Basic Concepts
Basic
definitions and few properties of stationary time series are given in this
section.
Definition 2.1:
Probability space: A probability space is a triplet (Ω, С, Р) where
(1). Ω is a set
of all possible results of an experiment;
(2). C is class
of subsets of Ω (called events) forming a s- algebra, i.e.
Definition 2.2:
A time series: Let (Ω, С, Р)
be a probability space let T be an index set. A real valued time
series is a real valued function X(t ,ω) defined on T x Ω such that for each
fixed t Є T, X( t, ω) is a random variable on (Ω, С, Р).
The function
X(t, ω) is written as X(ω) or X t and a time series
considered as a collection {X t : t Є T}, of random
variables [14].
Definition 2.3:
Stationary time series: The plot of a time series over a time interval
[t, t+h]
may sometimes closely resemble a plot at another interval [s, s+h]. This
implies that there is temporal homogeneity in the behavior of the series, which
is called stationary. For example, the number of personal bankruptcies may be
stationary in monthly data. This means that the time series between January to
March in one year may resemble June to August of another year. A stationary
series should have no discernible trends. For precise definition of stationary,
some concepts from probability theory, which are developed later, are needed.
An imprecise operational definition of stationary time series is as follows:
When the mean: E(Xt), the variance: Var(Xt) and all auto-covariances of
specified lags(say h): Cov(Xt, Xt+h) do not depend on t, the time at which they
are measured, we have a stationary time series.
A process whose
probability structure does not change with time is called stationary. Broadly
speaking a time series is said to be stationary, if there is no systematic
change in mean i.e. no trend and there is no systematic change in variance.
Definition 2.4:
Stationary time series: A process whose probability structure does not change
with time is called stationary. Broadly speaking a time series is said to be
stationary, if there is no systematic change in mean i.e. no trend and there is
no systematic change in variance.
Weak Stationary
or Covariance Stationary Time Series A weaker concept of stationary allows the
joint distributions to change somewhat over time, but requires that E(Xt),
Var(Xt) do not change. Also, Cov(Xt, Xt+h) is required to be a function of lag
length only, not depend on time t at which it is measured. Many stationary
series are also called covariance stationary, wide-sense stationary, or second
order stationary in the literature. For the multivariate normal joint
distribution, weak and strict stationarity are equivalent.
Definition 2.5:
Strictly stationary time series: A time series is called strictly stationary, if their
joint distribution function satisfy
Where, the
equality must hold for all possible sets of indices ti and (ti +
h) in the index set. Further the joint distribution depends only on the
distance h between the elements in the index set and not on their actual
values.
In other words Strict
stationary time series is a stronger concept where the properties are unaffected
by a change of time origin. It requires that the joint distribution function F
F[ x(t1),
x(t2), … x(tn)] = F[ x(t1+h), x(t2+h),
… x(tn+h)] … (2)
for all choices
of time points x1 to xn and for all h. If the
first two moments of strictly stationary process exist, that is they are finite
it is also covariance stationary and it satisfies the three conditions (1).
Stationary implies stable relations and the object of any theory is to obtain stable
relationship among variables.
Definition 2.6:
Weak Stationary or Covariance Stationary time series: A weaker concept of stationary
allows the joint distributions to change somewhat over time, but requires that
E(Xt), Var(Xt) do not change. Also, Cov(Xt, Xt+h) is required to be a function
of lag length only, not depend on time t at which it is measured. Many
stationary series are also called covariance stationary, wide-sense stationary,
or second order stationary in the literature. For the multivariate normal joint
distribution, weak and strict stationarity are
equivalent.
For a random
process to be weakly stationary – also called covariance stationary – there are
three requirements:
Proof: Proof
follows from definition (2.4).
In usual cases
above equation (2) is used to determine that a time series is stationary i.e.
there is no trend.
Definition 2.5: Auto-covariance function: The
covariance between {X t} and
the
function ¡ (h) is called the auto covariance function.
Definition 2.6:
The auto correlation function: The correlation between observation which are separated
by h time unit is called auto-correlation at lag h. It is given by
where μ is
mean.
Remark
2.1: For a
stationary time series the variance at time (t + h) is same as that at time t.
Thus, the auto correlation at lag h is
Remark
2.2: For h = 0, we
get, ρ (0) = 1.
For application,
attempts have been made to establish that prices of potatoes at certain
districts of Marathwada satisfy equation (1) and (8).
Theorem
2.2: If the time
series {X t: t Є T}, is weakly or covariance stationary then
The
function ¡ is called the auto covariance function of the process
{X t: t Є T} and
¡ (h),
for given h, lag h auto covariance of the time series {Xt: t Є T}.
Proof: Let rÎT any element of T. Since the
time series {Xt: t Є T} is weakly stationary, we have, for t, t + h
Є T such that t £ t + h,
Theorem
2.3: The covariance
of a real valued stationary time series is an even function of h.
i.
e. ¡ (h) = ¡(-h).
Proof: We assume that without loss of
generality, E {X t} = 0, then since the series is stationary we
get, E{X t X t + h} = ¡(h) , for all
t and t + h contained in the index set. Therefore if we set t0 =
t1 - h,
Consider first
two random samples X1, X2 … Xn and
Y1, Y2, … Yn and the quantity
Since the
second derivative with respective to a is 
(assuming a non-trivial sample of one repeated constant value), we see that
this value of a in fact minimizes (10).
Since (11) =
(10) and (10) ³ 0, putting the value of a in eqn. (11) gives us
This is known
as the cauchy – Schwartz inequality.
Taking the
square root of both sides, we have
Note that the
resulting right-hand side will be made larger if we sum those squares over all
indices from 1 to n. Thus,
Theorem
2.5: Let X t’s be independently and identically
distributed with E(X t) = μ
and var(Xt)
= σ2
then
This process is
stationary in the strict sense.
3. TESTING
PROCEDURE
3.1: Inference
concerning slope (b1): We set up null hypothesis for test statistic for
testing H0:
b1 = 0 Vs H1: b1 > 0 for a = 0.05 percent level using t distribution with degrees
of freedom is equal to n – 2 were considered.
3.2: Example
of time series: prices of
potatoes per Kg data of five districts namely Aurangabad, Parbhani, Osmanabad,
Beed and Nanded in Marathwada region were collected. The data were collected
from Socio Economic Review and District Statistical Abstract; Directorate
Economics and Statistics Government of Maharastra Bombay and Maharastra
Quarterly Bulletin of Economics and Statistics; Directorate of Economics and
Statistics Government of Maharastra, Bombay [2, 3]. Hence we have
five dimensional time series ti, i = 1, 2, 3, 4, 5 districts and
whole Marathwada region respectively.
Over the years
many scientists have analyzed rainfall, temperature, humidity, agricultural
area, production and productivity of region of Maharastra state, [1, 4,
5, 6, 7, 8, 9, 10, 11, 16]. Most of them have treated the time series for
each of the revenue districts as independent time series and tried to examine
the stability or non-stability depending upon series. Most of the times non-
stability has been concluded, and hence possibly any sort of different
treatment was possibility never thought of. In this investigation we treat the
series first and individual series. The method of testing intercept (b0 =
0) and regression coefficient (b1 = 0), Hooda R.P. [14] and
for testing correlation coefficient Bhattacharya G.K. and Johanson R.A.,
[12].
The regression
analysis tool provided in MS-Excel was used to compute b0, b1,
corresponding SE, t-values for the coefficients in regression models. Results
are reported in table 4.1B and table 4.2C. Elementary statistical analysis is
reported in table–4.1A. It is evident from the values of CV that there is no
any scatter of values around the mean indicating that all the series are having
trend.
Table 4.1B shows
that the model,
where,
1. X t are
the prices of potatoes per Kg series.
2. t
is the time in(years) variable.
3. e is
a random error term normally distributed as mean 0 and variance s 2 .
Xt is the prices of potatoes per Kg the dependent variable and
time t in (years) is the independent variable.
Values
of auto covariance computed for various values of h are given in table
4.2A. prices
of potatoes per Kg values for different districts were input as a matrix
to the software. Defining
¡(h) = cov(A,
B) were computed for various values of h. Since the time series constituted of
27 values, at least 10 values were included in the computation. The relation
between ¡(h) and h were examined using model, table-4.2C,
the testing
shows that, both the hypothesis b0 = 0 and b1 =
0 test is positive for both the district. Table-4.2C was obtained by regressing
values of ¡(h) and h, using “Data Analysis Tools” provided in MS
Excel. Table 4.2A formed the input for table 4.2C. In other wards, ¡(h)
are all non zero in two districts, in the area of jawar series of two districts
trend were found showing that X t, X t + h are
dependent in prices of potatoes per Kg series of these districts and there is a
trend in that series. Hence in these districts prices of potatoes per Kg series
X t is not stationary it is having a trend.
