Christopher Dougherty
EC220 - Introduction to econometrics (chapter 2)Slideshow: testing a hypothesis relating to a regression coefficient
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 2). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/128/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
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:
Alternative hypothesis
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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:
Alternative hypothesis
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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:
Alternative hypothesis
Example model: p = 1 + 2w + u
Null hypothesis:
Alternative hypothesis:
4
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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:
Alternative hypothesis
Example model: p = 1 + 2w + u
Null hypothesis:
Alternative hypothesis:
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
1.0 1.10.90.80.70.6 1.2 1.3 1.4
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
8
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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)
probability densityfunction of b2
b2
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
9
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
10
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
11
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
12
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
In terms of the general case, the estimate is one standard deviation below the hypothetical value.
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
13
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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%.
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
14
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
15
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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.
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
16
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
17
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Under the null hypothesis, the estimate is between 2 and 3 standard deviations below the mean.
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
18
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
19
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
20
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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.
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
21
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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%
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
22
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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%
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2 2+sd 2+2sd2-sd2-2sd 2+3sd2-3sd2-4sd 2+4sd00000 0 0 0 0
2.5% 2.5%
23
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
2.5% 2.5%
24
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
The second definitely would.
1.0 1.10.90.80.70.6 1.2 1.3 1.4
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
2.5% 2.5%
25
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
The third also would lead to rejection..
1.0 1.10.90.80.70.6 1.2 1.3 1.4
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =1.0 is true (standard deviation equals 0.1 taken as given)
26
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
The 2.5% tails of a normal distribution always begin 1.96 standard deviations from its mean.
2.5%2.5%
probability densityfunction of b2
b2
Distribution of b2 under the null hypothesis H0: 2 =2 is true (standard deviation taken as given)
0
2+1.96sd2-1.96sd 202-sd 2+sd00 00
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
Decision rule (5% significance level):reject
(1) if (2) if
(1) if (2) if
probability densityfunction of b2
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
Decision rule (5% significance level):reject
(1) if (2) if
(1) if (2) if
(1) if (2) if
probability densityfunction of b2
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
Decision rule (5% significance level):reject
(1) if (2) if
(1) if (2) if
(1) if (2) if
s.d.
022
b
z
probability densityfunction of b2
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Then the decision rule is to reject the null hypothesis if z is greater than 1.96 in absolute terms.
2.5%2.5%
Decision rule (5% significance level):reject
(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
probability densityfunction of b2
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
Decision rule (5% significance level):reject
(1) if (2) if
(1) if z > 1.96 (2) if z < -1.96
s.d.
022
b
z
probability 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
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
Decision rule (5% significance level):reject
(1) if (2) if
(1) if z > 1.96 (2) if z < -1.96
s.d.
022
b
z
probability 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
acceptance region for b2:
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%
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
1.0 1.10.90.80.70.6 1.2 1.3 1.4
probability densityfunction of b2
b2
Decision rule (5% significance level):reject
(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%
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
1.0 1.10.90.80.70.6 1.2 1.3 1.4
probability densityfunction of b2
b2
Decision rule (5% significance level):reject
(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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b2
Decision rule (5% significance level):reject
(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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
b22+1.96sd2-1.96sd 202-sd 2+sd00 00
acceptance region for b20220 :H 0
220 :H
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
reject reject
probability densityfunction of b2
b22+1.96sd2-1.96sd 202-sd 2+sd00 00
acceptance region for b20220 :H 0
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
Type I error: rejection of H0 when it is in fact true.
Probability of Type I error: in this case, 5%
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
reject reject
probability densityfunction of b2
b22+1.96sd2-1.96sd 202-sd 2+sd00 00
acceptance region for b20220 :H 0
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
Type I error: rejection of H0 when it is in fact true.
Probability of Type I error: in this case, 5%
Significance level of the test is 5%.
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
reject reject
probability densityfunction of b2
b22+1.96sd2-1.96sd 202-sd 2+sd00 00
acceptance region for b20220 :H 0
220 :H
Type I error: rejection of H0 when it is in fact true.
Probability of Type I error: in this case, 5%
Significance level of the test is 5%.
We can of course reduce the risk of making a Type I error by reducing the size of the rejection region.
40
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
2.5%2.5%
reject reject
probability densityfunction of b2
b22+1.96sd2-1.96sd 202-sd 2+sd00 00
acceptance region for b20220 :H 0
220 :H
Type I error: rejection of H0 when it is in fact true.
Probability of Type I error: in this case, 5%
Significance level of the test is 5%.
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
The rejection region now becomes the upper and lower 0.5% tails
2.5%2.5%
probability densityfunction of b2
b22+1.96sd2-1.96sd 202-sd 2+sd00 00
reject 0220 :H acceptance region for b2 reject 0
220 :H
Decision rule (1% significance level):reject
(1) if (2) if
(1) if z > 2.58 (2) if z < -2.58
43
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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%
probability densityfunction of b2
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
acceptance region for b2:
s.d. 58.2s.d. 58.2 022
02 b
58.258.2 z
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
0.5%0.5%
probability densityfunction of b2
b22+2.58sd2-2.58sd 202-sd 2+sd00 00
Type I error: rejection of H0 when it is in fact true.
Probability of Type I error: in this case, 1%
Significance level of the test is 1%.
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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%
probability densityfunction of b2
b2
Decision rule (1% significance level):reject
(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
acceptance region for b2:258.1742.0 2 b
258.12 b 742.02 b
0.1: 20 H
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
1.0 1.10.90.80.70.6 1.2 1.3 1.4
0.5%0.5%
probability densityfunction of b2
b2
Decision rule (1% significance level):reject
(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
acceptance region for b2:258.1742.0 2 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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
probability densityfunction of b2
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
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
Reject H0 at 5% level but not 1% level
Reject H0 at 1% level (and also 5% 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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Reject H0 at 1% level (and also 5% level)
Reject H0 at 5% level but not 1% level
Reject H0 at 5% level but not 1% level
Reject H0 at 1% level (and also 5% 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
The middle of the diagram indicates what you would report. You would not report the phrases in parentheses.
50
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Reject H0 at 1% level (and also 5% level)
Reject H0 at 5% level but not 1% level
Reject H0 at 5% level but not 1% level
Reject H0 at 1% level (and also 5% 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
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Reject H0 at 1% level (and also 5% level)
Reject H0 at 5% level but not 1% level
Reject H0 at 5% level but not 1% level
Reject H0 at 1% level (and also 5% 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
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
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Reject H0 at 1% level (and also 5% level)
Reject H0 at 5% level but not 1% level
Reject H0 at 5% level but not 1% level
Reject H0 at 1% level (and also 5% 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
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.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
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.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25