An Anatomy of the Business Cycle

repec.nep.mac

n?u=RePEc:red:sed016:1641&r=mac

An Anatomy of the Business Cycle

George-Marios Angeletos

MIT

Fabrice Collard

University of Bern

Harris Dellas

University of Bern

[extremely preliminary and incomplete; do not quote or distribute]

Abstract

We develop a new method for dissecting the comovement of macroeconomic variables over the

business cycle. We use this to show that the data is consistent with models in which the forces

(i.e., shocks and propagation mechanisms) that drive the fluctuations in output, investment,

hours, and unemployment are strongly connected with one another, while also being relatively

disconnected from those that drive the fluctuations in productivity, inflation, and interest rates.

We document a similar disconnect between inflation and the labor share. We explain why these

findings are at odds with existing macroeconomic models of either the RBC or the NK variety,

and discuss how they provide guidance for future theoretical research.


1 Introduction

We develop a new method that can provide a new anatomy of the business cycle comovements

of key macroeconomic variables. We then show how this method can guide the development of

macroeconomic theory by discriminating among different structural interpretations of the data.

Our method consists of three steps. First, we run a VAR (or a VECM) on nine macroeconomic

variables: GDP, unemployment, consumption, investment, hours, labor productivity, the labor

share, inflation, and the federal funds rate. 1 Write this VAR as

X = A(L)X + u

where X is the vector of the aforementioned variables and u are the residuals. Next, for any

variable of interest x, we construct an “x-factor” as the linear combination of the VAR residuals

u that accounts for most of the variation in variable x at business-cycle frequencies (between 6

and 32 quarters). Finally, we characterize the patterns of comovement in the data by deriving the

estimated effects of each of the identified factors on all the variables of interest in terms of IRFs,

conditional variances, and conditional correlations.

Before we explain the rational for this method, let us first note that it can be thought of as

a variant of principle components or dynamic factor analysis: the key differences are that (i) our

constructed “factors” need not be orthogonal to one another and (ii) each one of them focuses

exclusively on spanning as much of the volatility of one particular variable at certain frequencies

as possible (as opposed to standard factors, which seek to strike a balance across all variables and

all frequencies). For instance, the GDP-factor turns out to be highly correlated with the “hoursand

unemployment-factor, but not very much so with the inflation-factor. We will use the term

business cycle factor to refer to any of the output, investment, employment and unemployment

factors.

Our method is also related to the Structural VAR literature in the following regard: in the

jargon of that literature, our identified factors is a particular kind of “structural shock”, simply

because any such shock in that literature is defined as a linear combination of the VAR residuals.

The differences here are (i) that the criterion used to identify the factor is the maximization of the

volatility of a particular variable at particular frequencies, as opposed to, say, a zero long- or shortrun

restriction and (ii) the identified shock in the data may or may not have a direct theoretical

counterpart within a model. This last observation also explains why we opt to refer to our empirical

constructs as “factors” rather than as “shocks”; we reserve the latter term for theoretical objects

as contrasted to empirically constructed objects.

We find this methodology useful for two reasons. The first emanates from a desire to achieve for

parsimony. Suppose a theorist aspires to develop a model in which a single shock, or mechanism,

explains most of the business cycle in the data; think, for example, either of the baseline RBC

model, in which technology is assumed to be the main driver of business cycles, or of the traditional

1 GPD is total GDP; hours, labor productivity, and the labor share are in the non-farm business sector.

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Keynesian view according to which “aggregate demand” represents the main driver. Our GDP-,

hours-, and investment-factors may the help detect key empirical properties that a “useful” model

ought to possess in terms of conditional co-movements for the variables of interest. From this

perspective, one can think of our method as helping recover the “main” or “primary” source of the

business cycle in the data.

The second justification is it provides an instructive anatomy of commovement. By inspecting

the IRFs and variance contributions of any given factor in the data, as well as by varying the

targets in the identification of the factor, we learn something about the conditional comovement

across variables as well as across frequencies in the data. By replicating the methodology on artificial

data generated by any model, whether of the parsimonious type or the medium-scale DSGE type,

we similarly learn something about the corresponding comovement patterns implied by the model.

By comparing the former with the latter, we can assess the usefulness of any given model in offering

a successful structural interpretation of the observed business-cycle phenomena.

A partial summary of the main findings, which also helps illuminate the motives for undertaking

these exercises, follows.

We find that for output, employment, investment and unemployment, the factor that accounts

for the bulk of fluctuations in any one variable also accounts for the bulk of fluctuations in any other;

more precisely, it accounts for more than half of the business cycle variability in these variables.

Moreover, it gives rise to a realistic covariance pattern, with output, consumption, employment,

unemployment and investment moving in tandem.

Having established that this factor accounts for the bulk of business cycle fluctuations we make

further use of it by investigating its effects on the other variables contained in the VAR; and also

its contribution to long term aggregate fluctuations. Finally, we construct the factors that account

for the maximum business cycle volatility in the remaining VAR variables (interest rates, wages

etc).

Turning to the effects of the output based business cycle factor to long term fluctuations (80

+ quarters) we find that they are quite limited. 2 .. Consequently, viewing our results through a

Blanchard and Quah, 1989 or Gali, 1996, type of lenses that emphasizes a distinction between

shocks with transitory and permanent effects one could claim that they support the notion that

the business cycle and the long term are driven by completely distinct forces.

There is also valuable information contained in the patterns induced by the business cycle factor

on the remaining variables in the VAR. We find that it moves the nominal interest rate procyclically,

a fact that suggests, at most a weak association with monetary policy shocks. It also has a delayed,

procyclical effect on the real wage and inflation. Models that rely on aggregate demand variation

as the source of business cycle fluctuations in a world of real wage and nominal price rigidity are

capable of replicating these patterns. But while this seems encouraging for the New Keynesian

model, our empirical strategy uncovers additional patterns that pose a challenge to this model. In

2 The reverse is also true. The factor that accounts the most for the volatility of output over frequencies corresponding

to 80 plus quarters contributes very little to the volatility of output over business cycle frequencies

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particular, we find that the business cycle factor explains very little of the business cycle volatility

in the real wage rate and inflation. Relatedly, the factors that account for the bulk of fluctuations

in these variables only make minor contributions to the business cycle. These two findings taken

together suggest that a Phillips curve type of a relationship involving an inflation-unemployment

trade off may not play a central role in macroeconomic fluctuations.

The preceding discussion has interpreted our empirical findings under the lenses of parsimonious

models where the bulk of the business cycle is driven by a single shock or mechanism. In the second

part of the paper, we illustrate how our empirical methodology can be useful even if one has in

mind richer DSGE models in which business cycles are the combination of multiple forces, such as

Smets and Wouters.

As anticipated before, we do so by replicating our empirical methodology on artificial data

generated by the model and by comparing the results of this exercise with those from the data.

When we do so, we find that the model does a fairly good job in replicating the co-movement of real

quantities in terms of our identified GDP-, hours- and investment factors, but predicts the opposite

movements in inflation and interest rate than those found in the data. Furthermore, the model fails

to capture the relevant conditional co-movements between inflation (or the nominal interest rate)

and the labor share. Because the latter is consider a key empirical proxy for marginal costs in the

context of the NKPC, and because the co-movement of real activity and inflation more generally

is at the core of the Keynesian mechanism, both of the documented failures appear to call into

question the usefulness of this kind of models.

The rest of the paper is organized as follows. Section 1 contains the empirical analysis, along

with certain lessons for theory. Section 2 uses our approach to evaluate the Smets and Wouters

model.

2 Empirical analysis

Our method represents a particular application of principal components–dynamic factor approach

(Sims and Sargent, 1977, Stock and Watson, 2005). As in this approach, we use a small number

of VAR-based shocks, or factors, to capture the variation in the data. In our version, and very

much like the approach pioneered by Uhlig (2003), we focus on the single factor that has the largest

contribution to the volatility of a particular variable(s) at particular frequencies.

More specifically, we run a VAR or a VECM on ten macroeconomic variables over the 1960-2007

period: GDP, consumption, hours, investment, , inflation, the federal funds rate, and .In the VAR

we use levels and four lags. In the VECM we impose the theory implied cointegration restrictions 3 ,

namely that .. We construct the sought-after composite shock (factor) by taking the linear combination

of the VAR residuals that maximizes the sum of the volatilities of some macroeconomic

variable (the target variable) over the frequencies corresponding to 6 to 32 quarters. 4 .

3 The VECM results do not differ when we estimate rather than impose the cointegrating relations.

4 See Appendix A for a description of the data, the estimation and the technical details of the construction of the

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Table 1 reports the contribution of the business cycle factor, that was constructed by targeting

the volatility of GDP, to the volatility of various macroeconomic variables in three different

ranges: 6-32 quarters (business cycle), 32-80 (medium term) and 80-inf (long term). Figure 1 reports

the corresponding impulse response functions (IRFs) for these variables in the business cycle

frequencies.

Table 1: Y t factor: Variance Decomposition

Y t C t I t u t h t s w t Y t/h t ξt w W t R t π t r t

VAR(4)

6-32 Quarters 76.52 29.04 66.80 50.13 53.83 43.68 29.47 37.70 11.14 23.23 8.68 14.24

32-80 Quarters 39.66 35.01 11.52 34.61 37.01 24.57 13.67 22.39 32.76 6.03 22.00 13.89

80-∞ Quarters 7.58 8.38 6.81 7.35 7.75 6.45 7.76 6.84 8.22 7.57 6.09 6.09

VECM (Theo. Rest.)

6-32 Quarters 62.42 44.33 57.92 55.29 52.79 39.02 33.04 45.13 17.20 36.08 23.78 26.82

32-80 Quarters 32.24 22.24 34.36 37.57 26.31 16.68 7.80 21.74 12.23 40.32 19.02 24.14

80-∞ Quarters 6.89 6.89 6.89 8.60 8.23 7.09 6.89 7.26 6.89 10.23 6.87 10.91

What are the main properties of the identified factor? First, it captures more than one-half of

the volatility of output, hours, unemployment and investment at business-cycle frequencies. It also

gives rise to a realistic business cycle, with the aforementioned variables as well as consumption

all moving in tandem. These findings seem to justify our use of the term “primary business-cycle

shock” for this factor.

Second, it explains less but still a sizable fraction of the volatility in consumption and labor

productivity but much less of inflation, the real wage and the real interest rate. It moves the

nominal, real interest rate as well as the wage rate and inflation procyclically but there is some

delay in the response of the last two variables, in particular that of inflation: inflation remains

flat early on and then starts rising, reaching its peak after the real variables have peaked. While

an increase in economic activity that is followed –with a lag– by an increase in inflation is a well

documented stylized fact in the literature, which has been often attributed to monetary policy

shocks, the procyclical response of the nominal interest rate with regard to this shock contradicts

this interpretation. Moreover, the fact that this shock explains very little of inflation fluctuations

and the shock that accounts for the bulk of inflation volatility does not matter for the business cycle

(see below) casts doubts that the Philips curve plays an essential role for understanding business

cycles.

Third, this factor matters considerably less in the long term. We will –tentatively– claim that

this property implies that what drives the business cycle is distinct from what drives the economy

in the long term. We provide further evidence for this below using composite shocks constructed

composite shock. We drop the post-2007 data because we wish to abstract from the financial phenomena that have

played a distinct role in the recent recession.

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Figure 1: Y t factor, IRFs

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Output

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Consumption

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VAR;

VECM; Shaded area: 68% HPDI.

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over alternative frequencies.

The identified main business cycle factor has similar properties independent of whether it is

derived by targeting the volatility of GDP, hours, investment or unemployment (see top rows of

Table 2 and 2). And also when targeting the covariance of output with consumption, investment,

hours worked and unemployment instead of their variances (see Table 4 and figure 3). But the

properties of the factor differ when it is constructed on the basis of some other variable, such as

inflation and the wage rate; see bottom rows of tables 2 and 3. As we hinted above, this may

prove problematic for theories that emphasize variation in inflation and/or the real wage rate as

key features of the business cycle.

Table 2: Variance Decomposition, 6–32 Quarters, Alternative Factors (VAR)

Y t C t I t u t h t s w t Y t/h t ξt w W t R t π t r t

Y t 76.52 29.04 66.80 50.13 53.83 43.68 29.47 37.70 11.14 23.23 8.68 14.24

I t 65.00 17.84 79.13 54.42 50.27 39.23 23.64 33.16 10.22 32.06 9.45 17.63

u t 53.53 21.58 59.50 70.07 47.73 29.26 16.28 33.83 7.41 35.31 10.94 20.68

h t 54.38 16.11 52.17 45.65 74.12 36.25 21.63 58.86 13.01 23.30 5.05 18.46

ξt w 40.53 18.99 36.89 36.25 67.63 27.81 23.46 62.21 16.92 17.35 4.62 16.82

C t 35.31 57.88 19.13 19.15 18.27 13.03 15.78 20.08 8.76 6.58 12.73 6.88

s w t 40.44 16.40 32.49 24.18 34.69 65.82 28.10 32.29 17.95 11.15 4.25 9.48

Y t/h t 22.99 16.00 14.85 7.55 6.90 13.94 68.50 15.04 29.29 4.10 6.52 2.43

W t 2.74 3.89 2.83 2.20 6.92 15.36 31.44 10.79 81.32 1.83 6.46 6.17

R t 28.24 26.31 32.07 37.49 25.94 10.84 9.54 21.88 4.21 81.17 21.16 49.94

π t 9.37 18.45 6.65 10.41 7.18 4.28 4.55 8.41 7.44 15.85 85.06 18.42

rr t 13.35 13.45 13.73 18.37 14.26 9.54 6.02 14.54 3.67 48.85 10.91 71.75

On the basis of the information presented in table 1 we have claimed that what drives the

business cycle is disconnected from what drives long term fluctuations. We subject this claim to

further scrutiny by computing the composite shock that targets output volatility in the range 80-inf

quarters (and also the one over medium term frequencies, 32 − 80 quarters) and checking what this

long term factor contributes much to the business cycle.

Table 5 indicates that the long term factor explains everything at the long term but very little

at the business cycle frequencies, thus confirming the claim made earlier that the short and the

long term are driven by distinct forces. Interestingly, the long term factor also accounts for little of

the short term volatility in productivity and consumption. The fact that the business cycle factor

accounts for more of the business cycle volatility in consumption than the long term factor probably

suggests a low degree of consumption smoothing.

In all other respects, the long term factor has plausible properties, see Figure 4. It is associated

with a jump in current labor productivity, which is followed by sustained, gradual, further productivity

gains. The wage rate and consumption follow a path similar to that of productivity. There

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Table 3: Variance Decomposition, 6–32 Quarters, Alternative Factors (VECM)

Y t C t I t u t h t s w t Y t/h t ξt w W t R t π t r t

Y t 62.42 44.33 57.92 55.29 52.79 39.02 33.04 45.13 17.20 36.08 23.78 26.82

I t 57.87 33.34 62.79 58.19 51.36 37.61 27.34 41.33 17.46 43.57 23.36 34.69

u t 53.97 34.22 56.72 64.47 56.20 35.74 22.92 48.88 12.74 60.26 29.76 48.58

h t 54.21 36.13 52.21 58.93 61.68 37.52 24.95 56.41 12.06 50.71 27.25 40.28

ξt w 47.90 40.54 44.41 52.97 58.53 34.81 27.22 59.07 11.98 49.68 31.25 36.82

C t 47.23 61.19 38.26 36.98 37.54 22.09 27.11 39.19 17.91 25.95 25.99 16.33

s w t 34.78 22.79 30.17 31.61 34.74 57.93 29.43 34.38 18.68 24.65 15.30 19.07

Y t/h t 27.09 24.50 20.44 17.34 18.34 20.56 67.47 23.45 25.57 13.00 15.62 8.76

W t 9.52 10.92 10.12 6.12 4.92 12.87 26.21 6.02 73.69 3.73 7.59 5.60

R t 42.13 33.61 45.17 54.02 46.76 26.55 19.93 42.43 9.79 76.06 38.59 63.14

π t 26.40 28.30 21.53 27.70 26.46 11.74 13.95 27.26 12.50 36.89 74.37 12.35

rr t 31.17 20.29 37.21 42.21 35.50 25.87 16.14 29.91 9.04 61.70 12.00 77.59

Figure 2: IRFS, Alternative Factors

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Y t; I t; h t; u t; ξ w t ; Shaded area: 68% HPDI.

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Table 4: Variance Decomposition, Comovements

Y t C t I t u t h t s w t Y t/h t ξt w W t R t π t r t

6–32 Quarters Identification: Output

6-32 Quarters 76.52 29.04 66.80 50.13 53.83 43.68 29.47 37.70 11.14 23.23 8.68 14.24

32-80 Quarters 39.66 11.52 35.01 37.01 34.61 24.57 6.03 22.00 13.67 32.76 13.89 22.39

80-∞ Quarters 7.58 6.81 8.38 7.75 7.35 6.45 7.57 6.09 7.76 8.22 6.09 6.84

6–32 Quarters Identification: Comovement of (C t, I t, u t, h t) with Output

6-32 Quarters 75.28 31.90 63.34 46.60 52.35 45.05 30.93 38.51 12.82 17.63 10.21 11.19

32-80 Quarters 43.13 12.30 37.48 36.61 38.37 27.85 6.43 24.72 16.01 24.78 12.14 18.75

80-∞ Quarters 11.28 9.93 12.06 9.14 8.75 8.28 10.96 7.28 11.16 6.49 6.26 5.61

Figure 3: IRF, Comovements

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VAR(4), Output volatility;

VAR(4), Maximize contribution to comovements; Shaded area: 68% HPDI.

9


is no significant effect on the unemployment rate. Inflation and the nominal interest rate decline

in a persistent fashion while the real rate experiences the opposite pattern.

The analysis so far has computed factors aiming at accounting for the maximal volatility in any

particular variable. Are their properties sufficient to establish the term disconnect? It is conceivable,

for instance, that many other factors exist (as other rotations of the VAR residuals) that account

for the bulk (but not the maximum) of business cycle fluctuations and have very different properties

from the maximal volatility computed factor. In order to address this possibility we consider the set

of factors that can account for more than 50% of the business volatility in output (and similarly, the

set of those that account for more than 50% of the long term volatility) and compute the average

variance decomposition as well as the average IRFs from that set. This exercise does not alter any

of the results reported so far.

Table 5: Variance Decomposition, Output Target Over Alternative Frequencies

Y t C t I t u t h t s w t Y t/h t ξt w W t R t π t r t

6–32 Quarters Identification

6-32 Quarters 76.52 29.04 66.80 50.13 53.83 43.68 29.47 37.70 11.14 23.23 8.68 14.24

32-80 Quarters 39.66 11.52 35.01 37.01 34.61 24.57 6.03 22.00 13.67 32.76 13.89 22.39

80-∞ Quarters 7.58 6.81 8.38 7.75 7.35 6.45 7.57 6.09 7.76 8.22 6.09 6.84

32–80 Quarters Identification

6-32 Quarters 44.72 39.39 35.77 35.47 30.93 23.22 23.76 27.79 13.65 31.24 30.71 12.99

32-80 Quarters 65.03 56.79 50.25 47.49 41.82 34.35 17.48 35.12 35.67 29.93 34.87 18.78

80-∞ Quarters 16.32 15.18 17.63 24.69 14.92 16.32 17.84 13.24 18.89 21.42 16.93 24.94

80-∞ Quarters Identification

6-32 Quarters 12.78 17.23 9.54 8.15 11.78 8.77 6.73 11.91 7.44 8.35 10.24 6.35

32-80 Quarters 20.22 28.61 13.49 11.48 17.71 13.40 12.85 19.16 15.34 11.37 15.64 11.21

80-∞ Quarters 99.66 99.65 97.18 40.70 57.34 87.94 98.26 76.58 97.28 46.28 49.03 30.12

We now turn to the examination of the properties of the factor that maximizes the contribution

to the volatility of the nominal interest rates (table 2 and figure 5). There are two interesting

patterns worth reporting: The R-factor makes a substantial contribution to the business cycle; and

it does not look like a policy rule shock because it moves R in the same direction as Y.

Consumption factor

What can be learned from the consumption targeting factor? Note that for this factor, a

consumption boom is associated with a drop in inflation. This is inconsistent with NK models that

let discount-rate, news, or noise shocks generate realistic comovement only by having monetary

policy inflate the economy.

Unemployment factor The factor that drives the business cycle movements in unemployment

has a modest contribution to the volatility of consumption and a negligible one on that of the real

wage and inflation. It also has a delayed effect on inflation. This pattern could arise from a shift

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Figure 4: IRF, Output Target Over Alternative Frequencies

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0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Labor Share

5 10 15 20

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

Labor Productivity

5 10 15 20

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

Labor Wedge

5 10 15 20

0.5

0.4

0.3

0.2

0.1

0.0

0.1

Wage

5 10 15 20

0.15

0.10

0.05

0.00

0.05

0.10

Nom. Int. Rate

5 10 15 20

0.08

0.06

0.04

0.02

0.00

0.02

0.04

0.06

0.08

0.10

Inflation

5 10 15 20

0.10

0.08

0.06

0.04

0.02

0.00

0.02

0.04

Real Int. Rate

5 10 15 20

6-32 Quarters Ident.; 32-80 Quarters Ident.; 80-∞ Quarters Ident.; Shaded area: 68% HPDI.

11


Figure 5: IRFS, Targeting Nominal Variables, VARs

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

Output

5 10 15 20

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

0.5

Consumption

5 10 15 20

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Investment

5 10 15 20

0.3

0.2

0.1

0.0

0.1

0.2

0.3

Unemployment

5 10 15 20

0.6

0.4

0.2

0.0

0.2

0.4

Hours Worked

5 10 15 20

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Labor Share

5 10 15 20

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

Labor Productivity

5 10 15 20

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

Labor Wedge

5 10 15 20

0.8

0.6

0.4

0.2

0.0

0.2

0.4

Wage

5 10 15 20

0.25

0.20

0.15

0.10

0.05

0.00

0.05

0.10

Nom. Int. Rate

5 10 15 20

0.10

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Inflation

5 10 15 20

0.20

0.15

0.10

0.05

0.00

0.05

0.10

Real Int. Rate

5 10 15 20

Y t; R t; π t; W t; Shaded area: 68% HPDI.

12


in the demand for labor when the supply of labor is flat (high real wage rigidity).

Real interest rate factor It moves only nominal interest rates in the same direction, and also

contributes much to the volatility of this variable. But it does not do much else. On impact, it

makes inflation and the real rate move in opposite directions.

Inflation factor A shock that drives inflation down (in a persistent manner) gives rise to an

economic expansion (a positively sloped Philipps curve). But this factor does not play an important

role in the business cycle.

A wage factor This factor is associated with a decrease in unemployment and an increase in

output but it also decreases employment. Nonetheless, it matters very little for the business cycle.

3 Evaluating the Smets-Wouters’ Model

This section illustrates how our previous method to characterize comovements across main aggregates

over the business cycle can be used to shed light on the mechanisms at work in a particular

model. For this illustration, we use the Smets and Wouters (2007) model. We generate 1000 artificial

sets of time series for output, consumption, investment, hours worked, the labor share, the

nominal interest rate and the inflation rate from the model. For each set of set, we estimate the

same VAR as in the data and use our methodology to recover comovements between macroeconomic

variables at business cycle frequencies. We then compare these comovements, as generated by the

model, to those we obtained from the data.

Figure 6 reports the average across the 1000 draws of the response of aggregates to the business

cycle factor as identified by output. Table 6 compares the average contribution of the business

cycle factor to the volatility of aggregates from the model to those obtained in the data. The model

does a fairly good job at accounting for the propagation of the business cycle factor to output,

consumption, investment, hours worked. The impulse responses, as obtained from simulation of

the model, are in line with those obtained from the data, creating the right positive comovements

among these variables over the business cycle. Likewise, the contribution of the business cycle

shock to the volatility of these variables is also in line with the data at business cycle frequencies.

The model however tends to overestimate the persistence of the business cycle factor, as can be

seen from its contribution to the volatility of output, consumption, investment and hours worked

at medium and low frequencies. The properties of the models also deteriorate when one examines

the implications of the business cycle factor for inflation and interest rates. The data indicate

that following a positive shift in the business cycle factor, both inflation and the interest rates

increase. The model has the opposite implication: the business cycle factor is initially deflationary

and associated with a pronounced drop in the interest rates. The model also fails to account for the

contribution of the factor to both inflation and interest rate volatility. Finally the model cannot

explain the dynamics of wages and the labor share. In other words, while the model seems to do

a good job in terms of the real business cycle component, it fails to explain the comovements of

the latter with the nominal side –a failure that seems to speak to the core of the New Keynesian

13


model.

Figure 6: Y t Factor, IRF

1.0

0.8

0.6

0.4

0.2

0.0

0.2

Output

5 10 15 20

0.4

0.3

0.2

0.1

0.0

0.1

0.2

Consumption

5 10 15 20

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

Investment

5 10 15 20

0.6

0.4

0.2

0.0

0.2

Hours Worked

5 10 15 20

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Labor Share

5 10 15 20

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

Labor Productivity

5 10 15 20

0.5

0.4

0.3

0.2

0.1

0.0

0.1

Wage

5 10 15 20

0.15

0.10

0.05

0.00

0.05

0.10

Nom. Int. Rate

5 10 15 20

0.08

0.06

0.04

0.02

0.00

0.02

0.04

0.06

0.08

0.10

Inflation

5 10 15 20

0.10

0.08

0.06

0.04

0.02

0.00

0.02

0.04

Real Int. Rate

5 10 15 20

Data;

Model; Shaded area: 68% HPDI.

Table 6: Y t Factor, Variance Decomposition

Y t C t I t h t s w t Y t/h t W t R t π t r t

Data

6-32 Quarters 76.52 29.04 66.80 53.83 43.68 29.47 11.14 23.23 8.68 14.24

32-80 Quarters 39.66 11.52 35.01 34.61 24.57 6.03 13.67 32.76 13.89 22.39

80-∞ Quarters 7.58 6.81 8.38 7.35 6.45 7.57 7.76 8.22 6.09 6.84

Model

6-32 Quarters 68.12 25.34 49.73 42.50 20.46 54.49 22.52 13.96 13.93 7.10

32-80 Quarters 51.05 22.80 45.13 35.02 13.42 46.40 26.89 11.17 12.18 8.10

80-∞ Quarters 24.71 13.02 23.49 13.22 10.16 26.94 25.90 9.74 10.37 7.33

Table 7 replicate the exercise when the business cycle factor is identified using investment, and

illustrate another aspect of the model. While the data indicate that the business cycle factor, as

identified relying on output, investment or hours worked, has very similar implications for the main

aggregate variables, the model shows some discrepancy. First, the dynamics of consumption is

affected by this shift in the identification. But more importantly the contribution of the business

cycle factor to the volatility of the main aggregates is sensitive to a change in the definition of

the factor. For instance, the data suggests that whether output, investment or hours worked is

used to identify the business cycle, the business cycle factor explains accounts for about 70% of the

volatility of output and investment and about 50% of that of hours worked. In the model, using

investment instead of output to identify the factor leads to a 40% decrease of its contribution to

output volatility (from 70 to 40%). Its contribution to the volatility of hours worked shows a 33%

14


drop. A similar phenomenon occurs when hours worked are now used to identify the business cycle

shock. This indicates that in the model, there is a certain disconnect between the forces that drive

investment fluctuations from those that drive output and hours fluctuations. In contrast, such a

disconnect does not appear in the data.

Table 7: I t Factor, Variance Decomposition

Y t C t I t h t s w t Y t/h t W t R t π t r t

Data

6-32 Quarters 65.00 17.84 79.13 50.27 39.23 23.64 10.22 32.06 9.45 17.63

32-80 Quarters 35.82 11.80 41.06 31.57 25.62 5.54 12.32 44.20 21.34 25.59

80-∞ Quarters 5.44 4.85 6.75 6.11 5.53 5.36 5.55 15.63 8.81 12.38

Model

6-32 Quarters 41.18 9.89 82.51 31.07 9.37 27.92 17.46 13.90 7.55 10.17

32-80 Quarters 28.28 12.17 61.80 22.50 9.03 25.34 20.56 16.59 9.22 13.59

80-∞ Quarters 12.89 8.82 33.25 9.43 7.85 15.48 18.85 9.82 8.09 8.31

Figure 7 repeats the preceding experiment, but we now identify the factor that maximizes the

explained volatility of the nominal interest rate. By doing this, we hope shedding light on the

role of the nominal propagation mechanism, in particular monetary policy, over the business cycle.

Figure 7 indicates that the model does a fairly good job at matching the comovements between the

nominal interest rate and inflation. But Figure 7 also reveals that the model cannot account for the

comovements between these variables and the labor share. This failure again speaks to the core of

the New Keynesian model in so far as the labor share gives a measure of the real marginal cost in

the model. Figure 8 reports the response of aggregate variables to the labor share factor, and hence

mirrors the previous experiment. The figure delivers a message very similar. While the model labor

share, as in the data, co-moves positively with the real variables, it is negatively correlated with

the nominal variables, therefore confirming the failure observed in the previous experiment.

15


Figure 7: Nominal Factors, IRF

0.25

0.20

0.15

0.10

0.05

0.00

0.05

0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Nom. Int. Rate

5 10 15 20

Nom. Int. Rate

5 10 15 20

Data;

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

0.02

0.04

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

(a) R t Factor

Inflation

5 10 15 20

(b) π t Factor

Inflation

5 10 15 20

Model; Shaded area: 68% HPDI.

0.15

Labor Share

0.10

0.05

0.00

0.05

0.10

0.15

0.20

0.25

5 10 15 20

0.25

Labor Share

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

5 10 15 20

Figure 8: s w t

Factor, IRF

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

Output

5 10 15 20

Labor Productivity

5 10 15 20

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Consumption

5 10 15 20

Wage

5 10 15 20

2.0

1.5

1.0

0.5

0.0

0.5

1.0

0.15

0.10

0.05

0.00

0.05

0.10

Investment

5 10 15 20

Nom. Int. Rate

5 10 15 20

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.04

0.02

0.00

0.02

0.04

0.06

0.08

0.10

Hours Worked

5 10 15 20

Inflation

5 10 15 20

0.4

0.2

0.0

0.2

0.4

0.6

0.8

0.10

0.08

0.06

0.04

0.02

0.00

0.02

0.04

Labor Share

5 10 15 20

Real Int. Rate

5 10 15 20

Data;

Model; Shaded area: 68% HPDI.

16

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