Parameter Constraints & Significance

Setting the values of one or more parameters for a GARCH model or applying constraints to the range of permissible values can be useful.

The art of the econometrician consists in finding the set of assumptions which are both sufficiently specific and sufficiently realistic to allow him to take the best possible advantage of the data available to him. Edmond Malinvaud, “Statistical Methods of Econometrics”

If you have insight into the values or ranges for specific parameters then you can use the setfixed() and setbounds() functions to set these before fitting the model.

Reference Model

Let’s start by creating a reference model using a Standard GARCH model with Skewed Student-t Distribution fore residuals and an AR(1) model for the mean.

specification <- ugarchspec( 
  mean.model = list(armaOrder = c(1, 0)),
  variance.model = list(model = "sGARCH"),
  distribution.model = "sstd"
)
specification


*---------------------------------*
*       GARCH Model Spec          *
*---------------------------------*

Conditional Variance Dynamics 	
------------------------------------
GARCH Model		: sGARCH(1,1)
Variance Targeting	: FALSE 

Conditional Mean Dynamics
------------------------------------
Mean Model		: ARFIMA(1,0,0)
Include Mean		: TRUE 
GARCH-in-Mean		: FALSE 

Conditional Distribution
------------------------------------
Distribution	:  sstd 
Includes Skew	:  TRUE 
Includes Shape	:  TRUE 
Includes Lambda	:  FALSE 

Fit that to the data and check the coefficients.

fit <- ugarchfit(data = TATASTEEL, spec = specification)

        Estimate  Std. Error  t value Pr(>|t|)
mu         0.001       0.001    2.500    0.012
ar1       -0.031       0.025   -1.242    0.214
omega      0.000       0.000    1.471    0.141
alpha1     0.038       0.005    7.270    0.000
beta1      0.951       0.007  134.613    0.000
skew       1.045       0.036   28.858    0.000
shape      5.072       0.657    7.721    0.000

Neither of the values for ar1 or omega appear to be significantly different to zero.

Restricted GARCH: Specified Parameter Values

Let’s test the setfixed() function. Restrict the model by imposing fixed values for ar1 and omega.

restricted <- specification
setfixed(restricted) <- list(ar1 = 0, omega = 0)
restricted


*---------------------------------*
*       GARCH Model Spec          *
*---------------------------------*

Conditional Variance Dynamics 	
------------------------------------
GARCH Model		: sGARCH(1,1)
Variance Targeting	: FALSE 

Conditional Mean Dynamics
------------------------------------
Mean Model		: ARFIMA(1,0,0)
Include Mean		: TRUE 
GARCH-in-Mean		: FALSE 

Conditional Distribution
------------------------------------
Distribution	:  sstd 
Includes Skew	:  TRUE 
Includes Shape	:  TRUE 
Includes Lambda	:  FALSE 

Fit the restricted model.

fit <- ugarchfit(data = TATASTEEL, spec = restricted)

        Estimate  Std. Error  t value Pr(>|t|)
mu         0.001       0.001    2.242    0.025
ar1        0.000          NA       NA       NA
omega      0.000          NA       NA       NA
alpha1     0.030       0.004    7.257    0.000
beta1      0.969       0.004  237.205    0.000
skew       1.041       0.034   30.432    0.000
shape      5.723       0.717    7.978    0.000

Note that the values for ar1 and omega are those prescribed but that the remaining parameters have been fit to the data. Since these parameters are not being fit to the data we get neither a \(t\) nor \(p\)-value.

Restricted GARCH: Specified Parameter Ranges

Suppose that, rather than specific values, you want to constrain a parameter to values in a particular range.

ranges <- specification
setbounds(ranges) <- list(shape = c(10, 100))

That range encompasses the value obtained for shape in the reference model.

Estimate the bound constrained model.

fit <- ugarchfit(data = TATASTEEL, spec = ranges)

coef(fit)

NULL

Hmmmm. There are no coefficients? What has gone wrong here?

This is a somewhat subtle problem. The values assigned to the model parameters are obtained by numerical optimisation. The optimisation procedure assumes initial values for the model parameters and then iteratively refines those values until it converges on an optimal solution. Where do those initial values come from? They are chosen empirically as “good” (but also random) values. What has happened here is that the initial value for shape lies outside the range specified by setbounds(), so the optimisation algorithm finds that the initial set of parameter values doesn’t satisfy the constraints. And because the initial parameters are invalid it cannot proceed. 💡 The default initial values are generally very good but they can fail when constraints are imposed!

We can fix this by using setstart() to set an initial value for the shape parameter that lies within the permitted range.

setstart(ranges) <- list(shape = 50)

This can be done for multiple parameters. Now fit the model and check the coefficients.

fit <- ugarchfit(data = TATASTEEL, spec = ranges)

coef(fit)

           mu           ar1         omega        alpha1         beta1 
 1.475421e-03 -2.510039e-02  7.619083e-06  3.601476e-02  9.483590e-01 
         skew         shape 
 1.039648e+00  1.000000e+01 

Aha! Valid parameters. And the value obtained for shape lies within the specified range.