Econometrics Regression Analysis with Time Series Data ...docentes.fe.unl.pt/~azevedoj/Web...

Examples

EconometricsRegression Analysis with Time Series Data:

Examples

Joao Valle e Azevedo

Faculdade de EconomiaUniversidade Nova de Lisboa

Spring Semester

Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 1 / 15

Examples

Time Series Analysis

Using simply OLS

Would need to assume that TS.1 through TS.6 holdOr TS.1’ through TS.5’ hold...Unlikely

Inflationt = β0 + β1Unemployment + ut


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Testing for Absence of AR(1) Serial Correlation in theerrors with Strict Exogeneity

Regress residuals of previous equation on past residualsEvidence of Serial Correlation!

Evidence of Serial Correlation!


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Testing for Absence of AR(1) Serial Correlation in theerrors without Strict Exogeneity

Regress residuals of previous equation on past residuals and regressorsStill evidence of Serial Correlation! F test is for the null thatcoefficient associated with lagged residual is zero


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Alternative Specification - Augmented Phillips Curve

Inflationt − Inflationt−1 = α0 + α1(Unemploymentt − NaturalRate∗) + ut


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Regress residuals of previous equation on lagged residuals andregressors


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Cochrane - Orcutt (Example)

Assume that TS.1 through TS.4 holdBut TS.5 fails and have AR(1) in the error term

Inflationt = β0 + β1Unemploymentt + ut


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Step 1 - Estimate ρ

Will transform equation to correct for serial correlationyt = β0 + β1xt + ut into:

yt − ρyt−1 = (1− ρ)β0 + β1(xt − ρxt−1) + et

for t ≥ 2, but need to estimate ρ FGLS

Regress residuals of previous equation on past residuals


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Step 2 Transform variables with estimated ρ and applyOLS to transformed equation, t ≥ 2


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Comparison of OLS and Cochrane-Orcutt

5.51 = (1− Est.ρ)Est.β0 so Est.β0 = 12.9


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What if Strict Exogeneity (TS.3) fails?

Want to use Robust Standard Errors and assume only TS.1’, TS.2’ andTS.3’ hold

Estimate the model with OLS to get residuals ut , the standarddeviation of the regression, σ, and the ”usual” standard errors”se(β1”)

Run the auxiliary regression of xt1 on xt2, ..., xtk and get the residuals,rt

Form at = rt ut

Choose a g - typically y the integer part of n1/4

Compute

ν =n∑

t=1

a2t + 2

g∑h=1

[1− h/(g + 1)]

( n∑t=h+1

at at−h

)Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 11 / 15

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What if Strict Exogeneity (TS.3) fails? (Cont.)

Then,

Robust se(β1) = [”se(β1)”/σ]2√ν

I and similarly for any βj


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What if Strict Exogeneity (TS.3) fails? (Cont.)

With Eviews, can choose Newey-West HAC Robust standard Errors

In this case, no big difference between the (wrong!!) OLS standarderrors and the Robust standard errors

Inflationt = β0 + β1Unemploymentt + ut

Dependent Variable : INF Observations: 49


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Still, Unemployment and Inflation may contain a UnitRoot, Weak dependence and Stationarity fail

So, take first-differences to both unemployment and Inflation. Can alsoget rid of serial correlation. Let’s hope TS.1’ through TS.5’ now hold inthe model:

∆Inflationt = β0 + β1∆Unemployment + ut


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Regress residuals of previous equation on past residuals and regressors

No evidence of Serial Correlation! F test is for the null thatcoefficient associated with lagged residual is zero


Econometrics Regression Analysis with Time Series Data ...docentes.fe.unl.pt/~azevedoj/Web...

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