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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.

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

probabilities of events

cointegrating var models
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 models6
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 models7
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 models8
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.
slide9

** 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
slide10

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

slide11

------------------------------------------------------------------------------------------------------------------------------------------------------------

beta | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

_ce1 |

lm1 | 1 . . . . .

lmt | (dropped)

lcpi | (dropped)

loilp | (dropped)

ltcpn | .215578 .0697673 3.09 0.002 .0788365 .3523194

lxt | (dropped)

ltipp906bn | (dropped)

lgdp | -4.554976 .6489147 -7.02 0.000 -5.826825 -3.283127

_cons | 57.02687 . . . . .

-------------+----------------------------------------------------------------

_ce2 |

lm1 | (dropped)

lmt | 1 . . . . .

lcpi | -.0317544 .0087879 -3.61 0.000 -.0489784 -.0145304

loilp | -.0780758 .0255611 -3.05 0.002 -.1281746 -.027977

ltcpn | (dropped)

lxt | (dropped)

ltipp906bn | (dropped)

lgdp | -2.519458 .1105036 -22.80 0.000 -2.736041 -2.302875

_cons | 26.26122 . . . . .

------------------------------------------------------------------------------

slide12

*** 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"

slide13

** 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
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 forecasting15
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 forecasting16
Probability Forecasting
  • Future and parameter uncertainty
    • Let’s consider the standard linear regression model:

Where

probability forecasting17
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 forecasting18
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).
slide19

** 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)

slide20

** 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)

● ● ●

● ● ●

● ● ●

slide21

****** 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

● ● ●

● ● ●

● ● ●

slide22

**** 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 economy24

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

Gustavo Sánchez

April 2009