Dougherty 4th Edition [PDF] [EPUB] Introduction to Econometrics FIFTH EDITION Christopher. Dougherty London School of Economics and. Introduction to. Econometrics. FIFTH EDITION. Christopher Dougherty. London School of Economics and Political Science. OXFORD. UNIVERSITY PRESS. Introduction to Econometrics by Christopher Dougherty () Christopher Dougherty. Introduction To Econometrics Dougherty Solutions Manual Pdf.

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Dougherty: Introduction to Econometrics 5e Christopher Dougherty for the module "EC Elements of Econometrics" which Chapter 1 (PDF, Size: KB). ronaldweinland.info - Free Download Introduction Econometrics Dougherty Introduction To Econometrics Christopher Dougherty 5th Edition Pdf. introduction to econometrics christopher dougherty. Preview ronaldweinland.info The Last Black Unicorn Tiffany Haddish.

All rights reserved. Contents Preface 0. Econometrics is the application of statistical methods to the quantification and critical assessment of hypothetical economic relationships using data. It is with the aid of econometrics that we discriminate between competing economic theories and put numerical clothing onto the successful ones.

Then there is a checklist of learning outcomes anticipated as a result of studying the chapter in the textbook, doing the exercises in the subject guide, and making use of the corresponding resources on the website.

This consists of new topics that may be included in the next edition of the textbook. The second part of each chapter consists of additional exercises, followed by answers to the starred exercises in the text and answers to the additional exercises.

Preface read the Overview section from the Review chapter of the subject guide read the Review chapter of the textbook and do the starred exercises refer to the subject guide for answers to the starred exercises in the text and for additional exercises check that you have covered all the items in the learning outcomes section in the subject guide. You should repeat this process for each of the numbered chapters.

Note that the subject guide chapters have the same titles as the chapters in the text. Here you will find PowerPoint slideshows that provide a graphical treatment of the topics covered in the textbook, data sets for practical work and statistical tables.

However, I strongly recommend that you do study all the slideshows as well. Some do not add much to the material in the textbook, and these you can skim through quickly. Some, however, provide a much more graphical treatment than is possible with print and they should improve your understanding. Some present and discuss regression results and other hands-on material that could not be included in the text for lack of space, and they likewise should be helpful.

The student versions of such applications are adequate for doing all, or almost all, the exercises and of course are much cheaper than the professional ones. If you do not have access to a commercial econometrics application, you should use gretl. This is a sophisticated application almost as powerful as the commercial ones, and it is free. See the gretl manual on the OUP website for further information.

Whatever you do, do not be tempted to try to get by with the regression engines built into some spreadsheet applications, such as Microsoft Excel. Online study resources There are three major data sets on the website. You will find on the website versions in the formats used by Stata, EViews and gretl. If you are using some other application, you should download the text version comma-delimited ASCII and import it.

Answers to all of the exercises are provided in the relevant chapters of this subject guide. The exercises for the CES data set cover Chapters 1—10 of the text. For Chapters 11—13, you should use the Demand Functions data set, another major data set, to do the additional exercises in the corresponding chapters of this subject guide. Again you should download the data set in the appropriate format.

For these exercises, also, answers are provided. The third major data set on the website is the Educational Attainment and Earnings Function data set, which provides practical work for the first 10 chapters of the text and Chapter No answers are provided, but many parallel examples will be found in the text. You have probably already logged in to the Student Portal in order to register!

As soon as you registered, you will automatically have been granted access to the VLE, Online Library and your fully functional University of London email account. It forms an important part of your study experience with the University of London and you should access it regularly. Preface Electronic study materials: The printed materials that you receive from the University of London are available to download, including updated reading lists and references.

A student discussion forum: This is an open space for you to discuss interests and experiences, seek support from your peers, work collaboratively to solve problems and discuss subject material. Videos: There are recorded academic introductions to the subject, interviews and debates and, for some courses, audio-visual tutorials and conclusions.

Study skills: Expert advice on preparing for examinations and developing your digital literacy skills. Some of these resources are available for certain courses only, but we are expanding our provision all the time and you should check the VLE regularly for updates. If you are having trouble finding an article listed in a reading list, try removing any punctuation from the title, such as single quotation marks, question marks and colons.

For further advice, please see the online help pages: www.

The mathematics requirement is a basic understanding of multivariate differential calculus. With regard to statistics, you must have a clear understanding of what is meant by the sampling distribution of an estimator, and of the principles of statistical inference and hypothesis testing.

This is absolutely essential. Application of linear algebra to econometrics find that most problems that students have with introductory econometrics are not econometric problems at all but problems with statistics, or rather, a lack of understanding of statistics.

There are no short cuts. If you do not have this background knowledge, you should put your study of econometrics on hold and study statistics first. Otherwise there will be core parts of the econometrics syllabus that you do not begin to understand. In addition, it would be helpful if you have some knowledge of economics.

However, although the examples and exercises relate to economics, most of them are so straightforward that a previous study of economics is not a requirement. No answers are provided.

For Chapters 11— There are three major data sets on the website. A student discussion forum: This is an open space for you to discuss interests and experiences. This is absolutely essential.

These provide advice on how each examination question might best be answered. To access the majority of resources via the Online Library you will either need to use your University of London Student Portal login details. The mathematics requirement is a basic understanding of multivariate differential calculus. Study skills: Expert advice on preparing for examinations and developing your digital literacy skills. There are recorded academic introductions to the subject. Feedback forms.

Recorded lectures: For some courses. Preface Electronic study materials: The printed materials that you receive from the University of London are available to download. If you are having trouble finding an article listed in a reading list. With regard to statistics. Some of these resources are available for certain courses only. For further advice. There are no short cuts. It is provided for the benefit of those students who intend to take a further course in econometrics.

It is not part of the syllabus for the examination. Application of linear algebra to econometrics find that most problems that students have with introductory econometrics are not econometric problems at all but problems with statistics.

The primer assumes that such basic study has already been undertaken. The present course is ambitious. Candidates should answer eight out of ten questions in three hours: Because of this we strongly advise you to always check both the current Regulations for relevant information about the examination.

If you do not have this background knowledge. Otherwise there will be core parts of the econometrics syllabus that you do not begin to understand. In addition. A calculator may be used when answering questions on this paper and it must comply in all respects with the specification given with your Admission Notice. There are many excellent texts and there is no point in duplicating them. For its purposes.

The primer does not attempt to teach it. Please note that subject guides may be used for several years. It is intended to show how the econometric theory in the text can be handled with this more advanced mathematical approach. Preface up-to-date information on examination and assessment arrangements for this course where available. Overview Review: Random variables and sampling theory 0. The Review chapter of my textbook is not a substitute. They are central to econometric analysis and if you have not encountered them before.

It has the much more limited objective of providing an opportunity for revising some key statistical concepts and results that will be used time and time again in the course. There are many excellent textbooks that offer a first course in statistics. You should also be able to explain why they are important. Each chapter has a set of additional exercises.

Define its probability density function. Construct a similar table for the case where they are two standard deviations apart.

The concepts of efficiency. The answers to them are provided at the end of the chapter after the answers to the starred exercises in the text. Preface 0. Find the probability distribution for X. Answers to the starred exercises in the textbook AR. Given a sample of observations. What are the consequences of erroneously performing a one-sided test when a two-sided test would have been appropriate?

The appropriate alternative hypothesis is therefore H1: An investigator wishes to test H0: To simplify the analysis. The table shows the 36 possible outcomes. The probabilities have been written as fractions. Given a sample of n independent observations. The probability distribution is derived by counting the number of times each outcome occurs and dividing by The actual figure is Given that the largest values of X 2 have the highest probabilities.

The table is based on Table R. It is a good idea to guess the outcome before doing the arithmetic. In this case it is not easy to make a guess. Note that four decimal places have been used in the working. The population variance is 1.

This is to eliminate the possibility of the estimate being affected by rounding error. If the calculated value does not conform with the guess. In this case. Preface R. Thus the variance is The last-digit discrepancy between this figure and that in Exercise R.

This is the counterpart of Exercise R. When the sample size increases. Because it is improving in this important sense. Preface Answer: Is it correct to describe the estimator as becoming more efficient?

The estimator with the smallest variance is said to be the most efficient. But it is the wrong use of the term. You cannot use efficiency as suggested in the question because you are comparing the variances of the same estimator with different sample sizes.

Efficiency is a comparative concept that is used when you are comparing two or more alternative estimators. Answers to the starred exercises in the textbook R. It may be assumed that the distribution is known to have variance equal to one.

The figure shows the potential distribution of X conditional on H0 being true. This implies that the probability of it lying 1. He decides to reject H0 if X lies in the central 5 per cent of the distribution the tinted area in the figure.

Hence the right-hand 5 per cent rejection region begins 1. For the 1 per cent test. This means that it is located 1. For the 5 per cent test. He wishes to test the null hypothesis H0: The cumulative probability for 0. The difference in their powers must therefore tend to zero. According to the standard normal distribution table. See Figure R. A researcher has a sample of observations with sample mean X. It is the same. By construction. The figure shows that the power of the test approaches one asymptotically.

The probability of not rejecting H0 when it is false will be lower. The greater the difference between the true value and the hypothetical mean. The figure shows the power functions for the test using the conventional upper and lower 2.

Preface b Explain in intuitive terms why his test is unwise. If the null hypothesis is true. The second figure is for the case where the true value is lower than the hypothetical value. Issues relating to Type II errors are irrelevant when the null hypothesis is true. To use the obvious technical term. The vertical axis is the power of the test.

The first figure has been drawn for the case where the true value is greater than the hypothetical value. The following discussion assumes that you are performing a 5 per cent significance test. This is an extreme example of a very bad test procedure. The reason that the central part of the conditional distribution is not used as a rejection region is that it leads to problems when the null hypothesis is false. This would be a surprising outcome.

Fail to reject H0 at the 5 per cent level. There are 24 degrees of freedom. Answers to the starred exercises in the textbook 1. Reject H0 at the 1 per cent level. If H0 is true. Reject H0 at the 5 per cent level but not the 1 per cent level.

Taking a sample of 25 firms. In principle we should stick to our guns and fail to reject H0. Reject at the 1 per cent level and 0. We can now reject H0 at the 5 per cent level. For example. No change. If it is decided that it cannot.

The other is that the model is misspecified in some way and the misspecification is responsible for the unexpected sign. Explain whether she might have been justified in performing one-sided tests in cases a — d. Here there is a problem because the coefficient has the unexpected sign.

One is that the justification for a one-sided test is incorrect not very likely in this case. Using the plim rules. Show that the variance of Z does not tend to zero as n tends to infinity and that therefore Z is an inconsistent estimator. The weights sum to unity. Show that: To put it formally. This will tend to zero as n tends to infinity. The estimator will be consistent for the same reason as explained in b.

Given the symmetry of the distribution of X. Explain verbally whether or not each estimator is 1 unbiased. The estimator is therefore consistent. The first figure shows the distributions of the estimators a and b for 1. Call the maximum value of X in the sample Xmax and the minimum value Xmin. For sample size ? If the mean square error is used to compare the estimators. Answers to the starred exercises in the textbook The table gives the means and variances of the distributions as computed from the results of the simulations.

In such a case the optimal properties of the sample mean are no longer guaranteed and it may be eclipsed by a score statistic such as the largest observation in the sample.

How large does n have to be for b to be preferred to a using the mean square error criterion? The crushing superiority of b over a may come as a surprise. The underlying reason in this case is that we are estimating a boundary parameter. Since the length of the rectangle is 2. We will encounter superconsistent estimators again when we come to cointegration in Chapter However we will derive this formally.

Answers to the additional exercises AR. Two-sided tests Under the false H0: One-sided tests Under H0: Under H0. Hence the table is: Table R. Hence the variance is 0. From Exercise AR.

Under H1: If the true value is 0. A Type II error therefore occurs if X is more than 0. Issues relating to Type II error do not arise because the null hypothesis is true. The risk of a Type I error is. A Type II error therefore occurs if X is less than 0. The power functions for one-sided and two-sided tests are shown in the first figure below. The power is not automatically zero for true values that are negative because even for these it is possible that a sample might have a mean that lies in the right tail of the distribution under the null hypothesis.

If the true value is negative. That for the two-sided test is the same as that in the first figure. Preface If the true value is positive.

We know that: The chapter continues by showing how the coefficients should be interpreted when the variables are measured in natural units. Chapter 1 Simple regression analysis 1. Give an interpretation of the coefficients. Provide an interpretation of the coefficients.

Simple regression analysis In addition. Summary statistics for the data are also provided. Additional exercises. GDP per capita. Simple regression analysis A1. We will write the fitted models for the two specifications as: Give a mathematical demonstration that the value of R2 in such a regression is zero. Answers to the starred exercises in the textbook. Literally the regression implies that. The regressions: By definition: Explain intuitively why this should be so.

Show that the regression coefficients will automatically satisfy the following equations: The intercept.

Confirm that the estimates of the intercept and slope coefficient are as should be expected from the changes in the units of measurement. The intuitive explanation is that the regressions break down income into predicted wages and profits and one would expect the sum of the predicted components of income to be equal to its actual level.

The slope coefficient is not affected by demeaning: The slope coefficient and intercept for the regression in metric units. This result generalises to the multiple regression case.

It implies: Simple regression analysis The regression output confirms that the calculations are correct subject to rounding error in the last digit. Evaluate the outcome if the slope coefficient were estimated using 1. The numerator of the sample correlation coefficient for Yb and u b can be decomposed as follows. The reason for the fall in R2 is the huge increase in the total sum of squares.

If the explanatory variable were income. Explain why R2 is lower than in the regression reported in Exercise 1. Hence the correlation is zero. Answers to the additional exercises by 1. The explained sum of squares is actually higher than that in Exercise 1. In the respondents were aged 27— Housing has the largest coefficient. Was this due to the relatively heavy becoming even heavier.

Note that this is an instance where the constant term can be given a meaningful interpretation and where it is as of much interest as the slope coefficient. The explanation is that this is a very poorly specified earnings function and that. All the slope coefficients are highly significant. For those that did. The interpretation of the slope coefficient is obviously highly implausible. The R2 indicates that weight accounts for 71 per cent of the variance in weight.

Later on. The regression output indicates that weight in was approximately equal to weight in plus 17 pounds. The negative intercept has no possible interpretation.

The intercept: The new intercept is: Note that R2 is very low. Hence we find an apparent positive association between earnings and height in a simple regression. R2 is unit-free and so it is not possible for the overall fit of a relationship to be affected by the units of measurement. Hence RSS is unchanged. Hence the new slope coefficient is: The residual in observation i in the new regression.

Answers to the additional exercises males earn more than females. Males also tend to be taller. Note that this makes sense intuitively. As in Exercise A1. Answers to the additional exercises A1. It is a fundamental chapter because much of the rest of the text is devoted to extending the least squares approach to handle more complex models. The third criterion.

We are equally concerned with assessing the performance of our regression techniques and with developing an understanding of why they work better in some circumstances than in others. The chapter is a long one and you should take your time over it because it is essential that you develop a perfect understanding of every detail. In particular. Chapter 2 Properties of the regression coefficients and hypothesis testing 2.

Chapter 2 is the starting point for this objective and is thus equally fundamental. In Section 2. Properties of the regression coefficients and hypothesis testing 2.

Given a sample of n observations. There must be a shorter proof. Give an intuitive explanation. X may be assumed to be a nonstochastic variable. P We saw in Exercise A2. What can be said in this case about the efficiency of the estimator in these two cases. Explain the implications of the difference in the variances. The true model is: One researcher fits: In view of the true model: Subtracting the second equation from the first. Y and X. Note that.

The second researcher included an intercept in the specification. Properties of the regression coefficients and hypothesis testing A2. Hence show that: The corresponding values that Y would take. Would dropping the unnecessary intercept lead to a gain in efficiency?

The presence of the disturbance term in the model causes the actual values of Y in a sample to be different. The values of X are fixed and are as shown in the figure. Four of them. X1 to X4. The fifth. The solid black circles depict a typical sample of observations.

Properties of the regression coefficients and hypothesis testing Discuss the advantages and disadvantages of dropping the observation corresponding to X5 when regressing Y on X. Given a sample of observations on Y. If you keep the observation in the sample. Answers to the starred exercises in the textbook Hence: Xj Answer: In Exercise 1.

What are the implications for the efficiency of the estimator? If Z happens to be an exact linear function of X. This means that the residuals are the same.

What should he report: Fail to reject at the 5 per cent level. We can now reject H0 at the 1 per cent level but not at the 0. We can then perform a one-sided test.

Not affected by the change. Reject H0 at the 5 per cent level but not at the 1 per cent level. Reject at the 0. It would be illogical for an individual with greater skills to be paid less on that account. One is that the justification for a one-sided test is incorrect. With 48 degrees of freedom. There are 48 degrees of freedom.

The objective of training is to impart skills. If you think that one-sided tests are justified. In principle we should ignore this and fail to reject H0. Here there is a problem because the coefficient has an unexpected sign and is large enough to reject H0 at the 5 per cent level with a two-sided test. Answers to the starred exercises in the textbook He is prepared to test the null hypothesis H0: Hence the t statistic is unaffected by the transformation.

Another possible reason for a coefficient having an unexpected sign is that the model is misspecified in some way. Properties of the regression coefficients and hypothesis testing Apprenticeships are the classic example.

How will the F statistic for this regression be related to the F statistic for the original regression? See also Exercises 1. See also Exercise 2. In either case. For a simple regression the F statistic is the square of the t statistic on the slope coefficient. With n equal to We saw in Exercise 1. The 95 per cent confidence interval is therefore: We see that we cannot quite reject the null hypothesis H0: In Exercise 2. Each of the exercises below relates to a simple regression.

Obviously a one-sided t test. In general. The F statistic is equal to the square of the t statistic and. Xi Xi Xi Xi Hence: Answers to the additional exercises 2. Returning to the general case. It is the OLS estimator in this case. The F test is equivalent to the t test on the slope coefficient.

The F statistic is Since this is a simple regression model. Thus using a criterion such as mean square error. This is because the estimators are then identical. There is little point performing a t test on the intercept. The reason for this is that the estimators are now identical: In principle the estimate of an extra 41 cents of hourly earnings for every extra inch of height could have been a purely random result of the kind that one obtains with nonsense models.

Of course. The F test is equivalent. This is given by: Thus the new standard error is given by: Hence F is unchanged. Hence the new standard error is given by: We saw in Exercise A1. Since RSS is unchanged. Xi However. If the second researcher had fitted 3 instead of 2. Answers to the additional exercises Demonstration that the residuals are the same: Hence the estimator will be the same.

Demonstration that R2 in 2 is equal to R2 in 1: It follows that dropping the unnecessary intercept would not have led to a gain in efficiency. Using 3. Demonstration that the OLS estimator of the variance of the disturbance term in 2 is equal to that in 1: Exercise 1.

Hence the F statistic will be the same and R2 will be the same. This said, in practice one would wish to check whether it is sensible to assume that the model relating Y to X for the other observations really does apply to the observation corresponding to X5 as well. This question can be answered only by being familiar with the context and having some intuitive understanding of the relationship between Y and X.

Specific topics are treated with reference to a model with just two explanatory variables, but most of the concepts and results apply straightforwardly to more general models. The chapter begins by showing how the least squares principle is employed to derive the expressions for the regression coefficients and how the coefficients should be interpreted.

It continues with a discussion of the precision of the regression coefficients and tests of hypotheses relating to them. Next comes multicollinearity, the problem of discriminating between the effects of individual explanatory variables when they are closely related. The chapter concludes with a discussion of F tests of the joint explanatory power of the explanatory variables or subsets of them, and shows how a t test can be thought of as a marginal F test.

You should know the expression for the population variance of a slope coefficient in a multiple regression model with two explanatory variables. Provide an interpretation of the regression coefficients and perform appropriate tests. Delete observations where expenditure on your category is zero.

She fits the relationship 1 for a sample of 25 manufacturing enterprises, and 2 for a sample of services enterprises. The table provides some data on the samples. The researcher finds that the standard error of the coefficient of K is 0. Explain the difference quantitatively, given the data in the table. He decides to relate Y , gross hourly earnings in , to S, years of schooling, and PWE, potential work experience, using the semilogarithmic specification:. PWE is defined as age — years of schooling — 5.

Since the respondents were all aged 43 in , this becomes:. The researcher finds that it is impossible to fit the model as specified. Stata output for his regression is reproduced below:.

Explain why the researcher was unable to fit his specification. Explain how the coefficient of S might be interpreted. This is an extension of one of the useful results in Section 1. If the model is: For observation i we have: He attempts to fit the following model: Explain why he is unable to fit this equation.

Give both intuitive and technical explanations. How might he resolve the problem? There is exact multicollinearity since there is an exact linear relationship between W , NW and the constant term. As a consequence it is not possible to tell whether variations in E are attributable to variations in W or variations in NW, or both.

One way of dealing with the problem would be to drop N W from the regression. The mean value of I in the sample is 2. He fits the regression standard errors in parentheses: Explain why the t tests and F test have different outcomes. Although there is not an exact linear relationship between W and O, they must have a very high negative correlation because the mean value of I is so small.

Hence one would expect the regression to be subject to multicollinearity, and this is confirmed by the results. The t statistics for the coefficients of W and O are only 1. Both of these effects are very highly significant.

The intercept has no plausible interpretation. At first sight it may seem surprising that SIZE has a significant negative effect for some categories. The reason for this is that an increase in SIZE means a reduction. Effectively poorer. Answers to the additional exercises in expenditure per capita. One explanation of the negative effects could be economies of scale.

The purpose of this specification is to test whether household size has an effect on expenditure per capita on food consumed at home.

Now SIZE has a very significant negative effect. Expenditure per capita on food consumed at home increases by 4. To determine the true direct effect.

R2 is lower than in Exercise A3. To investigate this further. SIZE has a positive effect. In the case of SHEL. Another might be family composition — larger families having more children. In the case of DOM. As might be expected. Multiple regression analysis this is not plausible in the case of some. Answers to the additional exercises results for TOB are puzzling. The net effect.

Other things being equal. S Data Factors manufacturing services manufacturing services sample sample sample sample Number of 25 0. Multiple regression analysis A3.

Thus the coefficient of schooling estimates the proportional increase in earnings associated with an additional year of schooling.

An extra year of schooling implies one fewer year of potential work experience. Sometimes an apparently nonlinear model can be linearised by taking logarithms. If you plot earnings on schooling. Often the real reason for preferring a nonlinear specification to a linear one is that it makes more sense theoretically. Because they can be fitted using linear regression analysis. The chapter shows how the least squares principle can be applied when the model cannot be linearised.

Chapter 4 Transformations of variables 4. It should be skipped on first reading because it makes use of material on maximum likelihood estimation. It generalises with no substantive changes to the multiple regression model.

Assuming that u is iid independently and identically distributed N 0. Transformations of variables explain the role of the disturbance term in a nonlinear model explain how in principle a nonlinear model that cannot be linearised may be fitted perform a transformation for comparing the fits of models with linear and logarithmic dependent variables. To keep the mathematics uncluttered.

An expanded version is offered here. The two models are actually special cases of the more general model: Hence the likelihood function for the parameters is: Further material and the log-likelihood is: Under the null hypothesis.

One simple procedure is to perform a grid search. RSS was In this case we are testing whether the log-likelihood under the restriction is significantly smaller than the unrestricted log-likelihood. For the linear and logarithmic specifications. Example 0 -1 As with all tests. The maximum likelihood estimate is 0.

The logarithmic specification is clearly much to be preferred. An alternative model specification. Note that it is not possible to test the two hypotheses directly against each other. Believing there to be diminishing returns to experience. What should be the strategy of the researcher for determining which of the four specifications has the best fit?

Additional exercises A4. Compare the RSS for these equations. Transformations of variables A4. Using your EAWE data set. Use your category of expenditure from the CES data set. Determine how the regression coefficients are related to those of the original regression.

The mean values of S. Give an interpretation of the regression output. Note that: Let the fitted regression be: As a consequence. Demonstrate that this was not a coincidence. This is a special case of the transformation in Exercise 4. Answers to the starred exercises in the textbook Answer: Nothing of substance is affected since the change amounts only to a fixed constant shift in the measurement of the explanatory variable.

R2 must also be the same: In such a situation. Stata drops one of the variables responsible.. Explain the regression results Iteration Transformations of variables 4. The figure below represents the relationship. The specification in Exercise 4. As in Exercise 4. The other parameters do not have straightforward interpretations.

The revised specification indicates that the adverse effect is more evenly spread and is more enduring. Comparing this figure with that for Exercise 4. The specification is an extension of that for Exercise 4.

Provide an interpretation of the regression results and compare it with that for Exercise 4. The table gives the results for all the categories of expenditure. In addition to testing the null hypothesis that the elasticity is equal to zero.

The size elasticity is significantly negative. The specification is equivalent to that in Exercise 4. Answers to the additional exercises A4.

This follows from the fact that the theoretical coefficient. The specification differs from that in Exercise A4. Writing the latter again as: Note that the estimates of the income elasticity are identical to those in Exercise 4. Determine whether 2 is a reparameterised or a restricted version of 1.

Let u bi be the residual in 3 and vbi the residual in 4. Given the relationship, it is also for the test of H0: The tests are equivalent since both of them reduce the model to log Y depending only on an intercept and the disturbance term. Explain whether R2 would be the same for the two regressions. R2 will be different because it measures the proportion of the variance of the dependent variable explained by the regression, and the dependent variables are different.

In 2 it is the proportion of the variance of log Y explained by the regression. Thus, although related, they are not directly comparable. In 1 RSS has dimension the squared units of Y. In 2 it has dimension the squared units of log Y. Typically it will be much lower in 2 because the logarithm of Y tends to be much smaller than Y. The specifications with the same dependent variable may be compared directly in terms of RSS or R2 and hence two of the specifications may be eliminated immediately.

RSS for the scaled regressions will then be comparable. The differences are actually surprisingly large and suggest that some other factor may also be at work. One possibility is that the data contain many outliers, and these do more damage to the fit in linear than in logarithmic specifications. Iteration messages have been deleted.

The latter is very soundly rejected by the likelihood-ratio test. However, rewriting the model as: Since no individual in the sample had fewer than 8 years of schooling, the perverse sign of the estimate illustrates only the danger of extrapolating outside the data range. It makes better sense to evaluate the implicit coefficient for an individual with the mean years of schooling, The positive sign of the coefficient of SA suggests that schooling and cognitive ability have mutually reinforcing effects on earnings.

One way of avoiding nonsense parameter estimates is to measure the variables in question from their sample means. The predict command saves the fitted values from the most recent regression, assigning them the variable name that follows the command, in this case YHAT.

Somewhat surprisingly, its coefficient is not significant. Multicollinearity is responsible for the failure to detect nonlinearity hear.

Although the intercept dummy may appear artificial and strange at first sight, and the slope dummy even more so, you will become comfortable with the use of dummy variables very quickly.

The key is to keep in mind the graphical representation of the regression model. We have seen that the logarithm of earnings is more satisfactory than earnings as the dependent variable in a wage equation. Fitting the semilogarithmic specification, we obtain:. In Exercise A1. The output below shows the result of adding the dummy variable to the specification, to control for sex.

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