EC3090 Econometrics Junior Sophister 20072008

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EC3090 Econometrics Junior Sophister 2007-2008
Topic 1 Statistical Review
Reading Wooldridge, Appendix C1-C6 Gujarati,
Appendix A7, A8
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Topic 1 Statistical Review

• 1. Populations, Parameters and Random Sampling
• Use statistical inference to learn something
  about a population
• Population Complete group of agents, e.g. the
  population of students studying JS Econometrics
• Typically only observe a sample of data
• Random sampling Drawing random samples from a
  population
• Know everything about the distribution of the
  population except for one parameter
• Use statistical tools to say something about the
  unknown parameter
• Estimation and hypothesis testing

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Topic 1 Statistical Review

• 2. Estimators and Estimates
• Given a random sample drawn from a population
  distribution that depends on an unknown parameter
  ?, an estimator of ? is a rule that assigns each
  possible outcome of the sample a value of ?
• Examples
• Estimator for the population mean
• Estimator for the variance of the population
  distribution
• An estimator is given by some function of the
  r.v.s
• This yields a (point) estimate which is itself a
  r.v.
• Distribution of estimator is the sampling
  distribution
• Criteria for selecting estimators

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Topic 1 Statistical Review

• 3. Finite sample properties of estimators
• Unbiasedness
• An estimator of ? is unbiased if
  for all values of ?
• i.e., on average the estimator is correct
• If not unbiased then the extent of the bias is
  measured as
• Extent of bias depends on underlying
  distribution of population and estimator that is
  used
• Choose the estimator to minimise the bias
• Illustration
• Example

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Topic 1 Statistical Review

• 3. Finite sample properties of estimators
• Efficiency
• What about the dispersion of the distribution of
  the estimator?
• i.e, how likely is it that the estimate is close
  to the true parameter?
• Useful summary measure for the dispersion in the
  distribution is the sampling variance.
• An efficient estimator is one which has the
  least amount of dispersion about the mean i.e.
  the one that has the smallest sampling variance
• If and are two unbiased estimators
  of ?, is efficient relative to when
  for all ?, with strict inequality for at
  least one value of ?.

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Topic 1 Statistical Review

• 3. Finite sample properties of estimators
• Efficiency
• What if estimators are not unbiased?
• Estimator with lowest Mean Square Error (MSE) is
  more efficient
• Example
• Compare the small sample properties of the
  following estimates of the population mean

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Topic 1 Statistical Review

• 4. Asymptotic Properties of Estimators
• How do estimators behave if we have very large
  samples as n increases to infinity?
• Consistency
• How far is the estimator likely to be from the
  parameter it is estimating as the sample size
  increases indefinitely.
• is a consistent estimator of ? if for every
  egt0
• This is known as convergence in probability
• The above can also be written as
• Illustration
• Example

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Topic 1 Statistical Review

• 4. Asymptotic Properties of Estimators
• Consistency (continued)
• Sufficient condition for consistency
• Bias and the variance both tend to zero as the
  sample size increases indefinitely. That is

Law of Large numbers an important result in
statistics When estimating a population
average, the larger n the closer to the true
population average the estimate will be
Properties of probability limits Examples
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Topic 1 Statistical Review

• 4. Asymptotic Properties of Estimators
• Asymptotic Efficiency
• Compares the variance of the asymptotic
  distribution of two estimators
• A consistent estimator of ? is
  asymptotically efficient if its asymptotic
  variance is smaller than the asymptotic variance
  of all other consistent estimators of ?

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Topic 1 Statistical Review

• 4. Asymptotic Properties of Estimators
• Asymptotic Normality
• An estimator is said to be asymptotically
  normally distributed if its sampling distribution
  tends to approach the normal distribution as the
  sample size increases indefinitely.
• The Central Limit Theorem average from a random
  sample for any population with finite variance,
  when standardized, has an asymptotic normal
  distribution.

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Topic 1 Statistical Review

• 5. Approaches to parameter estimation
• Method of Moments (MM)
• Moment Summary statistic of a population
  distribution (e.g. mean, variance)
• MM replaces population moments with sample
  counterparts
• Examples
• Estimate population mean µ with
• (unbiased and consistent)
• Estimation population variance s2 with
• (consistent but biased)

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Topic 1 Statistical Review

• 5. Approaches to parameter estimation
• Maximum Likelihood Estimation (MLE)
• Let Y1,Y2,,Yn be a random sample from a
  population distribution defined by the density
  function f(Y?)
• The likelihood function is the joint density of
  the n independently and identically distributed
  observations given by

The log likelihood is given by
The likelihood principle Choose the estimator of
? that maximises the likelihood of observing the
actual sample (illustrate by example)
MLE is the most efficient estimate but correct
specification required for consistency
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Topic 1 Statistical Review

• 5. Approaches to parameter estimation
• Least Squares Estimation
• Minimise the sum of the squared deviations
  between the actual and the sample values
• Example Find the least squares estimator of the
  population mean
• (Note Dont forget Second Order Condition)

The least squares, ML and MM estimator of the
population mean is the sample average
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Topic 1 Statistical Review

• 6. Interval Estimation and Confidence Intervals
• How do we know how accurate an estimate is?
• A confidence interval estimates a population
  parameter within a range of possible values at a
  specified probability, called the level of
  confidence, using information from a known
  distribution the standard normal distribution
• Let Y1,Y2,,Yn be a random sample from a
  population with a normal distribution with mean µ
  and variance s2 YiN(µ,s2)
• The distribution of the sample average will be
• Standardising
• Using what we know about the standard normal
  distribution we can construct a 95 confidence
  interval

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Topic 1 Statistical Review

• 6. Interval Estimation and Confidence Intervals
• Re-arranging

What if s unknown? An unbiased estimator of s
95 confidence interval given by
Example
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Topic 1 Statistical Review

• 7. Hypothesis Testing
• Hypothesis statement about a popn. developed for
  the purpose of testing
• Hypothesis testing procedure based on sample
  evidence and probability theory to determine
  whether the hypothesis is a reasonable statement.
• Steps
• 1. State the null (H0 ) and alternate (HA )
  hypotheses
• Note distinction between one and two-tailed
  tests
• 2. State the level of significance
• Probability of rejecting H0 when it is true
  (Type I Error)
• Note Type II Error failing to reject H0 when
  it is false
• Power of the test 1-Pr(Type II error)
• 3. Select a test statistic
• Based on sample info., follows a known
  distribution)
• 4. Formulate decision rule
• Conditions under which null hypothesis is
  rejected. Based on critical value from known
  probability distribution.
• 5. Compute the value of the test statistic, make
  a decision, interpret the results.

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Topic 1 Statistical Review

• 7. Hypothesis Testing
• Example 1
• A packaging device is set to fill detergent
  packets with a mean weight of 150g. The standard
  deviation is known to be 5.0g. A random sample of
  25 boxes is checked and are found to have a mean
  weight of 152.5g. Can we conclude that the
  machine is producing a mean weight of more than
  150g?
• Example 2
• The personnel department of a company has
  developed an aptitude test for screening
  potential employees. The person who devised the
  test predicted that the mean mark attained would
  be 100. From a random sample of 13 applicants a
  mean mark of 96 was recorded with a standard
  deviation of 52. At the 1 significance level
  determine whether the mean mark is in fact equal
  to 100.

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Topic 1 Statistical Review

• 7. Hypothesis Testing
• P-value
• Alternative means of evaluating decision rule
• Probability of observing a sample value as
  extreme as, or more extreme than the value
  observed when the null hypothesis is true
• If the p-value is greater than the significance
  level, H0 is not rejected
• If the p-value is less than the significance
  level, H0 is rejected
• If the p-value is less than
• 0.10, we have some evidence that H0 is not true
• 0.05 we have strong evidence that H0 is not true
• 0.01 we have very strong evidence that H0 is not
  true