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Transcript
4-1-2004
Lecture Two
1
I. Inertial Versus Causal Models
A causal model is based on economic theory and can be expressed in the form of
"structural" or, alternatively, as reduced form equations. An example might be a demand and
supply model for the price of gold. The factors affecting demand might include the rate of inflation
in the consumer price index, ∆I/I, and an index of world political stability, s, as well as income, y,
and price, p:
q = d - e*p + f*y + g*∆I/I - h*s + u1,
where d, e, f, g, and h are parameters and u1 is an error term. Supply is given by the aggregate
curve for all producers of their marginal cost of production, mc, which in turn depends upon the
quantity supplied, q, and an index of real wages, w:
mc = a + b*q +c*w + u2,
where a, b, and c are parameters and u2 is an error term. The quantity demanded equals the
quantity supplied at the value where price equals marginal cost:
mc = p.
Quantity and price are jointly determined by these "structural" demand and supply curves. These
equations can be solved to express these endogenous variables in terms of the exogenous
variables. For example, the expression, or reduced form, for quantity is:
q =1/(1 + e*b) [d - e*a - e*c*w + f*y + g*∆I/I - h*s + u1 - e*u2]
In order to estimate structural equations it is necessary to be sufficiently knowledgeable about
the causal factors to correctly specify the equations. It is also necessary to have access to the data
describing the behavior of the exogenous variables.
An alternative approach is to specify an inertial model, sometimes referred to in time series
analysis as a structural model, not to be confused with terminology used for structural causal
models. A simple example of an inertial model for the price of gold might be a time trend, t, and an
error term, u:
p(t) = t + u(t) .
Notice that the value of price in the previous period is:
4-1-2004
Lecture Two
2
p(t-1) = *(t-1) + u(t-1),
and the difference in price, ∆p, defined as:
∆p = p(t) - p(t-1)
is:
∆p = + u(t) - u(t-1) ,
i. e. differencing removes the trend. The simple time trend model requires much less information or
data to estimate but is less informative than the structural model. Thus there is a tradeoff. Inertial
models are often useful if the investigator is limited by data and/or time.
II. Structural(Inertial) Models of Time Series
A simple concept underlying the structural approach to time series is to conceive of the time
series as composed of components, (1) trend, (2) cyclical, (3) seasonal, and (4) irregular or residual.
A simple additive formulation would be:
time series(t) = trend(t) + cycle(t) + seasonal(t) + irregular(t) .
One could construct the time series as the sum of these components, i.e. a synthesis of the
components. Of course one needs to model each component. A simple model of the price of gold
was used in the example above where price was the sum of a linear trend plus an irregular or error
term. Much of this course will be concerned with developing alternative approaches to modeling
these components.
Often the only data the investigator has is the time series itself. From the time series, one can
try and develop representations of the components. This is called decomposition, the inverse of
synthesis. A simple example of the concept involved can be illustrated with a back of the envelope
technique using a monthly time series for the unemployment rate and a Buys-Ballot table. The latter
is simply an array of the data with month heading the columns and years heading the rows. An
average for a row is the annual average and is a measure of the two components: cycle plus trend,
having averaged out the seasonal and the irregular. An average for a column is a measure of the
seasonal component, having averaged over trend, cycle and irregular. Thus we have created
representations of some of the components, or combinations of components from the time series
itself, an example of decomposition using a Buys-Ballot table.
4-1-2004
Lecture Two
3
Civilian Unemployment Rate
Year Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
AA
10.1 10.4
10.8
10.8
9.7
82
8.6
8.9
9.0
9.3
9.4
9.6
9.8
9.8
83
10.4
10.4
10.3
10.2
10.1
10.1
9.4
9.5
9.2
8.8
8.5
8.3
9.6
84
8.0
7.8
7.8
7.7
7.4
7.2
7.5
7.5
7.3
7.4
7.2
7.3
7.5
85
7.4
7.2
7.2
7.3
7.2
7.3
7.4
7.1
7.1
7.2
7.0
7.0
7.2
86
6.7
7.2
7.1
7.2
7.2
7.2
7.0
6.9
7.0
7.0
6.9
6.7
7.0
87
6.6
6.6
6.5
6.4
6.3
6.2
6.1
6.0
5.9
6.0
5.9
5.8
6.2
88
5.8
5.7
5.6
5.5
5.6
5.4
5.4
5.6
5.4
5.3
5.4
5.3
5.5
89
5.4
5.1
5.0
5.3
5.2
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
90
5.3
5.3
5.3
5.4
5.3
5.3
5.5
5.6
5.7
5.7
5.9
6.1
5.5
91
6.3
6.5
6.8
6.6
6.8
6.8
6.7
6.8
6.7
6.8
6.9
7.2
6.7
92
7.1
7.4
7.3
7.3
7.5
7.7
7.5
7.5
7.5
7.3
7.3
7.3
7.4
93
7.1
7.0
7.0
7.0
6.9
6.9
6.8
6.7
6.7
6.7
6.5
6.4
6.8
94
6.7
6.6
6.5
6.4
6.1
6.1
6.1
6.0
5.8
5.7
5.6
5.4
6.1
Av
7.0
7.1
7.0
7.1
7.0
7.0
7.0
7.0
6.9
6.9
6.9
6.8
7.0
Source: United States Department of Commerce, Business Conditions Digest and Survey of
Current Business .
4-1-2004
Lecture Two
4
A nnual A verage of t he Civilian Unemployment Rate
10
9
8
Perc ent
7
6
5
1982
1984
1986
1988
1990
1992
1994
Y ear
Civilian Unemployment Rate
Monthly A v erage f or Thirteen Y ears : Civ ilian Unemploy ment Rate
8
7
6
5
Perc ent
4
3
2
1
0
2
4
6
8
10
12
Month
A v erage
4-1-2004
Lecture Two
5
III. Deterministic and Stochastic Time Series
A deterministic time series is a sequence of values, indexed by a linear index, in this case
time, t, where the values are not random. An example is :
y = 2*t
Deterministic Series
Time
Y
0
0
1
2
2
4
3
6
4
8
This example of a trended deterministic time series is growing or evolving and hence is called
evolutionary. It is also easily forecastable.
A. Random Variables and Stochastic Sequences
A continuous random variable, x, takes on values in some range with liklihood determined by
its density function, f(x). An example of such a density function is the uniform, with f(x) = 1 for x
taking values in the range zero to one, i.e. 0≤x≤1.The distribution function, F(x) is the integral of the
density function :
x
F(x) = 
 f(x)dx ,
0
4-1-2004
Lecture Two
6
and for the uniform distribution the distribution function is:
F(x) = x.
Density of the uniform distribution
f(x)
1
0
x
1
Distribution Function of the Uniform
1
F(x)
0
0
x
1
The expected value of a random variable is:
∞
E[x] = 
 f(x)*x dx
0
which for a variable distrbuted uniformly on the interval zero to one is one half. The variance of a
random variable is:
4-1-2004
Lecture Two
7
E[x - Ex]2,
which in the case of the uniform variable above is 1/12. .
A uniformly distributed random variable can be simulated using EViews.
File Menu: “New”
New Object Window: Workfile: “Uniform”
Workfile Range: “Undated”
Start Observations: “1”
End Observations: “10”
Workfile Menu: GENR
Equation: “x=RND”
Workfile: Select Object “x”
Workfile Menu: View: “Open Selection”
Series x Window: View: “Spreadsheet”
Series x Window: View: “Line Graph”
Series x Window: View: “Histogram and Stats”
_______________________________________________________________
The command “Spreadsheet” lists the ten independent and uniformly distributed observations, the
first five in the first row followed by the second five in the next row:
_______________________________________________________________
1
2
3
4
5
0.738304
0.052492
0.912626
0.707297
0.509690
0.866146
Modified: 1 10 // x=rnd
0.627857
0.077609
0.631123
0.224799
Ten Observations from the Uniform Distribution
_______________________________________________________
The command “Histogram and Stats” can be used to obtain descriptive statistics for these
ten observations on the uniform variable:
Histogram and statistics for 10 observations from the Uniform distribution.
4-1-2004
Lecture Two
8
6
Series: X
Sample 1 10
Observations 10
5
4
3
2
1
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.534794
0.629490
0.912626
0.052492
0.312699
-0.479597
1.652887
Jarque-Bera
Probability
1.139486
0.565671
0
0.00
0.25
0.50
0.75
1.00
Note that the sample mean of 0.535 is close to the expectation of 0.5 and that the standard
deviation of 0.3127 implies a sample variance of 0.098, not too far from the expectation of 0.083.
Note also that the minimum and maximum observations fall within the expected range.
The sample of ten observations generated from the uniform distribution is a sequence
identically distributed and independent of one another, i.e. the second observation is
independent of the first and third observations, etc.. If we index these observations by time, we
have a stochastic time series. The command “Line Graph” produces the trace of the time series:
4-1-2004
Lecture Two
9
1.0
0.8
0.6
0.4
0.2
0.0
1
2
3
4
5
6
7
8
9
10
X
Trace of the time series of ten observations, Uniform distribution
_____________________________________________________________
This time series stays within the bounds of zero and one and hence is stationary, not
evolutionary. A series of 100 observations can be generated in a similar fashion. Using the
command “Histogram and Stats”, this sample's density function can be illustrated.
Histogram and statistics, 100 observations, Uniform distribution
4-1-2004
Lecture Two
10
10
Series: X
Sample 1 100
Observations 100
8
6
4
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.469755
0.469832
0.980651
0.020844
0.258724
0.071820
1.955055
Jarque-Bera
Probability
4.635594
0.098490
2
0
0.00
0.25
0.50
0.75
1.00
________________________________________________________________
IV. Stationary Stochastic Process
Recall the generation of 100 observations from the uniform distribution. If we consider this
a sample at a point in time the order of the observations does not matter. However, if we index
these observations by time in the order they were drawn, this time index fixes the order and we
have a sequence of observations or time series. The trace of this time series follows:
4-1-2004
Lecture Two
11
1.0
0.8
0.6
0.4
0.2
0.0
10
20
30
40
50
60
70
80
90 100
X
Uniform distribution time series
______________________________________________________________
By design and construction, this time series is strictly stationary. The first fifty observations are a
sequence of identically distributed observations independent from one another. The same can be
said of the second fifty observations. So the halves of this series are conceptually
indistinguishable. This expectation is borne out in practice by comparing the descriptive
characteristics of the two samples which provide estimates of the parameters of the parent
uniform distribution. The “Sample” command in EViews can be used to select the first fifty
4-1-2004
Lecture Two
12
observations and the “Histogram and Stats” command yields the following descriptive statistics for
this sample:
8
Series: X
Sample 1 50
Observations 50
6
4
2
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.486257
0.501053
0.961211
0.020844
0.267334
-0.114270
1.852169
Jarque-Bera
Probability
2.853636
0.240072
0
0.0
0.2
0.4
0.6
0.8
1.0
Similarly, the second sample can be selected, Sample 51 100, and the Histogram and Stats
command generates the descriptive statistics for this sample:
4-1-2004
Lecture Two
13
8
Series: X
Sample 51 100
Observations 50
6
4
2
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.453253
0.417859
0.980651
0.039033
0.251435
0.269753
2.113176
Jarque-Bera
Probability
2.244840
0.325491
0
0.0
0.2
0.4
0.6
0.8
1.0
SMPL range: 51 - 100
Number of observations: 50
The difference in the sample means is 0.033. The variance of the sample mean is:
[Var x]/n, and the variance of the difference in sample means is Var(mean 2 - mean1) = Var mean2
+ Var mean1, since the samples are independent, i.e. equals 0.071467/50 + 0.06322/50 =
0.00143 + 0.00126 = 0.00269. The null hypothesis of no difference in the sample means, i.e. an
expected difference of zero can be tested by the statistic: (0.033-0)/√0.00269 = 0.033/0.0519 =
0.636, so the difference is insignificant.