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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

10.1 (a)

Like cross-sectional observations, we can assume that most timeseries observations are independently distributed.

Do you Agree or Disagree?


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

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.

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

Example of clustering:

10.1(b)

The OLS estimator in a time series regression is unbiased underthe first three Gauss-Markov assumptions.



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

10.1(c)

A trending variable cannot be used as the dependent variable inmultiple regression analysis.



Suppose your model yt = !0 + !1xt + ut looks like this

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

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.


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


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

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

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

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

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


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

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

10.6(b)

Explain the regression you would run to estimate %h


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

10.6(c)

The Polynomial distribute lag model is a restricted version of thegeneral model. How many restriction are imposed? How would youtest these?


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.

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

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