Cointegrating VAR Models and Probability Forecasting: Applied to a Small Open Economy

Cointegrating VAR Models and Probability Forecasting:Applied to a Small Open Economy Gustavo Sánchez April 2009

Summary • VEC and Cointegrating VAR Models • Estimate Parameters • Probability Forecasting • Simulate Forecasts • Summary Statistics to estimate probabilities of events

Point Forecast and Confidence Interval

Cointegrating VAR models • Based on the vector error correction (VEC) model specification. • The specification assumes that the economic theory characterizes the long-run equilibrium behavior • The short-run fluctuations represent deviations from that equilibrium. • The short-run and long-run (economic) concepts are linked to the statistical concept of stationarity.

Cointegrating VAR models Reduced form for a VEC model Where: • I(1) Endogenous variables • Matrices containing the long-run adjustment coefficients and coefficients for the cointegrating relationships • Matrix with coefficients associated to short-run dynamic effects • Vectors with coeficients associated to the intercepts and trends • Vector with innovations

Cointegrating VAR models Reduced form for a VEC model • Identifying α and β requires r2 restrictions (r: number of cointegrating vectors). • Johansen FIML estimation identifies α and β by imposing r2 atheoretical restrictions.

Cointegrating VAR models • Garrat et al. (2006) describe the Cointegrating VAR approach: • Use economic theory to impose restrictions to identify αβ. • Exact identification is not necessarily achieved by the theoretical restrictions. • Test whether the overidentifying restrictions are valid.

** Restrictions on VEC system ** • *** Restrictions on Beta lm1 *** • constraint 1 [_ce1]lm1=1 • . . . • constraint 6 [_ce1]ltipp906bn=0 • *** Restrictions on Beta lmt *** • constraint 8 [_ce2]lmt=1 • . . . • constraint 11 [_ce2]ltipp906bn=0 • *** Restrictions on alpha *** • constraint 12 [D_loilp]l._ce1=0 • constraint 13 [D_loilp]l._ce2=0 • ** VEC specification ** • veclm1 lmt lcpi loilp ltcpn lxt ltipp906bn lgdp/// • if tin(1991q1,2008Q4), lags(2) rank(2) /// • bconstraints(1/11) aconstraints(12/13) /// • noetable

Vector error-correction model Sample: 1991q1 - 2008q4 No. of obs = 72 AIC = -15.80442 Log likelihood = 659.9591 HQIC = -14.6589 Det(Sigma_ml) = 1.51e-18 SBIC = -12.92697 Cointegrating equations Equation Parms chi2 P>chi2 ------------------------------------------- _ce1 2 50.19532 0.0000 _ce2 3 1639.412 0.0000 ------------------------------------------- Identification: beta is overidentified Identifying constraints: ( 1) [_ce1]lm1 = 1 ( 2) [_ce1]lmt = 0 ( 3) [_ce1]lxt = 0 ( 4) [_ce1]loilp = 0 ( 5) [_ce1]lcpi = 0 ( 6) [_ce1]ltipp906bn = 0 ( 7) [_ce2]lm1 = 0 ( 8) [_ce2]lmt = 1 ( 9) [_ce2]lxt = 0 (10) [_ce2]ltcpn = 0 (11) [_ce2]ltipp906bn = 0

*** Point Forecast *** fcast compute y_, step(4) keep y_lm1 y_lmt y_lcpi /// y_loilp y_ltcpn y_lxt /// y_ltipp906bn y_lgdp quarter keep if tin(2009q1,2009q4) save "filename"

** Residuals from the VEC equations ** foreach x of varlist lm1 lmt lxt loilp /// ltcpn lcpi /// ltipp906bn lgdp { predict res_`x'if e(sample), /// residuals /// equation(D_`x') }

Probability Forecasting • It is basically an estimation of the probability that a single or joint event occurs. • We could define the event in terms of the levels of one or more variables, for one or more future time periods. • It is associated to the uncertainty inherent to the predictions produced by regression models.

Probability Forecasting • This methodology can be applied to a wide diversity of models. Our focus here is on the predictions from a cointegrating VAR model. • In general, forecasting based on econometric models are subject to: • Future uncertainty • Parameters uncertainty • Model uncertainty • Measurement and policy uncertainty

Probability Forecasting • Future and parameter uncertainty • Let’s consider the standard linear regression model: Where

Probability Forecasting • Future and parameter uncertainty • For example, for σ2 known we could simulate ; j=1,2,…,J ; s=1,2,…,S Where: j-th random draw from s-th random draw from which is independent from the random draw for

Probability Forecasting • Computations for VAR cointegrating models • Let’s consider the VEC model • Non-Parametric Approach • Simulated errors are drawn from in sample residuals • 2. The Choleski decomposition for the estimated Var-Cov matrix of the error term is used in a two-stage procedure combined with the simulated errors in (1).

** Matrix for Simulation (First Stage, Pag.167) ** matrix sigma=e(omega) /* V-C Matrix of the residuals */ matrix P=cholesky(sigma) mkmat res_lm1res_lmt res_lxt res_loilp /// res_ltcpn res_lcpi /// res_lgdp res_ltipp906bn /// if tin(1991q1,2008q4), /// matrix(res) matrix invP_res=inv(P)*res' matrix invP_rs1=invP_res‘ svmat invP_rs1,names(col)

** Program for Residual Resampling ** program mysim_np, rclass preserve bsample 4 if tin(1991q1,2008q4) /* 4 frcst. per. */ mkmat IP_R_D_lm1 IP_R_D_lm IP_R_D_lcpi /// IP_R_D_loilp IP_R_D_ltcpn IP_R_D_lxt /// IP_R_D_ltipp906bn IP_R_D_lgdp, /// matrix(IP_R) matrix PE_tr=P*IP_R' matrix PE=PE_tr' svmat PE,names(col) ● ● ● ● ● ● ● ● ●

****** Simulation ****** simulate “varlist", rep(###) /// saving("filename",replace): /// mysim_np command: mysim_np s_lm1_1: r(res_lm1_1) s_lm1_2: r(res_lm1_2) ● ● ● ● ● ● ● ● ● s_lgdp_3: r(res_lgdp_3) s_lgdp_4: r(res_lgdp_4) Simulations (###) ─┼─ 1 ─┼─ 2 ─┼─ 3 ─┼─4 ─┼─ 5 .................................................... 50 ● ● ● ● ● ● ● ● ●

**** Probability Forecasting **** generate dgdp=gdp/gdp2008*100-100 /// if year==2009 & /// replication>0 generate inf=cpi/cpi2008*100-100 /// if year==2009 & /// replication>0 generate gdp_n__inf45=cond(dgdp<0 & inf>45,1,0) proportion gdp_n__inf35

Cointegrating VAR Models and Probability Forecasting:Applied to a Small Open Economy Gustavo Sánchez April 2009

Cointegrating VAR Models and Probability Forecasting: Applied to a Small Open Economy