11

Takehome OneTakehome One

20082008

22

3 month treasury bill rate3 month treasury bill rate

33

5 year Treasury5 year Treasury

44

0

4

8

12

16

20

55 60 65 70 75 80 85 90 95 00 05

GS5 TB3MS

3 month bill and 5 year treasury: April 1953-April 2008

55

66

A measure of the term structureA measure of the term structure

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TERM

term = GS5 - TB3MS

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GS5 TB3MS TERM

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1. You should try this so that you know at least one way of obtaining time series from FRED. If you have difficulty, an Excel file called Takeone, is available on the class page.2. Generate a time series called term that is the difference between GS5 and TB3MS.3. Is term stationary, i.e. are GS5 and TB3ms co-integrated?4. Is term normally distributed?5. Estimate your best autoregressive model for term.6. Estimate your best ARMA model for term through April 2007 and see how well a forecast for this model fits the next 12 months.7. Re-estimate your best model for term through April 2008 and forecast for the remaining months of 2008.

Questions: Takehome OneQuestions: Takehome One

1010

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2 4 6 8 10 12 14 16

Series: GS5Sample 1953:04 2008:04Observations 661

Mean 6.240393Median 5.850000Maximum 15.93000Minimum 1.850000Std. Dev. 2.756402Skewness 0.978683Kurtosis 3.850276

Jarque-Bera 125.4318Probability 0.000000

GS5: Rate for Five Year Treasury

Histogram and Stats for Five Histogram and Stats for Five YearYear

1111

1212

Unit Root test for GS5Unit Root test for GS5

1313

Histogram and Stats for TermHistogram and Stats for Term

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-2 -1 0 1 2 3 4

Series: TERMSample 1953:04 2008:04Observations 661

Mean 1.146051Median 1.130000Maximum 4.330000Minimum -2.250000Std. Dev. 0.957974Skewness -0.045212Kurtosis 3.247015

Jarque-Bera 1.905692Probability 0.385642

Histogram and Stats for Term = GS5 - TB3ms

1414

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Co-integrationCo-integration

1*TS5 – 1*TB3MS = Term1*TS5 – 1*TB3MS = Term

Evolutionary Stationary

1717

Modeling TermModeling Term

PACFACF

1818

SpecificationSpecification

PACF(u) AR(p)PACF(u) AR(p)ACF(u) MA(q) ACF(u) MA(q)

1919

Best AR ModelBest AR ModelAr(1) ar(2) ar(3) ser = 0.307Ar(1) ar(2) ar(3) ser = 0.307Ar(1) ar(2) ar(3) ar(4) ser = 0.305Ar(1) ar(2) ar(3) ar(4) ser = 0.305Ar(1) ar(2) ar(3) ar(4) ar(5) ser = 0.3048Ar(1) ar(2) ar(3) ar(4) ar(5) ser = 0.3048Ar(1) ar(2) ar(3) ar(4) ar(6) ser = 0.3045Ar(1) ar(2) ar(3) ar(4) ar(6) ser = 0.3045

2020

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SpecificationSpecificationAr(1) ar(2) : look at residualsAr(1) ar(2) : look at residualsAr(1) ar(2) ar(3) : look at residualsAr(1) ar(2) ar(3) : look at residualsAr(1) ar(2) ar(3) ma(3) : look at Ar(1) ar(2) ar(3) ma(3) : look at

residualsresidualsAr(1) ar(2) ar(3) ma(3) ma(9) : look at Ar(1) ar(2) ar(3) ma(3) ma(9) : look at

residualsresidualsADD MA(15)ADD MA(15)ADD MA(20)ADD MA(20)ADD MA(21), ser = 0.295ADD MA(21), ser = 0.295

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ValidationValidationCorrelogram of residualsCorrelogram of residualsActual, fitted & residual graphActual, fitted & residual graphSerial correlation testSerial correlation testHistogram of residualsHistogram of residuals

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-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Series: ResidualsSample 1953:07 2008:04Observations 658

Mean -3.44E-05Median -0.014282Maximum 1.957743Minimum -1.938455Std. Dev. 0.292944Skewness -0.307950Kurtosis 11.78426

Jarque-Bera 2125.959Probability 0.000000

ar(1) ar(2) ar(3) ma(3) ma(9) ma(15) ma(20) ma(21)

3232

Within Sample ForecastingWithin Sample Forecasting

Re-estimate model from 1953:04 -Re-estimate model from 1953:04 -2007:042007:04

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07:05 07:07 07:09 07:11 08:01 08:03

TERMF ± 2 S.E.

Forecast: TERMFActual: TERMSample: 2007:05 2008:04Include observations: 12

Root Mean Squared Error 0.301040Mean Absolute Error 0.208092Mean Abs. Percent Error 60.36766Theil Inequality Coefficient 0.261678 Bias Proportion 0.148157 Variance Proportion 0.158776 Covariance Proportion 0.693066

In sample forecast: 2007:04-In sample forecast: 2007:04-2008:042008:04

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Sample: 2005:01 – 2008:04

Quick menu: show

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In sample forecastIn sample forecast

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TERMFORECAST

FORECAST+2*SEFFORECAST-2*SEF

In sample forecast

3737

Out of sample forecastOut of sample forecastProcs: expand 1953:04 – 2008:12Procs: expand 1953:04 – 2008:12Sample 1953:04 – 2008:12Sample 1953:04 – 2008:12

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Out of Sample ForecastOut of Sample Forecast

0

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08:05 08:06 08:07 08:08 08:09 08:10 08:11 08:12

TERMF ± 2 S.E.

Forecast 2008:05-2008:12

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Out of Sample ForecastOut of Sample Forecast

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TERMFORECAST

FORECAST+2*SEFFORECAST-2*SEF

Out of sample forecast 2008:04-2008:12

4141

ARCHARCH

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Series: ResidualsSample 1953:07 2008:04Observations 658

Mean -3.44E-05Median -0.014282Maximum 1.957743Minimum -1.938455Std. Dev. 0.292944Skewness -0.307950Kurtosis 11.78426

Jarque-Bera 2125.959Probability 0.000000

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TERM RESQ

ARCH: Noisy Residuals when term goes negative

ARCH: when Inverted Term ARCH: when Inverted Term StructureStructure

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5 yr: 3.233 m: 1.86Term; 1.37

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Estimate ARCH/GARCHEstimate ARCH/GARCH

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DiagnosticsDiagnosticsCorrelogram of standardized Correlogram of standardized

residuals residuals Actual, fitted, residual graphActual, fitted, residual graphcorrelogram of standardized correlogram of standardized

residuals squaredresiduals squaredLM ARCH testLM ARCH test

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Arch LM TestArch LM Test

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Series: Standardized ResidualsSample 1953:07 2008:04Observations 658

Mean 0.027893Median 0.028664Maximum 3.760420Minimum -4.166763Std. Dev. 1.000265Skewness 0.016000Kurtosis 4.028786

Jarque-Bera 29.04587Probability 0.000000

Standardized Residuals from GARCH Model

Histogram of Standardized Histogram of Standardized ResidualsResiduals

5151

Estimate of Conditional Variance Estimate of Conditional Variance hh

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TERM GARCH01

Estimate of Conditional variance h

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Estimate of a Simpler Model with Estimate of a Simpler Model with ARCHARCH

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Residual Actual Fitted

Ordinary Residuals from Ar(1) ar(2) ar(3) ar(4) ARCH model

Ordinary residuals from ARFOUR, Ordinary residuals from ARFOUR, ARCHARCH

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AppendixAppendix

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Alternative model #1Alternative model #1

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Residuals from modelResiduals from model

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Series: ResidualsSample 1953:11 2008:04Observations 654

Mean 0.019485Median 0.007140Maximum 2.185045Minimum -2.026377Std. Dev. 0.294567Skewness -0.145333Kurtosis 13.77294

Jarque-Bera 3164.837Probability 0.000000

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Alternative model #2Alternative model #2

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Series: ResidualsSample 1953:06 2008:04Observations 659

Mean -9.24E-05Median -0.010673Maximum 2.113270Minimum -1.736855Std. Dev. 0.294642Skewness -0.025338Kurtosis 11.88345

Jarque-Bera 2166.966Probability 0.000000

Residulas from (2,1,q) model

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Autoregressive Conditional Autoregressive Conditional HeteroskedasticityHeteroskedasticity

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RESSQ

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TERM

Residuals squared from (2,1,q) model

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Series: Standardized ResidualsSample 1953:07 2008:04Observations 658

Mean 0.041163Median 0.028847Maximum 3.840148Minimum -4.563053Std. Dev. 0.999768Skewness -0.040181Kurtosis 4.379016

Jarque-Bera 52.31493Probability 0.000000

Residuals from Arch-Garch Model