Raising the bar

Chapter 11. Maximum likelihood method

Unit 11.4. Questions for repetition

$latex \quad$1. Using a sample on the horizontal axis and several candidate densities above it, explain the visual idea behind the maximum likelihood method. Why does the density that "covers" the observed points well correspond to a higher likelihood?

$latex \quad$2. State the three steps of the ML algorithm. At which step does the assumption of independent observations enter, and why is this assumption needed?

$latex \quad$3. Define the likelihood function and the log-likelihood function. Why is it usually more convenient to maximize the log-likelihood than the likelihood itself?

$latex \quad$4. Derive the ML estimator of the parameter $latex \mu$ of an exponential distribution. Verify the second-order condition.

$latex \quad$5. Derive the ML estimator of the parameter $latex p$ of a Bernoulli distribution and show that it equals the sample mean.

$latex \quad$6. For the linear model $latex y_i = \beta_1 + \beta_2 x_i + u_i$ with i.i.d. normal errors of variance $latex \sigma^2$, write down

$latex \quad$$latex \quad$(a) the density of one observation,

$latex \quad$$latex \quad$(b) the likelihood and the log-likelihood function,

$latex \quad$$latex \quad$(c) the first-order conditions with respect to $latex \beta_1, \beta_2, \sigma^2$.

How do the ML estimators of $latex \beta_1, \beta_2$ relate to the OLS estimators from Chapter 10?

$latex \quad$7. Distinguish between models and methods. In what sense can OLS and ML be applied to the same linear model and yield the same estimators of the slope coefficients?

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