Vector Autogregression and Impulse Response Functions

Transcription

1 Chapter 8 Vector Autogregression and Impulse Response Functions 8.1 Vector Autogregressions Consider two sequences {y t } and {z t }, where the time path of {y t } is affected by current and past realizations of the {z t } sequence. Moreover, the time path of the {z t } sequence is also affected by current and past realizations of the {y t } sequence. Assuming that both variables are stationary we model them symmetrically, for example, in the following system: y t = b 10 + b 12 z t + γ 11 y t 1 + γ 12 z t 1 + ε yt (8.1) z t = b 20 + b 21 y t + γ 21 y t 1 + γ 22 z t 1 + ε zt (8.2) where {ε yt } and {ε zt } are uncorrelated white-noise disturbances, and the standard deviations of ε yt and ε zt are σ y and σ z, respectively. We say that Equations 8.1 and 8.2 are first-order vector autoregression (VAR) because the longest lag is equal to one. These are not reduced-form equations because of the existence of contemporaneous effects. Using matrix algebra we can write the system in its compact form or where 1 b12 b 21 1 B= yt z t = b10 b 20 γ11 γ + 12 γ 21 γ 22 yt 1 z t 1 + εyt Bx t = Γ 0 +Γ 1 x t 1 + ε t (8.3) 1 b12, x b 21 1 t = yt z t γ11 γ Γ 1 = 12, ε γ 21 γ t = 22, Γ 0 = εyt ε zt b10 b 20 ε zt 89

2 90 8 Vector Autogregression and Impulse Response Functions An also familiar form to express the VAR is in its standard from x t = A 0 + A 1 x t 1 + e t (8.4) which is obtained by premultiplying the compact form by B 1, and where A 0 = B 1 Γ 0, A 1 = B 1 Γ 1, and e t = B 1 ε t. This last equation can also be written as: y t = a 10 + a 11 y t 1 + a 12 z t 1 + e 1t (8.5) z t = a 20 + a 21 y t 1 + a 22 z t 1 + e 2t (8.6) where each a i0 is the element i of the vector A 0, a i j is the element in row i and column j of the matrix A 1, and e it is element i of vector e t. These equations 8.5 and 8.6 are called VAR standard form, while equations 8.1 and 8.2 are the referred to as the structural VAR. The variance/covariance matrix of the shocks e 1t and e 2t is defined as var(e1t ) cov(e Σ = 1t,e 2t ) cov(e 1t,e 2t ) var(e 2t ) and because the elements in Σ do not depend on time, we can also write matrix Σ as σ 2 Σ = 1 σ 12 σ 21 σ2 2 where obviously σ 2 1 = var(e 1t), σ 2 2 = var(e 2t), and σ 21 = σ 21 = cov(e 1t,e 2t ). Recall that in the multivariate GARCH we were modeling this matrix Σ = H to depend on time, without focusing on the mean equations of y t and z t Stationarity Conditions For the same reason that in the simple first-order autoregressive model we needed that the autoregressive term be less than one, the VAR processes also need to some stationarity conditions to hold. Let s use the lag operators to rewrite Equations 8.5 and 8.6 as or y t = a 10 + a 11 Ly t + a 12 Lz t + e 1t (8.7) z t = a 20 + a 21 Ly t + a 22 Lz t + e 2t (8.8) (1 a 11 L)y t = a 10 + a 12 Lz t + e 1t (8.9) (1 a 22 L)z t = a 20 + a 21 Ly t + e 2t (8.10)

3 8.1 Vector Autogregressions 91 Solving for y t and z t we obtain 1 y t = a 10(1 a 22 )+a 12 a 20 +(1 a 22 L)e 1t + a 12 e 2t 1 (1 a 11 L)(1 a 22 L) a 12 a 21 L 2 (8.11) z t = a 20(1 a 11 )+a 21 a 10 +(1 a 11 L)e 2t + a 21 e 1t 1 (1 a 11 L)(1 a 22 L) a 12 a 21 L 2 (8.12) where as long as a 12 and a 21 are not both equal to zero, Equations 8.11 and 8.12 both have the same characteristic equation. Convergence requires that the root of the polynomial(1 a 11 L)(1 a 22 L) a 12 a 21 L 2 must lie outside the unit circle. For a formal treatment for the stability conditions see Appendix 6.1 in Enders (2010) Dynamics in Simulated VAR Models To obtain an intuition of the dynamics of VAR models, we will simulate four different sequences y t and z t for different values of a i j (i = 1,2, j = 0,1,2) in Equations 8.5 and 8.6. The first two are stationary, while the last two are not stationary. 1. The processes y1 t and z1 t are generated with y1 t = 0.7y1 t z1 t 1 + e 1t (8.13) z1 t = 0.2y1 t z1 t 1 + e 2t (8.14) where the roots of the polynomial (1 a 11 L)(1 a 22 L) a 12 a 21 L 2 are 1.11 and 2.0. Because both lie outside the unit circle, the system is stationary. 2. The processes y2 t and z2 t are generated with y2 t = 0.5y2 t 1 0.2z2 t 1 + e 1t (8.15) z2 t = 0.2y2 t z2 t 1 + e 2t (8.16) where the roots of the polynomial(1 a 11 L)(1 a 22 L) a 12 a 21 L 2 are and Again, because both lie outside the unit circle, this system is also stationary. The upper panel in Figure 8.1 illustrates the first two processes (y1 t and z1 1 ), while the lower panel illustrates the other two (y2 t and z2 t ). The Stata code to generate the processes and the graphs is: clear set obs 150 set seed gen time=_n tsset time gen white1=invnorm(uniform()) set seed gen white2=invnorm(uniform()) gen y1 = 0 1 For the details, see Enders (2010) page 301.

4 92 8 Vector Autogregression and Impulse Response Functions Simulated Stationary VAR Processes y1 and z time y1 z1 y2 and z time y2 z2 Fig. 8.1 Simulated Stationary VAR Processes gen z1 = 0 forvalues iter=2/150 { replace y1 = 0.7*l.y *l.z1 + white1 if time == iter replace z1 = 0.2*l.y *l.z1 + white2 if time == iter } twoway line y1 z1 time, m(o) c(l) scheme(sj) /// ytitle( "y1 and z1" ) saving(one, replace) gen y2 = 0 gen z2 = 0 forvalues iter=2/150 { replace y2 = + 0.5*l.y2-0.2*l.z2 + white1 if time == iter replace z2 = - 0.2*l.y *l.z2 + white2 if time == iter } twoway line y2 z2 time, m(o) c(l) scheme(sj) /// ytitle( "y2 and z2" ) saving(two, replace) gr combine one.gph two.gph, col(1) /// iscale(0.7) fysize(100) /// title( "Simulated Stationary VAR Processes" ) The upper panel of the figure shows that there is a tendency for the sequences to move together. Because a 21 is positive, large realizations of y1 t induce a large realization of z t+1. Likewise, because a 12 is positive, a large realization of z t induce a large a large realization of y t+1. Here the two series are positively correlated,

5 8.1 Vector Autogregressions 93 Cross-correlations of y1 and z Cross-correlogram Lag Fig. 8.2 Cross correlogram for the simulated processes y1 t and z1 t which can be easily checked by looking at the cross correlogram in Figure 8.2 obtained using Equation In the lower panel a 21 and a 12 are both negative, so positive realizations of y t can be associated with negative realizations of z t+1 and vise versa. In this case the two series are negatively correlated The processes y3 t and z3 t are generated with 4. The processes y4 t and z4 t are generated with The Stata code to obtain cases 3 and 4 is: y3 t = 0.5y3 t z3 t 1 + e 1t (8.17) z3 t = 0.5y3 t z3 t 1 + e 2t (8.18) y4 t = 0.5y4 t z4 t 1 + e 1t (8.19) z4 t = y4 t z4 t 1 + e 2t (8.20) gen y3 = 0 gen z3 = 0 forvalues iter=2/150 { replace y3 = + 0.5*l.y *l.z3 + white1 if time == iter 2 Please see at the end of the chapter for the Stata codes to generate the cross correlogram in Figure 8.2.

6 94 8 Vector Autogregression and Impulse Response Functions Simulated Nonstationary VAR Processes y3 and z time y3 z3 y4 and z time y4 z4 Fig. 8.3 Simulated Nonstationary VAR Processes replace z3 = + 0.5*l.y *l.z3 + white2 if time == iter } twoway line y3 z3 time, m(o) c(l) scheme(sj) /// ytitle( "y3 and z3" ) saving(three, replace) gen y4 = 0 gen z4 = 0 forvalues iter=2/150 { replace y4 = *l.y *l.z4 + white1 if time == iter replace z4 = *l.y *l.z4 + white2 if time == iter } twoway line y4 z4 time, m(o) c(l) scheme(sj) /// ytitle( "y4 and z4" ) saving(four, replace) gr combine three.gph four.gph, col(1) /// iscale(0.7) fysize(100) /// title( "Simulated Nonstationary VAR Processes" ) Figure 8.3 shows two processes with unit root. There is little tendency in these series (upper and lower panels) to revert to a constant long-run value. The upper panel shows a multivariate generalization of the random walk model. The lower panel shows how the value of a 10 = 0.5 acts as a drift for both of the series. The drift appears as a deterministic trend that makes these series nonstationary, along with the generalized random walk process.

7 8.1 Vector Autogregressions Estimation Forecasting Consider the simple first-order model presented in Equation 8.4 and reproduced here x t = A 0 + A 1 x t 1 + e t. (8.21) Once the coefficients A 0 and A 1 are estimated using data until period T, it is easy to obtain the one-step ahead forecast as: E T x T+1 = A 0 + A 1 x T. (8.22) Likewise, the two-step ahead forecast can be obtain recursively using: E T x T+2 = A 0 + A 1 E T x T+1 = A 0 + A 1 A 0 + A 1 x T. (8.23) However, because usually VAR models are overparameterized, this forecast may be unreliable. One alternative is to drop statistically significant coefficients and estimate the remaining model using Seemingly Unrelated Regressions (SUR) Identification There are basically two representations for the VAR. The structural VAR as presented in Equations 8.1 and 8.2 and the standard form VAR, as presented in Equations 8.5 and 8.6. Because of the feedback in the VAR process, the structural VAR cannot be directly estimated. Notice that z t is correlated with the error term ε yt and that y t is correlated with the error term ε zt. This is a problem because standard estimation techniques require the regressors to the uncorrelated with the error term. However, this problem does not arise in the standard form VAR, where OLS can be used to obtain estimates of A 0 and A 1. Then the key question is whether we can use the OLS estimated of A 0 and A 1 to retrieve the structural VAR estimates. The short answer is no, unless we are willing to impose restrictions on the structural VAR equations. The reason is simple, in the structural VAR we need to estimate eight coefficients (b 10, b 20, b 12, b 21, γ 11, γ 12, γ 21, and γ 22 ) and there are two standard deviations (σ y and σ z ), while in the standard form VAR there are only six coefficient estimates (a 10, a 20, a 11, a 12, a 21, and a 2 ) and we calculate three additional values (var(e 1t ), var(e 2t ), cov(e 1t,e 2t )). That is, the standard form has 9 parameters and the structural needs 10. Hence, we say that the structural VAR is underidentified. A simple identification strategy is to impose a restriction such as b 21 = 0, so that the structural VAR becomes y t = b 10 + b 12 z t + γ 11 y t 1 + γ 12 z t 1 + ε yt (8.24) z t = b 20 + γ 21 y t 1 + γ 22 z t 1 + ε zt (8.25)

8 96 8 Vector Autogregression and Impulse Response Functions This means that z t has a contemporaneous effect on y t, but y t can only affect the {z t } sequence with one lag. The resulting system is exactly identified. B 1 is 1 b12 B= 0 1 Premultiplying the structural VAR system by B 1 yields yt 1 b12 b10 1 b12 γ11 γ = + 12 z t γ 21 γ 22 or yt z t = b12 b 12 b 20 b 20 b 20 yt 1 γ11 + b + 12 γ 21 γ 12 b 12 γ 22 γ 21 γ 22 z t 1 1 b yt 1 z t 1 After estimating A 0 and A 1 we have the following nine equations: εyt ε zt εyt b + 12 ε zt a 10 = b 12 b 12 b 20 (8.26) a 20 = b 20 (8.27) a 11 = γ 11 + b 12 γ 21 (8.28) a 12 = γ 12 b 12 γ 22 (8.29) a 21 = γ 21 (8.30) a 22 = γ 22 (8.31) e 1t = ε yt b 12 ε zt (8.32) e 2t = ε zt (8.33) var(e 1 ) = σ 2 y + b 2 12σ 2 z (8.34) var(e 2 ) = σ 2 z (8.35) cov(e 1,e2) = b 12 σ 2 z (8.36) that can be used to obtain b 10, b 12, γ 11, γ 12, b 20, γ 21, γ 22, σ 2 y, and σ 2 z. ε zt 8.2 The Impulse Response Function Is we write the standard form VAR in matrix form we have yt a10 a11 a = + 12 yt 1 + z t a 21 a 22 z t 1 a 20 Recall that every autoregressive process has a moving-average representation. The same is true for VAR, where the moving-average representation is called the vector moving average (VMA). The idea in the moving average representations is to write the current values of y t and z t in terms of the current and past values of the shocks. e1t e 2t

9 8.2 The Impulse Response Function 97 The moving-average representation of the above system is: 3 yt z t = ȳt + z t i=0 i a11 a 12 e1t i a 21 a 22 e 2t i While these VMA are written in terms of e 1t i and e 2t i, we can use e t = B 1 ε t to write the VMA in terms of ε yt and ε zt : yt z t = ȳt z t b 12 b 21 i=0 a11 a 12 a 21 a 22 i 1 b12 b 21 1 A simplified way of writing this moving-average representation is: yt ȳt φ11 (i) φ = + z t 12 (i) εyt i φ 21 (i) φ 22 (i) z t i=0 where φ jk (i) are just elements of the φ i matrix: φ i = A i 1 1 b 12 b 21 1 b12 b 21 1 ε zt i εyt The moving-average representation is important because it allows examining the interaction between the {y t } and {z t } sequences. The coefficients φ i are used to generate the effects of shocks ε yt and ε zt on the entire time paths of the{y t } and{z t } sequences. The impact multipliers are the four elements φ jk (0). Likewise, φ ik (1) are the one-period responses and so on. The impulse response functions are the four sets of coefficients φ 11 (i), φ 12 (i), φ 21 (i), and φ 22 (i). They are usually presented using a plot of φ jk (i) against i. To illustrate the impulse response functions, we will use the same simulated sequences as before. That is, the processes y1 t and z1 t generated with ( y1t z1 t ) = ( ) ( y1t 1 and the processes y2 t and z2 t are generated with ( ) ( ) y2t = z2 t z1 t 1 ( y2t 1 z2 t 1 ) + ) + ( e1t e 2t ) ( e1t To keeps things simple we impose the restrictions b 12 = b 21 = 0, such that the structural VAR is the same as the standard form VAR. Figure 8.4 shows the responses of the sequences {y1 t } and {z1 t } to e 1t and e 2t. Likewise, Figure 8.5 shows the responses of the sequences{y2 t } and{z2 t } to e 1t and e 2t. Notice the symmetry in the impulse response functions. This is just coming from the symmetry in the VARs. e 2t ) ε zt 3 This one is obtained by backward iteration and assuming that the stability conditions are met.

10 98 8 Vector Autogregression and Impulse Response Functions Impulse Response Functions Response to e1t shock Response to e2t shock time y1 z time y1 z1 Fig. 8.4 Impulse Response Functions for the Simulated Processes y1 and z1 8.3 Estimation in Stata Vector Autoregressions Models Estimation of VAR models in Stata is initially a simple task, but you have to make sure you understand the options because Stata may impose different identifying restrictions in the structural VAR. Consider the following example use clear tsset var dln_inv dln_inc if qtr>=tq(1961q2), lags(1/2) Vector autoregression Sample: 1961q2-1982q4 No. of obs = 87 Log likelihood = AIC = FPE = 2.80e-07 HQIC = Det(Sigma_ml) = 2.23e-07 SBIC = Equation Parms RMSE R-sq chi2 P>chi dln_inv dln_inc Coef. Std. Err. z P> z 95% Conf. Interval dln_inv dln_inv L

11 8.3 Estimation in Stata 99 Impulse Response Functions Response to e1t shock Response to e2t shock time time y2 z2 y2 z2 Fig. 8.5 Impulse Response Functions for the Simulated Processes y2 and z2 L dln_inc L L _cons dln_inc dln_inv L L dln_inc L L _cons There is a number of selection-order statistics to assist in fitting the VAR of the correct order. A useful command in Stata is varsoc, which computes the following four information criteria: the final prediction error (FPE), the Akaike s information criterion (AIC), the Hannan and Quinn information criterion (HQIC), and the Schwarzs Bayesian information criterion(sbic). varsoc dln_inv dln_inc Selection-order criteria Sample: 1961q2-1982q4 Number of obs = lag LL LR df p FPE AIC HQIC SBIC e * * * e-07* *

12 100 8 Vector Autogregression and Impulse Response Functions.06 varbasic, dln_inc, dln_inc varbasic, dln_inc, dln_inv varbasic, dln_inv, dln_inc varbasic, dln_inv, dln_inv step 95% CI orthogonalized irf Graphs by irfname, impulse variable, and response variable Fig. 8.6 Impulse Response Functions e e e Endogenous: dln_inv dln_inc Exogenous: _cons Impulse Response Function Note that Stata reports the standard form VAR and will not be able to obtain the impulse response function (IRF) without further assumptions. The following command will estimate the same VAR and will provide the orthogonalized IRF: set scheme sj varbasic dln_inv dln_inc, lags(1/2) The orthogonalized impulse-response functions is imposing the cholesky decomposition in the structure of the errors. In simple words, it is setting b 21 = 0 when going from the standard form VAR to the structural VAR. Notice that the order in which you place the variables in the varbasic command is important because a different order will impose a different restriction on the structure of the errors. Try estimating: varbasic dln_inc dln_inv, lags(1/2) and you will see that the standard form VAR as reported in the output is exactly the same as before, however, the IFR are different because of the different restriction. If you are willing to impose b 21 = b 12 = 0, then the IRF can be obtained using: irf graph irf

13 8.3 Estimation in Stata varbasic, dln_inc, dln_inc varbasic, dln_inc, dln_inv varbasic, dln_inv, dln_inc varbasic, dln_inv, dln_inv step 95% CI impulse response function (irf) Graphs by irfname, impulse variable, and response variable Fig. 8.7 Impulse Response Functions The resulting IRF is presented in Figure 8.7. Obviously, in this case the order is not important and by construction the size on the shock is one. Stata can estimate different structural VAR depending on the restrictions you are willing to impose, please see the manual for more details Stability Conditions If you want to test whether the stability conditions hold, Stata has the command varstable that will calculate the eigenvalues for the stability conditions. Consider the following example using the stationary processes {y1 t } and {z1 t } that we simulated before var y1 z1, lags(1/2) (output omitted) varstable Eigenvalue stability condition Eigenvalue Modulus All the eigenvalues lie inside the unit circle. VAR satisfies stability condition.

14 102 8 Vector Autogregression and Impulse Response Functions Now, consider the nonstationary processes{y4 t } and{z4 t } that we simulated earlier var y4 z4, lags(1/2) (output omitted) varstable Eigenvalue stability condition Eigenvalue Modulus At least one eigenvalue is at least 1.0. VAR does not satisfy stability condition. Note that when the system is not stationary, the IRF will not dissipate over time because by definition shocks will have a permanent effect on the series. Finally, including exogenous variables in the estimation is simple. var dln_inc dln_inv, lags(1/2) exog(dln_consump) (output omitted) The key assumptions are: (1) We know the exact form in which the exogenous variable enters the system. (2) We role out any feedback from the endogenous variable to the exogenous variable Granger Causality Consider the following model: y t = a 10 + z t = a 20 + p i=1 p i=1 a 11 (i)l i y t + a 21 (i)l i y t + p i=1 p i=1 a 12 (i)l i z t + e 1t (8.37) a 22 (i)l i z t + e 2t (8.38) One causality test is whether the lags of one variable enter into the equation for another variable. In the two-equation model above we say that {y t } does not Granger cause {z t } if and only if all the coefficients of the lags of y t on the z t equation are equal to zero. Hence, if {y t } does not improve the forecasting performance of {z t }, then{y t } does not Granger cause{z t }. Under the assumption that all the VAR variables are stationary, a direct way to test Granger causality is use the standard F-test of the restriction a 21 (1)=a 21 (2)=a 21 (3)= =a 21 (p)=0 (8.39) Consider the following example in Stata: use var dln_inv dln_inc dln_consump

15 8.4 Supporting.do files 103 (output omitted) vargranger Granger causality Wald tests Equation Excluded chi2 df Prob > chi dln_inv dln_inc dln_inv dln_consump dln_inv ALL dln_inc dln_inv dln_inc dln_consump dln_inc ALL dln_consump dln_inv dln_consump dln_inc dln_consump ALL which can also be carried out using a simple F-test: test dln_invl.dln_inc dln_invl2.dln_inc ( 1) dln_invl.dln_inc = 0 ( 2) dln_invl2.dln_inc = 0 chi2( 2) = 0.56 Prob > chi2 = A large p-value is evidence against the null. 8.4 Supporting.do files For Figures 8.3 and 8.4: clear set obs set seed gen time=_n tsset time gen white1=invnorm(uniform()) set seed gen white2=invnorm(uniform()) gen y1 = 0 gen z1 = 0 forvalues iter=2/ { replace y1 = 0.7*l.y *l.z1 + white1 if time == iter replace z1 = 0.2*l.y *l.z1 + white2 if time == iter } xcorr y1 z1, lags(40) gen y2 = 0 gen z2 = 0 forvalues iter=2/ { replace y2 = + 0.5*l.y2-0.2*l.z2 + white1 if time == iter replace z2 = - 0.2*l.y *l.z2 + white2 if time == iter } xcorr y2 z2, lags(40) Figure 8.8 presents the cross correlogram for the simulated processes y2 t and z2 t.

16 104 8 Vector Autogregression and Impulse Response Functions Cross-correlations of y2 and z Cross-correlogram Lag Fig. 8.8 Cross correlogram for the simulated processes y2 t and z2 t