l01 - Overview Linear regression model. k Yj 1 X 1 j 2 X 2...

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Overview — Page 1 Overview Linear regression model. Y X X X X E Y X j j j k kj j i ij j i k j j 1 1 2 2 1 ... ( | ) i partial slope coefficient (also called partial regression coefficient, metric coefficient, or just regression coefficient). It represents the change in E(Y) associated with a one-unit increase in X i when all other IVs are held constant. the intercept. Geometrically, it represents the value of E(Y) where the regression surface (or plane) crosses the Y axis. Substantively, it is the expected value of Y when all the IVs equal 0. the deviation of the value Y j from the mean value of the distribution given X. This error term may be conceived as representing (1) the effects on Y of variables not explicitly included in the equation, and (2) a residual random element in the dependent variable. Parameter estimation. In most situations, we are not in a position to determine the population parameters directly. Instead, we must estimate their values from a finite sample from the population. The sample regression model is written as Y a b X b X b X e a b X e Y e j j j k kj j i ij j i k j j 1 1 2 2 1 ... where a is the sample estimate of and b k is the sample estimate of k . Hypothesis testing. We use the sample to test hypotheses about the population parameters. Some typical hypotheses: H 0 : 1 = 0 H A : 1 <> 0 One of the Betas = 0. Use a T test or an F test. H 0 : 1 = 2 = 3 ... = k = 0 H A : At least one beta <> 0 Test of whether any Betas differ from 0. Equivalent to a test of R 2 = 0. F test H 0 : 2 = 3 ... = k = 0 H A : At least one of the above betas <> 0 Test whether a subset of the betas = 0. Same as a test of R 2 constrained = R 2 unconstrained . F test H 0 : 1 = 2 H A : 1 <> 2 Test whether two betas are =. F test, or maybe a t-test. H 0 : (1) = (2) H A : (1) <> (2) Test whether the betas are the same in two different populations, e.g. is the model for men the same as the model for women? F test
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Overview — Page 2 Assumptions and violations of assumptions Assumptions concerning correct model specification . One of the most critical assumptions you make is that the model is correctly specified, i.e. it really is the case that Y X j i ij j i k 1 . Incorrect model specification can result in biased parameter estimates and violations of other assumptions. The following assumptions all follow from the requirement that the model be correctly specified. A good theory, and an accurate understanding of what a
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l01 - Overview Linear regression model. k Yj 1 X 1 j 2 X 2...

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