A Companion to Econometric Analysis of Panel Data - Softcover

Baltagi, Badi H.

Softcover

ISBN 10: 0470744030 ISBN 13: 9780470744031

Verlag: John Wiley & Sons, 2009

Inhaltsangabe

This book is a companion to Baltagi’s (2008) leading graduate econometrics textbook on panel data entitled Econometric Analysis of Panel Data, 4^th Edition.

The book guides the student of panel data econometrics by solving exercises in a logical and pedagogical manner, helping the reader understand, learn and apply panel data methods. It is also a helpful tool for those who like to learn by solving exercises and running software to replicate empirical studies. It works as a complementary study guide to Baltagi (2008) and also as a stand alone book that builds up the reader’s confidence in working out difficult exercises in panel data econometrics and applying these methods to empirical work.

The exercises start by providing some background information on partitioned regressions and the Frisch-Waugh-Lovell theorem. Then it goes through the basic material on fixed and random effects models in a one-way and two-way error components models: basic estimation, test of hypotheses and prediction. This include maximum likelihood estimation, testing for poolability of the data, testing for the significance of individual and time effects, as well as Hausman's test for correlated effects. It also provides extensions of panel data techniques to serial correlation, spatial correlation, heteroskedasticity, seemingly unrelated regressions, simultaneous equations, dynamic panel models, incomplete panels, measurement error, count panels, rotating panels, limited dependent variables, and non-stationary panels.

Über diesen Titel

Über die Autorin bzw. den Autor

Badi H. Baltagi is Distinguished Professor of Economics, and Senior Research Associate at the Center for Policy Research, Syracuse University. He is a fellow of the Journal of Econometrics, a recipient of the Multa and Plura Scripsit Awards from Econometric Theory, and the Journal of Applied Econometrics Distinguished Authors Award.

The book guides the student of panel data econometrics by solving exercises in a logical and pedagogical manner, helping the reader understand, learn and apply panel data methods. It is also a helpful tool for those who like to learn by solving exercises and running software to replicate empirical studies. It works as a complementary study guide to Baltagi (2008) and also as a stand alone book that builds up the reader’s confidence in working out difficult exercises in panel data econometrics and applying these methods to empirical work.

The exercises start by providing some background information on partitioned regressions and the Frisch-Waugh-Lovell theorem. Then it goes through the basic material on fixed and random effects models in a one-way and two-way error components models: basic estimation, test of hypotheses and prediction. This include maximum likelihood estimation, testing for poolability of the data, testing for the significance of individual and time effects, as well as Hausman's test for correlated effects. It also provides extensions of panel data techniques to serial correlation, spatial correlation, heteroskedasticity, seemingly unrelated regressions, simultaneous equations, dynamic panel models, incomplete panels, measurement error, count panels, rotating panels, limited dependent variables, and non-stationary panels.

The book provides several empirical examples that are useful to applied researchers, illustrating them using Stata and EViews showing the reader how to replicate these studies. The data sets are provided on the Wiley web site: www.wileyeurope.com/college/baltagi.

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A Companion to Econometric Analysis of Panel Data

By Badi Baltagi

John Wiley & Sons

Copyright © 2009 Badi Baltagi
All right reserved.
ISBN: 978-0-470-74403-1

Chapter One

Partitioned Regression and the Frisch-Waugh-Lovell Theorem

This chapter introduces the reader to important background material on the partitioned regression model. This should serve as a refresher for some matrix algebra results on the partitioned regression model as well as an introduction to the associated Frisch-Waugh-Lovell (FWL) theorem. The latter is shown to be a useful tool for proving key results for the fixed effects model in Chapter 2 as well as artificial regressions used in testing panel data models such as the Hausman test in Chapter 4.

Consider the partitioned regression given by

y = X + u = [X.sub.1][.sub.1] + [X.sub.2][.sub.2] + u (1.1)

where y is a column vector of dimension (n x 1) and X is a matrix of dimension (n x k). Also, X = [[X.sub.1], [X.sub.2]] with [[X.sub.1] and [[X.sub.2] of dimension (n x [k.sub.1]) and (n x [k.sub.2]), respectively. One may be interested in the least squares estimates of [.sub.2] corresponding to [[X.sub.2], but one has to control for the presence of [[X.sub.1] which may include seasonal dummy variables or a time trend; see Frisch and Waugh (1933) and Lovell (1963). For example, in a time-series setting, including the time trend in the multiple regression is equivalent to detrending each variable first, by residualing out the effect of time, and then running the regression on these residuals. Davidson and MacKinnon (1993) denote this result more formally as the FWL theorem.

The ordinary least squares (OLS) normal equations from (1.1) are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Exercise 1.1 (Partitioned regression). Show that the solution to (1.2) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the projection matrix on [X.sub.1], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution

Write (1.2) as two equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solving for [[??].sub.1,OLS] in terms of [[??].sub.2,OLS] by multiplying the first equation by [([X.sup.'.sub.1][X.sub.1]).sup.-1], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting [[??].sub.1,OLS] in the second equation, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Collecting terms, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], y as given in (1.3). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the orthogonal projection matrix of [X.sub.1] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] generates the least squares residuals of each column of [X.sub.2] regressed on all the variables in [X.sub.1]. In fact, if we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

using the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is idempotent. For a review of idempotent matrices, see Abadir and Magnus (2005, p.231). This implies that [[??].sub.2,OLS] can be obtained from the regression of [??] on [[??].sub.2]. In words, the residuals from regressing y on [X.sub.1] are in turn regressed upon the residuals from each column of [X.sub.2] regressed on all the variables in [X.sub.1]. If we premultiply (1.1) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and use the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

Exercise 1.2 (The Frisch-Waugh-Lovell theorem). Prove that:

(a) the least squares estimates of [.sub.2] from equations (1.1) and (1.5) are numerically identical;

(b) the least squares residuals from equations (1.1) and (1.5) are identical.

Solution

(a) Using the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is idempotent, it immediately follows that OLS on (1.5) yields [[??].sub.2,OLS] as given by (1.3). Alternatively, one can start from (1.1) and use the result that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where [P.sub.X] = [X(X' X).sup.-1] X' and [P.sub.X] = [I.sub.n] - [P.sub.X]. Premultiplying (1.6) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and using the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

But [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using this fact along with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the last term of (1.7) drops out yielding the result that [[??].sub.2,OLS] from (1.7) is identical to the expression in (1.3). Note that no partitioned inversion was used in this proof. This proves part (a) of the FWL theorem. To learn more about partitioned and projection matrices, see Chapter 5 of Abadir and Magnus (2005).

(b) Premultiplying (1.6) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and using the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

Note that [[??].sub.2,OLS] was shown to be numerically identical to the least squares estimate obtained from (1.5). Hence, the first term on the right-hand side of (1.8) must be the fitted values from equation (1.5). Since the dependent variables are the same in equations (1.8) and (1.5), [P.sub.x] y in equation (1.8) must be the least squares residuals from regression (1.5). But [P.sub.x] y is the least squares residuals from regression (1.1). Hence, the least squares residuals from regressions (1.1) and (1.5) are numerically identical. This proves part (b) of the FWL theorem. Several applications of the FWL theorem will be given in this book.

Exercise 1.3 (Residualing the constant). Show that if [X.sub.1] is the vector of ones indicating the presence of a constant in the regression, then regression (1.8) is equivalent to running ([y.sub.i] - [bar.y]) on the set of variables in [X.sub.2] expressed as deviations from their respective sample means.

Solution

In this case, X = [[l.sub.n], [X.sub.2]] where n is a vector of ones of dimension n. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a matrix of ones of dimension n. But [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [J.sub.n] y/n = [bar.y]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has a typical element ([y.sub.i] - [bar.y]). From the FWL theorem, [[??].sub.2,OLS] can be obtained from the regression of ([y.sub.i] - [bar.y]) on the set of variables in X2 expressed as deviations from their respective means, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From the solution of Exercise 1.1, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the vector of sample means of the independent variables in [X.sub.2].

Exercise 1.4 (Adding a dummy variable for the ith observation). Show that including a dummy variable for the ith observation in the regression is equivalent to omitting that observation from the regression. Let y = X + [D.sub.i] [gamma] + u, where y is n x 1, X is n x k and [D.sub.i] is a dummy variable that takes the value 1 for the ith observation and 0 otherwise. Using the FWL theorem, prove that the least squares estimates of and [gamma] from this regression are [??].sub.OLS] = [(X*'X*).sup.-1] X*'y* and [[??].sub.OLS] = [y.sub.i] - [x.sup.'.sub.i][[??].sub.OLS], where X* denotes the X matrix without the ith observation, y* is the y vector without the ith observation and ([y.sub.i], [x.sup.'.sub.i]) denotes the ith observation on the dependent and independent variables. Note that [[??].sub.OLS] is the forecasted OLS residual for the ith observation obtained from the regression of y* on X*, the regression which excludes the ith observation.

Solution

The dummy variable for the ith observation is an n x 1 vector [D.sub.i] = (0, 0, . . . , 1, 0, . . ., 0) of zeros except for the ith element which takes the value 1. In this case, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is a matrix of zeros except for the ith diagonal element which takes the value 1. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an identity matrix except for the ith diagonal element which takes the value zero. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] returns the vector y except for the ith element which is zero. Using the FWL theorem, the OLS regression

y = X + [D.sub.i][gamma] + u

yields the same estimates as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which can be rewritten as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The OLS normal equations yield [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the ith OLS normal equation can be ignored since it gives 0'[[??].sub.OLS] = 0. Ignoring the ith observation equation yields (X*' X*)[[??].sub.OLS] = X*'y*, where X* is the matrix X without the ith observation and y* is the vector y without the ith observation. The FWL theorem also states that the residuals from [??] on [??] are the same as those from y on X and Di. For the ith observation, [[??].sub.i] = 0 and [[??].sub.i] = 0. Hence the ith residual must be zero. This also means that the ith residual in the original regression with the dummy variable [D.sub.i] is zero, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Rearranging terms, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In other words, [[??].sub.OLS] is the forecasted OLS residual for the ith observation from the regression of y* on X*. The ith observation was excluded from the estimation of [[??].sub.OLS] by the inclusion of the dummy variable [D.sub.i].

The results of Exercise 1.4 can be generalized to including dummy variables for several observations. In fact, Salkever (1976) suggested a simple way of using dummy variables to compute forecasts and their standard errors. The basic idea is to augment the usual regression in (1.1) with a matrix of observation-specific dummies, i.e., a dummy variable for each period where we want to forecast:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where [delta]' = (', [gamma]'). X* has in its second part a matrix of dummy variables, one for each of the [T.sub.o] periods for which we are forecasting.

Exercise 1.5 (Computing forecasts and forecast standard errors)

(a) Show that OLS on (1.9) yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [[??].sub.o] = [X.sub.o][??]. In other words, OLS on (1.9) yields the OLS estimate of without the [T.sub.o] observations, and the coefficients of the [T.sub.o] dummies, i.e., [??], are the forecast errors.

(b) Show that the first n residuals are the usual OLS residuals e = y - X]??] based on the first n observations, whereas the next [T.sub.o] residuals are all zero. Conclude that the mean square error of the regression in (1.10), [s.sup.*2], is the same as [s.sup.2] from the regression of y on X.

(c) Show that the variance-covariance matrix of [??] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

where the off-diagonal elements are of no interest. This means that the regression package gives the estimated variance of [??] and the estimated variance of the forecast error in one stroke.

(d) Show that if the forecasts rather than the forecast errors are needed, one can replace [y.sub.o] by zero, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in (1.9). The resulting estimate of [gamma] will be [[??].sub.o] = [X.sub.o][??], as required. The variance of this forecast will be the same as that given in (1.11).

Solution

(a) From (1.9) one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The OLS normal equations yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [X.sub.o][[??].sub.OLS] + [[??].sub.OLS] = [y.sub.o]. From the second equation, it is obvious that [[??].sub.OLS] = [y.sub.o] - [X.sub.o][[??].sub.OLS]. Substituting this in the first equation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which upon cancellation gives [[??].sub.OLS] = [(X' X).sup.-1] X' y. Alternatively, one could apply the FWL theorem using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This means that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Premultiplying (1.9) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equivalent to omitting the last [T.sub.o] observations. The resulting regression is that of y on X, which yields [[??].sub.OLS] = [(X'X).sup.-1] X'y as obtained above.

(b) Premultiplying (1.9) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the last [T.sub.o] observations yield zero residuals because the observations on both the dependent and independent variables are zero. For this to be true in the original regression, we must have [y.sub.o] - [X.sub.o] [[??].sub.OLS] - [[??].sub.OLS] = 0. This means that [[??].sub.OLS] = [y.sub.o] - [X.sub.o][[??].sub.OLS] as required. The OLS residuals of (1.9) yield the usual least squares residuals

[e.sub.OLS] = y - X][??].sub.OLS]

(Continues...)

Excerpted from A Companion to Econometric Analysis of Panel Databy Badi Baltagi Copyright © 2009 by Badi Baltagi. Excerpted by permission.
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Details zur Ausgabe

• Verlag: John Wiley & Sons
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 0470744030
• ISBN 13: 9780470744031
• Einband: Tapa blanda
• Anzahl der Seiten: 308
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar