5.1 Goals, Framework, and Learning Objectives

5.1.1 Class Announcements

  • You’re done with probability theory. Yay!
  • You’re also done with your first test. Double Yay!
  • We’re going to have a second test in a few weeks. Then we’re done testing for the semester Yay?

5.1.2 Learning Objectives

At the end of this week, students will be able to

  1. Understand what iid sampling is, and evaluate whether the assumption of iid sampling is sufficiently plausible to engage in frequentist modeling.
  2. Appreciate that with iid sampling, summarizing functions of random variables are, themselves, random variables with probability distributions and values that they obtain.
  3. Recall the definition of an estimator,
  4. Recall definition of an estimator, state and understand the desirable properties of estimators, and evaluate whether an estimator possesses those desirable properties.
  5. Distinguish between the concepts of {expectation & sample mean}, {variance & unbiased sample variance estimator, sampling-based variance in the sample mean}.

5.1.3 Roadmap

5.1.3.1 Where We’re Going – Coming Attractions

  • We’re going to start bringing data into our work
  • First, we’re going to develop a testing framework that is built on sampling theory and reference distributions: these are the frequentist tests.
  • Second, we’re going to show that OLS regression is the sample estimator of the BLP. This means that OLS regression produces estimates of the BLP that have known convergence properties.
  • Third, we’re going combine the frequentist testing framework with OLS estimation to produce a full regression testing framework.

5.1.3.2 Where We’ve Been – Random Variables and Probability Theory

Statisticians create a model (also known as the population model) to represent the world. This model exists as joint probability densities that govern the probabilities that any series of events occurs at the same time. This joint probability of outcomes can be summarized and described with lower-dimensional summaries like the expectation, variance, covariance. While the expectation is a summary that contains information on about one marginal distribution (i.e. the outcome we are interested in) we can produce predictive models that update, or condition the expectation based on other random variables. This summary, the conditional expectation is the best possible (measured in terms of minimizing mean squared error) predictor of an outcome. We might simplify this conditional expectation predictor in many ways; the most common is to simplify to the point that the predictor is constrained to be a line or plane. This is known as the Best Linear Predictor.

5.1.3.3 Where we Are

  • We want to fit models – use data to set their parameter values.
  • A sample is a set of random variables
  • Sample statistics are functions of a sample, and they are random variables
  • Under iid and other assumptions, we get useful properties:
    • Statistics may be consistent estimators for population parameters
    • The distribution of sample statistics may be asymptotically normal