Analysis of Cross Section and Panel Data

1
Analysis of Cross Section and Panel Data

• Yan Zhang
• School of Economics, Fudan University
• CCER, Fudan University

2
Introductory Econometrics A Modern
Approach

• Yan Zhang
• School of Economics, Fudan University
• CCER, Fudan University

3
Analysis of Cross Section and Panel Data

• Part 1. Regression Analysis on Cross Sectional
  Data

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Chap 2. The Simple Regression ModelPractice
for learning multiple Regression

• Bivariate linear regression model
• the slope parameter in the relationship
  between y and x holding the other factors in u
  fixed it is of primary interest in applied
  economics.
• the intercept parameter, also has its uses,
  although it is rarely central to an analysis.

5
More Discussion

• A one-unit change in x has the same effect
  on y, regardless of the initial value of x.
• Increasing returns wage-education (f. form)
• Can we draw ceteris paribus conclusions about how
  x affects y from a random sample of data, when we
  are ignoring all the other factors?
• Only if we make an assumption restricting how the
  unobservable random variable u is related to the
  explanatory variable x

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Classical Regression Assumptions

• Feasible assumption if the intercept term is
  included
• Linearly uncorrelated zero conditional
  expectation
• Meaning
• ???
• PRF (Population Regression Function) sth. fixed
  but unknown

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OLS

• Minimize uu
• sample regression function (SRF)
• The point is always on the OLS regression
  line.

??????
PRF
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OLS

• Coefficient of determination
• the fraction of the sample variation in y that is
  explained by x.
• the square of the sample correlation coefficient
  between and
• Low R-squareds

9
Units of Measurement

• If one of the dependent variables is multiplied
  by the constant cwhich means each value in the
  sample is multiplied by cthen the OLS intercept
  and slope estimates are also multiplied by c.
• If one of the independent variables is divided or
  multiplied by some nonzero constant, c, then its
  OLS slope coefficient is also multiplied or
  divided by c respectively.
• The goodness-of-fit of the model, R-squareds,
  should not depend on the units of measurement of
  our variables.

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Function Form

• Linear Nonlinear
• Logarithmic dependent variable
• A
• Percentage change in y, semi-elasticity
• an increasing return to edu.
• Other nonlinearity diploma effect
• Bi-Logarithmic
• A
• a
• Constant elasticity
• Change of units of measurement
• P45, error b0b0log(c1)-b1log(c2)

• Bi-Logarithmic
• A
• a
• Constant elasticity
• Change of units of measurement
• P45, error
• b0b0log(c1)-b1log(c2)
• Be proficient at interpreting the coef.

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Unbiasedness of OLS Estimators

• Statistical properties of OLS
• ????????????????OLS?? ?????
• Assumptions
• Linear in parameters (f. form advanced methods)
• Random sampling (time series data nonrandom
  sampling)
• Zero conditional mean (unbiased biased
  spurious cor)
• Sample Variation in the independent variables
  (colinearity)
• Theorem (Unbiasedness)
• Under the four assumptions above, we have

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Variance of OLS Estimators

• ?????? ???,??? ???? ???
• Assumptions
• Homoskedasticity
• Error variance
• A larger means that the distribution of the
  unobservables affecting y is more spread out.
• Theorem (Sampling variance of OLS estimators)
• Under the five assumptions above

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Variance of y given x

• Conditional mean
• and variance of y
• Heteroskedasticity

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What does depend on?

• More variation in the unobservables affecting y
  makes it more difficult to precisely estimate
• The more spread out is the sample of xi -s, the
  easier it is to find the relationship between
• E(y x) and x
• As the sample size increases, so does the total
  variation in the xi. Therefore, a larger sample
  size results in a smaller variance of the
  estimator

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Estimating Error Variance

• Errors (Disturbances) and Residuals
• Errors , population
• Residuals , estimated f.
• Theorem (The unbiased estimator of )
• Under the five assumptions above, we have
• standard error of the regression (SER)
• Estimating the standard deviation in y after the
  effect of x has been taken out.
• Standard Error of

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Regression through the Origin

• Regression through the Origin
• Pass through
• E.g. income tax revenue income
• The estimator of OLS
• only if 0
• if the intercept 0, then is a
  biased estimator of

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Chap 3. Multiple Regression AnalysisEstimation

• Advantages of multiple regression analysis
• build better models for predicting the dependent
  variable.
• E.g.
• generalize functional form.
• Marginal propensity to consume
• Be more amenable to ceteris paribus analysis
• Chap 3.2
• Key assumption
• Implication other factors affecting wage are not
  related on average to educ and exper.
• Multiple linear regression model

the ceteris paribus effect of xj on y
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Ordinary Least Square Estimator

• SPF
• OLS
• Minimize
• F.O.C
• ceteris paribus interpretations
• Holding fixed, then
• Thus, we have controlled for the variables
  when estimating the effect of x1 on y.

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Holding Other Factors Fixed

• The power of multiple regression analysis is that
  it provides this ceteris paribus interpretation
  even though the data have not been collected in a
  ceteris paribus fashion.
• it allows us to do in non-experimental
  environments what natural scientists are able to
  do in a controlled laboratory setting keep other
  factors fixed.

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OLS and Ceteris Paribus Effects

• Step of OLS
• (1) the OLS residuals from a multiple
  regression
• of x1 on
• (2) the OLS estimator from a simple
  regression
• of y on
• measures the effect of x1 on y after x2,,
  xk have been partialled or netted out.
• Two special cases in which the simple regression
  of y on x1 will produce the same OLS estimate on
  x1 as the regression of y on x1 and x2.

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Goodness-of-fit

• also equal the squared correlation coef.
  between the actual and the fitted values of y.
• R never decreases, and it usually increases when
  another independent variable is added to a
  regression.
• The factor that should determine whether an
  explanatory variable belongs in a model is
  whether the explanatory variable has a nonzero
  partial effect on y in the population.

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Regression through the origin

• the properties of OLS derived earlier no longer
  hold for regression through the origin.
• the OLS residuals no longer have a zero sample
  average.
• can actually be negative.
• to calculate it as the squared correlation
  coefficient
• if the intercept in the population model is
  different from zero, then the OLS estimators of
  the slope parameters will be biased.

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The Expectation of OLS Estimator

• Assumptions(???????????????)
• Linear in parameters
• Random sampling
• Zero conditional mean
• No perfect co-linearity
• none of the independent variables is constant
• and there are no exact linear relationships among
  the independent variables
• Theorem (Unbiasedness)
• Under the four assumptions above, we have

rank (X)K
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Notice 1 Zero conditional mean

• Exogenous Endogenous
• Misspecification of function form (Chap 9)
• Omitting the quadratic term
• The level or log of variable
• Omitting important factors that correlated with
  any independent v.
• ???????????????,?????????,??????
• Measurement Error (Chap 15, IV)
• Simultaneously determining one or more x-s with y
  (Chap 16, ?????)

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Omitted Variable Bias The Simple Case

• ProblemExcluding a relevant variable or
  Under-specifying the model(???????????(??)??????)
• Omitted Variable Bias (misspecification analysis)
• The true population model
• The underspecified OLS line
• The expectation of
• The Omitted variable bias

??3.2???x1?x2??
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Omitted Variable Bias Nonexistence

• Two cases where is unbiased
• The true population model
• is the sample covariance between x1 and x2
  over the sample variance of x1
• If , then
  ?????x2??,?????????,?x2???????????????
• Summary of Omitted Variable Bias
• The expectation of
• The Omitted variable bias

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The Size of Omitted Variable Bias

• Direction Size
• A small bias of either sign need not be a cause
  for concern.
• Unknown Some idea
• we usually have a pretty good idea about the
  direction of the partial effect of x2 on y, that
  is, the sign of
• in many cases we can make an educated guess about
  whether x1 and x2 are positively or negatively
  correlated.
• E.g. (Upward/downward Bias biased toward zero)

??!
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Omitted Variable Bias More General Cases

• Suppose x2 and x3 are uncorrelated, but that x1
  is correlated with x3.
• Both and will normally be biased. The
  only exception to this is when x1 and x2 are also
  uncorrelated.
• Difficult to obtain the direction of the bias in
  and
• Approximation if x1 and x2 are also uncor.

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Notice 2 No Perfect Collinearity

• An assumption only about x-s, nothing about the
  relationship between u and x-s
• Assumption MLR.4 does allow the independent
  variables to be correlated they just cannot be
  perfectly correlated. Ceteris Paribus
  effect
• If we did not allow for any correlation among the
  independent variables, then multiple regression
  would not be very useful for econometric
  analysis.
• Significance

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Cases of Perfect Collinearity

• When can independent variables be perfectly
  collinear softwaresingular
• Nonlinear functions of the same variable is not
  an exact linear f.
• Not to include the same explanatory variable
  measured in different units in the same
  regression equation.
• More subtle ways
• one independent variable can be expressed as an
  exact linear function of some or all of the other
  independent variables. Drop it
• Key

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Notice 3 Unbiase

• the meaning of unbiasedness
• an estimate cannot be unbiased an estimate is a
  fixed number, obtained from a particular sample,
  which usually is not equal to the population
  parameter.
• When we say that OLS is unbiased under
  Assumptions MLR.1 through MLR.4, we mean that the
  procedure by which the OLS estimates are obtained
  is unbiased when we view the procedure as being
  applied across all possible random samples.

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Notice 4 Over-Specification

• Inclusion of an irrelevant variable or
  over-specifying the model
• does not affect the unbiasedness of the OLS
  estimators.
• including irrelevant variables can have
  undesirable effects on the variances of the OLS
  estimators.

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Variance of The OLS Estimators

• Adding Assumptions
• Homoskedasticity
• Error variance
• A larger means that the distribution of the
  unobservables affecting y is more spread out.
• Gauss-Markov assumptions (for cross-sectional
  regression) Assumption 1-5
• Theorem (Sampling variance of OLS estimators)
• Under the five assumptions above

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More about

• The stastical properties of y on x(x1, x2, ,
  xk)
• Error variance
• only one way to reduce the error variance to add
  more explanatory variablesnot always possible
  and desirable
• The total sample variations in xj SSTj
• Increase the sample size

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Multi-collinearity(?????)

• The linear relationships among the independent v.
• ???????xj?????(????)
• If k2
• the proportion of the total variation in
  xj that can be explained by the other independent
  variables
• High (but not perfect)
  correlation between two or more of the in
  dependent variables is called multicollinearity.

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Micro-numerosity problem of small sample size

• High
• Low SSTj
• one thing is clear everything else being equal,
  for estimating j, it is better to have less
  correlation between xj and the other x-s.
• How to solve the multicollinearity?
• Increase sample size
• Dropping some v.? ???????????????,??????

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Notice The influence of multicollinearity

• A high degree of correlation between certain
  independent variables can be irrelevant as to how
  well we can estimate other parameters in the
  model.
• E.g.

• Importance for economistscontrolling v.

????
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Variances in Misspecified Models
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Whether or Not to Include x2 Two Favorable
Reasons

• The choice of whether or not to include a
  particular variable in a regression model can be
  made by analyzing the tradeoff between bias and
  variance..
• However, when2 0, there are two favorable
  reasons for including x2 in the model.
• any bias in does not shrink as the sample
  size grows
• The variance of estimators both shrink to zero as
  n increase
• Therefor, the multicollinearity induced by adding
  x2 becomes less important as the sample size
  grows. In large samples, we would prefer

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Estimating Standard Errors of the OLS
Estimators
????
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EFFICIENCY OF OLS THE GAUSS-MARKOV THEOREM

• BLUE
• Best smallest variance
• linear
• unbiased

????(1)????????????????(2)??G-M????????,?BLUE???
???????????(???)????????????,???????????
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Classical Linear Model AssumptionsInference
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???????????

• Jeffrey M. Wooldridge, Introductory
  EconometricsA Modern Approach, Chap 2-3.