In this paper we consider a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models where the eigenvalues of the conditional covariance matrix are time-varying. The proposed dynamics of the eigenvalues is based on applying the general theory of dynamic conditional score models as proposed by Creal, Koopman and Lucas (2013) and Harvey (2013). We denote the obtained GARCH model with dynamic conditional eigenvalues (and constant conditional eigenvectors) as the-GARCH model. We provide new results on asymptotic theory for the Gaussian QMLE, and for testing of reduced rank of the (G)ARCH loading matrices of the time-varying eigenvalues. The theory is applied to US data, where we …nd that the eigenvalue structure can be reduced similar to testing for the number in factors in volatility models.

The primary contribution of this paper is to establish that the long-swings behavior observed in the market price of Danish housing since the 1970s can be understood by studying the interplay between short-term expectation formation and long-run equilibrium conditions. We introduce an asset market model for housing based on uncertainty rather than risk, which under mild assumptions allows for other forms of forecasting behavior than rational expectations. We test the theory via an I(2) cointegrated VAR model and find that the long-run equilibrium for the housing price corresponds closely to the predictions from the theoretical framework. Additionally, we corroborate previous findings that housing markets are well characterized by short-term momentum forecasting behavior. Our conclusions have wider relevance, since housing prices play a role in the wider Danish economy, and other developed economies, through wealth effects.

In this paper we consider a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models where the eigenvalues of the conditional covariance matrix are time-varying. The proposed dynamics of the eigenvalues is based on applying the general theory of dynamic conditional score models as proposed by Creal, Koopman and Lucas (2013) and Harvey (2013). We denote the obtained GARCH model with dynamic conditional eigenvalues (and constant conditional eigenvectors) as the -GARCH model. We provide new results on asymptotic theory for the Gaussian QMLE, and for testing of reduced rank of the (G)ARCH loading matrices of the time-varying eigenvalues. The theory is applied to US data, where we …nd that the eigenvalue structure can be reduced similar to testing for the number in factors in volatility models.