Structural modelling: Causality, exogeneity and unit roots Andrew P. Blake CCBS/HKMA May 2004.

Structural modelling: Causality, exogeneity and unit roots

Andrew P. Blake

CCBS/HKMA May 2004

What do we need to do with our data?

• Estimate structural equations (i.e. understand what’s happening now)

• Forecast (i.e. say something about what’s likely to happen in the future)

• Conduct scenario analysis (i.e. perform simulations) to inform policy

What do we need to know?

• Inter-relationships between variables– Causality in the Granger sense– Exogeneity

• Concepts

– Unit roots• Spurious regression

• Role of pre-testing

• Appropriate single equation methods

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X Y

Period t Period t+1

xt

yt yt+1

xt+1

Inter-relationships between variables

How best to estimate an equation?

• Single equation structural model (estimated by OLS)

• Single equation reduced form (IV/OLS)

• Structural system (estimated by TSLS, 3SLS or by a system method - SUR, FIML)

• Unrestricted VAR (OLS)

• VECM (FIML)

xt is autoregressive

Period t Period t+1

xt

yt yt+1

xt+1

xt has an autoregressive representation

Period t Period t+1

xt

yt yt+1

xt+1

xt has an ARMA representation

11

11

111

1

,

1

ttttt

tttt

ttt

ttt

ttt

xxso

xy

xy

yy

yx

Structural system

Reduced form

}

Period t Period t+1

xt

yt yt+1

xt+1

Granger Causality

Period t Period t+1

xt

yt yt+1

xt+1

Vector autoregressions (VARs)

Needs to be modelled to have

a structural interpretation

Granger causality

• If past values of y help to explain x, then y Granger causes x

• Statistical concept

• A lack of Granger causality does not imply no causal relationship

GC tested by an unrestricted VAR

• Definition of Granger Causality:– y does not Granger cause x if a12=b12=...=0– x does not Granger cause y if a21=b21=...=0

• NB. x and y could still affect each other in the same period or via unmeasured common shocks to the error terms.

tttttt

tttttt

ybxbyaxay

ybxbyaxax

...

...

222221122121

212211112111

Eviews Granger causality test resultNull Hypothesis F-Statistic Probability

x does not Granger Cause y F1 P1

y does not Granger Cause x F2 P2

• The closer P1 is to zero, the less the likelihood of accepting the null that x does not Granger cause y.

• (P1<0.10 : at least 90% confident that s1 Granger causes s2).

• P1 should be less than 0.10 for us to be reasonably confident that x Granger causes y.

y is a leading indicator of x if

• y Granger causes x;

• x does not Granger cause y;

• and y is weakly exogenous.

Leading indicators

Long term trends of money and prices in UK

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5.0

10.0

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30.0

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n y

ear

earl

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sm

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thed

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rices l

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Broad Money Prices

Criticisms of Granger causality

• Granger causality can be assessed using an unrestricted VAR - not tied to any particular theory

• How would you explain to your governor when it goes wrong?

• It depends on the choice of lags, data frequency and variables in VAR

Exogeneity

• Engle et al. (1983)– Separate parameters into two groups– Those that matter, those that don’t

• These are endogenous and weakly exogenous variables

• In practice a bit more complicated than that

Exogeneity (cont.)

• Correct assumptions of exogeneity simplify modeling, reduce computational expense and aid interpretation

• But incorrect assumptions may lead to inefficient or inconsistent estimates and misleading forecasts

Exogeneity (cont.)

• A variable is exogenous if it can be taken as given without losing information for the purpose at hand

• This varies with the situation

• We do not want the independent variables to be correlated with the regressors

• If they are, the estimates will be biased

Period t Period t+1

xt

yt yt+1

xt+1

Relationships between variables

• We do not want the black arrows

• We need to understand the red arrows

Both demand and supply shocks

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0 1 2 3 4 5 6

P

Q

OLS is unable to identify either the demand or supply curve

Only supply shocks

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P

Q

We can identify the demand schedule using OLS

Weak exogeneity

• Is y weakly exogenous with respect to x?• Do values of current x affect current y?• Are x and y both affected by a common

unmeasured third variable?• Does the range of possible values for the

parameters in the process that determines x affect the possible values of those that determine y

Weak exogeneity: example 1

• Money demand function:

• Would you estimate this as a single equation using OLS?

• Very unlikely that money does not affect real output or the nominal interest rate

ttt rym

Weak exogeneity: example 2

• Uncovered interest parity:

• Tests of UIP have performed very poorly, but ...

• No risk premia and monetary policy might react to exchange rate changes

*1tE ttt rre

Interest rate differentials

Exchange rate change

Question: how would you test for exogeneity in UIP?

Weak exogeneity: example 3

• In UK consumption had been forecast using single-equation ECM

• But relationship broke down in late 1980s

• Problem was that possibility that wealth reactions to disequilibrium had been ignored

11

11

...

...

tt

tttt

xy

xxyy

Single Equation ECM

Dynamic terms

Long run

Vector ECMS

Halfway between structural VARs and unrestricted VARs

ECMyxxy

ECMyxyx

tttt

tttt

21221212

11121111

...

...

Strong exogeneity

• Necessary for forecasting

• Is y strongly exogenous to x?– Is y weakly exogenous to x– Does x Granger cause y?

• Need the answers to be yes and no respectively

Strong exogeneity: example

First order VAR, ‘core’ and non-‘core’ inflation:

Given a forecast of {yt} can we forecast {xt}?

• If y is not strongly exogenous to x, feedback problems

', ,1 ttttt-t yxzAzz

Super exogeneity

Necessary for policy/scenario analysis. Is y super exogenous to x?

• Is y weakly exogenous to x?

• Is the relationship between x and y invariant?

Need the answers to be yes to both

Invariance

• The process driving a variable does not change in the face of shocks

• Linked to ‘deep parameters’

• Example: the Lucas critique

Testing for weak exogeneity: orthogonality test

• Estimate a reduced form (marginal model) for x, regress x on any exogenous variables of the system

• Take residuals from this reduced form and put them into the structural equation for y

• If they are significant then x is not weakly exogenous with respect to the estimation of c10

Testing for weak exogeneity with respect to c(lr)

• Estimate a reduced form (marginal model) for x: regress x on exogenous variables of system, including lagged ECM term involving x and y

• Test if coefficient of ECM term is significant• If it is, then x is not weakly exogenous with

respect to the estimation of long-run coeff, c(lr)• Consequence is that estimate is inefficient

Stationarity

• Why should we test whether series are stationary?• A non-stationary time series implies that shocks

never die out• The mean, variance and higher moments depend

on time• Standard statistics do not have standard

distributions• Problem of spurious regression

Non-stationarity

• Start with the following expression

yt = + yt-1 + ut u, 2• Substitute recursively:

yt = n + n yt-n + n-1jut-j

• The variable will be non-stationary if =E(y)=t

Var(y) = Var(n-1ut-j - t) = t 2

• Displays time dependency

Non-stationarity (cont.)

t is a stochastic trend• The series drifts upwards or downwards

depending on sign of ; increases if positive• Stationary series tend to return to its mean value

and fluctuate around it within a more-or-less constant range

• Non-stationary series has a different mean at different points in time and its variance increases with the sample size

Non-stationarity (cont.)

• Mean and variance increase with time

• yt = n + n yt-n +n-1jut-j

• If = then shocks never die out

• If | |<1 as n, then y is like a finite MA

• What do non-stationary series look like?

• Could show made-up series (with and without drift)

Difference vs trend stationarity• Compare previous equation with

yt = a + b t + ut

E(y) = a + b t

var(y) = 2

• b t - deterministic trend

• But stationary around a trend

E(y - b t) = a

Difference vs trend stationarity (2)

• Compare two generated series

• Stationary around trend

• Difference stationary are non-constant around a trend

• But can be difficult to tell apart

• Also difficult to tell series with AR coefficients 1 and 0.95

Difference vs trend stationary

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X Z

Difference vs trend stationarity

• Can you tell the difference?

xt = 1 + xt-1 + 0.6 ut

zt = 1 + 0.15 t + 0.8 et

• Can you tell the difference with a near-unit root?

Unit root vs near-unit root

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X W

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X W

Testing for unit roots

• Dickey-Fuller test

• Write

yt = yt-1 + et

as

yt - yt-1 = (-1)yt-1 + et

Null: Coefficient on lagged value 0, vs < 0

Dickey-Fuller tests

• Test akin to t-test but distributions not standard• Depends if series contains constant and/or trends• Must incorporate this into DF test• Augmented DF test - use lags of dependent

variable to remove serial correlation• All of these must be checked against relevant DF

statistic• But introducing extra variables reduces power

Unit versus near-unit roots

• Thus difficult to tell the difference between two series over small samples

• Low power of ADF tests (sample of 400)

x: ADF statistic -0.77048 p-value 0.8258

w: ADF statistic -6.90130 p-value 0.0000

• Small sample (40 observations)

x: ADF statistic 0.39323 p-value 0.9804

w: ADF statistic -0.49216 p-value 0.8828

Stationarity in non-stationary time series

• A variable is integrated of order d - I(d) - if it musto be differenced d times for stationarity

• The required number of differences depends on the number of unit roots a series has

• For example, an I(1) variable needs to be differenced once to achieve stationarity: it has only one unit root

Spurious regressions• Trends in data can lead to spurious correlation

between variables: there appears to be meaningful relationships

• What is present are uncorrelated trends

• Time trend in a trend-stationary variable can be removed by regressing variable on time

• Regression model then operates with stationary series with constant means and variances (standard t and F test inferences)

Spurious regressions

• Regressing a non-stationary variable on a time trend generally does not yield a stationary variable (it must be differenced) i.e. taking trend away does not lead to stationarity

• Using standard regression techniques with non-stationary data can lead to the problem of spurious regression involving invalid inference based on usual t and F tests

Spurious regressions• Consider the following DGP:

yt = yt-1 + ut u , 1

xt = xt-1 + et e , 1• y and x are uncorrelated, but estimating

yt = a + b xt + vt

we find that we can reject b = 0.

• Why? Non-stationary data => v non-stationary gives problems with t and F stats

• Also find high R2 and low DW (G&N 1974)

Spurious RegressionsDependent Variable: YMethod: Least SquaresDate: 03/31/03 Time: 18:28Sample: 1900:1 2003:4Included observations: 416

Variable Coefficient Std. Error t-Statistic Prob.

X 0.964478 0.001112 867.6800 0.0000

R-squared 0.997879 Mean dependent var 202.9399Adjusted R-squared 0.997879 S.D. dependent var 120.3730S.E. of regression 5.543177 Akaike info criterion 6.265414Sum squared resid 12751.63 Schwarz criterion 6.275103Log likelihood -1302.206 Durbin-Watson stat 0.023766

Spurious regression

• Why do we find significant coefficients?

• What will happen if we estimate a spurious regression with the variables in first differences?

• What ‘economic problem’ do we encounter if we only use differenced variables in economics?

• We lose information about the long-run

Spurious RegressionDependent Variable: DYMethod: Least SquaresDate: 03/31/03 Time: 18:36Sample(adjusted): 1900:2 2003:4Included observations: 415 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C 0.989704 0.016085 61.52980 0.0000DX -0.005194 0.012185 -0.426235 0.6702

R-squared 0.000440 Mean dependent var 0.984475Adjusted R-squared -0.001981 S.D. dependent var 0.211713S.E. of regression 0.211922 Akaike info criterion -0.260386Sum squared resid 18.54827 Schwarz criterion -0.240973Log likelihood 56.03014 F-statistic 0.181676Durbin-Watson stat 1.752192 Prob(F-statistic) 0.670159

Cointegration (definition)

• In general, regressing two I(d) variables, d>0, leads to the problem of spurious regression

• Assume two I(d) variables and estimate:

• If is a vector such that t is I(d-b) then we say that y and x are co-integrated of order CI(d,b)

ttt xy

What is cointegration?

• If two (or more) series have an equilibrium relationship in the long run even though the series contain stochastic trends they move together such that a (linear) combination of them is stationary

• Cointegration resembles a long-run equilibrium and differences from the relationship are akin to disequilibrium

• Trivially, a stationary model must be cointegrated but may not co-break

Modelling the short-run

• Are we ever in the long run?

• How do we model the short run?

• Problem of using only differenced data and the loss of long-run information

• Assume

• In steady state has little meaning for the long run

ttt xy 0 tt xy

Modelling short run

• Assume

yt = xt + yt-1 + xt-1 + t, , 2

• If a LR relationship exists

yt = + xt

• We can write

yt = xt - (1- )(yt-1 - - xt-1 ) + t

• (1- ) is speed of adjustment

• Implications for the sign of ECM

Modelling the short-run• There are some issues about the estimation

of • Stock (1987) shows that OLS is fine, is

super-consistent; the estimator converges to its true value at a faster rate when a series is I(1) than when it is I(0)

• However, there is significant of bias in small samples

Testing strategies• Perron’s suggestion:

– start with regression with constant and trend

– proceed trying to reduce unnecessary paramaters

– if we fail to reject parameters continue testing until we are able to reject the hypothesis of a unit root

• In the end we should use common sense and economics– If there should not be a unit root - probably a

break

Cointegration and single equations

• When looking at single equations it is easy to test for cointegration– Engle and Granger two-step procedure– Engle-Granger-Yoo three-step approach

• What if there is more than a single cointerating relationship?– Need a system approach– VECMs

Modelling strategies• Understand the data

– Do whatever tests necessary to be sure of using appropriate models

• Understand the limitations of individual methods– By not taking limitations into account a rejection does not

necessarily imply that the hypothesis is false

• Use appropriate methods for different problems

EXOGENEITY• Banerjee, A, D.F. Hendry and G.E. Mizon (1996) “The econometric analysis of economic policy”, Oxford Bulletin of

Economics and Statistics 58(4), 573-600

• Ericsson, N.R. and J.S. Irons (eds) (1994) Testing Exogeneity. Advanced Texts in Econometrics. Oxford University Press.

• Lindé, J. (2001) “Testing for the Lucas Critique: A quantitative investigation”, American Economic Review 91(4), 986-1005.

• Monfort, A and R. Rabemananjara (1990) “From a VAR model to a structural model, with an application to the wage-price spiral”, Journal of Applied Econometrics 5, 203-227

• Urbain, J.P. (1995) “Partial versus full system modelling of cointegrated systems: An empirical illustration”, Journal of Econometrics 69(1), 177-210.

• Boswijk, P. and J.P. Urbain (1997) “Lagrange Multiplier tests for weak exogeneity: A synthesis”, Econometric Reviews 16(1), 21-38.

• Charezma, W.W and D.F. Deadman, (1997) New Directions in Econometric Practice, Edward Elgar, Second Edition.

• Urbain, J.P. (1992) “On weak exogeneity in error correction models”, Oxford Bulletin of Economics and Statistics 54(2), 187-207.

MODELLING AND FORECASTING SHORT-TERM DATA

• Jondeau, É., H. Le Bihan and F. Sédillot (1999) Modelling and Forecasting the French Consumer Price Index Components, Banque de France Working paper 68.

• Clements, M. P. and D.F. Hendry (1999) Forecasting non-stationary economic time series. MIT Press.

• Bardsen, G and P.G. Fisher (1996) On the roles of economic theory and equilibria in estimating dynamic econometric models-with an application to wages and prices in the United Kingdom, Essays in Honour of Ragnar Frisch.

VARS

• Levtchenkova, S., A.R. Pagan and J.C. Robertson (1998) “Shocking stories”, Journal of Economic Surveys 12(5), 507-532.