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Objectives

Objectives. Lecture 2. By the end of this lecture students will: understand the simple linear regression model understand the logic behind the method of ordinary least squares (OLS) estimation CLRM topic is a review of the material covered in 316-206 Quantitative Methods 2 References:

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Objectives

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  1. Objectives Lecture 2 • By the end of this lecture students will: • understand the simple linear regression model • understand the logic behind the method of ordinary least squares (OLS) estimation • CLRM topic is a review of the material covered in 316-206 Quantitative Methods 2 • References: • Gujarati., chapter 2 • also pp 100-105

  2. The Model • True Model (Population Regression Function) • Yi is the ith observation of the dependent variable • Xi is the ith observation of the independent or explanatory variable • ui is the ith observation of the error or disturbance term •  and  are unknown parameters (coefficients) • i = 1,..., n observations

  3. The Model • Estimated model (Sample Regression Function) • where • are the estimators of the coefficients (parameters) • are the i estimated residuals • Objective: estimate the PRF on basis of SRF

  4. The Model • In terms of the SRF the observed Yi can be written as • in terms of the PRF Yi can be written as • Given the SRF is an approximation to the PRF can we make this approximation as close as possible?

  5. The Model Y Yi Yi ui E(Y|Xi) E(Y|Xi) X Xi

  6. Ordinary Least Squares (OLS) • OLS obtains the estimators and minimising the sum of squared residuals (RSS) with respect to  and : • Partially differentiate RSS w.r.t. the coefficients

  7. Ordinary Least Squares (OLS) • The first order conditions are: • (1) • (2)

  8. Ordinary Least Squares (OLS) • Solving (1) and (2) simultaneously gives • and • where and are the sample means

  9. Ordinary Least Squares (OLS) • Example: Melbourne house prices • Y price of house in A$’000 • X distance of house from GPO in kilometres • A=area of house in square metres • Interpret the results

  10. Ordinary Least Squares (OLS) • What type of data is this? • Calculate the missing values of Y-hat and u-hat in the table

  11. Ordinary Least Squares (OLS) • CHECK THE ANSWERS YOURSELVES!

  12. Summary and conclusions • PRF vs SRF • OLS methodology • Next lecture • Gujarati p65 – 76 (Classical Assumptions) • Gujarati p107 – 112 (Normality Assumption)

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