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Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: t tests Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/141/http://learningresources.lse.ac.uk/141/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

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1 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > t crit or t < –t crit The diagram summarizes the procedure for performing a 5% significance test on the slope coefficient of a regression under the assumption that we know its standard deviation.

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2 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > t crit or t < –t crit This is a very unrealistic assumption. We usually have to estimate it with the standard error, and we use this in the test statistic instead of the standard deviation. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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3 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > t crit or t < –t crit Because we have replaced the standard deviation in its denominator with the standard error, the test statistic has a t distribution instead of a normal distribution. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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4 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > t crit or t < –t crit Accordingly, we refer to the test statistic as a t statistic. In other respects the test procedure is much the same. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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5 We look up the critical value of t and if the t statistic is greater than it, positive or negative, we reject the null hypothesis. If it is not, we do not. s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > t crit or t < –t crit t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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6 Here is a graph of a normal distribution with zero mean and unit variance normal t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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7 A graph of a t distribution with 10 degrees of freedom (this term will be defined in a moment) has been added. normal t, 10 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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8 When the number of degrees of freedom is large, the t distribution looks very much like a normal distribution (and as the number increases, it converges on one). normal t, 10 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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9 Even when the number of degrees of freedom is small, as in this case, the distributions are very similar. normal t, 10 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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10 Here is another t distribution, this time with only 5 degrees of freedom. It is still very similar to a normal distribution. normal t, 5 d.f. t, 10 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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11 So why do we make such a fuss about referring to the t distribution rather than the normal distribution? Would it really matter if we always used 1.96 for the 5% test and 2.58 for the 1% test? normal t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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12 The answer is that it does make a difference. Although the distributions are generally quite similar, the t distribution has longer tails than the normal distribution, the difference being the greater, the smaller the number of degrees of freedom. normal t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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13 As a consequence, the probability of obtaining a high test statistic on a pure chance basis is greater with a t distribution than with a normal distribution. normal t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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14 This means that the rejection regions have to start more standard deviations away from zero for a t distribution than for a normal distribution. normal t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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15 The 2.5% tail of a normal distribution starts 1.96 standard deviations from its mean. normal -1.96 t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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16 The 2.5% tail of a t distribution with 10 degrees of freedom starts 2.33 standard deviations from its mean. normal -2.33 t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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17 That for a t distribution with 5 degrees of freedom starts 2.57 standard deviations from its mean. normal -2.57 t, 10 d.f. t, 5 d.f. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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18 For this reason we need to refer to a table of critical values of t when performing significance tests on the coefficients of a regression equation. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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19 At the top of the table are listed possible significance levels for a test. For the time being we will be performing two-sided tests, so ignore the line for one-sided tests. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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20 Hence if we are performing a (two-sided) 5% significance test, we should use the column thus indicated in the table. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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21 The left hand vertical column lists degrees of freedom. When estimating the population mean using the sample mean, the number of degrees of freedom is defined to be the number of observations minus 1. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 When estimating the population mean using the sample mean, the number of degrees of freedom = n – 1. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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22 Thus, if there are 20 observations in the sample, as in the sales tax example we will discuss in a moment, the number of degrees of freedom would be 19 and the critical value of t for a 5% test would be 2.093. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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23 Note that as the number of degrees of freedom becomes large, the critical value converges on 1.96, the critical value for the normal distribution. This is because the t distribution converges on the normal distribution. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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24 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > t crit or t < –t crit t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN Hence, referring back to the summary of the test procedure,

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25 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : = 0 if t > 2.093 or t < –2.093 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN we should reject the null hypothesis if the absolute value of t is greater than 2.093.

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26 If instead we wished to perform a 1% significance test, we would use the column indicated above. Note that as the number of degrees of freedom becomes large, the critical value converges to 2.58, the critical value for the normal distribution. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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27 For a samplr of 20 observations, the critical value of t at the 1% level is 2.861. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% 16.31412.70631.82163.657318.31636.62 22.9204.3036.9659.92522.32731.598 32.3533.1824.5415.84110.21412.924 42.1322.7763.7474.6047.1738.610 52.0152.5713.3654.0325.8936.869 ………………… 181.7342.1012.5522.8783.6103.922 191.7292.0932.5392.8613.5793.883 201.7252.0862.5282.8453.5523.850 ………………… 6001.6471.9642.3332.5843.1043.307 1.6451.9602.3262.5763.0903.291 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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28 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : = 0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 1% significance test: reject H 0 : = 0 if t > 2.861 or t < –2.861 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN So we should use this figure in the test procedure for a 1% test.

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We now consider an example of a t test. 29 Example: A certain city abolishes its local sales tax on consumer expenditure. A survey of 20 households shows that, in the following month, mean household expenditure increased by $160 and the standard error of the increase was $60. We wish to determine whether the abolition of the tax had a significant effect on household expenditure. We take as our null hypothesis that there was no effect: H 0 : = 0. The test statistic is The critical values of t with 19 degrees of freedom are 2.09 at the 5 percent significance level and 2.86 at the 1 percent level. Hence we reject the null hypothesis of no effect at the 5 percent level but not at the 1 percent level. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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30 Example: A certain city abolishes its local sales tax on consumer expenditure. A survey of 20 households shows that, in the following month, mean household expenditure increased by $160 and the standard error of the increase was $60. We wish to determine whether the abolition of the tax had a significant effect on household expenditure. We take as our null hypothesis that there was no effect: H 0 : = 0. The test statistic is The critical values of t with 19 degrees of freedom are 2.09 at the 5 percent significance level and 2.86 at the 1 percent level. Hence we reject the null hypothesis of no effect at the 5 percent level but not at the 1 percent level. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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31 Example: A certain city abolishes its local sales tax on consumer expenditure. A survey of 20 households shows that, in the following month, mean household expenditure increased by $160 and the standard error of the increase was $60. We wish to determine whether the abolition of the tax had a significant effect on household expenditure. We take as our null hypothesis that there was no effect: H 0 : = 0. The test statistic is The critical values of t with 19 degrees of freedom are 2.09 at the 5 percent significance level and 2.86 at the 1 percent level. Hence we reject the null hypothesis of no effect at the 5 percent level but not at the 1 percent level. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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32 Example: A certain city abolishes its local sales tax on consumer expenditure. A survey of 20 households shows that, in the following month, mean household expenditure increased by $160 and the standard error of the increase was $60. We wish to determine whether the abolition of the tax had a significant effect on household expenditure. We take as our null hypothesis that there was no effect: H 0 : = 0. The test statistic is The critical values of t with 19 degrees of freedom are 2.09 at the 5 percent significance level and 2.86 at the 1 percent level. Hence we reject the null hypothesis of no effect at the 5 percent level but not at the 1 percent level. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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33 Example: A certain city abolishes its local sales tax on consumer expenditure. A survey of 20 households shows that, in the following month, mean household expenditure increased by $160 and the standard error of the increase was $60. We wish to determine whether the abolition of the tax had a significant effect on household expenditure. We take as our null hypothesis that there was no effect: H 0 : = 0. The test statistic is The critical values of t with 19 degrees of freedom are 2.09 at the 5 percent significance level and 2.86 at the 1 percent level. Hence we reject the null hypothesis of no effect at the 5 percent level but not at the 1 percent level. t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN

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Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.11 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 11.07.25

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