Forecasting with Bayesian Vector Autoregression

1. Forecasting with Bayesian Vector Autoregression Student: Ruja Cătălin Supervisor: Professor Moisă Altăr

2. 1. Objectives • To apply the BVAR methodology to a group of Romanian macroeconomic time series. • To show an improvement in forecast performance compared to the unrestricted VAR.

3. 2. Improving forecast performance • The BVAR methodology allows a tradeoff between oversimplification and overfitting. • oversimplification – univariate (ARIMA) models which cannot capture the important interaction between variables. • overfitting – unrestricted VAR which tend to incorporate useless or misleading relationships (due to small sample size and large number of coefficients to be estimated).

4. 3. The Bayesian approach • The Bayesian approach allows the use of prior beliefs, in the form of probabilities, about which of the possible models will forecast best. • An extensive system of confidence for the priors is specified. • A statistical procedure is used to revise this priors in light of the evidence in the data.

5. 4. The BVAR methodology • Consider an unrestricted, time-varying, p order VAR representation: (1) n-vector of variables (n x n) matrix of polynomials.

6. 5. Specifying the priors • The scalar representation for the i th component of : (2) • The prior for coefficient vector: (3) is generated as a function of the hyperparameters vector π.

7. (i-1)*p+1elements 6.a The Minnesota priors Litterman (1985), Kadiyala and Karlsson (1997), Kenny, Meyler and Quinn (1998), Ritschl and Woiteck (2002) (4) (5) For the constant term a flat (diffuse) prior is specified.

8. 6.b The Minnesota prior • Changes in the hyperparameters which lead to smaller (larger) variances of coefficients are referred to as tightening (loosening) the prior. Thus, in the limit, as the prior is tightened around its mean each equation takes the form of a random walk whit drift fit to the data: (6)

9. 7. Specifications Doan, Litterman and Sims (1984), Racette and Raynauld (1992), Ballabriga (1997), Ritschl and Woiteck (2002) • The coefficient evolve over time according with: (7) The matrix Vis proportional with A1, with π4 as the proportionality factor.

10. 8.a The state space representation • The general state space representation of the dynamics of yt,an (n x 1) vector of variables (the signal vector): (8) state equation (9) signal equation (r x 1) state vector (k x 1) vector of exogenous or predetermined variables.

11. 8.b The state space representation (10) (11) (12) the state equation (13) the signal equation Doan, Litterman and Sims (1984)

12. 9. Forecast performance • The root mean square error (14) • The log-determinant of the matrices of summed cross-product of s-step ahead out-of-sample forecast (15)

13. 10. Data • The variables included in the model are: • the industrial output index IPI (index as against previous month) • ROL/USD exchange rate ER (average monthly nominal exchange rate) • consumer price index CPI (index as against previous month) • the net nominal wage and salary earnings NW (monthly averages) • the monetary aggregate M2 (monthly averages).

14. (i-1)*p+1elements 11.a Adapted Minnesota priors (16) (17)

15. 11.b Adapted Minnesota priors Doan, Litterman and Sims (1984), Litterman (1985), and Kadiyala and Karlsson (1997) (18)

16. 12.a Estimation results • Hyperparameters values • Root mean square error

17. 12.b Forecasting performance • The overall measures of fit

18. 13. Conclusions This paper assesses the opportunity of using the Bayesian Vector Autoregression for forecasting a group of series for the Romanian economy. Using the methodology developed by Doan, Litterman and Sims (1984), Litterman (1984, 1985) it is found that BVAR performs better than the unrestricted vector autoregression. This conclusion is similar to that of Doan, Litterman and Sims (1984), and more recently Racette and Raynauld (1992) and Ballabriga, Alvarez and Jareno (2000).