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Statistics Recap: Sampling Distributions and Estimation, Slides of Data Analysis & Statistical Methods

Alliance UniversityData Analysis & Statistical Methods

An overview of statistical concepts related to sampling distributions and estimation, including the concept of biased and unbiased estimators, the gauss-markov theorem, and the importance of holding constant only included independent variables when interpreting estimated slope coefficients. It also includes a reference to an assignment with instructions and due date.

Typology: Slides

2012/2013

Uploaded on 02/07/2013

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Recap
•Suppose the population of students at Marietta College =
1200
•The model is
–Y = β0 + β1 X1 + β2 X2 + e
•Y = GPA
• X1 = hours of study
• X2 = IQ score
•We don’t see the true βs
•We choose a sample of 50 students and estimate β^s
•Are our β^s the same as true βs?
–No
•What if we chose another sample of 50 observations?
–We will get different β^s
•Most likely
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Recap

  • Suppose the population of students at Marietta College = 1200
  • The model is
    • Y = β 0 + β 1 X 1 + β 2 X 2 + e
      • Y = GPA
      • X 1 = hours of study
      • X 2 = IQ score
  • We don’t see the true βs
  • We choose a sample of 50 students and estimate β^s
  • Are our β^s the same as true βs?
    • No
  • What if we chose another sample of 50 observations?
    • We will get different β^s
      • Most likely

The sampling distribution of the

estimated coefficients

  • Displays the values of all possible β^s that we can get if we draw an infinite number of samples from the population to estimate our equation using a given procedure.
  • If the error term is normally distributed  the estimated coefficients are normally distributed too

Biased/ Unbiased Estimator

  • Is a method of estimation which results in β^s that belong to a distributions whose means are equal to the true βs

Best (most efficient) Estimator

  • Is a method of estimation whose β^s belong to distributions with the lowest possible variances.

The Gauss- Markov Theorem

  • Given assumptions 1 through 6, the OLS estimator is BLUE (Best Linear Unbiased Estimator)

Important note on the meaning of

the estimated slope coefficients

  • Suppose the estimated model is
  • Y^ = β^ 0 + β^ 1 X 1 + β^ 2 X (^2)
    • β^ 1 measures the effect of 1 more unit of X 1 on Y^, holding X 2 constant and ignoring the effects of other relevant but omitted independent variables. - Key: you can only hold an independent variable constant if it is included in your model. If X 3 is another relevant variable and it is excluded from your model, you can’t hold it constant.