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• 1 Simple linear regression With One Regressor: Estimation

• 1.1.1.2 The OLS estimator

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• What does the OLS estimator do?

  It minimizes the squared difference between the actual values of Y_i and the predicted Y's.

• 1.3 Maximum likelihood estimation

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• Explain the general idea of the ML estimation

  ML estimation "chooses" the coefficients that will most likely have "produced" a given dataset.

• Explain the differences between OLS and ML estimation

  2 differences;
  A. The objective function is different;
  For OLS: the sum of squared errors (or least squares?)
  For ML: the joint density function
  
  B. Minimize/maximize
  For OLS; we want the smallest errors so we aim to minimize the value of the objective function
  For ML; we want to find the highest probability, so we aim to maximize the value of the joint density function

• 1.3.1 The likelihood function in general

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• Show how to turn the density function into a likelihood function.

  All you gotta do is switch the conditional statement and data set around;
  Here is the joint pdf conditional on θ:
  f(x1, ..., xn|θ)
  
  Here is the likelihood function that comes from it:
  L(θ|x1, ..., xn)
  
  Simply switched around

• Provide the order of progression starting from: "a  random variable x, conditional on a set of parameters, θ" and ending at: the log-likelihood function

  1. Define the function: f(x|θ)
  2. Consider the joint density function
  3. Turn density function into likelihood function
  4. Take the log

• The big picture of ML estimation is to maximize the joint density function (which we have transformed into a log-likelihood function). Show the steps involved to maximize the function.

  1. Take the derivative with respect to the estimator, θ.
  2. Set our derivative equal to 0.
  3. Solving for the parameter estimate gives its ML estimate

• After the 3 steps involved in maximizing the joint density(log-likelihood) function, what is one (optional) additional step we are sometimes asked to take and what is its purpose?

  We have already calculated the first derivative, sometimes we check the second derivative to "verify that we have found a maximum"

• 1.3.2 ML estimation of sample mean & variance, normal pdf

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• For the following model:xi = µ + ei          ei ∼ N(0, σ2 ) Estimate the sample mean

  This should be the answer (see image):
  

• 2 Simple linear regression with One regressor: Inference

• 2.1 LS assumptions for the linear model

• In what way specifically, do the classical assumptions from week 1, differ from the more realistic assumptions we discuss in week 2?

  1. Regressors are no longer assumed to be fixed
  2. Variances are no longer assumed to be constant
  3. Error terms are no longer assumed to be normally distributed

• 2.2.1 Unbiasedness

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• Starting with the (regression) formula for both Yi and Y-bar, prove unbiasedness of the B1 estimator

  Abc