Applied Econometrics Techniques 153400119

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Applied Econometrics Techniques 153400119

• Lecture 3
• Multiple Regression
• Recap and
• Asymptotic Properties of OLS

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Reading

• Recap Ch2,3,4 of Wooldridge
• Asymptotics Ch5 Wooldridge

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Assumed knowledge

• Interpretation of the OLS parameters
• GM assumptions and theorem
• Properties of OLS unbiasedness, efficiency
• Hypothesis testing
• Sampling distributions
• Normality assumptions and tests
• t, F-tests, confidence intervals etc.
• Linear restrictions
• Functional forms
• Omitted Variables bias

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OLS parameters marginal effect

• Ceteris paribus assumption
• Consider the following model

• OLS here gives the same estimate for b1 as the
  following regression

• Where ri1 is the residual from the regression

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Example using Crime data

• Using the Crime data CRIME.dta
• Consider model

• Then run

• Notice that

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Properties of the OLS estimators
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Gauss Markov Assumptions

• Population model is linear in parameters y
  b0 b1x1 b2x2 bkxk u
• We can use a random sample of size n, (xi1,
  xi2,, xik, yi) i1, 2, , n
• None of the xs is constant, and there are no
  exact linear relationships (perfect MC)
• E(ux1, x2, xk) 0, implying that all of the
  explanatory variables are exogenous
• Homoskedasticity Var(ux1, x2, xk)s2

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The Conditional Distribution
Marginal distribution of u
Joint distribution of x and u
u
Conditional distribution of u given x 3 divide
by f(x3) 0.540
Marginal distribution of x
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Conditional Expectations
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OLS Variance

• We have a measure of central tendency but not
  spread
• Here we use GM1-5 (GM5 homoskedasticity) to
  derive the variance

• High R2j or low Sx (data variation) are
  problematic
• The former suggests MC
• Trade-off between adding and taking away
  variables
• Omitting induces bias, including raises variance
• Use MSE to compare ?.

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E.g. A Misspecified model

• If we estimate

• Parameter variance is

• As opposed to the population model

• The former is biased, the latter has a lower
  variance.

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Estimation of OLS s2

• The following estimator is unbiased

• But biased for the sample analogue

• The unbiased estimator is

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GM Theorem

• If GM 1-5 hold then OLS is not only unbiased, but
  also it is efficient.
• Best Linear Unbiased Estimator among all unbiased
  linear estimators.
• If any of the GM assumptions fail, this is not
  true
• We have used homoskedasticity for this

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Inference

• GM 6 Normality population error is independent
  of the explanatory variables and normally
  distributed with mean and variance s2
• Much stronger assumption Before
• GM4 E(ux1, x2, xk) 0
• GM5 homoskedasticity
• GM 6 implies
• GM 4 and 5
• OLS is the best of all unbiased estimators

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Normality Jarque-Bera test in STATA

• Not true by and large
• t and F stats are invalid where this is not the
  case

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sktest Birth-weight Data

• STATA undertakes a test of kurtosis and skewness
  as well as a combined JB test.

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Testing linear hypothesis in STATA

• Consider the following model (using Murder data)

• Test the hypothesis that b1 and b2 are equal
• Test the hypothesis that b1 and b2 0 Exclusion
  restrictions
• Use the test command for Wald test

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Economic vs Statistical Significance

• Consider the following model of participation
  rates in pension plans

• Run regression in STATA
• The coefficient on totemp is -0.00013 increase
  the size of the firm by 10000, participation rate
  decreases by 1

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Consistency of OLS
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Consistency

• Under the Gauss-Markov assumptions OLS is BLUE,
  but in other cases it wont always be possible to
  find unbiased estimators
• In those cases, we may settle for estimators
  that are consistent, meaning as n ? 8, the
  distribution of the estimator collapses to the
  parameter value

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Sampling Distributions as n ?
n3
n1 lt n2 lt n3
n2
n1
b1
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Consistency of OLS

• Under the first 4 GM assumptions, the OLS
  estimator is consistent (and unbiased)
• Consistency can be proved for the simple
  regression case in a manner similar to the proof
  of unbiasedness
• Will need to take probability limit (plim) to
  establish consistency

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A Weaker Assumption

• For unbiasedness, we assumed a zero conditional
  mean E(ux1, x2,,xk) 0
• For consistency, we can have the weaker
  assumption of zero mean and zero correlation
  E(u) 0 and Cov(xj,u) 0, for j 1, 2, , k
• Without this assumption, OLS will be inconsistent!

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Conditional Expectations
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Deriving the Inconsistency

• Just as we could derive the omitted variable
  bias earlier, now we want to think about the
  inconsistency, or asymptotic bias, in this case

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Asymptotic Bias (cont)

• direction of the asymptotic bias found just as
  for bias in omitted variables case
• Main difference is that asymptotic bias uses the
  population variance and covariance, while bias
  uses the sample counterparts
• Remember, inconsistency is a large sample
  problem it doesnt go away as add data

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Large Sample Inference

• Under the CLM assumptions, the sampling
  distributions are normal ? t and F distributions
  for testing
• This exact normality was due to assuming the
  population error distribution was normal
• This assumption of normal errors implied that
  the distribution of y, given the xs, was normal
  as well

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Large Sample Inference (cont)

• Easy to come up with examples for which this
  exact normality assumption will fail
• take the data in wage.dta
• Normality assumption not needed to conclude OLS
  is BLUE, only for inference

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Non-normal variables
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Central Limit Theorem

• Based on the central limit theorem, we can show
  that OLS estimators are asymptotically normal
• Asymptotic Normality implies that P(Zltz)?F(z) as
  n ??, or P(Zltz) ? F(z)
• The central limit theorem states that the
  standardized average of any population with mean
  m and variance s2 is asymptotically N(0,1), or

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Asymptotic Normality
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Asymptotic Normality (cont)

• Because the t distribution approaches the normal
  distribution for large df, we can also say that

• Note that while we no longer need to assume
  normality with a large sample, we do still need
  homoskedasticity

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Asymptotic Standard Errors

• If u is not normally distributed, we sometimes
  will refer to the standard error as an asymptotic
  standard error, since

• So, we can expect standard errors to shrink at a
  rate proportional to the inverse of vn

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Lagrange Multiplier statistic

• Once we are using large samples and relying on
  asymptotic normality for inference, we can use
  more than t and F stats
• The Lagrange multiplier or LM statistic is an
  alternative for testing multiple exclusion
  restrictions
• Because the LM statistic uses an auxiliary
  regression its sometimes called an nR2 stat

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LM Statistic (cont)

• Suppose we have a standard model, y b0 b1x1
  b2x2 . . . bkxk u and our null hypothesis
  is
• H0 bk-q1 0, ... , bk 0
• First, we just run the restricted model

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LM Statistic (cont)

• With a large sample, the result from an F test
  and from an LM test should be similar
• Unlike the F test and t test for one exclusion,
  the LM test and F test will not be identical

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Asymptotic Efficiency

• Estimators besides OLS will be consistent
• However, under the Gauss-Markov assumptions, the
  OLS estimators will have the smallest asymptotic
  variances
• We say that OLS is asymptotically efficient
• Important to remember our assumptions though, if
  not homoskedastic, not true