Functional%20Form%20and%20Dynamic%20Models

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Functional Form and Dynamic Models
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Introduction

• Discuss the importance of functional form
• Examine the Ramsey Reset Test for Functional Form
• Describe the use of lags in econometric models
• Evaluate the Koyk transformation as a means of
  overcoming some of the problems of lagged
  variables

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Functional Form

• A further assumption we make about the
  econometric model is that it has the correct
  functional form.
• This requires the most appropriate variables in
  the model and that they are in the most suitable
  format, i.e. logarithms etc.
• One of the most important considerations with
  financial data is that we need to model the
  dynamics appropriately, with the most appropriate
  lag structure.

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Functional Form

• It is important we include all the relevant
  variables in the model, if we exclude an
  important explanatory variable, the regression
  has omitted variable bias. This means the
  estimates are unreliable and the t and F
  statistics can not be relied on.
• Equally there can be a problem if we include
  variables that are not relevant, as this can
  reduce the efficiency of the regression, however
  this is not as serious as the omitted variable
  bias.
• The Ramsey Reset test can be used to determine if
  the functional form of a model is acceptable.

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Ramsey Reset Test for Functional Form

• This test is based on running the regression and
  saving the residual as well as the fitted
  values.
• Then run a secondary regression of the residual
  on powers of these fitted values.

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Ramsey Reset Test

• The R-squared statistic is taken from the
  secondary regression and the test statistic
  formed TR-squared.
• It follows a chi-squared distribution with (p-1)
  degrees of freedom.
• The null hypothesis is the functional form is
  suitable.
• If a TR-squared statistic of 7.6 is obtained and
  we had up to the power of 3 in the secondary
  regression, then the critical value for
  chi-squared (2) is 5.99, 7.7gt5.99 so reject the
  null and the functional form is a problem.

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Lagged Variables

• A possible source of any problem with the
  functional form is the lack of a lagged structure
  in the model.
• One way of overcoming autocorrelation is to add a
  lagged dependent variable to the model.
• However although lagged variables can produce a
  better functional form, we need theoretical
  reasons for including them.

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Inclusion of Lagged variables

• Inertia of the dependent variable, whereby a
  change in an explanatory variable does not
  immediately effect the dependent variable.
• The overreaction to news, particularly common
  in asset markets and often referred to as
  overshooting, where the asset overshoots its
  long-run equilibrium position, before moving back
  towards equilibrium
• To allow the model to produce dynamic forecasts.

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Types of Lag

• Autoregressive refers to lags in the dependent
  variable
• Distributed lag refers to lags of the explanatory
  variables
• Moving average refers to lags in the error term
  (covered later).

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ARDL Models

• An Autoregressive Distributed lag model or ARDL
  model refers to a model with lags of both the
  dependent and explanatory variables. An ARDL(1,1)
  model would have 1 lag on both variables

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Differenced Variables

• A differenced or change variable is used to
  model the change in a variable from one time
  period to the next. This type of variable is
  often used with lagged variables to model the
  short run.

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The long-run static equilibrium

• In econometrics the long and short run are
  modelled differently. (later we will use
  cointegration to model this).
• The long-run equilibrium is defined as when the
  variables have attained some steady-state values
  and are no longer changing.
• In the long-run we can ignore the lags as

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Long-Run

• To obtain the long-run steady-state solution from
  any given model we need to
• - Remove all time subscripts, including lags
• - Set the error term equal to its expected
  value of 0.
• - Remove the differenced terms
• - Arrange the equation so that all x and y
  terms are on the same side.

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Long-run

• For example given the following model, we can use
  the previous rules to form a long-run
  steady-state solution

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Potential Problems with Lagged Variables

• The main problem is deciding how many lags to
  include in a model.
• The use of lagged dependent variables can produce
  some econometric problems.
• With a number of lags, there can be problems of
  multicollinearity between the lags
• There can be difficulties with interpreting the
  coefficients on the lags and offering a
  theoretical reason for their inclusion

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Koyck Distribution

• The Koyck distribution is a general dynamic model
  with a number of applications.
• The distribution has the lagged values of the
  explanatory variables declining geometrically. In
  the case of one explanatory variable it follows
  the following form

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Koyck Distribution

• The previous model can not be estimated using the
  usual OLS techniques as
• - There would almost certainly be
  multicolliinearity
• - There would be multiple estimates of the ß
  and d parameters, so it would be impossible to
  identify its real value.

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Koyck Transformation

• It is possible to obtain a model which is easier
  to estimate by performing the Koyck
  transformation.
• This requires the equation from earlier to be
  lagged and multiplied by d, so the dependent
  variable is now y(t-1).
• By subtracting this second equation from the
  first, all the lagged values of x cancel out.

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Koyck Transformation

• The Koyck transformation produces the following
  model

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Koyck Transformation

• The transformed Koyck model produces estimates of
  ß and d, which can then be used to produce
  estimates of the coefficients in the original
  Koyck distribution.
• This model allows both the short and long run to
  be analysed separately, the previous model is the
  short run, in the long run we ignore the lags and
  error term to produce the following long-run
  model.

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Koyck Model

• The long-run model is as follows

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Koyck Model

• Although this transformed model appears better
  than the original model it suffers from a
  problem.
• The lagged dependent variable (y) is now an
  explanatory variable and in the new error term
  there is a lagged error term (u).
• Given that both these terms appear in the
  original Koyck distribution in non-lagged form
  they must be related.
• This means the fourth Gauss-Markov assumption is
  failed, leading to biased and inconsistent OLS
  estimates as

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Koyck Model

• To obtain unbiased estimates of the parameters in
  the transformed Koyck model, we need to use an
  Instrumental Variable (IV) technique. (This will
  be covered later).
• Alternatively we could use a non-linear method to
  estimate the original Koyck distribution,
  although this too requires an alternative
  technique to OLS.

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Koyck Transformation

• Given the following estimates from a model of
  income (y) on stock prices (s), we can use them
  to interpret the original Koyck distribution on
  which they are based

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Koyck Transformation

• The previous estimates can be used to produce
  values for all the original parameters, which can
  then be inputted into the original Koyck
  distribution

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Long-run

• These estimates can also be used to produce the
  long-run solution as follows

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Conclusion

• It is important to ensure the functional form of
  the econometric model is correct.
• This may require the inclusion of lags.
• The use of lags and differenced variables allows
  the examination of the short-run dynamic
  properties of the model.
• The Koyck distribution is a general model for
  examining the dynamics.