Test for Zero Median of Errors in an Arma–garch Model

Abstract

Because the ARMA–GARCH model can generate data with some important properties such as skewness, heavy tails, and volatility persistence, it has become a benchmark model in analyzing financial and economic data. The commonly employed quasi maximum likelihood estimation (QMLE) requires a finite fourth moment for both errors and the sequence itself to ensure a normal limit. The self-weighted quasi maximum exponential likelihood estimation (SWQMELE) reduces the moment constraints by assuming that the errors and their absolute values have median zero and mean one, respectively. Therefore, it is necessary to test zero median of errors before applying the SWQMELE, as changing zero mean to zero median destroys the ARMA–GARCH structure. This paper develops an efficient empirical likelihood test without estimating the GARCH model but using the GARCH structure to reduce the moment effect. A simulation study confirms the effectiveness of the proposed test. The data analysis shows that some financial returns do not have zero median of errors, which cautions the use of the SWQMELE.

Testing the Intercept of a Balanced Predictive Regression Model

“…Then, we have uniformly for as . As did in the proof of Theorem 1 in [ 32 ], we can show by the Taylor expansion that where , based on ( A4 ) and Lemmas 1 and 2. Using this, we have that the minimizer of must lie in .…”

“…For convenience, denote with being the solution to with respect to for fixed . The following proof is similar to that of Theorem 1 in [ 32 ]. Let with .…”

A Unified Test for the AR Error Structure of an Autoregressive Model

“…Based on Lemmas A1 and A2, we can show by using similar techniques as in Theorem 1 of [28] as n → ∞. ε t (µ, φ) = X t − µ − φX t−1 , X t−1 = X t−1 + ∑ p i=1 ψ i,0 X t−i−1 and γ 0 = 1 + ∑ p i=1 ψ i,0 .…”