gauss-qardl-library

This repository houses the materials to install and use the GAUSS quantile cointegration code by Jin Seo Cho.

What is GAUSS?

GAUSS is an easy-to-use data analysis, mathematical and statistical environment based on the powerful, fast and efficient GAUSS Matrix Programming Language. GAUSS is a complete analysis environment with the built-in tools you need for estimation, forecasting, simulation, visualization and more.

What is the GAUSS QARDL library?

The GAUSS QARDL library is a collection of GAUSS codes developed by Jin Seo Cho. The raw codes provided by Jin Seo Jo have been modified to:

1. Make use of GAUSS structures.
2. Use the internal quantileFit procedure.
3. Include new comments and explanations in the example files.
4. Use up-to-date graphing tools in the example.

Note: The QARDL library no longer requires the QREG library. It uses the internal quantileFit procedure instead.

Getting Started

Installing

GAUSS 20+ The GAUSS QARDL library can be installed and updated directly in GAUSS using the GAUSS package manager.

GAUSS 19+ The GAUSS QARDL library can be easily installed using the GAUSS Application Installer, as shown below:

1. Download the zipped folder qardl_1.0.0.zip from the QARDL Library Release page.

2. Select Tools > Install Application from the main GAUSS menu.
   

3. Follow the installer prompts, making sure to navigate to the downloaded qardl_1.0.0.zip.

4. Before using the functions created by qardl you will need to load the newly created qardl library. This can be done in a number of ways:

• Navigate to the Library Tool Window and click the small wrench located next to the qardl library. Select Load Library.
  
• Enter library qardl in the Program Input/output Window.
• Put the line library qardl; at the beginning of your program files.

Note: I have provided the individual files found in qardl_1.0.0.zip for examination and review. However, installation should always be done using the qardl_1.0.0.zip from the release page and the GAUSS Application Installer.

The qardl Procedure Returns

Estimated P and Q Orders

These are the obtained QARDL orders obtained by the information criterion.

• The demo.e program estimates a p order of 2 and a q order of 1.

Estimated p order
=========================================================
       2.0000000
=========================================================
Estimated q order
=========================================================
       1.0000000

Long-run parameter estimate (β)

• These are the long-run parameters given from the lowest percentile. In the demo.e program there are two explanatory variables and three quantiles: 0.25, 0.50 and 0.75. The following results are printed for β.

• These are stored in the qardlOut structure in the qardlOut.bigBt element.

=========================================================
Long-run parameter estimate (Beta)
=========================================================

       6.6645846
       6.6668972
       6.6659552
       6.6666716
       6.6652370
       6.6663398

• The first two values, 6.6645846 and 6.6668972, are the long-run parameters of the first and second parameters, respectively, at the 0.25 percentile.
• The next two estimates, 6.6659552 and 6.6666716, are the long-run parameters of the first and second parameters, respectively, at the 0.50 percentile.
• The final two estimates, 6.6652370 and 6.6663398, are the long-run parameters of the first and second parameters, respectively, at the 0.75 percentile.

Covariance matrix estimate of long-run parameter (β):

• This is the estimated covariance of the long-run parameters.
• These are stored in the qardlOut structure in the qardlOut.bigbt_cov element.

The demo.e program prints the following 6x6 covariance matrix:

=========================================================
Covariance matrix estimate of long-run parameter (Beta)
=========================================================

   107.147    -17.451     58.427     -9.516     39.498     -6.433
   -17.451     15.212     -9.516      8.295     -6.433      5.608
    58.427     -9.516     95.579    -15.567     64.614    -10.524
    -9.516      8.295    -15.567     13.570    -10.524      9.174
    39.498     -6.433     64.614    -10.524    101.923    -16.601
    -6.433      5.608    -10.524      9.174    -16.601     14.471

Short-run parameter estimate (Φ)

• These are the short-run parameters given from the lowest percentile first.

• These are stored in the qardlOut structure in the qardlOut.phi element.

• The demo.e program estimates an autoregressive order (p) of 2 and has three percentiles (0.25, 0.50, and 0.75). This results in 6 short-run parameter estimates. It prints the following results:

=========================================================
Short-run parameter estimate (Phi)
=========================================================

      0.25537159
   -0.0043015969
      0.26163588
   -0.0069863046
      0.26073101
   -0.0063757138

• The first two values (0.25537159 and -0.0043015969) are the short-run parameters of the first and second lagged dependent variables, respectively at the percentile of 0.25.
• The next two estimates (0.26163588 and -0.0069863046) are the short-run parameters of the first and second lagged dependent variables, respectively at the percentile of 0.50.
• The final two estimates (0.26073101 and -0.0063757138) are the short-run parameters of the first and second lagged dependent variables, respectively at the percentile of 0.75.

Covariance matrix estimate of short-run parameter (Φ)

• This is the estimated covariance of the short-run parameters.

• These are stored in the qardlOut structure in the qardlOut.phi_cov element.

• The demo.e program estimates the following covariance for Φ :

=========================================================
Covariance matrix estimate of short-run parameter (Phi)
=========================================================

     0.238     -0.130      0.129     -0.070      0.087     -0.047
    -0.130      0.083     -0.070      0.045     -0.047      0.030
     0.129     -0.070      0.211     -0.115      0.142     -0.077
    -0.070      0.045     -0.115      0.074     -0.077      0.050
     0.087     -0.047      0.142     -0.077      0.225     -0.122
    -0.047      0.030     -0.077      0.050     -0.122      0.079

Short-run parameter estimate (γ)

• These are the short-run parameters given from the lowest percentile.

• These are stored in the qardlOut structure in the qardlOut.gamma element.

• The demo.e program has 2 variables and three percentiles (0.25, 0.50, and 0.75). This results in 6 short-run parameter estimates. It prints the following results:

=========================================================
Short-run parameter estimate (Gamma)
=========================================================

     4.991
     4.993
     4.968
     4.969
     4.972
     4.973

• The first two values (4.9913074 and 4.9930394) are the short-run parameters of the first and second variables, respectively at the percentile of 0.25.
• The next two estimates (4.9684725 and 4.9690065) are the short-run parameters of the first and second explanatory variables, respectively at the percentile of 0.50.
• The final two estimates (4.9698987 and 4.9707210) are the short-run parameters of the first and second explanatory variables, respectively at the percentile of 0.75.

Covariance matrix estimate of short-run parameter (γ)

• This is the estimated covariance of the short-run γ estimates.

• This is stored in the qardlOut structure in the qardlOut.gamma_cov element.

• The demo.e program estimates the following covariance for γ:

=========================================================
Covariance matrix estimate of short-run parameter (Gamma)
=========================================================

     2.774      2.775      1.506      1.506      1.017      1.017
     2.775      2.776      1.506      1.506      1.017      1.017
     1.506      1.506      2.454      2.454      1.659      1.659
     1.506      1.506      2.454      2.455      1.659      1.659
     1.017      1.017      1.659      1.659      2.618      2.619
     1.017      1.017      1.659      1.659      2.619      2.619

Wald Testing

The QARDl library provides the functions for performing Wald tests. The wtestlrb, wtestlrp, and wtestlrg procedures. The procedures perform Wald tests for β, φ, and γ, respectively. Each test returns the test statistic and it's corresponding p-value.

Setting up the hypothesis tests

Each Wald test procedure requires 5 inputs to summarize the estimated parameters, the desired tests, and the data.

Input	Description
coeff	The β, φ or γ parameter estimates stored in qardlOut.beta, qardlOut.phi, or qardlOut.gamma elements, respectively.
cov	The estimated parameter covariance, stored in qardlOut.beta_cov, qardlOut.phi_cov, or qardlOut.gamma_cov, respectively.
bigR	The R restriction matrix in the null hypothesis.
smlr	The r restriction matrix in the null hypthesis.
data	The dataset used to estimate the coefficients.

Constructing the restriction matrices

The Wald tests in the QARDL library test the null hypothesis:

$H_0 : R \beta = r$

against the alternative

$H_A : R \beta \neq r$

Example

As an example, consider the Wald test of β that is performed in the demo.e program. This test looks at the null hypothesis:

$H_0 : \beta_1(0.25) = \beta_1(0.50) = \beta_1(0.75)$

To do this we set :

bigR = { 1 0 -1 0 0 0, 0 0 1 0 -1 0};
smlr = {0, 0};

Wald test results

The demo.e program prints the following results for the Wald tests:

=========================================================
 Wald test (Beta) and its p-value
=========================================================
     2.220      0.330
=========================================================
 Wald test (Phi) and its p-value
=========================================================
     2.061      0.357
=========================================================
 Wald test (Gamma) and its p-value
=========================================================
     2.356      0.308
=========================================================

These p-values suggest that we cannot reject the null hypothesis and there is not evidence for asymmetries in β₁.

Rolling QARDL Testing

The rollingQARDL procedure computes the QARDL regression for a for a rolling fixed window. The window is fixed at 10% of the time series length.

The rollingQARDL inputs

The rollingQARDL procedure requires the same inputs as the qardl procedure and one additional input, the waldTestRestrictions structure. The waldTestRestrictions structure contains 6 members:

Member	Description
waldR.bigR_gamma	Matrix, R restriction matrix in the null hypothesis for the test for γ.
waldR.smlr_gamma	Matrix, r restriction matrix in the null hypothesis for the test for γ.
waldR.bigR_phi	The R matrix in the null hypothesis for the test for φ.
waldR.smlr_phi	The r restriction matrix in the null hypothesis for the test for φ.
waldR.bigR_beta	The R matrix in the null hypothesis for the test for β.
waldR.smlr_beta	The r restriction matrix in the null hypothesis for the test for β.

The rollingQARDL outputs

The rollingQARDL procedure has one output, the rollingQARDLOut output structure. The rollingqardlOut structure has 6 elements. In each element, the estimates for separate quantiles (τ) are stored in individual columns, while each row corresponds to the separate estimation window.

Member	Description
rqaOut.bigbt	An array of beta estimates which contains the estimates for each of the independent variables on a separate plane.
rqaOut.bigbt_se	An array of standard error estimates which contains the se estimates for each of the independent variables on a separate plane. .
rqaOut.phi	An array of phi estimates which contains the estimates for each lagged independent variable on a separate plane.
rqaOut.phi_se	An array of standard error estimates which contains the se estimates for each lagged independent variable on a separate plane.
rqaOut.gamma	An array of gamma estimates which contains the estimates for each of the independent variables on a separate plane.
rqaOut.gamma_se	An array of standard error estimates which contains the se estimates for each of the independent variables on a separate plane.

More about the GAUSS QARDL library can be found in the blog, The Quantile Autoregressive-Distributed Lag Parameter Estimation and Interpretation in GAUSS.

Authors

Eric Clower
Aptech Systems, Inc