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

**An** **An**atomy **of** **the** **Business** **Cycle**

George-Marios **An**geletos

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[extremely preliminary and incomplete; do not quote or distribute]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 ano**the**r, 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 **the**se

findings are at odds with existing macroeconomic models **of** ei**the**r **the** RBC or **the** NK variety,

and discuss how **the**y provide guidance for future **the**oretical research.

1 Introduction

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

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

macroeconomic **the**ory 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 ano**the**r and (ii) each one **of** **the**m 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 **the**oretical

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

constructs as “factors” ra**the**r than as “shocks”; we reserve **the** latter term for **the**oretical 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 **the**orist aspires to develop a model in which a single shock, or mechanism,

explains most **of** **the** business cycle in **the** data; think, for example, ei**the**r **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.

2

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, whe**the**r **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 **of**fering

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

**the**se 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 o**the**r;

more precisely, it accounts for more than half **of** **the** business cycle variability in **the**se 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

fur**the**r use **of** it by investigating its effects on **the** o**the**r 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 **the**y 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 **the**y 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 **the**se 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

3

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 **the**se variables only make minor contributions to **the** business cycle. These two findings taken

toge**the**r suggest that a Phillips curve type **of** a relationship involving an inflation-unemployment

trade **of**f 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. Fur**the**rmore, **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 **the**ory. 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** **the**ory 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 ra**the**r 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**

4

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 **the**se 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 **the**re is some

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

flat early on and **the**n 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 **of**ten 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 fur**the**r 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.

5

Figure 1: Y t factor, IRFs

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Output

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Consumption

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Investment

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Unemployment

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Inflation

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VAR;

VECM; Shaded area: 68% HPDI.

6

over alternative frequencies.

The identified main business cycle factor has similar properties independent **of** whe**the**r it is

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

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

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

properties **of** **the** factor differ when it is constructed on **the** basis **of** some o**the**r 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 **the**ories 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

fur**the**r 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 o**the**r 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, fur**the**r productivity

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

7

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.

8

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 **the**ir properties sufficient to establish **the** term disconnect? It is conceivable,

for instance, that many o**the**r factors exist (as o**the**r 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

10

Figure 4: IRF, Output Target Over Alternative Frequencies

0.8

0.6

0.4

0.2

0.0

0.2

Output

5 10 15 20

0.6

0.5

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

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

0.20

0.25

Unemployment

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

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. None**the**less, 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 **the**n compare **the**se 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 **the**se variables over **the** business cycle. Likewise, **the** contribution **of** **the** business cycle

shock to **the** volatility **of** **the**se 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 o**the**r 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 ano**the**r 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 whe**the**r 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, **the**re 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 **the**se 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, **the**refore 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