econ 240 c

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Econ 240 CEcon 240 C

Lecture 3Lecture 3

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Part IPart I

• Modeling Economic Time Series

33

Total Returns to Standard and Total Returns to Standard and Poors 500, Monthly, 1970-2003Poors 500, Monthly, 1970-2003

0

1000

2000

3000

4000

5000

70 75 80 85 90 95 00

SPRETURN

Total Returns for the Standard and Poors 500

Source: FRED http://research.stlouisfed.org/fred/

44

Analysis (Decomposition)Analysis (Decomposition)

• Lesson one: plot the time series

55

Model One: Random WalksModel One: Random Walks

• Last time we characterized the logarithm of total returns to the Standard and Poors 500 as trend plus a random walk.

• Ln S&P 500(t) = trend + random walk = a + b*t + RW(t)

66

Trace of ln S&P 500(t) Trace of ln S&P 500(t)

4

5

6

7

8

9

0 100 200 300 400 500

TIME

LNS

P50

0

Logarithm of Total Returns to Standard & Poors 500

77

Analysis(Decomposition)Analysis(Decomposition)

• Lesson one: Plot the time series

• Lesson two: Use logarithmic transformation to linearize

88

Ln S&P 500(t) = trend + RW(t)Ln S&P 500(t) = trend + RW(t)

• Trend is an evolutionary process, i.e. depends on time explicitly, a + b*t, rather than being a stationary process, i. e. independent of time

• A random walk is also an evolutionary process, as we will see, and hence is not stationary

99


• This model of the Standard and Poors 500 is an approximation. As we will see, a random walk could wander off, upward or downward, without limit.

• Certainly we do not expect the Standard and Poors to move to zero or into negative territory. So its lower bound is zero, and its model is an approximation.

1010


• The random walk model as an approximation to economic time series– Stock Indices– Commodity Prices– Exchange Rates

1111

Model Two: White NoiseModel Two: White Noise

• Last time we saw that the difference in a random walk was white noise.

)()( tWNtRW

1212


• How good an approximation is the white noise model?

• Take first difference of ln S&P 500(t) and plot it and look at its histogram.

1313

Trace of ln S&P 500(t) – ln S&P(t-1)Trace of ln S&P 500(t) – ln S&P(t-1)

-0.3

-0.2

-0.1

0.0

0.1

0.2

70 75 80 85 90 95 00

DLNSP500

Trace of lnsp500 - lnsp500(-1)

1414

Histogram of Histogram of ln S&P 500(t) – ln S&P(t-1)ln S&P 500(t) – ln S&P(t-1)

0

20

40

60

80

100

-0.2 -0.1 0.0 0.1

Series: DLNSP500Sample 1970:02 2003:02Observations 397

Mean 0.008625Median 0.011000Maximum 0.155371Minimum -0.242533Std. Dev. 0.045661Skewness -0.614602Kurtosis 5.494033

Jarque-Bera 127.8860Probability 0.000000

1515

The First Difference of ln S&P The First Difference of ln S&P 500(t)500(t)

ln S&P 500(t)=ln S&P 500(t) - ln S&P 500(t-1) ln S&P 500(t) = a + b*t + RW(t) -

{a + b*(t-1) + RW(t-1)} ln S&P 500(t) = b + RW(t) = b + WN(t)

• Note that differencing ln S&P 500(t) where both components, trend and the random walk were evolutionary, results in two components, a constant and white noise, that are stationary.

1616

Analysis(Decomposition)Analysis(Decomposition)

• Lesson one: Plot the time series

• Lesson two: Use logarithmic transformation to linearize

• Lesson three: Use difference transformation to reduce an evolutionary process to a stationary process

1717


• Kurtosis or fat tails tend to characterize financial time series

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The Lag Operator, ZThe Lag Operator, Z

• Z x(t) = x(t-1)• Zn x(t) = x(t-n)• RW(t) – RW(t-1) = (1 – Z) RW(t) = RW(t) = WN(t)• So the difference operator, can be written in

terms of the lag operator, = (1 – Z)

1919

Model Three: Model Three: Autoregressive Time Series of Autoregressive Time Series of

Order OneOrder One• An analogy to our model of trend plus

shock for the logarithm of the Standard Poors is inertia plus shock for an economic time series such as the ratio of inventory to sales for total business

• Source: FRED http://research.stlouisfed.org/fred/

2020

Trace of Inventory to Sales, Trace of Inventory to Sales, Total Business Total Business

1.30

1.35

1.40

1.45

1.50

1.55

1.60

92 93 94 95 96 97 98 99 00 01 02 03

RATIOINVSALE

Ratio of Inventory to Sales, Monthly, 1992:01-2003:01

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AnalogyAnalogy

• Trend plus random walk:

• Ln S&P 500(t) = a + b*t + RW(t)

• where RW(t) = RW(t-1) + WN(t)

• inertia plus shock

• Ratioinvsale(t) = b*Ratioinvsale(t-1) + WN(t)

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Model Three: Autoregressive of Model Three: Autoregressive of First OrderFirst Order

• Note: RW(t) = 1*RW(t-1) + WN(t)

• where the coefficient b = 1

• Contrast ARONE(t) = b*ARONE(t-1) + WN(t)

• What would happen if b were greater than one?

2323

Using Simulation to Explore Using Simulation to Explore Time Series BehaviorTime Series Behavior

• Simulating White Noise:

• EVIEWS: new workfile, irregular, 1000 observations, GENR WN = NRND

2424

Trace of Simulated White Noise:Trace of Simulated White Noise:100 Observations100 Observations

-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100

WN

Simulated White Noise

2525

Histogram of Simulated White Histogram of Simulated White NoiseNoise

0

20

40

60

80

100

120

-3 -2 -1 0 1 2 3 4

Series: WNSample 1 1000Observations 1000

Mean 0.008260Median -0.003042Maximum 3.782479Minimum -3.267831Std. Dev. 1.005635Skewness -0.047213Kurtosis 3.020531


2626

Simulated ARONE ProcessSimulated ARONE Process

• SMPL 1 1, GENR ARONE = WN

• SMPL 2 1000

• GENR ARONE =1.1* ARONE(-1) + WN

• Smpl 1 1000

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Simulated Unstable First Order Simulated Unstable First Order Autoregressive Process Autoregressive Process

-20000

-15000

-10000

-5000

0

10 20 30 40 50 60 70 80 90 100

ARONE

First 100 Observations of ARONE = 1.1*Arone(-1) + WN

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First 10 Observations of ARONEFirst 10 Observations of ARONE

obs WN ARONE

1 -1.204627 -1.2046272 -1.728779 -3.0538693 1.478125 -1.8811314 -0.325830 -2.3950735 -0.593882 -3.2284636 0.787438 -2.7638727 0.157040 -2.8832198 -0.211357 -3.3828989 -0.722152 -4.44334010 0.775963 -4.111711

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Model Three: AutoregressiveModel Three: Autoregressive

• What if b= -1.1?

• ARONE*(t) = -1.1*ARONE*(t-1) + WN(t)

• SMPL 1 1, GENR ARONE* = WN

• SMPL 2 1000

• GENR ARONE* = -1.1*ARONE*(-1) + WN

• SMPL 1 1000

3030

Simulated Autoregressive, b=-1.1Simulated Autoregressive, b=-1.1

-400

-200

0

200

400

5 10 15 20 25 30 35 40 45 50 55 60

ARONESTAR

Simulated First Order Autoregressive Process, b = -1.1

3131

Model Three: ConclusionModel Three: Conclusion

• For Stability ( stationarity) -1<b<1

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Part IIPart II

• Forecasting: A preview of coming attractions

3333

Ratio of Inventory to SalesRatio of Inventory to Sales

• EVIEWS Model: Ratioinvsale(t) = c + AR(1)

• Ratioinvsale is a constant plus an autoregressive process of the first order

• AR(t) = b*AR(t-1) + WN(t)

• Note: Ratioinvsale(t) - c = AR(t), so

• Ratioinvsale(t) - c = b*{ Ratioinvsale(t-1) - c} + WN (t)

3434

Ratio of Inventory to SalesRatio of Inventory to Sales

• Use EVIEWS to estimate coefficients c and b.

• Forecast of Ratioinvsale at time t is based on knowledge at time t-1 and earlier (information base)

• Forecast at time t-1 of Ratioinsale at time t is our expected value of Ratioinvsale at time t

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One Period Ahead ForecastOne Period Ahead Forecast

• Et-1[Ratioinvsale(t)] is:

• Et-1[Ratioinvsale(t) - c] =

• Et-1[Ratioinvsale(t)] - c =

• Forecast - c = b*Et-1[Ratioinvsale(t-1) - c] + Et-1[WN(t)]

• Forecast = c + b*Ratioinvsale(t-1) -b*c + 0

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Dependent Variable: RATIOINVSALEMethod: Least SquaresDate: 04/08/03 Time: 13:56Sample(adjusted): 1992:02 2003:01Included observations: 132 after adjusting endpoints

Convergence achieved after 3 iterations

Variable Coefficient Std. Error t-Statistic Prob.

C 1.417293 0.030431 46.57405 0.0000AR(1) 0.954517 0.024017 39.74276 0.0000

R-squared 0.923954 Mean dependent var 1.449091

Adjusted R-squared 0.923369 S.D. dependent var 0.046879

S.E. of regression 0.012977 Akaike info criterion -5.836210

Sum squared resid 0.021893 Schwarz criterion -5.792531

Log likelihood 387.1898 F-statistic 1579.487

Durbin-Watson stat 2.674982 Prob(F-statistic) 0.000000

Inverted AR Roots .95

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How Good is This Estimated How Good is This Estimated Model?Model?

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

1.30

1.35

1.40

1.45

1.50

1.55

1.60

93 94 95 96 97 98 99 00 01 02 03

Residual Actual Fitted

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Plot of the Estimated ResidualsPlot of the Estimated Residuals

0

5

10

15

20

25

-0.050 -0.025 0.000 0.025

Series: ResidualsSample 1992:02 2003:01Observations 132

Mean -2.74E-13Median 0.000351Maximum 0.042397Minimum -0.048512Std. Dev. 0.012928Skewness 0.009594Kurtosis 4.435641


3939

Forecast for Ratio of Inventory Forecast for Ratio of Inventory to Sales for February 2003to Sales for February 2003

• E2003:01 [Ratioinvsale(2003:02)= c - b*c + b*Ratioinvsale(2003:02)

• Forecast = 1.417 - 0.954*1.417 + 0.954*1.360

• Forecast = 0.06514 + 1.29744

• Forecast = 1.36528

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How Well Do We Know This How Well Do We Know This Value of the Forecast?Value of the Forecast?

• Standard error of the regression = 0.0130

• Approximate 95% confidence interval for the one period ahead forecast = forecast +/- 2*SER

• Ratioinvsale(2003:02) = 1.36528 +/- 2*.0130

• interval for the forecast 1.34<forecast<1.39

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Trace of Inventory to Sales, Trace of Inventory to Sales, Total Business Total Business

1.30

1.35

1.40

1.45

1.50

1.55

1.60

92 93 94 95 96 97 98 99 00 01 02 03

RATIOINVSALE

Ratio of Inventory to Sales, Monthly, 1992:01-2003:01

4242

Lessons About ARIMA Lessons About ARIMA Forecasting ModelsForecasting Models

• Use the past to forecast the future

• “sophisticated” extrapolation models

• competitive extrapolation models– use the mean as a forecast for a stationary time

series, Et-1[y(t)] = mean of y(t)

– next period is the same as this period for a stationary time series and for random walks, Et-1[y(t)] = y(t-1)

– extrapolate trend for an evolutionary trended time series, Et-1[y(t)] = a + b*t = y(t-1) + b

econ 240 c

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