1. Properties of the LS Estimates Inference for Individual Coefficients
Lecture 8
Review of Lecture 7
Properties of the LS Estimates
Inference for Individual Coefficients
Procedure for Solving Problems
An Example
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ST3131, Lecture 8

2. Review: Interpret the Regression Equation1
Straight line 2
Plane
>= 3
Hyper-plane
Interpret the Regression Coefficients
Interpretation 1: from the view of response change
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3. Interpretation 2:
The MLR coefficients and the associated p SLR coefficients are not the same unless all predictor
variables are uncorrelated. For example,
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4. Properties of LS Estimators
We first show that the LS estimators are Linear combinations of Y1, Y2, …, Yn
First note that b=(n-1)Cov(X,Y) is linear combination of Y1,Y2,…Yn.
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5. 3). BLUE estimators (Best Linear Unbiased Estimators)
It follows that
Properties:
1). Linearity
2). Unbiased
3). BLUE estimators (Best Linear Unbiased Estimators)
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6. Variances of the Estimates
In particular, when p=1, we have
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7. Noise variance estimator and Standard Errors
4) Normality
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8. Inferences for Individual Coefficients
Hypothesis Testing
Confidence Interval
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9. Procedure for Solving Problems
Step 1. List the given conditions from the question statement.
Step 2. Express the question in mathematical formula.
Step 3. Plug-in the known conditions or results from other parts of the problem.
Step 4. Simplify the results.
Step 5. State your conclusion in ordinary languages.
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10. n=30. Answer the following questions:
Example
For the Supervisor Performance Data, you are given the regression coefficient table for
the MLR model:
n=30. Answer the following questions:
(1). Find SSE and its degrees of freedom
(2). Find the proportion of the variability in Y explained by Regression
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11. Regression Analysis: Y versus X1, X2, X3, X4, X5, X6
The regression equation is
Y = X X X X X X6
Predictor Coef SE Coef T P
Constant X X X X X X
S = R-Sq = 73.3% R-Sq(adj) = 66.3%
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12. (4) Find the effect of X1 and its estimated variance
(3). Find the total variability of Y and the variability explained by Regression
(4) Find the effect of X1 and its estimated variance
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13. (6). Without any calculation, can you give the intercept of the fitted line? What is it?
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17. been explained by the regression?
(9e). Based on the reduced model, how much of the total variability in Y has not
been explained by the regression?
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18. After-class Questions:
Can you interpret the exact meaning of the un-biasedness of a LS estimate?
Why can we say the LS estimates are best linear unbiased estimates?
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