Lecture 10 - Lecture 10 Simple Regression Copyright 2012...

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© Copyright 2012, Team MGMT1050 Lecture 10 Lecture 10 Simple Regression
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© Copyright 2012, Team MGMT1050 Bivariate Statistics Bivariate Statistics > So far, we have been dealing with statistics of individual variables > We also have learned statistics that allow us to compare variables
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© Copyright 2012, Team MGMT1050 Interactions Interactions Sometimes two variables appear related: > smoking and lung cancers > height and weight > years of education and income > engine size and gas mileage > GMAT scores and MBA GPA > house size and price
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© Copyright 2012, Team MGMT1050 Interactions Interactions > Some of these variables would appear to positively related & others negatively > If these were related, we would expect to be able to derive a linear relationship: y = a + bx > where, b is the slope, and a is the intercept
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© Copyright 2012, Team MGMT1050 Linear Relationships Linear Relationships > We will be deriving linear relationships from bivariate (two-variable) data > Our symbols for the true regression relationship will be: term Error Intercept ˆ Slope ˆ x y ˆ or x y 0 1 1 0 1 0 ε β β β + β = ε + β + β =
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© Copyright 2012, Team MGMT1050 Understanding the Two Forms Understanding the Two Forms > The first relationship describes the line in terms of the actual data points, none of which will be on the line, hence the error term > The second relationship describes the line itself, based on the predicted values of y given x, hence no error term > When using the equation of the line empirically, the error term is not actually used x y ˆ or x y 1 0 1 0 β + β = ε + β + β =
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© Copyright 2012, Team MGMT1050 Estimating a Line Estimating a Line > The symbols for the estimated linear relationship are: > b 1 is our estimate of the slope, β 1 > b 0 is our estimate of the intercept, β 0 x b b y ˆ 1 0 + =
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© Copyright 2012, Team MGMT1050 Example Example > Consider the following example comparing the returns of Consolidated Moose Pasture stock (CMP) and the TSX 300 Index > The next slide shows 25 monthly returns
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© Copyright 2012, Team MGMT1050 Example Data Example Data TSX CMP TSX CMP TSX CMP x y x y x y 3 4 -4 -3 2 4 -1 -2 -1 0 -1 1 2 -2 0 -2 4 3 4 2 1 0 -2 -1 5 3 0 0 1 2 -3 -5 -3 1 -3 -4 -5 -2 -3 -2 2 1 1 2 1 3 -2 -2 2 -1
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© Copyright 2012, Team MGMT1050 Example Example > From the data, it appears that a positive relationship may exist Most of the time when the TSX is up, CMP is up Likewise, when the TSX is down, CMP is down most of the time Sometimes, they move in opposite directions > Let’s graph this data
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© Copyright 2012, Team MGMT1050 Graph Of Data Graph Of Data -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 TSX CMP
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© Copyright 2012, Team MGMT1050 Example Summary Statistics Example Summary Statistics > The data do appear to be positively related > Let’s derive some summary statistics about these data: Mean s 2 s TSX 0.00 7.25 2.69 CMP 0.00 6.25 2.50
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© Copyright 2012, Team MGMT1050 Observations Observations > Both have means of zero and standard deviations just under 3 The data was deliberately created to have means of zero for both variables for the demonstration of the deviations on next three slides > However, each data point does not have simply one deviation from the mean, it deviates from both means > Consider Points A, B, C and D on the next graph
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© Copyright 2012, Team MGMT1050 Graph of Data Graph of Data -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 TSX CMP A B C D
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