2 Week 1 Live online lecture

Learning objectives. After this week you will be able to

to be written;
to be written;
to be written.

Problems we aim to tackle this week

Estimating the average return and risk, as measured by standard deviation, to investing in equities.

Estimating the average default rate of corporate bonds by grade

2.1 Structure

Population Parameters, Estimators, and Estimands
1. Population vs. sample
2. Clarify the difference between the population and a sample.
  1. Population parameters are fixed, while sample statistics vary from sample to sample.
  2. Discuss issues in sampling and, in particular, the importance of random sampling.
  3. Make students aware of important types of non-random sampling encountered in finance, including survivorship bias and sample selection bias.
3. Parameters
  1. A parameter is a fixed but unknown quantity that we would like to know. Although parameters are unknown, they are not random.
4. Estimator, estimate, and estimand
  1. An estimator is a well-defined rule or procedure used to estimate an unknown quantity. In many cases, estimators are specific functions, such as the mean. Because an estimator is a function of random variables, it is also a random variable. Estimators exist before any data are observed. We can study estimators to understand their properties under different assumptions.
  1. An estimate is a realization of an estimator. It is the result of applying the estimator to a sample of data. The data sample is itself a realization of the random variables contained in the estimator. Estimates are numbers and are therefore non-random.
  2. An estimand is the quantity that an estimator aims to estimate. For example, when the estimator is the sample mean,
    
    \[ \frac{1}{n}\sum_{i=1}^{n}X_i, \]
    
    the estimand is the expected value, \(\mathbb{E}[X]\).
  3. The word statistic is sometimes confusingly used to mean either an estimator or an estimate. For example, the mean is often called a statistic. Similarly, many reputable newspapers report statistics that are numerical values—that is, estimates.
Method of Moments Estimation
1. Definition of a moment
  1. Introduce the sample analogue principle, moving from the population moment
    
    \[ \mathbb{E}[X] \]
    
    to its sample analogue,
    
    \[ \frac{1}{n}\sum_{i=1}^{n}X_i. \]
2. First moment: the mean
3. Higher central moments: variance, skewness, and kurtosis
  1. Define the general method of moments estimator for higher-order moments.
  2. Use the plug-in principle for skewness and kurtosis.
    1. Skewness and kurtosis are ratios of moments. The most common way to estimate quantities of this type is to use the plug-in principle, under which each quantity in the ratio is replaced by its estimator.
4. Simultaneous estimation of multiple moments: mean and variance
  1. Set up a generic expression showing how two parameters are estimated jointly using the method of moments.
5. Finite-sample properties under IID assumptions
  1. Define a finite-sample property and preview the use of asymptotic analysis.
    1. Bias: demonstrate the properties of the expectation operator and use the identically distributed assumption to simplify the expression.
    2. Variance of the mean: demonstrate the properties of the variance operator. Remind students that independence implies zero covariance and that identically distributed variables have the same variance.
    3. Derive the covariance matrix of a vector of estimators to show that multiple statistics calculated from the same data may be correlated.
  1. Standard error vs. standard deviation
    1. Define the standard error.
    2. Show that a standard error is analogous to a standard deviation, but applies to an estimator rather than to a population variable.
    3. Clarify that standard errors usually decrease as the sample size increases, while standard deviations generally do not.
6. Beyond bias: efficiency
  1. Define the mean squared error of an estimator.
    1. Mean squared error can be used to compare estimators that may have different biases and variances.
  2. Note that when an estimator is unbiased, its mean squared error is equal to its variance.
  3. When multiple unbiased estimators are available, it is natural to compare them based on their variances.
  4. Optional: Compare a weighted mean with the usual sample mean under IID assumptions. Show that the weighted mean is less efficient than the usual mean in this setting. Then explain that, if the IID assumption is relaxed and the errors have different variances, a weighted estimator may be more efficient than the usual estimator. Explain the intuition: place greater weight on observations that are more accurate, with smaller error variances, and less weight on less informative observations.
7. Optional: Uniform parameter estimation
  
  Use the method of moments to estimate the range parameter of
  
  \[ X \sim U(0,a), \]
  
  using
  
  \[ \mathbb{E}[X] = \frac{a}{2}. \]
8. Data examples