Essays in macroeconomic forecasting with linear and non-linear time series models using Bayesian techniques

This PhD thesis comprises three essays which explore novel approaches to modelling and forecasting macroeconomic time series using Bayesian techniques. In short, the first essay examines the impact of long-run expectations in a conditional forecasting setting. The second essay proposes the extension of the non-linear class of threshold autoregressive models with a parsimonious set of time-varying parameters. Finally, the third essay proposes multivariate time series forecasting methods which incorporate time-frequency analysis. In the first essay, I investigate whether incorporating survey-based long-run expectations via the steady-state prior in Bayesian vector autoregressions (BVARs) can improve conditional forecasts. The long-run behaviour of the conditional forecasts is disciplined by employing the steady-state prior, whose mean is set equal to the Survey of Professional Forecasters' long-run expectations. Using US real-time data since 1980 in an out-of-sample forecast evaluation (1997-2013) and assuming the realization of short-term interest rates is known ex-ante, I find that the steady-state prior improves accuracy for long-term yields consistently and further improves GDP and unemployment forecasts with the inclusion of stochastic volatility relative to a benchmark-Minnesota prior BVAR. However, under a model’s reality check assuming that the future paths of the variables in the VAR, except for one, are known ex-ante, survey expectations of unemployment improve conditional forecasts accuracy in the period following the financial crisis, whereas, for the CPI and the FFR, they do not. Finally, for homoscedastic steady-state prior BVARs, the hyperparameters are estimated for each forecasting exercise separately by marginal likelihood maximization employing particle swarm optimisation, and it is found that the financial crisis has a minor impact on their optimal values. In the second essay, I propose the extension of the logistic smooth transition autoregressive model in the univariate setting by allowing the threshold and the constant to be time-varying. Using Monte Carlo simulation, I show that the model's parameters can be estimated successfully via a combination of particle filtering and Markov chain Monte Carlo algorithms. In the empirical application part, using US data for GDP, CPI, FFR, and 10-year bond yield, the proposed model outperforms the linear benchmark model in an out-of-sample forecasting exercise over the period 1985-2018 on many occasions. Following the Great Recession, the time variation of the results shows stronger evidence in favour of the proposed model, and in many instances, the difference versus the linear benchmark model is found to be significant. The inclusion of stochastic volatility does not improve the forecasting performance except for the FFR at the 1-quarter ahead forecast horizon. Finally, in the third essay, by employing wavelet analysis which allows the time localisation of time series frequency characteristics, I extend the existing literature on wavelet-based univariate time series modelling and forecasting into a multivariate setting and under a Bayesian framework both for single and mixed-frequency time series. Regarding the singlefrequency time series, the first approach uses discrete wavelet transform (DWT) based denoising and subsequent conventional Bayesian vector autoregression (BVAR) forecasting. The second approach employs the Haar Maximal Overlap Discrete Wavelet Transform (MODWT), which is a time series multiscale additive decomposition describing fluctuations over different frequency bands. Separate scale BVARs are formed across each scale, and forecasts are estimated by aggregating the separate scale forecasts. The third approach extends the Multiscale Autoregressive model of Renaud et al. (2003) into the Multiscale BVAR employing a Minnesota-inspired prior, which allows a varying degree of shrinkage across different scales, as well as an SSVS prior. In an out-of-sample forecasting exercise using US macroeconomic variables, the three forecasting approaches are found to outperform a conventional BVAR on many occasions. In particular, wavelet-based denoising forecasting presents merits for density forecasts, and the Multiscale BVAR outperforms the benchmark across all variables for medium to long-term forecasts. Regarding modelling mixed-frequency time series using wavelet analysis, the proposed wavelet-based MF-VAR model, which comprises separate scale MF-VARs in a single system, exhibits increased in-sample forecast accuracy for known monthly time series in a statistical sense compared to the standard MFVAR; however, this behaviour reverses during recessionary periods like the latest COVID-19 recession.This PhD thesis comprises three essays which explore novel approaches to modelling and forecasting macroeconomic time series using Bayesian techniques. In short, the first essay examines the impact of long-run expectations in a conditional forecasting setting. The second essay proposes the extension of the non-linear class of threshold autoregressive models with a parsimonious set of time-varying parameters. Finally, the third essay proposes multivariate time series forecasting methods which incorporate time-frequency analysis. In the first essay, I investigate whether incorporating survey-based long-run expectations via the steady-state prior in Bayesian vector autoregressions (BVARs) can improve conditional forecasts. The long-run behaviour of the conditional forecasts is disciplined by employing the steady-state prior, whose mean is set equal to the Survey of Professional Forecasters' long-run expectations. Using US real-time data since 1980 in an out-of-sample forecast evaluation (1997-2013) and assuming the realization of short-term interest rates is known ex-ante, I find that the steady-state prior improves accuracy for long-term yields consistently and further improves GDP and unemployment forecasts with the inclusion of stochastic volatility relative to a benchmark-Minnesota prior BVAR. However, under a model’s reality check assuming that the future paths of the variables in the VAR, except for one, are known ex-ante, survey expectations of unemployment improve conditional forecasts accuracy in the period following the financial crisis, whereas, for the CPI and the FFR, they do not. Finally, for homoscedastic steady-state prior BVARs, the hyperparameters are estimated for each forecasting exercise separately by marginal likelihood maximization employing particle swarm optimisation, and it is found that the financial crisis has a minor impact on their optimal values. In the second essay, I propose the extension of the logistic smooth transition autoregressive model in the univariate setting by allowing the threshold and the constant to be time-varying. Using Monte Carlo simulation, I show that the model's parameters can be estimated successfully via a combination of particle filtering and Markov chain Monte Carlo algorithms. In the empirical application part, using US data for GDP, CPI, FFR, and 10-year bond yield, the proposed model outperforms the linear benchmark model in an out-of-sample forecasting exercise over the period 1985-2018 on many occasions. Following the Great Recession, the time variation of the results shows stronger evidence in favour of the proposed model, and in many instances, the difference versus the linear benchmark model is found to be significant. The inclusion of stochastic volatility does not improve the forecasting performance except for the FFR at the 1-quarter ahead forecast horizon. Finally, in the third essay, by employing wavelet analysis which allows the time localisation of time series frequency characteristics, I extend the existing literature on wavelet-based univariate time series modelling and forecasting into a multivariate setting and under a Bayesian framework both for single and mixed-frequency time series. Regarding the singlefrequency time series, the first approach uses discrete wavelet transform (DWT) based denoising and subsequent conventional Bayesian vector autoregression (BVAR) forecasting. The second approach employs the Haar Maximal Overlap Discrete Wavelet Transform (MODWT), which is a time series multiscale additive decomposition describing fluctuations over different frequency bands. Separate scale BVARs are formed across each scale, and forecasts are estimated by aggregating the separate scale forecasts. The third approach extends the Multiscale Autoregressive model of Renaud et al. (2003) into the Multiscale BVAR employing a Minnesota-inspired prior, which allows a varying degree of shrinkage across different scales, as well as an SSVS prior. In an out-of-sample forecasting exercise using US macroeconomic variables, the three forecasting approaches are found to outperform a conventional BVAR on many occasions. In particular, wavelet-based denoising forecasting presents merits for density forecasts, and the Multiscale BVAR outperforms the benchmark across all variables for medium to long-term forecasts. Regarding modelling mixed-frequency time series using wavelet analysis, the proposed wavelet-based MF-VAR model, which comprises separate scale MF-VARs in a single system, exhibits increased in-sample forecast accuracy for known monthly time series in a statistical sense compared to the standard MFVAR; however, this behaviour reverses during recessionary periods like the latest COVID-19 recession