1-3a Cross-Sectional Data 5 1-3b Time-Series Data 7 1-3c Pooled Cross-Sections 8 1-3d Panel or Longitudinal Data 9 1-3e A Comment on Data Structures 10 1-4 Causality and the Concept of Ceteris Paribus. 3-3b Omitted variable bias: the simple case 78 3-3c Omitted variable bias: More general cases 81 3-4 The variance of the oLS estimators 81 .
Limited Dependent Variable Models and Sample Selection Corrections 524
Advanced Time Series Topics 568
Carrying Out an Empirical Project 605
Organized for Today’s Econometrics Instructor
MLR.1: Introduce the population model and interpret the population parameters (which we hope to estimate). Assumption MLR.5 (homoscedasticity) is added for the Gauss-Markov theorem and for the usual OLS variance formulas to be valid.
New to This Edition
After introducing assumptions MLR.1 to MLR.3, one can discuss the algebraic properties of ordinary least squares—that is, the properties of OLS for a particular set of data. Assumption MLR.6 (normality), first introduced in Chapter 4, is added to round out the classical linear model assumptions.
Targeted at Undergraduates, Adaptable for Master’s Students
The treatment of unobserved effects panel data models in Chapter 14 has been expanded to include more of a discussion of unbalanced panel data sets, including how the fixed effects, random effects, and correlated random effects approaches can still be used. I also include discussion of some subtle problems that can arise when using clustered standard errors when the data are obtained from a random sampling scheme.
Design Features
Data Sets—Available in Six Formats
Updated Data Sets Handbook
Instructor Supplements
Instructor’s Manual with Solutions
You will find instructional slides for each chapter in this edition, including the advanced chapters in Part 3. The PowerPoint® slides are available for convenient download in the password-protected instructor-only section of the book's companion website at http:// login.cengage.com.
Scientific Word Slides
Test Bank
Student Supplements
MindTap
Aplia
Student Solutions Manual
Suggestions for Designing Your Course
Acknowledgments
His careful reading of the manuscript and keen eye for detail have greatly improved this sixth edition. This book is dedicated to my wife, Leslie Papke, who contributed materially to this publication by writing the initial versions of the Science Word slides for the chapters in Part 3; she then used the slides in her public policy course.
About the Author
We will use these data later in a time series analysis of the effect of the minimum wage on employment. We will use methods and insights from cross-sectional analysis in the remainder of the text.
Summary
For the first investment, you know exactly what the return is at the time of purchase because you know the starting price of the three-month T-bill along with its face value. According to the expectations hypothesis, the expected return from the second investment, given all information at the time of investment, should be equal to the return from buying a three-month Treasury bill.
Key Terms
The expectation hypothesis from financial economics states that, given all the information available to investors at the time of investing, the expected return on two investments is the same. Therefore, there is uncertainty in this investment for someone with a three-month investment horizon.
Problems
Computer Exercises
Part 1
Sometimes, the explanatory variable explains a significant portion of the sample variation in the dependent variable. Finally, equation (2.44) may not capture all of the nonlinearity in the relationship between wages and schooling.
It turns out that the variance of the OLS estimators can be calculated under the assumptions SLR.1 to SLR.4. Assumption SLR.4 concerns the expected value of u, while assumption SLR.5 concerns the variance of u (both depending on x). Recall that we established the unbiasedness of OLS without assumption SLR.5: The homoscedasticity assumption plays no role in showing that b^0 and b^1 are unbiased.
This makes sense, since more variation in the unobservables that affect y makes it harder to estimate b1 precisely. On the other hand, more variability in the independent variable is preferred: as the variability in xi increases, the variance of b^1 decreases. Thus, there are only n22 degrees of freedom in the OLS residuals as opposed to n degrees of freedom in the errors.
C4 Use the data in WAGE2 to estimate a simple regression that explains monthly wage (salary) in terms of IQ score (IQ). i) Find the mean salary and mean IQ in the sample. Report the estimated equation in the usual way, including sample size and R square. iv). What is the greatest number of executions. by OLS and report the results in the usual way, including sample size and R-squared. iii) Interpret the slope coefficient reported in part (ii).
APPEndix 2A
In Section 3-2, we demonstrate how to estimate the parameters in the multiple regression model using the method of ordinary least squares. The general multiple linear regression (MLR) model (also called the multiple regression model) can be written in the population as. The term "linear" in a multiple linear regression model means that equation (3.6) is linear in the parameters, ie.
This means that the sample mean, y, "explains" more of the variation in yi than the explanatory variables. In the sample (and therefore in the population), none of the independent variables is constant and there is no exact linear relationship between the independent variables. On the other hand, Assumption MLR.4—by far the more important of the two—constrains the relationship between the unobserved factors in u and the explanatory variables.
As it turns out, we have done almost all of the work to remove the bias in the simple regression estimator of b| to deduce. Assumption MLR.5 means that the variance in the error term, u, conditional on the explanatory variables, is the same for all combinations of outcomes of the explanatory variables. Assumption MLR.5 states that the variance of y, given x, does not depend on the values of the independent variables.
Remember that the subscript j simply denotes any of the independent variables (such as education or poverty level). In the general case, R2j is the proportion of the total variation in xj that can be explained by the other independent variables appearing in the equation. For example, an R2j of 5.9 means that 90% of the sample variation xj can be explained by the other independent variables in the regression model.
3-4c Estimating s 2 : Standard Errors of the OLS Estimators
The variance formula in (3.55) depends on the values of xi1 and xi2 in the sample, which provides the best scenario for b|. But the expression in equation (3.55) does not account for the increase in error variance because it will treat both regressors as non-random. This can be compared to errors ui which have n degrees of freedom in the sample.).
Theorem 3.3
In the rare case that an interrupt is not evaluated, the number of parameters is reduced by one.). We also know what an unbiased estimator is: in the present context, an estimator of, say, b|. In other words, in the class of unbiased linear estimators, OLS has the smallest variance (under the five Gauss-Markov assumptions).
R2 is the proportion of sample variation in the dependent variable explained by the independent variables and serves as a measure of goodness-of-fit. 7 Which of the following can cause OLS estimators to be biased? ii) Omission of an important variable. iii). Why should we remove one of the variables of the tax part from the equation? ii) Give a careful interpretation of b1.
APPEndix 3A
In Section 4-1, we begin by finding the distributions of the OLS estimators under the added assumption that the population error is normally distributed. In Section 3-4, we obtained the variances of the OLS estimators under Gauss-Mark assumptions. According to the CLM assumptions of MLR.1 to MLR.6, depending on the sample values of the independent variables.
Theorem 4.1
If u is a complicated function of the unobserved factors, then the CLT argument does not really apply. As we will see in Chapter 5 - and this is important - non-normality of the errors is not a serious problem with large sample sizes. In Chapter 5, we will show that the normality of the OLS estimators is still approximately true in large samples even without the normality of the errors.
Theorem 4.2
We have put "the" in quotation marks because, as we will soon see, a more general form of the t-statistic is needed to test other hypotheses about bj. When the alternative is two-sided, we are interested in the absolute value of the t-statistic. That is, the p-value is the significance level of the test when we use the value of the test statistic, 1.85 in the above example, as the critical value for the test.
The p-value nicely summarizes the strength or weakness of the empirical evidence against the null hypothesis. General multiple linear constraints can be tested using the sum of squared residuals of the form of the F statistic.
Theorem 5.1
But we can prove Theorem 5.1 without problems in the case of the simple regression model. First, OLS is found to be biased (but consistent) under assumption MLR.4r if E1u0x1, p, xk2 depends on any of xj. The exact normality of the OLS estimators depends crucially on the normality of the distribution of the error, u, in the population.
Theorem 5.2
However, it turns out that, to obtain a usable test statistic, we must include all independent variables in the regression. But you should be aware of the LM statistic as it is used in applied work. We also showed that the LM statistic can be used instead of the F statistic for testing exclusion restrictions.
Theorem 5.3
Why does your answer contradict the assumption of a normal distribution for score. ii) Explain what happens in the left tail of the histogram. If y has a normal distribution in the population, the sample skewness measure for the standardized values should not be significantly different from zero. i) First use the dataset 401KSUBS, keep only observations with fsize51. What are the smallest and largest values in the sample. ii) Consider the linear model.
APPEndix 5A
Another way to think about it is that the error in the equation with bwghtlbs as the dependent variable has a standard deviation that is 16 times the standard deviation of the original error. Recall that the key independent variable is nox, a measure of nitrogen oxide in the air above each community. Using the simple algebraic properties of the exponential and logarithmic functions gives the exact percentage change in the predicted y angle.
