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
Inter-relationships between
variables
Period t
Period t+1
xt
xt+1
yt
yt+1
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
xt+1
yt
yt+1
xt has an autoregressive
representation
Period t
Period t+1
xt
xt+1
yt
yt+1
xt has an ARMA representation
xt  yt   t
}
yt  yt 1   t
 yt 1 
1

x
t 1
Structural system
  t 1 

yt  xt 1   t 1    t

so, xt  xt 1   t   t   t 1 Reduced form
Granger Causality
Period t
Period t+1
xt
xt+1
yt
yt+1
Vector autoregressions (VARs)
Period t
Period t+1
xt
xt+1
yt
yt+1
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
xt  a11 xt 1  a12 yt 1  b11 xt 2  b12 yt 2  ...   t
yt  a21 xt 1  a22 yt 1  b21 xt 2  b22 yt 2  ...   t
• 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.
Eviews Granger causality test result
Null Hypothesis
x does not Granger Cause y
y does not Granger Cause x
F-Statistic Probability
F1
F2
P1
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.
Leading indicators
y is a leading indicator of x if
• y Granger causes x;
• x does not Granger cause y;
• and y is weakly exogenous.
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% on year earlier, smoothed, prices lagged 6
quarters
Long term trends of money and prices
in UK
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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
Relationships between variables
Period t
Period t+1
xt
xt+1
yt
yt+1
• We do not want the black arrows
• We need to understand the red arrows
Both demand and supply shocks
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Q
OLS is unable to identify either the demand or supply curve
Only supply shocks
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We can identify the demand schedule using OLS
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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:
mt    yt  rt
• Would you estimate this as a single equation
using OLS?
• Very unlikely that money does not affect
real output or the nominal interest rate
Weak exogeneity: example 2
• Uncovered interest parity:
E t et 1  rt  rt
*
• Tests of UIP have performed very poorly,
but ...
• No risk premia and monetary policy might
react to exchange rate changes
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
Single Equation ECM
yt  yt 1  xt  xt 1  ...
Dynamic terms
...    yt 1  xt 1 
Long run
Vector ECMS
xt   1yt   11xt 1   12 yt 1  ...   1 ECM
yt   2 xt   21xt 1   22 yt 1  ...   2 ECM
Halfway between structural VARs and
unrestricted VARs
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:
zt  Azt-1   t , zt  xt , yt '
Given a forecast of {yt} can we forecast {xt}?
• If y is not strongly exogenous to x, feedback
problems
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-1jut-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-1jut-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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60
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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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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
y t = a + b xt + v t
we find that we can reject b = 0.
• Why? Non-stationary data => v nonstationary gives problems with t and F stats
• Also find high R2 and low DW (G&N 1974)
Spurious Regressions
Dependent Variable: Y
Method: Least Squares
Date: 03/31/03 Time: 18:28
Sample: 1900:1 2003:4
Included observations: 416
Variable
X
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Coefficient
Std. Error
t-Statistic
Prob.
0.964478
0.001112
867.6800
0.0000
0.997879
0.997879
5.543177
12751.63
-1302.206
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
202.9399
120.3730
6.265414
6.275103
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 Regression
Dependent Variable: DY
Method: Least Squares
Date: 03/31/03 Time: 18:36
Sample(adjusted): 1900:2 2003:4
Included observations: 415 after adjusting endpoints
Variable
C
DX
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
Std. Error
t-Statistic
Prob.
0.989704
-0.005194
0.016085
0.012185
61.52980
-0.426235
0.0000
0.6702
0.000440
-0.001981
0.211922
18.54827
56.03014
1.752192
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.984475
0.211713
-0.260386
-0.240973
0.181676
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:
yt    xt   t
• 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)
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
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 yt  xt   t
• In steady state yt  xt  0 has little
meaning for the long run
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 wageprice 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.