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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

Period tPeriod t+1 xtxt ytyt y t+1 x t+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)

x t is autoregressive Period tPeriod t+1 xtxt ytyt y t+1 x t+1

x t has an autoregressive representation Period tPeriod t+1 xtxt ytyt y t+1 x t+1

x t has an ARMA representation Structural system Reduced form }

Period tPeriod t+1 xtxt ytyt y t+1 x t+1 Granger Causality

Period tPeriod t+1 xtxt ytyt y t+1 x t+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.

Eviews Granger causality test result Null Hypothesis F-Statistic Probability x does not Granger Cause yF1 P1 y does not Granger Cause xF2 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

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 tPeriod t+1 xtxt ytyt y t+1 x t+1 Relationships between variables We do not want the black arrows We need to understand the red arrows

Both demand and supply shocks

OLS is unable to identify either the demand or supply curve

Only supply shocks

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

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

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 Dynamic terms Long run

Vector ECMS 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: Given a forecast of {y t } can we forecast {x t }? 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 y t =  +  y t-1 + u t u ,  2  Substitute recursively: y t =  n +  n y t-n +  n-1  j u t-j The variable will be non-stationary if  =  E(y)=  t Var(y) = Var(  n-1 u t-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 y t =  n +  n y t-n +  n-1  j u t-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 y t = a + b t + u t 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

Difference vs trend stationarity Can you tell the difference? x t = 1 + x t-1 + 0.6 u t z t = 1 + 0.15 t + 0.8 e t Can you tell the difference with a near-unit root?

Unit root vs near-unit root

Testing for unit roots Dickey-Fuller test Write y t =  y t-1 + e t as y t - y t-1 = (  -1)y t-1 + e t 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 -0.77048p-value 0.8258 w: ADF statistic -6.90130 p-value 0.0000 Small sample (40 observations) x: ADF statistic 0.39323p-value 0.9804 w: ADF statistic -0.49216p-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: y t = y t-1 + u t u  , 1  x t = x t-1 + e t e  , 1  y and x are uncorrelated, but estimating y t = a + b x t + v t 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 R 2 and low DW (G&N 1974)

Spurious Regressions

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

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)

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

Modelling short run Assume y t =  x t +  y t-1 +  x t-1 +  t,  ,  2  If a LR relationship exists y t =  +  x t We can write  y t =  x t - (1-  )(y t-1 -  -  x t-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.

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