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ebusiness

  1. 1. Undergraduate Econometrics Wei-Chih Chen SIBA, SHUFE Fall 2010Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 1 / 10
  2. 2. Hypothesis Testing The hypothesis testing problem (for the mean): make a provisional decision, based on the evidence at hand, whether a null hypothesis is true, or instead that some alternative hypothesis is true. That is, test H0 : E (Y ) = µY ,0 vs. H1 : E (Y ) > µY ,0 (1-sided, >) H0 : E (Y ) = µY ,0 vs. H1 : E (Y ) < µY ,0 (1-sided, <) H0 : E (Y ) = µY ,0 vs. H1 : E (Y ) 6= µY ,0 (2-sided) Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 2 / 10
  3. 3. Hypothesis Testing Some terminology for testing statistical hypotheses: p-value = probability of drawing a statistic (e.g. Y ) at least as adverse to the null as the value actually computed with your data, assuming that the null hypothesis is true. The signi…cance level of a test is a pre-speci…ed probability of incorrectly rejecting the null, when the null is true. Calculating the p-value based on Y : act act p value = PrH 0 [jY µY 0 j > jY µY 0 j], where Y is the value of Y actually observed (nonrandom) Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 3 / 10
  4. 4. The p-value In practice, σY is unknown - it must be estimated Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 4 / 10
  5. 5. Estimation of the Variance n 1 Σi =1 (Yi 2 1 n The sample variance: sY = Y )2 q The sample stadard deviation: sY = 2 sY sY The standard error of Y : SE(Y )= pn The p-value when σY is unknown: see note Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 5 / 10
  6. 6. The t Statistic Y µY 0 The t-statistic or t-ratio: t = SE (Y ) When n is large, the distribution of the t-statistic is approximated by the standard normal distribution (CLT) Suppose the signi…cance level is 5%. We reject H0 if jt act j > 1.96 When n is small and Y is normally distributed, the t-statistic has a Student t distribution with n 1 degrees of freedom Suppose the signi…cance level is 5% and n = 10. We reject H0 if jt act j > 2.26 (check the table) Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 6 / 10
  7. 7. The Link between the p-value and the Signi…cance Level The signi…cance level is prespeci…ed. For example, if the prespeci…ed signi…cance level is 5%, you reject the null hypothesis if jt j > 1.96 equivalently, you reject if p 0.05. Often, it is better to communicate the p-value than simply whether a test rejects or not – the p-value contains more information than the “yes/no” statement about whether the test rejects. Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 7 / 10
  8. 8. Con…dence Intervals A 95% con…dence interval for µY is an interval that contains the true value of µY in 95% of repeated samples. A 95% con…dence interval can always be constructed as the set of values of µY not rejected by a hypothesis test with a 5% signi…cance level. 95% con…dence interval for µY = fY 1.96 SE (Y )g (see note) What is random here? The values of Y1 , . . . , Yn and thus any functions of them, including the con…dence interval. The con…dence interval it will di¤er from one sample to the next. The population parameter, µY , is not random, we just don’ know it. t Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 8 / 10
  9. 9. Di¤erences-of-Means Estimation Suppose we want to test the hypothesis that two means are equal: µm µf = 0 Y Yf Ym Yf The t-statistic: t= r m2 = SE (Y m Y f ) , where sm and sf2 are 2 sm s2 nm + nf f sample variances If both nm and nf are large, the t-statistic has a standard normal distribution If nm and nf are small, the t-statistic doesn’ have a Student t t distribution, even if the population distribution of Y in the two groups is normal! There is a statistic testing this hypothesis that has a normal 2 distribution, the “pooled variance” t-statistic – spooled However the pooled variance t-statistic is only valid if the variances of the normal distributions are the same in the two groups. Would you expect this to be true, say, for men’ v. women’ wages? s s Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 9 / 10
  10. 10. Summary From the two assumptions of: 1 simple random sampling of a population, that is, {Yi , i = 1, . . . , n} are i.i.d. 2 0 < E (Y 4 ) < ∞ We developed, for large samples (large n): Theory of estimation (sampling distribution of Y ) Theory of hypothesis testing (large-n distribution of t-statistic and computation of the p-value) Theory of con…dence intervals (constructed by inverting test statistic) Wei-Chih Chen (SIBA, SHUFE) Undergraduate Econometrics Fall 2010 10 / 10

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