Non-Stationarity and Unit Roots

8/13/2019 Non-Stationarity and Unit Roots

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VAR is used when we are dealing with

stationary variables where we may have

an endogeneity problem.

When we are dealing with non-stationarydata we must use an alternative method:

Co-integration


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Revision of Stationarity


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Quick revision of non-stationary series:

Consider the series: Yt= Yt-1+ ut

This series is stationary if ||


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Example: Yt= Yt-1+Ut


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Difference Stationary with drift.

If the series includes an intercept (or driftterm), 0: Yt= 0+ Yt-1+ ut

Again this series is non-stationary [=1 again] Each period the series changes by a certain

amount (0) plus a random amount (ut) i.e. thereis a trend in the series!!

But if we look atYt Yt= YtYt-1= 0+ ut

This series is stationary but instead of fluctuating around

E(ut)=0, it fluctuates around E(0+ ut) = E(0) = 0


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Example: Yt= 0+ Yt-1+Ut


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Trend Stationary

If the series also includes a time trend, 1:

Yt= 0+ 1t + Yt-1+ ut This series is non-stationary even if non-stationary.

To make this series stationary, we de-trend theoriginal series! Yt - 1t= 0+ Yt-1+ ut , which is stationary if (||


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Example: Yt= 0+ 1t + Yt-1+Ut


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Cointegration and Error

Correction Models


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OLS Regression with non-stationary data:

The spurious regression problem

Imagine we have two series (Y and X) which are

non-stationary.

If both these series display a trend (either

deterministic or stochastic), the series will behighly correlated with each other, even if there

is no true relationship between them

Thus if we carry out an OLS regression of Yand X we will find that X seems to explain a

good portion of Y.


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Silly example

To give a stupid example:

Suppose we run a regression with an index of shares as

our Y variable and your height as the X variable.

Well equity indices usually grew over the past 25 years.

Your height would have also increased over the last 25

years

So OLS would find a significant relationship between height

and Equities

But the key point is in reality shares dont increase wheneveryour height increases!!!

=> It was a false result due to both series tending to increase

over time!


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Explanation of result

Recall that OLS coefficients can be

interpreted as how much Y changes on

average when X changes by one unit

It is only based on the degree of associationnot causation!!

Since both tend to increase there is positiveassociationbut there is no causation!!!


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The spurious regression problem

What our regression tells us:

As X increases, Y increases

=> X and Y are related to each other

What is really going on:

As t increase, X and Y are both increasing

X and Y are both related to t, but may not be related

to each other really.


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The spurious regression problem(contd.)

Our model will appear to fit well A high R2and high t ratios indicates that the explanatory

power of the regression is very high suggesting (falsely)avery good result.

In this case the trend in both variables is related, but not

explicitly modelled, causing autocorrelation. But as thetrends in the two variables is related, the explanatorypower is high.

Granger and Newbold (1974) proposed the followingrule of thumb for detecting spurious regressions: If the

R-squared statistic is larger than the DW (DurbinWatson) statistic, or if R-squared 1 then theregression is spurious. Note DW statistic measure autocorrelation


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A Possible Solution to Spurious regression

problem

Since the problem is caused by stochastic ordeterministic trends, the obvious way to solve theproblem is to get rid of the trend: Stochastic Trend =>

difference the data => stationary

Deterministic Trend =>

De-trend the data

Both stochastic and deterministic trends: =>

take first difference and then de-trend.

Stationary series dont have trends => problemsolved!

Or is it?..............


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Problems with this solution: If we have differenced the series, we are now looking at the

relationship between changesin the variables rather than in thelevels.

The variables in this form may not be in accordance with the

original theory

This model could be omitting important long-run information,

differenced variables are usually thought of as representing theshort-run. [since it is only the change since the last period]

This model may not have the correct functional form.


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Question: So, given that differencing maybe undesirable, is there a way to estimate

regressions involving non-stationary

variables but allowing us to keep thevariables in levels?

Answer: Yes, if there is an

equilibrium relationship betweenthe variables, otherwise No!


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Cointegration The basic idea:

If theory tells us that there is some equilibrium relationship betweenthe variables, then their stochastic trends must cancel out.

Why?

Well if they didnt, the stochastic trend in one of the variables would

take us away from the equilibrium and we may never return (since

the series is non-stationary it doesnt have to return to its previouslevel!)

Ex. Imagine house prices are related only to annual rents. Then the price in a

period should on average be a certain number of times the annual rent (say

20 times here!)

If, over time, rents are increasing due to a stochastic trend, then for there to be an

equilibrium relationship, house prices must also increase by the stochastic trend! Otherwise the series diverge.

We cant have an equilibrium where rents are increasing but house prices remain

constant!


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Cointegration

Now we will look at it in terms of Ytand some explanatoryvariables X1tand X2t.

Suppose the model for ytis correctly specified as

Yt = 0+ 1X1t+ 2X2t+ et

Where Yt, X1tand X2tare non-stationary series. For there to be an equilibrium relationship in this model, Ytcant

diverge indefinitelyfrom the explained part of the equation It can diverge for a while, as long as it will eventually return

i.e. Ytcant diverge indefinitely from 0+ 1X1t+ 2X2t Ex. simple housing model, applied to the recent history in Ireland

House prices had been increasing quicker than rents, i.e. diverging from theirequilibrium value. This was unsustainable according to our simple theory.

Now house prices have began to fall, restoring the equilibrium relationshipalthoughthey currently may have further to fall.

In reality, there are more variables at play than just rent.


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Example of cointegrated series:Time series of consumption

and income


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Cointegration

Ytcant diverge indefinitely from 0+ 1X1t+

2X2t Think of 0+ 1X1t+ 2X2tas the equilibrium value of Yt.

So what does this mean? Well, if we look at the difference between Ytand

the explained part, Yt(0+ 1X1t+ 2X2t).

For there to be an equilibrium, this musteventually return to 0. (i.e. the equilibrium must be

restored)

i.e. Yt(0+ 1X1t+ 2X2t) must be stationary!!!


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Cointegration

Linear Combinations of Integrated Variables

So: Yt(0+ 1X1t+ 2X2t) must be stationary!

But: et= Yt(0+ 1X1t+ 2X2t) [since Yt = 0+ 1X1t+ 2X2t+ et]

Thus, etmust be stationary if the theory underlying our

specification is correct Not stationary => our theory must be incorrect

Not related, no equilibrium relationship between them

If ethas a stochastic trend there will be no tendency for theequilibrium relationship between y

t, x

1t& x

2tto be restored

Remember if ethas a stochastic trend, shocks in ethave apermanent, albeit random effect

Crucial insight: Equilibrium theories involving non stationary variablesrequire the existence of a combination of the variables that isstationary


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Cointegration

In the long runequilibrium yt- 0- 1x1t- 2x2t = 0

And the equilibrium error et= the deviation from equilibriumwhich is stationary et= yt- 0- 1xt- 2x2t

In this case the variables yt, x1t& x2tare said to becointegrated of order CI(d,b) d -> amount of times variables have to be differenced to make them

stationary [i.e. the variables are integrated of order d I(d)]

b -> the reduction in integration resulting from the cointegration.[This may be a little confusing, in our example d=1 because thevariables were I(1), after cointegration our e is I(0) so b=(1-0) >b=1]

Usually we deal with variables that are CI(1,1) The vector = (1, 0, 1,2) is said to be the cointegrating

vector


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Cointegration

Linear Combinations of Integrated Variables

Notes: In general if = (1,0,1, 2) is a cointegrating vector then

= (, 0, 1, 2) is also a cointegrating vector. In other words, the cointegrating vector is not unique.

In estimation we need to normalize the cointegrating vector

by fixing one of the coefficients at unity [Here we fixed thecoefficient on Ytto be 1]].

All variables must be integrated of the same order. If variables are integrated of different orders then they cannot be

cointegrated.

A series with a unit root and a stationary series (different orders) Cant be an equilibrium relationship b/ them I(1) moving randomly, I(0)

isnt

If there are n cointegrated variables, there can be up to n-1linearly independent cointegrating vectors


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To get an idea what we mean by this

suppose house prices tend to be 5 times

rents

i.e. House price =5*Rent

Then 2*House price = 10*Rent

And 3*House price = 15*Rent

So [1,5] , [2,10] and [3,15] would all becointegrating vectors!

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Non-Stationarity and Unit Roots