8 Nonparametric test. Question 1: Are (expected) value of x and y the same?

Transcription

1 Econometrics A: Tokyo International University 2017 autumn Satoshi OHIRA 26 8 Nonparametric test Question 1: Are (expected) value of x and y the same? One of the simplest way to answer the question is to compare mean of x and y. We, however, have to compare two distributions. ( two statistics) mean/median comparison can assume normal dist cannot t test compare two mean Wilcoxon rank sum test compare two median We are not sure if we can assume normality when 1. sample size is quite small and/or 2. there is/are outlier(s). Nonparametric test, for example Wilcoxon test, is applicable in the case. Question 2: Are x and y related? Correlation coefficient is a typical statistic. correlation can assume normal dist cannot correlation coefficient rank correlation coefficient Question 3: Are x and y from the same distribution? Kolmogorov-Smirnov test

2 Econometrics A: Tokyo International University 2017 autumn Satoshi OHIRA 27 data: per capita GDP and human capital in 2010 countryname region pgdp hc Haiti Latin America & Caribbean Cambodia East Asia & Pacific Timor-Leste East Asia & Pacific Myanmar East Asia & Pacific Lao PDR East Asia & Pacific Solomon Islands East Asia & Pacific Vietnam East Asia & Pacific Kiribati East Asia & Pacific Nicaragua Latin America & Caribbean Honduras Latin America & Caribbean Bolivia Latin America & Caribbean Papua New Guinea East Asia & Pacific Philippines East Asia & Pacific Mongolia East Asia & Pacific Guatemala Latin America & Caribbean Micronesia, Fed. Sts. East Asia & Pacific Vanuatu East Asia & Pacific Tuvalu East Asia & Pacific Guyana Latin America & Caribbean Indonesia East Asia & Pacific Marshall Islands East Asia & Pacific Paraguay Latin America & Caribbean Samoa East Asia & Pacific El Salvador Latin America & Caribbean Tonga East Asia & Pacific Fiji East Asia & Pacific Belize Latin America & Caribbean China East Asia & Pacific Ecuador Latin America & Caribbean Jamaica Latin America & Caribbean Peru Latin America & Caribbean Thailand East Asia & Pacific Dominican Republic Latin America & Caribbean St. Vincent and the Grenadines Latin America & Caribbean Colombia Latin America & Caribbean Dominica Latin America & Caribbean Grenada Latin America & Caribbean Panama Latin America & Caribbean St. Lucia Latin America & Caribbean Costa Rica Latin America & Caribbean Suriname Latin America & Caribbean Palau East Asia & Pacific Mexico Latin America & Caribbean Malaysia East Asia & Pacific Brazil Latin America & Caribbean Uruguay Latin America & Caribbean Antigua and Barbuda Latin America & Caribbean Chile Latin America & Caribbean Venezuela, RB Latin America & Caribbean St. Kitts and Nevis Latin America & Caribbean Barbados Latin America & Caribbean Trinidad and Tobago Latin America & Caribbean Bahamas, The Latin America & Caribbean Korea, Rep. East Asia & Pacific Aruba Latin America & Caribbean Hong Kong SAR, China East Asia & Pacific New Zealand East Asia & Pacific Brunei Darussalam East Asia & Pacific Japan East Asia & Pacific Singapore East Asia & Pacific Australia East Asia & Pacific Curacao Latin America & Caribbean..

3 Econometrics A: Tokyo International University 2017 autumn Satoshi OHIRA mean/median comparison. ttest pgdp, by(region) Two-sample t test with equal variances Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] East Asi Latin Am combined diff diff = mean(east Asi) - mean(latin Am) t = Ho: diff = 0 degrees of freedom = 59 Ha: diff < 0 Ha: diff!= 0 Ha: diff > 0 Pr(T < t) = Pr( T > t ) = Pr(T > t) = We are not sure if we can assume pgdp follows normal distribution. Density 0 5.0e e 04 East Asia & Pacific Latin America & Caribbean pgdp Graphs by Region. ranksum pgdp, by(region) Two-sample Wilcoxon rank-sum (Mann-Whitney) test region obs rank sum expected East Asia & Latin Americ combined unadjusted variance adjustment for ties adjusted variance Ho: pgdp(region==east Asia \& Pacific) = pgdp(region==latin America \& Caribbean) z = Prob > z =

4 Econometrics A: Tokyo International University 2017 autumn Satoshi OHIRA 29 The result is different if we restrict our sample (pgdp<10000).. ranksum pgdp if pgdp<10000, by(region) Two-sample Wilcoxon rank-sum (Mann-Whitney) test region obs rank sum expected East Asia & Latin Americ combined unadjusted variance adjustment for ties adjusted variance Ho: pgdp(region==east Asia \& Pacific) = pgdp(region==latin America \& Caribbean) z = Prob > z = the result of t test is. ttest pgdp if pgdp<10000, by(region) Two-sample t test with equal variances Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] East Asi Latin Am combined diff diff = mean(east Asi) - mean(latin Am) t = Ho: diff = 0 degrees of freedom = 42 Ha: diff < 0 Ha: diff!= 0 Ha: diff > 0 Pr(T < t) = Pr( T > t ) = Pr(T > t) = correlation. pwcorr pgdp hc, sig pgdp hc pgdp hc The linear relationship between two variables is not clear.

5 Econometrics A: Tokyo International University 2017 autumn Satoshi OHIRA 30 pgdp Human capital index, see note hc. reg pgdp hc Source SS df MS Number of obs = 40 - F(1, 38) = Model e e+09 Prob > F = Residual e R-squared = Adj R-squared = Total e Root MSE = pgdp Coef. Std. Err. t P> t [95% Conf. Interval] hc _cons The linear relationship between the ranks data of two variables is clearer. rank of (hc) rank of (pgdp). spearman pgdp hc Number of obs = 40 Spearman s rho = Test of Ho: pgdp and hc are independent Prob > t = ktau pgdp hc Number of obs = 40 Kendall s tau-a = Kendall s tau-b = Kendall s score = 456 SE of score = Test of Ho: pgdp and hc are independent Prob > z = (continuity corrected)

6 Econometrics A: Tokyo International University 2017 autumn Satoshi OHIRA comparison of distribution. ksmirnov pgdp, by(region) Two-sample Kolmogorov-Smirnov test for equality of distribution functions Smaller group D P-value East Asia & Paci: Latin America & : Combined K-S: The first line tests the hypothesis that East Asia & Pacific contains smaller values than Latin America & Caribbean. The largest difference between the distribution functions is The approximate p-value for this is 0.014, which is not significant. The second line tests the hypothesis that East Asia & Pacific contains larger values than Latin America & Caribbean. The largest difference between the distribution functions in this direction is The approximate p- value for this small difference is Finally, the approximate p-value for the combined test is These approximations are not good for small samples (n < 50).. ksmirnov pgdp, by(region) exact Two-sample Kolmogorov-Smirnov test for equality of distribution functions Smaller group D P-value Exact East Asia & Paci: Latin America & : Combined K-S: