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\textit{Wooldridge, Introductory Econometrics, 4th ed.}

\bigskip \textbf{Chapter 1:\ Nature of Econometrics and Economic Data}

What do we mean by {\it econometrics}? Econometrics is the field of economics in
which statistical methods are developed and applied to estimate economic
relationships, test economic theories, and evaluate plans and policies
implemented by private industry, government, and supranational
organizations. Econometrics encompasses forecasting--not only the
high-profile forecasts of macroeconomic and financial variables, but also
forecasts of demand for a product, likely effects of a tax package, or the
interaction between the demand for health services and welfare reform.

Why is econometrics separate from mathematical statistics? Because most
applications of statistics in economics and finance are related to the use
of non-experimental data, or observational data. The fundamental techniques
of statistics have been developed for use on experimental data: that
gathered from controlled experiments, where the design of the experiment and
the reliability of measurements from its outcomes are primary foci. In
relying on observational data, economists are more like astronomers, able to
collect and analyse increasingly complete measures on the world (or
universe) around them, but unable to influence the outcomes.

This distinction is not absolute; some economic policies are in the nature
of experiments, and economists have been heavily involved in both their
design and implementation. A good example within the last five years is the
implementation of welfare reform, limiting individuals' tenure on the
welfare rolls to five years of lifetime experience. Many doubted that this
would be successful in addressing the needs of those welfare recipients who
have low job skills; but the reforms have been surprisingly successful, as a
recent article in \textit{The Economist} states, at raising the employment
rate among this cohort. Economists are also able to carefully examine the
economic consequences of massive regime changes in an economy, such as the
transition from a planned economy to a capitalist system in the former
Soviet bloc. But fundamentally applied econometricians observe the data, and
use sophisticated techniques to evaluate their meaning.

We speak of this work as empirical analysis, or empirical research. The
first step is the careful formulation of the question of interest. This will
often involve the application or development of an economic model, which may
be as simple as noting that normal goods have negative price elasticities,
or exceedingly complex, involving a full-fledged description of many aspects
of a set of interrelated markets and the supply/demand relationships for the
products traded (as would, for instance, an econometric analysis of an
antitrust issue, such as \textit{U.S. v Microsoft). }Economists are often
attacked for their imperialistic tendencies--applying economic logic to
consider such diverse topics as criminal behavior, fertility, or
environmental issues--but where there is an economic dimension, the
application of economic logic and empirical research based on econometric
practice may yield valuable insights. Gary Becker, who has made a career of
applying economics to non-economic realms, won a Nobel Prize for his
efforts.Crime, after all, is yet another career choice, and for high school
dropouts who don't see much future in flipping burgers at minimum wage, it
is hardly surprising that there are ample applicants for positions in a drug
dealer's distribution network. In risk-adjusted terms (gauging the risk of
getting shot, or arrested and successfully prosecuted...) the risk-adjusted
hourly wage is many times the minimum wage. Should we be surprised by the
outcome?

Irregardless of whether empirical research is based on a formal economic
model or economic intuition, the hypotheses about economic behavior must be
transformed into an econometric model that can be applied to the data. In an
economic model, we can speak of functions such as $Q=Q\left( P,Y\right) ;$
but if we are to estimate the parameters of that relationship, we must have
an explicit functional form for the $Q$ function, and determine that it is
an appropriate form for the model we have in mind. For instance, if we were
trying to predict the efficiency of an automobile in terms of its engine
size (displacement, in in$^{3}$ or liters), Americans would likely rely on a
measure like $mpg$ -- miles per gallon. But the engineering relationship is
not linear between $mpg$ and displacement; it is much closer to being a
linear function if we relate gallons per mile $\left( gpm=1/mpg\right) $ to
engine size. The relationship will be curvilinear in $mpg$ terms, requiring
a more complex model, but nearly linear in $gpm$ vs displacement. An
econometric model will spell out the role of each of its variables: for
instance, 
\[
gpm_{i}=\beta _{0}+\beta _{1}displ_{i}+\epsilon _{i}
\]
would express the relationship between the fuel consumption of the $i^{th}$
automobile to its engine size, or displacement, as a linear function, with
an additive error term $\epsilon _{i}$ which encompasses all factors not
included in the model. The $parameters$ of the model are the $\beta $ terms,
which must be estimated via statistical methods. Once that estimation has
been done, we may test specific $hypotheses$ on their values: for instance,
that $\beta _{1}$ is positive (larger engines use more fuel), or that $\beta
_{1}$ takes on a certain value. Estimating this relationship for Stata's $%
auto.dta$ dataset of 74 automobiles, the predicted relationship is 
\[
\widehat{gpm}_{i}=0.029+0.011displ_{i}
\]
where displacement is measured in hundreds of in$^{3}.$ This estimated
relationship has an ``$R^{2}"$ value of 0.59, indicating that 59\% of the
variation of $gpm$ around its mean is ``explained'' by displacement, and a
root-mean-squared error of 0.008 (which can be compared to $gpm$'s mean of
0.050, corresponding to about 21 $mpg$).

\textbf{The structure of economic data}

We must acquaint ourselves with some terminology to describe the several
forms in which economic and financial data may appear. A great deal of the
work we will do in this course will relate to \textbf{cross-sectional} data:
a sample of units (individuals, families, firms, industries, countries...)
taken at a given point in time, or in a particular time frame. The sample is
often considered to be a random sample of some sort when applied to
microdata such as that gathered from individuals or households. For
instance, the official estimates of the U.S. unemployment rate are gathered
from a monthly survey of individuals, in which each is asked about their
employment status. It is not a count, or census, of those out of work. Of
course, some cross sections are not samples, but may represent the
population: e.g. data from the 50 states do not represent a random sample of
states. A cross-sectional dataset can be conceptualized as a spreadsheet,
with variables in the columns and observations in the rows. Each row is
uniquely identified by an observation number, but in a cross-sectional
datasets the ordering of the observations is immaterial. Different variables
may correspond to different time periods; we might have a dataset containing
municipalities, their employment rates, and their population in the 1990 and
2000 censuses.

The other major form of data considered in econometrics is the\textbf{\ time
series}: a series of evenly spaced measurements on a variable. A time-series
dataset may contain a number of measures, each measured at the same
frequency, including measures derived from the originals such as lagged
values, differences, and the like. Time series are innately more difficult
to handle in an econometric context because their observations almost surely
are interdependent across time. Most economic and financial time series
exhibit some degree of persistence. Although we may be able to derive some
measures which should not, in theory, be explainable from earlier
observations (such as tomorrow's stock return in an efficient market), most
economic time series are both interrelated and autocorrelated--that is,
related to themselves across time periods. In a spreadsheet context, the
variables would be placed in the columns, and the rows labelled with dates
or times. The order of the observations in a time-series dataset matters,
since it denotes the passage of equal increments of time. We will discuss
time-series data and some of the special techniques that have been developed
for its analysis in the latter part of the course.

Two combinations of these data schemes are also widely used: \textbf{pooled
cross-section/time series} (CS/TS)\ datasets and \textbf{panel}, or \textbf{%
longitudinal}, data sets. The former (CS/TS) arise in the context of a
repeated survey--such as a presidential popularity poll--where the
respondents are randomly chosen. It is advantageous to analyse multiple
cross-sections, but not possible to link observations across the
cross-sections. Much more useful are panel data sets, in which we have
timeseries of observations on the same unit: for instance, $C_{i,t}$ might
be the consumption level of the $i^{th}$ household at time $t.$ Many of the
datasets we commonly utilize in economic and financial research are of this
nature: for instance, a great deal of research in corporate finance is
carried out with Standard and Poor's COMPUSTAT, a panel data set containing
20 years of annual financial statements for thousands of major U.S.
corporations. There is a wide array of specialized econometric techniques
that have been developed to analyse panel data; we will not touch upon them
in this course.

\textbf{Causality and ceteris paribus}

The hypotheses tested in applied econometric analysis are often posed to
make inferences about the possible causal effects of one or more factors on
a response variable: that is, do changes in consumers' incomes ``cause''
changes in their consumption of beer? At some level, of course, we can never
establish causation--unlike the physical sciences, where the interrelations
of molecules may follow well-established physical laws, our observed
phenomena represent innately unpredictable human behavior. In economic
theory, we generally hold that individuals exhibit rational behavior; but
since the econometrician does not observe all of the factors that might
influence behavior, we cannot always make sensible inferences about
potentially causal factors. Whenever we ``operationalize'' an econometric
model, we implicitly acknowledge that it can only capture a few key details
of the behavioral relationship, and is leaving many additional factors
(which may or may not be observable) in the ``pound of \textbf{ceteris
paribus}.'' ceteris paribus--literally, other things equal--always underlies
our inferences from empirical research. Our best hope is that we might
control for many of the factors, and be able to use our empirical findings
to ascertain whether systematic factors have been omitted. Any econometric
model should be subjected to diagnostic testing to determine whether it
contains obvious flaws. For instance, the relationship between $mpg$ and $%
displ$ in the automobile data is strictly dominated by a model containing
both $displ$ and $displ^{2}$, given the curvilinear relation between $mpg$
and $displ.$ Thus the original linear model can be viewed as unacceptable in
comparison to the polynomial model; this conclusion could be drawn from
analysis of the model's residuals, coupled with an understanding of the
engineering relationship that posits a nonlinear function between $mpg$ and $%
displ.$

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