would this be evidence against the null

hypothesis 2= 1.0?

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


10


No, it is not. It is lower than 1.0, but, because there is a disturbance term in the model, we would not expect to get an estimate exactly equal to 1.0.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


11


If the null hypothesis is true, we should frequently get estimates as low as 0.9, so there is no real conflict.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


12


In terms of the general case, the estimate is one standard deviation below the hypothetical value.


b2


0


13


If the null hypothesis is true, the probability of getting an estimate one standard deviation or more above or below the mean is 31.7%.


b2


0


14


Now suppose that in the price inflation/wage inflation model we get an estimate of 1.4. This clearly conflicts with the null hypothesis.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


15


1.4 is four standard deviations above the hypothetical mean and the chance of getting such an extreme estimate is only 0.006%. We would reject the null hypothesis.


b2


0


16


Now suppose, with the price inflation/wage inflation model, that the sample estimate is 0.77. This is an awkward result.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


17


Under the null hypothesis, the estimate is between 2 and 3 standard deviations below the mean.


b2


0


18


There are two possibilities. One is that the null hypothesis is true, and we have a slightly freaky estimate.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


19


The other is that the null hypothesis is false. The rate of price inflation is not equal to the rate of wage inflation.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


20


The usual procedure for making decisions is to reject the null hypothesis if it implies that the probability of getting such an extreme estimate is less than some (small) probability p.


b2


0


21


For example, we might choose to reject the null hypothesis if it implies that the probability of getting such an extreme estimate is less than 0.05 (5%).

2.5%2.5%


b2


0


22


According to this decision rule, we would reject the null hypothesis if the estimate fell in the upper or lower 2.5% tails.

2.5%2.5%


b2


0


2.5% 2.5%

23


If we apply this decision rule to the price inflation/wage inflation model, the first estimate of

2 would not lead to a rejection of the null hypothesis.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


2.5% 2.5%

24


The second definitely would.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


2.5% 2.5%

25


The third also would lead to rejection..

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


26


The 2.5% tails of a normal distribution always begin 1.96 standard deviations from its mean.

2.5%2.5%


b2


0

2+1.96sd2-1.96sd 202-sd 2+sd00 00


2.5%2.5%

Decision rule (5% significance level):reject

(1) if (2) ifprobability densityfunction of b2

b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

0220 :H

Thus we would reject H0 if the estimate were 1.96 standard deviations (or more) above or below the hypothetical mean.

27


2.5%2.5%


(1) if (2) if

(1) if (2) if


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

s.d. 96.1022 b s.d. 96.10

22 b

0220 :H

We would reject H0 if the difference between the sample estimate and hypothetical value were more than 1.96 standard deviations.

28


2.5%2.5%


(1) if (2) if

(1) if (2) if

(1) if (2) if


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

s.d. 96.1022 b s.d. 96.10

22 b

96.1s.d./)( 022 b 96.1s.d./)( 0

22 b

0220 :H

We would reject H0 if the difference, expressed in terms of standard deviations, were more than 1.96 in absolute terms (positive or negative).

29


2.5%2.5%


(1) if (2) if

(1) if (2) if

(1) if (2) if

s.d.

022

b

z


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

s.d. 96.1022 b s.d. 96.10

22 b

96.1s.d./)( 022 b 96.1s.d./)( 0

22 b

0220 :H

We will denote the difference, expressed in terms of standard deviations, as z.

30

31


Then the decision rule is to reject the null hypothesis if z is greater than 1.96 in absolute terms.

2.5%2.5%


(1) if (2) if

(1) if (2) if

(1) if (2) if

(1) if z > 1.96 (2) if z < -1.96

s.d.

022

b

z


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

s.d. 96.1022 b s.d. 96.10

22 b

96.1s.d./)( 022 b 96.1s.d./)( 0

22 b

0220 :H


2.5%2.5%


(1) if (2) if

(1) if z > 1.96 (2) if z < -1.96

s.d.

022

b

z


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

0220 :H

The range of values of b2 that do not lead to the rejection of the null hypothesis is known as the acceptance region.

32

acceptance region for b2:

s.d. 96.1s.d. 96.1 022

02 b


2.5%2.5%


(1) if (2) if

(1) if z > 1.96 (2) if z < -1.96

s.d.

022

b

z


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

s.d. 96.1022 b s.d. 96.10

22 b

0220 :H


s.d. 96.1s.d. 96.1 022

02 b

96.196.1 z

The limiting values of z for the acceptance region are 1.96 and -1.96 (for a 5% significance test).

33

2.5% 2.5%


1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


(1) if (2) ifs.d. 96.1022 b s.d. 96.10

22 b

0.1: 20 H

We will look again at the decision process in terms of the price inflation/wage inflation example. The null hypothesis is that the slope coefficient is equal to 1.0.

34

2.5% 2.5%


1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


(1) if (2) if

(1) if (2) if

s.d. 96.1022 b s.d. 96.10

22 b

0.1 96.10.12 b 0.1 96.10.12 b

0.1: 20 H

We are assuming that we know the standard deviation and that it is equal to 0.1.

35

2.5% 2.5%

36


The acceptance region for b2 is therefore the interval 0.804 to 1.196. A sample estimate in this range will not lead to the rejection of the null hypothesis.

1.0 1.10.90.80.70.6 1.2 1.3 1.4


b2


(1) if (2) if

(1) if (2) if

(1) if (2) if

s.d. 96.1022 b s.d. 96.10

22 b

0.1 96.10.12 b 0.1 96.10.12 b

196.12 b 804.02 b

0.1: 20 H

acceptance region for b2:196.1804.0 2 b

37


Rejection of the null hypothesis when it is in fact true is described as a Type I error.

2.5%2.5%

Type I error: rejection of H0 when it is in fact true.

reject reject


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

acceptance region for b20220 :H 0

220 :H


2.5%2.5%

reject reject


b22+1.96sd2-1.96sd 202-sd 2+sd00 00


220 :H

With the present test, if the null hypothesis is true, a Type I error will occur 5% of the time because 5% of the time we will get estimates in the upper or lower 2.5% tails.

38


Probability of Type I error: in this case, 5%


2.5%2.5%

reject reject


b22+1.96sd2-1.96sd 202-sd 2+sd00 00


220 :H

The significance level of a test is defined to be the probability of making a Type I error if the null hypothesis is true.

39



Significance level of the test is 5%.


2.5%2.5%

reject reject


b22+1.96sd2-1.96sd 202-sd 2+sd00 00


220 :H




We can of course reduce the risk of making a Type I error by reducing the size of the rejection region.

40


2.5%2.5%

reject reject


b22+1.96sd2-1.96sd 202-sd 2+sd00 00


220 :H




For example, we could change the decision rule to “reject the null hypothesis if it implies that the probability of getting the sample estimate is less than 0.01 (1%)”.

41

42


The rejection region now becomes the upper and lower 0.5% tails

2.5%2.5%


b22+1.96sd2-1.96sd 202-sd 2+sd00 00

reject 0220 :H acceptance region for b2 reject 0

220 :H


(1) if (2) if

(1) if z > 2.58 (2) if z < -2.58

43


The 0.5% tails of a normal distribution start 2.58 standard deviations from the mean, so we now reject the null hypothesis if z is greater than 2.58, in absolute terms.

0.5%0.5%


b22+2.58sd2-2.58sd 202-sd 2+sd00 00

s.d.

022

b

z

s.d. 58.2022 b s.d. 58.20

22 b

0220 :H


s.d. 58.2s.d. 58.2 022

02 b

58.258.2 z


0.5%0.5%


b22+2.58sd2-2.58sd 202-sd 2+sd00 00




reject 0220 :H acceptance region for b2 reject 0

220 :H

Since the probability of making a Type I error, if the null hypothesis is true, is now only 1%, the test is said to be a 1% significance test.

44

45


In the case of the price inflation/wage inflation model, given that the standard deviation is 0.1, the 0.5% tails start 0.258 above and below the mean, that is, at 0.742 and 1.258.

1.0 1.10.90.80.70.6 1.2 1.3 1.4

0.5%0.5%


b2


(1) if (2) if

(1) if (2) if

(1) if (2) if

s.d. 58.2022 b s.d. 58.20

22 b

0.1 58.20.12 b 0.1 58.20.12 b


258.12 b 742.02 b

0.1: 20 H


1.0 1.10.90.80.70.6 1.2 1.3 1.4

0.5%0.5%


b2


(1) if (2) if

(1) if (2) if

(1) if (2) if

s.d. 58.2022 b s.d. 58.20

22 b

0.1 58.20.12 b 0.1 58.20.12 b


258.12 b 742.02 b

0.1: 20 H

The acceptance region for b2 is therefore the interval 0.742 to 1.258. Because it is wider than that for the 5% test, there is less risk of making a Type I error, if the null hypothesis is true.

46

47


This diagram compares the decision-making processes for the 5% and 1% tests. Note that if you reject H0 at the 1% level, you must also reject it at the 5% level.

0.5%0.5%

5% and 1% acceptance regions compared

5%: -1.96 < z < 1.96

1%: -2.58 < z < 2.58

5% level

1% level


b22 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0

s.d.

022

b

z

s.d. 58.2s.d. 58.2 022

02 b

s.d. 96.1s.d. 96.1 022

02 b


0.5%0.5%

5% and 1% acceptance regions compared

5%: -1.96 < z < 1.96

1%: -2.58 < z < 2.58

5% level

1% level


b22 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0

s.d.

022

b

z

s.d. 58.2s.d. 58.2 022

02 b

s.d. 96.1s.d. 96.1 022

02 b

Note also that if b2 lies within the acceptance region for the 5% test, it must also fall within it for the 1% test.

48


The diagram summarizes the possible decisions for the 5% and 1% tests, for both the general case and the price inflation/wage inflation example.

49

Reject H0 at 1% level (and also 5% level)

Reject H0 at 5% level but not 1% level



Do not reject H0 at 5% level (or at 1% level) 1.000

0.804

0.742

1.196

1.258

Price inflation/wage inflation example

General case Decision

s.d. 58.202

02

s.d. 96.102

s.d. 96.102

s.d. 58.202







0.804

0.742

1.196

1.258



s.d. 58.202

02

s.d. 96.102

s.d. 96.102

s.d. 58.202

The middle of the diagram indicates what you would report. You would not report the phrases in parentheses.

50







0.804

0.742

1.196

1.258



s.d. 58.202

02

s.d. 96.102

s.d. 96.102

s.d. 58.202

If you can reject H0 at the 1% level, it automatically follows that you can reject it at the 5% level and there is no need to say so. Indeed, you would look ignorant if you did.

51







0.804

0.742

1.196

1.258



s.d. 58.202

02

s.d. 96.102

s.d. 96.102

s.d. 58.202

Likewise, if you cannot reject H0 at the 5% level, that is all you should say. It automatically follows that you cannot reject it at the 1% level and you would look ignorant if you said so.

52







0.804

0.742

1.196

1.258



s.d. 58.202

02

s.d. 96.102

s.d. 96.102

s.d. 58.202

You should report the results of both tests only when you can reject H0 at the 5% level but not at the 1% level.

53

Copyright Christopher Dougherty 2011.

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refer to the author.

The content of this slideshow comes from Section 2.6 of C. Dougherty,

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

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11.07.25

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