[Note for bibliographic reference: Melberg, Hans O. (1997), Advanced Econometrics
for Beginners - A review of Maddala (1992), http://www.oocities.org/hmelberg/papers/970519.htm]
Advanced Econometrics for Beginners
A review of Maddala (1992)
by Hans O. Melberg
Introduction to Econometrics
G. S. Maddala
Macmillan Publishing Company
New York, 1992 (2nd ed, first 1988)
631 pages, ISBN: 0-02-374545-2
Introduction
In the preface to Introduction to Econometrics Maddala writes that "There has
been many important developments in econometrics during the last two decades, but
introductory books in the field still deal mostly with what econometrics was in the 1960s.
The present book is meant to familiarize students (and researchers) with some of these
developments ..." (p. xv). To assess whether the book fulfils this aim, I shall first
present the virtues of the book. Next, in the second part of the review, I discuss its
shortcomings.
1 Virtues
This book has two major virtues. First, Maddala is very good at criticising and showing
the weaknesses of the concepts presented. Second, the book deals with many new and
important topics such as exogeneity, unit roots, cointegration, error correction models,
model selection and specification testing. I shall examine these in turn.
1.1 Critical attitude
Whatever one may think about this book, one cannot argue that it is uncritical of
traditional econometrics. I shall give several examples, starting with the problem of
significance.
Most econometricians know that there is nothing sacred about the 1% and 5% significance
levels. Moreover, they know that it is easy to make a variable significant by using a very
large sample. Lastly, we know that statistical significance should not be confused with
economic or practical significance - or even worse causal significance. Yet, somehow this
knowledge does not filter through to students. One source of this problem is that
textbooks do not discuss the issues in any detail. Maddala's book is an exception. He
explicitly faces these problems and thereby he gives the students a feeling for the
limitations of statistics (see pp. 30-32. See also p. 495).
A less known problem, is that of testing for multicollinearity (that the explanatory
variables are highly related so it is difficult to distinguish between their effects).
Sometimes high standard errors of the regression coefficients is taken to indicate
multicollinearity. However, as Maddala shows, it is possible to have high standard errors
without multicollinearity, and low standard errors even when there really is
multicollinearity (p. 271). To understand this consider the formulas for the variance of
the regressors in a the case of two explanatory variables. We have:
Var (b1) = var (e) / S11 (1-r212)
Var (b2) = var (e) / S22 (1-r212)
Cov (b1,b2) = Var (e) r212 / S12
(1-r212)
As we can see there are three factors which affect the standard error of a regressor:
1. Var (e)
2. S11 (for b1)
3. r212
So, even if we have multicollinearity (a high r212), the standard
error may be low if Var (e) is low and S11 is high). Similarly, we may have
little multicollinearity (a low r212), but a high standard error due
to a high Var (e) and a low S11. Thus, using the standard error as a diagnostic
for multicollinearity can be deceptive. Moreover, it is also deceptive to focus on the
size of r212 as a measure of how big the problem of
multicollinearity is (since it ignores the size of S11 and Var(e)). The case is
even worse when you have more than three variables, as Maddala shows in a highly
instructive example (p. 272). With three variables we may have low r212,
r213, and r223, and still have great
multicollinearity problems (For example, when x3 = x1 + x2).
In sum, the standard error is not a good indicator of the existence of multicollinearity
or its seriousness.
A third example of Maddala's critical attitude, is revealed by his discussions of
various statistical terms. I have already mentioned his discussion of the term
"significant." However, Maddala also objects to the term "the null
hypothesis" (p. 81); He wants to replace the term "rational expectations"
with "model consistent expectations" (p. 433); And he thinks the concept of
"Granger causality" should be replaced with "precedence" since it does
not really imply causality as we use the term in most other contexts (p. 397). Whether one
agrees with this or not, the discussion itself reveals a critical mind at work.
One could go on to give many more examples of how Maddala develops a critical attitude
in the reader. For example, he discusses the difficulties of interpreting dummy variables
(p 310); the problems of using polynomial lags (p. 426); and the traditional solutions to
autocorrelation (p. 244). However, I hope I have showed enough examples to convince the
reader that Maddala does not simply uncritically present econometric procedures.
1.2 New topics
Being critical is good, but often easy. It is much more difficult to integrate a long list
of fragmented critical remarks into a systematic critique. Finally, it is even more
difficult to construct an alternative methodology which is better than the procedures
being criticised. In a previous review I criticised Wonnacott and Wonnacott for presenting
only fragmentary criticism. Maddala cannot be accused of the same. Much of his criticism
is build on a particular view of econometrics. Moreover, he provides an alternative
methodology. Both issues will now be discussed in more detail.
Maddala's view of econometrics is presented in a figure (p. 7). The key to this view is
its emphasis on feedback - how theory affects data and how data affects theory formation.
Traditional econometrics was mainly concerned with estimation of a given model, not the
feedback between theory and data. Autocorrelation, for example, was viewed as a problem
for estimation (since the estimators were no longer best, linear, unbiased estimators -
BLUE). Viewed in this light, any mechanical procedure that could reduce autocorrelation
would "solve" the problem. However, this does not explicitly consider the
feedback process between econometrics and economic theory. Autocorrelation is a sign that
something that something is missing from our theory. Thus, we should revise our theory,
not simply find a "mechanical" fix for the problem. Thinking about this kind of
feedback led econometricians to focus on the problem of model selection (how to
choose between rival models i.e. not just how to best estimate a given model) and specification/diagnostic
checking (checking the adequacy of the model - not just estimating it). It is this
focus which unifies some of the criticism of traditional econometrics.
How should we choose between rival models? Should we simply look at the R2
(the coefficient of determination - indicating the percentage of variation explained by
the variables)? The problem with this is that if you use the data to find the model (you
simply determine which variables to include by choosing those that which gives you the
highest R2), it would be circular to use R2 as a selection
criterion. The data cannot be used to test the adequacy of a model when the data has been
used to derive this model! Instead, Hendry suggests the following criteria for a good
model: 1. Data admissible (for example, it should not predict negative prices); 2. Theory
consistent (no "mechanical" fixes without justification based on economic
theory); 3. Weakly exogenous regressors (more on this later); 4. Ecompassing (A model is
better than another if it explains everything the first model explains and more -
especially why the first was sometimes wrong); 5. Stability (the parameters should be
stable); 6. Data coherent (errors should be random).
This may seem abstract, but - as I will now try to show - this approach has led to many
new and interesting developments in econometrics.
Take, for example, the second criterion - theory consistency. As traditional
econometrics became increasingly aware of the problem of spurious correlation (when two
variables are correlated but not causally related), they began to difference the data.
That is, instead of regressing y on x, they regressed yt - yt-1 on xt
- xt-1. There is little wrong with that, but economic theory usually has little
to say about how the difference of variables are related. Typically, economic theory tells
us something about the long-run relationship - the equilibrium to which the economy is
supposed to gravitate toward. For example, monetarist think that inflation - at least in
the long run - is always caused by the money supply. In the short run, however, the
relationship is not so clear-cut. Hence, a regression of differences need not reveal a
significant relationship, while a regression in levels would. The same problem appears in
the consumption function. In the long run we may consume a constant proportion of our
income, but in the short run consumption often deviates from this desired ratio. A
regression in differences may solve some technical estimation problems, but these models
are not inspired by theory, they do not always have obvious theoretical interpretations,
and we loose information about levels.
A constructive alternative to the traditional approach, is the new emphasis on Error
Correction Models. This is a regression in differences, but it also includes a long term
equilibrium. Consider first the following equation:
yt = a0 + a1 yt-1 + b0 xt
+ b1 xt-1 + et
(Note how this model resembles the unrestricted model of traditional solutions to
autocorrelation. See my review of Wonnacott and Wonnacott for more on this.)
Subtract yt-1 and b0 xt-1 from both the left and right
hand side of the model, and rearrange. We get:
yt - yt-1 = a0 + b0 (xt - xt-1)
+ (a - 1) yt-1 + (b0 + b1) xt-1 + et
Define s = (b0 + b1) / (1-a1). We then get:
yt - yt-1 = a0 + b0 (xt - xt-1)
+ (1 - a) (yt-1 - s xt-1) + et
This is an error correction model. It is, as we can see, based on differences, but -
and this is the key for the present discussion - it also has a steady-state, long run
equilibrium. This long run equilibrium is:
yt = s x + constant
And if the (testable) restriction a1 + b0+ b1 = 1
holds, the long run equilibrium is:
yt = x + constant
As an example of applied work using this model, is the consumption function estimated
by Davidson, Hendry, Srba and Yeo (Economic Journal, 1978), based on quarterly data
from 1958 to 1975 (d here symbolizes delta):
d4ct = 0.48 d4yt - 0.23 d1d4yt
+ 0.09 (c - y)t-4 + 0.006 DUM - 0.12 d4pt -0.31 d1d4
pt
The details need not bother us here (all the coefficients were significant, R2
= 0.85, DW = 2.0 and the standard error was 0.0062). The point I wanted to emphasise, was
that there is a long run solution in this model - ensured by the term (c - y)t-4.
It is time to sum up. I started with the question of how to choose between different
models. I then presented Hendry's list of six criteria - one of which was theory
consistency. I then showed how ECM models - unlike the traditional difference models -
included a long run equilibrium i.e. it more consistent with theory than the traditional
difference models. This shows how the new approach to econometrics has created good
alternative to the traditional models. Finally, this example illustrates how important the
issue is, and that it deserves space in a textbook like Maddala's Introduction to
Econometrics.
Another recent development, is the renewed interest in exogeneity - and Maddala is one
of the few authors who spend more than a few sentences on exogeneity. The standard
definition of an exogenous variable is one that is not correlated with the error term. As
Maddala explains, this approach is unsatisfactory because it is arbitrary (we simply
decide a priori which variables we want to call exogenous, and two different
researchers may decide to use different variables as exogenous); It excludes - in order to
achieve identification - some variables that should be included (the Liu critique);
Finally, the coefficients in the equation will not be independent of the exogenous
variables if people are rational forward looking individuals (The Lucas critique). The
question is then what we can do about this.
The basic answer is to distinguish between different types of exogeneity and use tests
to see which concept applies in a concrete situation. Leamer, for example, has suggested
that we should distinguish between exogeneity in the sense of predeterminedness (i.e. when
the variable is independent of the contemporaneous and future errors in the equation) and
strict exogeneity (predeterminedness plus independence of past errors too). Engle, Hendry
and Richard want to divide the concept of exogeneity into three: weak, strong and
superexogeneity. The distinction follows from the view that exogeneity is only relevant if
we first ask "Exogenous for what?" (which variables). The concepts are somewhat
technical (but relatively easy. For more see Maddala pp. 392-393). The important point to
note, however, is that only weak exogeneity is required for efficient estimation, while
superexogeneity is required if we want to use our model to conduct policy predictions. We
may then use various test to determine what kind of exogeneity we have, such as the
Hausmann test.
The last point I want to discuss, is how the new approach to econometrics has widened
our toolbox. When we start focusing on choosing the right model, we need to develop a
different set of tests than those needed for estimating a model. Maddala's textbook
reflects this. He explains a (too?) large number of new tests used to examine the quality
of various models. Particularly important, of course, is it to examine the residuals of
the regression since the residuals may reveal problems such as autocorrelation,
heteroskedasticity, instability and many other problems indicating a need to rethink the
model (remember Hendry's data coherence criteria for choosing between rival models).
Maddala discusses this, and he makes the useful distinction between the residuals from the
traditional regression, the predicted residuals (residuals from observations not used to
estimate the model), studentizised residuals (predicted residual divided by its standard
error), BLUS residuals (constructed to have zero mean, be uncorrelated and constant
variance) and recursive residuals (see p. 481 for more on all of these). Once again we see
how the new approach to econometrics really do have concrete and alternative proposals.
I hope I have convinced the reader that the new approach to econometrics has produced
many new and interesting developments, and that Maddala's discussion of this at a
relatively accessible level is a great quality of the book. There are many more topics
which might be mentioned. For example, is it best to start with a very general model and
simply it using data-based criteria, or should we start with a simple model and gradually
increase its complexity? (p. 493ff) What is cointegration, unit-roots, integrated and
stationary time series? (p. 577ff) What is the difference between
trend-stationary-processes and difference-stationary processes, and what is the importance
of distinguishing between the two? (p. 259) Maddala presents all these topics without
being too technical, thus making it accessible (but not without effort) to undergraduates.
2 Weak points
One basic problem with this book is that the author cannot make up his mind whether he is
writing a textbook for first time students, or an introduction to recent developments for
students who already know some econometrics. The consequence of this is a long book trying
to please both camps. For example, to satisfy the beginner students, he has included a
chapter on introductory statistics. This chapter is redundant for those who know some
econometrics, while it is too brief for the beginner student. Another example is the many
appendices using matrix algebra. These are too complicated for the beginner, while the
advanced reader is bored with two explanations of the same material - one without and one
with matrix algebra. Lastly, the advanced reader will not be pleased with the often
repeated phrase "this topic/proof is beyond the scope of this book" (see, for
example, p. 75, 162, 230, 367, 368, 403, 476, 483, 526, 584). In this way the book
sometimes falls between two chairs - frightening the beginner and boring the more advanced
student.
One might argue that the instructor/the reader is free to skip the chapters he thinks
are too easy or advanced. This is true, and I suspect this is what will happen. Students
will be assigned one standard textbook (not Maddala), and some chapters from Maddala will
be used in addition to this (such as chapter 12 on Diagnostic Checking, Model
Selection, and Specification Testing and chapter 14 on Vector Autoregression, Unit
Roots, and Cointegration). This is perfectly acceptable, but somewhat unfortunate. It
is unfortunate because Maddala could have written an excellent introduction to
econometrics for beginners. By cutting the appendices and reducing the number of topics,
this could have been a perfect critical introduction to econometrics. On the other
hand, by eliminating the elementary statistics and enlarging the discussion of new topics,
this could have been a very good book for students who have already done introductory
econometrics but who wants an update on recent developments. As it is we have two books in
one, and the sum of this is not as excellent as the separate books would have been if they
were isolated.
Pedagogically the book is average. The language is clear and concise, but at least
three improvements could be made. First, to make much more extensive use of figures. For
example, Wonnacott and Wonnacott make great use of figures when explaining the problem of
simultaneous equations and autocorrelation; Maddala does not - though, he occasionally
uses figures very effectively (see for example p. 90). Second, the discussion of theory
could have been much better integrated with empirical examples and well-known real-life
situations. For example, Maddala has a good discussion of type I and type II errors (pp.
30-31), but he nowhere presents the intuitive simple explanation that a type I error
occurs when we convict an innocent person, while a type II error is equivalent to failing
to convict a person who is guilty. Similarly, Maddala's presentation of the consequences
of omitting relevant variables is theoretical (p. 161ff) - unlike Wonnacott and Wonnacott
who presents a wonderfully intuitive example (involving yield, rainfall and temperature.
See WW p. 96). Of course, Maddala also presents examples, but the point is that these are
isolated from the theory. His strategy is to present a theory and then to give several
examples. A better approach, I think is to first present an example showing the problem
and then present the general theory. Third, and last, the layout of the book could be
improved. The book is very uniform with the same fonts and backgrounds everywhere. A
better alternative would be to isolate some of the many examples in (optional) boxes with
dark background, to highlight important material in frames (as he does in the beginning,
but for some reason stops doing later on), and to present some of the data-material in a
smaller fonts.
On a more positive note, Maddala has included very good summaries after each chapter,
and he generates interest by presenting authors with rival views (conflict is always a
good way to fight boredom!). He also gives excellent references for further reading on
almost every single topic in the book.
Finally, I have some small quibbles. I did not understand why he discussed
superexogeneity before strong-exogeneity (pp. 392-3). Intuitively it should be the other
way around since superexogeneity is the "strongest" concept. I am also sceptical
of the utility of discussing the method of moments as a method for estimating the
regression coefficients. The last square method is certainly needed, but why spend time on
the method of moments? I also think it would be better to use the notation of deviation
(small x, not big) right away, at least before deriving the variances and covariance of
the regression coefficients (p. 77). Another small point is the discussion of slope
dummies which is made unnecessary complicated by the inclusion of intercept dummies (p.
313). It is better to keep the two separate before bringing them together. There is also,
I think, a small contradiction between the advice of using the reverse regression to
discover discrimination (p. 72) , and the later recommendation only to use the reverse
regression when this makes sense in terms of the direction of causation (p. 75) - salaries
do not cause qualifications, but the regression is still informative. Lastly, I was a bit
surprised at Maddala's somewhat non-standard formula for the t-statistic (with the square
root of n in the numerator and not in the denominator as is more common). However, these
are minor quibbles.
Altogether
Maddala has produced a good book. It is critical, comprehensive, and it deals with many
new developments in a way which many undergraduates may understand. Nevertheless, I would
not recommend it as a textbook for beginners with little background in
statistics/econometrics. It is simply a bit too frightening for beginners - sometimes too
brief, yet overall too long, and sometimes too advanced. Still, beginners could benefit
from individual chapters, and the book is good (but long) as a critical review after
a basic course in econometrics. As such this book is recommended.
[Note for bibliographic reference: Melberg, Hans O. (1997), Advanced Econometrics for
beginners - A review of Maddala (1992), http://www.oocities.org/hmelberg/papers/970519.htm]