Vector Autoregressions (VARs): Operational Perspectives

Transcription

1 Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), Macroeconomericians do 4 hings: 1. Describe and summarize macroeconomic daa, 2. Make macroeconomic forecass, 3. Quanify wha we know abou he rue srucure of he macroeconomy, 4. Advise or ac as policymakers. Following he problems of he 1970s, none of he srucural models or univariae ime series approaches seemed rusworhy. VARs arose in his vacuum. VARs come in hree varieies: 1. Reduced Form 2. Recursive 3. Srucural A reduced-form VAR expresses each variable as a linear funcion of is own pas values and he pas values of all oher variables being considered and a serially uncorrelaed error erm. 1

2 In heory, he VAR uses all available or relevan pas values. In pracice, frequenly he Akaike (AIC) or Bayes (BIC) informaion crieria are used. The error erms are viewed as surprises movemens in he variables afer aking is pas ino accoun. If he differen variables are correlaed wih each oher, hen he error erms will also be correlaed across equaions. A recursive VAR consrucs he error erms in each regression o be uncorrelaed wih he error erm in he preceding equaion. This is done by adding carefullyseleced conemporaneous values as regressors. Esimaion of each equaion by OLS produces residuals ha are uncorrelaed across equaions. The recursive VAR amouns o esimaing he reduced form, hen compuing he Cholesky facorizaion of he reduced form VAR covariance marix. (See he book by Lukepohl, 1993). Unforunaely he resuls depend on he order of he variables. Changing he order changes he VAR equaions, coefficiens, and residuals, and here are n! recursive VARs possible considering he possible reorderings. A srucural VAR uses economic heory o sor ou conemporaneous links among he variables. Srucural VARs require idenifying assumpions ha esablish causal links among variables. These produce insrumenal variables. 2

3 Sock and Wason offer his example of a srucural VAR based on a Taylor rule: R = r * ( π π *) 1.25( u u *) + lagged values of R, π, u + ε The aserisked values are desired values and bar values are 4 quarer railing averages. This equaion becomes he ineres rae equaion in he srucural VAR. Firs he reduced form VAR and a recursive VAR are esimaed o summarize he co-movemens of he hree series involved. Second, he reduced form VAR is used o forecas he variables. Third, he srucural VAR is used o esimae he effec of a policy-induced change in he fed funds rae on inflaion and unemploymen. Sandard pracice is o repor Granger-causaliy ess, impulse responses, and forecas error variance decomposiions. (These are more informaive o undersanding he relaionships han he VAR regression coefficiens or R 2 saisics.) 3

4 Granger-Causaliy Tes Dependen Variable Regressor π u R π u R These are p-values for F-saisics for join ess on lags. So unemploymen helps predic inflaion (2% level), bu fed funds does no help predic inflaion (27% level). Here is a variance decomposiion for he recursive VAR orders as π, u, R. ( , quarerly). The variance decomposiion (forecas error decomposiion) is he percenage of he variance of he error made in forecasing a variable due o a specific shock a a specific ime horizon. Variance decomposiion of R. Var. Decomp. In Percenage Poins Forecas Horizon Fcs. Sandard Error π u R This suggess ha 75% of he error in he forecas of he fed funds rae 12 quarers ou is due o inflaion and unemploymen shocks in he recursive VAR. 4

5 Impulse responses race ou he response of curren and fuure values of each of he variables o a one-uni increase in he curren value of one of he VAR errors. I is a oneperiod shock which revers o zero immediaely. These make more sense in he conex of a model wih uncorrelaed errors across equaions. In hese we see he effec of a 1% change in each variable as i works hrough he recursive VAR sysem wih he 5

6 coefficiens esimaed from acual daa. Also ploed are ±1 sandard deviaion error bands, yielding roughly 66% confidence inervals. The reduced-form VAR model can also be used o ierae forward o forecas. Sock and Wason hen replace he ineres rae equaion wih wo forms of he Taylor rule (one backward looking and one forward looking), and compare impulse responses of moneary policy shocks. 6

7 Assessmen VARs are good a capuring co-movemens of muliple ime series. Granger-causaliy ess, impulse response funcions and variance decomposiions are well-acceped and widely used. Small VARs have become he benchmarks agains which new forecasing sysems are judged. Sims (1993) allows for ime-varying parameers o capure imporan drifs in coefficiens. Adding variables involves coss. A 9-variable, four lag VAR as 333 unknown coefficiens (including inerceps). Esimaion of all of hese requires resricions. Bayesian approaches have helped conrol he number of parameers in large VAR models. Srucural inference is ougher. A lo of he success of hese models depends upon evaluaion of shocks. VAR shocks reflec omied variables. If he omied variables (facors or informaion) correlae wih included variables, hen he esimaes will conain omied variable bias. Also, if agens are forward looking, impulse responses may sugges bizarre causal responses. Changing policy rules may lead o misspecificaion in consan parameer srucural VARs jus as hey migh in sandard muli-equaion srucural models. Researchers also seem o be aemping o raionalize a specific causal relaionship in order o be able o jusify a 7

8 paricular recursive ordering so ha heir srucural VAR collapses o a recursive VAR, which makes analysis easier. Wih regard o forecas error variances, Spencer (JMCB, 1989), finds: Ordering of variables: ordering of he variables is criically imporan. I is of greaer imporance for emporally aggregaed daa since he conemporaneous correlaion of he pre-orhogonalized aggregaed daa is likely o be greaer. There is less problem for monhly daa han for quarerly, semi-annual, and annual daa. Trend removal: he mehod of derending can make a subsanial difference o variance decomposiion resuls. Lag lengh: In a mrpy model, a second year of lags o he VAR gives increased esimaes of he imporance of money in explaining indusrial producion. Adding lags also seems o improve he sabiliy of resuls across orderings. Level of emporal aggregaion: While aggregaion may reduce noise in series, i increases con. correlaion. 8