EC3090 Econometrics Junior Sophister 20092010

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EC3090 Econometrics Junior Sophister 2009-2010
Topic 3 The Multiple Regression Model
Reading Wooldridge, Chapter 3 Gujarati and
Porter, Chapter 7
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Topic 3 The Multiple Regression Model

• 1. The model with two independent variables
• Say we have information on more variables that
  theory tells us may influence Y
• ß0 measures the average value of Y when X1 and
  X2 are zero
• ß1 and ß2 are the partial regression
  coefficients/slope coefficients which measure the
  ceteris paribus effect of X1 and X2 on Y,
  respectively
• Key assumption
• For k independent variables

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Topic 3 The Multiple Regression Model

• 2. OLS estimation of the multiple regression
  model
• Simultaneously choose the values of the unknown
  parameters of the population model that minimise
  the sum of the squared residuals
• The first order conditions are given by the k1
  equations
• ..
• ..
• Note these equations can also be obtained using
  MM estimation

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Topic 3 The Multiple Regression Model

• 2. OLS estimation of the multiple regression
  model
• Consider the case where k2. OLS requires that
  we minimise
• The first order conditions are given by the 3
  equations
• Solve simultaneously to find OLS parameter
  estimator
• (Illustrate on board)

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Topic 2 The Simple Regression Model

• 2. OLS estimation of the multiple regression
  model
• Algebraic Properties
• 1.
• 2.
• 3.
• 4. is always on the regression line

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Topic 3 The Multiple Regression Model

• 3. Interpreting the coefficients of the Multiple
  Regression Model
• OLS slope coefficients depend on the
  relationship between each of the individual
  variables and Y and on the relationship between
  the Xs (illustrate)
• In the two-variable example re-write the OLS
  estimator for ß1 as
• where are the OLS residuals from a simple
  regression of X1 on X2.
• Thus, gives the pure effect of X1 on Y,
  i.e., netting out the effect of X2.
• Predicted Values
• Residuals
• If model under-predicts Y
• If model over-predicts Y

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Topic 3 The Multiple Regression Model

• 3. Interpreting the coefficients of the Multiple
  Regression Model
• Relationship between simple and multiple
  regression estimates.
• where the coefficients are OLS estimates from
• The inclusion of additional regressors will
  affect the slope estimates
• But where
• 1.
• 2.

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Topic 3 The Multiple Regression Model

• 4. Goodness-of-Fit in the Multiple Regression
  Model
• How well does regression line fit the
  observations?
• As in simple regression model define
• SSTTotal Sum of Squares
• SSEExplained Sum of Squares
• SSRResidual Sum of Squares
• Recall SST SSE SSR ? SSE ? SST and SSE gt 0
• ? 0 ? SSE/SST ? 1
• R2 never decreases as more independent variables
  are added use adjusted R2

Includes punishment for adding more variables to
the model
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Topic 3 The Multiple Regression Model

• 5. Properties of OLS Estimator of Multiple
  Regression Model
• Gauss-Markov Theorem
• Under certain assumptions known as the
  Gauss-Markov assumptions the OLS estimator will
  be the Best Linear Unbiased Estimator
• Linear estimator is a linear function of the
  data
• Unbiased
• Best estimator is most efficient estimator,
  i.e., estimator has the minimum variance of all
  linear unbiased estimators

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Topic 3 The Multiple Regression Model

• 5. Properties of OLS Estimator of Multiple
  Regression Model
• Assumptions required to prove unbiasedness
• A1 Regression model is linear in parameters
• A2 X are non-stochastic or fixed in repeated
  sampling
• A3 Zero conditional mean
• A4 Sample is random
• A5 Variability in the Xs and there is no
  perfect collinearity in the Xs
• Assumptions required to prove efficiency
• A6 Homoscedasticity and no autocorrelation

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Topic 3 The Multiple Regression Model

• 6. Estimating the variance of the OLS estimators
• Need to know dispersion (variance) of sampling
  distribution of OLS estimator in order to show
  that it is efficient (also required for
  inference)
• In multiple regression model
• Depends on
• a) s2 the error variance (reduces accuracy of
  estimates)
• b) SSTk variation in X (increases accuracy of
  estimates)
• c) R2k the coefficient of determination from a
  regression of Xk on all other independent
  variables (degree of multicollinearity reduces
  accuracy of estimates)
• What about the variance of the error terms ?2?

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Topic 3 The Multiple Regression Model

• 7. Model specification
• Inclusion of irrelevant variables
• OLS estimator unbiased but with higher variance
  if Xs correlated
• Exclusion of relevant variables
• Omitted variable bias if variables correlated
  with variables included in the estimated model
• True Model
• Estimated Model
• OLS estimator
• Biased
• Omitted Variable Bias