11

Power 2Power 2

Econ 240CEcon 240C

22

Lab 1 RetrospectiveLab 1 Retrospective

• Exercise:– GDP_CAN = a +b*GDP_CAN(-1) + e– GDP_FRA = a +b*GDP_FRA(-1) + e

33

44

55

66

Data in ExcelData in ExcelYear 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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StackingStacking

• So for stacking, the data start with 1951

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Data in ExcelData in Excelyear GDP_CAN GDP_CAN(-1) C_CAN

1989 17758 17394 1

1990 17308 17758 1

1991 16444 17308 1

1992 16413 16444 1

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StackingStacking• 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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StackingStacking• 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

1111

1212

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OutlineOutline

• 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 ConceptsUnivariate 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

1515Conceptual time series Conceptual time series

components modelcomponents 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?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 behaviorCyclical 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 SynthesisSimulation 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 noiseSimulated 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

2020The first ten values of simulated white noise, wn(t)

Value draw = time index-0.628093683959 1-0.627803051549 20.00723255412415 31.94192735344 4-1.10119663665 50.514236967572 6-0.843129585702 7-0.0153352207678 81.25353192311 91.48589824393 10

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Trace (plot) of first 100 values of wn(t)Trace (plot) of first 100 values of wn(t)

No obvious timeDependence, i.e.Stationary, notTrended, notseasonal

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Histogram and Statistics, 1000 Obs.Histogram and Statistics, 1000 Obs.

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IndependenceIndependence• 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

2424

Independence: conceptualIndependence: 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

2525

Independence: conceptualIndependence: 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(t-u) – 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

2626

Independence: ConceptualIndependence: 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

2727

Independence: ConceptualIndependence: 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 wn, 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.

2828Theoretical Autocorrelation Theoretical Autocorrelation

Function: White NoiseFunction: White Noise

Theoretical Autocorrelation Function: White Noise

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

Lag

Rh

o

2929What use is the autocorrelation What use is the autocorrelation

function in practice?function in practice?

• Estimated Autocorrelations in EViews

3030

3131

1000 observations ofWhite Noise

3232

AnalysisAnalysis

• Breaking down the structure of an observed time series, i.e. modeling it

• Example: weekly closing price of gold, Handy & Harmon, $ per ounce

3333

PRICE OF GOLDPRICE OF GOLDDate 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

370

380

390

400

410

420

430

440

450

460

0 10 20 30 40 50 60

Week

$/o

z

3535

3636

Price of gold doesNot look like white noise

3737

What now?What now?

• How about week to week changes in the price of gold?

• In EViews: Gen dgold = gold –gold(-1)

3838

3939

4040

4141

4242

Changes in the price of goldChanges 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?

4343

The price of goldThe 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(t-2) =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 goldThe 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

4545

The price of goldThe 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)

4646

The first difference of a random walkThe first difference of a random walk

• The first difference of a random walk is white noise

4747

Random walk plus trendRandom 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

4848

The time series is too short for the constantTo be significant

4949

Simulated Random walkSimulated Random walk

• EViews, sample 1 1, gen rw = wn

• Sample 2 1000, gen rw = rw(-1) + wn

5050

Simulated random walkSimulated random walktime White noise Random walk

1 -0.628094 -0.628094

2 -0.627803 -1.255897

3 0.007233 -1.248664

4 1.941927 0.693263

5151

-40

-30

-20

-10

0

10

20

30

200 400 600 800 1000

RW WN

5252

5353

5454

Random walkRandom walk

• Is a random walk evolutionary or stationary?

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Random walkRandom 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

5656

Variance of an infinite rw(t)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) = ∞

• So the variance of an infinitely long random walk is not bounded, but infinite, and a random walk can go wandering off.

5757

Random walk modelRandom 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.

5858

QuestionQuestion• What does the autocovariance function of

an infinite random walk look like plotted against lag?

lag0

rw, rw

5959

Recall the autocorrelation functionFor a finite sample of a simulatedRandom walk decays slowly

6060

SummarySummary• 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