In this thesis, we propose two Gaussian hidden Markov models: univariate Gaussian hidden Markov models with covariate and bivariate Gaussian hidden Markov models. After that they are applied to stock market returns to inspect the return-beta relationship stated in
the capital asset pricing model (CAPM). The relationship is examined under 3 definitions of regimes: market regimes, idiosyncratic regimes and co-regimes. Results show that betas are larger under bullish market regime compared to bearish. Although no consistent patterns in beta are discovered under different idiosyncratic regimes and co-regimes, for each stock the betas do seem to vary considerably across regimes. Our model is also able to capture volatility clustering exhibited in return series.