Dougherty С. Introduction to Econometrics, 3Ed

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PROPERTIES OF THE REGRESSION COEFFICIENTS

the 5 percent level, it automatically follows that you will not reject at the 1 percent level, and again

you would look ignorant if you said so. The only time when you should quote both results is when

you reject the null hypothesis at the 5 percent level but not at the 1 percent level.

What Happens if the Standard Deviation of b

is Not Known

So far we have assumed that the standard deviation of b

is known, which is most unlikely in practice.

It has to be estimated by the standard error of b

, given by (3.29). This causes two modifications to the

test procedure. First, z is now defined using s.e.(b

) instead of s.d.(b

), and it is referred to as the t

statistic:

)(s.e.

−

(3.49)

Second, the critical levels of t depend upon what is known as a t distribution instead of a normal

distribution. We will not go into the reasons for this, or even describe the t distribution

mathematically. Suffice to say that it is a cousin of the normal distribution, its exact shape depending

on the number of degrees of freedom in the regression, and that it approximates the normal

distribution increasingly closely as the number of degrees of freedom increases. You will certainly

have encountered the t distribution in your introductory statistics course. Table A.2 at the end of the

text gives the critical values of t cross-classified by significance level and the number of degrees of

freedom.

The estimation of each parameter in a regression equation consumes one degree of freedom in the

sample. Hence the number of degrees of freedom is equal to the number of observations in the sample

minus the number of parameters estimated. The parameters are the constant (assuming that this is

specified in the regression model) and the coefficients of the explanatory variables. In the present case

of simple regression analysis, only two parameters,

and

, are estimated and hence the number of

degrees of freedom is n – 2. It should be emphasized that a more general expression will be required

when we come to multiple regression analysis.

The critical value of t, which we will denote t

crit

, replaces the number 1.96 in (3.45), so the

condition that a regression estimate should not lead to the rejection of a null hypothesis H

crit

)(s.e.

≤

−

≤−

(3.50)

Hence we have the decision rule: reject H

crit

).(.

bes

−

, do not reject if

crit

).(.

bes

≤

−

where

).(.

bes

−

is the absolute value (numerical value, neglecting the sign) of t.

PROPERTIES OF THE REGRESSION COEFFICIENTS

Examples

. reg EARNINGS S

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 1, 568) = 65.64

Model | 3977.38016 1 3977.38016 Prob > F = 0.0000

Residual | 34419.6569 568 60.5979875 R-squared = 0.1036

---------+------------------------------ Adj R-squared = 0.1020

Total | 38397.0371 569 67.4816117 Root MSE = 7.7845

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

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

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

S | 1.073055 .1324501 8.102 0.000 .8129028 1.333206

_cons | -1.391004 1.820305 -0.764 0.445 -4.966354 2.184347

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

In Section 2.6 hourly earnings were regressed on years of schooling using data from the United States

National Longitudinal Survey of Youth with the output shown above. The first two columns give the

names of the variables, here just S and the intercept (Stata denotes this as

_cons

) and the estimates of

their coefficients. The third column gives the corresponding standard errors. Let us suppose that one

of the purposes of the regression was to confirm our intuition that earnings are affected by education.

Accordingly, we set up the null hypothesis that

is equal to 0 and try to refute it. The corresponding

t statistic, using (3.49), is simply the estimate of the coefficient divided by its standard error:

10.8

1325.0

0731.1

)(s.e.

−

(3.51)

Since there are 570 observations in the sample and we have estimated two parameters, the

number of degrees of freedom is 568. Table A.2 does not give the critical values of t for 568 degrees

of freedom, but we know that they must be lower than the corresponding critical values for 500, since

the critical value is inversely related to the number of degrees of freedom. The critical value with 500

degrees of freedom at the 5 percent level is 1.965. Hence we can be sure that we would reject H

the 5 percent level with 568 degrees of freedom and we conclude that schooling does affect earnings.

To put this test into words, with 568 degrees of freedom the upper and lower 2.5 percent tails of

the t distribution start approximately 1.965 standard deviations above and below its mean of 0. The

null hypothesis will not be rejected if the regression coefficient is estimated to lie within 1.965

standard deviations of 0. In this case, however, the discrepancy is equivalent to 8.10 estimated

standard deviations and we come to the conclusion that the regression result contradicts the null

hypothesis.

Of course, since we are using the 5 percent significance level as the basis for the test, there is in

principle a 5 percent risk of a Type I error, if the null hypothesis is true. In this case we could reduce

the risk to 1 percent by using the 1 percent significance level instead. The critical value of t at the 1

percent significance level with 500 degrees of freedom is 2.586. Since the t statistic is greater than

this, we see that we can easily reject the null hypothesis at this level as well.

Note that when the 5 percent and 1 percent tests lead to the same conclusion, there is no need to

report both, and indeed you would look ignorant if you did. Read carefully the box on reporting test

results.

PROPERTIES OF THE REGRESSION COEFFICIENTS

This procedure of establishing a relationship between a dependent and an explanatory variable by

setting up, and then refuting, a null hypothesis that

is equal to 0 is used very frequently indeed.

Consequently all serious regression applications automatically print out the t statistic for this special

case, that is, the coefficient divided by its standard error. The ratio is often denoted "the" t statistic. In

the regression output, the t statistics for the constant and slope coefficient appear in the middle

column.

However, if the null hypothesis specifies some nonzero value of

, the more general expression

(3.50) has to be used and the t statistic has to be calculated by hand. For example, consider again the

price inflation/wage inflation model (3.39) and suppose that the fitted model is (standard errors in

parentheses):

= –1.21 + 0.82w (3.52)

(0.05) (0.10)

If we now investigate the hypothesis that price inflation is equal to wage inflation, our null

hypothesis is that the coefficient of w is equal to 1.0. The corresponding t statistic is

80.1

10.0

00.182.0

)(s.e.

−=

−

(3.53)

If there are, say, 20 observations in the sample, the number of degrees of freedom is 18 and the

critical value of t at the 5 percent significance level is 2.101. The absolute value of our t statistic is

less than this, so on this occasion we do not reject the null hypothesis. The estimate 0.82 is below our

hypothesized value 1.00, but not so far below as to exclude the possibility that the null hypothesis is

correct. One final note on reporting regression results: some writers place the t statistic in parentheses

under a coefficient, instead of the standard error. You should be careful to check, and when you are

presenting results yourself, you should make it clear which you are giving.

The Reject/Fail-to-Reject Terminology

In this section it has been shown that you should reject the null hypothesis if the absolute

value of the t statistic is greater than t

crit

, and that you fail to reject it otherwise. Why "fail

to reject", which is a clumsy expression? Would it not be better just to say that you

accept the hypothesis if the absolute value of the t statistic is less than t

crit

The argument against using the term accept is that you might find yoursel

"accepting" several mutually exclusive hypotheses at the same time. For instance, in the

price inflation/wage inflation example, you would not reject a null hypothesis

= 0.9, or a null hypothesis H

= 0.8. It is logical to say that you would not

reject these null hypotheses, as well as the null hypothesis H

= 1 discussed in the

text, but it makes little sense to say that you simultaneously accept all three hypotheses.

In the next section you will see that one can define a whole range of hypotheses which

would not be rejected by a given experimental result, so it would be incautious to pic

out one as being "accepted".

PROPERTIES OF THE REGRESSION COEFFICIENTS

Reporting the Results of

tests

Suppose you have a theoretical relationship

+ u

and your null and alternative hypotheses are H

, H

≠

. Given an

experimental estimate b

, the acceptance and rejection regions for the hypothesis for the

5 percent and 1 percent significance levels can be represented in general terms by the left part

of Figure 3.9.

The right side of the figure gives the same regions for a specific example, the price

inflation/wage inflation model, the null hypothesis being that

is equal to 1. The null

hypothesis will not be rejected at the 5 percent level if b

lies within 2.101 standard errors o

1, that is, in the range 0.79 to 1.21, and it will not be rejected at the 1 percent level if b

lies

within 2.878 standard deviations of 1, that is, in the range 0.71 to 1.29.

From Figure 3.9 it can be seen that there are three types of decision zone:

1. where b

is so far from the hypothetical

that the null hypothesis is rejected at both

the 5 percent and the 1 percent levels,

2. where b

is far enough from the hypothetical

for the null hypothesis to be rejected

at the 5 percent but not the 1 percent level,

3. where b

is close enough to the hypothetical

for the null hypothesis not to be

rejected at either level.

From the diagram it can be verified that if the null hypothesis is rejected at the 1 percen

level, it is automatically rejected at the 5 percent level. Hence in case (1) it is only necessary

to report the rejection of the hypothesis at the 1 percent level. To report that it is rejected also

at the 5 percent level is superfluous and suggests that you are not aware of this. It would be a

bit like reporting that a certain high-jumper can clear two metres, and then adding that the

athlete can also clear one and a half metres.

In case (3), likewise, you only need to make one statement, in this case that the

hypothesis is not rejected at the 5 percent level. It automatically follows that it is not rejected

at the 1 percent level, and to add a statement to this effect as well would be like saying that

the high-jumper cannot clear one and a half metres, and also reporting that the athlete cannot

clear two metres either.

Only in case (2) is it necessary (and desirable) to report the results of both tests.

Note that if you find that you can reject the null hypothesis at the 5 percent level, you

should not stop there. You have established that the null hypothesis can be rejected at that

level, but there remains a 5 percent chance of a Type I error. You should also perform the

test at the 1 percent level. If you find that you can reject the null hypothesis at this level, this

is the outcome that you should report. The risk of a Type I error is now only 1 percent and

your conclusion is much more convincing. This is case (1) above. If you cannot reject at the

1 percent level, you have reached case (2) and you should report the results of both tests.

PROPERTIES OF THE REGRESSION COEFFICIENTS

Figure 3.9.

Reporting the results of a

test (no need to report

conclusions in parentheses)

p Values

The fifth column of the output above, headed

|t|

, provides an alternative approach to reporting

the significance of regression coefficients. The figures in this column give the probability of

obtaining the corresponding t statistic as a matter of chance, if the null hypothesis H

= 0 were true.

A p value of less than 0.01 means that the probability is less than 1 percent, which in turn means that

the null hypothesis would be rejected at the 1 percent level; a p value between 0.01 and 0.05 means

that the null hypothesis would be rejected at the 5 percent, but not the 1 percent level; and a p value of

0.05 or more means that it would not be rejected at the 5 percent level.

The p value approach is more informative than the 5 percent/1 percent approach, in that it gives

the exact probability of a Type I error, if the null hypothesis is true. For example, in the earnings

function output above, the p values for the intercept is 0.445, meaning that the probability of obtaining

a t statistic of 0.764 or greater, in absolute terms, on a pure chance basis is in this case 44.5 percent.

Hence the null hypothesis that the intercept is 0 would not be rejected at any sensible significance

level. In the case of the slope coefficient, the p value is 0.0000, meaning that the probability of

obtaining a t statistic as large as 8.102, or larger, as a matter of chance is less than 0.005 percent.

Hence we would reject the null hypothesis that the slope coefficient is 0 at the 1 percent level. Indeed

we would reject it at the 0.1 percent level – see below. Choice between the p value approach and the 5

percent/1 percent approach appears to be entirely conventional. The medical literature uses p values,

but the economics literature generally uses 5 percent/1 percent.

0.1 Percent Tests

If the t statistic is very high, you should check whether you can reject the null hypothesis at the 0.1

percent level. If you can, you should always report the result of the 0.1 percent test in preference to

that of the 1 percent test because it demonstrates that you are able to reject the null hypothesis with an

even smaller risk of a Type I error.

Reject

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

Reject

at 5% level but not at 1% level

Reject

at 5% level but not at 1% level

Reject

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

Do not reject

at 5% level (or at 1% level)

DECISION

PRICE INFLATION/

WAGE INFLATION

EXAMPLE

GENERAL CASE

1.29

1.21

1.00

0.79

0.71

crit,1%

×s.e.

crit,5%

×s.e.

crit, 5%

×s.e.

crit,1%

×s.e.

PROPERTIES OF THE REGRESSION COEFFICIENTS

Exercises

3.7

Give more examples of everyday instances in which decisions involving possible Type I and

Type II errors may arise.

3.8

Before beginning a certain course, 36 students are given an aptitude test. The scores, and the

course results (pass/fail) are given below:

student

test

score

course

result

student

test

score

course

result

student

test

score

course

result

1 30 fail 13 26 fail 25 9 fail

2 29 pass 14 43 pass 26 36 pass

3 33 fail 15 43 fail 27 61 pass

4 62 pass 16 68 pass 28 79 fail

5 59 fail 17 63 pass 29 57 fail

6 63 pass 18 42 fail 30 46 pass

7 80 pass 19 51 fail 31 70 fail

8 32 fail 20 45 fail 32 31 pass

9 60 pass 21 22 fail 33 68 pass

10 76 pass 22 30 pass 34 62 pass

11 13 fail 23 40 fail 35 56 pass

12 41 pass 24 26 fail 36 36 pass

Do you think that the aptitude test is useful for selecting students for admission to the course,

and if so, how would you determine the pass mark? (Discuss the trade-off between Type I and

Type II errors associated with the choice of pass-mark.)

3.9

A researcher hypothesizes that years of schooling,

, may be related to the number of siblings

(brothers and sisters),

SIBLINGS,

according to the relationship

SIBLINGS

She is prepared to test the null hypothesis

= 0 against the alternative hypothesis

≠

0 at the 5 percent and 1 percent levels. She has a sample of 60 observations. What should she

report

1. if

= 0.20, s.e.(

) = 0.07?

2. if

= –0.12, s.e.(

) = 0.07?

3. if

= 0.06, s.e.(

) = 0.07?

4. if

= 0.20, s.e.(

) = 0.07?

3.10*

A researcher with a sample of 50 individuals with similar education but differing amounts of

training hypothesizes that hourly earnings,

EARNINGS

, may be related to hours of training,

TRAINING

, according to the relationship

EARNINGS

TRAINING

He is prepared to test the null hypothesis

= 0 against the alternative hypothesis

≠

0 at the 5 percent and 1 percent levels. What should he report

PROPERTIES OF THE REGRESSION COEFFICIENTS

1. if b

= 0.30, s.e.(b

) = 0.12?

2. if b

= 0.55, s.e.(b

) = 0.12?

3. if b

= 0.10, s.e.(b

) = 0.12?

4. if b

= –0.27, s.e.(b

) = 0.12?

3.11

Perform a t test on the slope coefficient and the intercept of the educational attainment function

fitted using your EAEF data set, and state your conclusions.

3.12

Perform a t test on the slope coefficient and the intercept of the earnings function fitted using

your EAEF data set, and state your conclusions.

3.13*

In Exercise 2.1, the growth rate of employment was regressed on the growth rate of GDP for a

sample of 25 OECD countries. Perform t tests on the slope coefficient and the intercept and

state your conclusions.

3.8 Confidence Intervals

Thus far we have been assuming that the hypothesis preceded the empirical investigation. This is not

necessarily the case. Usually theory and experimentation are interactive, and the earnings function

regression provides a typical example. We ran the regression in the first place because economic

theory tells us to expect earnings to be affected by schooling. The regression result confirmed this

intuition since we rejected the null hypothesis

= 0, but we were then left with something of a

vacuum, since our theory is not strong enough to suggest that the true value of

is equal to some

specific number. However, we can now move in the opposite direction and ask ourselves the

following question: given our regression result, what hypotheses would be compatible with it?

Obviously a hypothesis

= 1.073 would be compatible, because then hypothesis and

experimental result coincide. Also

= 1.072 and

= 1.074 would be compatible, because the

difference between hypothesis and experimental result would be so small. The question is, how far

can a hypothetical value differ from our experimental result before they become incompatible and we

have to reject the null hypothesis?

We can answer this question by exploiting the previous analysis. From (3.50), we can see that

regression coefficient b

and hypothetical value

are incompatible if either

crit

−

)(s.e.

crit

−<

−

)(s.e.

(3.54)

that is, if either

–

> s.e.(b

)

crit

or b

–

< –s.e.(b

)

crit

(3.55)

that is, if either

– s.e.(b

)

crit

or b

+ s.e.(b

)

crit

(3.56)

It therefore follows that a hypothetical

is compatible with the regression result if both

PROPERTIES OF THE REGRESSION COEFFICIENTS

– s.e.(

)

crit

≤

+ s.e.(

)

crit

≥

(3.57)

that is, if

satisfies the double inequality

– s.e.(

)

crit

≤

+ s.e.(

)

crit

(3.58)

Any hypothetical value of

that satisfies (3.58) will therefore automatically be compatible with

the estimate

, that is, will not be rejected by it. The set of all such values, given by the interval

between the lower and upper limits of the inequality, is known as the

confidence interval

for

Note that the center of the confidence interval is

itself. The limits are equidistant on either

side. Note also that, since the value of

crit

depends upon the choice of significance level, the limits will

also depend on this choice. If the 5 percent significance level is adopted, the corresponding

confidence interval is known as the 95 percent confidence interval. If the 1 percent level is chosen,

one obtains the 99 percent confidence interval, and so on.

A Second Interpretation of a Confidence Interval

When you construct a confidence interval, the numbers you calculate for its upper and

lower limits contain random components that depend on the values of the disturbance

term in the observations in the sample. For example, in inequality (3.58), the upper limit

+ s.e.(

)

crit

Both

and s.e.(

) are partly determined by the values of the disturbance term, and

similarly for the lower limit. One hopes that the confidence interval will include the true

value of the parameter, but sometimes it will be so distorted by the random element that it

will fail to do so.

What is the probability that a confidence interval will capture the true value of the

parameter? It can easily be shown, using elementary probability theory, that, in the case

of a 95 percent confidence interval, the probability is 95 percent, provided that the model

is correctly specified and that the Gauss–Markov conditions are satisfied. Similarly, in

the case of a 99 percent confidence interval, the probability is 99 percent.

The estimated coefficient [for example,

in inequality (3.58)] provides a point

estimate of the parameter in question, but of course the probability of the true value being

exactly equal to this estimate is infinitesimal. The confidence interval provides what is

known as an

interval estimate

of the parameter, that is, a range of values that will include

the true value with a high, predetermined probability. It is this interpretation that gives

the confidence interval its name (for a detailed and lucid discussion, see Wonnacott and

Wonnacott, 1990, Chapter 8).

PROPERTIES OF THE REGRESSION COEFFICIENTS

Since t

crit

is greater for the 1 percent level than for the 5 percent level, for any given number of

degrees of freedom, it follows that the 99 percent interval is wider than the 95 percent interval. Since

they are both centered on b

, the 99 percent interval encompasses all the hypothetical values of

the 95 percent confidence interval and some more on either side as well.

Example

In the earnings function output above, the coefficient of S was 1.073, its standard error was 0.132, and

the critical value of t at the 5 percent significance level was about 1.97. The corresponding 95 percent

confidence interval is therefore

1.073 – 0.132

1.97

≤

1.073 + 0.132

1.97 (3.59)

that is,

0.813

≤

1.333 (3.60)

We would therefore reject hypothetical values above 1.333 and below 0.813. Any hypotheses within

these limits would not be rejected, given the regression result. This confidence interval actually

appears as the final column in the output above. However, this is not a standard feature of a regression

application, so you usually have to calculate the interval yourself.

Exercises

3.14

Calculate the 99 percent confidence interval for

in the earnings function example (b

= 1.073,

s.e.(b

) = 0.132), and explain why it includes some values not included in the 95 percent

confidence interval calculated in the previous section.

3.15

Calculate the 95 percent confidence interval for the slope coefficient in the earnings function

fitted with your EAEF data set.

3.16*

Calculate the 95 percent confidence interval for

in the price inflation/wage inflation example:

= –1.21 + 0.82w

(0.05) (0.10)

What can you conclude from this calculation?

3.9 One-Tailed

Tests

In our discussion of t tests, we started out with our null hypothesis H

and tested it to see

whether we should reject it or not, given the regression coefficient b

. If we did reject it, then by

implication we accepted the alternative hypothesis H

≠

PROPERTIES OF THE REGRESSION COEFFICIENTS

Thus far the alternative hypothesis has been merely the negation of the null hypothesis. However,

if we are able to be more specific about the alternative hypothesis, we may be able to improve the

testing procedure. We will investigate three cases: first, the very special case where there is only one

conceivable alternative true value of

, which we will denote

; second, where, if

is not equal to

, it must be greater than

; and third, where, if

is not equal to

, it must be less than

In this case there are only two possible values of the true coefficient of X,

and

. For sake of

argument we will assume for the time being that

is greater than

Suppose that we wish to test H

at the 5 percent significance level, and we follow the usual

procedure discussed earlier in the chapter. We locate the limits of the upper and lower 2.5 percent

tails under the assumption that H

is true, indicated by A and B in Figure 3.10, and we reject H

if the

regression coefficient b

lies to the left of A or to the right of B.

Now, if b

does lie to the right of B, it is more compatible with H

than with H

; the probability of

it lying to the right of B is greater if H

is true than if H

is true. We should have no hesitation in

rejecting H

in favor of H

However, if b

lies to the left of A, the test procedure will lead us to a perverse conclusion. It tells

us to reject H

in favor of H

, even though the probability of b

lying to the left of A is negligible if H

is true. We have not even drawn the probability density function that far for H

. If such a value of b

occurs only once in a million times when H

is true, but 2.5 percent of the time when H

is true, it is

much more logical to assume that H

is true. Of course once in a million times you will make a

mistake, but the rest of the time you will be right.

Hence we will reject H

only if b

lies in the upper 2.5 percent tail, that is, to the right of B. We

are now performing a one-tailed test, and we have reduced the probability of making a Type I error to

2.5 percent. Since the significance level is defined to be the probability of making a Type I error, it is

now also 2.5 percent.

As we have seen, economists usually prefer 5 percent and 1 percent significance tests, rather than

2.5 percent tests. If you want to perform a 5 percent test, you move B to the left so that you have 5

Figure 3.10.

Distribution of

under

and

probability density

function of