BEE2006 Statistics and Econometrics

BEE2006 Statistics and Econometrics

Tutorial 3: Further Issues in using OLS with Time series data

February 2013

Question 11.1

Let {xt:t = 1,2, ....} be a covariance stationary process define γh=Cov(xt,xt+h) for h≥0

Show that

Corr(xt,xt+h) = γh

γ₀

Main Points:

A stochastic process {xt:t = 1,2, ....} is covariance stationary if and only if

1 E(xt) =µfor allt = 1,2, ...

2 Var(xt) =σ² for allt= 1,2, ...

3 γh=Cov(xt,xt+h) for h≥0 andt = 1,2,3, ...

theCov(xt,xt+h) is independent oft

Why Stationary is important:

1 Simplifies the central limit theorem

2 If we want to study the relationship between yt andxt, we want to make “say something” that will be robust across time

Recall that:

Corr(xt,xt+h) = Cov(xt,xt+h)

!Var(xt)Var(xt+h)

by definition Cov(xt,xt+h) =γh

Var(xt) =Cov(xt,xt) =γ₀ Var(xt+h) =Cov(xt+h,xt+h) =γ₀

Therefore:

Corr(xt,xt+h) = √_Var^Cov(xt,xt+h)

(xt)Var(xt+h)

= √^γh γ₀γ₀

= ^γh

γ0

Question 11.2 (i)

Let {et :t =−1,0,1,2, ...} be a sequence of independent, identically distributed random variables with mean zero and variance one.

Define a stochastic process by

xt =et−1

2et−1+ 1

2et−2 for t = 1,2, ...

Find E(xt) and Var(xt)

From the description of et, we have thatE(et) = 0:

E(xt) = E(et)−¹2E(et−1) +¹₂E(et−2)

= 0 + 0 + 0

= 0

Most importantly we have thatE(xt) is independent of time

From the description of et, Var(et) = 1

Cov(et,es) = 0 for allt $=s since by the iid assumptions Var(xt) = Var(et)−¹₄Var(et−1) + ¹₄Var(et−2)

= 1 +¹₄ +¹₄

= ³₂

Most importantly we have thatVar(xt) is independent of time

Question 11.2 (ii-iii)

Show that Corr(xt,xt+1) =−1/2 andCorr(xt,xt+2) = 1/3 Find the general expression for Corr(xt,xt+h) for h>2

Firstly

Corr(xt,xt+1) = Cov(xt,xt+1)

!Var(xt)Var(xt+1)

Var(xt) =Var(xt+1) = ³₂ Cov(xt,xt+1) =Cov

"

et−1

2et−1+1

2et−2,et+1− 1 2et+1

2et−1

#

Cov(xt,xt+1) =Cov

"

et,−1 2et

# +Cov

"

−1 2et−1,1

2et−1

#

Cov(xt,xt+1) =−1

2(1)−1

4(1) =−3 4

Therefore

Corr(xt,xt+1) = Cov(xt,xt+1)

!Var(xt)Var(xt+1) = −³₄

$3 2

%₃

2

&

=−1 2

Again

Corr(xt,xt+1) = Cov(xt,xt+2)

!Var(xt)Var(xt+2)

Var(xt) =Var(xt+2) = ³₂ Cov(xt,xt+2) =Cov

"

et−1

2et−1+1

2et−2,et+2− 1

2et+1+1 2et

#

Cov(xt,xt+2) =Cov

"

et,1 2et

#

Cov(xt,xt+2) = 1 2(1)

Therefore

Corr(xt,xt+2) = Cov(xt,xt+2)

!Var(xt)Var(xt+2) =

1 2

$3 2

%₃

2

&

= 1 3

In General

Corr(xt,xt+h) = Cov(xt,xt+h)

!Var(xt)Var(xt+h) = 0 where h>2

Var(xt) =Var(xt+h) = ³₂ Cov(xt,xt+h) =Cov

"

et−1

2et−1+1

2et−2,et+h−1

2et+h−1+1 2et+h−2

#

Cov(xt,xt+h) = 0

Question 11.2 (iv)

Is {xt}an asymptotically uncorrelated process?

Notice that:

Corr(xt,xt+1)$= 0 Corr(xt,xt+2)$= 0 Corr(xt,xt+3) = 0 ...

Corr(xt,xt+h) = 0 for allh >2

So it’s obvious that Corr(xt,xt+s)→0 ass → ∞. Therefore {xt} is an asymptotically uncorrelated process.

The{xt} process is weakly dependent

Question 11.3 (i)

Suppose that a time series process {yt} is generated by yt =z+et ∀t= 1,2, ....

where {et} is aiid sequence with mean zero and variance σe². The random variablez does not change over time; it has a mean of zero and variance of σz². Assume that eachet is uncorrelated with z

Find the expected value and variance of yt.

Quick recap

E(et) = 0, Var(et) =σe² andCov(et,es) = 0 for all t$=s E(z) = 0 and Var(z) =σz²

Corr(et,z) = 0 for allt = 1,2,3, ....

Therefore

E(yt) =E(z) +E(et) = 0 furthermore

Var(yt) =Var(z) +Var(et) + 2Cov(et,z) =σe²+σ²z

Question 11.3 (ii)

Is {yt}covariance stationary?

Recall that this requires

E(yt) , Var(yt) does not depend ont

Cov(yt,yt+h) forh≥1, depends onh and not ont

Now we only need to find Cov(yt,yt+h) for h≥1 Whenh = 1 then

Cov(yt,yt+1) =Cov(z+et,z+et+1) =Cov(z,z) =σz²

Whenh = 2 then

Cov(yt,yt+2) =Cov(z+et,z+et+2) =Cov(z,z) =σz²

Whenh ≥1 then

Cov(yt,yt+h) =Cov(z+et,z+et+h) =Cov(z,z) =σ²z

Therefore {yt} is covariance stationary

Question 11.3 (iii)

Show that

Corr(yt,yt+h) = σ²z

σz²+σe²

for allt andh

Notice that the key work here is “for all t and h”it doesn’t make sense to consider the correlation between Corr(yt,yt). Therefore once agains we consider for all h≥1

Corr(yt,yt+h) = Cov(yt,yt+h)

!Var(yt)Var(yt+h) = σ²z

σz²+σe²

Question 11.3 (iv)

Does the {yt}satisfy the intuitive requirement for being asymptotically uncorrelated?

is {yt}a weakly dependent process?

Recall that:

Corr(yt,yt+h) = σ²z

σz²+σe²

but

Corr(yt,yt+h)→ σ²z

σz²+σe²

as h→ ∞

therefore {yt}is not a weakly dependent process.

Question 10.6

Suppose yt follows a second order FDL model:

yt =α₀+δ₀zt+δ₁zt−1+δ₂zt−2+ut

let z^∗ denote the equilibrium value ofzt andy^∗ be the equilibrium value of yt such that

y^∗ =α₀+δ₀z^∗+δ₁z^∗+δ₂z^∗+ut

show that the change in y^∗, due to a change inz^∗, equals the long-run propensity times the change in z^∗

∆y^∗=LRP×∆(z^∗)

The economy is at equilibrium such that yt =y^∗ andzt =z^∗ If there is no change in z^∗ then we have it that:

y^∗=α₀+δ₀z^∗+δ₁z^∗+δ₂z^∗+ut

The economy is at equilibrium such that yt =y^∗ andzt =z^∗ Assume that at timet there is a permanent change inz^∗ by∆ yt=α₀+δ₀(z^∗+∆) +δ₁z^∗+δ₂z^∗+ut

yt+1 =α₀+δ₀(z^∗+∆) +δ₁(z^∗+∆) +δ₂(z^∗) +ut

yt+2 =α₀+δ₀(z^∗+∆) +δ₁(z^∗+∆) +δ₂(z^∗+∆) +ut

...

yt+h=α₀+δ₀(z^∗+∆) +δ₁(z^∗+∆) +δ₂(z^∗+∆) +ut for h ≥2

Therefore in the longrun:

ynew^∗ =α₀+(δ₀+δ₁+δ₂)∆+(δ₀+δ₁+δ₂)z^∗+ut =y^∗+(δ₀+δ₁+δ₂)∆ intuitively the long run equilibrium changes from y^∗ to

y^∗+ (δ₀+δ₁+δ₂)∆ therefore

∆y^∗ = (δ₀+δ₁+δ₂)∆z^∗