Autoregressive Modeling and Impulse Responses
4.3 Impulse Response Analysis
4.3.1 Stationary VAR Processes
If the process ytis I(0), the effects of shocks in the variables of a given system are most easily seen in its Wold moving average (MA) representation
yt = 0ut+ 1ut−1+ 2ut−2+ · · · , (4.9)
where 0= IK and the s =
s j=1
s− jAj, s = 1, 2, . . . , (4.10)
can be computed recursively, as in (3.47), from the reduced-form coefficients of the VAR in levels specified in (4.5). The coefficients of this representation may be interpreted as reflecting the responses to impulses hitting the system.
The (i, j)th elements of the matrices s, regarded as a function of s, trace out the expected response of yi,t+sto a unit change in yj t, holding constant all past values of yt. Since the change in yi t, given{yt−1, yt−2, . . .}, is measured by the innovation ui t, the elements of s represent the impulse responses of the components of ytwith respect to the utinnovations. In the presently considered I(0) case, s→ 0 as s → ∞. Hence, the effect of an impulse is transitory as it vanishes over time. These impulse responses are sometimes called forecast error impulse responses because the ut’s are the 1-step ahead forecast errors.
Occasionally, interest centers on the accumulated effects of the impulses.
They are easily obtained by adding up the smatrices. For example, the accu-mulated effects over all periods, the total long-run effects, are given by
=∞
s=0
s= (IK − A1− · · · − Ap)−1. (4.11)
This matrix exists if the VAR process is stable (see the stability condition in (3.2)).
A critique that has been raised against forecast error impulse responses is that the underlying shocks are not likely to occur in isolation if the components of ut
are instantaneously correlated, that is, ifuis not diagonal. Therefore, orthog-onal innovations are preferred in an impulse response analysis, as mentioned in Section 4.2. One way to get them is to use a Choleski decomposition of the covariance matrixu. IfB is a lower triangular matrix such thatu = BB, the orthogonalized shocks are given byεt = B−1ut. Hence, we obtain the following from (4.9):
yt = 0εt+ 1εt−1+ · · · , (4.12)
wherei = iB (i= 0, 1, 2, . . .). Here 0= B is lower triangular, and thus an ε shock in the first variable may have an instantaneous effect on all the variables, whereas a shock in the second variable cannot have an instantaneous impact on y1t but only on the other variables, and so on. Thus, we have a Wold causal chain. Given that theε shocks are instantaneously uncorrelated (orthogonal), the corresponding impulse responses are often referred to as orthogonalized impulse responses.
Because many matricesB exist that satisfy BB= u, using a Choleski de-composition approach is to some extent arbitrary unless there are good reasons
for a particular recursive structure specified by a givenB. As mentioned in Sec-tion 4.1, ifB is found by a lower triangular Choleski decomposition, choosing a different ordering of the variables in the vector ytmay produce different shocks.
Hence, the effects of a shock may depend on the way the variables are arranged in the vector yt. In view of this difficulty, Sims (1981) has recommended try-ing various triangular orthogonalizations and checktry-ing the robustness of the results with respect to the ordering of the variables if no particular ordering is suggested by subject matter theory. Using information based on the latter leads to SVAR models, of course.
As discussed earlier, in an SVAR such as (4.3), the residuals are represented asBεtandεtis a (K × 1) vector of structural shocks with (diagonal) covariance matrixE(εtεt)= ε, which is often specified to be an identity matrix. In any case, the structural shocks are instantaneously uncorrelated. In theAB-model the relation to the reduced form residuals is given by Aut = Bεt. Therefore, the impulse responses in a general SVAR model may be obtained from (4.12) withj = jA−1B. If restrictions on the long-run effects are available, they may be placed on = A−1B, where is the matrix specified in (4.11). For example, one may want to impose the restriction that some shocks do not have any long-run effects. This is achieved by setting the respective elements of the long-run impact matrix = 0+ 1+ · · · equal to zero.
As an example we consider the model suggested by Blanchard & Quah (1989). On the basis of a simple economic model, Blanchard and Quah have identified supply shocks as having persistent effects on output whereas demand shocks are transitory. Suppose in a VAR model for yt = (Qt, Ut), where Qt
denotes the log of output and Utis the unemployment rate, we wish to identify innovations that can be interpreted as supply shocks and demand shocks,εt = (εts, εtd). Because we have K = 2 variables and can specify A = I2, we need K (K − 1)/2 = 1 restriction to identify the structural shocks from the VAR residuals. The effects of these shocks on the output growth ratesQt and the unemployment Utare obtained from theimatrices. Accumulating them gives the effects on Qtand the accumulated unemployment rate. Thus, the restriction that demand shocks have no long-run impact on output can be imposed by constraining the (1,2)-element of the matrix = ∞
i=0ito be equal to zero.
Estimation of models with long-run restrictions is discussed in Section 4.4.
4.3.2 Impulse Response Analysis of Nonstationary VARs and VECMs Although the Wold representation does not exist for nonstationary cointe-grated processes, it is easy to see from Section 3.6 that the s impulse re-sponse matrices can be computed in the same way as in (4.10) based on VARs with integrated variables or the levels version of a VECM [L¨utkepohl (1991, Chapter 11) and L¨utkepohl & Reimers (1992)]. In this case, the s
may not converge to zero as s → ∞; consequently, some shocks may have
permanent effects. Of course, one may also consider orthogonalized or accu-mulated impulse responses. Because the s ands may not approach zero as s→ ∞, the total accumulated impulse responses will generally not exist, however. Recall that for cointegrated systems the matrix (IK − A1− · · · − Ap) is singular. From Johansen’s version of Granger’s Representation Theorem [see Johansen (1995a)] it is known, however, that if yt is generated by a reduced-form VECMyt = αβyt−1+ 1yt−1+ · · · + p−1yt−p+1+ ut, it has the following MA representation:
yt =
t i=1
ui+ ∗(L)ut+ y0∗,
where = β⊥
α⊥(IK − p−1 i=1 i)β⊥
−1
α⊥, ∗(L)= ∞
j=0∗jLj is an infinite-order polynomial in the lag operator with coefficient matrices∗j that go to zero as j → ∞. The term y0∗ contains all initial values. Notice that has rank K− r if the cointegrating rank of the system is r. It represents the long-run effects of forecast error impulse responses, whereas the∗j’s contain transitory effects.
Because the forecast error impulse responses based on and the ∗j’s are subject to the same criticism as for stable VAR processes, appropriate shocks have to be identified for a meaningful impulse response analysis. If utis replaced byA−1Bεt, the orthogonalized “short-run” impulse responses may be obtained as∗jA−1B in a way analogous to the stationary VAR case. Moreover, the long-run effects ofε shocks are given by
A−1B. (4.13)
This matrix has rank K − r because rk() = K − r and A and B are nonsingu-lar. Thus, the matrix (4.13) can have at most r columns of zeros. Hence, there can be at most r shocks with transitory effects (zero long-run impact), and at least k∗= K − r shocks have permanent effects. Given the reduced rank of the matrix, each column of zeros stands for only k∗ independent restrictions.
Thus, if there are r transitory shocks, the corresponding zeros represent k∗r in-dependent restrictions only. To identify the permanent shocks exactly we need k∗(k∗− 1)/2 additional restrictions. Similarly,r(r − 1)/2 additional contempo-raneous restrictions identify the transitory shocks [see, e.g., King et al. (1991)].
Together these are a total of k∗r+ k∗(k∗− 1)/2 + r(r − 1)/2 = K (K − 1)/2 restrictions. Hence, assumingA= IK, we have just enough restrictions to iden-tifyB.
For example, in King et al. (1991) a model is considered for the log of private output (qt), consumption (ct), and investment (it). Using economic theory, King et al. inferred that all three variables should be I(1) with r= 2 cointegration relations and only one permanent shock. Because k∗ = 1, the permanent shock
is identified without further assumptions (k∗(k∗− 1)/2 = 0). For identification of the transitory shocks, r (r− 1)/2 = 1 further restriction is needed. Suppose a recursive structure of the transitory shocks is assumed such that the second transitory shock does not have an instantaneous impact on the first one. Placing the permanent shock first in theεtvector, these restrictions can be represented as follows in the foregoing framework:
B = ∗ 0 0
∗ 0 0
∗ 0 0
and B= ∗ ∗ ∗
∗ ∗ 0
∗ ∗ ∗
,
where asterisks denote unrestricted elements. BecauseB has rank 1, the two zero columns represent two independent restrictions only. A third restriction is placed onB, and thus we have a total of K (K− 1)/2 independent restrictions as required for just-identification.
In some situations A may also be specified differently from the identity matrix. In any case, long-run restrictions imply in general nonlinear restrictions onA,B, or both. To illustrate the process of deriving structural restrictions from economic theory, we will discuss a more complex example next.
An example. Long-run identifying assumptions for theεtshocks are typically derived from economic theory. To illustrate this point, we briefly describe a simple macroeconomic model of the labor market used by Jacobson, Vredin &
Warne (1997) to investigate the effects of shocks to Scandinavian unemploy-ment. This model consists of a production function, a labor demand relation, a labor supply, and a wage-setting relation. All variables are expressed in natural logarithms. The production function relates output gd pt to employment et as follows:
gd pt = ρet+ θ1,t, (4.14)
whereρ measures the returns to scale. The quantity θ1,tis a stochastic technol-ogy trend that follows a random walk,
θ1,t = θ1,t−1+ εtgd p,
andεgd pt is the pure technology shock. Labor demand relates employment to output and real wages (w − p)t:
et = λgdpt− η(w − p)t+ θ2,t, (4.15)
with an error process
θ2,t = φdθ2,t−1+ εdt.
If|φd| < 1, the labor demand is stationary. In that case the pure labor demand innovationεdt has only temporary effects on employment. Jacobson et al. (1997)
assumedφd = 0 a priori, which implies that the labor demand shock has no long-run effects. Within a cointegration analysis, stationarity of labor demand can be tested; hence, the a priori assumption is not needed here. In the third equation of the model, the labor force ltis related to real wages according to
lt= π(w − p)t+ θ3,t. (4.16)
The exogenous labor supply trendθ3,tfollows a random walk θ3,t = θ3,t−1+ εst,
whereεtsis the underlying labor supply shock. Finally, we have the wage-setting relation
(w − p)t = δ(gdpt− et)− γ (lt− et)+ θ4,t (4.17) stating that real wages are a function of productivity (gd pt− et) and unemploy-ment (lt− et). The wage setting trendθ4,t can be stationary or nonstationary, as determined byφwin
θ4,t = φwθ4,t−1+ εwt .
If|φw| < 1, the wage setting trend is stationary. Again, results from empirical analysis can be used to determine whether wage setting is stationary.
Under standard assumptions for the shocksεgd p, εd, εs, andεwthe solution of the model (4.14)–(4.17) in terms of the variables used in the empirical analysis is given by
From this solution it is obvious that productivity, employment, unemployment, and real wages are driven by two random walks in productivity and labor supply.
As explained earlier, the labor demand and the wage setting component can be stationary or nonstationary. In terms of the common trends literature, there are at least two and at most four common trends in this model. This implies at
most two cointegration relations: a labor demand relation and a wage-setting relation. The model together with results from a cointegration analysis implies a set of identifying assumptions for the structural VECM. Suppose, for example, that two cointegration relations (r = 2) have been found in a four-dimensional VAR (K = 4) for productivity, employment, unemployment, and real wages.
Consequently, only k∗= K − r = 2 shocks may have permanent effects. We associate the technology, the labor demand, the labor supply, and the wage-setting shocks with the equations for productivity, employment, unemployment, and real wages, respectively, such thatεt = (εtgd p, εtd, εst, εwt ).
For stationarity of labor demand and wage-setting our model implies that labor demand and wage-setting shocks have no long-run impact on the system variables and, hence, the second and fourth columns of the long-run impact matrix A−1B are zero. To identify the two permanent shocks we have to impose k∗(k∗− 1)/2 = 1 additional restriction. Assuming constant returns to scale (ρ = 1) implies that productivity is only driven by productivity shocks εgd p in the long-run [see (4.18)]. Thus, if A= IK is assumed, these sets of restrictions can be expressed as follows in terms of the long-run impact matrix:
B =
∗ 0 0 0
∗ 0 ∗ 0
∗ 0 ∗ 0
∗ 0 ∗ 0
. (4.19)
Here unrestricted elements are again indicated by asterisks. Note that, owing to the reduced rank of, we cannot simply count the zeros in (4.19) to determine the number of restrictions imposed on the model. As explained at the beginning of this section, the two zero columns represent k∗r= 4 linearly independent restrictions only. Hence, the zeros in (4.19) stand for only five linearly inde-pendent restrictions. In addition, we need r (r− 1)/2 = 1 contemporaneous restriction to disentangle the effects of the two transitory shocks. For instance, we may choose the restriction that labor demand shocks do not affect real wages on impact, that is, we setB42 = 0. In our example, a typical problem within the SVAR modeling class arises: The theoretical model does not suggest con-temporaneous restrictions, and thus, defending this type of restriction may be difficult. In practice, a sensitivity analysis with respect to different contempo-raneous identifying assumptions can be useful. We will employ the presented theoretical model to derive identifying assumptions in a structural VECM for Canadian labor market data in Section 4.7.3.
To compute the impulse responses we need not only the reduced form pa-rameters but also the structural papa-rameters. How to estimate them will be dis-cussed in the next section before we consider inference for impulse responses in Section 4.5.