Econ 299 Chapter3c - 3.9 Properties of the OLS Estimator...

Info icon This preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
1 3.9 Properties of the OLS Estimator There exist a variety of methods to estimate  the coefficients of our model (b 1  and b 2 ) Why use Ordinary Least Squares (OLS)? 1) OLS minimizes the sum of squared errors,  creating a line that fits best with the  observations 2) With certain assumptions, OLS exhibits  beneficial statistical properties.  In particular,  OLS is BLUE.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 3.9.1 The OLS Estimator As seen before, our model is: Y i  = b + b 2 X i  +  є i Where b1 and b2 are not observed.  OLS estimates  these parameters through: X b Y b X X Y Y X X b i i i 2 1 2 2 ˆ ˆ ) ( ) )( ( ˆ - = - - - =
Image of page 2
3 3.9 The OLS Estimator These OLS estimates create a straight line going  through the “middle” of the estimates: Studying and Marks 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 Marks Studying
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
4 3.9.2 Fitted or Predicted Values From the above we see that often the actual data  points lie above or below the estimated line. Points on the line give us ESTIMATED y values for  each given x. The predicted or fitted y values are found using our x  data and our estimated b’s: i i X b b Y 2 1 ˆ ˆ + =
Image of page 4
5 3.9.2 Estimators Example Ols Estimation Qhat = 110/6 –(5/6)P For Price =4, Qhat = 110/6-(5/6)4 = (110-24)/6 = 14.3 Price 4 3 3 6 Pbar = 4 Quantity 10 15 20 15 Qbar=15
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
6 3.9.3 Estimating Errors or Residuals The estimated y values (yhat) are rarely equal to their  actual values (y).  The difference is the error term: Y Y E i i ˆ - =
Image of page 6
7 3.9.3 Estimators Example Ols Estimation Qhat = 110/6 –(5/6)P For Price =4, Qhat =  14.3 Ehat = Y – Yhat = 10-14.3 = -4.3 Price 4 3 3 6 Pbar = 4 Quantity 10 15 20 15 Qbar=15
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
8 3.9.4 Rationale for OLS Estimators Reason 1: OLS minimizes the sum of the squared errors,  providing the best fit line between the  observed data. Section 4 will use calculus to show that b1hat and  b2hat minimize squared errors.
Image of page 8
9 3.9.4 Rationale for OLS Estimators Reason 2: Under certain error term assumptions, OLS  estimators have good statistical properties. Of all linear and unbiased estimators, the OLS  estimators have the smallest variance.
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
10 3.9.5 Statistical Properties of OLS In our model: Y, the dependent variable, is made up of two  components: a) b + b 2 X i  – a non-random component that  indicates the effect of X on Y.  In this course,  X is non-random. b) Є i  – a random error term representing other  influences on Y.
Image of page 10
11 3.9.5 Statistical Properties of OLS Error Assumptions: a) E( є i ) = 0; we expect no error; we assume the  model is complete b) Var( є i ) =  σ 2 ; the error term has a constant  variance c) Cov( є i є j ) = 0; error terms from two different  observations are uncorrelated.  If the last error  was positive, the next error need not be  negative.
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
12 3.9.5 Statistical Properties of OLS
Image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern