Integration andcointegrationGranger & Newbold warned of the dangersof using ‘conventional’ methods onnon-stationary data. Methods have sincebeen developed for analysing relationsbetween integrated variables.

The random walk is an example of anintegrated process. the first difference Δy t ofan integrated process is stationary:

Δy t = y t − y t−1 = t, t is stationary.

(For the random walk the error is whitenoise, a stationary process.)

Such a y t is said to be integrated of order 1and denoted I1. If we have to differencetwice to achieve stationarity then the seriesis called I2. A stationary series is I0.● In econometrics I1 series are the

nonstationary processes most studied.

● Occasionally I2 series are studied–butthere is no widely accepted theory.

Checking for stationarityBecause non-stationary processes havesuch different properties from stationaryones, it is important to be able to distinguishbetween the two.

The correlogram is informative. Here is theEViews output for a sample of 100 from astationary AR1 with ρ = 0. 9. )

The autocorrelations/serial correlations dieaway quite quickly for a stationary processbut for a nonstationary process they do not.

Here is the EViews output for sample of 100from a random walk

The AC’s stay close to 1 for many periods.

There are also formal tests–the mostimportant is the Dickey-Fuller tests whichfirst became available in 1979. They arealso called unit root tests. See EViewsmanual ch 17 section on unit root tests.

In EViews the tests live in the Quick/SeriesStatistics/Unit root tests. Their appearancein that section reflects the view that peopleusing time series will want to investigate thestationarity or otherwise of the seriesbefore doing any complicated statisticalanalysis with them.

EViews presents many options

● Many variations of model and hypothesesare conceivable, so there are many differenttest statistics.

● We begin with the simplest situation of anAR1 and the question–is this a randomwalk (ρ = 1) or is it stationary (ρ < 1) ?

Consider the AR1 and the hypotheses

y t = ρy t−1 + t : t ∽ IN0,σ2

H0 : ρ = 1 : HA : ρ < 1.

The model can also be written in (first)differences. So, subtracting y t−1 from eachside of the equation, writing Δy t = y t − y t−1

and defining α = ρ − 1 we get

Δy t = αy t−1 + t : t ∽ IN0,σ2

H0 : α = 0 : H1 : α < 0

where we are just restating the originalhypotheses about ρ in terms of α.

● The natural way of testing whether α = 0 isto regress Δy t on y t−1 and do a t-test. Thatis, form the ratio

αest. s. e of

α

whereα is the least squares estimate of α

and reject the null for large negativevalues. If the null were α = −0. 1 (i.e.ρ = 0. 9 and the process were stationarythen the t-distribution associated with thenull hypothesis would be N0, 1 in large

samples. This fits in with the ordinaryregression case where the t-distribution

tends to the standard normal when thenumber of degrees of freedom tends toinfinity.

● However when ρ ≥ 1 the ‘t-statistic’ is notN0, 1 in large samples.

● Dickey & Fuller worked out the distributionof the ‘t-statistic’ for ρ = 1 α = 0 andtabulated it.

There are several variants of the basicDickey-Fuller test including for the cases

Δy t = γ + αy t−1 + t

Δy t = γ + αy t−1 + βt + t.

In the first case α = 0 ⇔ r. w. with drift andin the second γ = 0 ⇔ r. w. with drift & timetrend.

There is also an Augmented Dickey-Fuller(ADF) test in which additional terms in Δy t−i

are included in the regression to cover thepossibility that the process is a higher orderAR:

Δy t = γ + αy t−1 + β1Δy t−1 +. . .+βpΔy t−p + t

The test is of α = 0 as before. In thisgeneralised random walk Δy t is expressedin terms of past Δy ′s.

In this more general specification we askwhether the series contains a “unit root”.

In practice we have no good idea about theappropriate value of p, the order of the AR.To help choose, EViews presents a bundleof model selection criteria, AkaikeSchwarz, and others.

Here is EViews output for testing whetherthe nonstationary series considered abovehas a unit root.

The t-statistic is very small and we cannotreject the hypothesis that α = 0. So there isno evidence against the unit root nullhypothesis.

Here is EViews output for a sample from astationary process.

Here the prob values is very small and thenull hypothesis of a unit root is rejected.

Relations between integratedvariables–cointegration

Cointegrated variables are I1 variablesthat are tied together by a linearrelationship, but not rigidly.

The biggest name is Granger who receivedthe Nobel Prize in 2003 for his work oncointegration.

Possible instances of cointegration

● Aggregate consumption (C) and income(Y), for the former is roughly a constantproportion of the latter: C kY. Here 2variables are involved.

● Interest rates within a country. The US

Federal funds rate of interest and the Bondrate. (P of E 12.4.1 ) 2 variables again.

● The price levels in 2 countries and theexchange rate between their currencies:P1/P2 E, or ln P1 − ln P2 ln E.(Purchasing power parity.) 3 variables.

● The interest rates in two countries, theexchange rate of their currencies and thedomestic rates of inflation. 5 variables.(Covered interest rate parity.)

The first example is inspired by research onthe consumption function, one of thecomponents of a Klein-typemacroeconometric model.

The other examples are based on the ideaof arbitrage: if there are competitivemarkets and buyers and investors canmove freely between opportunities then therewards of those opportunities cannot getout of line for long periods. Arbitrageproduces a tendency towards equilibrium.

Formally cointegration obtains between x t

and y t if x t ∽ I1 and y t ∽ I1 andαx t + βy t ∽ I0.

x t and y t are described as cointegrated with

α,β the cointegrating vector.

In this plot of artificial series, the series x1t

and x2t are I1 (in fact random walks withdrift) but the combination x1t − x2t − 1 is I0,i.e. stationary

Further points

● The coefficient vector α,β is not uniquelydetermined: multiply this vector by ascalar–the combination with the newcoefficients is also cointegrated

● However there cannot be two linearlyindependent cointegrating relationshipsbetween a pair of I1 series

● The notion of cointegration extends to morethan two variables–we may consider say

three I1 series and find thatαx t + βy t + γz t is I0.

● It is possible to have more than onecointegrating relation: up to k − 1cointegrating relationships between k

variables.

Interpreting a cointegrating relationship

αx t + βy t ∽ I0

1. Common stochastic trend. Thevariables are cointegrated becausethey share a common I1 componentw t (a common ‘stochastic trend’). Longrun co-movements in the output ofdifferent countries may reflect sharedtechnical progress. Suppose

x t = γw t +x t

y t = δw t +y t

wherex t,

y t are I0 with zero mean

and w t is the I1 common trend. Notethat this trend is a random process: it isa stochastic trend–not a deterministictrend like a function of t.

2. Error correction model. (ECM) The ideais that the cointegrating relationship(obtained by setting the cointegratingcombination equal to 0) expressesequilibrium and that the two series arepulled towards this equilibriumrelationship. The series plotted earlierwere generated from a simple ECM,with cointegrating relationship

x1,t − x2,t = 1

Of course this is a special case of

β1x1,t + β2x2,t = β0

I generated the data from the process(using particular values for the α ′s andβ ′s :

△x1t = γ1 + α1β1x1,t−1 + β2x2,t−1 − β0 + 1t

△x2t = γ2 + α2β1x1,t−1 + β2x2,t−1 − β0 + 2t

where it ∼ IN0,σ2 and E1t,2s = 0 all t, s.

The change in a variable e.g. △x1t,reflects drift represented by γ1, therandom shock 1t and the extent of‘disequilibrium’ β1x1,t−1 + β2x2,t−1 − β0.This last term is often written as e t−1.

● The terms α1,α2 are called speed of

adjustment terms: they describe theshort-run dynamics of the process; the β ′s

in the cointegrating relationshipcharacterise the long run solution.

● Depending on the size of α1 and α2, thesystem is drawn slowly or rapidly towardsthe equilibrium. If α1 = α2 = 0 we haveindependent random walks with drift.

● An advantage of the ECM is that all thevariables are stationary: all the △x ′s ande t−1 are I0. This means that‘conventional’ methods can be used.