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Comparing the Forecasting Performance of Var, Bvar and U-Midas

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Comparing the Forecasting Performance of Var, Bvar and U-Midas COMPARING THE FORECASTING PERFORMANCE OF VAR, BVAR AND U-MIDAS Submitted by Alessio Belloni A thesis submitted to the Department of Statistics in partial fulfillment of the requirements for a one-year Master degree in Statistics in the Faculty of Social Sciences Supervisors: Sebastian Ankargren & Mattias Nordin Spring, 2017 ABSTRACT This paper aims to compare the forecasting performance of the widely used VAR and Bayesian VAR model to the unrestricted MIDAS regression. The models are tested on a real-time macroeconomic data set ranging from 2000 to 2015. The variables are mixed frequency data, specifically, predictions are made for GDP, using economic tendency indicator, unemployment and inflation as predicting variables. The baseline model of this analysis is a simple VAR, while it has great flexibility, this model risks to overfit the data and as a consequence makes unreliable predictions. The Villani Bayesian VAR is meant to solve this problem by introducing long run beliefs about the data structure and the steady state unconditional means of each se- ries. When facing mixed frequency data, both these approached aggregate at the lower level by discarding useful information. In this scenario, the unrestricted MIDAS model addresses this problem without losing high frequency information. The results show how both BVAR and U-MIDAS outperform VAR at every horizon, while there is no absolute winner among BVAR and U-MIDAS. Evidence suggests that U-MIDAS is superior for short horizons, specifically up to the 5th step ahead, which corresponds to one year and a quarter. Contents 1 Introduction 3 2 Theory 4 2.1 VAR........................................4 2.2 Bayesian VAR . .6 2.2.1 Steady State Prior . .7 2.3 Mixed Data Sampling . .8 3 Data 10 3.1 Vintages . 11 3.2 Economic Tendency Indicator . 11 3.3 GDP Growth . 11 3.4 Underlying Inflation Rate . 12 3.5 Unemployment . 12 4 Empirical Analysis 12 4.1 VAR(4) . 13 4.2 Villani Steady State . 13 4.3 U-MIDAS . 14 4.4 Root Mean Square Error and Mean Absolute Error . 14 4.5 Diebold-Mariano Test . 15 5 Results 15 5.1 Accuracy Measure . 16 5.2 Diebold-Mariano Test . 19 6 Conclusion 21 A Appendix 23 2 1 Introduction Forecasting macroeconomic variables plays a vital role in the policy makers’ decision concern- ing the future direction of the economy. When it comes to forecasting Swedish macroeconomic variables, one of the most successful and commonly used approaches to deal with this matter are the vector autoregressive models (Karlsson, 2013). The aim of this study is to compare the widely used vector autoregressive (VAR) and Bayesian vector autoregressive (BVAR) models with a relatively new approach referred to as Mixed Data Sampling (henceforth MIDAS), which is meant to address the possible dispersion of information when dealing with mixed data frequencies (Ghysels et al., 2007). This situation is commonly faced when dealing with macroeconomic and financial variables. The classic example is GDP, usually measured quarterly, while predictors such as unemploy- ment, inflation and economic sentiment indicators are sampled monthly. For the financial vari- ables, the gap among frequencies is even wider, since they are quite often reported daily. The common approach when facing mixed data frequency is to aggregate at the lower level with a potential loss of information. The aim of this paper is to compare the relative efficiency, in forecasting Swedish GDP, of the MIDAS regression model, against the VAR and BVAR models. The VAR model, relatively simple and well enstablished in the literature (Litterman, 1979) is considered as the baseline approach of this analysis. As a next step, the Villani steady-state Bayesian VARis implemented. This study shows how the latter deals with over-parametrization issues of the VAR model (Doan et al., 1983; Litterman, 1986; Villani, 2009). Finally, MIDAS regression, with its functional form, is presented as a way to not discard infor- mation when the data have a mixed frequency structure. The variables included to forecast Swedish GDP are the Economic Tendency Indicator (ETI), unemployment and inflation. The forecasting accuracy is measured in terms of root mean square error (RMSE) and mean absolute error (MAE). The dataset presents a mixed fre- quency structure and span from 2000 Q1 up to 2015 Q3. The analysis is based on real-time data set, meaning that we have vintages or snapshot of the data at a specific point in time. 3 The model used to forecast is built for every vintage, whereas to evaluate the forecasting ac- curacy the last vintage is used. This procedure is implemented because as a matter of fact, past vintages have higher uncertainty compared to the recent ones, which are revised and ad- justed. In this way, when modelling we are in a situation as close as possible to the one faced by forecasters - characterized by relatively high uncertainty, whereas when evaluating model accuracy, the vintage with all the available information at that poin in time is used - the last one. The results are limited to the present data and model specification. For the empirical anal- ysis, R is used as a software and BMR (Keith O’Hara, 2015) and Midasr (Ghysels et al., 2016) are the packages used for VAR-BVAR and MIDAS respectively. The structure of this paper is the following, Section 2 focuses on the theory behind the dis- cussed models, Section 3 describes the data structure and variables, in Section 4 the empirical analysis is presented and Section 5 shows the results. Finally, discussion and conclusion in Section 6. 2 Theory In the following section I briefly describe the theory behind the models I will be using. First in Section 2.1 I introduce the VAR model and its drawbacks, then in Section 2.2 I explain how the BVAR addresses some of these issues and finally in Section 2.3 I present the MIDAS model and the way it deals with mixed data frequencies. 2.1 VAR The vector autoregressive model (VAR) is one of the most succesful models for macroeco- nomics forecasting (Karlsson, 2013). The large use in economics has been motivated by Sims (1980) and the success of the VAR models stems from their simplicity, flexibility and ability to fit the data. Those features come from the rich parametrization of the VAR models, which brings the risk of overfitting the data and as a consequence produce imprecise inference and large uncertainty when it comes to forecasting (Karlsson, 2013). When we are not sure about the direction of the causality between variables, the VAR 4 approach is to treat each variable symmetrically, in the sense that (in the two variates case), we let x be affected by the past realization of z and we let z be affected by the past realization of x (Walter, 1948). Therefore with one lag, we have two equations such that: xt = b10 + b11xt−1 + γ11zt−1 + "yt (1) zt = b20 + b21zt−1 + γ21xt−1 + "zt; (2) where fxg and fzg are stationary proceses and "y and "z are white noise disturbances. The model from Equations (1) and (2) is a first order vector autoregressive process because the maximum lag length is 1. It appears crucial to remark that for just two variables and one lag, the number of parameters to estimate is 6. As a consequence when the number of variables and lags grow, the number of parameters to estimate increases drastically. This confirms what stated in the first part of this section regarding the large number of parameters of the VAR model. We will see how Bayesian statistics address this issue in the following section. Because of double causality, to estimate the parameters in Equations (1) and (2) with ordi- nary least square (OLS), we need to rewrite the model in matrix notation. Where for VAR(1) the final form is the following: Yt = A0 + A1Yt−1 + "t (3) In the literature this is referred to as VAR model in standard form (Enders, 2015). The general VAR(p) model with p lag, can be rewritten as: Yt = A0 + A1Yt−1 + ::: + ApYt−p + "t; t = 1; ::; :T (4) Where Yt denotes a (n × 1) vector of time series variables, A0 is a vector (n × 1) of intercepts, Ai are (n×n) matrices of coefficients and "t is a (n×1) random error term, centered around 0. The right hand side of Equation (4), contains only predetermined variable, therefore equation by equation can be estimated by OLS (Walter, 1948). The way the VAR model is used to forecast is fairly straightforward. The one step ahead forecast is the expected value of Yt+1, which in notation of Equation (4) is E[Yt+1jYt; Yt−1; :::] = A0 + A1Yt + ::: + ApYt−p+1. The two step ahead repeats the same procedure using t + 1 in the equation and the same iterative procedure is done for longer period forecasts. As mentioned 5 before VAR models might produce unreliable forecasts due to overparametrization, especially when including high order lags and several variables. One of the possible ways to deal with this issue is presented in the next section. 2.2 Bayesian VAR Essentially, the Bayesian approach deal with the overparametrization of the VAR model with the introduction of a prior, which contains information about the long run properties of the data, independent from the short run observed data. How the Bayes Theorem combines these two pieces of information, i.e. prior beliefs and observed data, is presented below. All the parameters, θ, are treated as random variables with their own probability distribu- tion. The prior distribution of the parameter, π(θ), represent the researcher’s beliefs and is independent of the observed data.
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