Week 2 - More on the Reliability Precision and Performance...

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More on the Reliability, Precision, and Performance of the regression model and its estimated parameters. As the least-squares coefficient/parameter estimates ( β j ’s) and the SRF’s ability to explain variation in the dependent variable (Y) can vary from sample to sample, what is needed are some sort of measures of reliability and precision. Let us review some of the more useful indices and tests.
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1. Standard error of β and related statistics Standard error ” of β the standard deviation of the sampling distribution of β… based on sample estimates from repeated samples of a given size n, and general noted as se(β j ) for any given β j (where j = 0, 1, …., k) and se(β 0 ) standard error of estimated coefficient for the y-intercept or constant term; and se(β j ) standard error of estimated slope coefficient associated with a j- th regressor (j=1,…, k)
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Hypothesis testing for estimated regression coefficients To assess whether an “estimated β j ” (β j ) differs significantly from a hypothesized value of β j , as designated under a stated null hypothesis. For example, consider a the two-tailed test: H o : β j = β j,H o H a : β j = β j,H o , (j=0, 1,…, k) Typically, the default test procedure assume β j,H o = 0. ^ ^ ^
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Test statistic : t-statistic t = ^ ^ β j - β j,H o se(β j ) …or when we are testing to see if the estimated beta coefficient is significantly different from a value of zero: t = ^ ^ β j se(β j ) …distributed as a t -distribution with n-k* degrees of freedom.
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t Probability density -t α/2 0 +t α/2 non-rejection region rejection region rejection region α/2 α/2 (reject H o ) (reject H o ) (fail to reject or accept H o ) 2-tailed test criterion: Since | t | > | t α/2 | , we must “reject” H o at the (1- α ) x 100% level of confidence. Note: Critical t-values are found for (n-k*) degrees of freedom, where n sample size; k number of regressors; k*=k+1 number of regression coefficents to be estimated including the intercept term or constant. t-distribution
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In general, the t-tests on the individual β’s allow us to evaluate the “explanatory power” and/or “statistical significance” of each individual explanatory variable in the model. Rule of thumb : the higher the t-value, the greater the contribution of a variable (X) to explain variation in a dependent variable Y.
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For the bi-variate model, it can be shown that the “standard error” of the estimated slope parameter β 1 is se( β 1 ) = σ 2 / Σ (X i - X ) 2 ^ ^ n [ Σ ε i 2 ] / (n-k*) i=1 σ 2 = ^ … the error variance (where k* =2). Recall that the square root of the error variance is the standard error of the estimate or “root mean square error” -- RMSE. where ^
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In our bi-variate example (the GPA model), β 1 = .12037 and se(β 1 ) = .0259 Under the null hypothesis: H o : β 1 = 0 H a : β 1 = 0 , …the t -statistic is t = ( β 1 0 ) / se( β 1 ), or t = β 1 / se( β 1 ) = .12037 / .0259 = 4.64 ^ ^ ~ ^ ^ ~ ^ ^
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Since t > t-critical ( α=.05, 6 d.f., two -tailed test): 4.64 > 2.447 then we must ‘reject” the null hypothesis H o : β 1 = 0 at the 95% confidence level in favor of the alternative hypothesis.
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