Bayesian models often involve a small set of hyperparameters determined by maximizing the marginal likelihood. Bayesian optimization is an iterative method where a Gaussian process posterior of the underlying function is sequentially updated by new function evaluations. We propose a novel Bayesian optimization framework for situations where the user controls the computational effort and therefore the precision of the function evaluations. This is a common in econometrics where the marginal likelihood is often computed by Markov chain Monte Carlo or importance sampling methods. The new acquisition strategy gives the optimizer the option to explore the function with cheap noisy evaluations and therefore find the optimum faster. The method is applied to estimating the prior hyperparameters in two popular models on US macroeconomic time series data: the steady-state Bayesian vector autoregressive (BVAR) and the time-varying parameter BVAR with stochastic volatility.

In this paper, we make use of Bayesian VAR (BVAR) models to conduct an out-of-sample forecasting exercise for CPIF inflation, the inflation target variable at the Riksbank in Sweden. The proposed BVAR models generally outperform simple benchmark models, the BVAR model used by the Riksbank as presented in Iversen et al. (Real-time forecasting for monetary policy analysis: the case of Sveriges Riksbank, Working Paper 16/318, Sveriges riksbank, Stockhol, 2016) and professional forecasts made by the National Institute of Economic Research in Sweden. Moreover, the BVAR models proposed in the present paper have better forecasting precision than both survey forecasts and the method suggested by Faust and Wright (in: Elliott, Timmermann (eds) Handbook of forecasting, 2013). The findings in this paper might be of value to analysts, policymakers and forecasters of the inflation in Sweden (and possibly other small open economies alike).

Time-dependent volatility clustering (or heteroscedasticity) in macroeconomic and financial time series has been analyzed for more than half a century. The inefficiencies it causes in various inference procedures are well known and understood. Despite this, heteroscedasticity is surprisingly often neglected in practical work. The correct way is to model the variance jointly with the other properties of the time series by using some of the many methods available in the literature. In the first two papers of this thesis, we explore a third option, that is rarely used in the literature, in which we first remove the heteroscedasticity and only then fit a simpler model to the homogenized data.

In the first paper, we introduce a filter that removes heteroscedasticity from simulated data without affecting other time series properties. We show that filtering the data leads to efficiency gains when estimating parameters in ARMA models, and in some cases to higher forecast precision for US GDP growth.

The work of the first paper is extended to the case of multivariate time series in Paper II. In this paper, the stochastic volatility model is used for tracking the latent evolution of the time series variances. Also in this scenario variance stabilization offers efficiency gains when estimating model parameters.

During the last decade, there has been an increasing interest in using large-scale VARs together with Bayesian shrinkage methods. The rich parameterization together with the need for simulations methods results in a computational bottleneck that either force concessions regarding the flexibility of the model or the size of the data set. In the last two papers, we address these issues with methods from the machine learning literature.  

In Paper III, we develop a new Bayesian optimization strategy for finding optimal hyperparameters for econometric models via maximization of the marginal likelihood. We illustrate that the algorithm finds optimal values fast compared to conventional methods. 

Finally, in Paper IV we present a fast variational inference (VI) algorithm for approximating the parameter posterior and predictive distribution of the steady-state BVAR. We show that VI produces results that are very close to those of the conventional Gibbs sampler but are obtained at a much lower computational cost. This is illustrated in both a simulation study and on US macroeconomic data.

In this paper new filters for removing unspecified form of heteroscedasticity are proposed. The filters build on the assumption that the variance of a pre-whitened time series can be viewed as a latent stochastic process by its own. This makes the filters flexible and useful in many situations. A simulation study shows that removing heteroscedasticity before fitting a model leads to efficiency gains and bias reductions when estimating the parameters of ARMA models. A real data study shows that pre-filtering can increase the forecasting precision of quarterly US GDP growth.

The growth rate of the gross domestic product (GDP) usually carries heteroscedasticity, asymmetry and fat-tails. In this study three important and significantly heteroscedastic GDP series are examined. A Normal, normal-mixture, normal-asymmetric Laplace distribution and a Student's t-Asymmetric Laplace (TAL) distribution mixture are considered for distributional fit comparison of GDP growth series after removing heteroscedasticity. The parameters of the distributions have been estimated using maximum likelihood method. Based on the results of different accuracy measures, goodness-of-fit tests and plots, we find out that in the case of asymmetric, heteroscedastic and highly leptokurtic data the TAL-distribution fits better than the alternatives. In the case of asymmetric, heteroscedastic but less leptokurtic data the NM fit is superior. Furthermore, a simulation study has been carried out to obtain standard errors for the estimated parameters. The results of this study might be used in e.g. density forecasting of GDP growth series or to compare different economies.

In this article, variance stabilizing filters are discussed. A new filter with nice properties is proposed which makes use of moving averages and moving standard deviations, the latter smoothed with the Hodrick-Prescott filter. This filter is compared to a GARCH-type filter. An ARIMA model is estimated for the filtered GDP series, and the parameter estimates are used in forecasting the unfiltered series. These forecasts compare well with those of ARIMA, ARFIMA, and GARCH models based on the unfiltered data. The filter does not color white noise.

Three important and significantly heteroscedastic gross domestic product series are studied. Omnipresent heteroscedasticity is removed and the distributions of the series are then compared to normal, normal mixture and normal-asymmetric Laplace (NAL) distributions. NAL represents a skewed and leptokurtic distribution, which is in line with the Aghion and Howitt [1] model for economic growth, based on Schumpeter's idea of creative destruction. Statistical properties of the NAL distributions are provided and it is shown that NAL fits the data better than the alternatives.

Schumpeter (1942) framkastade idén om att det är företagares innovationer som driver tillväxten. Aghion och Howitt (1992) specificerar en hypotes om kopplingarna innovationer → tillväxt → innovationer. Vi studerar tillväxtens frekvensfunktion för att ta reda på om den uppvisar likheter med Poissonfördelningen, den överföringsmekanism som antas i deras hypotes. Ett nytt filter introduceras för eliminering av heteroskedasticitet. Och visst, en blandning av en normal och Poisson- fördelningarna har den bästa tillpassningen till data! Vidare uppvisar aktiebörsindex samma sorts fördelning. Till slut redovisas några resultat om aktiebörsens roll i tillväxtprocessen.

This paper explores the connection between Dow Jones industrial average (DJIA) stock prices and the US GDP growth. Both series are heteroscedastic, making standard detrending procedures, such as Hodrick-Prescott or Baxter-King, inadequate. The results from these procedures are compared to the results from heteroscedasticity corrected data, thus the effect of the neglected heteroscedasticity is measured. The analysis is mainly done in the frequency domain but relevant time domain results are also reported.

The distribution of differences in logarithms of the Dow Jones stock index is compared to the Normal (N), Normal Mixture (NM) and a weighted sum of a normal and an asymmetric Laplace distribution (NAL). It is found that the NAL fits best. We came to this result by studying samples with high, medium and low volatility, thus circumventing strong heteroscedasticity in the entire series. The NAL distribution also fitted economic growth, thus revealing a new analogy between financial data and real growth.

Heteroscedasticity (or time-dependent volatility) in economic and financial time series has been recognized for decades. Still, heteroscedasticity is surprisingly often neglected by practitioners and researchers. This may lead to inefficient procedures. Much of the work in this thesis is about finding more effective ways to deal with heteroscedasticity in economic and financial data. Paper I suggest a filter that, unlike the Box-Cox transformation, does not assume that the heteroscedasticity is a power of the expected level of the series. This is achieved by dividing the time series by a moving average of its standard deviations smoothed by a Hodrick-Prescott filter. It is shown that the filter does not colour white noise.

An appropriate removal of heteroscedasticity allows more effective analyses of heteroscedastic time series. A few examples are presented in Paper II, III and IV of this thesis. Removing the heteroscedasticity using the proposed filter enables efficient estimation of the underlying probability distribution of economic growth. It is shown that the mixed Normal - Asymmetric Laplace (NAL) distributional fit is superior to the alternatives. This distribution represents a Schumpeterian model of growth, the driving mechanism of which is Poisson (Aghion and Howitt, 1992) distributed innovations. This distribution is flexible and has not been used before in this context. Another way of circumventing strong heteroscedasticity in the Dow Jones stock index is to divide the data into volatility groups using the procedure described in Paper III. For each such group, the most accurate probability distribution is searched for and is used in density forecasting. Interestingly, the NAL distribution fits best also here. This could hint at a new analogy between the financial sphere and the real economy, further investigated in Paper IV. These series are typically heteroscedastic, making standard detrending procedures, such as Hodrick-Prescott or Baxter-King, inadequate. Prior to this comovement study, the univariate and bivariate frequency domain results from these filters are compared to the filter proposed in Paper I. The effect of often neglected heteroscedasticity may thus be studied.

Normality is often mechanically and without sufficient reason assumed in econometric models. In this paper three important and significantly heteroscedastic GDP series are studied. Heteroscedasticity is removed and the distributions of the filtered series are then compared to Normal, Normal Mixture and Normal - Asymmetric Laplace (NAL) distributions. NAL represents a skewed and leptokurtic distribution, which is in line with the Aghion and Howitt (1992) model for economic growth, based on Schumpeter's idea of creative destruction. Statistical properties of the NAL distributions are provided and it is shown that NAL competes well with the alternatives.

In this paper variance stabilizing filters are discussed. A new filter with nice properties is proposed which makes use of moving averages and moving standard deviations, the latter smoothed with the Hodrick-Prescott filter. This filter is compared to a GARCH-type filter. An ARIMA model is estimated for the filtered GDP series, and the parameter estimates are used in forecasting the unfiltered series. These forecasts compare well with those of ARIMA, ARFIMA and GARCH models based on the unfiltered data. The filter does not colour white noise.