Economics 300 Econometrics Hypothesis Testingˇ the critical t-value for a one-tailed test at a...

Economics 300Econometrics

Hypothesis Testing

Dennis C. Plott

University of Illinois at ChicagoDepartment of Economics

ECON 300 Website

Fall 2014

Dennis C. Plott (UIC) ECON 300 – Fall 2014 1 / 25

http://www.dennisplott.com/econ-300---econometrics-fall-2014.html

Outline

Hypothesis TestingWhat Is Hypothesis Testing?The t-TestAlternative Measures of Statistical Significancet-Test Caveats

Dennis C. Plott (UIC) ECON 300 – Fall 2014 2 / 25

Outline


Dennis C. Plott (UIC) Hypothesis Testing ECON 300 – Fall 2014 3 / 25

What Is Hypothesis Testing?

Ï Hypothesis testing is used in a variety of settingsÏ The Food and Drug Administration (FDA), for example, tests new products before

allowing their saleÏ If the sample of people exposed to the new product shows some side effect

significantly more frequently than would be expected to occur by chance, the FDA islikely to withhold approval of marketing that product

Ï Similarly, economists have been statistically testing various relationships, for

example that between consumption and income

Ï Note here that while we cannot prove a given hypothesis (for example theexistence of a given relationship), we often can reject a given hypothesis(again, for example, rejecting the existence of a given relationship)


Classical Null and Alternative Hypotheses

Ï The researcher first states the hypotheses to be tested

Ï Here, we distinguish between the null and the alternative hypothesis:Ï Null hypothesis (H0): the outcome that the researcher does not expect (almost

always includes an equality sign)Ï Alternative hypothesis (H1 or Ha): the outcome the researcher does expect

Ï Example:Ï H0 :β≤ 0 (the values you do not expect)Ï H1 :β> 0 (the values you do expect)


Type I and Type II Errors

Ï Two types of errors possible in hypothesis testing:Ï Type I: Rejecting a true null hypothesisÏ Type II: Not rejecting a false null hypothesis

Ï Example: Suppose we have the following null and alternative hypotheses:Ï H0 :β≤ 0Ï H1 :β> 0Ï Even if the true β really is not positive, in any one sample we might still observe

an estimate of β that is sufficiently positive to lead to the rejection of the null

hypothesis


Rejecting a True Null Hypothesis Is a Type I Error

If β= 0, but you observe β that is very positive, you might reject a true nullhypothesis, H0 :β≤ 0, and conclude incorrectly that the alternative hypothesisH1 :β> 0 is true.


Type I and Type II Errors (cont.)

Ï Alternatively, it’s possible to obtain an estimate of β that is close enough tozero (or negative) to be considered “not significantly positive”

Ï Such a result may lead the researcher to “accept” the null hypothesis that β≤ 0when in truth β> 0

Ï This is a Type II Error; we have failed to reject a false null hypothesis


Failure to Reject a False Null Hypothesis Is a Type IIError

If β= 1, but you observe a β that is negative but close to zero, you might fail toreject a false null hypothesis, H0 :β≤ 0, and incorrectly ignore the fact that thealternative hypothesis, H1 :β> 0, is true.


Decision Rules of Hypothesis Testing

Ï To test a hypothesis, we calculate a sample statistic that determines when thenull hypothesis can be rejected depending on the magnitude of that samplestatistic relative to a pre-selected critical value (which is found in a statisticaltable)

Ï This procedure is referred to as a decision rule

Ï The decision rule is formulated before regression estimates are obtained

Ï The range of possible values of the estimates is divided into two regions, an“acceptance” (really, non-rejection) region and a rejection region

Ï The critical value effectively separates the “acceptance”/non-rejection regionfrom the rejection region when testing a null hypothesis


"Acceptance" and Rejection Regions for a One-SidedTest of β

For a one-sided test of H0 :β≤ 0 vs. H1 :β> 0, the critical value divides thedistribution of β (centered around zero on the assumption that H0 is true) into“acceptance” and rejection regions


"Acceptance" and Rejection Regions for a Two-SidedTest of β

For a two-sided test of H0 :β= 0 vs. H1 :β 6= 0, we divided the distribution of β intoan “acceptance” regions and two rejection regions


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The t-Test

Ï The t-test is the test that econometricians usually use to test hypotheses aboutindividual regression slope coefficients

Ï Tests of more than one coefficient at a time (joint hypotheses) are typically done

with the F-test, presented later

Ï The appropriate test to use when the stochastic error term is normallydistributed and when the variance of that distribution must be estimated

Ï Since these usually are the case, the use of the t-test for hypothesis testing has

become standard practice in econometrics


The t-Statistic

Ï For a simple (bivariate) regression:

yi =β0 +β1x1i +ui (1)

we can calculate t-values for each of the estimated parameters; i.e., the βs

Ï Specifically, the t-statistic for the kth (foreshadowing multivariate regression;i.e., several βs) coefficient is:

tk = βk −βH0

SE(βk)(2)

k = 1,2, . . . ,K


The Critical t-Value and the t-Test Decision Rule

Ï To decide whether to reject or not to reject a null hypothesis based on acalculated t-value, we use a critical t-value

Ï A critical t-value is the value that distinguishes the “acceptance” region fromthe rejection region

Ï The critical t-value, tc, is traditionally selected from a t-table depending on:Ï whether the test is one-sided or two-sided,Ï the level of Type I Error specified andÏ the degrees of freedom (defined as the number of observations minus the

number of coefficients estimated (including the constant) or N −K −1)


The Critical t-Value and the t-Test Decision Rule(cont.)

Ï The rule to apply when testing a single regression coefficient ends up beingthat you should:

Ï Reject H0 if |tk| > tc and if tk also has the sign implied by H1Ï Do not reject H0 otherwise

Ï Note that this decision rule works both for calculated t-values and criticalt-values for one-sided hypotheses around zero (or another hypothesized value


The Critical t-Value and the t-Test Decision Rule(cont.)

Ï As well as for two-sided hypotheses around zero, or another hypothesizedvalue:

H0 :βk = 0

H1 :βk 6= 0

Ï the critical t-value for a one-tailed test at a given level of significance is exactlyequal to the critical t-value for a two-tailed test at twice the level ofsignificance of the one-tailed test


One-Sided and Two-Sided t-Tests

The tc for a one-sided test at a given level of significance is equal exactly to the tc fora two-sided test with twice the level of significance of the one-sided test. Forexample, tc = 1.699 for a 10-percent two-sided and for a 5-percent one-sided test(for 29 degrees-of-freedom).


Choosing a Level of Significance

Ï The level of significance must be chosen before a critical value can be found

Ï The level of significance indicates the probability of observing an estimatedt-value greater than the critical t-value if the null hypothesis were correct

Ï It also measures the amount of Type I Error implied by a particular criticalt-value

Ï Which level of significance is chosen?Ï 5 percent is the convention, unless you know something unusual about the

relative costs of making Type I and Type II Errors


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Confidence Intervals

Ï A confidence interval is a range that contains the true value of an item aspecified percentage of the time

Ï It is calculated using the estimated regression coefficient, the two-sidedcritical t-value and the standard error of the estimated coefficient as follows:

Confidence Interval (CI) = β± tc · SE(β) (3)

Ï What is the relationship between confidence intervals and two-sidedhypothesis testing?

Ï If a hypothesized value fall within the confidence interval, then we cannotreject the null hypothesis


p-Values

Ï This is an alternative to the t-test

Ï A p-value, or marginal significance level, is the probability of observing at-score that size or larger (in absolute value) if the null hypothesis were true

Ï Graphically, it is two times the area under the curve of the t-distributionbetween the absolute value of the actual t-score and infinity.

Ï The p-value decision rule therefore is:Ï Reject H0 if p-value is less than the level of significance and if βk has the sign

implied by H1


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Limitations of the t-Test

Ï With the t-values being automatically printed out by computer regressionpackages, there is reason to caution against potential improper use of thet-test:

1. The t-Test Does Not Test Theoretical Validity:Ï If you regress the consumer price index on rainfall in a time-series regression and find

strong statistical significance does that also mean that the underlying theory is valid?Of course not!

2. The t-Test Does Not Test “Importance”:Ï The fact that one coefficient is “more statistically significant” than another does not

mean that it is also more important in explaining the dependent variable, but merelythat we have more evidence of the sign of the coefficient in question

3. The t-Test Is Not Intended for Tests of the Entire Population:Ï From the definition of the t-score, given by equation (2), it is seen that as the sample

size approaches the population (whereby the standard error will approach zero sincethe standard error decreases as N increases), the t-score will approach infinity.


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