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BEE2006: Statistics and Econometrics
Tutorial 2: Time Series - Regression Analysis and Further Issues
(Part 1)
February 1, 2013
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
10.1 (a)
Like cross-sectional observations, we can assume that most time
series observations are independently distributed.
Do you Agree or Disagree?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
Consider the following two models
Returni = β0 + β1 GDPi + ui
Returnt = β0 + β1 GDPt + ut
Returni is the stock market returns at time t of country i
Returnt is the stock market returns of country i at time t
GDPi is the GDP at time t of country i
GDPt is the GDP of country i at time t
Would it be natural to expect:
Corr (ui , us |GDP) = 0
∀i "= s
Corr (ut , us |GDP) = 0
∀t "= s
Suppose that if the stock market drastically decreased in
period t − 1 ( think about some oil shock ut−1 ), the
government afraid of recession actively intervenes and shocks
the stock market with some stimulus ut .
ut = α0 + α1 ut−1 + et
then we’ll have autocorrelation.
Would it be natural to expect:
!
"
ui ∼ N 0, σ 2
!
"
ut ∼ N 0, σ 2
A lot of research in time series is devoted to the idea of
Autoregressive conditional heteroskedasticity
2
2
2
2
σt2 = α0 + α1 et−1
+ .. + αq et−q
+ δ1 σt−1
+ ... + δp σt−p
Example of clustering:
10.1(b)
The OLS estimator in a time series regression is unbiased under
the first three Gauss-Markov assumptions.
Do you Agree or Disagree?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
The first three assumptions:
yt = β0 + β1 x1t + .... + βk xkt + ut
Assumption 1: Linear in Parameters
Assumption 2:
E (ut |X) = 0
t = 0, 1, 2, ..., n
E (ut |x1t , ...., xkt ) = E (u|xt ) = 0
Assumption 3: No perfect Collinearity
Corr (xjt , xit ) "= 1
j "= i
and
THEN THE OLS IS UNBIASED
t = 1, 2, 3, ..., n
10.1(c)
A trending variable cannot be used as the dependent variable in
multiple regression analysis.
Do you Agree or Disagree?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
Suppose your model yt = β0 + β1 xt + ut looks like this
There is obviously at time trend (upward) you should have consider
this model:
yt = β0 + β1 xt + β2 t + ut
Then β2 captures the changes in yt caused by xt isolating for
the time trend
10.1(d)
Seasonality is not an issue when using annual time series
observations.
With annual data, each time period represents a year and is
not associated with any seasons.
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
10.2
Let gGDPt denote the annual percentage change in gross domestic
product and let intt denote a short-term interest rate.
gGDPt = α0 + δ0 intt + δ1 intt−1 + ut
Assume that:
E (ut |intt , intt−1 , intt−2 , ..., int0 ) = 0
Cov (ut , intt ) = 0 for t, t − 1, t − 2, t − 3, ..., 0
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
Suppose that the Federal Reserve seeks to control interest rate by
the rule
intt = γ0 + γ1 (gGDPt−1 − 3) + vt
γ1 > 0
Corr (vt , ut ) = 0 for all t
Corr (vt , intt ) = 0 for all t
show that
Cov (ut−1 , intt ) "= 0
and as a consequence
E (ut |int) "= 0
since E (ut−1 |int) "= 0
From
gGDPt = α0 + δ0 intt + δ1 intt−1 + ut
we can get
gGDPt−1 = α0 + δ0 intt−1 + δ1 intt−2 + ut−1
then
intt = γ0 + γ1 (α0 + δ0 intt−1 + δ1 intt−2 + ut−1 − 3) + vt
Rearranging we have that
intt = (γ0 + γ1 α0 − 3γ1 )+γ1 δ0 intt−1 +γ1 δ1 intt−2 +γ1 ut−1 +vt
Now find
Cov (ut−1 , intt ) =
Cov (ut−1 , (γ0 + γ1 α0 − 3γ1 ) + γ1 δ0 intt−1 + γ1 δ1 intt−2 + γ1 ut−1 + vt )
Recall that:
Cov (ut−1 , intt−1 ) = 0
Cov (ut−1 , intt−2 ) = 0
Cov (ut−t , vt ) = 0
Cov (ut−1 , intt ) = Cov (ut−1 , γ1 ut−1 ) = γ1 V (ut−1 )
Assume that V (ut−1 ) = σ 2 homoskedasticity
Then
Cov (ut−1 , intt ) = γ1 σ 2 "= 0
since γ1 > 0
10.6(a)
Consider the following General Model:
yt = α0 + δ0 zt + δ1 zt−1 + δ2 zt−2 + δ3 zt−3 + δ4 zt−4 + ut
Now assume that we have a specific polynomial distribution
lag
δj = γ0 + γ1 j + γ2 j 2
where j are the quadratic lag. Eg. δ2 = γ0 + γ1 2 + γ2 22
Plug δj into the model and rewrite the model in terms of
parameter γh for h = 0, 1, 2
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
We Know that:
δ0 = γ0
δ1 = γ0 + γ1 + γ2
δ2 = γ0 + 2γ1 + 4γ2
δ3 = γ0 + 3γ1 + 9γ2
δ4 = γ0 + 4γ1 + 16γ2
Rewrite the model we get
yt = α0 + γ0 (x1t ) + γ1 (x2t ) + γ2 (x3t ) + ut
where
x1t = zt + zt−1 + zt−2 + zt−3 + zt−4
x2t = zt−1 + 2zt−2 + 3zt−3 + 4zt−4
x3t = zt−1 + 4zt−2 + 9zt−3 + 16zt−4
10.6(b)
Explain the regression you would run to estimate γh
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
Run the OLS estimation
yt = α0 + γ0 (x1t ) + γ1 (x2t ) + γ2 (x3t ) + ut
we will find γ̂h thereafter we can find
δ̂j = γ̂0 + γ̂1 j + γ̂2 j 2
10.6(c)
The Polynomial distribute lag model is a restricted version of the
general model. How many restriction are imposed? How would you
test these?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part
BEE2006:
1)
Statistics and Econometrics
Recall that the General Model: (Unrestricted Model)
yt = α0 + δ0 zt + δ1 zt−1 + δ2 zt−2 + δ3 zt−3 + δ4 zt−4 + ut
has 6 variables and the Polynomial Model (restricted Model)
yt = α0 + γ0 x1t + γ1 x2t + γ2 x3t + ut
only has 4 variable.
2 and the restricted
Simply run the restricted model and find the Rur
2
model to find Rr . There are hence:
Two restrictions, moving from the unrestricted to restricted
model
We don’t have to really concern ourselves about what the
restrictions might be but we know that there are two
restrictions
(R 2 −Ru2 )/2
Fstat = (1−Rur2 )/(n−6)
∼ F2,n−6
ur