Vector Autoregression

Transcription

1 Vector Autoregression Jamie Monogan University of Georgia February 27, 2018 Jamie Monogan (UGA) Vector Autoregression February 27, / 17

2 Objectives By the end of these meetings, participants should be able to: Explain the relationship between VAR and Granger causality. Weigh the advantages and disadvantages of the VAR approach. Specify and estimate a VAR model with OLS. Interpret a VAR model in terms of causal tests and impulse response analysis. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

3 Granger and VAR The direct Granger test is a bivariate case of vector autoregression. That is, it considers a vector (length 2) of all possible endogenous variables in a system of 2 variables and it controls expected autoregression in y (and x) by introducing lags of the dependent variables on the right hand side. The VAR setup is an extension of the idea to a system of k variables. We still consider each of the k variables from the system as a function of their own lags and lags of the other k-1 variables. We still do causal and exogeneity testing with F -ratios. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

4 The Attitude of VAR Modeling Equally good theory published side by side in the best journals is strikingly contradictory. But all developed and tested according to accepted econometric practice. Maybe we have to step back from the presumption that it is possible to specify the interrelationships of a group of variables and say, instead: What really matters, the question above all other questions, is causal ordering. We have no confidence in the evidence we bring to bear on this issue because it is so deeply embedded in assumptions about multivariate relationships that the data are not allowed to speak. Let us instead assume that we really know nothing about the structure of a set of variables which may even be unknowable and proceed modestly to a means to let the data tell us. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

5 The Controversy over VAR Modeling VAR: Structural models, while desirable, are impossible for complex systems. Reduced form models telling us about causal ordering is the most that we can hope for. In fact, in a world of endogenous relationships, structural models may be horribly misspecified. Anti-VAR: Structure which is to say, theory is all that matters, and VAR cannot produce structural estimates. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

6 Problems of VAR Analysis VAR models require the estimation of very large numbers of parameters (because they impose no theoretical restrictions). They are therefore radically inefficient. Like the Granger case, VAR coefficients should never be interpreted. (We estimate with OLS, so this is well known.) But it is useful as a theory-light way to let the data speak to order of causality questions, as well as subsidiary questions like required lag length. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

7 The VAR Model y t = c + L B s y t s + u t s=1 In effect, written entirely in terms of y because several dependent variables are presumed to be jointly endogenous. (You do have the option of specifying some exogenous predictors as well.) This is a system of equations, where there are as many equations as variables, k. y t is understood to be a vector (hence, the name vector autoregression ), the value of each of the variables at time t. B s is a matrix of regression parameters, k by k. (Note, this notion differs from how B has been used in this course.) Jamie Monogan (UGA) Vector Autoregression February 27, / 17

8 Vector Notation Note: Brandt & Williams use row-vector notation; Enders and I use column-vector notation. Consider the case of a three-variable system. We therefore wish to estimate the following model: y 1t c 1 L β (s) 11 β (s) 12 β (s) 13 y 1t s y 2t = c 2 + β (s) 21 β (s) 22 β (s) 23 y 2t s y 3t c 3 s=1 β (s) 31 β (s) 32 β (s) y 3t s 33 + Notice that the u t vector does not allow us to specify disturbance covariances, so we can estimate this system with OLS equations. u 1t u 2t u 3t Jamie Monogan (UGA) Vector Autoregression February 27, / 17

9 Determining Lag Length E(u t u t) = Σ To test restrictions such as lag specification: Consider a model, UR, with s lags and a restricted model, a proper subset, with s-1. Then: l = (T c)(log ˆΣ R log ˆΣ UR ) l is a likelihood ratio test, distributed as χ 2 where log Σ R is the log of the determinant of the error covariance of the restricted model and c = ks + 1 is a degrees of freedom correction, based on the number of variables times the number of lags in the unrestricted model. You also may want to evaluate your specification my plotting predicted values against the true series. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

10 The Moving Average Representation Instead of a Vector Autoregression (VAR) representation of our model, how about a Vector Moving Average (VMA) representation? Recall: A low order AR process MA( ) We use a low-order AR specification (remember, column notation): y t = c + B 1 y t 1 + B 2 y t 2 + e t y t = c + (B 1 L + B 2 L 2 )y t + e t Hence, we can redefine our VAR model in terms of an MA( ): y t = d + (I + C 1 L + C 2 L )e t where (I + C 1 L + C 2 L ) = (I B 1 L B 2 L 2... B p L p ) 1. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

11 Interpretation from the VMA model The individual moving average coefficients are defined as: C 1 = B 1 C 2 = B 1 C 1 + B 2 C 3 = B 1 C 2 + B 2 C 1 + B 3. C l = B 1 C l 1 + B 2 C l B p C l p Now we can estimate an empirical impulse response function from each of the innovations series to each of the variables in the system. If we then shock the innovations, a process called innovation accounting, we can then observe the multivariate causal flow. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

12 Impulse Response Analysis With impulse rseponse analysis we ask: For a one unit pulse shock to variable A, what are the expected dynamic consequences in the system? The right hand side of a VMA model consists of disturbances of various terms multiplied by coefficients. Each MA coefficient gives us the expected impact on the LHS variable for a shock at a particular lag of a particular variable. Simply plotting the coefficients will graphically display what we expect to happen over time. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

13 Causal Testing The clearest technique is to do F -tests on separate equations. (I would advise this.) Language: null hypothesis of exogeneity or noncausality. The vars package in R does a multi-equation F -test that asks if all coefficients for one variable are zero in the equations for all other variables. (Comparable to an F -test of coefficient equality in a seemingly unrelated regression.) The advantage to the multi-equation F is it evaluates whether a variable has any consequence in the whole system. The disadvantage is it does not speak to which other variables it causes. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

14 Testing for Serial Correlation Visual Inspections Residuals plots ACF / PACF Both produced by default in R Portmanteau tests (multivariate) Breusch-Godfrey Box-Ljung h Q h = T j=1 tr(ˆγ j 1 ˆΓ 0 ˆΓ j ˆΓ 1 0 ) where Q χ 2 (k 2 (h p)) where k is the number of endogenous variables, h the the number of lags for which autocorrelation is considered, and p is the order of the VAR model (i.e., number of lags of each variable on the right-hand side). ˆΓ j is the covariance matrix of residuals at time t with those at time t j. We also might do a small-sample correction. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

15 More on Lag Length For a VAR(p) model with T observations, k variables, and ˆΣ the determinant of the error covariance matrix: AIC(p) = T log ˆΣ + 2(k 2 p + k) BIC(p) = T log ˆΣ + log(t )(k 2 p + k) HQ(p) = T log ˆΣ + 2 log(log(t ))(k 2 p + k) These fit indices can be calculated for any model with a log-likelihood function. They allow for a probabilistic view of model selection. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

16 Software Stata var varlist, lags(#) exogenous(list2) R vars library var.model < VAR(dataSet, p=#, type= const, exogen=vectorname) plot(var.model) causality(var.model, cause= variable.name ) var.model.irf < irf(var.model, impulse = variable.name, response = c( var1, var2, var3 )) plot(var.model.irf) Jamie Monogan (UGA) Vector Autoregression February 27, / 17

17 Homework For Next Time Reading: Monogan Panel Data Analysis. In International Encyclopedia of Political Science. Beck and Katz What to Do (and Not to Do) with Time Series Cross-Section Data. American Political Science Review 89: Answer question #4 from page 186 of Political Analysis Using R. Suppose you estimated the following VAR model (the constants are zero). Write down the VMA model through two lags: [ y1t y 2t ] = [ ] [ y1t 1 y 2t 1 ] [ ] [ y1t 2 y 2t 2 ] [ e1t + e 2t ] Jamie Monogan (UGA) Vector Autoregression February 27, / 17

18 Homework Additional Material Jamie Monogan (UGA) Vector Autoregression February 27, / 17

19 Bayesian Vector Autoregression Basic Introduction to Bayesian Methods Likelihood-based inference assumes that population parameters are fixed, and the data are randomly observed given the parameters. From the data, we estimate which parameters are most likely to have produced the data we have observed. Bayesian inference assumes that the data are fixed, and the population parameters are random. Consider: much of classical inference relies on repeated-sample theory. Could you repeat a sample of what you are studying? If you are Gregor Mendel, then yes you can find more pea pods. If you are studying the time series of Bush s approval rating, then no. So if the parameters are random, what is their distribution? For θ the parameters and D the data, Bayes law tells us: p(θ D) = p(θ)p(d θ) p(d) Jamie Monogan (UGA) Vector Autoregression February 27, / 17

20 Bayesian Vector Autoregression Priors, Likelihoods, & Posteriors Using this rule, we can define our posterior distribution with the prior distribution and likelihood function: π(θ D) = Θ p(θ)l(θ D) p(θ)l(θ D) p(θ)l(θ D)dθ A few approaches to priors (non-exhaustive list): Flat Conjugate Elicited π(θ D) may be complex and hard to marginalize (w.r.t. each parameter). Hence, we turn to MCMC to numerically give us the distribution of our posterior. Short-term memory: useful because it will wander around values with the highest density. Jamie Monogan (UGA) Vector Autoregression February 27, / 17

21 Bayesian Vector Autoregression Bayesian Vector Autoregression: Advantages Imposes structure through priors: coefficients diminishing to zero. Greater ability to account for unit roots. Easier and more accurate assessment of uncertainty. Fairer assumptions about data. The Sims-Zha Model q(a) L(Y A)π(a 0 )φ(ã +, Ψ) Jamie Monogan (UGA) Vector Autoregression February 27, / 17

22 Bayesian Vector Autoregression Sims-Zha Priors How do we define Ψ? Diagonal elements define variance of VAR parameters: ψ l,j,i = ( λ0 λ 1 σ j l λ 3 Since l represents lag length, a larger lag implies a smaller variance. (Parameters are converging to zero.) λ 0 speaks to overall parameter variance, λ 1 speaks to the standard deviation one lag out, and λ 3 speaks to the rate of decay. Additional hyperparameters: λ 2 = 1, were it any different, then a variables own lags would carry different relative weight. The variance of the constant is defined as (λ 0 λ 4 ) 2. The variance of exogenous variables is defined as (λ 0 λ 5 ) 2. µ 5 & µ 6 not addressed in reading: techniques for allowing cointegration. Jamie Monogan (UGA) Vector Autoregression February 27, / 17 ) 2

23 Bayesian Vector Autoregression Uncertainty in Impluse Response Analysis Typical form for a confidence band: ĉ ij (t) ± δ ij (t) We can easily get an estimate of ĉ ij (t) in a frequentist perspective. δ is less straightforward. With Bayesian methods, we can sample from the posteriors of our estimates. Gaussian approximation: ĉ ij (t) ± z α σ ij (t) (σ from posterior of c) Pointwise quantiles: [c ij.α/2 (t), c ij.(1 α)/2 (t)] Likelihood-based eigenvector: ĉ ij + γ k,low, ĉ ij + γ k,high (accounts for serial correlation) Jamie Monogan (UGA) Vector Autoregression February 27, / 17