Handout1 (1)

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School
University of California, Berkeley **We aren't endorsed by this school
Course
A,RESEC 210
Subject
Statistics
Date
Aug 25, 2023
Pages
76
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ARE 210 Handout 1 Contents 1 Main Object of First Year Sequence 6 2 Probability Space 6 2.1 Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Conditional Probability 16 3.1 Bayesian Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Independence 22 4.1 Independence of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Independence of σ Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Random Variables and Distributions 23 5.1 Distribution of a Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.1.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 Independence of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Transformation of Independent Random Variables . . . . . . . . . . . . . . . . . . . 31 5.4 The Cumulative Distribution Function for Scalar R.V.s . . . . . . . . . . . . . . . . 32 5.5 Cumulative Distribution Function for Random Vectors . . . . . . . . . . . . . . . . 34 5.6 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.7 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.8 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.9 "Mixed" Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.10 Degenerate Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 These lecture notes are very much a work in progress. Please let me know of any errors or typos you find. Thanks to Shelley He for going through the notes and identifying errors. This Version: July 28, 2023 1
5.11 Independence via CDFs and Densities . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Transformations of Random Variables I 47 7 Joint Distributions I 53 8 Transformation of Random Variables II 55 9 The Quantile Function 65 10 Neyman-Rubin Notation 68 10.1 Treatments Staggered Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 10.1.1 When T > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A Optional Material 73 A.1 Constructing the Cantor Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A.2 Lebesgue Measure on the Power Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2
Figure 1: Bloom et al. ( 2013 ) APPENDIX TABLE A.II OLS AND IV E STIMATIONS OF THE E FFECT OF M ANAGEMENT P RACTICES ON P LANT P ERFORMANCE (1) (2) (3) (4) (5) (6) (7) (8) Specification OLS IV 2nd stage OLS IV 2nd stage OLS IV 2nd stage OLS IV 2nd stage Dependent variable Quality defects Quality defects Inventory Inventory Output Output TFP TFP Management i,t -0.558 -1.694** -0.404 -0.833** 0.119 0.310** 0.167 0.477** (0.440) (0.850) (0.259) (0.347) (0.099) (0.128) (0.176) (0.242) Specification IV 1st stage IV 1st stage IV 1st stage IV 1st stage Dependent variable Management Management Management Management Cumulative treatment i,t 0.018*** 0.017*** 0.019*** 0.019*** (0.002) (0.002) (0.002) (0.002) Small sample robustness Ibragimov-Mueller (95% CI) [ À 19.00,5.22] [ À 5.28,-1.18] [ À 4.77,0.11] [ À 1.50,-0.57] [0.27,1.38] [0.15,1.21] [ À 3.59,11.97] [0.27,1.90] IV Permutation Tests (95% CI) [ À 28.05,-0.18] [ À 6.85,0.37] [0.09,0.78] [0.44,3.91] First stage F -test 73.41 72.88 107.55 86.00 Time FEs 127 127 127 127 127 127 127 127 Plant FEs 20 20 18 18 20 20 20 20 Observations 1,807 1,807 2,052 2,052 2,393 2,393 1,831 1,831 Notes. All regressions use a full set of plant and calendar week dummies. Standard errors bootstrap clustered at the firm level. Quality defects is a log of the quality defects index (QDI). Inventory is the log of the tons of yarn inventory in the plant. Output is the log of the weaving production picks. Management is the adoption share of the 38 management practices listed in Online Appendix Table AI . Cumulative treatment is the cumulative weeks of since beginning the implementation phase in each plant (0 in the control groups and prior to the implementation phase). OLS reports results with plant estimations. IV reports the results where the management variable has been instrumented with weeks of cumulative treatment. Time FEs report the number of calendar week time fixed effects. Plant FEs reports the number of plant-level fixed effects. Two plants do not have any inventory on site, so no inventory data are available. Small sample robustness implements three different procedures (described in greater detail in Online Appendix B ) to address issues of plant heterogeneity, within plant (and firm) correlation, and small sample concerns, where 95% CI reports 95% confidence intervals. Ibragimov-Mueller estimates parameters firm by firm and then treats the estimates as a draw from independent (but not identically distributed) normal distributions. Permutation Test I reports the p -values for testing the null hypothesis that the treatment has no effect for the ITT parameter by constructing a permutation distribution of the ITT estimate using 1,000 possible permutations (out of 12,376) of treatment assignment. IV-Permutation tests implements a permutation test for the IV parameter using 1,000 possible permutations (out of 12,376) of treatment assignment. These tests have exact finite sample size. *** denotes 1%, ** denotes 5%, * denotes 10% significance. QUARTERLY JOURNAL OF ECONOMICS 48 at University of California, Los Angeles on January 15, 2013 http://qje.oxfordjournals.org/ Downloaded from 3
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