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**Financial Econometrics**

Introduction to Systems Approach

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Introduction Complete the description of Error Correction Models and assess an example of cointegration using PPP. Explain the problems associated with simultaneous equation estimation. Describe the order condition and why this is important when estimating a system of equations Introduce the Vector Autoregressive (VAR) approach to estimating simultaneous equation models.

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Error Correction Term The error correction term tells us the speed with which our model returns to equilibrium following an exogenous shock. It should be negatively signed, indicating a move back towards equilibrium, a positive sign indicates movement away from equilibrium The coefficient should lie between 0 and 1, 0 suggesting no adjustment one time period later, 1 indicates full adjustment The error correction term can be either the difference between the dependent and explanatory variable (lagged once) or the error term (lagged once), they are in effect the same thing.

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Example of ECM The following ECM was formed, using 60 observations:

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Example of an ECM The error correction term has a t-statistic of 4, which is highly significant supporting the cointegration result. The coefficient on the error correction term is negative, so the model is stable. The coefficient of -0.32, suggests 32% movement back towards equilibrium following a shock to the model, one time period later.

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**Potential Problems with Cointegration**

The ADF test often indicates acceptance of the null hypothesis (no cointegration), when in fact cointegration is present. The ADF test is best when we have a long time span of data, rather than large amounts of observations over a short time span. This can be a problem with financial data which tends to cover a couple of years, but with high frequency data (i.e. daily data) It is only really used for bi-variate cointegration tests, although it can be used for multivariate models, a different set of critical values is required.

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**Multivariate Approach to Cointegration**

A different approach to testing for cointegration is generally required when we have more then 2 endogenous variables in the model. If we assume all the variables are endogenous, we can construct a VAR and then test for cointegration. One of the most common approaches to multivariate cointegration is the Johansen Maximum Likelihood (ML) test. This test involves testing the characteristic roots or eigenvalues of the π matrix (coefficients on the lagged dependent variable).

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**Steps in Testing for Cointegration**

Test all the variables to determine if they are I(0), I(1) or I(2) using the ADF test. If both variables are I(1), then carry out the test for cointegration If there is evidence of cointegration, use the residual to form the error correction term in the corresponding ECM Add in a number of lags of both explanatory and dependent variables to the ECM Omit those lags that are insignificant to form a parsimonious model Use the ECM for dynamic forecasting of the dependent variable and assess the accuracy of the forecasts.

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**Simultaneous Equations**

When we run a regression using OLS, we assume the explanatory variables are exogenous If the explanatory variables are endogenous, the estimates produced by OLS are biased This means are estimator is not BLUE, therefore are t and F statistics are invalid. This is known as simultaneous equation bias However a potential problem occurs with deciding which variables are exogenous.

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**Measures to Overcome the Bias**

One way to overcome the bias is to form ‘reduced form’ equations, in which we rearrange our model, by a process of substitution, until all the explanatory variables are exogenous. Another method involves using instrumental variable techniques and involves finding exogenous variables to act as instruments for the endogenous variables. A popular way to overcome the problem in the finance literature is to estimate a system of equations, termed a vector autoregressive (VAR) approach.

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**Reduced-Form Equations**

When producing a reduced form equation, we have to ensure our equation is identified. This means that we can form the coefficients in our structural equation from the estimated reduced form coefficients. There are three forms of identification: - Exactly Identified – We can form unique values for the structural coefficients - Under Identified – It is not possible to form the structural coefficients - Over Identified – We produce more then one value for the structural coefficients

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**Order Condition of Identification**

To determine if an equation is identified, under-identified or over-identified, we need to apply the order condition We also need to test the rank condition in theory, however the order condition is usually adequate to ensure identifiability The easiest way of applying the order condition, is to use the following: In a model of M simultaneous equations in order for an equation to be identified, it must exclude at least M-1 variables (endogenous and exogenous)

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**Example of order condition**

The following equations have (1) as exactly identified, (2) as overidentified

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**Indirect Least Squares (ILS)**

This consists of three stages: Obtain the reduced form equations. Apply OLS to the reduced form-equations. Obtain estimates of the original structural coefficients from the estimated reduced-form coefficients. This approach assumes the original equations are exactly identified.

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**Instrumental Variables (IV)**

An instrumental variable can be interpreted as a ‘proxy’ for a variable which suffers from simultaneity. In effect the instrumental variable is highly correlated with the endogenous variable, but is uncorrelated with the error term. If such a variable exists, OLS can be used to estimate the model with the IV instead of the endogenous variable. One way to incorporate an IV into a model is to use two-stage-least squares.

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**Two-Stage-Least Squares**

Given the following model, where equation (1) is overidentified:

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**Two-Stage Least Squares- Stage 1**

To get rid of the correlation between the endogenous variable s(t) and the residual u(t) in equation (1), regress s(t) on all the pre-determined (exogenous) variables in the system. In this case it is just x(t) and s(t-1) as the explanatory variables. Obtain the fitted value of the variable s(t), which is uncorrelated with the residual. This can now be used as an instrument in the original equation.

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Stage 2 The final step involves regressing the initial equation (1), but using the fitted value for s(t) as an instrument for s(t) The fitted value is a good instrument because it is similar to s(t), but uncorrelated with the residual. The standard errors will need to be adjusted, however this is a fairly standard routine, that most computer software automatically completes.

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Features of 2SLS It can be directly applied to an equation in a system, without needing to take into account any other equations. It can be used for both exactly identified and overidentified equations. Easy to use, the only information required is what the exogenous variables are. Given that the S.E can be calculated, t-statistics can also be used. It is a large-sample technique.

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**The Vector Autoregressive Approach (VAR)**

To overcome the problems of endogenous variables, one way around the problem is to estimate a system of equations, All the equations are identified, as we exclude all the level variables as explanatory variables In this case there are as many equations as there are variables in the system, with each variable acting as a dependent variable. This dependent variable is then regressed against lags of the other variables in the system and itself. The lagged variables are pre-determined (exogenous) When specifying the VAR, it is important to decide on the optimal number of lags to include.

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**Model (Equities (s) and interest rates (r))**

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VAR Assumptions OLS estimation of a single equation in the Unrestricted VAR ****************************************************************************** Dependent variable is DR3 123 observations used for estimation from 1961Q2 to 1991Q4 Regressor Coefficient Standard Error T-Ratio[Prob] DR3(-1) [.390] DR3(-2) [.013] DR3(-3) [.297 DR3(-4) [.671] DR10(-1) [.287] DR10(-2) [.070] DR10(-3) [.296] DR10(-4) [.990] DTBILL(-1) [.137 DTBILL(-2) [.158] DTBILL(-3) [.091] DTBILL(-4) [.546] K E [.993 As well as the usual diagnostic tests……

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Main Uses of VARs This model is often used to link different markets, such as the bond and stock markets. Testing for causality between variables The main use is forecasting, due to the dynamic nature of this model, it can produce reasonable dynamic forecasts of the variables in the system When the VAR is adapted slightly, it can be turned into a Vector Error Correction Model (VECM), this can then be used for assessing long as well as short run relationships.

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Causality Tests Regression does not imply causality, to test for causality a lagged model needs to be used. ‘Granger Causality’ is the main approach to testing for causality, in general the term causality is not used, instead it is said A ‘Granger causes’ B. Granger Causality tests can also be used to determine if a variable is exogenous or not.

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Granger Causality To test for ‘Granger causality’ between two variables, simply run the following VAR:

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Granger Causality To determine if there is any evidence of causality from s to r, we conduct a F-test for joint significance of the lagged explanatory variables. If jointly significantly different to 0, then causality run from s to r. If the same process is carried out on the other equation and this time the lagged explanatory variables are insignificant, we say ‘s causes r’. If we carry out the tests on both equations and it appears s causes r and r causes s then we have bi-causality, i.e. causality runs in both directions, many argue this suggests an invalid relationship between the variables.

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Conclusion Simultaneous equation bias is a serious problem in econometric modelling We can to an extent overcome the problem by using a reduced-form equation approach, assuming the equation is identified exactly A further way around the problem is to use a VAR, where all the explanatory variables are lagged, therefore pre-determined.

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