Modeling and Forecasting Trends: I
(Reference: Chapter 4)
Background –
The unobserved components approach to
modeling and forecasting economic time
series assumes that the typical economic
time series, yt, is made up of the sum of
three independent components
 a time trend component
 a seasonal component
 an irregular or cyclical component.
That is:
yt = time trend + seasonal + cyclical
= Tt + St + Ct
The time trend refers to the “long-run”
average behavior of the series.
The seasonal refers to the annual predictable
cyclical behavior of the series associated
with weather patterns, holiday patterns, etc.,
The cyclical component refers to the
remainder of the series after the trend and
seasonal have been accounted for.
The assumption that these components are
determined independently means that
 each component is determined and
influenced by its own set of forces
and, consequently,
 each component can be studied
separately
The approach is called an “unobserved
components” approach because we do not
directly observe each of the three
components; we only get to observe their
sum. Our job will be to model and estimate
the various components and use these
estimates as the basis for forecasting the
components and their sum.
Whether the assumption underlying the
unobserved components approach, that the
trend, seasonal, and cyclical components are
determined independently, is plausible or
not is debatable and is, in fact, an issue of
some controversy among economists. There
are, for example, many macroeconomists
who argue that economic growth (trend) and
the business cycle (cyclical) are determined
by a common set of forces. We will likely
come back to this issue later in the course.
But, for now, we take the unobserved
components model as our starting point.
Modeling the Trend –
If we look at the HEPI time series or any
one of your time series, the first thing that
stands out us is the obvious tendency of the
series to grow (or, in some cases, to fall)
over time.
That is, it is immediately apparent from the
time series plot that the average change in
the series is positive (or, in some cases,
negative). This tendency is the series’s
trend.
The simplest model of the time trend is the
linear trend model –
Tt = β0 + β1t, t = 1,…,T
(T1 = β0 + β1, T2 = β0 + 2β1,…,TT = β0 + Tβ1)
That is, the trend component is a straight
line with intercept β0 and slope β1.
Note that β1 = dTt/dt and β1 = Tt –Tt-1. So,
β1 > 0 if y has a positive trend and β1< 0 if y
has a negative trend.
The intercept, as is often the case in
econometric models, does not have a
meaningful interpretation and its sign can be
positive or negative, regardless of the
trend’s sign.
In some cases, a linear trend is inadequate to
capture the trend of a time series. A natural
generalization of the linear trend model is
the polynomial trend model –
Tt = β0 + β1t + β2t2 + … + βptp
where p is a positive integer.
Note that
 the linear trend model is a special case of
the polynomial trend model (p=1)
 for economic time series we almost
never require p > 2. That is, if the linear
trend model is not adequate, the
quadratic trend model will usually
work:
Tt = β0 + β1t + β2t2
In the quadratic model, dTt/dt = β1+2tβ2
The Log Linear Trend Model
Another alternative to the linear trend model
is the log linear trend model, which is also
called the exponential trend model:
Tt = β0exp(β1t)
or, taking natural logs on both sides,
log(Tt) = log(β0) + β1t
so that the log of the trend component is
linear.
Note that for the log linear trend model
β1 = log(Tt) – log(Tt-1) = % change in T
So, in the linear trend model the change in T
is constant over time; in the quadratic trend
model the change in T has a linear trend and
in the log linear trend model the growth rate
that is constant over time.
These differences can help you decide
whether the linear, quadratic or log linear
trend model is more appropriate for your
data.
However, in practice, as you will see when
you look at your own data series, it is not
always obvious by simply looking at the
time series plot which form the trend model
should take – linear, log linear, quadratic?
Other?
Some help in this regard –
 experience; yours and others
 knowing what different kinds of trends
look like for different (β0,β1)’s
Note that in all of these models, the trend is
deterministic, i.e., perfectly forecastable.
For instance, in the linear trend model,
TˆT  h ,T , the forecast of TT+h made at time T is:
TˆT  h ,T = β0 + (T+h)β1 = TT+h
(Later in the course we will talk about
stochastic trend models, in which the trend
of the series is not perfectly forecastable.)
However, even if we correctly specify the
shape of the trend (linear, quadratic,
exponential, …), the parameters of the trend
model are unknown. So, in practice, we will
have to estimate these parameters, which
will introduce errors (called sampling or
estimation error) into our trend forecasts.
Estimating the Trend Model
Our assumption at this point is that our time
series, yt, can be modeled as
yt = Tt(β) + εt
where Tt is one of the trend models we
discussed above, β denotes the parameters of
the trend model, and εt denotes the other
factors (i.e., the seasonal and cyclical
components) that determine yt.
We don’t observe the β’s and so we will
need to estimate them in order to forecast
the trend (and, eventually, y).
The natural approach to estimating the trend
model is the least square approach – Choose
the β’s to minimize
T
[ y
t 1
t
 Tt (  )] 2
In the case of the linear or quadratic trends
this is a straightforward application of OLS
Linear Trend Model –
Choose β0,β1 to minimize
T
[ y
t 1
t
 0  1t )]2
That is, run a regression of yt on a
constant and t.
Quadratic Trend Model –
Choose β0,β1,β2 to minimize
T
[ y
t 1
t
 0  1t   2t 2 )]2
That is, run a regression of yt on a
constant, t, and t2.
It turns out that under the assumptions of the
unobserved components model, the OLS
estimator of the linear and quadratic trend
models is unbiased, consistent, and
asymptotically efficient. Further, standard
regression procedures can be applied to test
hypotheses about the ’s and construct
interval estimates. [This is true even though
the regression errors will generally be
serially correlated and heteroskedastic.]
Estimation of the exponential trend at first
seems more complicated because the
problem of minimizing
T
[ y
t 1
t
 0 e 1t )]2
is a nonlinear least squares problem which
must be minimized numerically.
However, this minimization problem can be
converted into an OLS regression problem.
First, instead of assuming that the original series
follows an additive unobserved components
model, i.e.,
y t = Tt + ε t
assume that it follows a multiplicative model,
i.e.,
yt = Ttexp(εt)
Recall that
1) log(xy) = log(x)+log(y)
and
2) 2) log(ex) = x
Then, taking logs on both sides:
log(yt) = log(Tt) + εt
and the least squares estimates of β0 and β1 are
found by minimizing
T
 [log y
t 1
t
b0  1t )]2
to get b̂0 and ˆ1 .
Then set ˆ0  exp( bˆ0 ) .