SHAZAM Testing for ARCH

### Testing for ARCH

The residuals from a preliminary OLS estimation can be tested for ARCH behaviour. Testing approaches are as follows.

• Tests for non-normality can be considered. If the normality assumption is used to describe the conditional error distribution then a property of ARCH is that the unconditional error distribution will be non-normal with high values for kurtosis.

• The autocorrelation structure of the residuals and the squared residuals can be inspected. An indication of ARCH is that the residuals will be uncorrelated but the squared residuals will show autocorrelation. Test statistics are given by Ljung-Box-Pierce portmanteau tests on the residuals and the squared residuals.

• A test based on the Lagrange multiplier (LM) principle can be applied. Consider the null hypothesis of no ARCH errors versus the alternative hypothesis that the conditional error variance is given by an ARCH(q) process. The test approach proposed in Engle [1982] is to regress the squared residuals on a constant and q lagged values of the squared residuals. From the results of this auxiliary regression, a test statistic is calculated as:
(N`-`q)·R2
There is evidence to reject the null hypothesis if the test statistic exceeds the critical value from a chi-square distribution with q degrees of freedom.

#### Example

The SHAZAM commands (filename: `ARCH1.SHA`) below generate some statistics for the exchange rate data set.

 ```SAMPLE 1 1974 READ (DMBP.txt) Y DAYDUM * Estimation results: Table 2, column 1 (Bollerslev and Ghysels, 1996) * The GF option provides coefficients of skewness and kurtosis as well * as the Jarque-Bera test for non-normality. OLS Y / RESID=E GF * Inspect the autocorrelation structure of the residuals ARIMA E * Inspect the autocorrelation structure of the squared residuals GENR E2=E*E ARIMA E2 * Calculate Lagrange multiplier test statistics for ARCH errors SET NODOECHO NOOUTPUT GEN1 NLAG=10 DIM LM NLAG PVALUE NLAG DO #=1,NLAG * Calculate the test statistic OLS E2 E2(1.#) GEN1 TESTVAL=\$N*\$R2 * Calculate a p-value DISTRIB TESTVAL / TYPE=CHI DF=# CDF=CDF GEN1 LM:#=TESTVAL GEN1 PVALUE:#=1-CDF ENDO * Print the results SAMPLE 1 NLAG GENR P=TIME(0) FORMAT(F8.0,F10.2,F10.4) PRINT P LM PVALUE / FORMAT STOP ```

The above commands show the use of a `DO`-loop for calculating LM test statistics for ARCH(q), q=1,2,...,10.

The SHAZAM output can be viewed. The results show the following test statistics based on the OLS residuals.

Statistic p-value
Skewness `-`0.25
Excess kurtosis 3.64
Jarque-Bera test   1102.9   < 0.0005
Q(20) 27.8 0.113
Q2(20) 507.6 < 0.0005
LM ARCH(1) 96.2 < 0.0005
LM ARCH(8) 185.3 < 0.0005
Note: Q(20) and Q2(20) are the Ljung-Box-Pierce portmanteau tests for up to twentieth order serial correlation in the residuals and the squared residuals respectively.

The Jarque-Bera test statistic provides clear evidence to reject the null hypothesis of normality for the unconditional distribution of the daily percentage exchange rate changes. The high value for excess kurtosis indicates that the distribution is characterized by leptokurtosis. Note that the SHAZAM calculations for skewness and kurtosis incorporate small sample adjustments (see the SHAZAM User's Reference Manual).

The sample autocorrelation function of the residuals shows no autocorrelation. The Q(20) test statistic does not reject the null hypothesis of uncorrelated price changes. However, the sample autocorrelation function of the squared residuals tells a different story. The high value for the Q2(20) test statistic suggests that conditional homoskedasticity can be rejected. The slow decline of the autocorrelation function of the squared residuals suggests that a GARCH(1,1) process may be suitable for describing the errors. That is, a low order ARCH process may not fully capture the time-varying volatility in the data.

The LM tests for ARCH(1) and ARCH(8) errors confirm the presence of ARCH effects in the data.

[SHAZAM Guide home]

#### SHAZAM output

```|_SAMPLE 1 1974
UNIT 88 IS NOW ASSIGNED TO: DMBP.txt
2 VARIABLES AND     1974 OBSERVATIONS STARTING AT OBS       1

|_* Estimation results: Table 2, column 1 (Bollerslev and Ghysels, 1996)
|_* The GF option provides coefficients of skewness and kurtosis as well
|_* as the Jarque-Bera test for non-normality.
|_OLS Y / RESID=E GF
OLS ESTIMATION
1974 OBSERVATIONS     DEPENDENT VARIABLE = Y
...NOTE..SAMPLE RANGE SET TO:      1,   1974

R-SQUARE =   0.0000     R-SQUARE ADJUSTED =   0.0000
VARIANCE OF THE ESTIMATE-SIGMA**2 =  0.22113
STANDARD ERROR OF THE ESTIMATE-SIGMA =  0.47024
SUM OF SQUARED ERRORS-SSE=   436.29
MEAN OF DEPENDENT VARIABLE = -0.16427E-01
LOG OF THE LIKELIHOOD FUNCTION = -1311.10

VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
NAME    COEFFICIENT   ERROR    1973 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
CONSTANT -0.16427E-01 0.1058E-01  -1.552     0.121-0.035     0.0000     1.0000

DURBIN-WATSON = 1.9805    VON NEUMANN RATIO = 1.9815    RHO =  0.00937
RESIDUAL SUM = -0.14766E-13  RESIDUAL VARIANCE =  0.22113
SUM OF ABSOLUTE ERRORS=   648.23
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.0000
RUNS TEST:  963 RUNS, 1028 POS,    0 ZERO,  946 NEG  NORMAL STATISTIC = -1.0508
COEFFICIENT OF SKEWNESS =  -0.2497 WITH STANDARD DEVIATION OF 0.0551
COEFFICIENT OF EXCESS KURTOSIS =   3.6399 WITH STANDARD DEVIATION OF 0.1101

JARQUE-BERA NORMALITY TEST- CHI-SQUARE(2 DF)= 1102.8823 P-VALUE= 0.000

GOODNESS OF FIT TEST FOR NORMALITY OF RESIDUALS - 60 GROUPS
OBSERVED  [statistics not shown]
EXPECTED  [statistics not shown]
CHI-SQUARE =  438.8139 WITH 57 DEGREES OF FREEDOM, P-VALUE= 0.000

|_* Inspect the autocorrelation structure of the residuals
|_ARIMA E
ARIMA MODEL
NUMBER OF OBSERVATIONS =1974
...NOTE..SAMPLE RANGE SET TO:     1,  1974

IDENTIFICATION SECTION - VARIABLE=E
NUMBER OF AUTOCORRELATIONS =  24
NUMBER OF PARTIAL AUTOCORRELATIONS =  12

0     0 0
SERIES  (1-B) (1-B  )  E

NET NUMBER OF OBSERVATIONS = 1974
MEAN=  -0.56921E-17   VARIANCE=   0.22113       STANDARD DEV.=   0.47024

LAGS                      AUTOCORRELATIONS                          STD ERR
1 -12    0.01 -.03 0.03 0.02 0.02 0.00 -.02 0.02 0.02 0.01 -.04 0.00   0.02
13 -24    0.00 0.07 -.01 0.01 -.04 -.01 0.03 -.05 -.02 0.03 0.00 0.07   0.02

MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS  (CHI-SQUARE)
LAG    Q    DF  P-VALUE       LAG    Q    DF  P-VALUE
1    0.17   1  .677          13    9.75  13  .714
2    1.44   2  .486          14   18.78  14  .174
3    3.75   3  .289          15   19.06  15  .211
4    4.54   4  .338          16   19.26  16  .256
5    5.15   5  .398          17   21.94  17  .187
6    5.16   6  .524          18   22.10  18  .228
7    5.68   7  .577          19   23.44  19  .218
8    6.21   8  .624          20   27.84  20  .113
9    6.73   9  .665          21   28.31  21  .132
10    6.97  10  .728          22   30.02  22  .118
11    9.75  11  .553          23   30.05  23  .148
12    9.75  12  .638          24   39.56  24  .024

LAGS                  PARTIAL AUTOCORRELATIONS                      STD ERR
1 -12    0.01 -.03 0.03 0.02 0.02 0.00 -.02 0.01 0.01 0.01 -.04 0.00   0.02

|_* Inspect the autocorrelation structure of the squared residuals
|_GENR E2=E*E
|_ARIMA E2
ARIMA MODEL
NUMBER OF OBSERVATIONS =1974
...NOTE..SAMPLE RANGE SET TO:     1,  1974

IDENTIFICATION SECTION - VARIABLE=E2
NUMBER OF AUTOCORRELATIONS =  24
NUMBER OF PARTIAL AUTOCORRELATIONS =  12

0     0 0
SERIES  (1-B) (1-B  )  E2

NET NUMBER OF OBSERVATIONS = 1974
MEAN=   0.22102       VARIANCE=   0.27504       STANDARD DEV.=   0.52445

LAGS                      AUTOCORRELATIONS                          STD ERR
1 -12    0.22 0.18 0.14 0.12 0.19 0.09 0.09 0.10 0.09 0.12 0.05 0.06   0.02
13 -24    0.08 0.07 0.11 0.08 0.08 0.06 0.07 0.07 0.07 0.07 0.08 0.09   0.03

MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS  (CHI-SQUARE)
LAG    Q    DF  P-VALUE       LAG    Q    DF  P-VALUE
1   96.42   1  .000          13  416.61  13  .000
2  157.16   2  .000          14  426.78  14  .000
3  196.75   3  .000          15  452.89  15  .000
4  227.47   4  .000          16  466.87  16  .000
5  297.74   5  .000          17  479.10  17  .000
6  314.13   6  .000          18  486.82  18  .000
7  328.68   7  .000          19  497.13  19  .000
8  347.55   8  .000          20  507.59  20  .000
9  364.77   9  .000          21  518.58  21  .000
10  392.98  10  .000          22  529.09  22  .000
11  397.53  11  .000          23  540.34  23  .000
12  404.93  12  .000          24  554.81  24  .000

LAGS                  PARTIAL AUTOCORRELATIONS                      STD ERR
1 -12    0.22 0.13 0.08 0.06 0.13 0.00 0.02 0.04 0.03 0.05 -.02 0.01   0.02

|_* Calculate Lagrange multiplier test statistics for ARCH errors
|_SET NODOECHO NOOUTPUT
|_GEN1 NLAG=10
|_DIM LM NLAG PVALUE NLAG
|_DO #=1,NLAG
|_*  Calculate the test statistic
|_OLS E2 E2(1.#)
|_GEN1 TESTVAL=\$N*\$R2
|_*  Calculate a p-value
|_DISTRIB TESTVAL / TYPE=CHI DF=# CDF=CDF
|_GEN1 LM:#=TESTVAL
|_GEN1 PVALUE:#=1-CDF
|_ENDO
|_* Print the results
|_SAMPLE 1 NLAG
|_GENR P=TIME(0)
|_FORMAT(F8.0,F10.2,F10.4)
|_PRINT P LM PVALUE / FORMAT
P              LM             PVALUE
1.     96.24    0.0000
2.    129.30    0.0000
3.    142.18    0.0000
4.    149.70    0.0000
5.    182.43    0.0000
6.    182.27    0.0000
7.    182.58    0.0000
8.    185.28    0.0000
9.    187.18    0.0000
10.    192.38    0.0000
|_STOP
```