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

1 / 24

# Cointegrating VAR Models and Probability Forecasting: Applied to a Small Open Economy - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Cointegrating VAR Models and Probability Forecasting: Applied to a Small Open Economy' - cybill

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

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

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

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 . . . . .

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

*** 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

Gustavo Sánchez

April 2009