# arx: Estimate an AR-X model with log-ARCH-X errors In gets: General-to-Specific (GETS) Modelling and Indicator Saturation Methods

## Description

Estimation by OLS, two-step OLS if a variance specification is specified: In the first the mean specification (AR-X) is estimated, whereas in the second step the log-variance specification (log-ARCH-X) is estimated.

The AR-X mean specification can contain an intercept, AR-terms, lagged moving averages of the regressand and other conditioning covariates ('X'). The log-variance specification can contain log-ARCH terms, asymmetry or 'leverage' terms, log(EqWMA) where EqWMA is a lagged equally weighted moving average of past squared residuals (a volatility proxy) and other conditioning covariates ('X').

## Usage

 ```1 2 3 4 5``` ```arx(y, mc=FALSE, ar=NULL, ewma=NULL, mxreg=NULL, vc=FALSE, arch=NULL, asym=NULL, log.ewma=NULL, vxreg=NULL, zero.adj=0.1, vc.adj=TRUE, vcov.type=c("ordinary", "white", "newey-west"), qstat.options=NULL, user.estimator=NULL, user.diagnostics=NULL, tol=1e-07, LAPACK=FALSE, plot=NULL) ```

## Arguments

 `y` numeric vector, time-series or `zoo` object. Missing values in the beginning and at the end of the series is allowed, as they are removed with the `na.trim` command `mc` logical. `TRUE` includes an intercept in the mean specification, whereas `FALSE` (default) does not `ar` either `NULL` (default) or an integer vector, say, `c(2,4)` or `1:4`. The AR-lags to include in the mean specification. If `NULL`, then no lags are included `ewma` either `NULL` (default) or a `list` with arguments sent to the `eqwma` function. In the latter case a lagged moving average of `y` is included as a regressor `mxreg` either `NULL` (default) or a numeric vector or matrix, say, a `zoo` object, of conditioning variables. Note that, if both `y` and `mxreg` are `zoo` objects, then their samples are chosen to match `vc` logical. `TRUE` includes an intercept in the log-variance specification, whereas `FALSE` (default) does not. If the log-variance specification contains any other item but the log-variance intercept, then vc is set to `TRUE` `arch` either `NULL` (default) or an integer vector, say, `c(1,3)` or `2:5`. The log-ARCH lags to include in the log-variance specification `asym` either `NULL` (default) or an integer vector, say, `c(1)` or `1:3`. The asymmetry (i.e. 'leverage') terms to include in the log-variance specification `log.ewma` either `NULL` (default) or a vector of the lengths of the volatility proxies, see `leqwma` `vxreg` either `NULL` (default) or a numeric vector or matrix, say, a `zoo` object, of conditioning variables. If both `y` and `mxreg` are `zoo` objects, then their samples are chosen to match `zero.adj` numeric value between 0 and 1. The quantile adjustment for zero values. The default 0.1 means the zero residuals are replaced by the 10 percent quantile of the absolute residuals before taking the logarithm `vc.adj` logical. If `TRUE` (default), then the log-variance intercept is adjusted by the estimate of E[ln(z^2)], where z is the standardised error. This adjustment is needed for the conditional scale to be equal to the conditional standard deviation. If `FALSE`, then the log-variance intercept is not adjusted `vcov.type` character vector, "ordinary" (default), "white" or "newey-west". If "ordinary", then the ordinary variance-covariance matrix is used for inference. If "white", then the White (1980) heteroscedasticity-robust matrix is used. If "newey-west", then the Newey and West (1987) heteroscedasticity and autocorrelation-robust matrix is used `qstat.options` `NULL` (default) or an integer vector of length two, say, `c(1,1)`. The first value sets the lag-order of the AR diagnostic test, whereas the second value sets the lag-order of the ARCH diagnostic test. If `NULL`, then the two values of the vector are set automatically `user.estimator` `NULL` (default) or a `list` with one entry, `name`, containing the name of the user-defined estimator `user.diagnostics` `NULL` (default) or a `list` with two entries, `name` and `pval`, see the `user.fun` argument in `diagnostics` `tol` numeric value (`default = 1e-07`). The tolerance for detecting linear dependencies in the columns of the regressors (see `qr` function). Only used if `LAPACK` is `FALSE` (default) `LAPACK` logical. If `TRUE`, then use LAPACK. If `FALSE` (default), then use LINPACK (see `qr` function) `plot` `NULL` or logical. If `TRUE`, then the fitted values and the residuals are plotted. If `NULL` (default), then the value set by `options` determines whether a plot is produced or not.

## Details

See Sucarrat and Escribano (2012)

## Value

A list of class 'arx'

## Author(s)

Genaro Sucarrat, http://www.sucarrat.net/

## References

Genaro Sucarrat and Alvaro Escribano (2012): 'Automated Financial Model Selection: General-to-Specific Modelling of the Mean and Volatility Specifications', Oxford Bulletin of Economics and Statistics 74, Issue no. 5 (October), pp. 716-735

Halbert White (1980): 'A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity', Econometrica 48, pp. 817-838

Whitney K. Newey and Kenned D. West (1987): 'A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix', Econometrica 55, pp. 703-708

Extraction functions (mostly S3 methods): `coef.arx`, `ES`, `fitted.arx`, `plot.arx`,
`print.arx`, `recursive`, `residuals.arx`, `sigma.arx`, `rsquared`,
`summary.arx`, `VaR` and `vcov.arx`

Related functions: `getsm`, `getsv`, `isat`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```##Simulate from an AR(1): set.seed(123) y <- arima.sim(list(ar=0.4), 70) ##estimate an AR(2) with intercept: arx(y, mc=TRUE, ar=1:2) ##Simulate four independent Gaussian regressors: xregs <- matrix(rnorm(4*70), 70, 4) ##estimate an AR(2) with intercept and four conditioning ##regressors in the mean: arx(y, mc=TRUE, ar=1:2, mxreg=xregs) ##estimate a log-variance specification with a log-ARCH(4) ##structure: arx(y, arch=1:4) ##estimate a log-variance specification with a log-ARCH(4) ##structure and an asymmetry/leverage term: arx(y, arch=1:4, asym=1) ##estimate a log-variance specification with a log-ARCH(4) ##structure, an asymmetry or leverage term, a 10-period log(EWMA) as ##volatility proxy, and the log of the squareds of the conditioning ##regressors in the log-variance specification: arx(y, arch=1:4, asym=1, log.ewma=list(length=10), vxreg=log(xregs^2)) ##estimate an AR(2) with intercept and four conditioning regressors ##in the mean, and a log-variance specification with a log-ARCH(4) ##structure, an asymmetry or leverage term, a 10-period log(EWMA) as ##volatility proxy, and the log of the squareds of the conditioning ##regressors in the log-variance specification: arx(y, mc=TRUE, ar=1:2, mxreg=xregs, arch=1:4, asym=1, log.ewma=list(length=10), vxreg=log(xregs^2)) ```

### Example output

```Loading required package: zoo

Attaching package: 'zoo'

The following objects are masked from 'package:base':

as.Date, as.Date.numeric

Date: Tue Oct 31 19:56:00 2017
Dependent var.: y
Method: Ordinary Least Squares (OLS)
Variance-Covariance: Ordinary
No. of observations (mean eq.): 68
Sample: 3 to 70

Mean equation:

coef std.error  t-stat p-value
mconst  0.013715  0.115112  0.1191 0.90553
ar1     0.323324  0.125262  2.5812 0.01211
ar2    -0.040814  0.124257 -0.3285 0.74361

Diagnostics:

Chi-sq df p-value
Ljung-Box AR(3)   3.8157196  3 0.28206
Ljung-Box ARCH(1) 0.0087708  1 0.92538
Jarque-Bera       0.3225323  2 0.85107

SE of regression   0.94884
R-squared          0.09647
Log-lik.(n=68)   -91.41661

Date: Tue Oct 31 19:56:01 2017
Dependent var.: y
Method: Ordinary Least Squares (OLS)
Variance-Covariance: Ordinary
No. of observations (mean eq.): 68
Sample: 3 to 70

Mean equation:

coef std.error  t-stat p-value
mconst -0.020030  0.117600 -0.1703 0.86532
ar1     0.314052  0.130041  2.4150 0.01875
ar2    -0.058036  0.129614 -0.4478 0.65591
mxreg1 -0.045191  0.126042 -0.3585 0.72118
mxreg2  0.108048  0.126116  0.8567 0.39494
mxreg3  0.159350  0.133675  1.1921 0.23785
mxreg4  0.145276  0.114408  1.2698 0.20898

Diagnostics:

Chi-sq df p-value
Ljung-Box AR(3)   2.92995  3 0.40255
Ljung-Box ARCH(1) 0.17116  1 0.67908
Jarque-Bera       0.68521  2 0.70992

SE of regression   0.95304
R-squared          0.14454
Log-lik.(n=68)   -89.71711

Date: Tue Oct 31 19:56:01 2017
Dependent var.: y
Method: Ordinary Least Squares (OLS)
No. of observations (variance eq.): 66
Sample: 1 to 70

Log-variance equation:

coef std.error  t-stat p-value
vconst -0.334531  0.471100  0.5042  0.4776
arch1   0.040073  0.128641  0.3115  0.7565
arch2  -0.076960  0.133758 -0.5754  0.5672
arch3  -0.124580  0.133945 -0.9301  0.3560
arch4  -0.058274  0.134204 -0.4342  0.6657

Diagnostics:

Chi-sq df  p-value
Ljung-Box AR(1)   5.81809  1 0.015862
Ljung-Box ARCH(5) 3.66769  5 0.598180
Jarque-Bera       0.50863  2 0.775447

SE of regression   0.97479
R-squared         -0.00002
Log-lik.(n=66)   -91.89688

Date: Tue Oct 31 19:56:01 2017
Dependent var.: y
Method: Ordinary Least Squares (OLS)
No. of observations (variance eq.): 66
Sample: 1 to 70

Log-variance equation:

coef std.error  t-stat p-value
vconst -0.123582  0.468464  0.0696 0.79193
arch1   0.246437  0.174802  1.4098 0.16376
arch2  -0.071564  0.131724 -0.5433 0.58895
arch3  -0.130442  0.131916 -0.9888 0.32672
arch4  -0.018810  0.134119 -0.1402 0.88893
asym1  -0.378539  0.221001 -1.7128 0.09191

Diagnostics:

Chi-sq df  p-value
Ljung-Box AR(1)   5.2578  1 0.021849
Ljung-Box ARCH(5) 1.4461  5 0.919207
Jarque-Bera       2.2276  2 0.328317

SE of regression   0.97479
R-squared         -0.00002
Log-lik.(n=66)   -95.13333

Date: Tue Oct 31 19:56:01 2017
Dependent var.: y
Method: Ordinary Least Squares (OLS)
No. of observations (variance eq.): 66
Sample: 1 to 70

Log-variance equation:

coef std.error  t-stat p-value
vconst        0.365119  0.665309  0.3012 0.58315
arch1         0.301905  0.187916  1.6066 0.11387
arch2        -0.036126  0.138548 -0.2607 0.79526
arch3        -0.087623  0.140498 -0.6237 0.53543
arch4         0.064056  0.146672  0.4367 0.66402
asym1        -0.404899  0.235561 -1.7189 0.09126
logEqWMA(10) -1.259386  0.996093 -1.2643 0.21145
vxreg1       -0.132749  0.173628 -0.7646 0.44780
vxreg2        0.041071  0.175363  0.2342 0.81570
vxreg3        0.161644  0.153319  1.0543 0.29636
vxreg4        0.058217  0.134571  0.4326 0.66699

Diagnostics:

Chi-sq df  p-value
Ljung-Box AR(1)   4.6148  1 0.031697
Ljung-Box ARCH(5) 5.0951  5 0.404380
Jarque-Bera       8.6671  2 0.013121

SE of regression    0.97479
R-squared          -0.00002
Log-lik.(n=66)   -100.00487

Date: Tue Oct 31 19:56:01 2017
Dependent var.: y
Method: Ordinary Least Squares (OLS)
Variance-Covariance: Ordinary
No. of observations (mean eq.): 68
No. of observations (variance eq.): 64
Sample: 3 to 70

Mean equation:

coef std.error  t-stat p-value
mconst -0.020030  0.117600 -0.1703 0.86532
ar1     0.314052  0.130041  2.4150 0.01875
ar2    -0.058036  0.129614 -0.4478 0.65591
mxreg1 -0.045191  0.126042 -0.3585 0.72118
mxreg2  0.108048  0.126116  0.8567 0.39494
mxreg3  0.159350  0.133675  1.1921 0.23785
mxreg4  0.145276  0.114408  1.2698 0.20898

Log-variance equation:

coef  std.error  t-stat p-value
vconst       -0.7557661  0.5189110  2.1212 0.14527
arch1         0.0847695  0.1567522  0.5408 0.59092
arch2         0.0368367  0.1393747  0.2643 0.79257
arch3        -0.0447772  0.1311476 -0.3414 0.73413
arch4        -0.1020461  0.1305049 -0.7819 0.43773
asym1        -0.4486150  0.2423142 -1.8514 0.06969
logEqWMA(10) -0.5774913  0.7910264 -0.7301 0.46857
vxreg1       -0.1238884  0.1260718 -0.9827 0.33023
vxreg2        0.0448394  0.1229734  0.3646 0.71684
vxreg3       -0.0076974  0.1144433 -0.0673 0.94663
vxreg4       -0.0165456  0.0967990 -0.1709 0.86493

Diagnostics:

Chi-sq df  p-value
Ljung-Box AR(3)   6.2873  3 0.098438
Ljung-Box ARCH(5) 2.6166  5 0.758844
Jarque-Bera       1.4104  2 0.494006

SE of regression   0.95304
R-squared          0.14454
Log-lik.(n=64)   -83.12129
```

gets documentation built on April 4, 2018, 5:03 p.m.