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Power 2
Econ 240C
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Lab 2 Retrospective
• Exercise:
– GDP_CAN = a +b*GDP_CAN(-1) + e
– GDP_FRA = a +b*GDP_FRA(-1) + e
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Data in Excel
Year
GDP_CAN
GDP_CAN(-1
C_CAN
1950
13049
1951
13384
13049
1
1952
14036
13384
1
1953
14242
14036
1
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Stacking
• So for stacking, the data start with 1951
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Data in Excel
year
GDP_CAN GDP_CAN(-1) C_CAN
1989
17758
1990
17308
17758
1
1991
16444
17308
1
1992
16413
16444
1
17394
1
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Stacking
• So the dependent variable starts with
gdp_can(1951) and goes through
gdp_can(1992). Then the next value in the
stack is gdp_fra(1951) and the data continues
ending with gdp_fra(1992).
• The independent variable for Canada starts
with gdp_can(1950) and goes through
gdp_can(1991). Then the rest of the stack is 42
zeros
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Stacking
• The independent variable for France starts
with a stack of 42 zeros. Then the next
observation is gdp_fra(1950), the following
is gdp_fra(1951) etc. ending with
gdp_fra(1991)
• The constant stack for Canada is 42 ones
followed by 42 zeros
• The constant term for France is 42 zeros
followed by 42 ones
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Outline
• Time Series Concepts
– Inertial models
– Conceptual time series components
– Simulation and synthesis
– Simulated white noise, wn(t)
– Spreadsheet, trace, and histogram of wn(t)
– Independence of wn(t)
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Univariate Time Series Concepts
• Inertial models: Predicting the future from
own past behavior
– Example: trend models
– Other example: autoregressive moving
average (ARMA) models
– Assumption: underlying structure and forces
have not changed
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Conceptual time series
components model
• Time series = trend + seasonal + cycle +
random
• Example: linear trend model
– Y(t) = a + b*t + e(t)
• Example linear trend with seasonal
– Y(t) = a + b*t + c1*Q1(t) + c2*Q2(t) + c3*Q3(t) + e(t)
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How to model the cycle?
• We have learned how to model:
– Trend: linear and exponential
– Seasonality: dummy variables
– Error: e.g. autoregressive
• How do you model the cyclical component?
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Cyclical time series behavior
• Many economic time series vary with the
business cycle
• Model the cycle using ARMA models
• That is what the first half of 240C is all
about
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Simulation and Synthesis
• Build ARMA models from noise, white
noise, in a process called synthesis
• The idea is to start with a time series of
simple structure, and build ARMA models
by transforming white noise
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Simulated white noise
• Generate a sequence of values drawn
from a normal distribution with mean zero
and variance one, i.e. N(0, 1)
• In EViews: Gen wn = nrnd
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The first ten values of simulated white noise, wn(t)
Value
-0.628093683959
-0.627803051549
0.00723255412415
1.94192735344
-1.10119663665
0.514236967572
-0.843129585702
-0.0153352207678
1.25353192311
1.48589824393
draw = time index
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10
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Trace (plot) of first 100 values of wn(t)
No obvious time
Dependence, i.e.
Stationary, not
Trended, not
seasonal
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Histogram and Statistics, 1000 Obs.
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Independence
• We know each drawn value is from the
same distribution, i.e. N(0,1)
• We know every value drawn is
independent from all other values
• So wn(t) should be iid, independent and
identically distributed
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Independence: conceptual
• Suppose the mean series, m(t), of white
noise is zero, i.e. E wn(t) = m(t) = 0
• This is a good suppose because every
generated value has expectation zero
since it is from N(0,1)
• Then E[wn(t)*wn(t-1)] = 0, i.e. a value is
independent from the previous or lagged
value
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Independence: conceptual
• In general: cov [wn(t)*wn(t-u)], where wn(t-u)
is lagged u periods from t is defined as
cov[wn(t)*wn(t-u)] = E{[wn(t) – Ewn(t)]*[wn(tu) – Ewn(t-u)]} = E [wn(t)*wn(t-u)], since E
wn(t) = 0
• This is called the autocovarince, i.e. the
covariance of white noise with lagged values
of itself
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Independence: Conceptual
• For every value of lag except zero, the
autocovarince function of white noise is
zero by independence
• At lag zero, the autocovariance of white
noise is just its variance, equal to one
cov [wn(t)*wn(t)] = E[wn(t)*wn(t)] =1
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Independence: Conceptual
• the autocovariance function can be
standardized, i,e, made free of units or scale,
by dividing by the variance to obtain the
autocorrelation function, symbolized for wn(t) by
rwn, wn (u) = cov [wn(t)*wn(t-u)/Var wn(t)
• In general, the autocorrelation function for a
time series depends both on time, t, and lag, u.
However, for stationary time series it depends
only on lag.
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Theoretical Autocorrelation
Function: White Noise
Theoretical Autocorrelation Function: White Noise
1.2
1
Rho
0.8
0.6
0.4
0.2
0
0
1
2
3
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Lag
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7
8
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What use is the autocorrelation
function in practice?
• Estimated Autocorrelations in EViews
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1000 observations of
White Noise
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Analysis
• Breaking down the structure of an
observed time series, i.e. modeling it
• Example: weekly closing price of gold,
Handy & Harmon, $ per ounce
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PRICE OF GOLD
Date
Week
Price
4/16/04
0
$400.85
4/23/04
1
$394.50
4/30/04
2
$388.50
5/07/04
3
$380.80
5/13/04
4
$376.50
5/20/04
5
$385.30
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Weekly Closing Price of Gold, Handy & Harmon, April 16, '04-March 24, '05
460
450
440
$/oz
430
420
410
400
390
380
370
0
10
20
30
Week
40
50
60
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Price of gold does
Not look like white
noise
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What now?
• How about week to week changes in the
price of gold?
• In EViews: Gen dgold = gold –gold(-1)
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Changes in the price of gold
• If changes in the price of gold are not
significantly different from white noise, then
we have a use for our white noise model:
dgold(t) = c + wn(t)
• Ignore the constant for the moment
• What sort of time series is the price of gold?
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The price of gold
• dgold(t) = gold(t) – gold(t-1) = wn(t)
• i.e. gold(t) = gold(t-1) + wn(t)
• Lag by one: dgold(t-1) = gold(t-1) – gold(t2) =wn(t-1)
• i.e., gold (t-1) = gold(t-2) + wn (t-1), so
gold(t) = wn(t) + wn(t-1)+ gold(t-2)
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The price of gold
• Keep lagging and substituting, and
• gold(t) = wn(t) + wn(t-1) + wn(t-2) + ….
• i.e. the price of gold is the current shock,
wn(t), plus last week’s shock, wn(t-1), plus
the shock from the week before that, wn(t-2)
etc.
• These shocks are also called innovations
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The price of gold
• This time series for gold, i.e. the sum of
current and previous shocks is called a
random walk, rw(t)
• So rw(t) = wn(t) + wn(t-1) + wn(t-2) + …
• Lagging by one:
• rw(t-1) = wn(t-1) + wn(t-2) + wn(t-3) + …
• So drw(t) = rw(t) –rw(t-1) = wn(t)
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The first difference of a random walk
• The first difference of a random walk is
white noise
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Random walk plus trend
• If the price of gold is trend plus a random
walk: gold(t) = a + b*t + rw(t), it is said to
be a random walk with drift
• Lagging by one, gold(t-1) = a + b*(t-1) +
rw(t-1)
• And subtracting, dgold(t) = b + drw(t), i.e.
• dgold(t) = constant + white noise
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The time series is too
short for the constant
To be significant
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Simulated Random walk
• EViews, sample 1 1, gen rw = wn
• Sample 2 1000, gen rw = rw(-1) + wn
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Simulated random walk
time
White noise
Random walk
1
-0.628094
-0.628094
2
-0.627803
-1.255897
3
0.007233
-1.248664
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1.941927
0.693263
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10
0
-10
-20
-30
-40
200
400
RW
600
WN
800
1000
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Random walk
• Is a random walk evolutionary or
stationary?
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Random walk
• Mean function for a random walk, m(t)
• m(t) = E[rw(t)] = E[ wn(t) +wn(t-1) + …]
• m(t) = 0 + 0 + 0 ….= 0
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Variance of an infinite rw(t)
• Var[rw(t)] = E[rw(t)*rw(t)]
• Var[rw(t)] =E{[wn(t) + wn(t-1) + wn(t-2)
…]*[wn(t) + wn(t-1) + wn(t-2) ….]
• Var rw(t) = s2 + s2 + s2 + ... = ∞
• So the variance of an infinitely long random walk
is not bounded, but infinite, and a random walk
can go wandering off.
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Random walk model
• The price of gold is bounded below by
zero and is not likely to go wandering off to
infinity either, so the random walk model is
an approximation for the price of gold.
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Question
• What does the autocovariance function of
an infinite random walk look like plotted
against lag?
grw, rw
0
lag
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Recall the autocorrelation function
For a finite sample of a simulated
Random walk decays slowly
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Summary
• We are now familiar with two time series,
white noise and random walks
• We have looked at the theoretical
autocorrelation functions, or are in the
process of doing so.
• We have simulated sample of both and
looked at their empirically estimated
autocorrelation functions, benchmarks for
identification