Interpreting Engle–Ng Sign Bias Tests for GARCH Models

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


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












```
Sign Bias Test
------------------------------------
                   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 ***
```

## Answer by Pleb (score 3)

https://quant.stackexchange.com/a/66461

#### 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.