# How to interpret Sign bias test in GARCH (1,1) and in GJR-GARCH?

Sign Bias Test
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t-value     prob sig
Sign Bias           2.7345 0.006333 ***
Negative Sign Bias  0.2329 0.815853
Positive Sign Bias  0.1626 0.870861
Joint Effect       12.1091 0.007019 ***

• Have you tried looking this up in the documentation? The rugarch package in R has a great vignette and good help files. I would start from the former. Aug 16 '21 at 10:23

### Take a look at the rugarch documentation:

At p. 28 the author describes the purpose of the sign bias test and how it is constructed:

The signbias calculates the Sign Bias Test of Engle and Ng (1993), and is also displayed in the summary. This tests the presence of leverage effects in the standardized residuals (to capture possible misspecification of the GARCH model), by regressing the squared standardized residuals on lagged negative and positive shocks as follows: $$z_t^2 = c_0 + c_1 \cdot I_{\{\varepsilon_{t-1}<0\}} + c_2 \cdot I_{\{\varepsilon_{t-1}<0\}}\varepsilon_{t-1} + c_3 \cdot I_{\{\varepsilon_{t-1}\geq0\}}\varepsilon_{t-1} + u_t$$ where I is the indicator function and ˆεt the estimated residuals from the GARCH process. The Null Hypotheses are $$H_0 : c_i = 0$$ (for $$i = 1, 2, 3$$), and that jointly $$H_0 : c_1 = c_2 = c_3 = 0$$.

On the same couple of pages, he goes through an example where he also interprets the sign bias values.

You can even find more information of the test in the above specified paper (In general, see section II of Engle and Ng (1993)). Around p. 1757 they briefly describe the intuition of the three tests:

The sign bias test considers the variable $$I_{\{\varepsilon_{t-1}<0\}}$$ a dummy variable that takes a value of one when $$\varepsilon_{t-1}$$ is negative- and zero otherwise. This test examines the impact of positive and negative return shocks on volatility not predicted by the model under consideration. The negative size bias test utilizes- the variable $$I_{\{\varepsilon_{t-1}<0\}}$$ . It focuses on the different effects that large and small negative return shocks have on volatility which is not predicted by the volatility model. The positive size bias test utilizes the variable $$I_{\{\varepsilon_{t-1}\geq0\}}\varepsilon_{t-1}$$ where $$I_{\{\varepsilon_{t-1}\geq0\}}$$ is defined as 1 minus $$I_{\{\varepsilon_{t-1}<0\}}$$. It focuses on the different impacts that large and small positive return shocks may have on volatility, which are not explained by the volatility model.

I have changed the notation in the above quote, to fit the notation of the documentation for the rugarch package. You can find the original notation at p. 1757.

In general, when working with the rugarch package, it is a good idea to read the documentation, when questioning the output. I hope this provide some insight.

• thanks a lot.. actually i want to interpret sign bias test in GARCH output where we take decision whether we should go for asymmetric garch models or not? Aug 16 '21 at 12:56
• thanks for your time and valuable comment. Aug 16 '21 at 14:24
• @Younis In my opinion, based on the accepted null hypothesis for the positive and negative sign biases, It wouldn't be necessary to fit an asymmetrical GARCH model (However, the output of the sign bias might change, based on your estimation period). It just implies that the values c2 and c3 are not statistically different from 0, in the above regression on the residuals. You can always try to fit an EGARCH (or GJR-GARCH) and observe how the sign test then becomes. If the answer helped you, then please consider accepting it by clicking the tick sign under the votes :-)
– Pleb
Aug 16 '21 at 15:34