Functional Form and Dynamic
Models
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
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.
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.
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.
yt     xt  ut
2
uˆt   0  1 yˆ t
3
  2 yˆ t
p
 ...   p 1 yˆ t
 vt
Ramsey Reset Test
• The R-squared statistic is taken from the
secondary regression and the test statistic
formed: T*R-squared.
• It follows a chi-squared distribution with (p-1)
degrees of freedom.
• The null hypothesis is the functional form is
suitable.
• If a T*R-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, 7.6>5.99 so reject the null and the
functional form is a problem.
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.
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.
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).
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:
yt   0  1xt   2 xt 1  3 yt 1  ut
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.
yt  yt  yt 1
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 steadystate values and are no longer changing.
• In the long-run we can ignore the lags as:
yt  yt 1  yt 2  y *
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.
Long-run
• For example given the following model, we
can use the previous rules to form a longrun steady-state solution:
yt   0  1xt   2 yt 1   3 xt 1  ut
0   0   2 y *  3 x *
 2 y*   0   3 x *
0 3
y*  

x*
2 2
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
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:
yt    xt  xt 1  2 xt 2  3 xt 3.....  ut
1    1
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
β and δ parameters, so it would be
impossible to identify its real value.
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 δ, 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.
Koyck Transformation
• The Koyck transformation produces the
following model:
yt   (1   )  xt  yt 1  ut  ut 1
The new constant is :  (1   )
The new error term is : (ut  ut 1 )  vt
Koyck Transformation
• The transformed Koyck model produces
estimates of β and δ, 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.
Koyck Model
• The long-run model is as follows:
y*   (1   )  x * y *
y*   

1 
x*
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:
Cov( xt ut )  0
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.
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:
sˆt  0.96  0.12 yt  0.54st 1
(0.76) (0.04) (0.18)
R  0.65, DW  1.87
2
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:
  0.12
  0.54
 (1   )  0.96
 * (1  0.54)  0.96
  0.96 / 0.46  2.09
Long-run
• These estimates can also be used to
produce the long-run solution as follows:
s*     /(1   ) y *
s*  2.09  0.12 /(1  0.54) y *
s*  2.09  0.26 y *
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.