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Chapter 1 Difference Equations
1. Time Series Econometrics Models
Objective of time series econometrics:
develop models capable of
• forecasting
• interpreting, and
• testing hypotheses
concerning economic and financial data.
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Suppose we observe 50 data values shown in
Figure 1.1 and are interested in forecasting the
subsequent values. Using methods discussed in
next several chapters, it is possible to decom-
pose this series into
• trend
• seasonal, and
• irregular
components shown in the lower panel of the
figure. Notice
• the trend component causes change in the
mean of the series, while
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• the seasonal component imparts a regular
cyclical pattern with peaks occurring every
12 units of time
In practice, the trend and seasonal components
will not be deterministic functions shown in
this figure. With economic and financial data,
it is typical to find that a series contains
• stochastic elements
in the trend, seasonal, and irregular compo-
nents. However, for the time being, we sidestep
this complication so that the projection of the
trend and seasonal components into periods
51 and beyond is straightforward. Notice
• the irregular component, while lacking a
well-defined pattern, is somewhat predictable
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• positive and negative values occur in runs;
the occurrence of a large value in any pe-
riod tends to be followed by another large
value
• short run forecasts will make use of this
positive correlation in the irregular compo-
nent
• over the entire span, however, the irregular
component past period 50 rapidly decays
to zero
Overall forecast, shown in the top part of the
figure, is the sum of each forecasted compo-
nent.
General methodology used to make such fore-
casts entails finding the equation of motion
driving a stochastic process and using that equa-
tion to predict subsequent outcomes.
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Let yt denote the data value in period t. Then
the example in Figure 1.1 implies we observed
y1, . . . , y50. For t = 1, . . . , 50, the equation of
motion used to construct components of the
yt series are
Trend: Tt = 1 + 0.1t,
Seasonal: St = 1.6 sin(tπ/6), and
Irregular: It = 0.7It−1 + t.
Each of these three equations is a difference
equation. In its most general form, a differ-
ence equation expresses the value of a variable
as a function of its own lagged values, time,
and other variables.
• trend and seasonal terms are both func-
tions of time
• irregular term is a function of its own lagged
value and of the stochastic variable t
5
The reason for introducing this set of equa-
tions is to make the point that
• time series econometrics is concerned with
the estimation of difference equations con-
taining stochastic components
The time series econometrician may estimate
the properties of a single series or a vector
containing many interdependent series.
Many economic theories have natural repre-
sentations as stochastic difference equations.
Moreover, many of these have testable impli-
cations concerning the time path of a key eco-
nomic variable. Consider the following three
examples:
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1. Random Walk Hypothesis about change
in stock price: In its simplest form it suggests
that
• Day-to-day changes in the price of a stock
should have a mean value of zero, because
• if it is known that a capital gain can be
made by buying a share on day t and sell-
ing it for an expected profit the next day,
efficient speculation will drive up the cur-
rent price
• Similarly, no one will want to hold a stock
if it is expected to depreciate
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Formally, the model asserts that
• the price of a stock should evolve accord-
ing to the stochastic difference equation
yt+1 = yt + t+1
or, ∆yt+1 = t+1, where
• yt=price of a share of stock on day t
• yt+1=price of that share of stock on day
t+1
• ∆yt+1 = yt+1 − yt, and
• t+1=random shock on day t+1 that has
an expected value of zero
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Now consider the more general stochastic dif-
ference equation:
∆yt+1 = α0 + α1yt + t+1.
The random walk hypothesis requires
• the testable restriction: α0 = α1 = 0, and
• given the information available in period t,
the mean of t+1 be zero
• evidence that t+1 is predictable invalidates
the random walk hypothesis
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2. Reduced-Form and Structural Equa-
tions: Often it is useful to collapse a system
of difference equations into separate single-
equation models. As an example, consider a
stochastic version of the Keynesian model of
aggregate output determination:
yt = ct + it (1)
ct = αyt−1 + ct 0<α<1 (2)
it = β(ct − ct−1) + it β>0 (3)
where
• yt, ct, and it denote real GDP, consumption,
and investment in period t, respectively
• yt, ct, and it are endogenous variables
• previous period’s GDP and consumption,
yt−1 and ct−1, are called predetermined or
lagged endogenous variables
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• ct and it are zero mean random distur-
bances for consumption and investment
What do the above equations tell us?
• Equation (1) says that aggregate output
(GDP) is the sum of consumption and in-
vestment spending
• Equation (2) asserts that consumption spend-
ing is proportional to the previous period’s
GDP plus a random disturbance
• Equation (3) illustrates the accelerator prin-
ciple: Investment spending is proportional
to the change in consumption (the idea
that growth in consumption necessitates
new investment spending), and
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• ct and it are error terms that represent
the portions of consumption and invest-
ment not explained by the behavioral equa-
tions of the model
We note that
• Equation (3) is a structural equation since
it expresses the endogenous variable it as
being dependent on the current realization
of another endogenous variable, ct
• A reduced-form equation is one that ex-
presses the value of a variable in terms of
its own lags, lags of other endogenous vari-
ables, current and past values of exogenous
variables, and disturbance term
12
• As formulated, the consumption function
is already in reduced form: current con-
sumption depends only on lagged income
and current stochastic disturbance ct
• Investment is not in reduced form because
it depends on current period consumption
To derive a reduced-form equation for invest-
ment, substitute (2) into the investment equa-
tion (3) to obtain
it = β(αyt−1 + ct − ct−1) + it
= αβyt−1 − βct−1 + β ct + β it.
We note that the reduced-form equation for
investment is not unique. We can lag (2) one
period to obtain: ct−1 = αyt−2 + ct−1. Using
this expression, the reduced form investment
equation can also be written as
it = αβyt−1 − β[αyt−2 + ct−1] + β ct + it
= αβ[yt−1 − yt−2] + β[ ct − ct−1] + it (4)
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Similarly, a reduced-form for GDP can be ob-
tained by substituting (2) and (4) into (1):
yt = αyt−1 + ct + αβ[yt−1 − yt−2]
+ β[ ct − ct−1] + it
= α[1 + β]yt−1 − αβyt−2 + [1 + β] ct
+ it − β ct−1 (5)
• Equation (5) is a univariate reduced-form
equation; yt is expressed solely as a func-
tion of its own lags and disturbance terms
• A univariate model is particularly useful for
forecasting since it enables us to predict a
series solely on its own current and past
realizations
• It is possible to estimate (5) using uni-
variate time series techniques explained in
Chapters 2 through 4
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• Once we have obtained estimates of α and
β, it is straightforward to use the observed
values of y1 through yt to predict future
values yt+1, yt+2, . . . .
Chapter 5 considers estimation of multivari-
ate models when all variables are treated as
jointly endogenous. It also discusses the re-
strictions needed to recover (i.e., identify) the
structural model from the estimated reduced-
form model.
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3. Error-Correction: Forward and Spot
Prices:
• Certain commodities and financial instru-
ments can be bought and sold on the spot
market for immediate delivery or for deliv-
ery at some specified future date
• For example, suppose that the price of a
particular foreign currency on the spot mar-
ket is st dollars and that the price of the
currency for delivery one period into the
future is ft dollars
• Now, consider a speculator who purchased
forward currency at the price ft dollars per
unit
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• At the beginning of period t+1, the specu-
lator receives the currency and pays ft dol-
lars per unit received
• Since spot foreign exchange can be sold at
st+1, the speculator can earn a profit or
loss of st+1 − ft per unit transacted
The Unbiased Forward Rate (UFR) hypothesis
asserts that
• Expected profits from such speculative be-
havior should be zero
• Formally, the hypothesis posits the follow-
ing relationship between forward and spot
exchange rates:
st+1 = ft + t+1 (6)
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where t+1 has a mean value of zero from the
perspective of time period t.
• In (6), the forward rate in period t, is an
unbiased estimate of the spot rate in period
t + 1.
Thus, suppose we collected data on the two
rates and estimated the regression
st+1 = α0 + α1ft + t+1.
• If we were able to conclude that α0 =
0, α1 = 1, and that the regression resid-
uals t+1 have a mean value of zero from
the perspective time period t, the UFR hy-
pothesis could be maintained.
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We note that
• The spot and forward markets are said to
be in long-run equilibrium when t+1 = 0
• Whenever st+1 turns out to be different
from ft, some sort of adjustment must oc-
cur to restore the equilibrium in the subse-
quent period
• Consider the adjustment process
st+2 = st+1 − α[st+1 − ft] + st+2, α > 0(7)
ft+1 = ft + β[st+1 − ft] + f t+1, β > 0(8)
where st+2 and f t+1 both have a mean
value of zero.
• Equations (7) and (8) illustrate the type of
simultaneous adjustment mechanism con-
sidered in Chapter 6
19
• This dynamic model is called an error-
correction model because the movement
of the variables in any period is related
to the previous period’s gap from long-run
equilibrium
• If the spot rate st+1 turns out to be equal
to the forward rate ft, (7) and (8) state
that the spot rate and forward rate are ex-
pected to remain unchanged
• If there is a positive gap between the spot
and forward rates so that st+1 − ft > 0, (7)
and (8) lead to the prediction that the spot
rate will fall and the forward rate will rise
• Conversely, if the gap is negative, so that
st+1 − ft < 0, (7) and (8) lead to the pre-
diction that the spot rate will rise and the
forward rate will fall
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