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Structural modelling: Causality, exogeneity and unit roots
Andrew P. Blake
CCBS/HKMA May 2004
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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
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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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-0.08
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94 95 96 97 98 99 00 01
X Y
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Period t Period t+1
xt
yt yt+1
xt+1
Inter-relationships between variables
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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)
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xt is autoregressive
Period t Period t+1
xt
yt yt+1
xt+1
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xt has an autoregressive representation
Period t Period t+1
xt
yt yt+1
xt+1
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xt has an ARMA representation
11
11
111
1
,
1
ttttt
tttt
ttt
ttt
ttt
xxso
xy
xy
yy
yx
Structural system
Reduced form
}
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Period t Period t+1
xt
yt yt+1
xt+1
Granger Causality
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Period t Period t+1
xt
yt yt+1
xt+1
Vector autoregressions (VARs)
Needs to be modelled to have
a structural interpretation
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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
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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
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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.
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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
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Long term trends of money and prices in UK
0.0
5.0
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20.0
25.0
30.0
% o
n y
ear
earl
ier,
sm
oo
thed
, p
rices l
ag
ged
6
qu
art
ers
Broad Money Prices
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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
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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
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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
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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
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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
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Both demand and supply shocks
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P
Q
OLS is unable to identify either the demand or supply curve
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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
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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
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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
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Interest rate differentials
Exchange rate change
Question: how would you test for exogeneity in UIP?
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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
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11
11
...
...
tt
tttt
xy
xxyy
Single Equation ECM
Dynamic terms
Long run
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Vector ECMS
Halfway between structural VARs and unrestricted VARs
ECMyxxy
ECMyxyx
tttt
tttt
21221212
11121111
...
...
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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
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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
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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
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Invariance
• The process driving a variable does not change in the face of shocks
• Linked to ‘deep parameters’
• Example: the Lucas critique
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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
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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
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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
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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
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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
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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)
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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
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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
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Difference vs trend stationary
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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?
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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
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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
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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
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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
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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)
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.