Examples
Econometrics
Regression Analysis with Time Series Data:
Examples
João Valle e Azevedo
Faculdade de Economia
Universidade Nova de Lisboa
Spring Semester
João 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 hold
Or TS.1’ through TS.5’ hold...Unlikely
Inflationt = β0 + β1 Unemployment + ut
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
2 / 15
Examples
Time Series Analysis
Testing for Absence of AR(1) Serial Correlation in the
errors with Strict Exogeneity
Regress residuals of previous equation on past residuals
Evidence of Serial Correlation!
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
3 / 15
Examples
Time Series Analysis
Testing for Absence of AR(1) Serial Correlation in the
errors without Strict Exogeneity
Regress residuals of previous equation on past residuals and regressors
Still evidence of Serial Correlation! F test is for the null that
coefficient associated with lagged residual is zero
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
4 / 15
Examples
Time Series Analysis
Alternative Specification - Augmented Phillips Curve
Inflationt − Inflationt−1 = α0 + α1 (Unemploymentt − NaturalRate ∗ ) + ut
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
5 / 15
Examples
Time Series Analysis
Testing for Absence of AR(2) Serial Correlation in the
errors without Strict Exogeneity
Regress residuals of previous equation on lagged residuals and
regressors
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
6 / 15
Examples
Time Series Analysis
Cochrane - Orcutt (Example)
Assume that TS.1 through TS.4 hold
But TS.5 fails and have AR(1) in the error term
Inflationt = β0 + β1 Unemploymentt + ut
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
7 / 15
Examples
Time Series Analysis
Step 1 - Estimate ρ
Will transform equation to correct for serial correlation
yt = β0 + β1 xt + 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
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
8 / 15
Examples
Time Series Analysis
Step 2 Transform variables with estimated ρ and apply
OLS to transformed equation, t ≥ 2
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
9 / 15
Examples
Time Series Analysis
Comparison of OLS and Cochrane-Orcutt
5.51 = (1 − Est.ρ)Est.β0 so Est.β0 = 12.9
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
10 / 15
Examples
Time Series Analysis
What if Strict Exogeneity (TS.3) fails?
Want to use Robust Standard Errors and assume only TS.1’, TS.2’ and
TS.3’ hold
Estimate the model with OLS to get residuals ût , the standard
deviation of the regression, σ̂, and the ”usual” standard errors
”se(β̂1 ”)
Run the auxiliary regression of xt1 on xt2 , ..., xtk and get the residuals,
r̂t
Form ât = r̂t ût
Choose a g - typically y the integer part of n1/4
Compute
ν̂ =
n
X
t=1
João Valle e Azevedo (FEUNL)
at2
X
g
n
X
+2
[1 − h/(g + 1)]
ât ât−h
h=1
t=h+1
Econometrics
Lisbon, May 2011
11 / 15
Examples
Time Series Analysis
What if Strict Exogeneity (TS.3) fails? (Cont.)
Then,
√
Robust se(β̂1 ) = [”se(β̂1 )”/σ̂]2 ν̂
I
and similarly for any β̂j
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
12 / 15
Examples
Time Series Analysis
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 standard
errors and the Robust standard errors
Inflationt = β0 + β1 Unemploymentt + ut
Dependent Variable : INF Observations: 49
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
13 / 15
Examples
Time Series Analysis
Still, Unemployment and Inflation may contain a Unit
Root, Weak dependence and Stationarity fail
So, take first-differences to both unemployment and Inflation. Can also
get rid of serial correlation. Let’s hope TS.1’ through TS.5’ now hold in
the model:
∆Inflationt = β0 + β1 ∆Unemployment + ut
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
14 / 15
Examples
Time Series Analysis
Testing for Absence of AR(1) Serial Correlation in the
errors without Strict Exogeneity
Regress residuals of previous equation on past residuals and regressors
No evidence of Serial Correlation! F test is for the null that
coefficient associated with lagged residual is zero
João Valle e Azevedo (FEUNL)
Econometrics
Lisbon, May 2011
15 / 15