# arch.test: ARCH Engle's Test for Residual Heteroscedasticity In aTSA: Alternative Time Series Analysis

## Description

Performs Portmanteau Q and Lagrange Multiplier tests for the null hypothesis that the residuals of a ARIMA model are homoscedastic.

## Usage

 `1` ```arch.test(object, output = TRUE) ```

## Arguments

 `object` an object from arima model estimated by `arima` or `estimate` function. `output` a logical value indicating to print the results in R console, including the plots. The default is `TRUE`.

## Details

The ARCH Engle's test is constructed based on the fact that if the residuals (defined as e[t]) are heteroscedastic, the squared residuals (e^2[t]) are autocorrelated. The first type of test is to examine whether the squares of residuals are a sequence of white noise, which is called Portmanteau Q test and similar to the Ljung-Box test on the squared residuals. The second type of test proposed by Engle (1982) is the Lagrange Multiplier test which is to fit a linear regression model for the squared residuals and examine whether the fitted model is significant. So the null hypothesis is that the squared residuals are a sequence of white noise, namely, the residuals are homoscedastic. The lag parameter to calculate the test statistics is taken from an integer sequence of 1:min(24,n) with step 4 if n > 25, otherwise 2, where n is the number of nonmissing observations.

The plots of residuals, squared residuals, p.values of PQ and LM tests will be drawn if `output = TRUE`.

## Value

A matrix with the following five columns:

 `order` the lag parameter to calculate the test statistics. `PQ` the Portmanteau Q test statistic. `p.value` the p.value for PQ test. `LM` the Lagrange Multiplier test statistic. `p.value` the p.value for LM test.

## Note

Missing values are removed before analysis.

Debin Qiu

## References

Engle, Robert F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50 (4): 987-1007.

McLeod, A. I. and W. K. Li. Diagnostic Checking ARMA Time Series Models Using Squared-Residual Autocorrelations. Journal of Time Series Analysis. Vol. 4, 1983, pp. 269-273.

## Examples

 ```1 2 3``` ```x <- rnorm(100) mod <- estimate(x,p = 1) # or mod <- arima(x,order = c(1,0,0)) arch.test(mod) ```

### Example output

```Attaching package: 'aTSA'

The following object is masked from 'package:graphics':

identify

ARIMA(1,0,0) model is estimated for variable: x

Conditional-Sum-of-Squares & Maximum Likelihood Estimation
Estimate    S.E t.value p.value Lag
MU       0.145 0.1117    1.30   0.197   1
AR 1     0.140 0.0987    1.42   0.158   1
-----
n = 100; 'sigma' = 0.9615573; AIC = 281.9674; SBC = 287.1778
------------------------------
Correlation of Parameter Estimates
MU   AR 1
MU     1.0000 -0.0104
AR 1  -0.0104  1.0000
------------------------------
Autocorrelation Check of Residuals
lag    LB p.value
[1,]   4  2.25   0.689
[2,]   8  3.82   0.873
[3,]  12 10.85   0.542
[4,]  16 13.83   0.611
[5,]  20 14.31   0.814
[6,]  24 16.60   0.865
------------------------------
Model for variable: x
Estimated mean: 0.1451695
AR factors: 1 + 0.1405 B**(1)
ARCH heteroscedasticity test for residuals
alternative: heteroscedastic

Portmanteau-Q test:
order    PQ p.value
[1,]     4  4.33   0.364
[2,]     8 10.77   0.215
[3,]    12 13.31   0.347
[4,]    16 18.21   0.312
[5,]    20 20.70   0.415
[6,]    24 29.32   0.208
Lagrange-Multiplier test:
order     LM p.value
[1,]     4 13.374 0.00389
[2,]     8  5.597 0.58754
[3,]    12  2.420 0.99639
[4,]    16  1.084 1.00000
[5,]    20  0.578 1.00000
[6,]    24  0.281 1.00000
```

aTSA documentation built on May 29, 2017, 11:44 a.m.