Introduction to univariate Nonstationary time series models Laura Mayoral Winter 2012, BGSE 1 Introduction • Most economic and business time series are nonsta- tionary and, therefore, the type of models that we have studied cannot (directly) be used.
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Introduction to univariate
Nonstationary time series
models
Laura MayoralWinter 2012, BGSE
1 Introduction
• Most economic and business time series are nonsta-tionary and, therefore, the type of models that wehave studied cannot (directly) be used.
The following graphs (from Wei, p. 68, 69) present two
economic time series that show strong patterns of nonsta-
tionarity. The first plot corresponds to a process whose
mean is non constant while the second displays a time
series where both the mean and the variance seem to
increase over time.
• This chapter introduces several approaches for mod-
eling nonstationary time series.
• Consequences of nonstationarity for statistical infer-
ence will also be analyzed.
2 Trends
Most economic series are trended. A process presenting
a non-constant mean might present serious problems for
estimation since multiple realizations of the time mean
dependent function are not available.
The usual approach is to consider models that after some
transformations become stationary.
Two popular cases:
• Trend stationary model,
Xt = α+ βt+ ψ(L)εt.
where ψ(L)εt is a stationary process. This process
is often called trend-stationary because if one sub-
tracts the trend component βt the result is a sta-
tionary process.
• Unit root process
Xt = Xt−1 + β + ψ (L) εt
where ψ (1) �= 0. Xt can also be written as (1− L)Xt =
β + ψ (L) εt and is said to be a unit root process
because L = 1 is a root of the autoregressive poly-
nomial.
[Some notation: (1− L) = ∆.]
The transformed process (1−L)Xt = ∆Xt = Xt−Xt−1 is stationary and describes the changes (or the
growth rate if Xt is in logs) in the series Xt.
2.1 Deterministic trend functions: Trend
Stationary models
• A natural way of modelling a trend component is by
using a sth-degree polynomial in powers of t :
τ(t) = β0 + β1t+ ...+ βsts,
so that the process {Xt} can be written as
Xt = τ (t) + ut, (1)
where ut is a covariance-stationary process.
• Polynomial trend functions are particularly useful when
the mathematical form of the true trend is unknown
since any analytic mathematical function can be ap-
proximated by a polynomial.
• Typically, linear functions (β0 + β1t) are employed
to represent the trend. The reason for considering
linear trends is that in some cases the original vari-
able presents ’proportional growth’ that can be cap-
tured by an exponential trend, that is, τ(t) = eβt.
Then, dXt/dt = βeβt = βXt. These series are
usually modelled in logarithms and then the trend
becomes linear since
log(τ(t)) = βt.
Estimation. (see Hamilton, Ch. 16). Consider the
model:
Xt = α+ βt+ ut,
where ut is a stationary process. Then, 1) the poly-
nomial function can be consistently estimated by
OLS and 2) standard t or F tests can be employed
to test whether β and/or α are different from zero.
2.2 Stochastic trend models: ARIMAmod-
els
It is possible that what is perceived as a trend is the re-
sult of the accumulation of small stochastic fluctuations.
The simplest model embodying a stochastic trend is the
random walk model. Let {Xt} be the random walk se-
quence, then
Xt = Xt−1 + εt, (2)
where {εt} is a white noise sequence. Assuming that
Xt = 0 for all t < 0 and that X0 is a fixed finite initial
condition then, by back-substitution,
Xt = X0 +t∑
i=1
εi.
and E (Xt) = X0 and var(Xt) = tσ2.∗ This process
has no trend.
To introduce a trend component it is only needed to
include a constant in (2) . The random walk with drift
model is
Xt = β +Xt−1 + εt (3)
and by back-substitution
Xt = β+(β+Xt−2+εt−1)+εt = ... = X0+βt+t∑
i=1
εi.
∗Notice that if the starting point is in the indefinite past rather thanat t = 0, then the mean and the variance are undefined.
More general models can be found by allowing the sto-
chastic component in (3) to be a stationary sequence
(1− L)Xt = β + ut,
ut = ψ (L) εt,
where ψ (1) �= 0.† If ut admits an ARMA(p,q) repre-
sentation, such that φ (L)ut = θ (L) εt, then Xt is
an ARIMA (Autoregressive Integrated Moving Average)
process
φp (L) (1− L)dXt = β + θq (L) εt, (4)
where {εt} is a white noise process with variance σ2 <
∞, φp (L) = 1−φ1L−...−φpLp and θ (L) = 1+θ1L+
... + θqLq are the autoregressive and moving average
polynomials, respectively, sharing no common factors and
with all their roots outside the unit circle.†This assumption is necessary to guarantee that the MA componentdoes not contain a unit root that would cancel out with the unitroot of the AR polynomial, in which case yt would be a stationaryprocess.
In this case d = 1 but, more generally, d will be a posi-
tive integer number and represents the number of times
Xt must be differenced to achieve a stationary transfor-
mation. Typically, d ∈ {0, 1, 2}. The case d = 0
corresponds to the ARMA case, studied in Chapter 2.
The term β is a deterministic component and plays differ-
ent roles for different values of d. If d = 0, β represents
a constant term such that the mean of {Xt} is given by
µ = β/(1−φ1− ...−φp). If d = 1, β is the coefficient
associated with a linear trend and if d = 2, Xt is the
coefficient associated to a quadratic term, t2.
• If the variable is in logs, a unit root implies that the