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MODELING LONG-RUN RELATIONSHIPS IN FINANCE

1. Nonstationarity and spurious regression

The series ty is a random walk with drift if 1t t ty yµ ε−= + + where tε is a white
noise process, often referred to as shocks or innovations. This specification can be
thought of as an AR(1) process

= + + , with AR parameter 1ρ = .1

recursive substitution,

1 1t j t t j t j t
y j yµ ε ε ε

= ⋅ + + + + +

It then follows that

As j → ∞ , the mean and variance are unbounded. Consequently, a random walk is not a
covariance stationary process. It is also called as stochastically non-stationary process.
To induce non-stationarity we need to difference the time series

= + + using

difference operator

1 1 1t t t t t
y y y yµ ε

y µ ε∆ = +

Obviously, differences series

y∆ are white noise now. In this case

y is called integrated

of order one, I(1) and the differenced series

y∆ is integrated of order zero, I(0).

It may be the case that differencing only one time does not make the series stationary.
E.g., to induce stationary in

= + − + we need to difference twice:

t t t t t t
y y y y yµ ε

− = + − − + or

y∆ is still unit root process. Second differencing

1 1 2 1 2 1 2
( ) ( ) ( ) ( )

t t t t t t t t t
y y y y y y y yµ ε

− − − − − − −
− − − = + − − − + or

y yµ ε µ ε

∆ ∆ = + ∆ ∆ + = +

induces stationarity. In this case

y is called integrated of order two, I(2).

1 Note, when 1ρ = − , we also have unit root process where shocks are alternating in
sign. This case is similar to when 1ρ = and it rarely observed in finance.

We may have I(k) processes in general, but processes with k>2 are not typical in finance
and economics. Most of financial and economic time series are I(1), some prices series
may be I(2).

Another potential source of nonstationarity is presence of time trend in the data. Time
series generated by

y tµ β ε= + +

are said to contain a deterministic trend, or time trend.

To induce stationarity we need to estimate the model and filter out time trend (use
residuals).

Using nonstationary variables in linear regression may (but does not always!) produce the
phenomenon of spurious regression. Spurious regression may have high R2 , and large test
statistics (small p-values) suggesting relationship between dependent and independent
variables, where in fact may be not related. Therefore before using potentially non-
stationary variables in the regression one needs to test for unit root.

2. Dickey-Fuller test and other tests for nonstationarity.

The AR(1) model:

= + can be reparameterised as

y yψ ε ψ ρ

The Dickey-Fuller test is a test of the hypothesis that

H0 : 0 (i.e. 1)ψ ρ= = “unit root” against the alternative
H1 : 0 (i.e. 1)ψ ρ< < “stationary process” The t-ratio or t-statistic associated with the estimated value of ψ is used to test this hypothesis by comparison with the appropriate critical value. Often the regression above is augmented with lags of ty∆ appearing on the right-hand side to account for serial correlation. In this case, the test is referred to as an augmented Dickey-Fuller test and is performed in exactly the same way as the Dickey-Fuller test. Note that under the null the test statistics is not distributed according to Student’s t distributions, because if presence of a non-stationary term. Special Dickey-Fuller test critical values need to be used. Moreover, if you assume that your equation contains either a drift term, or a drift term and a time trend different critical values need to be Augmented Dickey-Fullers (ADF) test ties to incorporate any autocorrelations in ε and take the form t t i t i t y y yψ α ε ∆ = + ∆ +∑ . Phillips and Perron (PP) have developed a more comprehensive theory of unit root nonstationarity. The tests are similar to ADF tests, but they incorporate an automatic correction to the DF procedure to allow for autocorrelated residuals. The tests usually give the same conclusions as the ADF tests, and the calculation of the test statistics is complex. PP and ADF tests perform poor when we are in the situation close to unit root, that is they are poor at deciding if 1ρ = or 0.95ρ = , especially with small sample sizes. KPSS test (Kwaitowski, Phillips, Schmidt and Shin, 1992) tries to address this problem by using the reversed null hypothesis: H0 : “stationary process” against the alternative H1 : “unit root” Both sets of test may be performed for robustness. 2. Cointegration and Common Trends Often we observe the tendency for financial time series, for example, interest rates, to move together over time, even though individually each time series is characterized as an I(1) process. When two I(1) series move together over time, it is possible that there is a long-run relationship between them. Further, we discuss how a long-run relationship among two time series may be detected and, if such a relationship is uncovered, how the long-run relationship is restored when the series deviate in the short- run from the long-run relationship. Suppose we have data on two nominal interest rate series, one a short-term interest rate denoted str and the other a long-term interest rate denoted tr . Typically, we find that both interest rate series are I(1) processes on the basis of, say, ADF tests. One possible explanation for this finding is provided by the following model for interest rates: r a i ε= + + (1) r a i ε= + + (2) = + + (3) i is the rate of inflation from t-1 to t and the error term in each equation is a white noise process. Both nominal interest rates are postulated to vary one-for-one with the rate of inflation as first suggested by . Both the interest rate series are I(1) because the rate of inflation i is I(1); specifically the rate of inflation is a random walk process with drift. Although both nominal interest rates are I(1) processes, they will move together because they share a common unit root component, namely, the unit root in inflation. We say that str and tr share a common stochastic trend, which is the unit root component of inflation. In this setup, the spread, defined as l st tr r− , is stationary; it is an I(0) process. To see this, subtract equation (1) from equation (2) to get: t t t tr r a a ε ε− = − + − (4) The spread is an I(0) process, that is, it is stationary, because the difference of two white noise error terms is stationary (that is, l st tε ε− , is stationary). The spread can be thought of as a linear combination of the long and short nominal interest rate. In particular, the stationary linear combination of the interest rate series is: We say that the spread is cointegrated with cointegrating vector, denoted β , as =   −  To summarize, because there is a linear combination of the interest rate series which is stationary, the two interest rates are cointegrated. The specific linear combination is shown by the cointegrating vector, which in this case is β ′ = −(1 1) . There is cointegration between the two interest rate series because they share a common stochastic trend or a common unit root component. If the random walk component in ltr were independent of the random walk component in str , the two interest rate series would not share a common stochastic trend and thus would not be cointegrated, in which case there will not exist a linear combination of the two interest rates which is stationary. However, that does not happen in our model because the two interest rates share the same unit root component, which is the unit root in inflation. Without loss of generality, we can define 2 1a a a= − and t t tε ε ε= − so that equation (4) can be written as: l st t tr r a ε− = + Because tε is a white noise process and can be thought of as a stationary deviation, the long-run relationship between ltr and tr is given as: l st tr r a− = The deviation from the long-run or cointegrating relationship is given by tε where t t tr r aε = − − . =   −  . Then the long-run equilibrium relationship can be expressed as: ( )1 1 lt t β ′  = − = and the deviations from the long-run equilibrium relationship are t tX aε β ′= − . Consider a more general setup. Consider two I(1) variables tY and tX that are generated by two independent random walks: 1 1t t tY Y ε−= + 1 2t t tX X ε−= + where the 1tε and 2tε are white noise processes. Suppose we estimate a regression of the 0t t tY Xα β ε= + + (7) which can be written as: 0t t tY Xβ α ε− = + We can interpret 0t tY Xβ α− = as the long-run relationship between tY and tX as long as tε is a stationary process. However, in this case, tε will be non-stationary because both tY and tX are independent non-stationary processes so that there will be no linear combination of them which will be stationary. Let us change the set-up and assume that: 1 1t t tY Xβ ε−= + (7a) 1 2t t tX X ε−= + (7b) In this case, both variables are I(1) processes. The variable tX is a random walk process and tY is non-stationary because it depends on the random walk process tX . Again, suppose we estimate a regression of the form: 0t t tY Xα β ε= + + (8) which can be written as: 0t t tY Xβ α ε− = + (9) In this case, we can interpret 0t tY Xβ α− = as the long-run relationship between tY and tX because tε is a stationary process. To see this, substitute (7a) and (7b) into t tY Xβ− − −− = + − + ( )t t t t t t Hence t tY Xβ− is a stationary series since it is given by 1 2t tε βε− . Thus tε in equation (8) or (9) is a stationary error term. In this case, the series are cointegrated because the unit root in tY is due to the unit root in tX so that there is a common unit root in both of the series. In other words, it is legitimate to estimate a regression of the form of equation (8) where an I(1) dependent variable is regressed on an I(1) independent variable as long as the variables are cointegrated so that the regression residual is stationary. 3. The Engle- for Cointegration In view of the discussion so far, a natural test for cointegration between two I(1) variables, tY and tX is to estimate the regression α β ε= + +0t t tY X (9a) by OLS and test whether the estimated regression residual is stationary. Specifically, denote the OLS residuals from this regression as te . Estimate the Augmented Dickey Fuller regression: t t i t i t e e eγ δ υ− − ∆ = + ∆ +∑ Since the 'te s are the residuals from a regression equation, the mean of the 'te s is zero so there is no need to include an intercept term in the ADF regression. The choice of n is determined on the basis of some information criteria, for example, the AIC criterion. If the estimated residuals exhibit serial correlation, then n is chosen to be larger then zero and the augmented form of the Dickey-Fuller regression is used. If we can reject the null hypothesis of a unit root in the residuals (that is, we can reject the null of 0γ = ), we conclude that the two variables are cointegrated and share a common stochastic trend. If we cannot reject the null hypothesis of a unit root in the residuals (that is, we cannot reject the null of 0γ = ), we conclude that the two variables are not cointegrated and that the random walk in one variable is independent of the random walk in the other variable. The regression of tY on a constant and tX and the subsequent testing for a unit root in the estimated regression residual is referred to as the Engle-Granger procedure. In the application of the ADF test to the regression residuals, it is not appropriate to use the standard Dickey-Fuller critical values in testing for a unit root. The reason is that the OLS estimates of 0α and β in equation (9a) are chosen to minimize the sum of squared residuals (that is, to make the residual variance as small as possible). Thus, the procedure is biased toward finding a stationary error process, since unit root processes have large variance in finite samples. The critical values to test the null hypothesis of a unit root in the residuals must take account of this bias. Appropriate critical values for the ADF test when it is used as a test for cointegration have been derived by Engle and Granger and refined by MacKinnon. These critical values depend on the number of variables used in the analysis and on the sample size. For the case of two variables, the critical values are shown in the table below. Table 1: Critical Values for the Engle- for Two Variables Sample Size (T) 1% 5% 10% 50 -4.123 -3.461 -3.130 100 -4.008 -3.398 -3.087 200 -3.954 -3.368 -3.067 500 -3.921 -3.350 -3.054 Note: The critical values are for bivariate cointegrating relations (with a constant in the cointegrating vector) estimated using the Engle-Granger procedure. By comparison, for example, the standard ADF critical value for an ADF regression with no constant term is -2.60 for T=100. The critical values in Table 1 are larger to take account of the fact that the testing procedure is biased in favour of cointegration. 4. Cointegration and Error Correction Engle and Granger have shown that if tY and tX are cointegrated, then there exists an error correction model which describes how tY and tX adjust in the short-run following a deviation from long-run equilibrium. (This is known as the Theorem). In an error-correction model, the short-term dynamics of the variables in the system are influenced by the deviation from equilibrium. In the interest rate model, a simple error correction model that could apply is: t s t t st t l t t lt ∆ = − − − + where 0, 0s lα α> > and ,st ltη η are white-noise disturbance terms. The short and long
term interest rates change in response to stochastic shocks (represented by stη and ltη )

and in response to the previous period’s deviation from long-run equilibrium. Everything
else being equal, if this deviation happened to be positive ( 1 1

t tr r a− −− − >0), the short-term

interest rate would rise and the long-term rate would fall. Long-run equilibrium is
attained when l st tr a r= + . In practice, we would estimate the cointegrating regression

l st t tr a rβ ε= + +

and test whether the residuals are stationary. Suppose we find that our OLS estimates of
a and β are, respectively, â and β̂ and that the estimated OLS residuals ( te ) are
stationary. This would agree with our theory provided β̂ is very close to one. Then the
estimated deviation from the long-run equilibrium relationship is given by:

ˆˆl st t te r a rβ= − −

and the error-correction model can be written as

t s t t str r r aα β η− −∆ = − − + (10)

t l t t ltr r r aα β η− −∆ = − − − + (11)

or equivalently as

t s t str eα η−∆ = + (12)

t l t ltr eα η−∆ = − + (13)

We can estimate equations (10) and (11), or equivalently equations (12) and (13), by OLS
since the independent variable 1 1 1ˆ ˆ

t t te r r aβ− − −= − − is stationary by construction. The OLS

estimates of sα and lα (denoted ˆsα and ˆlα ) are often referred to as the estimates of the
speed of adjustment parameters and measure how quickly str and

tr adjust to bring about

long-run equilibrium. For example, the larger ˆsα the larger the response of the short-term
interest rate to the previous period’s deviation from long-run equilibrium. Suppose for
argument sake that the OLS estimate of ˆ 0lα = and ˆ 0sα > . In this case, all the
adjustment to long-run equilibrium is through movements in the short-term interest rate
because the long-term interest rate does not adjust to last period’s equilibrium error. We
say that the long-term interest rate is weakly exogenous in this case. (It cannot be the case
that both ˆ 0lα = and ˆ 0sα = since that would imply both variables are governed by
independent random walks as they are not adjusting to any equilibrium relation and thus
cannot be cointegrated).
Finally, we can formulate a more general error-correction model by including
lagged changes of each interest rate into equations (12) and (13) or, for that matter,
equations (10) and (11):

t s t i t i i t i st

r e a r a rα η− − −

∆ = + ∆ + ∆ +∑ ∑ (14)

t l t i t i i t i lt

r e a r a rα η− − −

∆ = − + ∆ + ∆ +∑ ∑ (15)

Equations (14) and (15) can be estimated consistently by OLS and standard statistical
inference on the coefficient estimates can be performed because all the variables in the
error-correction model are stationary. The choice of the appropriate number of lags to use
in each case can be determined by some lag length criteria or by dropping insignificant
lags from the estimated equations.
To summarize, suppose we have two time series tY and tX . First test to see if
both series are I(1) processes by using, say, an ADF test. If both series are found to be
I(1), there is the possibility that they share a common unit root component in which case
they are cointegrated. Second, test for cointegration using the Engle-Granger procedure
by estimating a regression of the form

0t t tY a Xβ ε= + +

and save the estimated residuals. Denote the OLS parameter estimates as 0â and β̂ and
the estimated residuals as te . Perform an ADF test (without a constant) on the estimated
residuals. If the estimated residuals are found to be stationary, conclude that the two
series are cointegrated. The cointegrating vector is ˆ(1 )β− . The long-run equilibrium
relationship is:

0 ˆˆt tY a Xβ= +

and the deviation from long-run equilibrium is:

0 ˆˆt t te Y a Xβ= − −

Third, there exists an error-correction representation which shows how the variables
adjust to last period’s deviation from the long-run cointegrating relationship. Specifically,
the error correction model is of the form:

1 1 11, 12,

t y t i t i t yt

Y k e a X a Yα η−

∆ = + + ∆ + ∆ +∑ ∑ (16)

2 1 21, 1 22, 1

t x t i t i t xt

X k e a X a Yα η− − −

∆ = + + ∆ + ∆ +∑ ∑ (17)

The two equations in the error-correction model can be estimated consistently by OLS.
Standard statistical inference on the coefficient estimates can be performed because all
the variables that appear in equations (16) and (17) are stationary. Finally, if the OLS

estimate of yα is zero, tY is said to be weakly exogenous as it does not adjust to
deviations from the long-run equilibrium relationship. All of the adjustment to
equilibrium occurs through changes in tX since 0xα ≠ . (If both yα and xα were

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