THE

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• THE
• MULTIPLE REGRESSION MODEL

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MULTIPLE REGRESSION

• In a multiple regression we are trying to
  evaluate the cumulative effects that changes to
  more than one independent variable (x1, x2, x3,
  etc.) or will have on a dependent variable (y)

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Transformations to a Linear Model

• Multiple regression can used to evaluate models
  like
• y ?0 ?1 x1 ?2 x2 ?3 x12 ?4 x1 x2 ?5
  x1/x2 ?6 logx1 ?
• Define
• x3 x12
• x4 x1 x2
• x5 x1/x2
• x6 log x1
• Then the model becomes
• y ?0 ?1 x1 ?2 x2 ?3 x3 ?4 x4 ?5 x5
  ?6x6 ?

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GENERAL FORM OF A MULTIPLE REGRESSION MODEL

• Since we can make substitutions similar to those
  just described, the general multiple regression
  model can be expressed as
• y ?0 ?1 x1 ?2 x2 ?3 x3 . ?k xk ?

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THE REGRESSION APPROACH

• Hypothesize a form of the model
• Determine the best estimates for the ?s
• Assumptions about ?
• Testing the strength of the model
• Using the model for prediction/estimation

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Example

• It is felt that the price of a house in Laguna
  Hills is a function of its square footage, its
  lot size, and its age.
• A sample of 38 recent sales in Laguna Hills is
  taken.

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STEP 1 Hypothesizing a form of the model

• One variable -- scatterplot
• If it looks curved, hypothesize a higher order
  model and make transformations to a linear model
• More than one variable
• Simply HYPOTHESIZE make a best judgment as the
  form of the model
• Make appropriate substitution of variables so
  that the model is linear

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Laguna Hills Model

• There are three variables.
• Hypothesize
• y ?0 ?1x1 ?2x2 ?3x3 ?

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STEP 2 Determining the Best Estimates for the
?s

• Involves complicated matrix operations but still
  uses the method of least squares.
• Use computer (EXCEL) only
• But the best values for the ?s minimizes the sum
  of the squared errors between the actual values
  of y and the predicted values for y -- i.e. They
  minimize SSE.

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Using Excel to Get the bs

• Go to TOOLS/DATA ANALYSIS/REGRESSION

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STEP 3 Assumptions For ?

• For any given set of the xs
• ? has a normal distribution
• E(?) 0
• Also
• Errors are independent
• ? does vary between different values of the xs

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STEP 4Assessing the Strength of the Model

• Question 1 Can we conclude that at least one of
  the independent variables (xs) is useful in
  predicting y?
• Question 2 If yes, which of the independent
  variables (xs) are useful in predicting y?
• Question 3 What proportion of the overall
  variation in y is due to the changes in the xs?
• These are addressed in another module.

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STEP 5 Use the Model for Prediction/Estimation
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Prediction/Confidence Intervals

• These are possible
• but not easily with EXCEL
• Other Stat packages -- MINITAB, SPSS, SAS perform
  these calculations.

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Important Excel Note -- Inputting a Contiguous
Range for the Xs

• Suppose in this example we wished to regress
  Price on only Sq. Feet (column B) and Age (column
  D).
• These are not next to each other
• They must be next to each other for the
  regression module in Excel to work
• Highlight the data in column D and click CUT
• Click cell C1, which is where you want the data
  to begin, with right mouse key
• Click INSERT CUT CELLS

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Review

• Multiple regression is used when
• y is a function of more than one x
• y includes terms of x raised to a power
• This can be converted to a linear term
• Excel (or another stat package) is used to
  calculate the best estimates of the ?s
• The assumptions about the error term are the same
  
• ? is constant for all values of all the xs