1 Econ 240 C Lecture 3. 2 3 4 5 6 1 White noise inputoutput 1/(1 – z) White noise input output...

transcript

Econ 240 CEcon 240 C

Lecture 3Lecture 3

White noiseWhite noise

inputoutput

1/(1 – z)

White noise

inputoutput

Random walk

SynthesisSynthesis

1/(1 – bz)

White noise

inputoutput

First order autoregressive

Simulated Random walkSimulated Random walk

• Eviews, sample 1 1000, gen wn = nrnd

• EViews, sample 1 1, gen rw = wn

• Sample 2 1000, gen rw = rw(-1) + wn

200 400 600 800 1000

Simulated First Order Simulated First Order Autoregressive Process Autoregressive Process

• Eviews, sample 1 1000, gen wn = nrnd

• EViews, sample 1 1, gen arone = wn

• Sample 2 1000, gen arone = b*arone(-1) + wn

SystematicsSystematics

• b =1, random walk

• b = 0.9

• b = 0.5

• b = 0.1

• b = 0, white noise

200 400 600 800 1000

arone = 0.9*arone(-1) + wn

Arone(t) = 0.9*arone(t-1) = wn(t)Arone(t) = 0.9*arone(t-1) = wn(t)

Arone(t) = 0.5*arone(t-1) + wn(t)Arone(t) = 0.5*arone(t-1) + wn(t)

200 400 600 800 1000

arone = 0.5*arone(-1) + wn

Arone(t) = 0.1*arone(t-1) + wn(t)Arone(t) = 0.1*arone(t-1) + wn(t)

200 400 600 800 1000

arone = 0.1*aronne(-1) + wn

Moving AveragesMoving Averages

BloombergBloomberg

Time Series ConceptsTime Series Concepts

• Analysis and Synthesis

AnalysisAnalysis

• Model a real time seies in terms of its components

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

70 75 80 85 90 95 00

SPRETURN

Total Returns for the Standard and Poors 500

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

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

0 100 200 300 400 500

Logarithm of Total Returns to Standard & Poors 500

ModelModel

• Ln S&P500(t) = a + b*t + e(t)

• time series = linear trend + error

Dependent Variable: LNSP500Method: Least SquaresSample(adjusted): 1970:01 2003:02Included observations: 398 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C 4.049837 0.022383 180.9370 0.0000TIME 0.010867 9.76E-05 111.3580 0.0000

R-squared 0.969054 Mean dependent var 6.207030 Adjusted R-squared 0.968976 S.D. dependent var 1.269965 S.E. of regression 0.223686 Akaike info criterion -0.152131 Sum squared resid 19.81410 Schwarz criterion -0.132098 Log likelihood 32.27404 F-statistic 12400.61Durbin-Watson stat 0.041769 Prob(F-statistic) 0.000000

70 75 80 85 90 95 00

-0.50 -0.25 0.00 0.25 0.50

Series: ResidualsSample 1970:01 2003:02Observations 398

Mean 2.59E-15Median -0.036203Maximum 0.478857Minimum -0.494261Std. Dev. 0.223405Skewness 0.419110Kurtosis 2.377547

Jarque-Bera 18.07682Probability 0.000119

Time Series Components ModelTime Series Components Model

• Time series = trend + cycle + seasonal + error

• two components, trend and seasonal, are time dependent and are called non-stationary

SynthesisSynthesis

• The Box-Jenkins approach is to start with the simplest building block to a time series, white noise and build from there, or synthesize.

• Non-stationary components such as trend and seasonal are removed by differencing

First DifferenceFirst Difference

• Lnsp500(t) - lnsp500(t-1) = dlnsp500(t)

70 75 80 85 90 95 00

DLNSP500

-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

75 80 85 90 95 00

LNDNSROTM

Log of Rotterdam Inport Price: Dark Northern Spring

75 80 85 90 95 00

DLNDNSROTM

First Difference in Log of Rotterdam Import Price: Dark Northern Spring

-0.2 -0.1 0.0 0.1 0.2 0.3

Series: DLNDNSROTMSample 1971:08 2001:12Observations 365

Mean 0.002290Median 0.000000Maximum 0.277165Minimum -0.244453Std. Dev. 0.052324Skewness 0.141729Kurtosis 7.454779

Histogram: First Difference of Log of Rotterdam Import Price, DNS

Time seriesTime series

• A sequence of values indexed by time

Stationary time seriesStationary time series

• A sequence of values indexed by time where, for example, the first half of the time series is indistinguishable from the last half

Stochastic Stationary Time Stochastic Stationary Time SeriesSeries

• A sequence of random values, indexed by time, where the time series is not time dependent

Summary of ConceptsSummary of Concepts

• Analysis and Synthesis

• Stationary and Evolutionary

• Deterministic and Stochastic

• Time Series Components Model

White Noise SynthesisWhite Noise Synthesis

• Eviews: New Workfile– undated 1 1000

• Genr wn = nrnd

• 1000 observations N(0,1)

• Index them by time in the order they were drawn from the random number generator

200 400 600 800 1000

10 20 30 40 50 60 70 80 90 100

-3 -2 -1 0 1 2 3

Series: WNSample 1 1000Observations 1000

SynthesisSynthesis• Random Walk

• RW(t) -RW(t-1) = WN(t) = dRW(t)

• or RW(t) = RW (t-1) + WN(t)

• lag by one: RW(t-1) = RW(t-2) + WN(t-1)

• substitute: RW(t) = RW(t-2) + WN(t) + WN(t-1)

• continue with lagging and substitutingRW(t) = WN(t) + WN (t-1) + WN (t-2) + ...

Part IPart I

• Modeling Economic Time Series

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

70 75 80 85 90 95 00

SPRETURN

Total Returns for the Standard and Poors 500

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

Analysis (Decomposition)Analysis (Decomposition)

• Lesson one: plot the time series

Model One: Random WalksModel One: Random Walks

• we can characterize 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)

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

0 100 200 300 400 500

Logarithm of Total Returns to Standard & Poors 500

Analysis(Decomposition)Analysis(Decomposition)

• Lesson one: Plot the time series

• Lesson two: Use logarithmic transformation to linearize

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

• 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.

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

Model Two: White NoiseModel Two: White Noise

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

)()( tWNtRW

• 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.

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

70 75 80 85 90 95 00

DLNSP500

Trace of lnsp500 - lnsp500(-1)

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

-0.2 -0.1 0.0 0.1

Series: DLNSP500Sample 1970:02 2003:02Observations 397

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.

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

• Kurtosis or fat tails tend to characterize financial time series

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)

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/

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

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

RATIOINVSALE

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

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)

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?

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

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

10 20 30 40 50 60 70 80 90 100

Simulated White Noise

Histogram of Simulated White Histogram of Simulated White NoiseNoise

-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

Simulated ARONE ProcessSimulated ARONE Process

• SMPL 1 1, GENR ARONE = WN

• SMPL 2 1000

• GENR ARONE =1.1* ARONE(-1) + WN

• Smpl 1 1000

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

-20000

-15000

-10000

10 20 30 40 50 60 70 80 90 100

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

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

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

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

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

ARONESTAR

Simulated First Order Autoregressive Process, b = -1.1

Model Three: ConclusionModel Three: Conclusion

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

Part IIPart II

• Forecasting: A preview of coming attractions

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)

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 Ratioinvsale at time t is our expected value of Ratioinvsale at time t

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

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

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

93 94 95 96 97 98 99 00 01 02 03

Residual Actual Fitted

Plot of the Estimated ResidualsPlot of the Estimated Residuals

-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

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

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

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

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

RATIOINVSALE

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

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

1 Econ 240 C Lecture 3. 2 3 4 5 6 1 White noise inputoutput 1/(1 – z) White noise input output...

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