Dougherty - Hyp Testing

Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: t test of 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

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t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT

s.d. of b2 known

discrepancy between hypothetical value and sample estimate, in terms of s.d.:

s.d.

022 bz

0

The diagram summarizes the procedure for performing a 5% significance test on the slope coefficient of a regression under the assumption that we know its standard deviation.

1

5% significance test: reject H0: 2 = 2 if z > 1.96 or z < 1.96


s.d. of b2 known


s.d.

022 bz

s.d. of b2 not known

discrepancy between hypothetical value and sample estimate, in terms of s.e.:

s.e.

022 bt

0

This is a very unrealistic assumption. We usually have to estimate it with the standard error, and we use this in the test statistic instead of the standard deviation.

2



s.d. of b2 known


s.d.

022 bz



s.e.

022 bt

0

Because we have replaced the standard deviation in its denominator with the standard error, the test statistic has a t distribution instead of a normal distribution.

3



s.d. of b2 known


s.d.

022 bz



s.e.

022 bt

0 0

Accordingly, we refer to the test statistic as a t statistic. In other respects the test procedure is much the same.

4


5% significance test: reject H0: 2 = 2 if t > tcrit or t < tcrit

5


We look up the critical value of t and if the t statistic is greater than it, positive or negative, we reject the null hypothesis. If it is not, we do not.

s.d. of b2 known


s.d.

022 bz




s.e.

022 bt


0 0

6


Here is a graph of a normal distribution with zero mean and unit variance

0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

normal

7


A graph of a t distribution with 10 degrees of freedom (this term will be defined in a moment) has been added.

0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

normal t, 10 d.f.

8


0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

When the number of degrees of freedom is large, the t distribution looks very much like a normal distribution (and as the number increases, it converges on one).

normal t, 10 d.f.

9


0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Even when the number of degrees of freedom is small, as in this case, the distributions are very similar.

normal t, 10 d.f.

10


Here is another t distribution, this time with only 5 degrees of freedom. It is still very similar to a normal distribution.

0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

normal

t, 5 d.f. t, 10 d.f.

11


So why do we make such a fuss about referring to the t distribution rather than the normal distribution? Would it really matter if we always used 1.96 for the 5% test and 2.58 for the 1% test?

0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

normal t, 10 d.f. t, 5 d.f.

12


The answer is that it does make a difference. Although the distributions are generally quite similar, the t distribution has longer tails than the normal distribution, the difference being the greater, the smaller the number of degrees of freedom.

0

0.1

0.2

0.3

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

normal t, 10 d.f. t, 5 d.f.

13


As a consequence, the probability of obtaining a high test statistic on a pure chance basis is greater with a t distribution than with a normal distribution.

normal

0

0.1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

t, 10 d.f. t, 5 d.f.

14


This means that the rejection regions have to start more standard deviations away from zero for a t distribution than for a normal distribution.

normal

0

0.1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

t, 10 d.f. t, 5 d.f.

15


The 2.5% tail of a normal distribution starts 1.96 standard deviations from its mean.

normal

0

0.1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-1.96

t, 10 d.f. t, 5 d.f.

16


The 2.5% tail of a t distribution with 10 degrees of freedom starts 2.33 standard deviations from its mean.

normal

0

0.1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-2.33

t, 10 d.f. t, 5 d.f.

17


That for a t distribution with 5 degrees of freedom starts 2.57 standard deviations from its mean.

normal

0

0.1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-2.57

t, 10 d.f. t, 5 d.f.

18


For this reason we need to refer to a table of critical values of t when performing significance tests on the coefficients of a regression equation.

t Distribution: Critical values of t

Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05%

1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

19


At the top of the table are listed possible significance levels for a test. For the time being we will be performing two-sided tests, so ignore the line for one-sided tests.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

20


Hence if we are performing a (two-sided) 5% significance test, we should use the column thus indicated in the table.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

21


The left hand vertical column lists degrees of freedom. The number of degrees of freedom in a regression is defined to be the number of observations minus the number of parameters estimated.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

Number of degrees of freedom in a regression = number of observations number of parameters estimated.

22


In a simple regression, we estimate just two parameters, the constant and the slope coefficient, so the number of degrees of freedom is n - 2 if there are n observations.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

23


If we were performing a regression with 20 observations, as in the price inflation/wage inflation example, the number of degrees of freedom would be 18 and the critical value of t for a 5% test would be 2.101.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

24


Note that as the number of degrees of freedom becomes large, the critical value converges on 1.96, the critical value for the normal distribution. This is because the t distribution converges on the normal distribution.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291


s.d. of b2 known


s.d.

022 bz




s.e.

022 bt


0 0

Hence, referring back to the summary of the test procedure,

25


s.d. of b2 known


s.d.

022 bz




s.e.

022 bt

5% significance test: reject H0: 2 = 2 if t > 2.101 or t < 2.101

0 0

we should reject the null hypothesis if the absolute value of t is greater than 2.101.

26

27


If instead we wished to perform a 1% significance test, we would use the column indicated above. Note that as the number of degrees of freedom becomes large, the critical value converges to 2.58, the critical value for the normal distribution.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291

28


For a simple regression with 20 observations, the critical value of t at the 1% level is 2.878.



1 6.314 12.706 31.821 63.657 318.31 636.62 2 2.920 4.303 6.965 9.925 22.327 31.598 3 2.353 3.182 4.541 5.841 10.214 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291


s.d. of b2 known


s.d.

022 bz




s.e.

022 bt

1% significance test: reject H0: 2 = 2 if t > 2.878 or t < 2.878

0 0

So we should use this figure in the test procedure for a 1% test.

29


We will next consider an example of a t test. Suppose that you have data on p, the average rate of price inflation for the last 5 years, and w, the average rate of wage inflation, for a sample of 20 countries. It is reasonable to suppose that p is influenced by w.

30

Example: uwp 21


31

Example: uwp 21

You might take as your null hypothesis that the rate of price inflation increases uniformly with wage inflation, in which case the true slope coefficient would be 1.

1:;1: 2120 HH


32

Example: uwp 21

Suppose that the regression result is as shown (standard errors in parentheses). Our actual estimate of the slope coefficient is only 0.82. We will check whether we should reject the null hypothesis.

)10.0()05.0(82.021.1 wp

1:;1: 2120 HH


33

Example:

.80.110.0

00.182.0)(s.e. 2022

bb

t

uwp 21

We compute the t statistic by subtracting the hypothetical true value from the sample estimate and dividing by the standard error. It comes to 1.80.

)10.0()05.0(82.021.1 wp

1:;1: 2120 HH


34

Example:

.80.110.0

00.182.0)(s.e. 2022

bb

t

18freedom of degrees;20 n

uwp 21

There are 20 observations in the sample. We have estimated 2 parameters, so there are 18 degrees of freedom.

)10.0()05.0(82.021.1 wp

1:;1: 2120 HH


35

Example:

)10.0()05.0(82.021.1 wp

.80.110.0

00.182.0)(s.e. 2022

bb

t

18freedom of degrees;20 n101.2%5,crit t

1:;1: 2120 HHuwp 21

The critical value of t with 18 degrees of freedom is 2.101 at the 5% level. The absolute value of the t statistic is less than this, so we do not reject the null hypothesis.


36

uXY 21

In practice it is unusual to have a feeling for the actual value of the coefficients. Very often the objective of the analysis is to demonstrate that Y is influenced by X, without having any specific prior notion of the actual coefficients of the relationship.


37

uXY 21

In this case it is usual to define 2 = 0 as the null hypothesis. In words, the null hypothesis is that X does not influence Y. We then try to demonstrate that the null hypothesis is false.

0:;0: 2120 HH


38

)(s.e.)(s.e. 22

2

022

bb

bb

t

uXY 21

For the null hypothesis 2 = 0, the t statistic reduces to the estimate of the coefficient divided by its standard error.

0:;0: 2120 HH


39

)(s.e.)(s.e. 22

2

022

bb

bb

t 0:;0: 2120 HH

uXY 21

This ratio is commonly called the t statistic for the coefficient and it is automatically printed out as part of the regression results. To perform the test for a given significance level, we compare the t statistic directly with the critical value of t for that significance level.

. reg EARNINGS S

Source | SS df MS Number of obs = 540

-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000

Residual | 92688.6722 538 172.283777 R-squared = 0.1725

-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------

EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


40

Here is the output from the earnings function fitted in a previous slideshow, with the t statistics highlighted.

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


41

You can see that the t statistic for the coefficient of S is enormous. We would reject the null hypothesis that schooling does not affect earnings at the 0.1% significance level without even looking at the table of critical values of t.

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


42

The t statistic for the intercept is also enormous. However, since the intercept does not hve any meaning, it does not make sense to perform a t test on it.

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


43

The next column in the output gives what are known as the p values for each coefficient. This is the probability of obtaining the corresponding t statistic as a matter of chance, if the null hypothesis H0: = 0 is true.

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


44

If you reject the null hypothesis H0: = 0, this is the probability that you are making a mistake and making a Type I error. It therefore gives the significance level at which the null hypothesis would just be rejected.

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


45

If p = 0.05, the null hypothesis could just be rejected at the 5% level. If it were 0.01, it could just be rejected at the 1% level. If it were 0.001, it could just be rejected at the 0.1% level. This is assuming that you are using two-sided tests.

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


46

In the present case p = 0 to three decimal places for the coefficient of S. This means that we can reject the null hypothesis H0: 2 = 0 at the 0.1% level, without having to refer to the table of critical values of t. (Testing the intercept does not make sense in this regression.)

. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765

_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444

------------------------------------------------------------------------------


47

It is a more informative approach to reporting the results of test and widely used in the medical literature. However in economics standard practice is to report results referring to 5% and 1% significance levels, and sometimes to the 0.1% level.

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

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Dougherty - Hyp Testing