A generalized method of moments approach to estimating a “Structural vector autoregression”

PETER R. HAmLEY Rice

University

Houston,

Texas

CARL E. WALSH University

of California

at Santa

Crwz

Santa

Cruz,

California

Bank

of San

Francisco

and Federal

Reserve

San Francisco,

California

A Generalized Method of Moments Approach to Estimating a “Structural Vector Autoregression”* A generalized method of moments estimator is used to test a small structural model of the U.S. macroeconomy. As with vector autoregression studies, we view the observable variables as endogenous functions of a set of unobservable shocks. We assume expectations are formed rationally and model monetary policy as a regime rather than a series of discretionary interventions. We find money growth rate shocks to be the most important source of output fluctuations. Real shocks to the demand for capital, and portfolio shocks, are found to be important in explaining covariation between output growth and changes in inflation and interest rates.

This paper tests a small structural model of the U.S. macroeconomy similar to the model discussed in Hartley and Walsh (1991). While the econometric theory is not new, our application of it is novel. The results of our investigation of U.S. business cycles will be of interest to business cycle theorists. Our econometric analysis takes seriously the notion (emphasized in Lucas 1977 for example) that the key feature of business cycles is a stable pattern of covariation between many economic aggregates, both contemporaneously and over a small number of periods into the past. It has become customary to summarize these patterns of covariation using vector autoregressions involving a small number of variables and lags. Sims (1972, 1980), Nelson and Plosser (1982), Litterman and *The authors are grateful for valuable comments received t?om Bill Brown, Frank Vella, three anonymous referees of the Journal of Macroeconomics, and seminar participants at Texas A&M University, Monash University and the Issues in Macro and Financiul Economics, Spring Academic Conference, Federal Reserve Bank of Atlanta, April 1990.

Journal of Macroeconomics, Spring 1992, Vol. Copyright 0 1992 by Louisiana State University 0164-0704/92/$1.50

14, No. Press

2, pp.

199-232

199

Peter R. Hartley

and Carl E. Walsh

Weiss (1985) and others have attempted to relate the results from vector autoregression studies to various theories of the source, and method of propagation, of business cycles. The basic idea underlying these investigations is that lagged values of a variable thought to cause business cycles ought to explain subsequent movements in output to a significant extent. The implied propagation mechanism ought also to be consistent with the theory. A major difficulty with these analyses is that the estimated statistical relationships are reduced form. The same reduced form relationship can be consistent with a large number of alternative structural models, and it is not always clear how the evidence on the reduced form relates to different structural models. ’ A virtue of the vector autoregression methodology is that it explicitly recognizes that macroeconomic variables are all endogenous stochastic variables. An implication of the rational expectations approach to macroeconomics is that it is very difficult to justify classifying variables as exogenous and endogenous and then use exclusion restrictions to identify a structural model from a reduced form.’ Rational expectations theorizing has also encouraged us to think of observable fluctuations in variables as being the result of movements in a small set of underlying unobservable stochastic shocks. While relating our paper to a vector autoregression is a reasonable analogy, we do not in fact estimate a vector autoregression. Rather, we take from that literature the idea that the pattern of covariation between current and lagged values of variables comprises the data to be explained by a business cycle theory. Whereas a vector autoregression in effect examines the partial covariation between one variable and the lagged values of each of the variables in the vector under consideration, we focus on the ordinary covariation between these variables. More explicitly, the structural model implies that each of the observable variables can be expressed as a moving average of the same underlying unobservable exogenous shocks. Expressions for the variances and contemporaneous and lagged covariances between the variables can be derived from these moving average representations. Parameter values can then be chosen to maximize the fit between the theoretical and sample second moments. Using the theory on generalized method ‘Blanchard (1986) and Bemanke parameters in a vector autoregression ‘Other macroeconomic theories exogenous deterministic processes.

200

(1986) use alternative approaches to structural parameters. often represent policy variables,

to relate for

example,

the as

A Generalized

Method

of Moments Approach

of moments estimators, we can derive formal statistical tests of the adequacy of the theoretical model as an explanation of the observed pattern of covariation between the variables. Formal statistical tests are not the only way to judge the ability of a theory to explain the empirical evidence. An advantage of estimating a structural model is that the estimates have a clear economic interpretation and can be compared with evidence from alternative sources (such as cross-sectional microeconomic data sets). We outline our economic model in Section 1 of the paper. Section 2 focuses on the econometric methodology, comparing our approach to estimating a vector autoregression. The results are presented and discussed in detail in Section 3 of the paper. Our discussion of the results includes an informal comparison of our estimates with alternative estimates of comparable parameters obtained using different data or methods. We also discuss the implications of our results for theories of the origin and propagation of business cycle disturbances.

1. The Economic

Model

The structural model is related to a small macroeconomic model discussed in Hartley and Walsh (1991). This section of the paper summarizes the main features of the model. We refer the reader to the original paper for a more detailed discussion. The point of departure for the model is the explicit incorporation of a financial intermediary sector between households as lenders and firms as borrowers.3 The liabilities of financial intermediaries (“deposits”) are used by households as a medium of exchange. Cash can also be used to fknd transactions. The choice between cash or deposits depends on the nominal interest rate paid on deposits. Intermediaries are required to hold reserves of high powered money. We assume there are no excess reserves and that required reserves are a fixed proportion of deposits. The remaining assets of intermediaries consist of loans to firms. Since reserves are a constant fraction of deposits, proportional variation in the supply of loans will equal the proportional variation in deposits. The intermediary industry is assumed to be competitive in the sense that each “bank” chooses the level of deposits and loans to 3Hartley (1988, 1991) provide microeconomic foundations for the model discussed in this paper and in Hartley and Walsh (1991). A cash-in-advance approach is used to model the demand for the different types of monetary assets.

201

Peter R. Hartley

and Carl E. Walsh

maximize profits, taking the interest rates on deposits and loans as given. We assume there is a U-shaped cost of providing intermediary services. As a result of profit maximization, proportional deviations of the supply of loans or deposits about trend will be a function of the deviations about trend of the interest differential adjusted for reserve requirements 0: = Li = ~[&(l - p) - t-J + 24,.

(1)

Here p is the proportion of deposits required to be held as reserves yielding zero interest, r is the semi-elasticity of deposit (and loan) supply with respect to the adjusted interest differential, it is deviations about trend of the loan interest rate, r, is deviations about trend of the deposit interest rate, and u is a supply shock which could result from stochastic technological innovation or regulatory changes in the intermediary industry. Proportional deviations about trend in the demand for deposits are assumed to depend positively on deviations about trend of the nominal deposit interest rate, r,, and deviations about trend of the logarithm of real income, yt: 0:’ = yt + yr, + 2, .

(2)

In this equation, zt is a random shock to the demand for liquid assets. Since we assume there is no direct saving in equities, z may alternatively be interpreted as a shock to the desired level of savings or consumption. As an initial assumption, which we examine again later in the paper, we have taken the income elasticity of demand for deposits at unity. Equilibrium in the market for deposits will require r[&(l - p) - rt] + u, = yt + yr, + zt .

(3)

Deviations about trend in the demand for capital, and therefore the demand for loans, are assumed to depend negatively on deviations about trend of the real rate of interest charged on loans, 4 + pt - Etpt+l, where p, is deviations about trend of the logarithm of the price of output. Deviations about trend in the demand for loans are also assumed to depend positively on deviations about trend of the anticipated output next period as a proxy for the expected marginal product of capital. Thus, we write 202

A Generalized

L;’ = PIE,Y,+I-

Method

I%[& + P, -

of Moments

Approach

GP,+J+ et.

(4)

The random shock to the demand for loans, E,, corresponds to a random shock to the desired aggregate capital stock. Equilibrium in the market for loans will require that

PIE,Y,+,

-

I%.[4

+ P, -

E,P,+J+ et = yt + YC+ 2,.

(5)

The demand for high-powered money consists of the demand for currency by consumers and the demand for reserves by banks. Proportional deviations in the latter will in turn depend on proportional deviations in the demand for deposits. Thus, proportional deviations in the demand for high-powered money will be a weighted sum of the proportional deviations in demands for currency and deposits. The weight on currency demand (reserves) will be the ratio of currency holdings (reserves) to total high-powered money. The income elasticity of demand for currency as well as deposits is set to unity. Therefore, the income elasticity of demand for high-powered money also will be unity. We let 4 be the net interest semielasticity of the demand for high-powered money with respect to the interest rate paid on deposits. A z shock is assumed to affect the demand for high-powered money with coefficient 6. In summary, proportional deviations in demand for high-powered money can be written H: = p, + yt - +rt + 62, . We assume deviations supply of high-powered

about trend in the rate of growth money can be written

Hi - HE-, = e, + m, - mtel + pi, .

(‘3) of the

(7)

shocks e and m are assumed to be white noise so that m is to be interpreted as a temporary shock, and e a permanent shock, to the level of base money.4 In addition, (7) incorporates the assumption that the Federal Reserve alters the growth rate of high-powered money in response to variations in the loan interest rate. In-

The

4Thus we can think of an m shock as a temporary deviation of the monetary base from money growth targets. An e shock is a change in the target. To distinguish these different shocks requires future information which cannot be known in period 1.

203

Peter R. Hartley

and Carl E. Walsh

creases in the loan interest rate are assumed to result in a contemporaneous increase in the rate of growth of base money.5 Equilibrium in the market for high-powered money will therefore require that

Pt - Pt-l + yt - s-1 - +(r, - rt-1) = e, + m, - m,-l + pi, - 6(z, - z+~) .

(8)

The three asset market-clearing conditions (3), (5) and (8) could be solved for the three endogenous variables it, r, and p, if output yt were exogenous. We assume, however, that output is endogenous and responds to lagged changes in the capital stock and the current real interest rate on deposits according to yt =

aoyt-1

+ dL1Yt

- 44-l + n-1 - Ld

+ dr, + P, - GP,+J+ w-1 + Et.

(9)

The term ytel is included on the right-hand side of (9) to reflect a persistence of shocks to output. The next two terms and a4Et-1 reflect the current effect of lagged changes in the capital stock.6 The fourth term allows for intertemporal substitution to affect the current labor supply. The white noise shock, 5, represents a transitory shock to output and could reflect, for example, random technology shocks or the effects of weather. Thus (9) is a standard aggregate ‘A potentially serious simplifying assumption incorporated into (7) is that monetary policy over the entire sample period was conducted as a single regime, whereas several regimes have in fact been in effect (including a fixed versus flexible exchange rate regime and the targeting of interest rates versus monetary aggregates). As we note when discussing the results of the estimation, this simplifying assumption could account for the relatively poor performance of the model over part of the sample period. However, the small number of available observations makes it impractical to estimate the model over a number of sub-periods. A referee has also suggested that U.S. monetary policy over the sample period might more reasonably be modeled by

Hi - Hi-, = e, + A(L)(m, - WI.+~) + p&. While we agree with this suggestion, as we note below, increasing the number of lags in the structural equations is likely to greatly exacerbate the computational burden. %nce these two terms reflect the effect of lagged changes in the capital stock, an implicit restriction on the model is that the ratio of (I, to PI should equal the ratio of o2 to p2.

204

A Generalized

Method

of Moments

Approach

production function linked with hypotheses about the source of changes in factor supplies.7 We assume that, when forming their expectations of future prices and output, individuals and firms know all lagged variables and shocks and the current values of qt, T),, i, and r,. In particular, the number of currently observable endogenous variables is less than the number of underlying shocks. Agents therefore have to solve an information processing problem as they choose their supplies or demands in the current period. This assumption incorporates into our model the hypothesis that an initial confusion of one shock for another, and the subsequent revelation of the nature of past shocks, is an important ingredient in explaining the observed contemporaneous and lagged correlations between the endogenous variables. In our specification, agents know the current values of &, (z, - et), (z, - u,) and (SZ, + e, + m,) but not the separate values of z,, E,, u,, e, or ~TL,.~ The inclusion of a signal processing problem into the model has important econometric implications. The observing econometrician is also faced with the problem that the dimensionality of the space of shocks exceeds the dimensionality of the space of endogenous variables. This in turn complicates the use of maximum likelihood to estimate the underlying structural parameters and is one reason we have instead relied upon a generalized method of moments approach.g ‘Note that we could rearrange (9) to obtain a “Phillips curve” relating inflation to deviations in output about trend, changes in expected inflation and changes in interest rates. Equation (9) thus embodies the Lucas interpretation of the Phillips curve, but is modified to allow for changes in the capital stock. ‘An advantage of our specification is that we do not have to solve for the equilibrium moving average coefficients simultaneously with the projection coefficients. The principal source of this simplification is the assumption that agents observe the current value of output. One, possibly unsatisfactory, consequence of the assumption is that confusion between “monetary” and “real” shocks is de-emphasized in the current model. Instead, our model emphasizes confusion between temporary and permanent shocks. It could be argued that a confusion between temporary and permanent shocks is a more plausible source of real effects of monetary shocks than a confusion between nominal and real shocks. Whether a current shock is temporary or permanent is something which cannot be easily resolved on the basis of current information. ‘More generally, the problems for a maximum likelihood approach identified in the next section of the paper will apply to almost any attempt to use vector autoregression techniques to evaluate a model incorporating information processing by agents in the economy. An exception would arise if we are willing to assume the econometrician has data superior to that available to agents in the economy.

205

Peter R. Hartley

and Carl E. Walsh

As we discussed at greater length in our earlier paper, the four equations (3), (5), (8), and (9) can be solved for the four endogenous variables yt, p,, r, and i, as moving averages of the white noise shocks 5, E, z, u, m and e. In the present case, it is easier to solve for yt, r,, it and the rate of inflation Pt = p, - P,-~ (see the discussion in the appendix).

2. Estimation

Methodology

Let the vector of endogenous

variables

be gt where

x; = [Yt pt rt 41* Let the vector

of shocks be 2, where

2: = [&et ztut m 61. Then we show in the appendix ten

01)

that the solution for gt can be writ-

(152 for matrices of coefficients Il,, and II,, and Al and As (possibly complex) roots of a cubic equation less than 1.0 in modulus.” As demonstrated in the appendix, the coefficient matrices II, and Iii and the roots A1 and A2 will be non-linear functions of the elasticity and variance parameters. For empirical implementation of the model using U.S. quarterly data we need to take trends and seasonality into account. A preliminary time series analysis suggested that seasonal diiTerencing of the time series might be needed to induce stationarity. We

“‘To obtain a unique solution for tbe endogenous variables exactly two of the roots of the cubic have to have modulus less than 1.0. Computations were greatly simplified by the availability of analytical solutions for the roots. A more complicated lag structure in the original model would have produced a higher degree polynomial in the lag operator which would have to be factored in each evaluation of the objective iunction. The roots of such a polynomial could be obtained numerically but this would have considerably increased the computational burden.

206

A Generalized

Method

of Moments

Approach

therefore postulated that the shocks in s, each possessed a unit root at the seasonal frequency (4 quarters).” Specifically, we assumed the driving shocks took the form 2, = 5t-4 + gt with w, a vector white noise process, assumed also to have a diagonal contemporaoperaneous covariance matrix. l2 Applying the seasonal dfierence tor, A = (1 - L4), to (12) we obtain a model for AT* = x, - _x~-~ in terms of 0, which has coefficients identical to (12): Ax, = &o,

+ HIlstwl + c

[{n,(At; - A;)

ik2

(13) To account for trends in the data, we also related our theoretical model to the mean-corrected annual changes in the logarithm of output, the inflation rate and the two interest rates. Our analysis therefore allows for trends which take the form of an annual random walk with drift. At this juncture, we might think of estimating the model by postulating a joint probability distribution for the shocks in CG~and estimating the unknown structural parameters by maximum likelihood. One difhculty with this approach is that the postulated number of shocks in 0, exceeds the number of variables in Ag. Therefore, the Jacobian of the transformation from AZ, to the vector of shocks is singular. In economic terms, we do not have sufficient “Another modification of the model would allow a subset of the exogenous shocks to have a seasonal unit root. Then 0, = lt - + would still be stationary but it would be autocorrelated at the seasonal frequency. “We have implicitly assumed that each of the underlying exogenous shocks is an independent source of randomness for the economy, uncorrelated over time. While the primary function of this assumption is to reduce the number of parameters to be estimated, and in effect identify the remaining parameters in the model, we believe it is not an unreasonable starting assumption. Ideally, we would like to explain the contemporaneous and lagged correlations using the structural parameters alone. Building in correlation or autocorrelation via the assumed nature of the unobservable exogenous shocks weakens the sense in which the structure of the model has accounted for the evidence. The assumptions that the underlying exogenous shocks are contemporaneously uncorrelated and white noise could be relaxed by increasing the number of sample covariances included among the moments to be explained and expanding the number of estimated parameters. This would have considerably increased the computational burden. Furthermore, each additional lag included in the empirical covariances to be fitted reduces the number of available observations by 4.

207

Peter R. Hartley

and Carl E. Walsh

endogenous variables to identify values for each of the shocks in each period.r3 However, since the representation (13) is covariance stationary, we know from the Wold decomposition theorem that there is an alternative representation in terms of a vector of fundamental shocks, xt, which has the same dimensionality as Azt. The variance covariance matrix of the shocks, %, will be related to the matrices II, and II,, the roots Al and h, and the variances of the underlying structural shocks by a set of simultaneous matrix equations.14 In principle, we could estimate the parameters of the model by estimating the resulting vector ARIMA process. For example, we could postulate that the vector v is jointly normally distributed and estimate the parameters by maximum likelihood. Instead of carrying out the maximum likelihood procedure, we estimated the parameters using generalized methods of moments (GMM). The GMM procedure is less difficult to compute and more intuitive. The above procedure would also be more vulnerable to problems of heteroscedasticity than the GMM procedure, and would be open to misspecification bias if the distributional assumptions for x are inaccurate. In the GMM procedure, the variance-covariance matrices E[Ag,A$] and E[Ax,A$,] implied by the theoretical model are compared with the corresponding moments calculated from the data. We could attempt to fit a large number of covariance matrices of Ax, with lagged values of AZ, using the same parameter values. In r3As problem “We

noted above, this follows for agents as an important can write (13) as (1 - h&)(1

Thus if we we have [l + (A,+

let 2,

- A&)~, and

A,)“]II&JI;

from our ingredient

= IL&

Z. denote

desire to include in the structural

+ U-f, - (AI + the

covariance

- (A, + A,)[KIJJI;

+&nun;]

a signal-processing model.

AdJ.ol_wt-, - Et + c,v,-, . matrices

of g and

+ n&en:

x respectively,

= 2, + c&c;

and

to be solved for P, and the moving average coefficient matrix Cr. These equations yield a quadratic matrix equation to be solved for Cr. This equation has many solutions, and would have to be solved for a matrix Cr with eigenvalues less than 1 in modulus each time the likelihood function is evaluated.

208

A Generalized

Method

of Moments

Approach

order to both maximize the sample size and reduce the computational burden, however, we chose to concentrate on explaining the contemporaneous and once-lagged matricesI Thus, we want to choose the structural parameters to match the 10 distinct contemporaneous variances and covariances and 16 once-lagged auto and cross covariances to the corresponding sample moments for the meancorrected annual growth in output, and the mean-corrected annual differences in the rate of inflation and nominal deposit and loan interest rates. After imposing the identifying restrictions that the theoretical covariances between the underlying shocks are zero, we can write the vector of parameters to be estimated as

Denote the 26 x 1 vector of contemporaneous and once-lagged theoretical variances and covariances between the elements of AZ by B(b). The elements of 9 are the theoretical second moments stacked -in the following order:

Ayt Apt Art

by,

AP,

Ar,

Ai,

1

2

3

4

5

6 8

Ai,

Apt-1

Art- 1

Ai,-,

11

12

13

14

7

15

16

17

18

9

19

20

21

22

10

23

24

25

26

b-1

From the data, we have N observations on 26 cross products using the mean-corrected annual changes in the logarithm of output, the mean-corrected annual changes in the inflation rate (or equivalently the quarterly changes in the annual inflation rate), and the mean-corrected annual changes in the two interest rates:

“Limiting the sample not restrict higher order enough sample moments sample moments might mates, a reduced sample

moments to variances and first-order autocovariances does autocovariances to be zero. All that is necessary is that are chosen to identify the parameters. While additional have allowed us to obtain more efficient parameter estisize would have offset these potential gains in efficiency.

209

Peter A. Hartley

and Carl E. Walsh

(Y” - Y,-J(Y” - Y”4) ’ * * (Y” - Yn-&n-l P” - p,-3Pn

- pn-4) * . . (P, - pn-4&,-l

(m - rn-3(1;, - rn-4) . . . Cm - f-AL1 (i” - in+)&

- in+) . . . (in - in-&“-l

- &l-5) , - in-s) , - LJ

,

- in-s) .

Following the notation of Hansen (1982), we can write f (AZ,,, b) for the 26 X 1 vector of differences between the sample cross products in period n and the corresponding theoretical second moments given in 8(b). We have 26 variances and covariances to be used to yield estimates of 19 parameters. Under the null hypothesis, E[ f(Azn, b)] = 0. We form

g,(b)

= $$f

(Am,@,

(14)

n-1

which in our case is equivalent to the vector of differences between the empirical second moments and the *corresponding theoretical second moments. First round estimates b of the parameters b are obtained by minimizing the sum of squared errors gN(b)‘gN(@. l6 Following the arguments in Hansen (1982), Cumby, Huizinga, and Obstfeld (11983) and White and Domowitz (1984) we can conclude to a random vector that fi(b - I$ will converge in distribution with mean zero and covariance matrix

where

“We used GQOPT to carry out the minimization. to obtain good initial values. Final estimates were convergence criterion of 1 x IO-’ for the relative jective function. Helpful advice from Dick Quandt fully acknowledged.

210

The DFP algorithm was used obtained using GRADX with a change in the value of the obon the use of GQOPT is grate-

A Generalized and the matrix

x

S is defined

Method

of Moments

Approach

by

The parameter estimates 6 can be used to calculate 20 matrix of first partial derivatives

the 26

(17) to give us an estimate of 4. To estimate S, Hansen (1982) suggests that, in cases where S depends on only a finite number of autocovariances, that is, E[f(AgO, b)f(A~-~, S can be estimated

b)‘] = 0

for

consistently

Ijl > k + 1

for some integer k ,

by

However, in our case the number of non-zero autocorrelations in e f ml ‘t e, so the estimator suggested by Hansen f(Ag”, b) may not b will not be appropriate. To cope with this problem, we used the method suggested by Newey and West (1987), who propose we estimate S by

s,=~~+~w(j,J)rrz,+g,, (19) j=l

where

4, 1) = 1 - [j/Cl + VI is a linearly declining weighting function with J chosen to be a fimction of the sample size so that J(N) + m with N more slowly than N’j4 and

211

Peter R. Hartley

and Carl E. Walsh

Andrews (1987) has addressed the problem of choosing the appropriate value of J’. AHe shows that, if each of the 26 components of the errors f(As,, b), n = 1 . . . N, follows an AR(l) process with common autoregressive coefficient p, ] = (1.447~~N)“~ with c4 = 4p2/(1 - p”)” will satisfy a minimax criterion for the optimal lag length in the Newey-West covariance matrix estimator. In our case N = 92 so that ] works out to about 6.2 for p = 0.5 and 13.8 for p = 0.8. On the other hand, a straightforward modification of Andrews’ calculations -shows that if each of the 26 components of the errors f(Agn, b), n = 1 . . . N, follows an AR(4) process, with a common non-zero autoregressive coefficient p only at lag 4, then CY= 64p2/(1 - p2)2 . As we would expect, the lag length needs to be longer when the errors are expected to be autocorrelated further into the past. Thus, if p = 0.5 in the AR(4) case, J works out to about 15.6. To cover a range of possible processes” for the error terms we examined the results for three different lag lengths J = 8, 12 and 16. Hansen (1982) shows that the optimal GMIM estimator (in the sense that the asymptotic covariance matrix of b is as small as possible) is obtained by minimizing a weighted sum of squares, or a symmetric weighting matrix, W, which is a conf%wW&(b)~ f sistent estimator of S-l. If we let b be the parameter vector which minimizes this weighted sum of squares then vN(b - &) will converge in distribution to a random vector with mean zero and cowhich can be estimated by variance matrix (d&&J’, “These calculations are meant to provide only 1. There is no guarantee that the error terms are a covariance stationary time series process. GMM is the null hypothesis implies that the error terms

a rough guide to the choice of homoscedastic, let alone follow often applied in a context where are expectational errors. In our

case, the error terms reflect sampling and possibly measurement null hypothesis. In practice, seasonal ditferencing is a crude way ality in the original time series, so the error terms may contain moving average components at the seasonal frequency.

212

error under the to handle seasonautoregressive or

A Generalized

Method

(ci$.& .

of Moments

Approach (21)

Then following the suggestion in Hansen (1982), Section 4, we test the over-identifying restrictions by evaluating

which will converge in distribution to a chi-square random variable with r-q degrees of freedom where r = the number of moment conditions (26 in our case) and g the number of estimated parameters (19 or 18 in our case). The data we used to estimate the model, which are available on a quarterly basis from 196O:iu-1985:i, are the logarithm of U.S. real GNP for yt, the logarithm of the GNP deflator for p, and two interest rate series from the MPS data file. We used a weighted average of the interest rates paid by banks on various categories of deposits for r,, while for i, we used the average bank rate on short term commercial and industrial loans.” Since we lose 5 observations when we form annual differences in inflation rates we are left with a sample size of N = 92. Figure 1 graphs the mean-corrected annual changes in yt, P, = pt - ptml, r, and i,. These are the variables which were used to form the sample variances and covariantes.

3. Results When we first attempted to estimate the model, estimates of a: tended to 0 with a large standard error. In subsequent analysis, we dropped 5 from the model. This was the only modification made to the model as originally specified before the estimates reported below were derived. We were left with 18 parameters to estimate and a final chi-square statistic for testing the over-identifying restrictions which had 8 degrees of freedom. The sum of squares of the empirical variances and covariances was approximately 310.76. The parameter estimates which minimized the sum of squared differences between the empirical and theoretical second moments produced a sum of squared differences ‘*We are indebted to Steven Goldfeld for providing the interest rate data to us. The data are very similar to two series published in the Federal Reserve Bulletin which rely on a sample survey of banks. Unfortunately, these series were discontinued recently so our data set ends in the first quarter of 1985.

213

Peter R. Hartley

and Carl E. Walsh

Mean

Figure 1. Corrected Annual

Changes

approximately equal to 7.39. The parameter values which minimized the sum of squared differences were very close to the estimates which minimized the weighted sum of squared differences. This was true for each of the lag lengths we used to calculate the weighting matrix. While the weighting matrix was positive definite when we took J = 8, I2 or 16, in the case ] = 8 we obtained an estimated variance-covariance matrix for the parameters which was numerically singular. Accordingly, we discuss below only the results for J = 12 and J = 16. The statistic for testing the over-identifying restrictions was 7.7176 when J = 12 and 6.1094 when J = 16. In both cases, the statistic is asymptotically chi-square with 8 degrees of freedom. The corresponding P-values are approximately 0.4615 when J = 12 and 0.635 when J = 16. These values are quite consistent with the null hypothesis that the model can explain the observed sample variances and covariances. The final fit between the sample and estimated theoretical second moments is illustrated in Figure 2. The order of the moments in this figure is the same as the order of the elements of fi. The major deviation between the estimated theoretical and sample moments is the covariance between current output growth Ay, and the change in interest rates on bank loans A&-,. The sample 214

A Generalized

Figure

Method

of Moments Approach

2

value is negative while the theoretical value is positive. This sample moment is diEcult to explain in the context of the positive sample correlation between Ay, and Ar,-, and the high sample correlation between Ar and Ai. Parameter values which reproduce these latter correlations also tend to imply a positive correlation between Ayt and Ai,-,. Allowing for risk premia on bank loans, or fluctuations in excess reserves, might enable greater differences in the behavior of loan and deposit rates to be explained. Another potential diEculty is that deposit interest rates are an imperfect measure of the return on deposits. Variations in fees charged customers on checking accounts, or the value to depositors of the accounting services provided by bank accounts, could have affected the extent to which recorded deposit rates approximated the theoretical variable of interest. Table 1 presents the parameter estimates, and the estimated standard errors, for the model without 5 and for J = 12 and 16. Several of the estimated standard errors are quite high relative to the corresponding estimated parameter value. A possible explanation is that the estimator Szj in (19) involves the fourth moments of the change in log income, interest rates and the rate of inflation. 215

Peter R. Hartley

and Carl E. Walsh

If empirical variances and covariances are not very stable over time, the estimated sZj will be large, making the standard errors for the estimated parameters relatively large. Of course, if the sample second moments are very unstable over time we will not be able to reject many hypotheses about the data (the tests will have low power). Indeed, a premise underlying the analysis is that a reasonably stable pattern of covariation between the variables under consideration constitutes the evidence we are attempting to explain. Some of the parameter estimates are difficult to reconcile with the microeconomic evidence relating to the same parameters. The significantly negative coefficient on E,-,y, is inconsistent with the idea that this term represents the effect of expected period t output on the demand for capital in period t. In fact, the sizes of most of the coefficients in the output equation seem much too large. The estimated patterns of response of output to anticipated and unanticipated movements in real interest rates, however, are quite reasonable. The semi-elasticity of output growth with respect to an unanticipated shock to the Zagged change in real loan interest rates is given by - 10.8966, -1.1945, -0.1309, -0.0144, -0.0016 and then virtually zero over subsequent periods. Similarly, the semielasticity of output growth with respect to an unanticipated shock to the current change in real deposit interest rates is 16.4488, 1.8031, 0.1976, 0.0217, 0.0024 and then virtually zero over subsequent periods. On the other hand, the effects of anticipated shocks to real interest rates are much smaller. The corresponding semi-elasticities are -0.6644, -0.0728, -0.0080, -0.0009 for lagged real loan interest rates and 1.0029, 0.1099, 0.0121, 0.0013 for current real deposit interest rates. A significant influence on output of both unanticipated and anticipated real interest rate shocks is, of course, consistent with an IS-LM model of the macroeconomy, although in that model interest rates affect aggregate demand as opposed to aggregate supply. Our results are also consistent with the vector autoregression results of Sims (1980) and Litterman and Weiss (1985), suggesting a strong relationship between movements in interest rates and subsequent movements in output. Finally, it is worth noting that the parameter (r3 measures the intertemporal substitution effect. The estimated parameter is of the right sign and is estimated with a reasonably low standard error. Turning next to the loan market equilibrium equation, both pi and f& are estimated with the hypothesized positive signs. The coefficient on expected future output in the loan demand equation is both larger and estimated with a lower standard error than the 216

A Generalized

Method

of Moments

Approach

coefficient on the real loan interest rate. Contrary to our prior expectations, the deposit interest rate is estimated to affect the supply of loans negatively. The coefficient y also enters the deposit market equilibrium condition, and its negative value again suggests the deposit interest rate affects the demand for deposits negatively. As we noted above, measurement error relating to the deposit interest rate might be one factor which has affected the ability of the model to account for the evidence on the relationship between interest rates and the remaining variables. It might be thought that the large estimated value for T suggests that the marginal cost curve for individual banks is reasonably elastic and/or entry to the intermediation industry is rapid. However, 7 is a semi-elasticity of supply with respect to the interest differential adjusted for p. At the sample means for r, and i,, and using the estimated value of p, the estimated value of 7 corresponds to an industry elasticity of supply of approximately 1.566 with respect to the adjusted interest rate differential. Unfortunately, p is estimated with a negative, as opposed to the hypothesized positive sign and is significantly different from zero. The final equation in the model represents equilibrium in the market for high powered money. The positive estimated coefficient on the deposit interest rate implies that increases in deposit interest rates have the hypothesized negative effect on the demand for base money. However, the coefficient is not statistically significantly different from zero. With regard to the supply of base money, the Federal Reserve reaction to changes in loan interest rates is estimated to be positive, although again the coefficient is not statistically significantly different from zero. We should note that p, plays a role in altering the effect of real shocks. As has been emphasized by advocates of real business cycle models (for example, Long and Plosser 1983, and Prescott 1986), a reaction function of the form incorporated in this model can explain pro-cyclical movements in the expected rate of inflation and nominal interest rates when the sources of disturbance are real rather than nominal. We also emphasize, however, that reactions of the Federal Reserve of the type captured by p will in general enhance the effects of real shocks on real variables. In our model, anticipated money shocks are not neutral unless they are a once-and-for-all change in the level of base money. The non-neutrality of anticipated money shocks is discussed further in Hartley and Walsh (1991). The remaining parameter estimates in Table 1 are the estimated standard deviations of the fundamental shocks driving the 217

Peter R. Hartley

and Carl E. Walsh

TABLE 1. Parameter

Estimate 1.7979

-15.4016 10.8966 16.4488 -51.4132 1.2038 0.3783 -0.4972 63.8748 -0.0465 2.2912 2.2871 10.0207 0.4287 0.1043 7.9318 0.2757 1.1536

Std. Error (J = 12)

6.6754 1.5037 2.5191 5.9793

50.2254 0.2095 0.3544 0.2556 67.0966 0.006 2.5163 2.3426 19.6605

0.5718 0.1741 9.4617 4.9502 0.2927

Ratio 0.269

10.242 4.326 2.751 1.024

5.746 1.067 1.945 0.952

7.750 0.911 0.976 0.510

0.750 0.599 0.838 0.056 3.941

Std. Error (J = 16) 5.6139 1.3864

2.1574 5.5083 47.2729 0.1854 0.3259 0.2224 60.0852 0.0055 2.3316 2.2058 20.0167 0.5279 0.1838 8.8302 4.0751 0.2889

Ratio

0.320 11.157 5.051 2.986 1.088 6.493 1.161

2.236 1.663

8.455 0.983 1.037 0.501 0.812 0.567 0.898 0.068 3.993

system of equations. Only permanent shocks to money growth rates (e) are estimated to be statistically significantly different from zero. To gauge the economic significance of the various shocks (at the point estimates) we can look at the components of the variance of the endogenous variables which are due to each type of shock. These variance components depend on the estimated coefficients in a moving average representation of the endogenous variables in terms of the shocks, and the estimated variances of each of the shocks (recall that we have assumed the covariances between the underlying shocks are zero). The estimated moving average coefficients are discussed further later in this section. Figure 3 graphs the contribution of each shock to the variance of each of the endogenous variables. Consistent with a “monetarist” model of business cycles, money growth shocks are estimated to provide the largest contribution to variance in output, while shocks to the demand for capital and money demand are the greatest sources of variation in interest rates and of secondary importance for output variation. However, it should be remembered that, as a result of the non-zero Federal Reserve reaction coefficient, p, each of these 218

A Generalized

Method

of Moments

Approach

shocks will also be accompanied by (further) contemporaneous endogenous movements in the growth rate of base money which will be nonneutral. While banking sector shocks (u) have a large estimated standard deviation, the coefficients of these shocks in the moving average representations of the endogenous variables are very small so the net contributions of u shocks to variations in the endogenous variables also are very small. Figure 4 illustrates the contribution of each shock to the contemporaneous covariance between each of the endogenous variables. In contradiction with the predictions of the IS-LM model, money growth shocks, e, contribute to a positive covariance between output growth and interest rate movements, whereas shocks to the demand for capital, E, contribute to a negative covariance between the same endogenous variables. Money demand shocks, z, are the main source of a positive covariance between inflation and interest rates. Each of the shocks E, z and e contribute to the positive covariance between loan and deposit interest rates. The decomposition of the covariances between Ay, and lagged values of the endogenous variables in Figure 5 again illustrates the importance of money growth shocks in the estimated model. Shocks

Figure

3.

219

n e

cov(Ay.AP)

wv(AU.Ar)

q z

n u

wu(Ag.Ai)

Figure

q m

CCv(AP.Ar)

cdAP,Ai)

cov(Ar.Ai)

4.

CCV(Au.LAP)

cov(Ay.LAd

Figure

Be

5.

cov(Au.LAi)

A Generalized

Method

of Moments

Approach

to the demand for capital, E, and money demand shocks, z, also contribute a negative component to the covariance between output growth and lagged interest rates. Figure 6 suggests that money demand shocks are the most important influence on the covariation between changes in inflation and lagged changes in the endogenous variables. In each case, money demand shocks contribute positively to the covariance. Shocks to the demand for capital, E, also make a positive contribution to the covariance between changes in inflation and lagged growth in output and a negative contribution to the covariance between changes in inflation and lagged changes in interest rates. Figure 7 again shows that all three of the shocks to the demand for capital, money demand and money growth make important contributions to explaining the covariance between interest rates and the remaining variables. Money growth shocks are again particularly important for explaining the covariance between changes in interest rates and lagged growth in output. In an attempt to improve the fit of the model we tried two modifications of the structural equations. First, we added expected inflation to the asset demands in the second, third and fourth equations. The estimated coefficients on the expected inflation rates were

.o.4-l

cov(AP,LAy)

. .

... .................................. ....................... .............. ..............

cov(AP.LAP)

Figure

cov(AP.LAr)

cov(AP.LAi)

6.

221

Peter R. Hartley

and Carl E. Walsh

Figure

7.

not significantly different from zero and the additional terms did not appreciably alter the fit of the model. Our second modification produced a greater improvement in the fit of the model. We allowed the income elasticities of demand for monetary assets to take on a value other than unity. While the least squares fit of the resulting model was considerably better than the model with unitary income elasticities, the estimated variancecovariance matrix of the parameters in the weighted estimation was numerically singular. Evidently, adding the two additional parameters to be estimated resulted in under-identification of the model. In addition, freeing up the income elasticities of demand for monetary assets did not appreciably alter the troublesome parameter estimates in Table 1. In an attempt to further gauge the goodness of fit of the model, we examined the vector ARMA representation implied by the final parameter estimates. For the parameter estimates presented in Table 1, the corresponding values of the matrices II, and III1 are given in Table 2. These coefficients are the first two terms in the moving average representation of the endogenous variables in terms of the shocks 0. It is interesting to note that the effects on the endogenous variables of many of the shocks peak in the second period. In particular, money growth shocks have their maximum effect on output growth and interest rates in the period after the period of the

A Generalized TABLE

Method

of Moments Approach

2.

AY -0.0521 9.2106 0.0156 0.9154 0.9154 -1.0952 0.2170 0.0094 -0.1793 1.3505

Eo ZO UO % e0 E-1 z-1 u-1 m-1 eel

AP 0.1124 0.8476 -0.0486 0.0910 0.0910 -0.4066 3.6220 -0.0056 0.4353 -0.0027

Ar -0.5065 2.5476 -0.0049 0.2032 0.2032 2.0533 9.7404 0.0060 0.9686 0.6479

Ai - -0.4810 2.5683 - .0.0194 0.2064 0.2064 1.9305 9.2387 0.0059 0.9157 0.6345

shock. Money demand shocks also have their peak effect on inflation and interest rates in the period after the shock first occurs. Exogenous shocks to the demand for loans, E, change sign in their effect on inflation and interest rates between the first and second periods, but the largest absolute effect again occurs in the period after the shock first occurs. From the second period on, the effects of the shocks decline according to the roots of the difference equation system governing the solution for the moving average coefficients. For the estimated parameter values in Table 1, the corresponding roots are a complex conjugate pair with modulus 0.48398 and argument 0.22058 radians. These roots imply a second order AR representation for Ay with coefficient on Ayt-i of -0.9445 and on Ayt+ of 0.2342. These coefficients are similar to the coefficients one obtains from a univariate ARIMA analysis of the log of GNP. Since the roots are a complex conjugate pair, the effects of the shocks will die out gradually over time and in an oscillatory fashion. The period of the oscillations will be 21r/O.2206 = 28 quarters. This is quite a bit longer than the usual estimate of business cycle periodicity. While the residuals in the model are the differences between the sample and theoretical second moments graphed in Figure 2, we can transform the estimated model to obtain a time series of “estimated residuals.” These provide another useful diagnostic tool which can be used to judge the performance of the model.” As we noted above, the moving average representation (13) of the endogenous variables Ax, can be written “We

thank

an anonymous

referee

for

suggesting

this

to us.

223

Peter R. Hartley

and Carl E. Walsh

[l - (A, + A& + X,A,L2]&,

= s + C&-i

)

(23)

where v is white noise with the same dimensions as Ax,. Furthermore, if XV is the contemporaneous (4 X 4) covariance matrix of g and C, the contemporaneous (5 5) covariance matrix of w, then Ci satisfies the matrix equation

x

c,z,c;- z,c; + 2, = 0) with co = [l + (A, +

h2)211-I~orId

and 2, = l-Iⅈ

- (A, + A&-l&-&,

.

(24)

Estimates of the matrices ll, and 111,from Table 2, the estimates of the roots Ai and AS and the estimates of Z, from Table 1 can be used to evaluate I;,, and 2, in (24). The matrix quadratic in (24) can then be solved for Ci. There are many solutions of this equation for Ci but only one of them yields an invertible MA representation for (1 - A,L)(l - A&)Az~. If we set vl = ti = 0, Equation (23) can be iterated forward to supply an estimated set of residuals &, t = 3, . . . ) N. These are plotted in Figure 8. The true residuals are (not necessarily contemporaneously uncorrelated) white noise. The stochastic process followed by the estimated residuals (which will be autocorrelated by construction, but should be white noise asymptotically) obviously changes dramatically about 1980. If we inspect the numerical values, it is clear that the break actually occurs in the fourth quarter of 1979. This is when the Federal Reserve changed its operating procedure from targeting interest rates to targeting monetary aggregates. Our model assumed the Fed followed an interest rate smoothing rule. Hence, it is perhaps not surprising that the model fails to explain this period. It is a little surprising, however, that the break appears so clearly in the estimated residuals. A similar break in the behavior of the original mean corrected annual changes in the observable endogenous variables is much less evident in Figure 1. Furthermore, the filter ap-

A Generalized

Method

of Moments

Approach

40”“‘. . ” ” “. ““““‘I 62 63 64 66 66 67 66 66 70 71 72 73 74 76 76 n 76 76 60 61 62 63 64 66 66

Figure

8.

plied to the data to derive the residuals in Figure 8 was based on fitting only second moments over the entire sample period. The estimated residuals in Figure 8 and the high standard errors in Table 1 both suggest that the chi-square statistic for the overall goodness of fit of the model might not be a very powerful test of its adequacy. In effect, sample variances and covariances are estimated with so much noise that many hypotheses about their determinants are difhcult to reject on the basis of the evidence. On the other hand, the fact that the model produces such a clear break in estimated residuals at the time the Federal Reserve changes its operating procedure suggests that the model has successfully captured some of the structure of the economy during the period when the Federal Reserve acted to smooth interest rate fluctuations.

4. Concluding

Remarks

This paper has presented estimates of a small structural model of the U.S. economy where all the observable variables are modeled as endogenous functions of a set of unobservable shocks. Since 225

Peter R. Hartley

and Carl E. Walsh

market participants do not observe the shocks, they process currently observable endogenous variables to recover estimates of the underlying shocks. If agents later learn about the nature of the shocks affecting the economy, the subsequent effects of the shocks can differ considerably from their impact effects. This is one reason incomplete information has been emphasized as an important ingredient in explaining the origin of business cycles. Whereas many theoretical models have focused on a confusion between real and monetary shocks, we have instead emphasized confusion between temporary and permanent shocks. Including an information processing problem into the model has implications for the econometric analysis. Since the number of shocks exceeds the number of observable endogenous variables, the mapping from the space spanned by the shocks to the space spanned by the observable endogenous variables is not invertible. This considerably complicates the computation of maximum likelihood estimates. We have instead used a generalized method of moments (GMM) procedure which is more directly related to the notion of a business cycle as a stable pattern of covariation between current and lagged values of key aggregate variables. In addition, relative to the GMM procedure we have used, maximum likelihood is more vulnerable to problems of heteroscedasticity, and would be open to misspecification bias if the distributional assumptions for the error terms are inaccurate. There are many reasonable features of the estimated model. The parameter estimates provide some support for many of the competing theories of the source and propagation of business cycles. Consistent with IS-LM and real business cycle models, we found an important role for shocks to the demand for capital and a transmission of monetary shocks to real variables via interest rates. Intertemporal substitution in labor supply also appeared to play a role in affecting output growth. Consistent with a monetarist viewpoint, shocks to base money growth rates were found to be the most important sources of output variation. An implicit assumption in our estimation is that the structure of the economy and the parameters we have estimated remained constant over the sample period. Insofar as some of these key parameters are influenced by policy and technological change this may not be a reasonable assumption. In particular, the short-term change in Federal Reserve operating procedures at the end of 1979 seems to have significantly altered the variance of interest rates and their relationships with other variables. The ARMA residuals implied by 226

A Generalized

Method

of Moments Approach

our parameter estimates suggest that our model, with its assumption of interest rate smoothing behavior by the Federal Reserve, did not explain this period. Receiued: September Final uersion: April

1990 1991

References Andrews, Donald K. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Cowles Foundation for Research in Economics. Working Paper, Yale University, October 1987. Bernanke, Ben S. “Alternative Explanations for the Money-Income Correlation.” Carnegie-Rochester Conference Series on Public Policy 25 (1986): 49-100. Blanchard, 0. J. “Empirical Structural Evidence on Wages, Prices and Employment in the U.S.” MIT Working Paper, September 1986. Cumby, R. E., J. Huizinga, and M. Obstfeld. “Two-Step Two-Stage Least Squares Estimation in Models with Rational Expectations.” Journal of Econometrics 21 (1983): 333-55. Hansen, L. P. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica 50, no. 4 (July 1982): 102954. Hartley, Peter R. “The Liquidity Services of Money.” International Economic Review 29, no. 1 (February 1988): l-24. -. “Interest Rates in a Credit Constrained Economy.” Rice University, School of Social Sciences Working Paper, 1991. Hartley, Peter R., and Carl E. Walsh. “Inside Money and Monetary Neutrality.” Journal of Macroeconomics 13, no. 3 (1991): 395416. Litterman, Robert B., and Laurence M. Weiss. “Money, Real Interest Rates, and Output: A Reinterpretation of Postwar U.S. Data.” Econometrica 53, no. 1 (January 1985): 129-56. Long, John B., Jr., and Charles I. Plosser. “Real Business Cycles.” Journal of Political Economy 91, no. 1 (February 1983): 39-69. Lucas, R. E., Jr. “Understanding Business Cycles. “Carnegie-Rochester Conference Series on Public Policy 5 (1977): 7-29. Nelson, C. R., and Charles I. Plosser. “Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications.” Jour& of Monetary Economics 10, no. 2 (September 1982): 139-62. 227

Peter R. Hartley

and Carl E. Walsh

Newey, Whitney K., and Kenneth D. West. “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica 55, no. 3 (May 1987): 703-8. Prescott, E. C. “Theory Ahead of Business Cycle Measurement.” Federal Reserve Bank of Minneapolis Staif Report 102, February 1986. Sims, C. A. “Money, Income, and Causality.” The American Economic Review 62, no. 4 (September 1972): 540-52. “Macroeconomics and Reality.” Econometrica 48, no. 1 -. (January 1980): l-48. White, H., and I. Domowitz. “Nonlinear Regression with Dependent Observations.” Econometrica 52, no. 1 (January 1984): 14361. Appendix In this appendix we outline how to solve the structural model for a moving average representation of each of the endogenous variables in terms of the underlying unobservable shocks. We also discuss the derivation of the theoretical variances and covariances from the moving average representations. For convenience, we repeat below the basic equations of the model: it = olo~t-1 + 4Ll~t

- %[Ll + P+~- Et-d

+

PlEt~t+l

St ;

(Al)

- Pz[it+ p, - Etpt+J+ et = vt + v-, + 2, ;

w

r[it(l

OL4%-1

+

- p) - rt] + nt = yr + yr, + z, ;

643)

p, - n-1 + yr - b-1 - +(rt - rt-J = et + m, - m,-, + I.4 - WG - G-1) ;

644)

Now define pt = Pt - Pt-1 228

645)

A Generalized

Method

of Moments

Approach

and observe that EJt+, = Eipt+, - pt. Solve (A3) for i, as a function of yt, r,, and the shocks zt and u,. Substitute the solution into (A2) to obtain r, as a function of yt, E,Y,+~, E,P,+, and the shocks E,, z, and u,. Finally, substitute the solutions for i, and r, as functions of it> Etyt+l> Wt+l and the shocks E,, zt and u, into (Al) and (A4) to obtain two equations involving ytel, yt, Etmly,, E,Y,+~, Pt, E,-J’,, ETt+,,, and the shocks E,-~, utwlr ztPl, St, et, at, u,, m, and e,. Using these two equations, we can solve for yr and P, (and hence also r, and i,) as moving averages of the shocks

si = [E,E,2,u, mt4 .

646)

Let the moving average coefficients for the shocks be n(k, i, m) where the index k = 1, 6 corresponds to the elements of 2 as in (A6), i = 0, 1, 2, . . . , m corresponds to the lag of the shock and m = 1, 2, 3, 4 corresponds to the variable in the order

x: = [yt pt 5 41 so that, for example,

(A7)

we can write

it=i:c.4k

W)

P, = i

64%

, i,

k=l

1) Sk,t-i

i=O

and

k=l

i

n(k, i, 2) sk,t-i .

i=O

If we substitute (A8) and (A9) into the set of simultaneous difference equations for yt and P, we obtain a set of difference equations to be solved for the T coefficients. Note that for i 2 2, the set of difference equations will be homogeneous (with right-hand sides the zero vector). We can write the corresponding homogeneous difference equations for yt and P, in matrix notation as

229

Peter R. Hartley

and Carl E. Walsh

Note that P,-1 appears in neither equations

equation.

We can rewrite

which can be solved for the rates of inflation outputs

[1 P t+1 pt

the two

as functions

of the

1

=

a12bz2- a&2

all&

1

b12h - b&l1 b12czl - bz2cll azzbll

-

a12hl

a22cll

-

a12c21

*[1 Yt+1

Yt

.

(A=)

Yt-1

But if both of these equations must hold for all t, then the righthand side of the second equation updated one period must equal the right-hand side of the first equation. Thus, we are left with a third-order difference equation to be solved for yt:

For the solution for the IT coefficients to be stationary, we want the complementary equation for (A13) to have one root greater than 1.0 in modulus and two roots less than 1.0 in modulus. When carrying out the non-linear optimization using GQOPT, the admissible parameter space was chosen to guarantee that this condition on the roots of the cubic was valid. Given the solutions for the roots of the cubic, X1 and h2, we can express the coefficients IT@, i, 1) for i L 2 in terms of the coefficients ~(k, 0, 1) and a(k, 1, 1) as 7r(k, i, 1) =

T(k 1, 1) Al

+ dk

A27dk, -

0, 1) A’;

A2

1, 1) - M(k

0, 1) Ai 2.

A2 -

230

A,

6414)

A Generalized

Method

of Moments

Approach

We solve for the two initial coefficients using the system of difference equations for the contemporaneous and once-lagged shocks. These difference equations will not only have non-zero right-hand sides corresponding to the non-zero coefficients on the contemporaneous and once-lagged shocks appearing in (Al)-(A4). The expectational terms in those equations will also introduce the variances of the shocks as coefficients of the different shocks into (Al)-(A4) via the projection equations. These are the equations which relate rational forecasts of the unknown contemporaneous shocks to the observed combinations of contemporaneous shocks. In our model the projection equations will depend on the variances of the different shocks but will not depend on the moving average representations of the endogenous variables in terms of the shocks. The contribution of shock k to the covariance between the mth and nth components of A_x, is given by r(k, 1, m)r(k,

1, n)(2 - C2) + r(k, 0, m)n(k, 0, n)(C2 - C4) 01.02

- [n(k, 1, m)4k,

0, n) + n(k, 0, m)n(k, 1, n)](Cl

- C3)

01.02 +

[dk,

+ 2 Nk

1, m)G,

0, n) + m(k, 0, m)rr(k, 1, n)]Cl Dl.(l + CO)

0, mMk,

0, n)CO - T(k, 1, m)r(k, Dl.(l + CO)

times the variance ai where we have defined

1, n)

(-416)

the constants

co = -A,& : Cl = A, + AZ ) C2 = A:. + A;, C3 = A; + A;, C4 = A; + A; , Dl = (A, - AJ2 = C2 + 2C0 , 02 = (1 - A;)( 1 - A;) = 1 - C2 + CO2 . 231

Peter R. Hartley

and Carl E. Walsh

The contribution of shock k to the covariance between the mth component of Agt and the nth component of As,-, is the variance d times the coefficient IT&, 1, +(k,

1, n)Cl.(l

+ CO) - ~(k, 0, m)~~(k, 0, n)CO.(Cl

- C3)

01.02 +

+

+

232

[IT@, 1, m)gc,

0, ?I) + gc, 0, m)lT(k, 1, n)]CO.(2 - C2) 01.02

m(k, 1, m)+,

0, n)C2 - 24k, Dl.(l

lT(k, 0, m)lT(k, 0, n)CO.Cl Dl.(l

0, m)p(k, 1, n)CO

+ CO) - ?T(k, 1, +r(k, + CO)

1, n)Cl 617)