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

\bigskip \textbf{Chapter 10: Basic regression analysis with time series data}

We now turn to the analysis of time series data. One of the key assumptions
underlying our analysis of cross-sectional data will prove to be untenable
when we consider time series data; thus, we separate out the issues of time
series modelling from that of cross sections. How does time series data
differ? First of all, it has a natural ordering, that of calendar time at
some periodic frequency. Note that we are not considering here a dataset in
which some of the variables are dated at a different point in time: e.g. a
survey measuring this year's income, and (as a separate variable) last
year's income. In time series data sets, the observations are dated, and
thus we need to respect their order, particularly if the model we consider
has a \textbf{dynamic} specification (involving variables from more than one
point in time). What is a time series?\ Merely a sequence of observations on
some phenomenon observed at regular intervals. Those intervals may
correspond to the passage of calendar time (e.g. annual, quarterly, monthly
data) or they may reflect an economic process that is irregular in calendar
time (such as business-daily data). In either case, our observations may not
be available for every point in time (for instance, there are days when a
given stock does not trade on the exchange).

A second important difference between cross-sectional and time series data:
with the former, we can reaonably assume that the sample is drawn randomly
from the appropriate population, and could conceive of one or many alternate
samples constructed from the same population. In the case of time series
data, we consider the sequence of events we have recorded as a \textbf{%
realization} of the underlying process. We only have one realization
available, in the sense that history played out a specific sequence of
events. In an alternate universe, Notre Dame might have beaten BC in the Frozen Four. % the Yankees lost games 4 and 5 in regulation innings, and the Diamondbacks took the World Series. 
Randomness
plays a role, of course, just as it does in cross-sectional data; we do not
know what will transpire until it happens, so that time series data \textit{%
ex ante }are random variables. We often speak of a time series as a \textbf{%
stochastic process}, or time series process, focusing on the concept that
there is some mechanism generating that process, with a random component.

\textbf{Types of time series regression models}

Models used in a time series context can often be grouped into those sharing
common features. By far the simplest is a \textbf{static} model, such as 
\begin{equation}
y_{t}=\beta _{0}+\beta _{1}x_{1,t}+\beta _{2}x_{2,t}+u_{t}  \label{tsm}
\end{equation}

We may note that this model is the equivalent of the cross-sectional
regression model, with the $i$ subscript in the cross section replaced by $t$
in the time series context. Each observation is modeled as depending only on 
\textbf{contemporaneous} values of the explanatory variables. This structure
implies that all of the interactions among the variables of the model are
assumed to take place {\it immediately}: or, taking the frequency into account,
within the same time period. Thus, such a model might be reasonable when
applied to annual data, where the length of the observation interval is long
enough to allow behavioral adjustments to take place. If we applied the same
model to higher-frequency data, we might consider that assumption
inappropriate; we might consider, for instance, that a tax cut would not be
fully reflected by higher retail sales in the same month that it took
effect. An example of such a structure that appears in many textbooks is the
static Phillips curve: 
\begin{equation}
\pi _{t}=\beta _{0}+\beta _{1}UR_{t}+u_{t}  \label{pcurve}
\end{equation}
where $\pi _{t}$ is this year's inflation rate, and $UR_{t}$ is this year's
unemployment rate. Stating the model in this form not only implies that the
level of unemployment is expected to affect the rate of inflation
(presumably with a negative sign), but also that the entire effect of
changes in unemployment will be reflected in inflation within the
observation interval (e.g. one year).

In many contexts, we find a static model inadequate to reflect what we
consider to be the relationship between explanatory variables and those
variables we wish to explain. For instance, economic theory surely predicts
that changes in interest rates (generated by monetary policy) will have an
effect on firms' capital investment spending. At lower interest rates, firms
will find more investment projects with a positive expected net present
value. But since it takes some time to carry out these projects--equipment
must be ordered, delivered, and installed, or new factories must be built
and equipped--we would not expect that quarterly investment spending would
reflect the same quarter's (or even the previous quarter's) interest rates.
Presumably interest rates affect capital investment spending with a lag, and
we must take account of that phenomenon. If we were to model capital
investment with a static model, we would be omitting relevant explanatory
variables: the prior values of the causal factors. These omissions would
cause our estimates of the static model to be biased and inconsistent. Thus,
we must use some form of \textbf{distributed lag} model to express the
relationship between current and past values of the explanatory variables
and the outcome. Distributed lag models may take a finite number of lagged
values into account (thus the Finite Distributed Lag model, or FDL) or they
may use an infinite distributed lag: e.g. all past values of the $x$
variables. When an infinite DL\ model is specified, some algebraic
sleight-of-hand must be used to create a finite set of regressors.

A simple FDL\ model would be
\begin{equation}
f_{t}=\beta _{0}+\beta _{1}pe_{t}+\beta _{2}pe_{t-1}+\beta _{3}pe_{t-2}+u_{t}
\label{fdl}
\end{equation}
in which we consider the fertility rate in the population as a function of
the personal exemption, or child allowance, over this year and the past two
years. We would expect that the effect of a greater personal exemption is
positive, but realistically we would not expect the effect to be (only)
contemporaneous. Given that there is at least a 9-month lag between the
decision and the recorded birth, we would expect such an effect (if it
exists) to be largely concentrated in the $\beta _{2}$ and $\beta _{3}$
coefficients. Indeed, we might consider whether additional lags are
warranted. In this model, $\beta _{1}$ is the \textbf{impact effect}, or 
\textbf{impact multiplier} of the personal exemption, measuring the
contemporaneous change. How do we calculate $\partial f/\partial pe?$ That
(total) derivative must be considered as the effect of a one-time change in $%
pe$ that raises the exemption by one unit and leaves it permanently higher.
It may be computed by evaluating the \textbf{steady state} of the model:
that with all time subscripts dropped. Then it may be seen that the total
effect, or\textbf{\ long-run multiplier}, of a permanent change in $pe$ is $%
\left( \beta _{1}+\beta _{2}+\beta _{3}\right) .$ In this specification, we
presume that there is an impact effect (allowing for a nonzero value of $%
\beta _{1})$ but we are imposing the restriction that the entire effect will
be felt within the two year lag. This is testable, of course, by allowing
for additional lag terms in the model, and testing for their joint
significance. However the analysis of individual lag coefficients is often
hampered--especially at higher frequencies such as quarterly and monthly
data--by high autocorrelation in the series. That is, the values of the
series are closely related to each other over time. If this is the case,
then many of the individual coefficients in a FDL\ regression model may not
be distinguishable from zero. This does not imply, though, that the sum of
those coefficients (i.e. the long run multiplier) will be imprecisely
estimated. We may get a very precise value for that effect, even if its
components are highly intercorrelated.

One additional concern that will apply in estimating FDL\ models, especially
when the number of observations is limited. Each lagged value included in a
model results in the loss of one observation in the estimation sample.
Likewise, the use of a first difference ($\Delta y_{t}\equiv
y_{t}-y_{t-1})\; $on either the left or right side of a model results in the
loss of one observation. If we have a long time series, we may not be too
concerned about this; but if we were working with monthly data, and felt it
appropriate to consider 12 lags of the explanatory variables, we would lose
the first year of data to provide these starting values. Computer programs
such as Stata may be set up to recognize the time series nature of the data
(in Stata, we use the \texttt{tsset} command to identify the date variable,
which must contain the calendar dates over which the data are measured), and
construct lags and first differences taking these constraints into account
(for instance, a lagged value of a variable will be set to a missing value
where it is not available). In Stata, once a dataset has been established as
time series, we may use the operators \texttt{L.,D.} and \texttt{F.} to
refer to the lag, difference or lead of a variable, respectively: so \texttt{%
L.gdp} is last period's \texttt{gdp, D.gdp} is the first difference, and 
\texttt{F.gdp} is next year's value. These operators can also consider
higher lags, so \texttt{L2.gdp} is the second lag, and \texttt{L(1/4).gdp}
refers to the first four lags, using standard Stata ``numlist'' notation (%
\texttt{help numlist }for details).

\textbf{Finite sample properties of OLS}

How must we modify the assumptions underlying OLS\ to deal with time series
data?\ First of all, we assume that there is a linear model linking $y$ with
a set of explanatory variables, $\{x_{1}...x_{k}\},$ with an additive error $%
u, $for a sample of $t=1,...,n.$ It is useful to consider the explanatory
variables as being arrayed in a matrix $X= 
\begin{array}{ccc}
x_{1,1} & \cdots & x_{1,k}^{{}} \\ 
x_{2,1} & \cdots & x_{2,k} \\ 
\vdots & \cdots & \vdots \\ 
x_{n,1}^{{}} & \cdots & x_{n,k}
\end{array}
$ where$,$ like a spreadsheet, the rows are the observations (indexed by
time) and the columns are the variables (which may actually be dated
differently: e.g. $x_{2}$ may actually be the lag of $x_{1},$ etc.) To
proceed with the development of the finite sample properties of OLS, we
assume:

\begin{proposition}
For each $t,$ $E(u_{t}|X)=0,$ where $X$ is the matrix of explanatory
variables.
\end{proposition}

This is a key assumption, and quite a strong one: it states not only that
the error is contemporaneously uncorrelated with each of the explanatory
variables, but also that the error is assumed to be uncorrelated with
elements of $X$ at every point in time. The weaker statement of \textbf{%
contemporaneous exogeneity},\\ $E(u_{t}|x_{t,1},x_{t,2},...,x_{t,k})=0$ is
analogous to the assumption that we made in the cross-sectional context. But
this is a stronger assumption, for it states that the elements of $X$, past,
present, and future, are independent of the errors: or that the explanatory
variables in $X$ are \textbf{strictly exogenous. } It is important to note
that this assumption, by itself, says nothing about the correlations over
time among the explanatory variables (or their correlations with each
other), nor about the possibility that successive elements of $u$ may be
correlated (in which case we would say that $u$ is \textbf{autocorrelated}).
The assumption only states that the distributions of $u$ and $X$ are
independent.

What might cause this assumption to fail? Clearly, omitted variables and/or
measurement error are likely causes of a correlation between the regressors
and errors. But in a time series context there are other likely suspects. If
we estimate a static model, for instance, but the true relationship is 
\textbf{dynamic}--in which lagged values of some of the explanatory
variables also have direct effects on $y-$then we will have a correlation
between contemporaneous $x$ and the error term, since it will contain the
effects of lagged $x,$ which is likely to be correlated with current $x.$ So
this assumption of strict exogeneity has strong implications for the correct
specification of the model (in this case, we would need to specify a FDL\
model). It also implies that there cannot be correlation between current
values of the error process and {\it future} $x$ values:something that would be
likely in a case where some of the $x$ variables are policy instruments. For
instance, consider a model of farmers' income, dependent on (among other
factors) on government price supports for their crop. If unprecedented
shocks (such as a series of droughts), which are unpredictable and random
effects of weather on farmers' income, trigger an expansion of the
government price support program, then the errors today are correlated with
future $x$ values.

The last assumption we need is the standard assumption that the columns of $%
X $ are {\it linearly independent}: that is, there are no exact linear relations,
or \textbf{perfect collinearity}, among the regressors.

With these assumptions in hand, we can demonstrate that the OLS\ estimators
are unbiased, both conditional on $X$ and unconditionally. The random
assumption that allowed us to prove unbiasedness in the cross-sectional
context has been replaced by the assumption of strict exogeneity in the time
series context. We now turn to the interval estimates. As previously, we
assume that the error variance, conditioned on $X,$ is homoskedastic: $%
Var(u_{t}|X)=Var(u_{t})=\sigma ^{2},\forall t.$ In a time series context,
this assumption states that the error variance is constant over time, and in
particular not influenced by the $X$ variables. In some cases, this may be
quite unrealistic. We now add an additional assumption, particular to time
series analysis: that there is no\textbf{\ serial correlation} in the
errors: $Cov(u_{t},u_{s}|X)=Cov(u_{t},u_{s})=0,\forall t\neq s.$ This
assumption states that the errors are not \textbf{autocorrelated}, or correlated with one another, so that there is no systematic pattern in the
errors over time. This may clearly be violated, if the error in one period
(for instance, the degree to which the actual level of $y$ falls short of
the desired level) is positively (or negatively) related to the error in the
previous period. Positive autocorrelation can readily arise in a situation
where there is partial adjustment to a discrepancy, whereas negative
autocorrelation is much more likely to reflect ``overshooting,'' in which a
positive error (for instance, an overly optimistic forecast) is followed by
a negative error (a pessimistic forecast). This assumption has nothing to do
with the potential autocorrelation within the $X$ matrix; it only applies to
the error process. Why is this assumption only relevant for time series?\ In
cross sections, we assume random sampling, whereby each observation is
independent of every other. In time series, the sequence of the observations
makes it likely that if independence is violated, it will show up in
successive observations' errors.

With these additional assumptions, we may state the Gauss-Markov theorem for
OLS\ estimators of a time series model (OLS\ estimators are BLUE), implying
that the variances of the OLS\ estimators are given by: 
\begin{equation}
Var(b_{j}|X)=\frac{\sigma ^{2}}{\left[ SST_{j}\left( 1-R_{j}^{2}\right)
\right] }  \label{varb}
\end{equation}

where $SST_{j}$ is the total sum of squares of the $j^{th}$ explanatory
variable, and $R_{j}^{2}$ is the $R^{2}$ from a regression of variable $%
x_{j} $ on the other elements of $X.$ Likewise, the unknown parameter $%
\sigma ^{2}$ may be replaced by its consistent estimate, $s^{2}=\frac{SSR}{%
n-k-1},$ identical to that discussed previously.

As in our prior derivation, we will assume that the errors are normally
distributed: $u\sim N(0,\sigma ^{2}).$ If the above assumptions hold, then
the standard $t-$statistics and $F-$statistics we have applied in a
cross-sectional context will also be applicable in time series regression
models.

\textbf{Functional form, dummy variables, and index numbers}

We find that a logarithmic transformation is very commonly used in time
series models, particularly with series that reflect stocks, flows, or
prices (rather than rates). Many models are specified with the first
difference of log($y$), implying that the dependent variable is the growth
rate of $y.$ Dummy variables are also very useful to test for \textbf{%
structural change}. We may have \textit{a priori} information that indicates
that unusual events were experienced in particular time periods: wars,
strikes, or presidential elections, or a market crash. In the context of a
dynamic model, we do not want to merely exclude those observations, since
that would create episodes of missing data. Instead, we can ``dummy'' the
period of the event, which then allows for an intercept shift (or, with
interactions, for a slope shift) during the unusual period. The tests for
significance of the dummy coefficients permit us to identify the importance
of the period, and justify its special treatment. We may want to test that
the relationship between inflation and unemployment (the ``Phillips curve'')
is the same in Republican and Democratic presidential administrations; this
may readily be done with a dummy for one party, added to the equation and
interacted to allow for a slope change between the two parties' equations.
Dummy variables are also used widely in financial research, to conduct 
\textbf{event studies}: models in which a particular event, such as the
announcement of a takeover bid, is hypothesized to trigger ``abnormal''
returns to the stock. In this context, high-frequency (e.g. daily)\ data on
stock returns are analyzed, with a dummy set equal to 1 on and after the
date of the takeover bid announcement. A test for the significance of the
dummy coefficient allows us to analyze the importance of this event. (These models are explicitly discussed in EC327, Financial Econometrics).

Creation of these dummies in Stata is made easier by the\texttt{\ tin()}%
function (read:\ tee-in). If the data set has been established as a time
series via \texttt{tsset}, you may refer to natural time periods in
generating new variables or specifying the estimation sample. For instance, 
\texttt{gen prefloat = (tin(1959q1,1971q3)) }will generate a dummy for that
pre-Smithsonian period, and a model may be estimated over a subset of the
observations via \texttt{regress ... if tin(1970m1,1987m9). }

In working with time series data, we are often concerned with series
measured as index numbers, such as the Consumer Price Index, GDP\ Deflator,
Index of Industrial Production, etc. The price series are often needed to
generate real values from nominal magnitudes. The usual concerns must be
applied in working with these index number series, some of which have been
rebased (e.g. from 1982=100 to 1987=100) and must be adjusted accordingly
for a new base period and value. Interesting implications arise when we work
with ``real'' magnitudes, expressed in logs: for instance, labor supply is
usually modelled as depending on the real wage, $\left( \frac{w}{p}\right) .$
If we express these variables in logs, the log of the real wage becomes $%
\log w-\log p.$ Regressing the log of hours worked on a single variable, $%
\left( \log w-\log p\right) $, is a restricted version of a regression in
which the two variables are entered separately. In that regression, the
coefficients will almost surely differ in their absolute value. But economic
theory states that only the real wage should influence workers' decisions;
they should not react to changes in its components (e.g. they should not be
willing to supply more hours of labor if offered a higher nominal wage that
only makes up for a decrease in their purchasing power).

\bigskip 

\textbf{Trends and seasonality}

Many economic time series are \textbf{trending}: growing over time. One of
the reasons for very high $R^{2}$ values in many time series regressions is
the common effect of time on many of the variables considered. This brings a
challenge to the analysis of time series data, since when we estimate a
model in which we consider the effect of several causal factors, we must be
careful to account for the co-movements that may merely reflect trending
behavior. Many macro series reflect upward trends; some, such as the cost of
RAM\ for personal computers, exhibit strong downward trends. We can readily
model a \textbf{linear trend} by merely running a regression of the series
on $t,$ in which the slope coefficient is then $\partial y/\partial t.$ To
create a time trend in Stata, you can just \texttt{generate t = \_n}, where 
\texttt{\_n }is the observation number. It does not matter where a trend
starts, or the units in which it is expressed; a trend is merely a series
that changes by a fixed amount per time period. A linear trend may prove to
be inadequate for many economic series, which we might expect on a
theoretical basis to exhibit constant growth, not constant increments. In
this case, an \textbf{exponential trend} may readily be estimated (for
strictly positive $y)$ by regressing $\log y$ on $t.$ The slope coefficient
is then a direct estimate of the percentage growth rate per period. We could
also use a polynomial model, such as a \textbf{quadratic time trend},
regressing the level of $y$ on $t$ and $t^{2}.$

Nothing about trending economic variables violates our basic assumptions for
the estimation of OLS regression models with time series data. However, it
is important to consider whether significant trends exist in the series; if
we ignore a common trend, we may be estimating a \textbf{spurious regression}%
, in which both $y$ and the $X$ variables appear to be correlated because of
the influence on both of an omitted factor, the passage of time. We can
readily guard against this by including a time trend (linear or quadratic)
in the regression; if it is needed, it will appear to be a significant
determinant of $y.$ In some cases, inclusion of a time trend can actually
highlight a meaningful relationship between $y$ and one or more $x$
variables: since their coefficients are now estimates of their co-movement
with $y,$ \textit{ceteris paribus}: that is, net of the trend in $y.$

We may link the concept of a regression inclusive of trend to the common
practice of analyzing \textbf{detrended} data. Rather than regressing $y$ on 
$X$ and $t,$ we could remove the trend from $y$ and each of the variables in 
$X.$ How? Regress each variable on $t,$ and save the residuals (if desired,
adding back the original mean of the series). This is then the detrended $y,$
call it $y^{*},$ and the detrended explanatory variables $X^{*}$ (not
including a trend term). If we now estimate the regression of $y^{*}$ on $%
X^{*},$ we will find that the slope coefficients' point and interval
estimates are exactly equal to those from the original regression of $y$ on $%
X$ and $t.$ Thus, it does not matter whether we first detrend the series,
and run the regression, or estimate the regression with trend included.
Those are equvalent strategies, and since the latter is less burdensome, it
may be preferred by the innately lazy researcher.

Another issue that may often arise in time series data of quarterly, monthly
or higher frequency is \textbf{seasonality}. Some economic variables are
provided in \textbf{seasonally adjusted} form. In databanks and statistical
publications, the acronym SAAR\ (seasonally adjusted at annual rate) is
often found. Other economic series are provided in their raw form, often
labelled NSA, or not seasonally adjusted. Seasonal factors play an important
role in many series. Naturally, they reflect the seasonal patterns in many
commodities' measures: agricultural prices differ between harvest periods
and out-of-season periods, fuel prices differ due to winter demand for oil
and natural gas, or summer demand for gasoline. But there are seasonal
factors in many series we might consider with a more subtle interpretation.
Retail sales, naturally, are very high in the holiday period: but so is the
demand for cash, since shoppers and gift-givers will often need more cash at
that time. Payrolls in the construction industry will exhibit seasonal
patterns, as construction falls off in cold climates, but may be stimulated
by a mild winter. Many financial series will reflect the adjustments made by
financial firms to ``dress up'' quarter-end balance sheets and improve
apparent performance.

If all of the data series we are using in a model have been seasonally
adjusted by their producers, we may not be concerned about seasonality. But
often we will want to use some NSA\ series, or be worried about the
potential for seasonal effects. In this case, just as we dealt with trending
series by including a time trend, we should incorporate seasonality into the
regression model by including a set of \textbf{seasonal dummies}. For
quarterly data, we will need 3 dummies; for monthly data, 11 dummies; and so
on. If we are using business-daily data such as financial time series, we
may want to include ``day-of-week'' effects, with dummies for four of the
five business days.

How would you use quarterly dummies in Stata?\ First of all, you must know
what the time variable in the data set is: give the command \texttt{tsset}
to find out. If it is a quarterly variable, the \texttt{tsset} range will
report dates with embedded ``\texttt{q}''s. Then you may create one
quarterly dummy as\texttt{\ gen q1=(quarter(dofq(qtr))==1)} which will take
on 1 in the first quarter, and 0 otherwise. To consider whether series 
\texttt{income} exhibits seasonality, \texttt{regress income L(1/3).q1} and
examine the $F-$statistic. You could, of course, include any three of the
four quarter dummies; \texttt{L(0/2)} would include dummies for quarters 1,
2 and 3, and yield the same $F-$statistic. Note that inclusion of these
three dummies will require the loss of at least two observations at the
beginning of the sample. This form of seasonal adjustment will consider the
effect of each season to be linear; if we wanted to consider multiplicative
seasonality, e.g. sales are always 10\%\ higher in the fourth quarter, that
could be achieved by regressing $\log y$ on the seasonal dummies. A trend
could be included in either form of the regression to capture trending
behavior over and above seasonality; in the latter regression, of course, it
would represent an exponential (constant growth) trend.

Just as with a trend, we may either deseasonalize each series (by regressing
it on seasonal dummies, saving the residuals, and adding the mean of the
original series) and regress seasonally adjusted series on each other; or we
may include a set of seasonal dummies (leaving one out) in a regression of $%
y $ on $X,$ and test for the joint significance of the seasonal dummies. The
coefficients on the $X$ variables will be identical, in both point and
interval form, using either strategy.

\end{document}
