Building Econometric Models

Another Way of Writing ARCH Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• For illustration, consider an ARCH(1). Instead of the above, we can write
•  
• yt = 1 + 2x2t + ... + kxkt + ut , ut = vtt
• , vt  N(0,1)
•  
• The two are different ways of expressing exactly the same model. The first form is easier to understand while the second form is required for simulating from an ARCH model, for example.

Testing for “ARCH Effects”

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• 1. First, run any postulated linear regression of the form given in the equation
• above, e.g. yt = 1 + 2x2t + ... + kxkt + ut
• saving the residuals, .
• 2. Then square the residuals, and regress them on q own lags to test for ARCH
• of order q, i.e. run the regression
• where vt is iid.
• Obtain R2 from this regression
• 3. The test statistic is defined as TR2 (the number of observations multiplied by the coefficient of multiple correlation) from the last regression, and is distributed as a 2(q).

Testing for “ARCH Effects” (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• 4. The null and alternative hypotheses are
• H0 : 1 = 0 and 2 = 0 and 3 = 0 and ... and q = 0
• H1 : 1  0 or 2  0 or 3  0 or ... or q  0.
• If the value of the test statistic is greater than the critical value from the 2 distribution, then reject the null hypothesis.
• Note that the ARCH test is also sometimes applied directly to returns instead of the residuals from Stage 1 above.