Model Building, Estimation

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Model Building, Estimation Prediction
Topics Motivational Example Multicollinearity
(Redundancy) Avoiding Multicollinearity Stepwise
Procedure Some Caveats Estimation Prediction
with Regression Models
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Problem Scenario

• A real estate agent wanted to develop a model to
  predict the selling price of a home. The agent
  believed that the most important variable in
  determining the price of house are its size,
  number of bedrooms, and lot size. Accordingly he
  gathered relevant data on a random sample of 100
  recently sold homes.

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Problem Scenario Regression Model
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Contradicting conclusions From Regression Output

• Global F-test is highly significant
• At least one explanatory variable is useful in
  predicting home price
• All t-tests indicate none of the explanatory
  variables have significant marginal contribution
• Home price decreases as lot size increases

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Whats Going On Here?
High correlation among X-variables leads to
redundancy
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MultiCollinearity

• Exists when any X-variable can be expressed as a
  linear combination of the other X-variables this
  is root cause
• E.g. X1 0.9 X2 X3 0.4X2 0.3X1
• Another symptom includes unstable coefficient
  estimates from one sample to the next high
  variance of bi

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Implications of MultiCollinearity

• Interpretation of regression coefficients not
  meaningful in presence of multiC
• Is often a problem in models that contain
  interaction terms and quadratic terms
• E.g. X1, X2, X1X2, X12, X22
• Does not affect predictions
• Significant F-test with one or more significant
  t-tests does NOT imply multiC

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Avoiding Multicollinearity

• If chief purpose of the model is to predict Y
  with no interest in interpreting relationships,
  ignore multiC
• Inspect correlation matrix and choose X variables
  that have highest correlation with Y but low
  correlation with other Xs
• Eliminate insignificant X variables from model in
  stepwise fashion

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Avoidance Example

• H_Size has highest correlation with Price
• H_Size also highly correlated with Lot_Sz and
  Bedrooms
• Choose 1-variable model with H_Size only

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Manual Stepwise Regression Review

• Include all reasonable potential X variables in
  the model
• If all t-stat gt 2 or p-value lt .05, stop
  accept model
• Else, drop X variable with lowest t-stat or
  highest p-value and rerun regression
• Go back to step 2 and continue until all t-stat
  gt 2 or p-value lt .05

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Automatic Stepwise Regression

• Is a search process that adds or deletes
  variables at each step until no changes can
  improve the model
• Three variants available
• Forward
• Backward
• General

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Variants of Stepwise Regression

• Forward procedure begins with no explanatory
  variables in the model and successively adds one
  at a time until no new explanatory variable makes
  a significant contribution
• Typical criterion to enter p-value lt .05

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Variants of Stepwise Regression

• Backward procedure begins with all potential
  explanatory variables in the model and deletes
  them one at a time until further deletion would
  do more harm than good
• Typical criterion to leave p-value gt .05

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Variants of Stepwise Regression

• vGeneral procedure much like Forward variant but
  a variable that enters the model could be deleted
  in a later step
• Typical criterion to enter p-value lt .05
• Typical criterion to leave p-value gt .10

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Application of General Stepwise Predicting
Monthly Rent Payment
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Caveats with Stepwise

• Do NOT use automatic procedure mindlessly. Can
  get nonsensical models
• Does NOT guarantee best model
• Regard number of X variables given in final model
  as a guide to how many should be your target.
  Check for glaring omissions or inclusions

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Running Stepwise Regression in StatTools

• Name the data set in the usual way
• Place the cursor anywhere in the spreadsheet and
  click on the Regression Classification icon
  (3rd from right)
• Select Regression then arrow down to highlight
  the regression type forward, backward, (general)
  stepwise

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Running Stepwise Regression in StatTools

• By clicking insert a check mark in the box next
  to the Y variable (D) and X-variables (I)
• By clicking insert a check mark in the box next
  to the advanced option include detailed step
  information
• Accept the default radio button Use p-values,
  adjust criteria to enter or leave (if needed)
  then click O.K.

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Prediction Interval for a new Individual
Observation

• Best point estimate for response variable (Y)
  when explanatory variables take on given values
  is given by plugging given values into final
  regression equation

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Prediction Interval for a new Individual
Observation

• Account for sampling variation by expressing
  prediction interval

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Prediction Interval for a new Individual
Observation

• tmult value depends on level of confidence
• Typically 95 level employed
• Lower and upper prediction limits available in
  StatTools

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Example Prediction Interval for new Individual
Obs.

• A particular family not included in the study has
  the following description
• Family size 5, Located in NE

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Example Prediction Interval for new Individual
Obs.

• Rents home, First wage earner 50K
• Second wage earner 20K
• Avg. monthly util. 200, total debt 5K

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StatTools Prediction Interval for new obs.

• With 95 confidence we predict that the a new
  family fitting this description will pay between
  442 and 1,409 per month in rent

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Confidence Interval for Mean Obs. with Given
Characteristic

• Best point estimate for response variable (Y)
  same as for prediction interval given by
  plugging given values into final regression
  equation

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Confidence Interval for Mean Obs. with Given
Characteristic

• Sampling variation smaller for estimating mean
  than for predicting individual new observation

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Confidence Interval formula for mean Obs.

• Lower and upper confidence limits NOT available
  in StatTools
• Easy hand calculation in Excel

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Example Confidence Interval for mean of all Obs.

• Estimate mean monthly payment for all families
  having the following description
• Family size 5, Located in NE

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Example Confidence Interval for mean of all Obs.

• Rents home, First wage earner 50K
• Second wage earner 20K
• Avg. monthly util. 200, total debt 5K

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Excel Calculation of Confidence Interval for mean
obs.

• From StatTools, Yhat 925.35

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Excel Calculation of Confidence Interval for mean
obs.

• We are 95 confident that families fitting this
  description will pay on average between 905 and
  947 per month in rent

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Concluding Remarks about Use of Regression Model

• Some useful variables were found to predict and
  estimate monthly rent/mortage payments
• R2 of 53 implies there might be other useful
  variables not yet considered

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Concluding Remarks about Use of Regression Model

• Wide prediction interval due to large Se
• Need to investigate assumption violations