\documentclass{slides}

\usepackage{amssymb}
\usepackage{graphicx}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}
\newtheorem{proposition}{Proposition}
\begin{document}


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

\bigskip \textbf{Chapter 9: More on specification and data problems}

\textbf{\bigskip \medskip }

\textbf{Functional form misspecification}

We may have a model that is correctly specified, in terms of including the
appropriate explanatory variables, yet commit functional form
misspecification--in which the model does not properly account for the
relationship between dependent and observed explanatory variables. We have
considered this sort of problem when discussing polynomial models; omitting
a squared term, for instance, and constraining $\partial y/\partial x$ to be
constant (rather than linear in $x)$ would be a functional form
misspecification. We may also encounter difficulties of this sort with
respect to interactions among the regressors. If omitted, the effects of
those regressors will be estimated as constant, rather than varying as they
would in the case of interacted variables. In the context of models with
more than one categorical variable, assuming that their effects can be
treated as independent (thus omitting interaction terms) would yield the
same difficulty.

We may, of course, use the tools already developed to deal with these
problems, in the sense that if we first estimate a general model that allows
for powers, interaction terms, etc. and then ``test down'' with joint F
tests, we can be confident that the more specific model we develop will not
have imposed inappropriate restrictions along the way. But how can we
consider the possibility that there are missing elements even in the context
of our general model?

One quite useful approach to a general test for functional form
misspecification is Ramsey's \textbf{RESET} (regression specification error
test). The idea behind RESET\ is quite simple; if we have properly specified
the model, no nonlinear functions of the independent variables should be
significant when added to our estimated equation. Since the fitted, or
predicted values ($\hat{y})$ of the estimated model are linear in the
independent variables, we may consider powers of the predicted values as
additional regressors. Clearly the $\hat{y}$ values themselves cannot be
added to the regression, since they are by construction linear combinations
of the $x$ variables. But their squares, cubes,... are not. The RESET\
formulation reestimates the original equation, augmented by powers of $\hat{y%
}$ (usually squares, cubes, and fourth powers are sufficient) and conducts
an F-test for the joint null hypothesis that those variables have no
significant explanatory power. This test is easy to implement, but many
computer programs have it already programmed; for instance, in Stata one may
just specify \texttt{estat ovtest} (omitted variable test) after any regression,
and the Ramsey RESET\ will be produced. However, as Wooldridge cautions,
RESET\ should not be considered a general test for omission of relevant
variables; it is a test for misspecification of the relationship between $y$
and the $x$ values in the model, and nothing more.

\textbf{Tests against nonnested alternatives}

The standard joint testing framework is not helpful in the context of
``competing models,'' or nonnested alternatives. These alternatives can also
arise in the context of functional form: for instance,

\begin{eqnarray}
y &=&\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+u  \label{nonnested} \\
y &=&\beta _{0}+\beta _{1}\log x_{1}+\beta _{2}\log x_{2}+u  \nonumber
\end{eqnarray}
are nonnested models. The mechanical alternative, in which we construct an
artificial model that contains each model as a special case, is often not
very attractive (and sometime will not even be feasible). An alternative
approach is that of Davidson and MacKinnon. Using the same logic applied in
developing Ramsey's RESET, we can estimate each of the models in (\ref
{nonnested}), generate their predicted values, and include them in the other
equation. Under the null hypothesis that the first form of the model is
correctly specified, a linear combination of the logs of the $x$ variables
should have no power to improve it, and that coefficient should be
insignificant. Likewise, one can reestimate the second model, including the
predicted values from the first model. This testing strategy--often termed
the Davidson-MacKinnon ``\textbf{J test''}--may indicate that one of the
models is robust against the other.

There are no guarantees, though, in that applying the J test to these two
equations may generate zero, one, or two rejections. If neither hypothesis
is rejected, then the data are not helpful in ranking the models. If both
are rejected, we are given an indication that neither model is adequate, and
that a continued specification search should be conducted. If one rejection
is received, then the J\ test is definitive in indicating that one of the
models dominates (or subsumes) the other, and not vice versa. However, this
does not imply that the preferred model is well specified; again, this test
is against a very specific alternative, and does not deliver a ``clean bill
of health'' for the preferred model should one emerge.

\pagebreak
\textbf{Proxy variables}

So far, we have discussed issues of misspecification resulting from improper
handling of the $x$ variables. In many economic models, we are forced to
employ ``proxy variables'': approximate measures of an unobservable
phenomenon. For instance, admissions officers use SAT\ scores and high
school GPAs as proxies for applicants' ability and intelligence. No one
argues that standardized tests or grade point averages are actually
measuring aptitude, or intelligence; but there are reasons to believe that
the observable variable is well correlated with the unobservable, or \textbf{%
\ latent}, variable. To what extent will a model estimated using such
proxies for the variables in the underlying relationship be successful, in
terms of delivering consistent estimates of its parameters? First, of
course, it must be established that there is a correlation between the
observable variable and the latent variable. If we consider the latent
variable as having a linear relation to a measurable proxy variable, the
error in that relation must not be correlated with other regressors. When we
estimate the relationship including the proxy variable, it should be
apparent that the measurement error from the latent variable equation ends
up in the error term, as an additional source of uncertainty. This is an
incentive to avoid proxy variables where one can, since they will inexorably
inflate the error variance in the estimated regression. But usually they are
employed out of necessity, in models for which we have no ability to measure
the latent variable. If there are several potential proxy measures, they
might each be tested, to attempt to ascertain whether bias is being
introduced to the relationship.

In some cross-sectional relationships, we have the opportunity to use a
lagged value of the dependent variable as a proxy variable. For instance, if
we are trying to explain cities' crime rates, we might consider that there
are likely to be similarities---irregardless of the effectiveness of
anti-crime strategies---between current crime rates and last year's values.
Thus, a prior value of the dependent variable, understandably independent of
this year's value, may be a useful proxy for a number of factors that cannot
otherwise be quantified. This approach might often be used to deal with
factors such as ``business climate,'' in which some states or municipalities
are viewed as more welcoming to business; there may be many aspects to this
perception, some of them more readily quantifiable (such as tax rates), some
of them not so (such as local officials' willingness to negotiate
infrastructure improvements, or assist in funding for a new facility). But
in the absence of radical changes in localities' stance in this regard, the
prior year's (or decade's) business investment in the locality may be a good
proxy for those factors, perceived much more clearly by the business
decisionmakers than by the econometrician.

\pagebreak

\textbf{Measurement error}

We often must deal with the issue of measurement error: that the variable
that theory tells us belongs in the relationship cannot be precisely
measured in the available data. For instance, the exact marginal tax rate
that an individual faces will depend on many factors, only some of which we
might be able to observe: even if we knew the individual's income, number of
dependents, and homeowner status, we could only approximate the effect of a
change in tax law on his or her tax liability. We are faced, therefore, with
using an approximate measure, including some error of measurement, whenever
we might attempt to formulate and implement such a model. This is
conceptually similar to the proxy variable problem we have already
discussed, but in this case it is not a latent variable problem. There is an
observable magnitude, but we do not necessarily observe it. For instance,
reported income is an imperfect measure of actual income, while IQ\ score is
only a proxy for ability. Why is measurement error of concern? Because the
behavior we're trying to model--be it of individuals, firms, or
nations--presumably is driven by the actual measures, not our mismeasured
approximations of those factors. To the extent that we fail to capture the
actual measure, we may misinterpret the behavioral response.

If measurement error is observed in the dependent variable--for instance, if
the true relationship explains $y^{*},$ but we only observe $y=$ $%
y^{*}+\epsilon ,$ where $\epsilon $ is a meanzero error process, then $%
\epsilon $ becomes a component of the regression error term: yet another
reason why the relationship does not fit perfectly. We assume that $\epsilon 
$ is not systematic, in particular, that it is not correlated with the
independent variables $X.$ As long as that is the case, then this form of
measurement error does no real harm; it merely weakens the model, without
introducing bias in either point or interval estimates. If the magnitude of
the measurement error in $y$ is correlated with one or more of the $x$
variables, then we will have a problem of bias.

Measurement error in an explanatory variable, on the other hand, is a far
more serious problem. Say that the true model is 
\begin{equation}
y=\beta _{0}+\beta _{1}x_{1}^{*}+u  \label{measerr}
\end{equation}
but that $x_{1}^{*}$ is not observed; instead, we observe $%
x_{1}^{{}}=x_{1}^{*}+\epsilon _{1}.$ We can assume that $E(\epsilon _{1})=0$
with generality. But what should be assumed about the relationship between $%
\epsilon _{1}$ and $x_{1}^{*}?$ First, let us assume that $\epsilon _{1}$ is
uncorrelated with the observed measure $x_{1}^{{}}$ (that is, larger values
of $x_{1}^{{}}$ do not give rise to systematically larger (or smaller)
errors of measurement). This can be written as $Cov($ $\epsilon _{1},$ $%
x_{1}^{{}})=0.$ But if this is the case, it must be true that $Cov($ $%
\epsilon _{1},$ $x_{1}^{*})\neq 0:$ that is, the error of measurement must
be correlated with the actual explanatory variable $x_{1}^{*},$ so that we
can write the estimated equation (in which $x_{1}^{*}$ is replaced with the
observable $x_{1}^{{}})$ as 
\begin{equation}
y=\beta _{0}+\beta _{1}x_{1}^{{}}+\left( u-\beta _{1}\epsilon _{1}\right) 
\label{measerr2}
\end{equation}

Since both $u$ and $\epsilon _{1}$ have zero mean and are uncorrelated (by
assumption) with $x_{1},$ the presence of measurement error merely inflates
the error term: that is, $Var\left( u-\beta _{1}\epsilon _{1}\right) =\sigma
_{u}^{2}+\beta _{1}^{2}\sigma _{\epsilon _{1}}^{2},$ given that we have
assumed that $u$ and $\epsilon _{1}$ are uncorrelated with each other. Thus,
measurement error in $x_{1}^{*}$ does not negatively affect the regression
of $y$ on $x_{1}^{{}};$ it merely inflates the error variance, like
measurement error in the dependent variable.

However, this is not the case that we usually consider under the heading of 
\textbf{errors-in-variables}. It is perhaps more reasonable to assume that
the measurement error is uncorrelated with the true explanatory variable: $
Cov($ $\epsilon _{1},$ $x_{1}^{*})=0.$ If this is so, then $Cov($ $\epsilon
_{1},$ $x_{1}^{{}})=Cov(\epsilon _{1},\left( x_{1}^{*}+\epsilon _{1}\right)
)\neq 0$ by construction, and the regression (\ref{measerr2})\ will have a
correlation between its explanatory variable $x_{1}^{{}}$ and the composite
error term. The covariance of $\left( x_{1},u-\beta _{1}\epsilon _{1}\right)
=-\beta _{1}Cov(\epsilon _{1},$ $x_{1}^{{}})=-\beta _{1}\sigma _{\epsilon
_{1}}^{2}\neq 0,$ causing the OLS\ regression of $y$ on $x_{1}$ to be biased
and inconsistent. In this simple case of a single explanatory variable
measured with error, we can determine the nature of the bias:

\begin{eqnarray}
\texttt{plim} (b_{1}) &=&\beta _{1}+\frac{Cov\left( x_{1},u-\beta
_{1}\epsilon _{1}\right) }{Var(x_{1})}  \label{bias} \\
&=&\beta _{1}\left( \frac{\sigma _{x_{1}}^{2}}{\sigma _{x_{1}}^{2}+\sigma
_{\epsilon _{1}}^{2}}\right)  \nonumber
\end{eqnarray}
demonstrating that the OLS\ point estimate will be \textbf{attenuated}%
--biased toward zero--since the bracketed expression must be a fraction.
Clearly, in the absence of measurement error, $\sigma _{\epsilon
_{1}}^{2}\rightarrow 0,$ and the OLS\ coefficient becomes unbiased and
consistent. As $\sigma _{\epsilon _{1}}^{2}$ increases relative to the
variance in the (correctly measured) explanatory variable, the OLS\
coefficient becomes more and more unreliable, shrinking toward zero.

What can we conclude in a multiple regression equation, in which perhaps one
of the explanatory variables is subject to measurement error? If the
measurement error is uncorrelated to the true (correctly measured)
explanatory variable, then the result we have here applies: the OLS\
coefficients will be biased and inconsistent for all of the explanatory
variables (not merely the variable measured with error), but we can no
longer predict the direction of bias in general terms. Realistically, more
than one explanatory variable may be subject to measurement error (e.g. both
reported income and wealth may be erroneous).

We might be discouraged by these findings, but fortunately there are
solutions to these problems. The models in question, in which we suspect the
presence of serious errors of measurement, may be estimated by techniques
other than OLS\ regression. We will discuss those \textbf{instrumental
variable} techniques, which may also be used to deal with problems of
simultaneity, or two-way causality, in Chapter 15.

\end{document}
