Understanding Econometrics: Estimators, Linearity, Unbiasedness, Consistency, Efficiency, Slides of Econometrics and Mathematical Economics

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Econometrics

Chapter 3: Estimators and Distributions

Estimators

• A rule (or procedure) for constructing an estimate of a population parameter.
• Example:

Linearity

• A linear estimator may be expressed as a linear function of observable random variables.
• Lower computational cost (not as important today given high-speed and low-cost computing technology)

Unbiasedness

• Expected value of the estimator equals the population parameter:

Consistency

• Estimator converges to the value of the population parameter as the size of the sample rises

Unbiasedness vs. consistency

• Unbiased if the average value in an infinite number of samples equals the population parameter
• Consistent if the estimator converges to the population parameter as the size of the sample tends toward infinity

Efficiency

• An unbiased estimator is efficient if the variance of this estimator is less than or equal to the variance of any other unbiased estimator:

Sample statistics

• Sample mean:

Sample standard deviation

• Sample standard deviation:

Sample covariance and correlation

• Sample covariance and correlation:

Normal distribution

Central limit theorems

• A wide variety of central limit theorems suggest that the distribution of many random variables tends toward a normal distribution as the size of the sample rises
• In particular, the distribution of a sample mean converges to a normal distribution as the size of the sample approaches infinity (as long as the underlying distribution has a constant mean and variance).

Standard normal distribution

• Z transformation:

Standard normal distribution