# The importance of sectoral and aggregate shocks in business cycles.

THE IMPORTANCE OF SECTORAL AND AGGREGATE SHOCKS IN BUSINESS CYCLES

The theoretical literature on business cycles proposes numerous causes for their occurrence. This paper attempts to measure the relative importance of aggregate, whether real or nominal, and sectoral factors in generating real economic fluctuations, as well as to identify economic variables that are correlated with the various factors. Empirical results indicate that both aggregate and industry-level factors are statistically significant in explaining variations in output with the aggregate factor being the most important. Some evidence is presented that links the aggregate factor with monetary variables.

I. INTRODUCTION

Prior to the Keynesian revolution, a great deal of theoretical and empirical research was conducted on the business cycle.(1) With the acceptance of Keynesian ideas the emphasis of macroeconomic research shifted from the study of business cycles to the study of macroeconomic policies required to reduce economic fluctuations. This research program continued until the 1970s when oil price increases and the generally poor predictive performance of Keynesian models caused many macroeconomists to question this research program. As a result, some economists, such as Robert Lucas, Edward Prescott, and Thomas Sargent, began to employ stochastic, general equilibrium models to explain the main characteristics of the business cycle.

Movements in output or real economic activity arise from the interaction of economic shocks and internal mechanisms which propagate their effects. In this general framework, which is based on the work of Frisch [1933] and Slutsky [1937], unanticipated disturbances or shocks are spread through the propagation mechanism, resulting in serially correlated fluctuations in output. While most theories of economic fluctuations can be fitted into this general framework, there are disagreements over the form of the propagation mechanism and the source of the disturbances. Some models stress the role of aggregate demand shocks, while other models, such as the real business cycle models, focus on real technological shocks. Such technological shocks may be sector-specific and thus represent a set of disaggregate disturbances that cannot be viewed in the manner of aggregate shocks. There is little agreement as to the main sources of disturbances. Knowledge of the relative importance of sector-specific and aggregate shocks would be a significant development in theoretical work on business cycles.

This paper attempts to determine the relative importance of aggregate and disaggregate shocks in generating macroeconomic fluctuations. It extends Norrbin and Schlagenhauf [1988] in a number of important ways. The relative contribution of aggregate and sectoral shocks in generating real economic fluctuations is measured using disaggregated industrial production data, which we believe is a better measure of real economic activity. In addition, we employ a more detailed disaggregation of manufacturing and mining industries, covering a longer time period. The robustness of the results is more carefully identified. Finally, the methodology used to determine the relative importance of the shocks is substantially different in this paper.(2)

II. A TIME SERIES MODEL OF DISAGGREGATE OUTPUT

The work of Frisch [1933] and Slutsky [1937] suggests that movements in output can be decomposed into an impulse effect and a propagation effect. In this section, we present a multivariate time series model of output for the United States. This model is in the tradition of Frisch's and Slutsky's work and is sufficiently general to encompass most competing linear business cycle theories. The model is developed at a disaggregate industry level allowing a propagation of output changes between industries. In addition, output fluctuations can arise from national, industry-group specific, or idiosyncratic factors.

Consider an economy consisting of I industries indexed by i. Let [Y.sub.it] denote the change in the log of output in industry i. Output changes in a particular industry depend on previous output changes in the industry as well as on lagged output changes in all other industries. We represent the output change in industry j in period t-k as [Y.sub.j,t-k]. The propagation effect of an output change in industry j in period t-1 for industry i output in period t is represented by [[Pi].sub.ij,t-1]. In addition, the change in industry i output can be influenced by international conditions, [INT.sub.t], and by a disturbance or innovation term specific to that industry, [[Epsilon].sub.it]. That is, the change in output in industry i can be written as: (1) [Mathematical Expression Omitted].

The innovation or impulse term, [[Epsilon].sub.ij], can be decomposed into an influence common to all industries, influences specific to the industry-group or sector to which the industry belongs, and influences that are idiosyncratic. A factor which influences all industries is referred to as an aggregate or national factor. An example would be a change in a nominal demand variable, such as an unanticipated change in the money supply, as discussed by Lucas [1977]. A real supply shock, such as an oil price increase, would also qualify as an aggregate disturbance since the shock would be common to all industries. We will represent the aggregate factor as [X.sub.n,t]. Since each industry may respond differently to this factor, we represent the industry response to a change in the aggregate factor by the coefficient [Mathematical Expression Omitted]. Hence, the output response of an industry to a common shock is captured by [Mathematical Expression Omitted]. Output in an industry can also be influenced by factors specific to the industry-group to which the industry belongs. For example, economic activity in the industries that constitute the textile group may be influenced by one factor, while the industries constituting the transportation group may be influenced by a different one. We refer to this type of disturbance or factor as disaggregate, because it is specific to an industry-group. Suppose industry i belongs to industry-group ind. Let the sectoral factor for this industry-group be denoted by [X.sub.ind,t] and the response of real economic activity in industry i to this factor by [Mathematical Expression Omitted]. Hence, the impulse term [[Epsilon].sub.it] may be written as (2) [Mathematical Expression Omitted] where [u.sub.it] is the idiosyncratic error term. It is assumed, for identification purposes, that the factors [X.sub.N,t], [X.sub.ind,t], and [u.sub.it] are orthogonal to each other.

If equations (1) and (2) are combined, the change in output in industry i can be written as (3) [Mathematical Expression Omitted].

Let [Y.sub.t] denote the I x 1 column vector of the various individual industry-output changes. When we aggregate over all industries, equation (3) can be rewritten as: (4) [Mathematical Expression Omitted] where [X.sub.N,t] is a scalar representing the aggregate factors influencing all sectors, [X.sub.IND,t] is a (IND x 1) vector of industry-group factors where IND represents the number of industry-groups, [[Alpha].sub.N] and [[Alpha].sub.IND] are (I x 1) and (I x IND) coefficient matrices, respectively, and [u.sub.t] is the vector of idiosyncratic influences on industry-output change. The feedback coefficient matrix is represented by [[Pi].sub.k]. In equation (4) the dynamics of real economic fluctuations can be thought of as arising from the interaction between the internal propagation mechanisms as captured by the feedback coefficients and the influence of the aggregate and disaggregate factors.(3)

Most linear business cycle theories can be easily accommodated by the general framework represented by equation (4) by imposing restrictions on the propagation mechanism and the type of shocks allowed. To illustrate this point, we present a relatively general business cycle model that is easily extended or modified. We assume output in a particular industry is determined by a constant returns technology with capital and labor as inputs and represent this by (5) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the capital stock at the beginning of period t in industry i; [h.sub.i,t] is an index of the capital utilization rate in period t in industry i; [l.sub.i,t] is the labor input in period t in industry i; [m.sub.i,t] is an index of the labor utilization rate in period t in industry i; and [[Epsilon].sub.it] represents various impulses to industry i. When we aggregate over all industries, the output equation in vector form can be written as (5') [Mathematical Expression Omitted] where [Y.sub.t], [Mathematical Expression Omitted], [h.sub.t], [l.sub.t] and [m.sub.t] are vectors comprised of the appropriate industry variables, and [Theta] and (I - [Theta]) are coefficient matrices.

The evolution of the capital stock in an industry depends on the capital available in period t for that industry, the industry-specific depreciation rate of that stock, and the industry's gross investment. Hence, the capital stock evolution equation in vector notation is (6) [Mathematical Expression Omitted] where [Z.sub.t] is a (I x 1) vector of industry gross investment and D [approximately equal to] (I-[Delta]) is a diagonal coefficient matrix comprised of one minus the industry-specific depreciation rate. Gross investment is assumed to depend on output conditions in all industries as represented by the vector of outputs, and is specified as (7) [Z.sub.t] = [AY.sub.t].

Labor input used in industry i is assumed to depend on the labor input employed in that industry in the previous period, as well as the vector of output conditions. In vector notation, labor input is specified as (8) [Mathematical Expression Omitted] with [Mathematical Expression Omitted] being a diagonal matrix comprised of industry-specific adjustment coefficients, and [Mathematical Expression Omitted] is a coefficient matrix. From an industry-specific perspective, this equation can be viewed as a decision rule from a linear quadratic optimization problem with adjustment costs.

In order to close the model, structural equations for the indices of the capital utilization rate and the labor utilization rate are needed. We assume that the vector of capital utilization indices depends directly on how close the vector of output is to the vector of the natural rate of output of the industries. Hence, (9) [Mathematical Expression Omitted]. The vector of labor utilization indices is determined in a similar manner.

That is, (10) [Mathematical Expression Omitted]. In equations (9) and (10), the coefficient matrices are both diagonal matrices comprised of industry-specific coefficients. The vector that is comprised of the various industries' natural rate of output is defined as a function of the vector of previous output levels. (11) [Mathematical Expression Omitted]

If equations (6) through (11) are substituted into equation (5) and the resulting equation manipulated, the vector of industry-outputs can be written as (12) [Mathematical Expression Omitted]. This equation clearly indicates that the model specified by equations (5) through (11) can be expressed as a restricted version of equation (4). The primary restrictions are imposed on the [[Pi].sub.k] matrix, which captures the propagation effect. Equations (5) through (11) can be easily modified so as to incorporate other linear business cycle models. The result of these modifications is to change the propagation mechanism and/or the composition of [Epsilon.sub.t]. Hence, alternative business cycle models could be tested by examining restrictions on the feedback terms. In this paper we do not perform such tests, as the focus is on the impulse term, [[Epsilon].sub.t].

Unfortunately, the model represented by equation (4) cannot be estimated because it is over-parameterized. Such problems frequently occur in multivariate time series analysis. Consequently, we impose restrictions to reduce the number of parameters to be estimated. Our approach is to analyze three different sets of restrictions on [[Pi].sub.k], so that the robustness of our findings on the relative importance of the different types of shocks can be established.

We impose a set of restrictions that allows us to retain as much of the effect of lagged industry-output changes as possible. In order to allow some national feedback effects and industry-group feedback effects, we create composite variables to represent such effects. These composite variables are similar to those used in Norrbin and Schlagenhauf [1988]. A national feedback effect is constructed as a weighted average of industry-output change. The composite national output variable is (13) [Mathematical Expression Omitted] where the weights are the relative importance of each industry in the aggregate industrial production index (base 1967), but rescaled for the manufacturing and mining industries included in our sample.

A specific industry can be influenced by output changes in its industry-group. To model this, we create IND industry-group composite variables defined as [Y.sub.ind,t]. Each one of these variables represents a weighted average of the change in output of each industry in the industry-group. The weight, [Mathematical Expression Omitted], is the fraction of output in industry-group ind accounted for by industry i. That is, (14) [Mathematical Expression Omitted] where G is the number of industries in a given industry-group.

Given these restrictions and allowing for an own-lag feedback effect, the change in output in matrix form is specified as (15) [Mathematical Expression Omitted] where [[Beta].sub.1], [[Beta].sub.2k] are (I x 1) vectors, [[Beta].sub.3k] is an (I x IND) matrix and [[Beta].sub.4k] is an (I x I) diagonal matrix.

An alternative approach to restricting the feedback matrices is to use principal component analysis to limit the cross-dependencies between output changes in equation (4). Principal component analysis reduces the dimension of the data matrix for the I industry-output variables by determining the degree of independence that exists in the information provided by each industry-output variable. If we let [P.sub.m], m = 1,...,M represent the M principal components, the change in output in matrix form is specified as (16) [Mathematical Expression Omitted] where [P.sub.t-k] is a M x 1 vector of principal components, and [Mathematical Expression Omitted] is an (I x M) coefficient matrix.(4)

The third approach to restricting the industry feedback coefficient matrices follows Long and Plosser [1983]. They present a model in which the feedback coefficients are restricted to be industry cost shares from the 1967 input-output table for the United States. We follow their lead by setting the feedback coefficients equal to the input requirements from other industries and the own industry. This requires the construction of an input-output table appropriate for our disaggregation scheme. The industry-output model in matrix form for this restriction approach is (17) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is an (I x I) coefficient matrix restricted to equal the modified input-output coefficients.

III. ECONOMETRIC METHODOLOGY

The estimation of equations (15), (16), and (17) is complicated by the fact that the factors appearing in each equation are unobservable. To resolve this problem, we estimate the equations using the dynamic multiple indicator multiple cause (DYMIMIC) framework of Watson and Engle [1983].(5) In this section, we briefly describe this approach, which, in addition to providing maximum likelihood estimates of the parameters and the unobserved variables, also allows empirical investigation of observable variables that are associated with the unobserved factors. This latter feature is important as it allows the economist to identify the cause of these factors.

The DYMIMIC model, expressed in state-space form, can be written as (18) [Mathematical Expression Omitted] (19) [X.sub.t] = [Delta](L)[X.sub.t-1] + [Gamma](L)[Z2.sub.t] + [e.sub.t] and (20) [Mathematical Expression Omitted] where [Y.sub.t] is a p x 1 vector of observed variables; [X.sub.t] is an n x 1 vector of unobserved variables or factors; [Z1.sub.t] and [Z2.sub.t] are s x 1 and q x 1 vectors, respectively, of observable exogenous and lagged dependent variables; [v.sub.t] and [e.sub.t] are vectors of disturbances; and L is the lag operator. The specification of the Z1 vector depends on the restriction approach employed, while the Z2 vector contains the causal variables that aid in the interpretation of the factors.

This state-space model is estimated using general maximum likelihood techniques along with the Kalman filter recursive algorithm. Since missing observations are present, we employ the EM algorithm (see Dempster, Laird, and Rubin [1977]) to maximize the likelihood function. The algorithm iterates between an estimation stage and maximization stage until convergence is achieved. Based on an initial guess of the parameters, the E-step of the algorithm employs the Kalman 'smoother' to construct estimates of the missing observations conditional on the observed data and parameters. Once the unobserved state variables are estimated, the M-step calculates the estimates of all unknown parameters conditional on the entire data set. These parameter estimates are then used in conjunction with the Kalman filter to generate a new estimate of the state variables. This interactive process is continued until an overall convergence is achieved. It has been shown that this algorithm will always increase the value of the likelihood function until it converges to a local maximum. The convergence is guaranteed under suitable regularity conditions.(6) A more detailed explanation of this algorithm is available from the authors upon request.

IV. EMPIRICAL RESULTS

Before examining our results, we discuss the data base and the specification of the factor equations. Because of the large number of coefficients estimated in each of the time series models, a measure of the relative importance of the national and industry-group-specific factors in generating output fluctuations is constructed. After discussing these results, we identify the observable economic variables that are correlated with each of these factors.

Data

Disaggregated output is measured by seasonally adjusted disaggregate industrial production. The Federal Reserve Board reports monthly industrial output at the three, and in some cases, four-digit Standard Industrial Classification level for manufacturing and mining industries. This data, as reported by Data Resources, Inc., is used to measure quarterly industry-output, where quarterly output is defined as the average over the appropriate monthly industrial production indices. The sample covers the period 1955 to 1984.

In order to measure industry-specific factors, the various industrial output indices are separated into industry-groups. These industry-groups normally coincide with the various two-digit SIC classes for manufacturing and mining. This method of segmentation results in the creation of fourteen industry-groups. These groups are primary metals and manufacturing; transportation equipment; machines, instruments and miscellaneous manufacturing; lumber and furniture; stone, glass, and glass products; textile products; leather products; food and kindred products; tobacco manufactures; paper and allied product; printing and publishing; chemicals and refining; plastics; and mining. Each industry-group is composed of at least two industries. A total of forty-nine industries are aggregated into the fourteen industry-groups.(7)

The quarterly output data are nonstationary. Many empirical studies of the business cycle deal with this problem by using a time trend to capture a (long-term) growth component. Nelson and Plosser [1982] suggest this procedure is likely to confound the growth component and cyclical component in the series, overstating the magnitude and duration of the cyclical component and understating the importance of the growth component. Therefore, we first difference the log of output. This renders the series stationary without removing the growth component, which may be stochastic.

The international variable must also be specified. Output in a particular industry can be affected by international developments such as a change in world relative demand or a change in comparative advantage. To allow for a potential international influence, we created a trade-weighted average of industrial production for the United State's largest trading partners.(8)

Specification of the Unobserved Variable

According to the state-space formulation, each factor can be expressed as a function of previous factors and observable causal variables. For the aggregate or national factors, we allow the previous period (own) national factor, [X.sub.N,t-1], to be a determinant. If the aggregate factor is interpreted as an unobserved impulse that is national in origin, there may be some question on the appropriateness of specifying a serially correlated impulse in an environment where agents have rational expectations. We determine the issue empirically and find the autoregressive terms are not statistically significant when the composite variable restriction approach is employed. However, the autoregressive terms are significant when the other restriction approaches are employed. In an attempt to give an economic interpretation to the behavior of the aggregate factor, we allow for a measure of monetary and fiscal policy, as well as an aggregate real supply variable, oil prices, to enter the Z2 vector. The monetary policy variable is defined as the rate of change of real M1, while fiscal policy variable is defined as the rate of change of the high employment real surplus. The current value and four lags of each variable are included in this exogenous variable vector.(9)

Industry-group factors are specified to follow a simple autoregressive process with causal variables. Prescott [1986] has specified a serially correlated technological impulse in his work on real business cycles, and Schumpeter's [1939] work on business cycles implies that technological bursts should be autocorrelated. The specification of the causal variables for these factors is straightforward in principle. Theory suggests that taste and technological changes should account for the movement in an industry factor. Most empirical work on the real business cycle model uses 'Solow residuals' to measure technological change. We follow this approach and create industry-group Solow residuals.(10,11)

The use of Solow residuals to measure technological shocks has been questioned recently. For instance, McCallum [1989] notes that Solow's method assumes that current capital and labor are the only relevant inputs. If adjustment costs exist, then labor and/or capital hoarding might cause the estimated Solow residuals to overstate the technological shock variance. We have attempted to deal with this criticism of the use of Solow residuals that could result in these variables reflecting aggregate developments rather than disaggregate developments by filtering the industry-group measures of technological change for the state of the business cycle. This was done by regressing the Solow residual on a constant, a time trend, a time trend squared and on the deviation of U.S. output from the natural rate of output. This should eliminate aggregate labor and capital hoarding, and the residuals would be only influenced to the extent to which different adjustment costs exist between industry-groups.(12)

Empirical Results

Before any estimation can be conducted, the specification of the length of the lag of the output feedback terms has to be considered. As mentioned above, restrictions have to be placed on these feedback terms in order to estimate the model. When the input-output coefficients are employed as restrictions, it is obvious that the lag is one period. When the composite variable approach is used as a restriction approach, each composite variable is allowed a lag of four quarters. The same lag length is employed for each principal component when the feedback terms are restricted using principal components. The estimation period covers the period 1954:1 through 1984:4.

An obvious first step in the analysis is to identify whether there is any statistical basis for the introduction of disaggregate factors in explaining industry-specific output change. If only aggregate sources are found to be significant, this would support a single index explanation of the business cycle. In order to determine whether the national factor and the set of industry-group factors are statistically significant, we conducted a series of likelihood ratio tests. The results from these hypothesis tests for each method of restricting the output feedback coefficient matrix are presented in Table I.(13) As can be seen, the null hypothesis for each test is strongly rejected for each approach to restrict the feedback coefficient matrix. This indicates that both the national factor and the set of industry-group factors play a role in triggering economic fluctuations.

Once the statistical importance of the various unobserved variables is established, it is important to determine the relative importance of each type of factor in generating the business cycle. Because of the large number of coefficients estimated in each model, we summarize the results to emphasize the relative importance of the various factors by deriving the moving average representation of the state-space model. From this representation, the decomposition of the forecast error of output can be generated for various horizons. These measures can be used to evaluate the relative importance of the various factors.

The general state-space model is expressed in equations (18)-(20). The actual exogenous and predetermined variables that appear in the [Z1.sub.t] vector depend on the restriction approach employed.(14) We partition the [Z1.sub.t] vector into exogenous and predetermined components. If we denote the exogenous variables as [Mathematical Expression Omitted] and the corresponding coefficients as [[Lambda].sub.1](L), equation (18) can be expressed as (18') [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. When both sides of (18') are premultiplied by [Phi][(L).sup.-1] the statespace model can be reformulated as (21) [Mathematical Expression Omitted] (22) [X.sub.t] = [Delta](L)[X.sub.t-1] + [Gamma](L)[Z2.sub.t] + [e.sub.t].

By substituting (22) into (21), the moving average representation of the statespace model is (23) [Mathematical Expression Omitted] This moving average representation is conditioned on current and past [Mathematical Expression Omitted] and [Z2.sub.t]. If generating equations are postulated for [Mathematical Expression Omitted] and [Z2.sub.t], an unconditional moving average representation can be derived. We postulate univariate autogressive forecasting equations for each element in [Mathematical Expression Omitted] and [Z2.sub.t]. That is, (24) [Mathematical Expression Omitted] (25) B(L)[Z2.sub.t] = [[Epsilon].sub.2t] where A(L) and B(L) are diagonal matrices.(15)

Under the assumption that each exogenous variable can be modelled as a second-order autoregressive process, the moving average representations of (24) and (25) are (26) [Mathematical Expression Omitted] (27) [Mathematical Expression Omitted] where [q.sub.A1] = [A.sub.1] [q.sub.A2] = [q.sub.A1][A.sub.1] + [A.sub.2] [q.sub.Aj] = [q.sub.Aj-1][A.sub.1] + [q.sub.Aj-2][A.sub.2] for j [is greater than or equal to] 3 [q.sub.B1] = [B.sub.1] [q.sub.B2] = [q.sub.B1][B.sub.1] + [B.sub.2] [q.sub.Bj] = [q.sub.Bj-1][B.sub.1] + [q.sub.Bj-2][B.sub.2] for j [is greater than or equal to] 3. Combining (23) through (27) yields the unconditional moving average representation (28) [Mathematical Expression Omitted] where the matrices are [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] where j=0, 1, 2,...,j and [[Theta].sub.0] = 1 [Mathematical Expression Omitted] where [q.sub.A0] = 1 [Mathematical Expression Omitted] where [q.sub.B0] = 1 [Mathematical Expression Omitted] where k = 0, 1, 2,...,k and [[Theta].sub.0] = 1.

Once the moving average representation is generated, a meaningful decomposition of the variance of output can be derived by calculating the variance of the forecast error at various horizons. If we let [Mathematical Expression Omitted] then the j-period ahead forecast error of output has variance (29) [Mathematical Expression Omitted] [Mathematical Expression Omitted] for j = 1 [Mathematical Expression Omitted] [Mathematical Expression Omitted] for j = 2, 3,...

This equation indicates that an individual industry's variance can be segmented into an industry-specific or idiosyncratic component contained in Q; an error arising from the factor of the specific industry-group and the common aggregate factor which are contained in R; an error from forecasting the exogenous (Z1) variables, Var([[Epsilon].sub.1]); and the error from forecasting the observable (Z2) variables that enter the factor equations, Var([[Epsilon].sub.2]).

Because of the number of industries in our sample, further aggregation is required to simplify interpretation. We would like to determine how important each of the factors is from an economy-wide perspective. In order to address this question, the [Mathematical Expression Omitted] must be collapsed to an aggregate forecast error over all industries. We accomplished this by weighting the variance of the forecast error of each individual industry by the relative importance of that industry's output in total industrial output. The weight actually corresponds to each industry's weight in the base 1967 industrial production index, appropriately rescaled. If [W.sub.N] is a (I x 1) vector of these weights, then premultiplying both sides of (29) by [Mathematical Expression Omitted] and post-multiplying by [W.sub.N] reduces the variance matrix to a scalar. Dividing the contribution of each component of the forecast error of output by the aggregate forecast error will yield a measure of the relative importance of each component. Another question of interest concerns the relative importance of the factors from an industry-group perspective. This decomposition is calculated by creating a weighting matrix where the weights are the fraction of each industry's output in the particular industry-group output. If all fourteen industry-group weighting matrices are stacked columnwise, an industry-group weighting matrix, [W.sub.IND], can be constructed with dimension I x IND. By pre-multiplying equation (29) by the transpose of this weighting matrix and post-multiplying by this weighting matrix, the Var([Y.sub.t+j]) can be reduced to a fourteen by fourteen matrix that yields industry-group insights.

The results from these calculations are presented in Tables II and III for each feedback restriction approach. We first examine the percentage of forecast error variance explained by the various factors from a national perspective under the composite variable restriction approach. In order to facilitate the comparison between the aggregate and sectoral sources of forecast errors, we sum the contribution of the error term in the factor equation with the contribution of its causal variables. As can be seen in Table II, the aggregate factor accounts for 52.14 percent of the one-period-ahead variance of the forecast error while the industry-group factors account for 34.30 percent of the variance. The international influence explains 9.11 percent of the one-period-ahead forecast variance. As the forecast horizon increases, the aggregate factor diminishes somewhat in importance, and the international influence and the industry-specific residual increase in importance. For example, in the steady state, defined as twenty periods ahead, the aggregate factor accounts for 40.32 of aggregate variance of the forecast error. The fraction of the variance of the forecast error explained by the industry-group factors increases slightly to 35.09 percent. These results indicate that the national common factor is the most important factor.

Table III presents the decomposition, evaluated at the steady state, from an industry-group perspective. The conclusion concerning the relative ranking of the national and industry-group factors changes when an industry-group perspective is employed. For many industry-groups, the industry-group factor dominates the national factor in explaining industry-group output variation. Given the results presented in the national perspective summary, the industry-group results may seem contradictory. The explanation for the apparent contradiction is in the estimation methodology. In estimating the individual industry-output equations, the industry-group factors were assumed to be orthogonal to each other. This assumption tends to give conservative estimates of the importance of the disaggregate industry-groups' factors when creating a national summary perspective. However, in constructing an industry-group summary, the industry-group factor explains output variation common to a group of similar industries, thus allowing a potentially larger role for the industry factor within the industry-group. The percentage of variation explained by the national and industry-group factors do vary across industry-groups. The national factor is especially important for machines, refining, textiles, transportation, lumber and plastics while being relatively unimportant for tobacco, mining, and printing. In addition, as would be expected, the industry-group factor and/or industry-specific residual seem to play a more important role in these industries where the aggregate factor is not as important. For example, mining, metals, and food industries are often thought of as industries not dominated by aggregate developments. Our results suggest that is the case: over 69.6 percent of the variance of the forecast error of output in the steady state is explained by the industry-group factor in metals while 67 percent of the variance is accounted for by the industry-specific residual in mining. For the food industry, the industry-group factor and industry-specific residual account for approximately 75 percent of the steady state variance of the forecast error.

In order to examine the robustness of these results, similar decompositions are presented for the principal component restriction approach and the input-output restriction approach. As can be seen in Table II, the general conclusions drawn from the composite variable restriction approach hold for the other approaches. If the principal component restriction approach is employed, the aggregate factor and industry-group factors account for 54.54 percent and 33.40 percent of the one-period-ahead variance of the forecast error of output from a national perspective. At the steady state, the variance of the forecast error of output explained by the aggregate factor slightly falls to 52.89 percent and the industry-group factor declines to 32.22 percent. If the feedback coefficients are restricted by input-output coefficients, we continue to find that the national factor is the most important, while industry-group factors explain a substantial amount of the variance of the forecast error. For instance, in the steady state the aggregate factor and the industry-group factors account for 53.99 percent and 37.41 percent, respectively, of the variance of the forecast error of output. The one major difference is the importance of the international variable. The decreased importance of the international variable when an input-output restriction approach is employed suggests that this restriction confuses an influence from this variable with the propagation mechanism. Based on the findings presented in Table II, Burns and Mitchell's [1946] view of a single aggregate shock source of business cycles seems inappropriate given the prominent role that disaggregate or industry-group factors play. Furthermore, the importance of the industry-group factors does seem to indicate an important role for some forms of the real business cycle model.(16)

An analysis of the factor equations yields some insights into the explanation of the movements in these factors. Since the dimension of the empirical problem makes it impossible to calculate standard errors, we conducted statistical tests using likelihood ratio procedures. The relevant test statistics are presented in Table IV. We start by considering the national factor equation. In this equation, we introduce the current and previous four periods of a fiscal policy variable, a monetary policy variable and a real shock variable as potential observable variables that could be correlated with the national factor. Our fiscal policy variable, measured as the real value of the high employment surplus, is not statistically significant. The current change in the real value of M1 and the four lags of this variable represent monetary policy.(17) The concurrent monetary variable has the correct sign and is statistically significant. More important, when the monetary factors are considered as a group, the null hypothesis that they are statistically different from zero can not be rejected. The other causal variable that enters the national factor equation is the change in oil prices. Theoretically, we would expect that oil price increases would be negatively correlated with the aggregate factor. The estimated equation indicates that this is indeed the case. However, the likelihood ratio test indicates that this variable is not significant. Thus, the only observable variable that is correlated with the national factor is monetary policy changes.

Disaggregate real business cycle theories suggest that the industry-group factors should be correlated with relative price, taste, or technological changes among industries. Since most research in this area seems to focus on technological change as the driving variable, we construct a set of technological change measures for each industry-group. As discussed above, we constructed a set of industry-group 'Solow residuals' as a proxy for industry-group technological change. Table IV shows that we can reject the null hypothesis that as a group these variables are not statistically different from zero. This is an important empirical result as the finding of correlation between industry-group factors and industry-group technological shocks provides important evidence in support of the disaggregate real business cycle paradigm.

V. CONCLUSION

In recent years macroeconomists have renewed their interest in the area of business cycles. Research by Prescott and others suggests that cycles may be due to aggregate real technological shocks rather than to monetary shocks, which many economists had viewed as the causal factor. Long and Plosser [1983; 1987] have extended the real business cycle paradigm so that business cycles can result from disaggregate real impulses. This paper attempts to measure the relative contribution of aggregate and sectoral factors in generating real economic fluctuations. Furthermore, it attempts to identify the economic variables that are correlated with these factors. We measure the relative importance of the various factors in the context of a multivariate time series model of output change which allows for feedback influences, as well as an aggregate factor, industry-group factors, and an international effect.

There are three major findings. First, both the aggregate and industry-group factors are statistically significant in explaining variations in output. This suggests that the Burns and Mitchell [1946] view of a single aggregate shock as the source of business cycles is incomplete. Second, the aggregate factor accounts for largest amount of the forecast error variance of aggregate output. However, the industry-group factors explain a significant amount of the forecast error variance of output so as to give empirical context to the disaggregate form of the real business cycle model. Third, the aggregate factors are correlated with monetary variables. These findings are robust to various approaches to restrict the feedback effects. Overall, the results in this paper suggest that a complete model of the business cycle must allow for both aggregate and sectoral sources of macroeconomic fluctuations. [Tabular Data 1 to 4 Omitted] [Appendix Omitted]

(1)Haberler [1958] summarizes the business cycle theories in the pre-Keynesian period. More recently, Zarnowitz [1985] presents an excellent review of the theories and evidence on business cycles. (2)Long and Plosser [1987] examine several similar issues, but they made no attempt to determine the fraction of the unexplained variance due to sectoral shocks or other shocks. In this paper we a more general estimation strategy, dynamic factor analysis, to derive estimates of the relative importance of shocks to output. Our approach estimates the unobserved factors, thus allowing us to investigate the relationships between the factors and observable economic variables suggested by theory. (3)The appearance of the [INT.sub.t] variable requires additional comment as to why this variable is not treated as an unobserved variable or factor. If international influences are treated as a factor, it would be an effect common to each equation. With an aggregate factor already specified, a statistical identification problem arises similar to the rotation problem in simple factor analysis. Since the aggregate factor is thought of as domestic in origin, so as to be consistent with the single index model, we introduce an observable [INT.sub.t] variable to allow for international influences and avoid identification problems. In results not reported in the paper we estimate the model when [INT.sub.t] is deleted from the output equation and introduced as a regressor in the aggregate factor equation. The conclusions concerning the relative importance of the various factors remain unchanged. These results are available upon request from the authors. (4)The first two principle components are used in the estimation process. They account for approximately 65 percent of the variation in the [Y.sub.t] vector. (5)The DYMIMIC model is a type of index model. Sargent and Sims [1980] have used an index model to study business cycles. They employ a frequency domain method with unrestricted lag distributions. The Watson and Engle [1983] estimation approach is a time domain method. (6)We attempt to avoid local optimization results by examining various starting values and employing a very severe convergence criteria (i.e., .0000001). (7)Which industries are included in a specific industry-group is explained in a data appendix available from the authors. The construction of any data is also discussed in this appendix. (8)In order to capture the world economic situation, we created a world industrial production variable. This variable is based on a weighted average of ten countries' industrial production indices. The ten countries are the major trading partners and include Germany, Japan, France, the United Kingdom, Canada, Italy, Netherlands, Belgium, Sweden, and Switzerland. The weights are calculated as the fraction of each country's imports plus exports with the United States as a fraction of total imports and exports for these countries. (9)The money supply is defined as the seasonally adjusted value of M1 and is from the CITIBANK data tape (series FM1). For data prior to 1959, the series came from the Handbook of Cyclical Indicators, series 85, U.S. Department of Commerce, Bureau of Economic Analysis, 1984. The high employment deficit variable includes adjustments for automatic inflation effects. These data are from the article by Thomas M. Holloway entitled "The Cyclically Adjusted Federal Budget and Federal Debt: Revised and Updated Estimates," Survey of Current Business, March 1986. We follow Hamilton [1983] in constructing the price of oil. (10)Industry-group productivity shocks, or Solow residuals, can be defined as: %[Delta][A.sub.ind] = %[Delta][Q.sub.ind] - [[r.sub.ind][K.sub.ind])/([P.sub.ind][Q.sub.ind])] (%[Delta][K.sub.ind]) + [([W.sub.ind][L.sub.ind])/([P.sub.ind][Q.sub.ind])] (%[Delta][L.sub.ind]) where A is technology change, Q is output, r is the rental rate of capital, w is the wage rate, K is net capital stock, L is labor input, and ind refers to an specific industry-group. In a data appendix, we detail how we calculated these variables. (11)We also introduce the oil price as an explanatory variable in each of the industry-group factor equations. While this variable is significant in a few industry-group equations, conclusions on the relative importance of the factors and the statistical significance of Solow residuals are unaffected. (12)An alternative approach to dealing with the criticism of the Solow residuals as measures of technological change is to delete these variables from the various industry-group factor equations. In this case, the industry-group factors would be determined by a first-order autoregressive process. This specification would be similar to the one employed by Prescott [1986]. We estimated the various models where the Solow residuals were deleted from the industry-group factor equations and found that our results are not especially sensitive to the use of these Solow residuals. (13)A comment is required concerning the specification of the restricted model employed in the testing procedure. Usually the importance of a variable or a set of variables is tested by deleting such variables from the unrestricted model. In terms of the DYMIMIC model, deletion of a factor or set of factors in the indicator equations also requires that each coefficient in the transition equation be restricted for the factor or set of the factors being examined. This accounts for the large number of restrictions associated with each likelihood ratio statistic. (14)If the principal components approach is employed, the [Z1.sub.t] vector would be comprised of a constant, the international influence variables, [INT.sub.t], and the principal components that enter each output equation. On the other hand, with the composite variable restriction approach this vector is comprised of a constant, the international variable, lagged composite national output variables, [Y.sub.N,t-k], lagged composite industry-group variables, [Y.sub.IND,t-k] as well as lagged own predetermined endogenous variables, [Y.sub.t-k]. The composite variables are predetermined variables as they can be expressed as a fixed coefficient matrix times the lagged weighted predetermined variable. (15)The univariate forecasting equations were estimated with a constant. (16)All of the decompositions that have been considered are for the entire sample period. Given the change in international monetary regimes that occurred in the early 1970s, it is important to determine whether the relative importance of the factors is sensitive to this potential structural change. The appendix table presents the decomposition of variance of the forecast error from a national perspective using the composite variable restriction approach for the periods 1956 to 1971 and 1971 to 1984. Because the sample size is much smaller in these subsamples, while the number of parameters to be estimated remains unchanged, care must be taken in analyzing the results. As can be seen, the aggregate factor continues to be the most important. The industry-group factors also account for a substantial amount of the variance of the forecast error of output. In fact, in the 1971-1984 subsample, the industry-group factor accounts for approximately the same percentage of the variance of output as does the aggregate factor in the steady state. The major difference in the two samples is the increased importance of the international variable in the latter. However, the size of the international influence and the industry-specific residual relative to the results presented in the table below does raise concern over problems caused by overparameterization. Additional work has failed to identify whether the time period dependencies of the variance decomposition are due to overparameterization problems or a problem with the construction of the international variable. (17)An extension would be to decompose the demand management variables into expected and unexpected components. Testing variables in this form would be interesting given recent empirical work on testing some business cycle theories. In order to gain some insights on this issue, we replaced [DM.sub.t] with a money shock variable composed of the error term where [DM.sub.t] was regressed on four lags of itself, the high employment surplus, and the three-month treasury bill rate. The money surprise variables as a group are insignificant. However, the current unanticipated monetary variable by itself is marginally significant.

REFERENCES

Burns, A. F. and W. C. Mitchell. Measuring Business Cycles. New York: Columbia University Press for

the National Bureau of Economic Research, 1946. Dempster, A. P., N. M. Laird and D. B. Rubin. "Maximum Likelihood from Incomplete Data via the EM

Algorithm." Journal of the Royal Statistical Society, B 39, 1-8. Frisch, Ragnar. "Propagation Problems and Impulse Problems in Dynamic Economics," in Economic Essays

in Honor of Gustav Cassel, London: Allen and Unwin, 1933. Haberler, Gottfried. Prosperity and Depression, 2d ed. New York: United Nations, 1958. Hamilton, James D. "Oil and the Macroeconomy since World War II." Journal of Political Economy,

April 1983, 228-48. Long, J. B., Jr. and C. I. Plosser. "Real Business Cycles." Journal of Political Economy, February

1983, 39-69. ______. "Sectoral vs. Aggregate Shocks in the Business Cycle." American Economic Review, May 1987,

333-36. Lucas, R. E., Jr. "Understanding Business Cycles," in Stabilization of the Domestic and

International Economy, edited by K. Brunner and A. H. Meltzer. Carnegie-Rochester Conference

Series on Public Policy, 5, 1977, 7-29. McCallum, B. T. "Real Business Cycle Models," in Modern Business Cycle Theory, edited by R. J.

Barro. Cambridge: Harvard University Press, 1989. Nelson, C. R. and C. I. Plosser. "Trends and Random Walks in Macroeconomic Time Series: Some

Evidence and Implications." Journal of Monetary Economics, September 1982, 139-62. Norrbin, S. C. and D. E. Schlagenhauf. "An Inquiry into the Sources of Macroeconomic Fluctuations."

Journal of Monetary Economics, July 1988, 44-70. Prescott, E. C. "Theory Ahead of Business Cycle Measurement," in Real Business Cycles, Real Exchange

Rates and Actual Policies, edited by K. Brunner and A. H. Meltzer. Carnegie-Rochester Conference

Series on Public Policy, 25, Autumn 1986, 11-44. Priestly, M. B., T. S. Rao, and H. Tong. "Applications of Principal Component Analysis and Factor

Analysis in the Identification of Multi-variable Systems." IEEE Transactions on Automatic

Control, AC-19, 1974, 730-34. Sargent, Thomas J. and Christopher A. Sims. "Business Cycle Modeling without Pretending to Have Too

Much A Priority Economic Theory," in New Methods in Business Cycle Research, edited by C. A.

Sims. Minneapolis: Federal Reserve Bank of Minneapolis, 1980. Schumpeter, Joseph. Business Cycles: A Theoretical, Historical and Statistical Analysis of the

Capital Process. New York: McGraw-Hill, 1939. Slutsky, E. "The Summation of Random Causes as the Source of Cyclic Processes." Econometrica,

January 1937, 105-46. Solow, R. M. "Technical Change and the Aggregate Production Function." Review of Economic and

Statistics, August 1957, 312-20. Watson, M. W. and R. F. Engle. "Alternative Algorithms for Estimation of Dynamic MIMIC, Factor, and

Time Varying Coefficient Regression Models." Journal of Econometrics March 1983, 285-300. Zarnowitz, V. "Recent Work on Business Cycles in Historical Perspective: A Review of Theories and

Evidence." Journal of Economic Literature June 1985, 523-80.

STEFAN C. NORRBIN and DON E. SCHLAGENHAUF, Florida State University and Arizona State University, respectively. We would like to thank Art Blakemore, Joe Brada, and Richard Sweeney for useful comments and Rosalie Bondad for her research assistance. The comments of the referees are gratefully acknowledged. This research has been supported by grants from the Dean's Council of 100 at Arizona State University, the Research School of Social Sciences at Australia National University, and the Center for Research in Financial Systems at Arizona State University.

The theoretical literature on business cycles proposes numerous causes for their occurrence. This paper attempts to measure the relative importance of aggregate, whether real or nominal, and sectoral factors in generating real economic fluctuations, as well as to identify economic variables that are correlated with the various factors. Empirical results indicate that both aggregate and industry-level factors are statistically significant in explaining variations in output with the aggregate factor being the most important. Some evidence is presented that links the aggregate factor with monetary variables.

I. INTRODUCTION

Prior to the Keynesian revolution, a great deal of theoretical and empirical research was conducted on the business cycle.(1) With the acceptance of Keynesian ideas the emphasis of macroeconomic research shifted from the study of business cycles to the study of macroeconomic policies required to reduce economic fluctuations. This research program continued until the 1970s when oil price increases and the generally poor predictive performance of Keynesian models caused many macroeconomists to question this research program. As a result, some economists, such as Robert Lucas, Edward Prescott, and Thomas Sargent, began to employ stochastic, general equilibrium models to explain the main characteristics of the business cycle.

Movements in output or real economic activity arise from the interaction of economic shocks and internal mechanisms which propagate their effects. In this general framework, which is based on the work of Frisch [1933] and Slutsky [1937], unanticipated disturbances or shocks are spread through the propagation mechanism, resulting in serially correlated fluctuations in output. While most theories of economic fluctuations can be fitted into this general framework, there are disagreements over the form of the propagation mechanism and the source of the disturbances. Some models stress the role of aggregate demand shocks, while other models, such as the real business cycle models, focus on real technological shocks. Such technological shocks may be sector-specific and thus represent a set of disaggregate disturbances that cannot be viewed in the manner of aggregate shocks. There is little agreement as to the main sources of disturbances. Knowledge of the relative importance of sector-specific and aggregate shocks would be a significant development in theoretical work on business cycles.

This paper attempts to determine the relative importance of aggregate and disaggregate shocks in generating macroeconomic fluctuations. It extends Norrbin and Schlagenhauf [1988] in a number of important ways. The relative contribution of aggregate and sectoral shocks in generating real economic fluctuations is measured using disaggregated industrial production data, which we believe is a better measure of real economic activity. In addition, we employ a more detailed disaggregation of manufacturing and mining industries, covering a longer time period. The robustness of the results is more carefully identified. Finally, the methodology used to determine the relative importance of the shocks is substantially different in this paper.(2)

II. A TIME SERIES MODEL OF DISAGGREGATE OUTPUT

The work of Frisch [1933] and Slutsky [1937] suggests that movements in output can be decomposed into an impulse effect and a propagation effect. In this section, we present a multivariate time series model of output for the United States. This model is in the tradition of Frisch's and Slutsky's work and is sufficiently general to encompass most competing linear business cycle theories. The model is developed at a disaggregate industry level allowing a propagation of output changes between industries. In addition, output fluctuations can arise from national, industry-group specific, or idiosyncratic factors.

Consider an economy consisting of I industries indexed by i. Let [Y.sub.it] denote the change in the log of output in industry i. Output changes in a particular industry depend on previous output changes in the industry as well as on lagged output changes in all other industries. We represent the output change in industry j in period t-k as [Y.sub.j,t-k]. The propagation effect of an output change in industry j in period t-1 for industry i output in period t is represented by [[Pi].sub.ij,t-1]. In addition, the change in industry i output can be influenced by international conditions, [INT.sub.t], and by a disturbance or innovation term specific to that industry, [[Epsilon].sub.it]. That is, the change in output in industry i can be written as: (1) [Mathematical Expression Omitted].

The innovation or impulse term, [[Epsilon].sub.ij], can be decomposed into an influence common to all industries, influences specific to the industry-group or sector to which the industry belongs, and influences that are idiosyncratic. A factor which influences all industries is referred to as an aggregate or national factor. An example would be a change in a nominal demand variable, such as an unanticipated change in the money supply, as discussed by Lucas [1977]. A real supply shock, such as an oil price increase, would also qualify as an aggregate disturbance since the shock would be common to all industries. We will represent the aggregate factor as [X.sub.n,t]. Since each industry may respond differently to this factor, we represent the industry response to a change in the aggregate factor by the coefficient [Mathematical Expression Omitted]. Hence, the output response of an industry to a common shock is captured by [Mathematical Expression Omitted]. Output in an industry can also be influenced by factors specific to the industry-group to which the industry belongs. For example, economic activity in the industries that constitute the textile group may be influenced by one factor, while the industries constituting the transportation group may be influenced by a different one. We refer to this type of disturbance or factor as disaggregate, because it is specific to an industry-group. Suppose industry i belongs to industry-group ind. Let the sectoral factor for this industry-group be denoted by [X.sub.ind,t] and the response of real economic activity in industry i to this factor by [Mathematical Expression Omitted]. Hence, the impulse term [[Epsilon].sub.it] may be written as (2) [Mathematical Expression Omitted] where [u.sub.it] is the idiosyncratic error term. It is assumed, for identification purposes, that the factors [X.sub.N,t], [X.sub.ind,t], and [u.sub.it] are orthogonal to each other.

If equations (1) and (2) are combined, the change in output in industry i can be written as (3) [Mathematical Expression Omitted].

Let [Y.sub.t] denote the I x 1 column vector of the various individual industry-output changes. When we aggregate over all industries, equation (3) can be rewritten as: (4) [Mathematical Expression Omitted] where [X.sub.N,t] is a scalar representing the aggregate factors influencing all sectors, [X.sub.IND,t] is a (IND x 1) vector of industry-group factors where IND represents the number of industry-groups, [[Alpha].sub.N] and [[Alpha].sub.IND] are (I x 1) and (I x IND) coefficient matrices, respectively, and [u.sub.t] is the vector of idiosyncratic influences on industry-output change. The feedback coefficient matrix is represented by [[Pi].sub.k]. In equation (4) the dynamics of real economic fluctuations can be thought of as arising from the interaction between the internal propagation mechanisms as captured by the feedback coefficients and the influence of the aggregate and disaggregate factors.(3)

Most linear business cycle theories can be easily accommodated by the general framework represented by equation (4) by imposing restrictions on the propagation mechanism and the type of shocks allowed. To illustrate this point, we present a relatively general business cycle model that is easily extended or modified. We assume output in a particular industry is determined by a constant returns technology with capital and labor as inputs and represent this by (5) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the capital stock at the beginning of period t in industry i; [h.sub.i,t] is an index of the capital utilization rate in period t in industry i; [l.sub.i,t] is the labor input in period t in industry i; [m.sub.i,t] is an index of the labor utilization rate in period t in industry i; and [[Epsilon].sub.it] represents various impulses to industry i. When we aggregate over all industries, the output equation in vector form can be written as (5') [Mathematical Expression Omitted] where [Y.sub.t], [Mathematical Expression Omitted], [h.sub.t], [l.sub.t] and [m.sub.t] are vectors comprised of the appropriate industry variables, and [Theta] and (I - [Theta]) are coefficient matrices.

The evolution of the capital stock in an industry depends on the capital available in period t for that industry, the industry-specific depreciation rate of that stock, and the industry's gross investment. Hence, the capital stock evolution equation in vector notation is (6) [Mathematical Expression Omitted] where [Z.sub.t] is a (I x 1) vector of industry gross investment and D [approximately equal to] (I-[Delta]) is a diagonal coefficient matrix comprised of one minus the industry-specific depreciation rate. Gross investment is assumed to depend on output conditions in all industries as represented by the vector of outputs, and is specified as (7) [Z.sub.t] = [AY.sub.t].

Labor input used in industry i is assumed to depend on the labor input employed in that industry in the previous period, as well as the vector of output conditions. In vector notation, labor input is specified as (8) [Mathematical Expression Omitted] with [Mathematical Expression Omitted] being a diagonal matrix comprised of industry-specific adjustment coefficients, and [Mathematical Expression Omitted] is a coefficient matrix. From an industry-specific perspective, this equation can be viewed as a decision rule from a linear quadratic optimization problem with adjustment costs.

In order to close the model, structural equations for the indices of the capital utilization rate and the labor utilization rate are needed. We assume that the vector of capital utilization indices depends directly on how close the vector of output is to the vector of the natural rate of output of the industries. Hence, (9) [Mathematical Expression Omitted]. The vector of labor utilization indices is determined in a similar manner.

That is, (10) [Mathematical Expression Omitted]. In equations (9) and (10), the coefficient matrices are both diagonal matrices comprised of industry-specific coefficients. The vector that is comprised of the various industries' natural rate of output is defined as a function of the vector of previous output levels. (11) [Mathematical Expression Omitted]

If equations (6) through (11) are substituted into equation (5) and the resulting equation manipulated, the vector of industry-outputs can be written as (12) [Mathematical Expression Omitted]. This equation clearly indicates that the model specified by equations (5) through (11) can be expressed as a restricted version of equation (4). The primary restrictions are imposed on the [[Pi].sub.k] matrix, which captures the propagation effect. Equations (5) through (11) can be easily modified so as to incorporate other linear business cycle models. The result of these modifications is to change the propagation mechanism and/or the composition of [Epsilon.sub.t]. Hence, alternative business cycle models could be tested by examining restrictions on the feedback terms. In this paper we do not perform such tests, as the focus is on the impulse term, [[Epsilon].sub.t].

Unfortunately, the model represented by equation (4) cannot be estimated because it is over-parameterized. Such problems frequently occur in multivariate time series analysis. Consequently, we impose restrictions to reduce the number of parameters to be estimated. Our approach is to analyze three different sets of restrictions on [[Pi].sub.k], so that the robustness of our findings on the relative importance of the different types of shocks can be established.

We impose a set of restrictions that allows us to retain as much of the effect of lagged industry-output changes as possible. In order to allow some national feedback effects and industry-group feedback effects, we create composite variables to represent such effects. These composite variables are similar to those used in Norrbin and Schlagenhauf [1988]. A national feedback effect is constructed as a weighted average of industry-output change. The composite national output variable is (13) [Mathematical Expression Omitted] where the weights are the relative importance of each industry in the aggregate industrial production index (base 1967), but rescaled for the manufacturing and mining industries included in our sample.

A specific industry can be influenced by output changes in its industry-group. To model this, we create IND industry-group composite variables defined as [Y.sub.ind,t]. Each one of these variables represents a weighted average of the change in output of each industry in the industry-group. The weight, [Mathematical Expression Omitted], is the fraction of output in industry-group ind accounted for by industry i. That is, (14) [Mathematical Expression Omitted] where G is the number of industries in a given industry-group.

Given these restrictions and allowing for an own-lag feedback effect, the change in output in matrix form is specified as (15) [Mathematical Expression Omitted] where [[Beta].sub.1], [[Beta].sub.2k] are (I x 1) vectors, [[Beta].sub.3k] is an (I x IND) matrix and [[Beta].sub.4k] is an (I x I) diagonal matrix.

An alternative approach to restricting the feedback matrices is to use principal component analysis to limit the cross-dependencies between output changes in equation (4). Principal component analysis reduces the dimension of the data matrix for the I industry-output variables by determining the degree of independence that exists in the information provided by each industry-output variable. If we let [P.sub.m], m = 1,...,M represent the M principal components, the change in output in matrix form is specified as (16) [Mathematical Expression Omitted] where [P.sub.t-k] is a M x 1 vector of principal components, and [Mathematical Expression Omitted] is an (I x M) coefficient matrix.(4)

The third approach to restricting the industry feedback coefficient matrices follows Long and Plosser [1983]. They present a model in which the feedback coefficients are restricted to be industry cost shares from the 1967 input-output table for the United States. We follow their lead by setting the feedback coefficients equal to the input requirements from other industries and the own industry. This requires the construction of an input-output table appropriate for our disaggregation scheme. The industry-output model in matrix form for this restriction approach is (17) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is an (I x I) coefficient matrix restricted to equal the modified input-output coefficients.

III. ECONOMETRIC METHODOLOGY

The estimation of equations (15), (16), and (17) is complicated by the fact that the factors appearing in each equation are unobservable. To resolve this problem, we estimate the equations using the dynamic multiple indicator multiple cause (DYMIMIC) framework of Watson and Engle [1983].(5) In this section, we briefly describe this approach, which, in addition to providing maximum likelihood estimates of the parameters and the unobserved variables, also allows empirical investigation of observable variables that are associated with the unobserved factors. This latter feature is important as it allows the economist to identify the cause of these factors.

The DYMIMIC model, expressed in state-space form, can be written as (18) [Mathematical Expression Omitted] (19) [X.sub.t] = [Delta](L)[X.sub.t-1] + [Gamma](L)[Z2.sub.t] + [e.sub.t] and (20) [Mathematical Expression Omitted] where [Y.sub.t] is a p x 1 vector of observed variables; [X.sub.t] is an n x 1 vector of unobserved variables or factors; [Z1.sub.t] and [Z2.sub.t] are s x 1 and q x 1 vectors, respectively, of observable exogenous and lagged dependent variables; [v.sub.t] and [e.sub.t] are vectors of disturbances; and L is the lag operator. The specification of the Z1 vector depends on the restriction approach employed, while the Z2 vector contains the causal variables that aid in the interpretation of the factors.

This state-space model is estimated using general maximum likelihood techniques along with the Kalman filter recursive algorithm. Since missing observations are present, we employ the EM algorithm (see Dempster, Laird, and Rubin [1977]) to maximize the likelihood function. The algorithm iterates between an estimation stage and maximization stage until convergence is achieved. Based on an initial guess of the parameters, the E-step of the algorithm employs the Kalman 'smoother' to construct estimates of the missing observations conditional on the observed data and parameters. Once the unobserved state variables are estimated, the M-step calculates the estimates of all unknown parameters conditional on the entire data set. These parameter estimates are then used in conjunction with the Kalman filter to generate a new estimate of the state variables. This interactive process is continued until an overall convergence is achieved. It has been shown that this algorithm will always increase the value of the likelihood function until it converges to a local maximum. The convergence is guaranteed under suitable regularity conditions.(6) A more detailed explanation of this algorithm is available from the authors upon request.

IV. EMPIRICAL RESULTS

Before examining our results, we discuss the data base and the specification of the factor equations. Because of the large number of coefficients estimated in each of the time series models, a measure of the relative importance of the national and industry-group-specific factors in generating output fluctuations is constructed. After discussing these results, we identify the observable economic variables that are correlated with each of these factors.

Data

Disaggregated output is measured by seasonally adjusted disaggregate industrial production. The Federal Reserve Board reports monthly industrial output at the three, and in some cases, four-digit Standard Industrial Classification level for manufacturing and mining industries. This data, as reported by Data Resources, Inc., is used to measure quarterly industry-output, where quarterly output is defined as the average over the appropriate monthly industrial production indices. The sample covers the period 1955 to 1984.

In order to measure industry-specific factors, the various industrial output indices are separated into industry-groups. These industry-groups normally coincide with the various two-digit SIC classes for manufacturing and mining. This method of segmentation results in the creation of fourteen industry-groups. These groups are primary metals and manufacturing; transportation equipment; machines, instruments and miscellaneous manufacturing; lumber and furniture; stone, glass, and glass products; textile products; leather products; food and kindred products; tobacco manufactures; paper and allied product; printing and publishing; chemicals and refining; plastics; and mining. Each industry-group is composed of at least two industries. A total of forty-nine industries are aggregated into the fourteen industry-groups.(7)

The quarterly output data are nonstationary. Many empirical studies of the business cycle deal with this problem by using a time trend to capture a (long-term) growth component. Nelson and Plosser [1982] suggest this procedure is likely to confound the growth component and cyclical component in the series, overstating the magnitude and duration of the cyclical component and understating the importance of the growth component. Therefore, we first difference the log of output. This renders the series stationary without removing the growth component, which may be stochastic.

The international variable must also be specified. Output in a particular industry can be affected by international developments such as a change in world relative demand or a change in comparative advantage. To allow for a potential international influence, we created a trade-weighted average of industrial production for the United State's largest trading partners.(8)

Specification of the Unobserved Variable

According to the state-space formulation, each factor can be expressed as a function of previous factors and observable causal variables. For the aggregate or national factors, we allow the previous period (own) national factor, [X.sub.N,t-1], to be a determinant. If the aggregate factor is interpreted as an unobserved impulse that is national in origin, there may be some question on the appropriateness of specifying a serially correlated impulse in an environment where agents have rational expectations. We determine the issue empirically and find the autoregressive terms are not statistically significant when the composite variable restriction approach is employed. However, the autoregressive terms are significant when the other restriction approaches are employed. In an attempt to give an economic interpretation to the behavior of the aggregate factor, we allow for a measure of monetary and fiscal policy, as well as an aggregate real supply variable, oil prices, to enter the Z2 vector. The monetary policy variable is defined as the rate of change of real M1, while fiscal policy variable is defined as the rate of change of the high employment real surplus. The current value and four lags of each variable are included in this exogenous variable vector.(9)

Industry-group factors are specified to follow a simple autoregressive process with causal variables. Prescott [1986] has specified a serially correlated technological impulse in his work on real business cycles, and Schumpeter's [1939] work on business cycles implies that technological bursts should be autocorrelated. The specification of the causal variables for these factors is straightforward in principle. Theory suggests that taste and technological changes should account for the movement in an industry factor. Most empirical work on the real business cycle model uses 'Solow residuals' to measure technological change. We follow this approach and create industry-group Solow residuals.(10,11)

The use of Solow residuals to measure technological shocks has been questioned recently. For instance, McCallum [1989] notes that Solow's method assumes that current capital and labor are the only relevant inputs. If adjustment costs exist, then labor and/or capital hoarding might cause the estimated Solow residuals to overstate the technological shock variance. We have attempted to deal with this criticism of the use of Solow residuals that could result in these variables reflecting aggregate developments rather than disaggregate developments by filtering the industry-group measures of technological change for the state of the business cycle. This was done by regressing the Solow residual on a constant, a time trend, a time trend squared and on the deviation of U.S. output from the natural rate of output. This should eliminate aggregate labor and capital hoarding, and the residuals would be only influenced to the extent to which different adjustment costs exist between industry-groups.(12)

Empirical Results

Before any estimation can be conducted, the specification of the length of the lag of the output feedback terms has to be considered. As mentioned above, restrictions have to be placed on these feedback terms in order to estimate the model. When the input-output coefficients are employed as restrictions, it is obvious that the lag is one period. When the composite variable approach is used as a restriction approach, each composite variable is allowed a lag of four quarters. The same lag length is employed for each principal component when the feedback terms are restricted using principal components. The estimation period covers the period 1954:1 through 1984:4.

An obvious first step in the analysis is to identify whether there is any statistical basis for the introduction of disaggregate factors in explaining industry-specific output change. If only aggregate sources are found to be significant, this would support a single index explanation of the business cycle. In order to determine whether the national factor and the set of industry-group factors are statistically significant, we conducted a series of likelihood ratio tests. The results from these hypothesis tests for each method of restricting the output feedback coefficient matrix are presented in Table I.(13) As can be seen, the null hypothesis for each test is strongly rejected for each approach to restrict the feedback coefficient matrix. This indicates that both the national factor and the set of industry-group factors play a role in triggering economic fluctuations.

Once the statistical importance of the various unobserved variables is established, it is important to determine the relative importance of each type of factor in generating the business cycle. Because of the large number of coefficients estimated in each model, we summarize the results to emphasize the relative importance of the various factors by deriving the moving average representation of the state-space model. From this representation, the decomposition of the forecast error of output can be generated for various horizons. These measures can be used to evaluate the relative importance of the various factors.

The general state-space model is expressed in equations (18)-(20). The actual exogenous and predetermined variables that appear in the [Z1.sub.t] vector depend on the restriction approach employed.(14) We partition the [Z1.sub.t] vector into exogenous and predetermined components. If we denote the exogenous variables as [Mathematical Expression Omitted] and the corresponding coefficients as [[Lambda].sub.1](L), equation (18) can be expressed as (18') [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. When both sides of (18') are premultiplied by [Phi][(L).sup.-1] the statespace model can be reformulated as (21) [Mathematical Expression Omitted] (22) [X.sub.t] = [Delta](L)[X.sub.t-1] + [Gamma](L)[Z2.sub.t] + [e.sub.t].

By substituting (22) into (21), the moving average representation of the statespace model is (23) [Mathematical Expression Omitted] This moving average representation is conditioned on current and past [Mathematical Expression Omitted] and [Z2.sub.t]. If generating equations are postulated for [Mathematical Expression Omitted] and [Z2.sub.t], an unconditional moving average representation can be derived. We postulate univariate autogressive forecasting equations for each element in [Mathematical Expression Omitted] and [Z2.sub.t]. That is, (24) [Mathematical Expression Omitted] (25) B(L)[Z2.sub.t] = [[Epsilon].sub.2t] where A(L) and B(L) are diagonal matrices.(15)

Under the assumption that each exogenous variable can be modelled as a second-order autoregressive process, the moving average representations of (24) and (25) are (26) [Mathematical Expression Omitted] (27) [Mathematical Expression Omitted] where [q.sub.A1] = [A.sub.1] [q.sub.A2] = [q.sub.A1][A.sub.1] + [A.sub.2] [q.sub.Aj] = [q.sub.Aj-1][A.sub.1] + [q.sub.Aj-2][A.sub.2] for j [is greater than or equal to] 3 [q.sub.B1] = [B.sub.1] [q.sub.B2] = [q.sub.B1][B.sub.1] + [B.sub.2] [q.sub.Bj] = [q.sub.Bj-1][B.sub.1] + [q.sub.Bj-2][B.sub.2] for j [is greater than or equal to] 3. Combining (23) through (27) yields the unconditional moving average representation (28) [Mathematical Expression Omitted] where the matrices are [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] [Mathematical Expression Omitted] where j=0, 1, 2,...,j and [[Theta].sub.0] = 1 [Mathematical Expression Omitted] where [q.sub.A0] = 1 [Mathematical Expression Omitted] where [q.sub.B0] = 1 [Mathematical Expression Omitted] where k = 0, 1, 2,...,k and [[Theta].sub.0] = 1.

Once the moving average representation is generated, a meaningful decomposition of the variance of output can be derived by calculating the variance of the forecast error at various horizons. If we let [Mathematical Expression Omitted] then the j-period ahead forecast error of output has variance (29) [Mathematical Expression Omitted] [Mathematical Expression Omitted] for j = 1 [Mathematical Expression Omitted] [Mathematical Expression Omitted] for j = 2, 3,...

This equation indicates that an individual industry's variance can be segmented into an industry-specific or idiosyncratic component contained in Q; an error arising from the factor of the specific industry-group and the common aggregate factor which are contained in R; an error from forecasting the exogenous (Z1) variables, Var([[Epsilon].sub.1]); and the error from forecasting the observable (Z2) variables that enter the factor equations, Var([[Epsilon].sub.2]).

Because of the number of industries in our sample, further aggregation is required to simplify interpretation. We would like to determine how important each of the factors is from an economy-wide perspective. In order to address this question, the [Mathematical Expression Omitted] must be collapsed to an aggregate forecast error over all industries. We accomplished this by weighting the variance of the forecast error of each individual industry by the relative importance of that industry's output in total industrial output. The weight actually corresponds to each industry's weight in the base 1967 industrial production index, appropriately rescaled. If [W.sub.N] is a (I x 1) vector of these weights, then premultiplying both sides of (29) by [Mathematical Expression Omitted] and post-multiplying by [W.sub.N] reduces the variance matrix to a scalar. Dividing the contribution of each component of the forecast error of output by the aggregate forecast error will yield a measure of the relative importance of each component. Another question of interest concerns the relative importance of the factors from an industry-group perspective. This decomposition is calculated by creating a weighting matrix where the weights are the fraction of each industry's output in the particular industry-group output. If all fourteen industry-group weighting matrices are stacked columnwise, an industry-group weighting matrix, [W.sub.IND], can be constructed with dimension I x IND. By pre-multiplying equation (29) by the transpose of this weighting matrix and post-multiplying by this weighting matrix, the Var([Y.sub.t+j]) can be reduced to a fourteen by fourteen matrix that yields industry-group insights.

The results from these calculations are presented in Tables II and III for each feedback restriction approach. We first examine the percentage of forecast error variance explained by the various factors from a national perspective under the composite variable restriction approach. In order to facilitate the comparison between the aggregate and sectoral sources of forecast errors, we sum the contribution of the error term in the factor equation with the contribution of its causal variables. As can be seen in Table II, the aggregate factor accounts for 52.14 percent of the one-period-ahead variance of the forecast error while the industry-group factors account for 34.30 percent of the variance. The international influence explains 9.11 percent of the one-period-ahead forecast variance. As the forecast horizon increases, the aggregate factor diminishes somewhat in importance, and the international influence and the industry-specific residual increase in importance. For example, in the steady state, defined as twenty periods ahead, the aggregate factor accounts for 40.32 of aggregate variance of the forecast error. The fraction of the variance of the forecast error explained by the industry-group factors increases slightly to 35.09 percent. These results indicate that the national common factor is the most important factor.

Table III presents the decomposition, evaluated at the steady state, from an industry-group perspective. The conclusion concerning the relative ranking of the national and industry-group factors changes when an industry-group perspective is employed. For many industry-groups, the industry-group factor dominates the national factor in explaining industry-group output variation. Given the results presented in the national perspective summary, the industry-group results may seem contradictory. The explanation for the apparent contradiction is in the estimation methodology. In estimating the individual industry-output equations, the industry-group factors were assumed to be orthogonal to each other. This assumption tends to give conservative estimates of the importance of the disaggregate industry-groups' factors when creating a national summary perspective. However, in constructing an industry-group summary, the industry-group factor explains output variation common to a group of similar industries, thus allowing a potentially larger role for the industry factor within the industry-group. The percentage of variation explained by the national and industry-group factors do vary across industry-groups. The national factor is especially important for machines, refining, textiles, transportation, lumber and plastics while being relatively unimportant for tobacco, mining, and printing. In addition, as would be expected, the industry-group factor and/or industry-specific residual seem to play a more important role in these industries where the aggregate factor is not as important. For example, mining, metals, and food industries are often thought of as industries not dominated by aggregate developments. Our results suggest that is the case: over 69.6 percent of the variance of the forecast error of output in the steady state is explained by the industry-group factor in metals while 67 percent of the variance is accounted for by the industry-specific residual in mining. For the food industry, the industry-group factor and industry-specific residual account for approximately 75 percent of the steady state variance of the forecast error.

In order to examine the robustness of these results, similar decompositions are presented for the principal component restriction approach and the input-output restriction approach. As can be seen in Table II, the general conclusions drawn from the composite variable restriction approach hold for the other approaches. If the principal component restriction approach is employed, the aggregate factor and industry-group factors account for 54.54 percent and 33.40 percent of the one-period-ahead variance of the forecast error of output from a national perspective. At the steady state, the variance of the forecast error of output explained by the aggregate factor slightly falls to 52.89 percent and the industry-group factor declines to 32.22 percent. If the feedback coefficients are restricted by input-output coefficients, we continue to find that the national factor is the most important, while industry-group factors explain a substantial amount of the variance of the forecast error. For instance, in the steady state the aggregate factor and the industry-group factors account for 53.99 percent and 37.41 percent, respectively, of the variance of the forecast error of output. The one major difference is the importance of the international variable. The decreased importance of the international variable when an input-output restriction approach is employed suggests that this restriction confuses an influence from this variable with the propagation mechanism. Based on the findings presented in Table II, Burns and Mitchell's [1946] view of a single aggregate shock source of business cycles seems inappropriate given the prominent role that disaggregate or industry-group factors play. Furthermore, the importance of the industry-group factors does seem to indicate an important role for some forms of the real business cycle model.(16)

An analysis of the factor equations yields some insights into the explanation of the movements in these factors. Since the dimension of the empirical problem makes it impossible to calculate standard errors, we conducted statistical tests using likelihood ratio procedures. The relevant test statistics are presented in Table IV. We start by considering the national factor equation. In this equation, we introduce the current and previous four periods of a fiscal policy variable, a monetary policy variable and a real shock variable as potential observable variables that could be correlated with the national factor. Our fiscal policy variable, measured as the real value of the high employment surplus, is not statistically significant. The current change in the real value of M1 and the four lags of this variable represent monetary policy.(17) The concurrent monetary variable has the correct sign and is statistically significant. More important, when the monetary factors are considered as a group, the null hypothesis that they are statistically different from zero can not be rejected. The other causal variable that enters the national factor equation is the change in oil prices. Theoretically, we would expect that oil price increases would be negatively correlated with the aggregate factor. The estimated equation indicates that this is indeed the case. However, the likelihood ratio test indicates that this variable is not significant. Thus, the only observable variable that is correlated with the national factor is monetary policy changes.

Disaggregate real business cycle theories suggest that the industry-group factors should be correlated with relative price, taste, or technological changes among industries. Since most research in this area seems to focus on technological change as the driving variable, we construct a set of technological change measures for each industry-group. As discussed above, we constructed a set of industry-group 'Solow residuals' as a proxy for industry-group technological change. Table IV shows that we can reject the null hypothesis that as a group these variables are not statistically different from zero. This is an important empirical result as the finding of correlation between industry-group factors and industry-group technological shocks provides important evidence in support of the disaggregate real business cycle paradigm.

V. CONCLUSION

In recent years macroeconomists have renewed their interest in the area of business cycles. Research by Prescott and others suggests that cycles may be due to aggregate real technological shocks rather than to monetary shocks, which many economists had viewed as the causal factor. Long and Plosser [1983; 1987] have extended the real business cycle paradigm so that business cycles can result from disaggregate real impulses. This paper attempts to measure the relative contribution of aggregate and sectoral factors in generating real economic fluctuations. Furthermore, it attempts to identify the economic variables that are correlated with these factors. We measure the relative importance of the various factors in the context of a multivariate time series model of output change which allows for feedback influences, as well as an aggregate factor, industry-group factors, and an international effect.

There are three major findings. First, both the aggregate and industry-group factors are statistically significant in explaining variations in output. This suggests that the Burns and Mitchell [1946] view of a single aggregate shock as the source of business cycles is incomplete. Second, the aggregate factor accounts for largest amount of the forecast error variance of aggregate output. However, the industry-group factors explain a significant amount of the forecast error variance of output so as to give empirical context to the disaggregate form of the real business cycle model. Third, the aggregate factors are correlated with monetary variables. These findings are robust to various approaches to restrict the feedback effects. Overall, the results in this paper suggest that a complete model of the business cycle must allow for both aggregate and sectoral sources of macroeconomic fluctuations. [Tabular Data 1 to 4 Omitted] [Appendix Omitted]

(1)Haberler [1958] summarizes the business cycle theories in the pre-Keynesian period. More recently, Zarnowitz [1985] presents an excellent review of the theories and evidence on business cycles. (2)Long and Plosser [1987] examine several similar issues, but they made no attempt to determine the fraction of the unexplained variance due to sectoral shocks or other shocks. In this paper we a more general estimation strategy, dynamic factor analysis, to derive estimates of the relative importance of shocks to output. Our approach estimates the unobserved factors, thus allowing us to investigate the relationships between the factors and observable economic variables suggested by theory. (3)The appearance of the [INT.sub.t] variable requires additional comment as to why this variable is not treated as an unobserved variable or factor. If international influences are treated as a factor, it would be an effect common to each equation. With an aggregate factor already specified, a statistical identification problem arises similar to the rotation problem in simple factor analysis. Since the aggregate factor is thought of as domestic in origin, so as to be consistent with the single index model, we introduce an observable [INT.sub.t] variable to allow for international influences and avoid identification problems. In results not reported in the paper we estimate the model when [INT.sub.t] is deleted from the output equation and introduced as a regressor in the aggregate factor equation. The conclusions concerning the relative importance of the various factors remain unchanged. These results are available upon request from the authors. (4)The first two principle components are used in the estimation process. They account for approximately 65 percent of the variation in the [Y.sub.t] vector. (5)The DYMIMIC model is a type of index model. Sargent and Sims [1980] have used an index model to study business cycles. They employ a frequency domain method with unrestricted lag distributions. The Watson and Engle [1983] estimation approach is a time domain method. (6)We attempt to avoid local optimization results by examining various starting values and employing a very severe convergence criteria (i.e., .0000001). (7)Which industries are included in a specific industry-group is explained in a data appendix available from the authors. The construction of any data is also discussed in this appendix. (8)In order to capture the world economic situation, we created a world industrial production variable. This variable is based on a weighted average of ten countries' industrial production indices. The ten countries are the major trading partners and include Germany, Japan, France, the United Kingdom, Canada, Italy, Netherlands, Belgium, Sweden, and Switzerland. The weights are calculated as the fraction of each country's imports plus exports with the United States as a fraction of total imports and exports for these countries. (9)The money supply is defined as the seasonally adjusted value of M1 and is from the CITIBANK data tape (series FM1). For data prior to 1959, the series came from the Handbook of Cyclical Indicators, series 85, U.S. Department of Commerce, Bureau of Economic Analysis, 1984. The high employment deficit variable includes adjustments for automatic inflation effects. These data are from the article by Thomas M. Holloway entitled "The Cyclically Adjusted Federal Budget and Federal Debt: Revised and Updated Estimates," Survey of Current Business, March 1986. We follow Hamilton [1983] in constructing the price of oil. (10)Industry-group productivity shocks, or Solow residuals, can be defined as: %[Delta][A.sub.ind] = %[Delta][Q.sub.ind] - [[r.sub.ind][K.sub.ind])/([P.sub.ind][Q.sub.ind])] (%[Delta][K.sub.ind]) + [([W.sub.ind][L.sub.ind])/([P.sub.ind][Q.sub.ind])] (%[Delta][L.sub.ind]) where A is technology change, Q is output, r is the rental rate of capital, w is the wage rate, K is net capital stock, L is labor input, and ind refers to an specific industry-group. In a data appendix, we detail how we calculated these variables. (11)We also introduce the oil price as an explanatory variable in each of the industry-group factor equations. While this variable is significant in a few industry-group equations, conclusions on the relative importance of the factors and the statistical significance of Solow residuals are unaffected. (12)An alternative approach to dealing with the criticism of the Solow residuals as measures of technological change is to delete these variables from the various industry-group factor equations. In this case, the industry-group factors would be determined by a first-order autoregressive process. This specification would be similar to the one employed by Prescott [1986]. We estimated the various models where the Solow residuals were deleted from the industry-group factor equations and found that our results are not especially sensitive to the use of these Solow residuals. (13)A comment is required concerning the specification of the restricted model employed in the testing procedure. Usually the importance of a variable or a set of variables is tested by deleting such variables from the unrestricted model. In terms of the DYMIMIC model, deletion of a factor or set of factors in the indicator equations also requires that each coefficient in the transition equation be restricted for the factor or set of the factors being examined. This accounts for the large number of restrictions associated with each likelihood ratio statistic. (14)If the principal components approach is employed, the [Z1.sub.t] vector would be comprised of a constant, the international influence variables, [INT.sub.t], and the principal components that enter each output equation. On the other hand, with the composite variable restriction approach this vector is comprised of a constant, the international variable, lagged composite national output variables, [Y.sub.N,t-k], lagged composite industry-group variables, [Y.sub.IND,t-k] as well as lagged own predetermined endogenous variables, [Y.sub.t-k]. The composite variables are predetermined variables as they can be expressed as a fixed coefficient matrix times the lagged weighted predetermined variable. (15)The univariate forecasting equations were estimated with a constant. (16)All of the decompositions that have been considered are for the entire sample period. Given the change in international monetary regimes that occurred in the early 1970s, it is important to determine whether the relative importance of the factors is sensitive to this potential structural change. The appendix table presents the decomposition of variance of the forecast error from a national perspective using the composite variable restriction approach for the periods 1956 to 1971 and 1971 to 1984. Because the sample size is much smaller in these subsamples, while the number of parameters to be estimated remains unchanged, care must be taken in analyzing the results. As can be seen, the aggregate factor continues to be the most important. The industry-group factors also account for a substantial amount of the variance of the forecast error of output. In fact, in the 1971-1984 subsample, the industry-group factor accounts for approximately the same percentage of the variance of output as does the aggregate factor in the steady state. The major difference in the two samples is the increased importance of the international variable in the latter. However, the size of the international influence and the industry-specific residual relative to the results presented in the table below does raise concern over problems caused by overparameterization. Additional work has failed to identify whether the time period dependencies of the variance decomposition are due to overparameterization problems or a problem with the construction of the international variable. (17)An extension would be to decompose the demand management variables into expected and unexpected components. Testing variables in this form would be interesting given recent empirical work on testing some business cycle theories. In order to gain some insights on this issue, we replaced [DM.sub.t] with a money shock variable composed of the error term where [DM.sub.t] was regressed on four lags of itself, the high employment surplus, and the three-month treasury bill rate. The money surprise variables as a group are insignificant. However, the current unanticipated monetary variable by itself is marginally significant.

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STEFAN C. NORRBIN and DON E. SCHLAGENHAUF, Florida State University and Arizona State University, respectively. We would like to thank Art Blakemore, Joe Brada, and Richard Sweeney for useful comments and Rosalie Bondad for her research assistance. The comments of the referees are gratefully acknowledged. This research has been supported by grants from the Dean's Council of 100 at Arizona State University, the Research School of Social Sciences at Australia National University, and the Center for Research in Financial Systems at Arizona State University.

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Author: | Norrbin, Stefan C.; Schlagenhauf, Don E. |
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Publication: | Economic Inquiry |

Date: | Apr 1, 1991 |

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