1 AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH Part II: Theory and Estimation of Regression Models Chapter 5: Simple Regression Theory

2 Population Line: uiui Y i = E[Y i ]+u i Xi E[Y] = B 0 +B 1 X E[Y i ] = B 0 +B 1 X i

3 Population Line: eiei Y i = Y i + e i Xi E[Y] = B 0 +B 1 X Y i = B 0 +B 1 X i ^ ^^^ Estimated Line: Y = B 0 +B 1 X ^^^ uiui E[Y i ]

4 eiei Xi Y = B 0 +B 1 X ^^^ eiei eiei eiei eiei eiei

5 The Ordinary Least Squares (OLS) Method In the Ordinary Least Squares (OLS) method, the criterion for estimating β 0 and β 1 is to make the sum of the squared residuals (SSR) of the fitted regression line as small as possible i.e.: Minimize SSR = minimize = minimize

6 The Ordinary Least Squares (OLS) Method The OLS estimator (formulas) are: (5.12) (5.13)

7 The Ordinary Least Squares (OLS) Method The regression line estimated using the OLS method has the following key properties: 1. (i.e. the sum of its residuals is zero) 2. It always passes through the point 3. The residual values (e i ’s) are not correlated with the values of the independent variable (X i ’s)

8 Interpretation of the Regression Model Assume, for example, that the estimated or fitted regression equation is:  or  Y i = X i + e i

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10 Interpretation of the Regression Model The value of = 0.15 indicates that if the cotton price received by farmers this year increases by 1 cent/pound (i.e.  X=1), then this year’s cotton acreage is predicted to increase by 0.15 million acres (150,000 acres). Y i = X i + e i

11 Interpretation of the Regression Model The value of = 3.7 indicates that if the average cotton price received by farmers was zero (i.e. =0), the cotton acreage planted this year will be 3.7 million (3,700,000) acres; sometimes the intercept makes no practical sense. Y i = X i + e i

12 Measures of Goodness of Fit There are two statistics (formulas) that quantify how well the estimated regression line fits the data: 1. The standard error of the regression (SER) (Sometimes called the standard error of the estimate) 2. R 2 - coefficient of determination

13 Measures of Goodness of Fit The SER slightly differs from the standard deviation of the e i ’s (by the degrees of freedom): (5.20)

14 Measures of Goodness of Fit: The R 2  The term on the left measures the proportion of the total variation in Y not explained by the model (i.e. by X) Thus, the R 2 measures the proportion of the total variation in Y that is explained by the model (i.e. X)

15 Properties of the OLS Estimators The Gauss-Markov Theorem states the properties of the OLS estimators; i.e. of the: and They are unbiased : E[B 0 ]= and E[B 1 ]=

16 Properties of the OLS Estimators And if the dependent variable Y (and thus the error term of the population regression model, u i ) has a normal distribution, the OLS estimators have the minimum variance

17 Properties of the OLS Estimators BLUE – Best Linear Unbiased Estimator Unbiased => bias of β j = E(β j ) - β j = 0 Best Unbiased => minimum variance & unbiased Linear => the estimator is linear ^ ^