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\textit{Wooldridge, Introductory Econometrics, 5th ed.}

\bigskip \textbf{Chapter 2: The simple regression model}

Most of this course will be concerned with use of a regression model: a
structure in which one or more explanatory variables are considered to
generate an outcome variable, or dependent variable. We begin by considering
the {\it simple regression} model, in which a single explanatory, or independent,
variable is involved. We often speak of this as `two-variable' regression,
or `Y on X regression'. Algebraically, 
\begin{equation}
y_{i}=\beta _{0}+\beta _{1}x_{i}+u_{i}  \label{2vreg}
\end{equation}
is the relationship presumed to hold in the population for each observation $%
i.$ The values of $y$ are expected to lie on a straight line, depending on
the corresponding values of $x.$ Their values will differ from those
predicted by that line by the amount of the error term, or disturbance, $u,$
which expresses the net effect of all factors other than $x$ on the outcome $%
y-$that is, it reflects the assumption of {\it ceteris paribus}, or `all else equal'. 

We often speak of  $x$ as the `regressor' in this relationship; less commonly we speak of $y$
as the `regressand.' The coefficients of the relationship, $\beta _{0}$ and $%
\beta _{1},$ are the regression parameters, to be estimated from a sample.
They are presumed constant in the population, so that the effect of a
one-unit change in $x$ on $y$ is assumed constant for all values of $x.$

As long as we include an intercept in the relationship, we can always assume
that $E\left( u\right) =0,$ since a nonzero mean for $u$ could be absorbed
by the intercept term.

The crucial assumption in this regression model involves the relationship
between $x$ and $u.$ We consider $x$ a random variable, as is $u,$ and
concern ourselves with the conditional distribution of $u$ given $x.$ If
that distribution is equivalent to the unconditional distribution of $u,$
then we can conclude that there is no relationship between $x$ and $u-$%
which, as we will see, makes the estimation problem much more
straightforward. To state this formally, we assume that 
\begin{equation}
E\left( u\mid x\right) =E\left( u\right) =0  \label{zcm}
\end{equation}
or that the $u$ process has a \textbf{zero conditional mean}. This
assumption states that the unobserved factors involved in the regression
function are not related in any systematic manner to the observed factors.

For instance, consider a regression of individuals' hourly wage on the
number of years of education they have completed. There are, of course, many
factors influencing the hourly wage earned beyond the number of years of
formal schooling. In working with this regression function, we are assuming
that the unobserved factors--excluded from the regression we estimate, and
thus relegated to the $u$ term--are not systematically related to years of
formal schooling. This may not be a tenable assumption; we might consider
``innate ability'' as such a factor, and it is probably related to success
in both the educational process and the workplace. Thus, innate
ability--which we cannot measure without some proxies--may be positively
correlated to the education variable, which would invalidate assumption (\ref
{zcm}).

The \textbf{population regression function}, given the zero conditional mean
assumption, is 
\begin{equation}
E\left( y\mid x\right) =\beta _{0}+\beta _{1}x_{i}  \label{prf}
\end{equation}
This allows us to separate $y$ into two parts: the systematic part, related
to $x,$ and the unsystematic part, which is related to $u.$ As long as
assumption (\ref{zcm}) holds, those two components are independent in the
statistical sense. Let us now derive the least squares estimates of the
regression parameters.

Let $\left[ \left( x_{i},y_{i}\right) :i=1,...,n\right] $ denote a random
sample of size $n$ from the population, where $y_{i}$ and $x_{i}$ are
presumed to obey the relation (\ref{2vreg}). The assumption (\ref{zcm})
allows us to state that $E(u)=0,$ and given that assumption, that $%
Cov(x,u)=E(xu)=0,$ where $Cov(\cdot )$ denotes the covariance between the
random variables. These assumptions can be written in terms of the
regression error: 
\begin{eqnarray}
E\left( y_{i}-\beta _{0}-\beta _{1}x_{i}\right) &=&0  \label{normeqnpop} \\
E\left[ x_{i}\left( y_{i}-\beta _{0}-\beta _{1}x_{i}\right) \right] &=&0 
\nonumber
\end{eqnarray}

These two equations place two restrictions on the joint probability
distribution of $x$ and $u.$ Since there are two unknown parameters to be
estimated, we might look upon these equations to provide solutions for those
two parameters. We choose estimators $b_{0}$ and $b_{1}$ to solve the sample
counterparts of these equations, making use of the principle of the method
of moments: 
\begin{eqnarray}
n^{-1}\sum_{i=1}^{n}\left( y_{i}-b_{0}-b_{1}x_{i}\right) &=&0
\label{normeqn} \\
n^{-1}\sum_{i=1}^{n}x_{i}\left( y_{i}-b_{0}-b_{1}x_{i}\right) &=&0  \nonumber
\end{eqnarray}

the so-called normal equations of least squares. Why is this method said to
be ``least squares''?\ Because as we shall see, it is equivalent to
minimizing the sum of squares of the regression residuals. How do we arrive
at the solution? The first ``normal equation'' can be seen to be 
\begin{equation}
b_{0}=\bar{y}-b_{1}\bar{x}  \label{regconst}
\end{equation}

where $\bar{y}$ and $\bar{x}$ are the sample averages of those variables.
This implies that the regression line passes through the point of means of
the sample data. Substituting this solution into the second normal equation,
we now have one equation in one unknown, $b_{1}:$

\begin{eqnarray}
0 & = & \sum_{i=1}^{n}x_{i}\left( y_{i}-\left( \bar{y}-b_{1}\bar{x}\right)
-b_{1}x_{i}\right) \nonumber  \label{regslope} \\ \nonumber \\
\sum_{i=1}^{n}x_{i}\left( y_{i}-\bar{y}\right)
&=&b_{1}\sum_{i=1}^{n}x_{i}\left( x_{i}-\bar{x}\right)  \nonumber \\ \nonumber \\
b_{1} &=&\frac{\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) \left( y_{i}-\bar{y}%
\right) }{\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) ^{2}}  \nonumber \\ \nonumber \\
b_{1} &=&\frac{Cov(x,y)}{Var(x)} %\nonumber
\end{eqnarray}

where the slope estimate is merely the ratio of the sample covariance of the
two variables to the variance of $x$, which, must be nonzero for
the estimates to be computed. This merely implies that not all of the sample
values of $x$ can take on the same value. There must be diversity in the
observed values of $x$. These estimates--$b_{0}$ and $b_{1}-$are said to be
the \textbf{ordinary least squares (OLS)}\ estimates of the regression
parameters, since they can be derived by solving the least squares problem: 
\begin{equation}
\min_{b_{0},b_{1}}S=\sum_{i=1}^{n}e_{i}^{2}=\sum_{i=1}^{n}\left(
y_{i}-b_{0}-b_{1}x_{i}\right) ^{2}  \label{ols}
\end{equation}

Here we minimize the sum of squared residuals, or differences between the
regression line and the values of $y,$ by choosing $b_{0}$ and $b_{1}.$ If
we take the derivatives $\partial S/\partial b_{0}$ and $\partial S/\partial
b_{1}$ and set the resulting first order conditions to zero, the two
equations that result are exactly the OLS solutions for the estimated
parameters shown above. The ``least squares'' estimates minimize the sum of
squared residuals, in the sense that any other line drawn through the
scatter of $(x,y)$ points would yield a larger sum of squared residuals. The
OLS\ estimates provide the unique solution to this problem, and can always
be computed if (i) $Var(x)>0$ and (ii) $n\geq 2.$ The estimated OLS\
regression line is then 
\begin{equation}
\hat{y}_{i}=b_{0}+b_{1}x_{i}  \label{yhat}
\end{equation}

where the ``hat'' denotes the predicted value of $y$ corresponding to that
value of $x.$ This is the \textbf{sample regression function (SRF)},
corresponding to the population regression function, or PRF (\ref{prf}). The
population regression function is fixed, but unknown, in the population; the
SRF\ is a function of the particular sample that we have used to derive it,
and a different SRF\ will be forthcoming from a different sample. The
primary interest in these estimates usually involves $b_{1}=\partial
y/\partial x=\Delta y/\Delta x,$ the amount by which $y$ is predicted to
change from a unit change in the level of $x.$ This slope is often of
economic interest, whereas the constant term in many regressions is devoid
of economic meaning. In some cases, though, it has an economic interpretation.

For instance, a regression of major companies' CEO\
salaries on the firms' return on equity---a measure of economic
performance---yields the regression estimates 
\begin{equation}
\hat{S}=963.191+18.501r  \label{saleqn}
\end{equation}
where $S$ is the CEO's annual salary, in thousands of 1990 dollars, and $r$
is average return on equity over the prior three years, in per cent. This
implies that a one percent increase in ROE\ over the past three years is
worth \$18,501 to a CEO, on average. The average annual salary for the 209
CEOs in the sample is \$1.28 million, so the increment is about 1.4\% of
that average salary. The SRF\ can also be used to predict what a CEO\ will
earn for any level of ROE; points on the estimated regression function are
such predictions.

\textbf{Mechanics of OLS}

Some algebraic properties of the OLS\ regression line:

(1) The sum (and average) of the OLS\ residuals is zero: 
\begin{equation}
\sum_{i=1}^{n}e_{i}^{{}}=0  \label{sumres}
\end{equation}
which follows from the first normal equation, which specifies that the
estimated regression line goes through the point of means $\left( \bar{x},%
\bar{y}\right) $, so that the mean residual must be zero.

(2) By construction, the sample covariance between the OLS residuals and the
regressor is zero: 
\begin{equation}
Cov(e,x)=\sum_{i=1}^{n}x_{i}e_{i}^{{}}=0  \label{covex}
\end{equation}
This is not an assumption, but follows directly from the second normal
equation. The estimated coefficients, which give rise to the residuals, are
chosen to make it so.

(3) Each value of the dependent variable may be written in terms of its
prediction and its error, or regression residual:
\[
y_{i}=\hat{y}_{i}+e_{i} 
\]
so that OLS\ decomposes each $y_{i}$ into two parts: a fitted value, and a
residual. Property (3) also implies that $Cov(e,\hat{y})=0,$ since 
$\hat{y}$ is a linear transformation of $x,$ and linear transformations have
linear effects on covariances. Thus, the fitted values and residuals are
uncorrelated in the sample. Taking this property and applying it to the
entire sample, we define 

\pagebreak
\begin{eqnarray*}
SST &=&\sum_{i=1}^{n}\left( y_{i}-\bar{y}\right) ^{2} \\ \\
SSE &=&\sum_{i=1}^{n}\left( \hat{y}_{i}-\bar{y}\right) ^{2} \\ \\
SSR &=&\sum_{i=1}^{n}e_{i}^{2}
\end{eqnarray*}
as the Total sum of squares, Explained sum of squares, and Residual sum of
squares, respectively. Note that $SST$ expresses the total variation in $y$
around its mean (and we do not strive to ``explain'' its mean; only how it
varies about its mean). 

The second quantity, $SSE,$ expresses the variation
of the predicted values of $y$ around the mean value of $y$ (and it is
trivial to show that $\hat{y}$ has the same mean as $y).$ 

The third quantity, $SSR,$ is the same as the least squares criterion $S$ from (\ref
{ols}). (Note that some textbooks interchange the definitions of $SSE$ and $%
SSR,$ since both ``explained'' and ``error'' start with E, and
``regression'' and ``residual'' start with R). Given these sums of squares,
we can generalize the decomposition mentioned above into 
\begin{equation}
SST=SSE\;+\;SSR  \label{ssacct}
\end{equation}
or, the total variation in $y$ may be divided into that \textbf{explained}
and that \textbf{unexplained}, i.e. left in the residual category. To prove
the validity of (\ref{ssacct}), note that

\pagebreak
\begin{eqnarray*}
\sum_{i=1}^{n}\left( y_{i}-\bar{y}\right) ^{2} &=&\sum_{i=1}^{n}\left(
\left( y_{i}-\hat{y}_{i}\right) +\left( \hat{y}_{i}-\bar{y}\right) \right)
^{2} \\ \\
&=&\sum_{i=1}^{n}\left[ e_{i}+\left( \hat{y}_{i}-\bar{y}\right) \right] ^{2}
\\ \\
&=&\sum_{i=1}^{n}e_{i}^{2}+2\sum_{i=1}^{n}e_{i}^{{}}\left( \hat{y}_{i}-\bar{y%
}\right) + \\ & &\sum_{i=1}^{n}\left( \hat{y}_{i}-\bar{y}\right) ^{2} \\ \\
SST &=&SSR+SSE
\end{eqnarray*}
given that the middle term in this expression is equal to zero. But this
term is the sample covariance of $e$ and $y,$ given a zero mean for $e,$ and
by (\ref{covex})\ we have established that this is zero.

How good a job does this SRF\ do? Does the regression function explain a
great deal of the variation of $y,$ or not very much? That can now be
answered by making use of these sums of squares: 
\[
R^{2}=\left[ r_{xy}\right] ^{2}=\frac{SSE}{SST}=1-\frac{SSR}{SST} 
\]
The $R^{2}$ measure (sometimes termed the coefficient of determination)
expresses the percent of variation in $y$ around its mean ``explained'' by
the regression function. It is an $r$, or simple correlation coefficient,
squared, in this case of simple regression on a single $x$ variable. Since
the correlation between two variables ranges between -1 and +1, the squared
correlation ranges between 0 and 1. In that sense, $R^{2}$ is like a batting
average. 

In the case where $R^{2}=0,$ the model we have built fails to
explain any of the variation in the $y$ values around their mean--unlikely,
but it is certainly possible to have a very low value of $R^{2}.$ In the
case where $R^{2}=1,$ all of the points lie on the SRF. That is unlikely
when $n>2,$ but it may be the case that all points lie close to the line, in
which case $R^{2}$ will approach 1. We cannot make any statistical judgment
based directly on $R^{2},$ or even say that a model with a higher $R^{2}$
and the same dependent variable is necessarily a better model; but other
things equal, a higher $R^{2}$ will be forthcoming from a model that
captures more of $y^{\prime }s$ behavior. 

In cross-sectional analyses, where we are trying to understand the idiosyncracies of individual behavior, very low $R^{2}$ values are common, and do not necessarily denote a failure to
build a useful model. In time-series analyses, much higher $R^2$ values are often commonly observed, as many economic and financial variables move together over time.

Important issues in evaluating applied work: how do the quantities we have
estimated change when the units of measurement are changed? In the estimated
model of CEO salaries, since the $y$ variable was measured in thousands of
dollars, the intercept and slope coefficient refer to those units as well.
If we measured salaries in dollars, the intercept and slope would be
multiplied by 1000, but nothing would change. The correlation between $y$
and $x$ is not affected by linear transformations, so we would not alter the 
$R^{2}$ of this equation by changing its units of measurement. Likewise, if
ROE\ was measured in decimals rather than per cent, it would merely change
the units of measurement of the slope coefficient. Dividing $r$ by 100 would
cause the slope to be multiplied by 100. In the original (\ref{saleqn}),
with $r$ in percent, the slope is 18.501 (thousands of dollars per one unit
change in $r).$ If we expressed $r$ in decimal form, the slope would be
1850.1. A change in $r$ from 0.10 to 0.11 -- a one per cent increase in
ROE--would be associated with a change in salary of (0.01)(1850.1)=18.501
thousand dollars. Again, the correlation between salary and ROE\ would not
be altered. This also applies for a transformation such as $F=32+\frac{9}{5}%
C;$ it would not matter whether we viewed temperature in degrees $F$ or
degrees $C$ as a causal factor in estimating the demand for heating oil,
since the correlation between the dependent variable and temperature would
be unchanged by switching from Fahrenheit to Celsius degrees.

\textbf{Functional form}

Simple linear regression would seem to be a workable tool if we have a
presumed linear relationship between $y$ and $x,$ but what if theory
suggests that the relation should be nonlinear? It turns out that the
``linearity'' of regression refers to $y$ being expressed as a linear
function of $x-$but neither $y$ nor $x$ need be the ``raw data'' of our
analysis. For instance, regressing $y$ on $t$ (a time trend) would allow us
to analyse a linear trend, or constant growth, in the data. What if we
expect the data to exhibit exponential growth, as would population, or sums
earning compound interest? If the underlying model is 
\begin{eqnarray}
y &=&A\exp \left( rt\right)  \label{singlelog} \\
\log y &=&\log A+rt  \nonumber \\
y^{*} &=&A^{*}+rt \nonumber
\end{eqnarray}

so that the ``single-log'' transformation may be used to express a
constant-growth relationship, in which $r$ is the regression slope
coefficient that directly estimates $\partial \log y/\partial t.$ Likewise,
the ``double-log'' transformation can be used to express a
constant-elasticity relationship, such as that of a Cobb-Douglas function:

\begin{eqnarray}
y &=&Ax^{\alpha }  \label{doublelog} \\
\log y &=&\log A+\alpha \log x  \nonumber \\
y^{*} &=&A^{*}+\alpha x^{*}  \nonumber
\end{eqnarray}

In this context, the slope coefficient $\alpha $ is an estimate of the
elasticity of $y$ with respect to $x,$ given that $\eta _{y,x}=\partial \log
y/\partial \log x$ by the definition of elasticity. The original equation is
nonlinear, but the transformed equation is a linear function which may be
estimated by OLS\ regression.

Likewise, a model in which $y$ is thought to depend on $1/x$ (the reciprocal
model) may be estimated by linear regression by just defining a new
variable, $z,$ equal to $1/x$ (presuming $x>0).$ That model has an
interesting interpretation if you work out its algebra.

We often use a polynomial form to allow for nonlinearities in a regression relationship. For instance, rather than including only $x$ as a regressor, we may include $x$ and $x^2$. Although this relationship is linear in the parameters, it implies that $\frac{\partial Y}{\partial x} = \beta + 2 \gamma x$, so that the effect of $x$ on $Y$ now depends on the level of $x$ for that observation, rather than being a constant factor. 

\textbf{Properties of OLS\ estimators}

Now let us consider the properties of the regression estimators we have
derived, considering $b_{0}$ and $b_{1}$ as estimators of their respective
population quantities. To establish the unbiasedness of these estimators, we
must make several assumptions:

\begin{proposition}
SLR1: in the population, the dependent variable $y$ is related to the
independent variable $x$ and the error $u$ as 
\begin{equation}
y_{{}}=\beta _{0}+\beta _{1}x_{{}}+u_{{}}  \label{2vregbis}
\end{equation}
\end{proposition}

\begin{proposition}
SLR2: we can estimate the population parameters from a sample of size $n,$ $%
\{(x_{i},y_{i}),i=1,...,n\}.$
\end{proposition}

\begin{proposition}
SLR3: the error process has a zero conditional mean: 
\begin{equation}
E\left( u\mid x\right) =0.  \label{zcmbis}
\end{equation}
\end{proposition}

\begin{proposition}
SLR4: the independent variable $x$ has a positive variance: 
\begin{equation}
\left( n-1\right) ^{-1}\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) ^{2}>0.
\label{posvar}
\end{equation}
\end{proposition}

Given these four assumptions, we may proceed, considering the intercept and
slope estimators as random variables. For the slope estimator; we may
express the estimator in terms of population coefficients and errors: 
\begin{equation}
b_{1}=\frac{\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) \left( y_{i}-\bar{y}%
\right) }{\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) ^{2}}=\frac{%
\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) y_{i}}{s_{x}^{2}}
\label{regslopebis}
\end{equation}
where we have defined $s_{x}^{2}$ as the total variation in $x$ (not the
variance of $x).$ Substituting, we can write the slope estimator as:

\begin{eqnarray*}
b_{1} =\frac{\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) y_{i}}{s_{x}^{2}} \\
=\frac{\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) \left( \beta _{0}+\beta
_{1}x_{i}+u_{i}\right) }{s_{x}^{2}} 
\end{eqnarray*}
\begin{equation}{\small
= \frac{\beta _{0}\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) \\ +\beta
_{1}\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) x_{i} +\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) u_{i}}{s_{x}^{2}} }
\end{equation}
We can show that the first term in the numerator is algebraically zero,
given that the deviations around the mean sum to zero. The second term can
be written as $\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) ^{2}=s_{x}^{2},$ so
that the second term is merely $\beta _{1}$ when divided by $s_{x}^{2}.$
Thus this expression can be rewritten as: 
\[
b_{1}=\beta _{1}+\frac{1}{s_{x}^{2}}\sum_{i=1}^{n}\left( x_{i}-\bar{x}%
\right) u_{i} 
\]
showing that any randomness in the estimates of $b_{1}$ is derived from the
errors in the sample, weighted by the deviations of their respective $x$
values. Given the assumed independence of the distributions of $x$ and $u$
implied by (\ref{zcmbis}), this expression implies that: 
\[
E\left( b_{1}\right) =\beta _{1}, 
\]
or that $b_{1}$ is an unbiased estimate of $\beta _{1},$ given the
propositions above. The four propositions listed above are all crucial for
this result, but the key assumption is the independence of $x$ and $u.$

We are also concerned about the precision of the OLS\ estimators. To derive
an estimator of the precision, we must add an assumption on the distribution
of the error $u:$

\begin{proposition}
SLR5:\ (\textbf{homoskedasticity}): 
\[
Var\left( u\mid x\right) =Var(u)=\sigma ^{2}. 
\]
\end{proposition}

This assumption states that the variance of the error term is constant over
the population, and thus within the sample. Given (\ref{zcmbis}), the
conditional variance is also the unconditional variance. The errors are
considered drawn from a fixed distribution, with a mean of zero and a
constant variance of $\sigma ^{2}.$ If this assumption is violated, we have
the condition of heteroskedasticity, which will often involve the magnitude
of the error variance relating to the magnitude of $x,$ or to some other
measurable factor.

Given this additional assumption, but no further assumptions on the nature
of the distribution of $u$, we may demonstrate that:
\begin{equation}
Var\left( b_{1}\right) =\frac{\sigma ^{2}}{\sum_{i=1}^{n}\left( x_{i}-\bar{x}%
\right) ^{2}}=\frac{\sigma ^{2}}{s_{x}^{2}}  \label{slopevar}
\end{equation}
so that the precision of our estimate of the slope is dependent upon the
overall error variance, and is inversely related to the variation in the $x$
variable. The magnitude of $x$ does not matter, but its variability in the
sample does matter. If we are conducting a controlled experiment (quite
unlikely in economic analysis) we would want to choose widely spread values
of $x$ to generate the most precise estimate of $\partial y/\partial x.$

We can likewise prove that $b_{0}$ is an unbiased estimator of the
population intercept, with sampling variance: 
\begin{equation}
Var\left( b_{0}\right) =n^{-1}\frac{\sigma ^{2}\sum_{i=1}^{n}x_{i}^{2}}{%
\sum_{i=1}^{n}\left( x_{i}-\bar{x}\right) ^{2}}=\frac{\sigma
^{2}\sum_{i=1}^{n}x_{i}^{2}}{ns_{x}^{2}}  \label{intvar}
\end{equation}
so that the precision of the intercept depends, as well, upon the sample
size, and the magnitude of the $x$ values. These formulas for the sampling
variances will be invalid in the presence of heteroskedasticity--that is,
when proposition SLR5 is violated.

These formulas are not operational, since they include the unknown parameter 
$\sigma ^{2}.$ To calculate estimates of the variances, we must first
replace $\sigma ^{2}$ with a consistent estimate, $s^{2},$ derives from the
least squares residuals:
\begin{equation}
e_{i}=y_{i}-b_{0}-b_{1}x_{i},~i=1,...,n  \label{olsresid}
\end{equation}
We cannot observe the error $u_{i}$ for a given observation, but we can
generate a consistent estimate of the $i^{th}$ observation's error with the $%
i^{th}$ observation's least squares \textbf{residual}, $\hat{u}_{i}.$
Likewise, a sample quantity corresponding to the population variance $\sigma
^{2}$ can be derived from the residuals: 
\begin{equation}
s^{2}=\frac{1}{(n-2)}\sum_{i=1}^{n}e_{i}^{2}=\frac{SSR}{(n-2)}
\label{ssqr}
\end{equation}
where the numerator is just the least squares criterion, $SSR,$ divided by
the appropriate degrees of freedom. Here, two degrees of freedom are lost,
since each residual is calculated by replacing two population coefficients
with their sample counterparts. This now makes it possible to generate the
estimated variances and, more usefully, the \textbf{estimated standard error}
of the regression slope:

\[
s_{b_{1}}=\frac{s}{s_{x}} 
\]
where $s$ is the standard deviation, or standard error, of the disturbance
process (that is, $\sqrt{s^{2}}),$ and $s_{x}$ is $\sqrt{s_{x}^{2}}.$ It is
this estimated standard error that will be displayed on the computer
printout when you run a regression, and used to construct confidence
intervals and hypothesis tests about the slope coefficient. We can calculate
the estimated standard error of the intercept term by the same means.

\pagebreak
\textbf{Regression through the origin}

We could also consider a special case of the model above where we impose a
constraint that $\beta _{0}=0,$ so that $y$ is taken to be proportional to $%
x.$ This will often be inappropriate; it is generally more sensible to let
the data calculate the appropriate intercept term, and reestimate the model
subject to that constraint only if that is a reasonable course of action.
Otherwise, the resulting estimate of the slope coefficient will be biased.
Unless theory suggests that a strictly proportional relationship is
appropriate, the intercept should be included in the model.

\end{document}
