bee2006: statistics and econometrics -...
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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 1)BEE2006: Statistics and Econometrics
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10.1 (a)
Like cross-sectional observations, we can assume that most timeseries observations are independently distributed.
Do you Agree or Disagree?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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Consider the following two models
Returni = !0 + !1GDPi + ui
Returnt = !0 + !1GDPt + 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
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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 inperiod t # 1 ( think about some oil shock ut!1), thegovernment afraid of recession actively intervenes and shocksthe stock market with some stimulus ut .
ut = "0 + "1ut!1 + et
then we’ll have autocorrelation.
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Would it be natural to expect:
ui $ N!
0,#2"
ut $ N!
0,#2"
A lot of research in time series is devoted to the idea ofAutoregressive conditional heteroskedasticity
#2t = "0 + "1e
2t!1 + ..+ "qe
2t!q + $1#
2t!1 + ...+ $p#
2t!p
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Example of clustering:
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10.1(b)
The OLS estimator in a time series regression is unbiased underthe first three Gauss-Markov assumptions.
Do you Agree or Disagree?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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The first three assumptions:
yt = !0 + !1x1t + ....+ !kxkt + 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 t = 1, 2, 3, ..., n
THEN THE OLS IS UNBIASED
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10.1(c)
A trending variable cannot be used as the dependent variable inmultiple regression analysis.
Do you Agree or Disagree?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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Suppose your model yt = !0 + !1xt + ut looks like this
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There is obviously at time trend (upward) you should have considerthis model:
yt = !0 + !1xt + !2t + ut
Then !2 captures the changes in yt caused by xt isolating forthe time trend
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10.1(d)
Seasonality is not an issue when using annual time seriesobservations.
With annual data, each time period represents a year and isnot associated with any seasons.
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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10.2
Let gGDPt denote the annual percentage change in gross domesticproduct and let intt denote a short-term interest rate.
gGDPt = "0 + $0intt + $1intt!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 1)BEE2006: Statistics and Econometrics
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Suppose that the Federal Reserve seeks to control interest rate bythe rule
intt = %0 + %1 (gGDPt!1 # 3) + vt
%1 > 0
Corr (vt , ut) = 0 for all t
Corr (vt , intt) = 0 for all t
show thatCov (ut!1, intt) "= 0
and as a consequence
E (ut |int) "= 0
since E (ut!1|int) "= 0
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FromgGDPt = "0 + $0intt + $1intt!1 + ut
we can get
gGDPt!1 = "0 + $0intt!1 + $1intt!2 + ut!1
then
intt = %0 + %1 ("0 + $0intt!1 + $1intt!2 + ut!1 # 3) + vt
Rearranging we have that
intt = (%0 + %1"0 # 3%1)+%1$0intt!1+%1$1intt!2+%1ut!1+vt
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Now findCov (ut!1, intt) =
Cov (ut!1, (%0 + %1"0 # 3%1) + %1$0intt!1 + %1$1intt!2 + %1ut!1 + vt)
Recall that:
Cov (ut!1, intt!1) = 0Cov (ut!1, intt!2) = 0Cov (ut!t , vt) = 0
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Cov (ut!1, intt) = Cov (ut!1, %1ut!1) = %1V (ut!1)
Assume that V (ut!1) = #2 homoskedasticity
ThenCov (ut!1, intt) = %1#
2 "= 0
since %1 > 0
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10.6(a)
Consider the following General Model:
yt = "0 + $0zt + $1zt!1 + $2zt!2 + $3zt!3 + $4zt!4 + ut
Now assume that we have a specific polynomial distributionlag
$j = %0 + %1j + %2j2
where j are the quadratic lag. Eg. $2 = %0 + %12 + %222
Plug $j into the model and rewrite the model in terms ofparameter %h for h = 0, 1, 2
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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We Know that:
$0 = %0
$1 = %0 + %1 + %2
$2 = %0 + 2%1 + 4%2
$3 = %0 + 3%1 + 9%2
$4 = %0 + 4%1 + 16%2
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Rewrite the model we get
yt = "0 + %0 (x1t) + %1 (x2t) + %2 (x3t) + ut
wherex1t = 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
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10.6(b)
Explain the regression you would run to estimate %h
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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Run the OLS estimation
yt = "0 + %0 (x1t) + %1 (x2t) + %2 (x3t) + ut
we will find %̂h thereafter we can find
$̂j = %̂0 + %̂1j + %̂2j2
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10.6(c)
The Polynomial distribute lag model is a restricted version of thegeneral model. How many restriction are imposed? How would youtest these?
Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1)BEE2006: Statistics and Econometrics
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Recall that the General Model: (Unrestricted Model)
yt = "0 + $0zt + $1zt!1 + $2zt!2 + $3zt!3 + $4zt!4 + ut
has 6 variables and the Polynomial Model (restricted Model)
yt = "0 + %0x1t + %1x2t + %2x3t + ut
only has 4 variable.
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Simply run the restricted model and find the R2ur and the restricted
model to find R2r . There are hence:
Two restrictions, moving from the unrestricted to restrictedmodel
We don’t have to really concern ourselves about what therestrictions might be but we know that there are tworestrictions
Fstat =(R2
ur!R2u)/2
(1!R2ur )/(n!6) $ F2,n!6