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1、Chapter 5What Is Hypothesis Testing?Hypothesis testing and statistical inference allow us to answer questions about the real world from a sample.It is used in a variety of settings:FDA product testingTesting theories of Keynes and FriedmanIts almost impossible to prove a theory is “correct” with hyp

2、othesis testing.All that can be done with hypothesis testing is to state that a particular sample conforms to a particular hypothesis.What Is Hypothesis Testing? (continued)The first step in hypothesis testing is to state the hypotheses to be tested.This should be done before equation is estimated.T

3、he hypothesis test is broken into two hypotheses:null hypothesis, denoted “H0:”, typically is a statement of the values not expected.alternative hypothesis denoted “HA:”, typically is a statement of the values expected.What Is Hypothesis Testing? (continued)Example: If you expect a positive coeffici

4、entNull hypothesis: H0: 0(the values you do not expect)Alternative hypothesis: HA: 0(the values you expect)Example: If you expect a negative coefficientNull hypothesis: H0: 0Alternative hypothesis: HA: 0Case 1: Assume true is NOT positive but researchers estimate leads to the rejection of the null h

5、ypothesis: This is a Type I ErrorCase 2 Assume true is positive but researchers estimate leads to not rejecting the null hypothesis: This is a Type II ErrorWhat Is Hypothesis Testing? (continued)A decision rule is a method of deciding whether to reject a null hypothesis.Typically a decision rule inv

6、olves comparing a sample statistic with a critical value.A decision rule should be formulated before regression estimates are obtained.Divide the range of possible values of into two regions:1. “Acceptance” region2. Rejection regionWhat Is Hypothesis Testing? (continued)A critical value divides “acc

7、eptance” and rejection regions.For a two-tailed test, two critical values are selected.Rejection region measures probability of a Type I Error if the null is true.Unfortunately, decreasing the chance of a Type I Error means increasing the chance of a Type II Error.Figures 5.1 and 5.2 graph “acceptan

8、ce” and rejection regions for a one-sided test and two-sided tests, respectively.What Is Hypothesis Testing? (continued)What Is Hypothesis Testing? (continued)The t-TestThe t-test is used to test hypotheses about individual slope coefficients.It is the appropriate test when:1. The stochastic error t

9、erm is normally distributed.2. Variance of the distribution must be estimated.Since these are usually the case, the use of the t-test for hypothesis testing has e standard in econometrics.The t-Test (continued)We can calculate t-values for each estimated coefficient of a typical multiple regression

10、equation:t-statistic for the kth coefficient:Where: = the estimated regression coef. for the kth variable= the border value coef. for the kth variable= the estimated standard error of The t-Test (continued)Since most regression hypotheses test whether a particular regression coefficient is significa

11、ntly different from 0, is typically 0.Thus, the most-used form of the t-statistic es: which simplifies to:The t-Test (continued)Example: Consider Woodys restaurant from Section 3.2Testing H0: P 0, the t-statistic is:The t-Test (continued)Whether to reject or not to reject a null hypothesis is based

12、comparing calculated t-value to a critical t-value.Critical t-value is the value that distinguishes the “acceptance” region from the rejection region.The critical t-value, tc, is selected from a t-table based on:1. Whether the test is one-sided or two-sided2. Level of Type I Error specified3. Degree

13、s of freedomThe t-Test (continued)Once a critical t-value (tc) has been selected and calculated t-value (tk) obtained, apply the following decision rule:Reject H0 if |tk| tc and if tk has the sign implied by HA. Do not reject H0 otherwise.The t-Test (continued)This decision rule is used for:1. One-s

14、ided hypotheses around zero:H0: k 0H0: k 0HA: k 0HA: k SHA: k 0There are 29 degrees of freedom (N K - 1, or 33 3 - 1) so appropriate tc at 5-percent significance: 1.699.Decision rule: Reject H0 if |tP| 1.699 and if tP is positiveRecall tP = +4.88 Since |4.88| 1.699 and 4.88 is positive, reject H0.Th

15、e t-Test (continued)The level of significance indicates the probability of observing an estimated t-value greater than the critical value if the null hypothesis were correct.It measures probability of a Type I Error implied by a particular critical t-value.How should you choose a level of significan

16、ce?Most beginning econometricians assume the lower the level of significance the better.The t-Test (continued)Unfortunately, a low level of significance dramatically increases the probability of a Type II Error.Using a 5-percent level of significance is mended unless you know something unusual about

17、 the relative costs of making Type I or Type II Errors.If you can reject the null at the 5-percent level of significance, it is common to summarize and say:“The coefficient is statistically significant at the 5-percent level.”The t-Test (continued)The p-value, or marginal significance level, is an a

18、lternative to the t-test.A p-value for a t-score is the probability of observing a t-score that size or larger (in absolute terms) if the null hypothesis were true.Standard regression software packages calculate p-values automatically for every coefficient.Be careful! Virtually every package reports

19、 p-values for two-sided hypotheses.The t-Test (continued)How do you use a p-value to run a t-test?Apply p-value decision rule:Reject H0 if p-value 0HA: 2 0HA: 3 1.943 and if 2.1 is positive. For 2: Reject H0 if |2.8| 1.943 and if 2.8 is negative. For 3: Reject H0 if |-0.1| 1.943 and if -0.1 is negat

20、ive.Figure 5.4 illustrates all three es. Examples of t-Tests (continued)Examples of t-Tests (continued)Two-sided tests fall into two categories:1. Two-sided tests of whether an estimate coefficient is significantly different than zero, and2. Two-sided tests of whether an estimated coefficient is sig

21、nificantly different from a specific value.Examples of t-Tests (continued)Example: Two-sided t-test whether an estimated coefficient is statistically different than zero: Woodys restaurant locationStep 1: Set up null and alternative hypothesisH0: I = 0HA: I 0Examples of t-Tests (continued)Step 2: Ch

22、oose a level of significance and therefore a critical t-value.5-percent significance leveldegrees of freedom = 29 Step 3: Run regression and obtain an estimated t-value. Examples of t-Tests (continued)Step 4: Apply the decision rule by comparing the calculated t-value with the critical t-value in or

23、der to reject or not reject the null hypothesis. For I: Reject H0 if |2.37| 2.045In this case, you reject the null hypothesis that I equals zero because 2.37 is greater than 2.045. Examples of t-Tests (continued)For a two-sided t-test with a specific nonzero coefficient value, the null and alternati

24、ve hypotheses e:H0: HA: The estimated t-value is:The decision rule remains the same.Limitations of the t-TestsOne problem with the t-test is that it is easy to misuse.Three limitations:1. The t-test does not test theoretical validityCoefficients are statistically significant.The catch is that P is t

25、he consumer price index and C is the cumulative amount of rainfall in the United Kingdom!Limitations of the t-Tests (continued)2. The t-test does not test importance.With a t-score of 10.0, X1s is more statistically significant than X2.That means is there is more evidence X1s coefficient is positive

26、does not mean more important. Limitations of the t-Tests (continued)3. The t-test is not intended for tests of the entire population.The t-test helps make inferences about the true value of a parameter from a sample of the population.A unbiased coefficient calculated from entire population, already

27、measures the population value.A significant t-test adds nothing to this knowledge. If the sample size is large enough to approach the population, then the standard error will approach zero and the t-score will e:Confidence IntervalsConfidence intervals are a third way to do hypothesis testing.A conf

28、idence interval is a range of values that will contain the true value of a certain percentage of the time.The confidence interval formula:confidence interval = tc is the two-sided critical value of the t-statistic for the chose significance level.Confidence Intervals (continued)Example: Woodys resta

29、urant site locationA 90 percent confidence interval for I:Theres a 90 percent chance true I falls between 0.365 and 2.211.Confidence Intervals (continued)If testing:H0: I = 0HA: I 0We can reject null hypothesis at the 10-percent level because 0 is not in the confidence interval.If testing:H0: I = 1H

30、A: I 1We cannot reject null hypothesis at the 10-percent level because 1 is in the confidence interval.Confidence Intervals (continued)Confidence intervals are also very useful in telling how precise a coefficient estimate is.Example: Contractor Grace and value of a bathroomThe F-TestThe F-test is a

31、 formal hypothesis test designed to deal with a null hypothesis that contains multiple hypotheses or a single hypothesis about a group of coefficients.The way an F-test works is fairly ingenious:1. Translate the null hypothesis into constraints that will be placed on the equation.2. Estimate the con

32、strained equation with OLS and compare the fit of the constrained equation with the fit of the unconstrained equation.The F-Test (continued)The fits of the equations are compared with the general F-statistic:where: RSS= residual sum of squares from the unconstrained equation RSSM = residual sum of squares from the constrained equation M = number of constraints (N-K-1) = degrees of freedom in the unconstrained equation.The F-Test (continued)Decision rule is reject the null hypothesis if calculated F-value (F) is greater than the critical F-v