Takehome One

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Takehome OneTakehome One

20082008

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3 month treasury bill rate3 month treasury bill rate

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5 year Treasury5 year Treasury

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GS5 TB3MS

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

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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

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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

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Unit Root test for GS5Unit Root test for GS5

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Histogram and Stats for TermHistogram and Stats for Term

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Series: TERMSample 1953:04 2008:04Observations 661

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


Histogram and Stats for Term = GS5 - TB3ms

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Co-integrationCo-integration

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

Evolutionary Stationary

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Modeling TermModeling Term

PACFACF

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SpecificationSpecification

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

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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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Series: ResidualsSample 1953:07 2008:04Observations 658

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


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

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Within Sample ForecastingWithin Sample Forecasting

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

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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

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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

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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

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ARCHARCH

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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


Standardized Residuals from GARCH Model

Histogram of Standardized Histogram of Standardized ResidualsResiduals

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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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Alternative model #2Alternative model #2

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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



Residuals from Arch-Garch Model

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