STR: An Ox package for smooth transition regression modelling

News / changes:

2014-06-18: Numerical 2nd derivatives for standard errors was accidentally switched off. Fixed initial quasi-Hessian for MaxBFGS (Jurgen Doornik)
2013-08-12: Adjusted for Ox 7. (Jurgen Doornik)
13-3-06: Adjusted for Ox 4. (Jurgen Doornik)
19-10-04: Package available for download again. Small changes to the code and documentation. Package now supplied as zip file, with example code. (Jurgen Doornik)
27.08.01: Downloadable Oxpack compiled for Ox 3.00. File named STR2.oxo. (Ivar Pettersen)
27.08.01: The experimental long-run estimates procedure has been hidden. (Ivar Pettersen)

I. Introduction

Based on Professor Timo Teräsvirta's GAUSS code Ivar Pettersen has written a class of procedures, called STR2 which estimate and test STAR-models.

We recommend Teräsvirta's paper on STR-estimation ("Handbook of applied economic statistics", volume 155, edited by Ullah and Giles, 1998 by Marcel Dekker, Inc) as a starting point. In that paper, a UK House data file is used, which has been kindly made available to us by David Hendry. It is installed with STR2 in the form of ukhouse4.in7/ukhouse4.bn7.

Please note that this program is very much in its Beta-stage, and we accept no responsiblity for any estimation errors or other problems. Some of the text in the program is in Norwegian, this will be translate eventually. No support is given for this program. The appropriate forum to discuss problems is the ox-users discussion list.

II. Installation

Create the STR2 folder in ox\packages (in a default installation this would create c:\program files\ox\packages\STR2). Unzip str2_084.zip (or a newer version if available) into this folder.

Next, install STR2 within OxPack (this requires Ox Professional)
Start OxMetrics, enter the Run-menu, select OxPack. The oxpack box includes a menu called Package. From this menu, select the item called Add/Remove Package. The following dialog box should appear:

The upper field should contain the (absolute) address of the STR2 oxo-file: use the Browse button to find this. The lower field should hold the name of the class, in this case STR2. Remember that the field is case-sensitive. Press Add and the STR2 package should be available.

See ukhouse1.ox for an example of an Ox program that uses the STR2 package.

III. Some notes on the Ox source file STR2.ox

The InitX procedures calculate starting values. I originally had one Init procedure that handled all types of STR-models. I chose to divide this to avoid confusion and to gain some speed.

The STR-estimation uses not only the estimated starting values for gamma and c but also the parameter estimates. The program ought to have an option to set the parameter starting values to zero. And the possibility of user-provided starting values should be implemented.

Timo Teräsvirta's procedures are at the end of the program. They are direct translations from the GAUSS files. But I have changed them to work with zero indexing.

The Gradient procedure is fishy. The formulas are Timo Teräsvirtas concerning LSTR1 with one or two transition variables but the ESTR- and LSTR2- calculations are incorrect. I have not given enough attention to this given that I never use ESTR. Comments appriciated. The errors will not affect estimation results, but will make several tests inapplicable.

The program is tested against results from Teräsvirta's estimation of Hendry's housing model. Several model formulations have not been validated (ESTR, LSTR2 and models where a variable is modelled with two transition-variables). The estimates of these routines provide may be wrong. (Better: assume them to be wrong.)

IV. An example for the UK housing market using OxPack

Start OxPack, and select STR2 from the Package menu. Make sure the ukhouse4 data set is loaded in OxMetrics, and select Model/Formulate in OxPack.

IV.1. Model formulation

In the variable selection dialog (choose Model -> Formulate) there are 8 types of variables when right-clicking on a variable(s) in the selection:

Transvar is the button for the transition variable (the first transition variable if using two).
Y_variable selects the endogenous variable.
Lin_variable selects variables that should enter the linear part of the model only.
Nlin_variable selects variables entering both linearly and non-linearly
ZeroRestrictions stands for, well, zero-restrictions and makes sure a Nlin_variable enters non-linearly only.
Transvar2 has several functions depending on what kind of model the user estimates. It was, and still is, the button that selects a second transition variable.
Nlin_variable2 selects variables that should follow this second transition variable.

If the user chooses a model with one transition variable (LSTR1) the Transvar2 and Nlin_variable2 have other purposes. If using LSTR1 all variables marked with Transvar2 will be tested as potential transition variables (given that linearity tests are selected in the options box, this applies to both pre-estimation tests for non-linearity and post-estimation tests for remaining non-linearity). The Nlin_variable2 box is used to select variables to be tested for constancy. It can be used to pick variables entering the linear part only. Variables entering the non-linear part are selected in the options menu (I ran out of buttons so some ad hoc solutions are used).

The program can only handle one or two transition variables. The non-linear variables may be modelled with either one of the transition variables, or both at the same time.

Figure 1: The variable selection dialog box.

The screen capture shows all the options described above. To replicate the results, formulate the model as shown. Notice that all variables which play a role in the estimation are labelled with a letter. The example is typical for an estimation of a model with one transition variable, delph_1. The Constant and mphh1are modelled both linearly and non-linearly, and are labeled N. The variables labeled L are modelled linearly only.

IV.2. Model settings

Figure 2: Model settings

The model setting box selects which tests to perform, sets parameters for starting values grid search and chooses among output options. The entries are:

Print starting values is a setting which allows the user to see the results of the grid search estimation (always a good idea).
Linear tests (pre-estimation) chooses linearity tests. These tests are used to determine whether there is any non-linearity within the model and to choose between the STR-specifications.
Print tr-function prints the values of the estimated transition function to the output file.
Trend is tr.var1 is used to indicate whether trend is used as transition variable. The trend needs special attention concerning standardisation. The program could try to detect a trend, but I chose not to implement this given that the user might want to standardise it before the estimation.
gamma, C The next settings custumizes parameters for the grid search, focusing on the slope parameter gamma and the break point c only. The user supplies a starting value, a step lenght and the number of gammas to be tried. In the example above 100 gammas would be tried, starting at 0 ending at 99. There are two sets of settings for gamma. The second set applies when an additional transition variable is used.
The setting for c determines how many break points should be tried. Once the number of c's have been set the grid search will search between the min and max value of the transition variable. The Step C option overrides No of C in a way, letting the user supply the steplength. If this option is unequal to zero the search will start at the minimum value of the transition variable, use the given increment, where the number of tries still is determined by No of C. By using this options the user risks a search out of the upper boundaries of the transition variable.
There is an option to use Num2Derivative to calculate standard errors. This is recommended. I my experience standard errors are always under-estimated when using iteration techniques. Timo Teräsvirta scales the standard errors with a constant to compensate for this. Consult the Ox-manual for details about this procedure.
Residual Statistics reports a battery of tests for autocorrelastion. The procedure which calculates these tests is a direct translation of Teräsvirta's corresponding GAUSS-file
Draw tr-function and Draw residuals do the obvious. The residuals are drawn together with a set of residuals from a linear regression (in practice a regression where the variables are modelled linearly only).
Reset-test is self-explanatory
Test for remaining non-linearity. This allows the user to check whether the non-linearity within the model is adequately handeled. The tests are available when estimating models with one transition variable only. Testing for remaining non-linearity has an additional purpose. It is possible to check whether the linear variables in your model could contain remaining non-linearity due to an additional transition variable. Labelling variables with the transvar 2-button makes it possible for the user to test several variables. In general, when using models with one transition variable, the transvar 2-button selects variables which should be tested as potential transition variables, both for pre- and post-estimation purposes.
Print residuals prints the values of the STR-residuals to the output file
Testing for parameter constancy is available when using one transition variable only. To select variables in this test the user must use two different methods. To select variables in the linear part, use Nlin_variable 2, to select variables in the non-linear part use the appropriate box in the model settings dialog. Select variables by number. The numbering is based by the order in which the variables appear in the variable selection dialog. Ex: if I wanted to test the constancy of mphh1 I would write 2 in the box (see figure 1 above). Notice that the numbers are separated by ;. Both variables would be tested if I wrote 1;2 in the box.

IV.3. Model estimation

Press OK after making your selections in the model settings dialog takes you to the Estimate Model-box, as seen below:

Figure 3: Estimate Model

This is where one can choose the preferred method and select the sample. LSTR1_2 is the option for two transition variables. Make your selections and the estimations start. The options button lets the user choose convergence criteria for the iteration procedure.

Estimation output is:

STR2 package version 0.84 beta, object created on 13-03-2006
ukhouse4.in7 loaded from D:\OxMetrics4\Ox\packages\STR2\ukhouse4.in7

Cstep: 0.00391612
Best g, c and ssr 
       3.0000     0.086171     0.011502

Number positive:37
Sample: 1959(1) - 1982(2)
Frequency: 4

Transition variable: dlph_1


Number of observations is      94
Variable name, coeff estimate, std deviation, and t-value
Constant               0.404        0.1023         3.948
mphh1                 0.1292       0.02537         5.093
a2dly                 0.4944        0.1091         4.531
f13dlp                0.7176        0.1234         5.817
f13lmp                0.4442        0.1069         4.156
dvbanks               -2.189        0.5344        -4.096
d674                  -3.636         1.265        -2.875
Q1                 0.0009773      0.004815         0.203
Q2                   0.02355      0.004169         5.649
Q3                   0.01706      0.003947         4.322
lyh1                  0.3469       0.06287         5.518
dr1to1               -0.5019        0.2211         -2.27
r1toa1_3             -0.2108       0.08869        -2.377
dlph_2                0.2207       0.07756         2.845
Non-linear parameters: dlph_1
Constant               2.755          0.68         4.052
mphh1                 0.5534        0.1396         3.965



The search for starting values took 0.766 seconds :

Returncode from MaxBFGS: 0
Strong convergence
Returncode from Num2Derivative: 1
Sample: 1959(1) - 1982(2)
Frequency: 4

Transition variable: dlph_1


Number of observations is      94
Variable name, coeff estimate, std deviation, and t-value
GAMMA                  2.484          1.69          1.47
C1                    0.0872      0.009004         9.685
Constant               0.378        0.1209         3.127
mphh1                 0.1231       0.02985         4.122
a2dly                 0.4759        0.1187         4.008
f13dlp                0.6996        0.1328         5.269
f13lmp                0.4354        0.1104         3.942
dvbanks               -2.137        0.5425         -3.94
d674                   -3.61          1.28         -2.82
Q1                  0.001117      0.004895        0.2282
Q2                   0.02358      0.004222         5.585
Q3                   0.01743       0.00408         4.271
lyh1                  0.3409       0.06626         5.146
dr1to1               -0.5135        0.2273        -2.259
r1toa1_3              -0.208       0.09007        -2.309
dlph_2                0.2171       0.07927         2.738
Non-linear parameters: dlph_1
Constant               2.932         1.333         2.199
mphh1                 0.5889         0.271         2.173
Initial func value
     0.011502
SSR/function value  0.0114488
Inv std of tr var      
       35.517
SSR/number of obs.     0.000121796
AIC                    -8.63019
SBIC                   -8.14317
1000*Residual variance 0.150642
R-squared              
      0.84739
Std error of residuals 0.0122736

This is a version of Hendry's UK housing prices model. First the procedure tests for non-linearity. The next output is the result of the grid search. Based on 5000 estimations (50 c's and 100 gammas) we find some maximum. These results are fed into the iterative procedure as starting values. Hopefully the model converges, in my example it does, giving estimation results close to what Teräsvirta estimated. The program then tries to test for remaining non-linearity, but fails. This is caused by the two dummies d674 and dvbanks. The program tries to avoid dummy-variables when testing linearity, but failed to do so in this case because both dummies are formulated in a non-conventional way. The estimated transition function looks like this:

Figure 4: Estimated transition function and residuals

IV.3. Changing the model estimation

A variable selection box for estimating a model with two transition variables is shown below. The difference from above is that Q2 and Q3 are modelled non-linearly with trend as the second transition variable.

Figure 5: The variable selection dialog box.