{'fraction_non_alphanumeric': 0.023809523809523808, 'fraction_numerical': 0.007936507936507936, 'mean_word_length': 5.426751592356688, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Estimation of Cointegrated Systems with I(2) Processes

This paper considers the properties of systems likelihood procedures for cointegrated systems when the I(2) variables are present. Two alternative methods are proposed: one is based on the full system likelihood, whereas another is based on the subsystem likelihood. By eliminating all unit roots in the system by the use of prior information concerning the presence of unit roots, these procedures yield estimates whose asymptotic distributions are mixed normal, free from nuisance parameters, and median-unbiased. Both methods are extensions of a full system maximum likelihood procedure by Phillips (1991a) to I(2) models. Three cases of cointegration with I(2) variables are considered in order to cover a wide variety of cointegration relationships. A triangular ECM representation and the two ML estimates are derived for each case, and the asymptotics are discussed as well. The asymptotic efficiency concerning the two estimates are considered.