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AAEC 4302 STATISTICAL METHODS IN AGRICULTURAL RESEARCH

AAEC 4302 STATISTICAL METHODS IN AGRICULTURAL RESEARCH. Chapter 7(7.1 &7.2): Theory and Application of the Multiple Regression Model. Introduction. The multiple regression model aims to and must include all of the independent variables X1, X2, X3, …, Xk that are believed to affect Y

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AAEC 4302 STATISTICAL METHODS IN AGRICULTURAL RESEARCH

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  1. AAEC 4302 STATISTICAL METHODS IN AGRICULTURAL RESEARCH Chapter 7(7.1 &7.2): Theory and Application of the Multiple Regression Model

  2. Introduction • The multiple regression model aims to and must include all of the independent variables X1, X2, X3, …, Xk that are believed to affect Y • The multiple linear regression model is given by: Yi = β0 + β1X1i + β2X2i + β3X3i +…+ βkXki + ui where i=1,…,n represents the observations, k is the total number of independent variables in the model, β0, β1,…, βk are the parameters to be estimated, and ui is the disturbance term

  3. Example

  4. Y X2 Regression surface (plane) E[Y] = βo+β1X1+β2X2 Ui X2 slope Measured by β2 βo X1 slope measured by β1 X1

  5. Model Estimation

  6. Interpretation of the Coefficients ^ • The intercept βo estimates the value of Y when all of the independent variables in the model take a value of zero • In our example βo , is 144.94, which means that if : • Yi = 144.94+β1*(0)+β2*(0) + β3*(0)+β4*(0) • All the independent variables take the value of zero (price of beef is zero cents/lb, price of chicken is zero cents/lb, price of pork is zero cents/lb, and the income for US population is zero dollars/ per – year, then the estimated beef consumption will be 144.94 lbs/year). ^ ^ ^ ^ ^

  7. Interpretation of the Coefficients ^ ^ ^ • In a strictly linear model, β1, β2,..., βk are slopes of coefficients that measure the unit change in Y when the corresponding X (X1, X2,..., Xk) changes by one unit and the values of all of the other independent variables remain constant at any given level • Ceteris paribus (other things being equal)

  8. Interpretation of the Coefficients ^ • In our example: • β1= -0.00291. That means, if the price of beef increases by one cent/lb then the beef consumption will decrease by 0.00291 pounds per – year, ceteris paribus • β2= -0.116. That means, if the price of chicken increases by one cent/lb then the beef consumption will decrease by 0.116 pounds per – year, ceteris paribus (Does this result makes sense?) ^

  9. The Model’s Goodness of Fit i=1 i=1 • R2 = 1 - { ei2/ (Yi-Y)2} • A disadvantage of R2 is that it always increases in value as independent variables are added into the model

  10. The Model’s Goodness of Fit • The adjusted or corrected R2 : • R2 = 1  [{ei2/(n-k-1)}/{(Yi-Y)2/(n-1)}] • The R2 is always less than the R2, unless the R2 = 1 • Adjusted R2 lacks the same straightforward interpretation as the regular R2

  11. The Specification Question • Any variable that is suspected to directly affect Y should be included in the model • Excluding such a variable would likely cause the estimates of the remaining parameters to be “incorrect”; i.e. biased

  12. The Earnings Function • Multiple regression model of the earnings: EARNGSi = β0 + β1EDi + β2EXPi + ui • Cross-section data set, 100 observations EARNGSi =-6.179+0.978EDi+0.124EXPi R2 = 0.315 Adj.R2 = 0.298

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