Hypothesis_Testing

The University of Melbourne

Department of Economics

ECON90033Quantitative Analysis of Finance1

Hypothesis Testing and EViews p-values:

Suppose that we want to test a null hypothesis about a single parameter using its es-timated value(for example a mean or a regression coe?cient).We can do so using a t-test.To begin,suppose that the parameter to be estimated isβ.We must?rst specify a null hypothesis and an alternative hypothesis.

2tail test:

For a two tailed test,we want to test whetherβis a particular value or not.We?rst set the value ofβthat we want to test.We’ll call thisβ0to indicate that this will be the value ofβunder the null hypothesis.In a two tail test,the null and alternative hypotheses are:

H0:β=β0

H A:β=β0

We proceed by estimatingβ.We denote the estimated value as?β.This could for example be a sample mean estimate of the population mean,a least squared estimate of a regression coe?cient,or a maximum likelihood estimate of a model coe?cient, depending on the context.The estimate?βis usually accompanied by a standard error to indicate how precisely it is estimated.We denote this standard error as se(?β).This re?ects the fact the?βis a random variable with a sampling distribution.It will have di?erent values in di?erent samples.

We can then form the following test statistic by computing the standardised statistic whereby we subtract the hypothesisized valueβ0from the estimate?βand divide by its standard error:

t-stat=?β?β

0 se(?β)

Again,this test statistic is a random variable since it depends on?β,which is itself a random variable.To make inference and do hypothesis testing about the value ofβ,we must assume a distribution for the above test statistic.This distribution is based on the null hypothesis being true.Since,for an unbiased estimate,the test statistic has zero mean and is standardised by its standard error,the test statistic doesn’t depend on the

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units ofβ,and we can usually assume a standardised distribution(such as the standard normal distribution with mean of zero and variance of1,or the t distribution).

In large samples,the above test statistic is usually assumed to follow a standard normal distribution,based on arguments related to the central limit theorem and estimators becoming normal in the limit as the sample size increases.This result holds even if the model errors are not themselves normally distributed.

In small samples,a t distribution is often used instead of a normal distribution.How-ever,this requires us assuming that the model errors are normally distributed.The t-distribution will depend on the sample size T and the number of parameters k to be estimated.So,in a regression model with k?1regressors and an intercept,a t distri-bution with T?k degrees of freedom should be used when testing a hypothesis about one of the regression coe?cients.When testing a sample mean,a t distribution with T?1degrees of freedom should be used(1degree of freedom is used up to estimate the mean).As the sample size increases,in the limit the t distribution becomes iden-tical to the standard normal distribution.Even in moderately large samples,the two distributions are nearly the same.

Having decided on the distribution for the test statistic under the assumed null hypoth-esis,we can then?nd critical values and p-values for the test statistic.This distribution is based onβbeing equal toβ0as under the null hypothesis.For a two tail test,the

critical values for the test statistic(±t crit,α

2)will be the values(positive and negative)of

the test statistic that make the probability(if the null hypothesis is true)of getting an absolute value of the test statistic bigger than the critical values equal to the chosen sig-

ni?cance levelα,usually5%(i.e.P(|t-stat|>t crit,α

2)=α).With anαsigni?cance level

and a symmetric distribution such as the standard normal distribution or t distribution, this puts probability ofα

2

in each tail.So:

Pr(t-stat>t crit,α

2)=

α

2

Pr(t-stat

2)=

α

2

With a5%signi?cance level and a symmetric distribution such as the standard normal or t distribution,this puts2.5%probability in each tail of the distribution.For a standard normal distribution and a5%signi?cance level,the critical values are±t crit,0.025=±1.96.So,using the5%signi?cance level,we reject the null hypothesis in favour of the alternative if the actual t-stat is less than-1.96or greater than1.96(i.e.|t-stat|> 1.96).

Now,the p-value for a two tailed test and a symmetric distribution,is the probability (assuming the null hypothesis is true)of getting a test statistic larger in absolute value than the actual measured t-stat.If we de?ne t possible to be a possible value of the test statistic,then the p-value of an actual t-stat is:

p-value=Pr(|t possible|>|t-stat|)

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If the actual t-stat is positive,the p-value is:

p-value=Pr(t possible>t-stat)+Pr(t possible

p-value=Pr(t possible?t-stat)(2) The p-values that EViews generally reports are these two tail test p-values.In EViews, these p-values usually come from a t distribution for hypothesis tests of a single parame-ter.We can do hypothesis testing by comparing the reported p-value to the signi?cance level.If the p-value is less than the signi?cance level(usually5%)then we reject the null hypothesis.Otherwise we don’t.

1tail test:

Suppose we have some theoretical or strong prior reasoning(prior to looking at the data), that a parameterβshould be above or below a particular valueβ0(oftenβ0=0).We can then do1-tail hypothesis testing.This alters how we set up the null and alternative hypotheses.

If we suspect thatβshould be higher thanβ0(e.g.βis positive,β>0),we would formulate the null and alternative hypotheses as follows:

H0:β≤β0

H A:β>β0

In contrast,If we suspect thatβshould be less thanβ0(e.g.βis negative,β<0),we would formulate the null and alternative hypotheses as follows:

H0:β≥β0

H A:β<β0

1tailed test(β>β0):

Let’s consider?rst the case in which we test whetherβ>β0.If we suspect thatβshould be higher thanβ0(e.g.βis positive,β>0),we would formulate the null and alternative hypotheses as follows:

H0:β≤β0

H A:β>β0

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In this case,we calculate our test statistic:

t-stat=?β?β

0 se(?β)

There is1critical value(t crit,α)for this1tail test which is the value of the test statistic that leaves probabilityαin the right tail(whereαis the chosen signi?cance level). Therefore:

Pr(t-stat>t crit,α)=α

To do hypothesis testing,we compare the actual t-stat with the critical value at the chosen signi?cance level.If the t-stat is greater than t crit,α,we reject the null hypothesis in favour of the alternative hypothesis.If it is not(for example the test statistic is negative),we fail to reject.So,for example,at the5%signi?cance level and using the standard normal distribution for the test statistic,the critical value is t crit,0.05=1.645. So,if we observe a t-stat greater than1.645,we reject the null hypothesis.If we observe a negative t-stat,e.g.-4.50,we fail to reject.This shows that even though the negative t-stat of-4.50would have been signi?cant in the two tail test,it is not in the1tail test.

The p-values in the1tail test are the probabilities of getting a t-stat higher than the actual measured t-stat.So:

p-value=Pr(t possible>t-stat)

For symmetric distributions and using(1)we can see that if the t-stat is positive,then

the p-value is given by:

p-value=1

2

[2tail p-value](3)

Additionally,if the t-stat is negative and using(2),then the p-value is given by:

p-value=1?1

2

[2tail p-value](4)

Since EViews generally reports the2tailed p-value,we should use(3)and(4)to compute the correct1tail p-values before comparing with the signi?cance levelαto do our hypothesis testing.

1tailed test(β<β0):

Let’s now consider the second case in which we testβ<β0.If we suspect thatβshould be lower thanβ0(e.g.βis negative,β<0),we would formulate the null and alternative hypotheses as follows:

H0:β≥β0

H A:β<β0

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In this case,we calculate our test statistic:

t-stat=?β?β

0 se(?β)

There is1critical value(t crit,α)for this1tail test which is the value of the test statis-tic that leaves probabilityαin the left tail(whereαis the chosen signi?cance level). Therefore:

Pr(t-stat

To do hypothesis testing,we compare the actual t-stat with the critical value at the chosen signi?cance level.If the t-stat is less than t crit,α,we reject the null hypothesis in favour of the alternative hypothesis.If it is not(for example the test statistic is positive), we fail to reject.So,for example,at the5%signi?cance level and using the standard normal distribution for the test statistic,the critical value is t crit,0.05=?1.645.So,if we observe a t-stat lower than-1.645(i.e.more negative),we reject the null hypothesis. If we observe a positive t-stat,e.g.4.50,we fail to reject.This shows that even though the positive t-stat of4.50would have been signi?cant in the two tail test,it is not in the 1tail test.

The p-values in the1tail test are the probabilities of getting a t-stat lower than the actual measured t-stat.So:

p-value=Pr(t possible

For symmetric distributions and using(1)we can see that if the t-stat is positive,then

the p-value is given by:

p-value=1?1

2

[2tail p-value](5)

Additionally,if the t-stat is negative and using(2),then the p-value is given by:

p-value=1

2

[2tail p-value](6)

Again,since EViews generally reports the2tailed p-value,we should use(5)and(6)to compute the correct1tail p-values before comparing with the signi?cance levelαto do our hypothesis testing.

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