The Simple Linear Regression Model: Specification and Estimation Chapter 2 The Simple Linear Regression Model: Specification and Estimation Walter R. Paczkowski Rutgers University

2.3 Estimating The Regression Parameters 2.3.1 The Least Squares Principle → Simple linear regression model: → Least squares estimates of and : b1 and b2 The fitted regression line is The least squares residuals (vertical distance from each point to the fitted line) (2.5) (2.6)

2.3 Estimating The Regression Parameters Figure 2.7 The relationship among y, ê and the fitted regression line

2.3 Estimating The Regression Parameters Suppose any other fitted line Then we know that least squares line has smallest sum of squared residuals If and then SSE<SSE*

2.3 Estimating The Regression Parameters Least squares estimates for the unknown parameters β1 and β2 are obtained by minimizing the sum of squares function

2.3 Estimating The Regression Parameters The Least Squares Estimators are below To obtain the least squares estimates of β1 and β2, plug the sample values yi and xi into the normal equations, (2.7) and (2.8) (2.7) (2.8)

<Sample Question> When x=1,2,3 and y=1,2,6, find the fitted regression line, 1) Using Excel Program: Insert → chart → scatter plot → trend line (w/ and w/o intercept)

<Sample Question> 2) Using Excel Program: Excel Option → Data Analysis → Data → Regression Analysis

Data Analysis ToolPak for MS Office Excel Data Analysis ToolPak: MS Office Excel add-in program that is available when you install MS Office or Excel. To use the Analysis ToolPak in Excel, however, you need to load it first. - Click the Microsoft Office Button, and then click Excel Options. - Click Add-Ins, and then in the Manage box, select Excel Add-ins. - Click Go. - In the Add-Ins available box, select the Analysis ToolPak check box - And then click OK. If Analysis ToolPak is not listed in the Add-Ins available box, click Browse to locate it. After you load the Analysis ToolPak, the Data Analysis command is available in the Analysis group on the Data tab.

<Sample Question> 3) Using Econometric Methods based on normal equations, =>

2.8 Approximating Logarithms (Taylor series approximation) ☞ Let y1=1+x and y0=1, then as long as x is small,

2.8 Approximating Logarithms

2.8 Approximating Logarithms in the Log-Linear Model ☞ Let ln(y0) = β1 + β2x0 and ln(y1) = β1 + β2x1

2.8 Approximating Logarithms in the Log-Linear Model ex) ⇒ Standard ↑ 1 unit ↑ 10.57% ↑ ⇒ App. ⇒

2.8 Approximating Logarithms in the Log-Log and Log-Linear Model 1% ↑ in 1unit ↑ in % ↑ in ?% ↑ in → → → → → find % Multiply 100 : % find % Log-Log : Log-Linear : ⇒ ≈

2.8 Approximating Logarithms in the Linear-Log Model

2.8 Approximating Logarithms in the Linear-Log Model Suppose that output (y) is a function of only labor input (x), and output is given by ☜ Using the previous approximation interpretation, we know 10% change in x increases 50 units of output

Examples of Linear and Nonlinear Relationships

Ex) Linear Function ☞ Suppose that Q is output, L is labor input and K is capital input

Ex) Quadratic Function ☞ If β3 > 0, then the curve is U-shaped, and it represents average or marginal cost functions, with increasing marginal effects. ☞ If β3 < 0, then the curve is an inverted-U shape, and it represents total product curves & total revenue curves, with diminishing marginal effects. The derivative is zero when,

Ex) Cubic Function Cubic functions can be used for total cost and total production curves in economics. The derivative of total cost is marginal cost, and the derivative of total production is marginal production. If the “total” curves are cubic then the “marginal” curves are quadratic functions, a U-shaped curve for marginal cost, and an inverted-U shape for marginal production.