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**Financial Econometrics**

Introduction to Systems Approach

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Introduction Describe the essential elements of using financial data to do econometrics. 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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**Financial Econometrics**

Financial Econometrics uses all the econometric tools available, to study the properties of variables of interest to them, in particular: Time series properties of assets and asset returns Measurement of Risk Using systems of equations to forecast Assessing the impact of shocks to our variables Efficiency of financial markets

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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

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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 reduced-form equation from the system of equations. 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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**The Vector Autoregressive Approach (VAR)**

To overcome the problems of endogenous variables, one way around the problem is to estimate a system of equations 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

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

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**VAR Assumptions The lagged variables are pre-determined (exogenous)**

When specifying the VAR, we need to decide on the number of lags to include. Too many and we lose degrees of freedom, too few and we could have omitted variable bias. A popular method is to use either the Akaike or Schwarz-Bayesian criteria to determine the lag length

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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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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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