What is Econometrics
Introduction
1.1 What is Econometrics?
The term “econometrics” is believed to have
been crafted by Ragnar Frisch (1895-1973)
of
Norway, one of the three principle founders
of the Econometric Society, …rst editor of the journal Econometrica, and
co-winner of the
…rst
Nobel Memorial
Prize
in Economic
Sciences in 1969. It is therefore
…tting that we turn to Frisch’s own words in the introduction
to the …rst issue of Econometrica for an explanation of the discipline.
A word of explanation regarding
the term econometrics may be in order. Its
de…ni- tion is implied in the statement of the scope of the [Econometric]
Society, in Section
I of the Constitution, which reads:
“The
Econometric Society is an international society
for the advancement of economic theory in its relation to statistics and mathematics.... Its main object shall be to promote studies that aim at a uni…cation of the theoretical- quantitative and the empirical-quantitative approach to economic
problems....”
But there are several aspects of the quantitative approach to economics, and no single one
of these aspects, taken by itself, should be confounded with econometrics. Thus, econometrics is by no means
the
same as economic statistics.
Nor is it identical
with what
we call general economic
theory, although
a considerable portion of this theory
has a de…ninitely quantitative character. Nor should econometrics be taken as synonomous with the application of mathematics to economics. Experience
has
shown that each
of these three
view-points,
that of statistics, economic theory, and mathematics, is
a necessary, but not by itself a su¢cient, condition for a real understanding of the quantitative relations in modern economic life.
It is the uni…cation of all three that is powerful. And it is this uni…cation that constitutes econometrics.
Ragnar
Frisch,
Econometrica, (1933), 1, pp.
1-2.
This de…nition remains valid today, although
some terms have evolved
somewhat in their usage. Today,
we would say
that econometrics
is the uni…ed study
of economic models,
mathematical statistics, and economic
data.
Within the …eld of econometrics there are sub-divisions and specializations. Econometric theory concerns
the
development
of tools and methods,
and
the
study
of the properties
of econometric
methods. Applied econometrics is a term
describing
the
development
of quantitative
economic
models and the application of
econometric methods
to these models using economic
data.
1.2 The Probability Approach to Econometrics
The unifying
methodology of modern econometrics was articulated by Trygve Haavelmo (1911-
1999) of Norway,
winner
of the 1989
Nobel Memorial Prize in Economic
Sciences, in his seminal
1
paper “The probability approach in econometrics”, Econometrica (1944). Haavelmo argued that quantitative economic models must
necessarily
be probability
models (by
which today we would mean stochastic). Deterministic models are blatently inconsistent
with
observed economic quan- tities, and
it
is incoherent
to
apply
deterministic models to non-deterministic data. Economic
models should be explicitly designed to incorporate randomness; stochastic errors
should
not
be simply added to deterministic models to make them
random. Once
we acknowledge
that
an eco- nomic model is a probability model, it follows naturally
that an appropriate tool way to quantify,
estimate, and conduct inferences about the economy
is through the powerful theory of mathe- matical statistics. The
appropriate method for a quantitative economic analysis follows from the probabilistic construction of the economic model.
Haavelmo’s probability approach
was quickly embraced by the economics profession. Today no quantitative work in economics
shuns its fundamental vision.
While all economists embrace the probability
approach, there has been some evolution in its implementation.
The structural approach is the closest to Haavelmo’s original
idea. A probabilistic economic model is speci…ed, and the quantitative analysis performed under the assumption that
the economic model
is correctly speci…ed. Researchers often describe
this as “taking their model seriously.” The structural approach
typically leads to likelihood-based analysis,
including maximum likelihood and Bayesian estimation.
A criticism of the structural
approach is that it is misleading to treat an economic model as correctly
speci…ed. Rather, it is more accurate
to view a model as a useful abstraction or approximation. In
this case, how should we interpret
structural econometric analysis? The quasi- structural approach
to inference views a structural economic model as an approximation
rather than the truth. This theory has led to the concepts of the pseudo-true value (the
parameter value de…ned by the estimation
problem), the quasi-likelihood function, quasi-MLE,
and quasi-likelihood inference.
Closely related is the semiparametric approach. A probabilistic
economic model is partially
speci…ed but some features are left unspeci…ed.
This approach typically
leads to estimation methods such as least-squares and the Generalized Method of Moments.
The semiparametric approach dominates contemporary econometrics,
and is the main focus of this textbook.
Another branch of quantitative structural economics is the calibration approach. Similar to the quasi-structural approach, the
calibration approach interprets structural models as approx-
imations and hence inherently false. The di¤erence is that the calibrationist literature rejects mathematical statistics as inappropriate for approximate
models, and instead
selects parameters
by matching model and data moments
using non-statistical ad
hoc1 methods.
1.3 Econometric Terms and Notation
In a typical application, an econometrician has a set of repeated measurements
on a set of vari- ables. For example,
in a labor application
the variables could include
weekly earnings, educational attainment, age, and other descriptive characteristics. We call this information the data, dataset, or sample.
We use the term observations to
refer to the distinct repeated measurements on the variables. An individual observation often corresponds to a speci…c economic
unit, such as a person, household,
corporation, …rm, organization, country, state, city or other geographical region. An individual observation could also be a measurement at a point in time,
such as quarterly GDP
or a daily interest
rate.
Economists typically denote variables by the italicized roman characters y, x; and/or z: The convention in econometrics
is to use the
character y to denote
the variable
to be explained, while
1 Ad hoc means “for this purpose”
– a method designed for a speci…c problem – and
not based on a generalizable principle.
the characters x
and z are used to denote
the conditioning (explaining) variables.
Following mathematical convention, real numbers
(elements of the real line R) are written using lower case italics such as y, and vectors (elements
of Rk ) by lower case bold italics such as x; e.g.
0 x1 1
x = B x2 C
B C
xk
Upper case bold italics
such as X are used for matrices.
We typically denote the number of observations by the natural number
n; and subscript the variables by the index i to denote
the individual observation, e.g. yi; xi and zi. In some contexts we use indices other than i, such as in time-series
applications where the
index t is common, and in panel studies
we typically use the double index it to refer to individual i at a time period t.
The i’th observation
is the set (yi; xi; zi):
It is proper mathematical practice to use upper case
X for random
variables and
lower case x for realizations
or speci…c values.
This practice is not commonly followed in econometrics
because instead we use upper case
to denote matrices. Thus
the notation yi will in some places
refer to a random
variable, and in other places
a speci…c realization. Hopefully there
will be no confusion as the use should be evident from the context.
We typically use Greek letters
such as ; and 2 to denote unknown
parameters of an econo-
metric model,
and
will use boldface, e.g.
or , when
these
are
vector-valued.
Estimates are
typically denoted by putting a hat “^”, tilde “~” or bar
“-” over the corresponding
letter,
e.g.
^
and ~ are estimates of :
The
covariance
matrix
of an econometric
estimator will typically be written using the capital
boldface V ; often
with a subscript
to denote the estimator, e.g. V b
= var pn
b
as the
covariance matrix
for pn
b
: Hopefully without causing confusion, we will use the notation
V =
avar( b ) to denote the asymptotic covariance
matrix
of pn
b
(the variance of the
asymptotic distribution).
Estimates will be denoted by appending
hats
or tildes, e.g. estimate Vb is an of V .
Tags: ECONOMETRICS
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