Solution Manual - Introductory Econometrics: A Modern Approach 8th Edition - Wooldridge (Chapter 1 - 19)

Introductory Econometrics m




Jeffrey M. Wooldridge m m




Chapter 1 The Nature of Econometrics and Economic Data ............................................. 1
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Part 1
m Regression Analysis with Cross-Sectional Data ................................................. 1
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Chapter 2 The Simple Regression Model ......................................................................... 1
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Chapter 3 Multiple Regression Analysis: Estimation ....................................................... 2
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Chapter 4 m Multiple Regression Analysis: Inference .................................................... 4
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Chapter 5 Multiple Regression Analysis: OLS Asymptotics ............................................ 5
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Chapter 6 Multiple Regression Analysis: Further Issues .................................................. 6
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Chapter 7 Multiple Regression Analysis with Qualitative Information: Binary variables 8
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Chapter 8 Heteroskedasticity ........................................................................................... 9
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Chapter 9 More on Specification and Data problems ..................................................... 12
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Part 2
m Regression Analysis with Time Series Data ...................................................... 14
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Chapter 10m Basic Regression analysis with Time Series Data ...................................... 14
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Chapter 11m Further Issues in Using OLS with Time Series Data ................................... 16
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Chapter 12m Serial Correlation and Heteroskedasticity in Time Series Regression ........ 19
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Part 3
m Advanced Topics .............................................................................................. 23
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Chapter 13m Pooling Cross Sections across Time. Simple Panel Data Methods ............. 23
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Chapter 14m Advanced Panel Data Methods .................................................................. 25
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Chapter 15m Instrumental Variables Estimation and Two Stage Least Squares .............. 27
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Chapter 16m Simultaneous Equations Models ................................................................ 30
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Chapter 17m Limited Dependent Variable Models and Sample Selection Corrections 31
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Chapter 18
m m Advanced Time Series Topics ................................................................... 35
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Chapter 19m Carrying Out an Empirical Project ............................................................. 39
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Appendix: Some fundamentals of probability .................................................................... 42
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,Introductory Econometrics m Study Notes by Zhipeng Yan m m m m




Chapter 1 The Nature of Econometrics and Economic Data m m m m m m m




I. The goal of any econometric analysis is to estimate the parameters in the model
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and to test hypotheses about these parameters; the values and signs of the
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parameters determine the validity of an economic theory and the effects of
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certain policies.
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II. Panel data - advantages: m m m




1. Having multiple observations on the same units allows us to control certain
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unobserved characteristics of individuals, firms, and so on. The use of more than
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one observation can facilitate causal inference in situations where inferring
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causality would be very hard if only a single cross section were available.
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2. They often allow us to study the importance of lags in behavior or the result of
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decision making.
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Part 1 m Regression Analysis with Cross-Sectional Data m m m m




Chapter 2 The Simple Regression Model m m m m m m




I. Model: Y = b0 + b1x + u m m m m m m m




1. Population regression function (PRF): E(y|x) = b0 + b1x m m m m m m m m




2. systematic part of y: b0 + b1x m m m m m m




3. unsystematic part: u m m




II. Sample regression function (SRF): yhat = b0hat + b1hat*x m m m m m m m m




1. PRF is something fixed, but unknown, in the population. Since the SRF is
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obtained for a given sample of data, a new sample will generate a different slope
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and intercept.
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III. Correlation: it is possible for u to be uncorrelated with x while being m m m m m m m m m m m m




correlated with functions of x, such as x2.
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E(u|x) = E(u)  Cov(u, x) = 0. not vice versa.
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IV. Algebraic properties of OLS statistics m m m m




1. The sum of the OLS residuals is zero.
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2. The sample covariance between the (each) regressors and the residuals is zero.
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Consequently, the sample covariance between the fitted values and the residuals is
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zero.
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3. The point ( x, y ) is on the OLS regression line.
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4. the goodness-of-fit of the model is invariant to changes in the units of y or x.
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5. The homoskedasticity assumption plays no role in showing OLS estimators are
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unbiased.
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V. Variance
1. Var(b1) = var(u)/SSTx m m




a. more variation in the unobservables (u) affecting y makes it more difficult to
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precisely estimate b1.
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,Introductory Econometrics m Study Notes by Zhipeng Yan m m m m




b. More variability in x is preferred, since the more spread out is the sample of
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independent variables, the easier it is to trace out the relationship between E(y|x)
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and x. That is, the easier it is to estimate b1.
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2. standard error of the regression, standard error of the estimate and the root m m m m m m m m m m m m




1

2 m


mean squared error = u
(n  2)
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Chapter 3 Multiple Regression Analysis: Estimation m m m m




I. The power of multiple regression analysis is that is allows us to do in
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nonexperimental environments what natural scientists are able to do in a
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controlled laboratory setting: keep other factors fixed.
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II. Model: Y = b0 + b1x1 + b2x2 + u m m m m m m m m m




b  (v y ) /(v2 ) , where v is the OLS residuals from a simple regression of x1
n n
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1 i1 m i i1
m i1 i1
on x2. m




1. v is the part of x1 that is uncorrelated with x2, or v is x1 after the effects of x2 have
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been partialled out, or netted out. Thus, b1 measures the sample relationship
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between y and x1 after x2 has been partialled out.
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III. Goodness-of-fit
1. R2 = the squared correlation coefficient between the actual y and the fitted
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values yhat. m m




2. R2 never decreases because the sum of squared residuals never increases when
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additional regressors are added to the model.
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IV. Regression through the origin: m m m




1. OLS residuals no longer have a zero sample average.
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2. R2 can be negative. This means that the sample average “explains” more of the
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variation in the y than the explanatory variables.
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V. MLR Assumptions: m




A1: linear in parameters. A2:
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random sampling.
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A3: Zero conditional mean: E(u|x1, x2, …, xk) = 0
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When A3 holds, we say that we have Exogenous explanatory variables. If xj is
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correlated with u for any reason, then xj is said to be an endogenous explanatory
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variables.
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A4: No perfect collinearity.
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A1 – A4  unbiasedness of OLS
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VI. Overspecifying the model: m m




1. Including one or more irrelevant variables, does not affect the unbiasedness of the m m m m m m m m m m m m




OLS estimators. m m




2

, Introductory Econometrics m Study Notes by Zhipeng Yan m m m m




VII. Variance of OLS estimators: m m m




A5: homoskedasticity
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1. Gauss – Markov assumptions: A1 – A5 m m m m m m




2 2
2. Var(bj ) 
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, where R is from regressing xj on all other independent
SST j (1 R j)
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2 m m

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variables (and including an intercept). m m m m




a. The error variance, σ2, is a feature of the population, it has nothing to do with the
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sample size.
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b. SSTj: the total sample variation in xj: a small sample size  small value of SSTj
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 large var(bj) m m




c. R 2j : high correlation between two or more independent variables is called
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multicollinearity.
3. A high degree of correlation between certain independent variables can be
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irrelevant as to how well we can estimate other parameters in the model: Y
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= b0 + b1x1 + b2x2 + b3x3 + u, where x2 and x3 are highly correlated.
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The var(b2) and var(b3) may be large. But the amount of correlation between x2 and x3
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has no direct effect on var(b1). In fact, if x1 is uncorrelated with x2 and x3, then
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2
R12=0 and var(b1) =
m


m m m , regardless of how much correlation there is between x2
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SST1
and x3.
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If b1 is the parameter of interest, we do not really care about the amount of
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correlation between x2 and x3. m m m m




4. The tradeoff between bias and variance. m m m m m




If the true model is Y = b0 + b1x1 + b2x2 + u, instead, we estimate Y = b0 + b’1x1 + u
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a. when b2 is nonzero, b’1 is biased, b1 is unbiased, var(b’1)<var(b1); m m m m m m m m m m




b. when b2 is zero, b’1 is unbiased, b1 is unbiased, var(b’1)<var(b1)  a m m m m m m m m m m m m




higher variance for the estimator of b1 is the cost of including an
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irrelevant variable in a model; m m m m m




VIII. Estimating: standard errors of estimators. m m m m




1 2
1. Under A1-A5: E(σ’ ) = σ , where σ’2 =

2 2
u (σ’ is σhat)
m



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(n  k 1) m m m m





2. Standard deviation of bj’, sd(bj’) = m m m m m m




SST j (1 R2)j m m m m m




hat
3. Standard error of bj’: se(bj’) = m m m m m m




SST j (1 R2)j m m m m m




Standard error of bj’ is not a valid estimator of sd(bj’) if the errors exhibit
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heteroskedasticity. Thus, while the presence of heteroskedasticity does not cause
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bias in the bj’, it does lead to bias in the usual formula for Var(bj’), which then
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invalidates the standard errors.
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