Module08 - Statistical Modelling for Business QBUS2810...

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Statistical Modelling for Business QBUS2810 Module 8: Statistical Inference for MLR II Marcel Scharth Discipline of Business Analytics, The University of Sydney Business School 1 / 46
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MLR modules Multiple linear regression. Statistical inference for MLR I. Statistical inference for MLR II . 2 / 46
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Outline Module 8: Statistical Inference for MLR II Standardised coefficients. Inference for one coefficient. F-tests for multiple coefficients. Appendix Reference: Fox Section 6.2. 3 / 46
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Standardised coefficients Standard errors The standard error for b β j is SE ( b β j ) = SER q ( 1 - R 2 j ) ( n i =1 ( x ji - x j ) 2 ) , where the standard error of the regression SER = v u u t 1 n - p - 1 n X i =1 e 2 i estimates σ . 4 / 46
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Standardised coefficients Robust standard errors You should use robust standard errors if the assumption of constant error variance is not satisfied for the data. We do not discuss the technical details of robust standard errors for the MLR case. 5 / 46
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Standardised coefficients Sampling distribution in Gaussian MLR model If the errors are Gaussian, we can show that b β j - β j SE ( b β j ) t n - p - 1 6 / 46
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Standardised coefficients Sampling distribution (general case) If the distribution of the errors is unspecified but the sample size is sufficiently large, we use the CLT approximation: b β j - β j SE ( b β j ) N (0 , 1) or b β j - β j SE ( b β j ) t n - p - 1 7 / 46
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Topics Standardised coefficients. Inference for one coefficient . F-tests for multiple coefficients. Appendix 8 / 46
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Inference for one coefficient Hypothesis testing Relationship test: Two-sided : H 0 : β j = 0 H 1 : β j 6 = 0 One-sided : H 0 : β j = 0 H 1 : β j > 0 9 / 46
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Inference for one coefficient Hypothesis testing H 0 : β j = 0 H 1 : β j 6 = 0 Interpretation: If the null hypothesis is correct, there is no relationship between predictor j and the response conditional on the other predictors. Alternatively, we can say that variable j does not predict the response, after taking the other predictors into account. 10 / 46
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Inference for one coefficient Hypothesis testing General test: Two-sided : H 0 : β j = b H 1 : β j 6 = b One-sided : H 0 : β j = b H 1 : β j > b 11 / 46
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Inference for one coefficient Test statistic t b β j = b β j - b j SE ( b β j ) t n - p - 1 We can carry out hypothesis testing on β j using the standard t-statistic with degrees of freedom n - p - 1 . The test is a large sample approximation when the errors are non-Gaussian. 12 / 46
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Hypothesis testing Summary One-tailed (left) Two-tailed One-tailed (right) H 0 : β j = b H 0 : β j = b H 0 : β j = b H 1 : β j < b H 1 : β j 6 = b H 1 : β j > b Test statistic t b β j = b β j - b SE ( b β j ) t n - p - 1 (under H 0 and assumptions) Rejection region t b β j < - t n - p - 1 t b β j < - t n - p - 1 ,α/ 2 t b β j > t n - p - 1 and t b β 1 > t n - p - 1 ,α/ 2 p-value ( t b β j t n - p - 1 ) P ( t n - p - 1 < t b β j ) 2 × P ( t n - p - 1 > | t b β j | ) P ( t n - p - 1 > t b β j ) 13 / 46
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Inference for one coefficient Why the p-value does not measure coefficient importance t b β j = b β j SER h ( 1 - R 2 j ) ( n - 1) s 2 x j i 1 / 2 Interpretation, all else equal: A higher coefficient magnitude leads to a larger test statistic.
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